RINGS, MODULES, AND ALGEBRAS IN STABLE HOMOTOPY THEORY

A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May

Author addresses:

Purdue University Calumet, Hammond IN 46323 E-mail address: aelmendo@math.purdue.edu The University of Michigan, Ann Arbor, MI 48109-1003 E-mail address: ikriz@math.lsa.umich.edu The University of Chicago, Chicago, IL 60637 E-mail address: mandell@math.uchicago.edu The University of Chicago, Chicago, IL 60637 E-mail address: may@math.uchicago.edu Abstract. Let S be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category ℳ_{S} of “S-modules” whose derived category 𝒟_{S} is equivalent to the classical stable homotopy category. This allows a simple and algebraically manageable definition of “S-algebras” and “commutative S-algebras” in terms of associative, or associative and commutative, products R ∧_{S} R ⟶ R. These notions are essentially equivalent to the earlier notions of A_{∞} and E_{∞} ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of R-modules in terms of maps R ∧_{S} M ⟶ M. When R is commutative, the category ℳ_{R} of R-modules also has an associative, commutative, and unital smash product, and its derived category 𝒟_{R} has properties just like the stable homotopy category.

Working in the derived category 𝒟_{R}, we construct spectral sequences that specialize to give generalized universal coefficient and Künneth spectral sequences. Classical torsion products and Ext groups are obtained by specializing our constructions to Eilenberg-Mac Lane spectra and passing to homotopy groups, and the derived category of a discrete ring R is equivalent to the derived category of its associated Eilenberg-Mac Lane S-algebra.

We also develop a homotopical theory of R-ring spectra in 𝒟_{R}, analogous to the classical theory of ring spectra in the stable homotopy category, and we use it to give new constructions as MU-ring spectra of a host of fundamentally important spectra whose earlier constructions were both more difficult and less precise.

Working in the module category ℳ_{R}, we show that the category of finite cell modules over an S-algebra R gives rise to an associated algebraic K-theory spectrum KR. Specialized to the Eilenberg-Mac Lane spectra of discrete rings, this recovers Quillen’s algebraic K-theory of rings. Specialized to suspension spectra Σ^{∞}(ΩX)_{+} of loop spaces, it recovers Waldhausen’s algebraic K-theory of spaces.

Replacing our ground ring S by a commutative S-algebra R, we define Ralgebras and commutative R-algebras in terms of maps A ∧_{R} A ⟶ A, and we show that the categories of R-modules, R-algebras, and commutative R-algebras are all topological model categories. We use the model structures to study Bousfield localizations of R-modules and R-algebras. In particular, we prove that KO and KU are commutative ko and ku-algebras and therefore commutative S-algebras.

We define the topological Hochschild homology R-module T HH^{R}(A; M) of A with coefficients in an (A, A)-bimodule M and give spectral sequences for the calculation of its homotopy and homology groups. Again, classical Hochschild homology and cohomology groups are obtained by specializing the constructions to Eilenberg-Mac Lane spectra and passing to homotopy groups. iv

Contents

Introduction 1 Chapter I. Prologue: the category of 𝕃-spectra 9 1. Background on spectra and the stable homotopy category 9 2. External smash products and twisted half-smash products 11 3. The linear isometries operad and internal smash products 14 4. The category of 𝕃-spectra 18 5. The smash product of 𝕃-spectra 21 6. The equivalence of the old and new smash products 24 7. Function 𝕃-spectra 27 8. Unital properties of the smash product of 𝕃-spectra 30 Chapter II. Structured ring and module spectra 35 1. The category of S-modules 35 2. The mirror image to the category of S-modules 39 3. S-algebras and their modules 41 4. Free A_{∞} and E_{∞} ring spectra and comparisons of definitions 44 5. Free modules over A_{∞} and E_{∞} ring spectra 47 6. Composites of monads and monadic tensor products 50 7. Limits and colimits of S-algebras 52 Chapter III. The homotopy theory of R-modules 57 1. The category of R-modules; free and cofree R-modules 57

vi CONTENTS

2. Cell and CW R-modules; the derived category of R-modules 60 3. The smash product of R-modules 65 4. Change of S-algebras; q-cofibrant S-algebras 68 5. Symmetric and extended powers of R-modules 71 6. Function R-modules 73 7. Commutative S-algebras and duality theory 77 Chapter IV. The algebraic theory of R-modules 81 1. Tor and Ext; homology and cohomology; duality 82 2. Eilenberg-Mac Lane spectra and derived categories 85 3. The Atiyah-Hirzebruch spectral sequence 89 4. Universal coefficient and Künneth spectral sequences 92 5. The construction of the spectral sequences 94 6. Eilenberg-Moore type spectral sequences 97 7. The bar constructions B(M, R, N) and B(X, G, Y) 99 Chapter V. R-ring spectra and the specialization to MU 103 1. Quotients by ideals and localizations 103 2. Localizations and quotients of R-ring spectra 107 3. The associativity and commutativity of R-ring spectra 111 4. The specialization to MU-modules and algebras 114 Chapter VI. Algebraic K-theory of S-algebras 117 1. Waldhausen categories and algebraic K-theory 117 2. Cylinders, homotopies, and approximation theorems 121 3. Application to categories of R-modules 124 4. Homotopy invariance and Quillen’s algebraic K-theory of rings 128 5. Morita equivalence 130 6. Multiplicative structure in the commutative case 134 7. The plus construction description of KR 136 8. Comparison with Waldhausen’s K-theory of spaces 141 Chapter VII. R-algebras and topological model categories 145

CONTENTS vii

1. R-algebras and their modules 146 2. Tensored and cotensored categories of structured spectra 149 3. Geometric realization and calculations of tensors 153 4. Model categories of ring, module, and algebra spectra 159 5. The proofs of the model structure theorems 163 6. The underlying R-modules of q-cofibrant R-algebras 167 Chapter VIII. Bousfield localizations of R-modules and algebras 173 1. Bousfield localizations of R-modules 174 2. Bousfield localizations of R-algebras 178 3. Categories of local modules 181 4. Periodicity and K-theory 184 Chapter IX. Topological Hochschild homology and cohomology 187 1. Topological Hochschild homology: first definition 188 2. Topological Hochschild homology: second definition 192 3. The isomorphism between thh^{R}(A) and A ⊗ S^{1} 196 Chapter X. Some basic constructions on spectra 201 1. The geometric realization of simplicial spectra 201 2. Homotopical and homological properties of realization 204 3. Homotopy colimits and limits 209 4. Σ-cofibrant, LEC, and CW prespectra 211 5. The cylinder construction 214 Chapter XI. Spaces of linear isometries and technical theorems 221 1. Spaces of linear isometries 221 2. Fine structure of the linear isometries operad 224 3. The unit equivalence for the smash product of 𝕃-spectra 230 4. Twisted half-smash products and shift desuspension 232 5. Twisted half-smash products and cofibrations 235 Chapter XII. The monadic bar construction 239 1. The bar construction and two deferred proofs 239

viii CONTENTS

2. Cofibrations and the bar construction 242 Chapter XIII. Epilogue: The category of 𝕃-spectra under S 247 1. The modified smash products ◁_{L}, ▷_{L}, and ⋆_{L} 247 2. The modified smash products ◁_{R}, ▷_{R}, and ⋆_{R} 251 Bibliography 257

Introduction

The last thirty years have seen the importation of more and more algebraic techniques into stable homotopy theory. Throughout this period, most work in stable homotopy theory has taken place in Boardman’s stable homotopy category [6], or in Adams’ variant of it [2], or, more recently, in Lewis and May’s variant [37]. That category is analogous to the derived category obtained from the category of chain complexes over a commutative ring k by inverting the quasi-isomorphisms. The sphere spectrum S plays the role of k, the smash product ∧ plays the role of the tensor product, and weak equivalences play the role of quasi-isomorphisms. A fundamental difference between the two situations is that the smash product on the underlying category of spectra is not associative and commutative, whereas the tensor product between chain complexes of k-modules is associative and commutative. For this reason, topologists generally work with rings and modules in the stable homotopy category, with their products and actions defined only up to homotopy. In contrast, of course, algebraists generally work with differential graded k-algebras that have associative point-set level multiplications.

We here introduce a new approach to stable homotopy theory that allows one to do point-set level algebra. We construct a new category ℳ_{S} of S-modules that has an associative, commutative, and unital smash product ∧_{S}. Its derived category 𝒟_{S} is obtained by inverting the weak equivalences; 𝒟_{S} is equivalent to the classical stable homotopy category, and the equivalence preserves smash products. This allows us to rethink all of stable homotopy theory: all previous work in the subject might as well have been done in 𝒟_{S}. Working on the point-set level, in ℳ_{S}, we define an S-algebra to be an S-module R with an associative and unital product R ∧_{S} R ⟶ R; if the product is also commutative, we call R a commutative S-algebra. Although the definitions are now very simple, these are not new notions: they are refinements of the A_{∞} and E_{∞} ring spectra that were introduced over twenty years ago by May, Quinn, and Ray [47]. In general, the latter need not satisfy the precise unital property that is enjoyed by our new Salgebras, but it is a simple matter to construct a weakly equivalent S-algebra from an A_{∞} ring spectrum and a weakly equivalent commutative S-algebra from an E_{∞} ring spectrum.

It is tempting to refer to (commutative) S-algebras as (commutative) ring spectra. However, this would introduce confusion since the term “ring spectrum” has had a definite meaning for thirty years as a stable homotopy category level notion. Ring spectra in the classical homotopical sense are not rendered obsolete by our theory since there are many examples that admit no S-algebra structure. In any case, the term S-algebra more accurately describes our new concept. With our theory, and the new possibilities that it opens up, it becomes vitally important to keep track of when one is working on the point-set level and when one is working up to homotopy. In the absence (or ignorance) of a good point-set level category of spectra, topologists have tended to be sloppy about this. The dichotomy will run through our work. The terms “ring spectrum” and “module spectrum” will always refer to the classical homotopical notions. The terms “S-algebra” and “S-module” will always refer to the strict point-set level notions.

We define a (left) module M over an S-algebra R to be an S-module M with an action R ∧_{S} M ⟶ M such that the standard diagrams commute. We obtain a category ℳ_{R} of (left) R-modules and a derived category 𝒟_{R}. There is a smash product M ∧_{R} N of a right R-module M and a left R-module N, which is an Smodule. For left R-modules M and N, there is a function S-module F_{R}(M, N) that enjoys properties just like modules of homomorphisms in algebra. Each F_{R}(M, M) is an S-algebra. If R is commutative, then M ∧_{R} N and F_{R}(M, N) are R-modules, and in this case ℳ_{R} and 𝒟_{R} enjoy all of the properties of ℳ_{S} and 𝒟_{S}. Thus each commutative S-algebra R determines a derived category of R-modules that has all of the structure that the stable homotopy category has. These new categories are of substantial intrinsic interest, and they give powerful new tools for the investigation of the classical stable homotopy category.

Upon restriction to Eilenberg-Mac Lane spectra, our topological theory subsumes a good deal of classical algebra. For a discrete ring R and R-modules M and N, we have

Tor_{n}^{R}(M, N) ≅ π_{n}(HM ∧_{HR} HN) and Ext_{R}^{n}(M, N) ≅ π_{−n}F_{HR}(HM, HN).

Here ∧_{R} and F_{R} must be interpreted in the derived category; that is, HM must be a CW HR-module. Moreover, the algebraic derived category 𝒟_{R} is equivalent to the topological derived category 𝒟_{HR}.

In general, for an S-algebra R, approximation of R-modules M by weakly equivalent cell R-modules is roughly analogous to forming projective resolutions in algebra. There is a much more precise analogy that involves developing the derived categories of modules over rings or, more generally, DGA’s in terms of cell modules. It is presented in [34], which gives an algebraic theory of A_{∞} and E_{∞} k-algebras that closely parallels the present topological theory.

Upon restriction to the sphere spectrum S, the derived smash products M ∧_{S} N and function spectra F_{S}(M, N) have as their homotopy groups the homology and cohomology groups N_{∗}(M) and N^{∗}(M). This suggests the alternative notations

Tor_{n}^{R}(M, N) = π_{n}(M ∧_{R} N) = N_{n}^{R}(M)

and

Ext_{R}^{n}(M, N) = π_{−n}F_{R}(M, N) = N_{R}^{n}(M)

for R-modules M and N. When R is connective, there are ordinary homology and cohomology theories on R-modules, represented by Eilenberg-Mac Lane spectra that are R-modules, and there are Atiyah-Hirzebruch spectral sequences for the computation of generalized homology and cohomology theories on R-modules.

The realization of algebraic Tor and Ext groups via Eilenberg-Mac Lane spectra generalizes to spectral sequences

E_{p,q}^{2} = Tor_{p,q}^{R_{∗}}(M_{∗}, N_{∗}) ⟹ Tor_{p+q}^{R}(M, N)

and

E_{2}^{p,q} = Ext_{R^{∗}}^{p,q}(M^{∗}, N^{∗}) ⟹ Ext_{R}^{p+q}(M, N).

These specialize to give Künneth and universal coefficient spectral sequences in classical generalized homology and cohomology theories. There are also Eilenberg-Moore type spectral sequences for the calculation of E_{∗}(M ∧_{R}N) under appropriate hypotheses on R and E.

Thinking of 𝒟_{R} as a new stable homotopy category, where R is a commutative S-algebra, we can realize the action of an element x ∈ R_{n} on an R-module M as a map of R-modules x : Σ^{n}M ⟶ M. We define M/xM to be the cofiber of x, and we define the localization M[x^{−1}] to be the telescope of a countable iterate of desuspensions of x, starting with M ⟶ Σ^{−n}M. By iteration, we can construct quotients by sequences of elements and localizations at sequences of elements. We define R-ring spectra, associative R-ring spectra, and commutative R-ring spectra in the homotopical sense, with products A ∧_{R} A ⟶ A defined via maps in the derived category 𝒟_{R}, and it turns out to be quite simple to study when quotients and localizations of R-ring spectra are again R-ring spectra.

When we take R = MU, we find easy direct constructions as MU-modules of all of the various spectra (MU/X)[Y^{−1}] that are usually obtained by means of the Baas-Sullivan theory of manifolds with singularities or the Landweber exact functor theorem. When their homotopy groups are integral domains concentrated in degrees congruent to zero mod 4, these MU-modules all admit canonical structures of associative and commutative MU-ring spectra. Remarkably, it is far simpler to prove the sharper statements that apply in the derived category of MU-modules than the much weaker stable homotopy category level analogs that were obtainable before our theory.

Thinking of ℳ_{R} as a new category of point-set level modules, where R is again a commutative S-algebra, we can define R-algebras A via point-set level products A ∧_{R} A ⟶ A such that the appropriate diagrams commute. For example, F_{R}(M, M) is an R-algebra for any R-module M. These have all of the good formal properties of S-algebras. We repeat the dichotomy for emphasis: The terms “Rring spectrum” and “R-module spectrum” will always refer to the homotopical notions defined in the derived category 𝒟_{R}. The terms “R-algebra” and “R-module” will always refer to the strict, point-set, level notions.

We shall construct Bousfield localizations of R-modules at a given R-module E. In principle, this is a derived category notion, but we shall obtain precise point-set level constructions. Using different point-set level constructions, we shall prove that the Bousfield localizations of R-algebras can be constructed to be R-algebras and the Bousfield localizations of commutative R-modules can be constructed to be commutative R-algebras. In particular, the localization R_{E} of R at E is a commutative R-algebra, and we shall see that the category of R_{E}-modules plays an intrinsically central role in the study of Bousfield localizations.

As a very special case, this theory will imply that the spectra KO and KU that represent real and complex periodic K-theory can be constructed as commutative algebras over the S-algebras ko and ku that represent real and complex connective K-theory. Therefore KO and KU are commutative S-algebras, as had long been conjectured in the earlier context of E_{∞} ring spectra. Again, it is far simpler to prove the sharper ko and ku-algebra statements than to construct S-algebra structures directly.

For an R-algebra A, we define the enveloping R-algebra A^{e} = A ∧_{R} A^{op}, and we define the topological Hochschild homology of A with coefficients in an (A, A)- bimodule M to be the derived smash product

T HH^{R}(A; M) = M ∧_{A^{e}} A.

This is the correct generalization from algebra to topology since, if R is a discrete commutative ring and A is an R-algebra that is flat as an R-module, then the algebraic and topological Hochschild homology are isomorphic:

HH_{n}^{R}(A; M) ≡ Tor_{n}^{A^{e}}(M, A) ≅ Tor_{n}^{HA^{e}}(HM, HA) ≡ π_{n}(T HH^{R}(A; M)).

In general, for a commutative S-algebra R, an R-algebra A, and an (A, A)-bimodule M, there is a spectral sequence

E_{p,q}^{2} = HH_{p,q}^{R_{∗}}(M_{∗}, A_{∗}) ⟹ π_{p+q}(T HH^{R}(A; M))

under suitable flatness hypotheses. More generally, there are similar spectral sequences converging to E_{∗}(T HH^{R}(A; M)) for a commutative ring spectrum E.

There is also a point-set level version thh^{R}(A; M) of topological Hochschild homology. It is obtained by mimicking topologically the standard complex for the calculation of algebraic Hochschild homology. When M = A, this construction has particularly nice formal properties, as was observed in [52] and as we shall explain: it is isomorphic to the tensor A ⊗ S^{1}. A key technical point is that the derived category and point-set level definitions become equivalent after replacing R and A by suitable weakly equivalent approximations.

Our S-algebras and their modules are enough like ordinary rings and modules that we can construct the algebraic K-theory spectrum KR associated to an Salgebra R by applying Waldhausen’s S_{•}-construction to the category of finite cell R-modules. Applied to the Eilenberg-Mac Lane spectrum HR of a discrete ring R, this gives a new construction of Quillen’s algebraic K-theory. Applied to the suspension spectrum Σ^{∞}(ΩX)_{+}, this gives a new construction of Waldhausen’s algebraic K-theory of the space X. The resulting common framework for topological Hochschild homology and Quillen and Waldhausen algebraic K-theory opens up several new directions and appears to bring a number of standing conjectures within reach. We merely lay the foundations here.

The technical heart of our theory is the problem of keeping our formal point-set level constructions under homotopical control. While we shall show by essentially formal categorical arguments that our various categories of R-modules, R-algebras, and commutative R-algebras are cocomplete and complete, tensored and cotensored, topological model categories, this formal structure does not in itself address the problem: forgetful functors from more to less structured spectra rarely preserve cofibrant objects, and may well not do so even up to homotopy type. The problem requires deeper analysis, and a crucial aspect of our work is that our discussion of model categories gives sufficient control on the underlying homotopy types of cofibrant R-algebras and cofibrant commutative R-algebras to allow the calculational use of bar constructions and topological Hochschild homology complexes. This is also crucial to our proof that Bousfield localizations of R-algebras can be constructed as R-algebras.

Another tool in keeping homotopical control is the category of “tame” spectra. It is an intermediate category between the ground category of spectra, which is welldesigned for formal point-set level work but not for homotopical analysis, and the category of CW spectra, which is well-designed for homotopical analysis but not for formal work. Its homotopy category is symmetric monoidal under the smash product, and we can approximate any structured spectrum by a weakly equivalent tame structured spectrum by means of a “cylinder construction” defined using homotopy colimits. Actually, this tool will only be needed in Chapter I, since the smash product of S-modules turns out to better behaved under weak equivalences than the smash product of spectra.

The basic construction underlying all of our work is the “twisted half-smash product” A ⋉ E of a suitable space A and a spectrum E. This construction is defined with respect to a given map α from A to an appropriate space of linear isometries. We prove that a homotopy equivalence A′ ⟶ A, with homotopy inverse unrelated to α, induces a homotopy equivalence A′ ⋉ E ⟶ A ⋉ E when E is tame. This invariance statement is the technical lynchpin of our theory.

The construction of thh, of bar constructions needed in our work, and of functorial homotopy colimits of spectra all require geometric realizations of simplicial spectra. This raises another technical problem. To understand geometric realization homotopically, the given simplicial spectra must satisfy certain cofibration conditions, and it is hard to verify that a map of spectra is a cofibration (satisfies the homotopy extension property). The solution to this problem is basic to the homotopical understanding of cofibrant R-algebras and commutative R-algebras.

The reader interested in using our theory need not be concerned with these matters, and most of the technical proofs are deferred until the last few chapters. The first three chapters explain the foundations needed for the applications of the next three, which are independent of one another. Chapter VII explains the foundations needed for Chapters VIII and IX, which are independent of each other. Each chapter has its own brief introduction. References within a chapter are of the form “Lemma 3.4”; references to results in other chapters are of the form “I.3.4”.

Our work is not independent of earlier work: the groundwork was laid in [37], and all of our ring, module, and algebra spectra are spectra in the sense of Lewis and May with additional structure. Moreover, the technical lynchpin referred to above depends on the first author’s paper [19]. In [37], the focus was on equivariant stable homotopy theory, the study of spectra with actions by compact Lie groups G. We have chosen to write this book nonequivariantly in the interests of readability. However, we have kept a close eye on the equivariant generalization, and we have been careful to use only arguments that directly generalize to the equivariant setting. We state a metatheorem.

Theorem 0.1. All of the definitions and all of the general theory in this paper apply to G-spectra for any compact Lie group G.

This has been used by Greenlees and May [27] to prove a completion theorem for the calculation of M_{∗}(BG) and M^{∗}(BG) for any MU-module spectrum M. Some of that work, together with some of ours, was described in the announcement [21] and in the series of expository papers [22, 25, 26]. The last two of those papers give both equivariant and non-equivariant applications of the present theory to localizations and completions of R-modules at ideals in π_{∗}(R).

We warn the knowledgeable reader that this material has undergone several major revisions, and the final definitions and terminology are not those of earlier announcements and drafts. In particular, our S-modules enjoy a unital property that was not imposed on the S-modules, here called 𝕃-spectra, of the earlier versions written by Elmendorf, Kriz, and May alone. The fact that one can impose this unital property and still retain homotopical control is one of many new insights contributed by Mandell. This substantially sharpens and simplifies the theory. Paradoxically, however, one cannot impose such a unital property in the parallel algebraic theory of [34]. Therefore, to facilitate a comparison of the algebraic and topological theories, we run through a little of the previous variant of our theory in the last chapter.

The chapter on algebraic K-theory has not been previously announced and is entirely work of Mandell: it is part of his Chicago PhD thesis in preparation.

Two other Chicago students deserve thanks. Maria Basterra has carefully read several drafts and caught numerous soft spots of exposition. Jerome Wolbert has made many helpful comments, and his Chicago PhD thesis in preparation will analyze the new derived categories associated to the various K-theory spectra.

It is a pleasure to thank Mike Hopkins, Gaunce Lewis, and Jim McClure for many helpful conversations and e-mails. We owe a critical lemma, namely I.3.4, to Hopkins [31]. Although trivial to prove, it broke a psychological barrier and played a pivotal role in our thinking. We should acknowledge the pioneering work on A_{∞} ring and module spectra of Alan Robinson, which gave precursors of many of the results of Chapter IV. We learned the material of IX§3, on thh, from the application of our theory given in the paper [52] of McClure, Schwänzl, and Vogt. We learned many of the results on cocomplete and complete, tensored and cotensored, topological model categories in II§7 and VII§§2,4 from Hopkins and McClure. Their foresight and insight have been inspirational.

8 INTRODUCTION

CHAPTER I Prologue: the category of 𝕃-spectra

In this prologue, we construct a category whose existence was previously thought to be impossible by at least two of the authors: a complete and cocomplete category of spectra, namely the 𝕃-spectra, with an associative and commutative smash product. This contrasts with the category constructed by Lewis and the fourth author in [47, 37], whose smash product is neither associative nor commutative (before passage to homotopy categories), and with the category constructed by the first author in [19], which is neither complete nor cocomplete. We will also give a function 𝕃spectrum construction that is right adjoint to the new smash product. The category of 𝕃-spectra has all of the properties that we desire except that its smash product, denoted by ∧_{ℒ} , is not unital. It has a natural unit map λ : S ∧_{ℒ} M ⟶ M, which is often an isomorphism and always a weak equivalence.

The curtain will rise on our real focus of interest in the next chapter, where we will define an S-module to be an 𝕃-spectrum M such that λ : S ∧_{ℒ} M ⟶ M is an isomorphism. Restricting ∧_{ℒ} to S-modules and renaming it ∧_{S}, this will give us a symmetric monoidal category in which to develop stable topological algebra.

1. Background on spectra and the stable homotopy category

We begin by recalling the basic definitions in Lewis and May’s approach to the stable category. We first recall the definition of a coordinate-free spectrum; see [37, I§2] or [19, §2] for further details. A coordinate-free spectrum is a spectrum that takes as its indexing set, instead of the integers, the set of finite dimensional subspaces of a “universe”, namely a real inner product space U ≅ ℝ^{∞}. Thus, a spectrum E assigns a based space EV to each finite dimensional subspace V of U, with (adjoint) structure maps

σ̃_{V,W} : EV ⟶^{≅}Ω^{W −V} EW

10 I. PROLOGUE: THE CATEGORY OF 𝕃-SPECTRA

when V ⊂ W. Here W − V is the orthogonal complement of V in W and Ω^{W} X is the space of based maps F (S^{W} , X), where S^{W} is the one-point compactification of W. These maps are required to be homeomorphisms and to satisfy an evident associativity relation. A map of spectra f : E → E′ is a collection of maps of based spaces f_{V} : EV → E′V for which each of the following diagrams commutes:

[figure: EV; f_{V}; σ̃_{V,W}; E′V; σ̃′_{V,W}; Ω^{W −V} EW; Ω^{W −V} f_{W}; Ω^{W −V} E′W.]

We obtain the category 𝒮 U of spectra indexed on U. If we drop the requirement that the maps σ̃_{V,W} be homeomorphisms, we obtain the notion of a prespectrum and the category 𝒫U of prespectra. The forgetful functor 𝒮 U ⟶ 𝒫U has a left adjoint L, details of which are given in [37, App]. Functors on prespectra that do not preserve spectra are extended to spectra by applying the functor L. For example, for a based space X and a prespectrum E, we have the prespectrum E∧X specified by (E ∧X)(V) = EV ∧X. When E is a spectrum, the structure maps for this prespectrum level smash product are not homeomorphisms, and we understand the smash product E ∧X to be the spectrum L(E ∧X). For example, ΣE = E ∧S^{1}. Function spectra are easier. We set F (X, E)(V) = F (X, EV) and find that this functor on prespectra preserves spectra. For example, ΩE = F (S^{1}, E). The following result is discussed in [37, p.13].

Proposition 1.1. The category 𝒮 U is complete and cocomplete.

Proof. Limits and colimits are computed on prespectra spacewise. Limits preserve spectra, and colimits of spectra are obtained by use of the functor L.

A homotopy in the category of spectra is a map E ∧ I_{+} ⟶ E′, and we have cofiber and fiber sequences that behave exactly as in the category of spaces. The cofiber Cf of a map f : E ⟶ E′ of spectra is the pushout E′ ∪_{f} CE, where CE = E ∧ I. A cofibration of spectra is a map i : E ⟶ E′ that satisfies the homotopy extension property (HEP: a homotopy h : E ∧ I_{+} ⟶ F of a restriction of a map f : E′ ⟶ F extends to a homotopy h̃ : E ∧ I_{+} ⟶ F of f). The canonical maps E ⟶ CE and E′ ⟶ Cf are examples. The fiber F f of a map f : E′ ⟶ E is the pullback E′ ×_{f} P E, where P E = F (I, E). A fibration of spectra is a map p : E ⟶ E′ that satisfies the covering homotopy property (CHP: a homotopy h : F ∧ I_{+} ⟶ E′ of a projection p ∘ f, f : F ⟶ E, is covered by a homotopy h̃ : F ∧ I_{+} ⟶ E of f). The canonical maps P E ⟶ E and F f ⟶ E′ are examples.

2. EXTERNAL SMASH PRODUCTS AND TWISTED HALF-SMASH PRODUCTS 11

A map f of spectra is a weak equivalence if each of its component maps f_{V} is a weak equivalence of spaces. The stable homotopy category h̄𝒮 U is constructed from the homotopy category of spectra by adjoining formal inverses to the weak equivalences, a process that is made rigorous by CW approximation.

The V th space functor from spectra to spaces has a left adjoint that we shall denote by Σ_{V}^{∞}, or Σ_{n}^{∞} when V = ℝ^{n} [37, I§4]. Its definition will be recalled in X.4.5. When V = {0}, this is the suspension spectrum functor Σ^{∞}. For n ≥ 0, the sphere spectrum S^{n} is the suspension spectrum Σ^{∞}S^{n} of the sphere space S^{n}. For n > 0, the sphere spectrum S^{−n} is Σ_{n}^{∞}S^{0}. There are canonical isomorphisms Σ^{m}S^{n} ≅ S^{m+n} for m ≥ 0 and integers n and there are canonical isomorphisms Σ_{m}^{∞}S^{n} ≅ S^{n−m} for m ≥ 0 and n ≥ 0. Sphere spectra are used to define the homotopy groups of spectra, π_{n}(E) = h𝒮 U(S^{n}, E), and a map of spectra is a weak equivalence if and only if it induces an isomorphism of spectrum-level homotopy groups.

Although we shall not introduce different notations for space level and spectrum level spheres, we shall generally write S for the zero sphere spectrum, reserving the notation S^{0} for the two-point space.

The theory of cell and CW spectra is developed by taking sphere spectra as the domains of attaching maps [37, I§5]. The stable homotopy category h̄𝒮 U is equivalent to the homotopy category of CW spectra. It is important to remember that homotopy-preserving functors on spectra that do not preserve weak equivalences are transported to the stable category by first replacing their variables by weakly equivalent CW spectra.

2. External smash products and twisted half-smash products

The construction of our new smash product will start from the external smash product of spectra. This is an associative and commutative pairing

𝒮 U × 𝒮 U′ → 𝒮 (U ⊕ U′)

for any pair of universes U and U′. It is constructed by starting with the prespectrum level definition

(E ∧ E′)(V ⊕ V′) = EV ∧ E′V′.

The structure maps fail to be homeomorphisms when E and E′ are spectra, and we apply the spectrification functor L to obtain the desired spectrum level smash product. This external smash product is the one used in [19].

There is an associated function spectrum functor

F : (𝒮 U′)^{op} × 𝒮 (U ⊕ U′) ⟶ 𝒮 U

and an adjunction

𝒮 (U ⊕ U′)(E ∧ E′, E′′) ≅ 𝒮 U(E, F (E′, E′′))

for E ∈ 𝒮 U, E′ ∈ 𝒮 U′, and E′′ ∈ 𝒮 (U ⊕ U′); see [37, p. 69].

Now let ℐ denote the category whose objects are universes U and whose morphisms are linear isometries. Universes are topologized as the unions of their finite dimensional subspaces, and the set ℐ (U, U′) of linear isometries U → U′ is given the function space topology; it is a contractible space [37, II.1.5]. The category 𝒮 constructed in [19] augments to the category ℐ. Since ℐ fails to have limits and colimits (it even fails to have coproducts), 𝒮 suffers from the same defects.

In order to obtain smash products internal to a single universe U, we shall exploit the “twisted half-smash product”. The input data for this functor consist of two universes U and U′ (which may be the same), an unbased space A with a given structure map α : A → ℐ (U, U′), and a spectrum E indexed on U. The output is the spectrum A ⋉ E, which is indexed on U′. It must be remembered that the construction depends on α and not just on A, although different choices of α lead to equivalent functors on the level of stable categories [37, VI.1.14]. The intuition is that the twisted half-smash product is a generalization to spectra of the “untwisted” functor A_{+} ∧ X on based spaces X. This intuition is made precise by the following “untwisting formula” relating twisted half-smash products and shift desuspensions. It is a substantial technical strengthening of results in [37] and will be proven in XI§4.

Proposition 2.1. For any map A ⟶ ℐ (U, U′) and any n ≥ 0, there is an isomorphism of spectra

A ⋉ Σ_{n}^{∞}X ≅ Σ_{n}^{∞}(A_{+} ∧ X)

that is natural in spaces A over ℐ (U, U′) and based spaces X.

Observe that the functor Σ_{n}^{∞} implicitly refers to the universe U on the left and to the universe U′ on the right. The twisted-half smash product enjoys the following formal properties, among others; see [37, VI.3.1 and VI.1.5] or [19, 3.18 and 5.1].

Proposition 2.2. The following statements hold.

(i) There is a canonical isomorphism {id_{U}} ⋉ E ≅ E.

(ii) Let A → ℐ (U, U′) and B → ℐ (U′, U′′) be given; let B × A have the structure map given by the composite

[figure: B × A; ℐ (U′, U′′) × ℐ (U, U′); ∘; ℐ (U, U′′).]

Then there is a canonical isomorphism

(B × A) ⋉ E ≅ B ⋉ (A ⋉ E).

(iii) Let A → ℐ (U_{1}, U′_{1}) and B → ℐ (U_{2}, U′_{2}) be given; let A × B have the structure map given by the composite

[figure: A × B; ℐ (U_{1}, U′_{1}) × ℐ (U_{2}, U′_{2}); ⊕; ℐ (U_{1} ⊕ U_{2}, U′_{1} ⊕ U′_{2}).]

Let E_{1} and E_{2} be spectra indexed on U_{1} and U_{2} respectively. Then there is a canonical isomorphism

(A × B) ⋉ (E_{1} ∧ E_{2}) ≅ (A ⋉ E_{1}) ∧ (B ⋉ E_{2}).

(iv) For A → ℐ (U, U′), E ∈ 𝒮 U, and a based space X, there is a canonical isomorphism

A ⋉ (E ∧ X) ≅ (A ⋉ E) ∧ X.

The functor A ⋉ (?) is a left adjoint [37, VI.1.1], and its right adjoint will be used in our construction of function S-modules.

Proposition 2.3. For any space A over ℐ (U, U′), the functor A ⋉ (?) has a right adjoint, which is denoted by F [A,?) and called a twisted function spectrum.

The functor A ⋉ E is homotopy-preserving in E, and it therefore preserves homotopy equivalences in the variable E. However, it only preserves homotopies over ℐ (U, U′) in A. Nevertheless, it very often preserves homotopy equivalences in the variable A. This fact will be essential in keeping control over the homotopical behavior of our point-set level constructions. To state it in proper generality, we need the following notion of a well-behaved spectrum.

Definition 2.4. A prespectrum D is Σ-cofibrant if each of its structure maps σ : Σ^{W} DV ⟶ D(V ⊕ W) is a cofibration of based spaces. A spectrum E is Σ-cofibrant if it is isomorphic to one of the form LD, where D is a Σ-cofibrant prespectrum. A spectrum E is tame if it is homotopy equivalent to a Σ-cofibrant spectrum.

We shall discuss such spectra in X§4, where we shall see that all shift desuspensions of based spaces are Σ-cofibrant and that all CW spectra are tame. We shall show in X§5 that structured ring or module spectra can be approximated functorially by weakly equivalent Σ-cofibrant spectra with the same structure.

Theorem 2.5. Let E ∈ 𝒮 U be tame and let A be a space over ℐ (U, U′). If ϕ : A′ ⟶ A is a homotopy equivalence, then ϕ ⋉ id : A′ ⋉ E ⟶ A ⋉ E is a homotopy equivalence.

Proof. We may assume without loss of generality that E = LD for a Σcofibrant prespectrum D. By the untwisting result in Proposition 2.1, the conclusion certainly holds when E = Σ_{n}^{∞}X for a based space X. By [37, I.4.7] (see X.4.4 below), LD ≅ colim Σ_{n}^{∞}D_{n}, where the colimit is taken over a sequence of cofibrations of spectra. Since the functor A ⋉ (?) has a right adjoint, it commutes with colimits, and it also preserves cofibrations. The conclusion follows since the colimit of a sequence of homotopy equivalences is a homotopy equivalence when the source and target colimits are taken over sequences of cofibrations.

Corollary 2.6. Let E ∈ 𝒮 U be a spectrum that has the homotopy type of a CW spectrum and let A be a space over ℐ (U, U′) that has the homotopy type of a CW complex. Then A ⋉ E has the homotopy type of a CW spectrum.

Proof. We may assume without loss of generality that E is a CW spectrum and A is a CW complex, in which case A ⋉ E is a CW spectrum by [37, VI.1.11].

3. The linear isometries operad and internal smash products

For the rest of the paper, we restrict attention to a particular universe U; the reader is welcome to consider it as notation for ℝ^{∞}. We agree to write 𝒮 instead of 𝒮 U for the category of spectra indexed on U. Except where explicitly stated otherwise, all given spectra, whatever extra structure they may have, will be in 𝒮. We are especially interested in twisted half smash products defined in terms of the following spaces of linear isometries.

Notations 3.1. Let U^{j} be the direct sum of j copies of U and let ℒ (j) = ℐ (U^{j}, U). The space ℒ (0) is the point i, where i : {0} → U, and ℒ (1) contains the identity map 1 = id_{U} : U → U. The left action of Σ_{j} on U^{j} by permutations induces a free right action of Σ_{j} on the contractible space ℒ (j). Define maps

γ : ℒ (k) × ℒ (j_{1}) × ⋯ × ℒ (j_{k}) ⟶ ℒ (j_{1} + ⋯ + j_{k})

by

γ(g; f_{1}, …, f_{k}) = g ∘ (f_{1} ⊕ ⋯ ⊕ f_{k}).

The spaces ℒ (j) form an operad [44, p.1] with structural maps γ, called the linear isometries operad. Points f ∈ ℒ (j) give inclusions {f} ⟶ ℒ (j). The corresponding twisted half-smash product is denoted f_{∗}; it sends spectra indexed on U^{j} to spectra indexed on U. Applied to a j-fold external smash product E_{1}∧⋯∧E_{j}, it gives an internal smash product f_{∗}(E_{1} ∧ ⋯ ∧ E_{j}). All of these smash products become equivalent in the stable homotopy category h̄𝒮, but none of them are associative or commutative on the point set level. In fact, the following sharper version of this assertion holds.

3. THE LINEAR ISOMETRIES OPERAD AND INTERNAL SMASH PRODUCTS 15

Theorem 3.2. Let 𝒮_{t} ⊂ 𝒮 be the full subcategory of tame spectra and let h𝒮_{t} be its homotopy category. The internal smash products f_{∗}(E ∧ E′) determined by varying f ∈ ℒ (2) are canonically isomorphic in h𝒮_{t}, and h𝒮_{t} is symmetric monoidal under the internal smash product. For based spaces X and tame spectra E, there is a natural isomorphism E ∧ X ≃ f_{∗}(E ∧ Σ^{∞}X) in h𝒮_{t}.

Proof. The external and internal smash products of Σ-cofibrant spectra are Σ-cofibrant by results in X§4. By Theorem 2.5, for any f ∈ ℒ (j) and any spectra E_{i} ∈ 𝒮_{t}, the map

f_{∗}(E_{1} ∧ ⋯ ∧ E_{j}) ⟶ ℒ (j) ⋉ (E_{1} ∧ ⋯ ∧ E_{j})

induced by the inclusion {f} ⟶ ℒ (j) is a homotopy equivalence. Taking j = 2, this shows that the internal smash products obtained from varying f are homotopy equivalent. Replacing f by f ∘ σ, where σ ∈ Σ_{2} is the transposition, we obtain a natural homotopy equivalence

f_{∗}(E_{2} ∧ E_{1}) ⟶ ℒ (2) ⋉ E_{1} ∧ E_{2},

and this shows that the internal smash product is commutative up to homotopy. Similarly, for associativity, the inclusions of the points {f(1 ⊕ f)} and {f(f ⊕ 1)} in ℒ (3) induce natural homotopy equivalences

f_{∗}(E_{1} ∧ f_{∗}(E_{2} ∧ E_{3})) ⟶ ℒ (3) ⋉ (E_{1} ∧ E_{2} ∧ E_{3}) ⟵ f_{∗}(f_{∗}(E_{1} ∧ E_{2}) ∧ E_{3}).

It is natural to think of based spaces as spectra indexed on the universe {0}. Then i_{∗} and the suspension spectrum functor are both left adjoint to the zeroth space functor, hence i_{∗}X ≅ Σ^{∞}X. The map ℒ (2) ⟶ ℒ (1) that sends f to f ∘ (1 ⊕ i) and the inclusion {1} ⟶ ℒ (1) induce natural homotopy equivalences

f_{∗}(E ∧ Σ^{∞}X) ⟶ ℒ (1) ⋉ (E ∧ X) ⟵ E ∧ X.

Thus, up to natural isomorphisms, the internal smash product determined by f becomes commutative, associative, and unital with unit S = Σ^{∞}S^{0} on passage to h𝒮_{t}. The commutativity of coherence diagrams that is required for the assertion that h𝒮_{t} is symmetric monoidal (see [42, p. 180]) can be checked by an elaboration of these arguments.

The following consequence strengthens the assertion [37, I.6.1] that the stable homotopy category really is a stable category, in the sense that the suspension and loop functors Σ and Ω pass to inverse self-equivalences of h̄𝒮.

Corollary 3.3. For tame spectra E and f ∈ ℒ (2), there is a natural homotopy equivalence between ΩE and f_{∗}(E ∧ S^{−1}), and the unit η : E ⟶ ΩΣE and counit ε : ΣΩE ⟶ E of the (Σ, Ω)-adjunction are homotopy equivalences.

Proof. For based spaces X, Σ^{∞}X is naturally isomorphic to Σ(Σ_{1}^{∞}X) since the structural homeomorphism E_{0} ⟶ ΩE_{1} gives a natural isomorphism between their right adjoints. Therefore, for E ∈ 𝒮_{t}, there is a natural homotopy equivalence

E = E ∧ S^{0} ≃ f_{∗}(E ∧ Σ^{∞}S^{0}) ≅ f_{∗}(E ∧ Σ(Σ_{1}^{∞}S^{0})) ≅ Σ(f_{∗}(E ∧ S^{−1})),

where the last isomorphism is given by Proposition 2.2(iv). It follows that, on h𝒮_{t}, the functor Σ is an adjoint equivalence with inverse given by the functor f_{∗}(E ∧ S^{−1}). The rest is a formal consequence of the uniqueness of adjoints.

Note that only actual homotopy equivalences, not weak ones, are relevant to these results. For this and other reasons, h𝒮_{t} will be a technically convenient halfway house between h𝒮 and the stable homotopy category h̄𝒮, which is obtained from either of these homotopy categories by inverting the weak equivalences.

We can deduce that cofiber sequences give rise to long exact sequences of homotopy groups.

Corollary 3.4. Any cofiber sequence E⟶^{f}E′ ⟶ Cf of tame spectra gives rise to a long exact sequence of homotopy groups

⋯ ⟶ π_{q}(E) ⟶ π_{q}(E′) ⟶ π_{q}(Cf) ⟶ π_{q−1}(E) ⟶ · · · .

Therefore the natural map F f ⟶ ΩCf is a weak equivalence.

Proof. Consider the diagram

[figure: S^{q}; id; Σ^{−1}γ; S^{q}; α; CS^{q}; β; ΣS^{q}; id; γ; ΣS^{q}; Σα; E; f; E′; i; Cf; ΣE; Σf; ΣE′.]

Here α is given such that i ∘ α ≃ 0. A homotopy induces a map β such that the second square commutes. The usual cofiber sequence argument gives γ such that the right two squares homotopy commute. Since η_{E} : E ⟶ ΩΣE is a weak equivalence, there is a map Σ^{−1}γ : S^{q} ⟶ E, unique up to homotopy, such that

η_{E} ∘ Σ^{−1}γ ≃ Ωγ ∘ η_{S^{q}}.

Therefore

η_{E′} ∘ f ∘ Σ^{−1}γ = ΩΣf ∘ η_{E} ∘ Σ^{−1}γ ≃ ΩΣf ∘ Ωγ ∘ η_{S^{q}} ≃ ΩΣα ∘ η_{S^{q}} = η_{E′} ∘ α.

Since η_{E′} is a weak equivalence, this implies that f ∘ Σ^{−1}γ ≃ α. The long exact sequence follows by extending the given cofiber sequence to the right, as usual. The last statement follows by the five lemma and a comparison of our cofiber sequence with the fiber sequence associated to f. Details of this may be found in [37, pp 128-130]. For later use, observe that we only used that the maps η are weak equivalences, not that they are homotopy equivalences, in this proof.

It follows that cofiber sequences are essentially equivalent to fiber sequences. More precisely, the cofibrations and fibrations give “triangulations” of the stable homotopy category such that the negative of a cofibration triangle is a fibration triangle, and conversely [37, pp 128-130].

Corollary 3.5. Pushouts of tame spectra along cofibrations preserve weak equivalences. That is, for a commutative diagram of tame spectra

[figure: E; β; D; i; f; α; F; γ; E′; D′; i′; f′; F′]

in which i and i′ are cofibrations and α, β, and γ are weak equivalences, the induced map δ : E ∪_{D} F ⟶ E′ ∪_{D′} F′ of pushouts is a weak equivalence.

Proof. As for spaces, Ci is homotopy equivalent to E/D, the induced map F ⟶ E ∪_{D} F is a cofibration, and the induced map E/D ⟶ E ∪_{D} F/F is an isomorphism. The conclusion follows from the previous corollary by a diagram chase and the five lemma.

Proposition 3.6. If E is a CW spectrum and ϕ : F ⟶ F′ is a weak equivalence between tame spectra, then f_{∗}(id ∧ϕ) : f_{∗}(E ∧ F) ⟶ f_{∗}(E ∧ F′) is a weak equivalence.

Proof. The functor f_{∗}((?) ∧ F) preserves cofiber sequences. Therefore, by Corollary 3.5 and induction up the sequential filtration of E (see III.2.1), the result will hold for general E if it holds for E = S^{n}. When E = S, the conclusion holds by the unit equivalence f_{∗}(S ∧F) ≃ F of Theorem 3.2. For n > 0, we easily deduce isomorphisms

f_{∗}(S^{n} ∧ F) ≅ Σ^{n}f_{∗}(S ∧ F) and Σ^{n}f_{∗}(S^{−n} ∧ F) ≅ f_{∗}(S ∧ F)

from Proposition 2.2(iv). In view of Corollary 3.3, the result for E = S^{−n} and E = S^{n} therefore follows from the result for S.

It follows that for general spectra E and tame spectra F, the smash product E ∧ F in the stable homotopy category h̄𝒮 is represented by f_{∗}(ΓE ∧ F), where ΓE is a CW spectrum weakly equivalent to E. That is, we do not also have to apply CW approximation to F. The mild restriction to tame spectra serves to avoid pathological point-set behavior.

4. The category of 𝕃-spectra

We think of ℒ (j)⋉(E_{1}∧⋯∧E_{j}) as a canonical j-fold internal smash product. It is still not associative, but we shall construct a commutative and associative smash product by restricting to 𝕃-spectra and shrinking the fat out of the construction. To define 𝕃-spectra, we focus attention on a small part of the operad ℒ. Recall the notion of a monad in a category from [42, ch.VI] or [44, 2.1].

Notations 4.1. Let 𝕃 denote the monad in the category 𝒮 that is specified by 𝕃E = ℒ (1) ⋉ E; the product

μ : 𝕃𝕃E ≅ (ℒ (1) × ℒ (1)) ⋉ E ⟶ ℒ (1) ⋉ E = 𝕃E

is induced by the product γ : ℒ (1) × ℒ (1) ⟶ ℒ (1) and the unit

η : E ≅ {1} ⋉ E ⟶ ℒ (1) ⋉ E = 𝕃E

is induced by the inclusion {1} ⟶ ℒ (1) of the identity element.

Definition 4.2. An 𝕃-spectrum is an 𝕃-algebra M, that is, a spectrum M together with an action ξ : 𝕃M ⟶ M by the monad 𝕃. Explicitly, the following diagrams are required to commute:

[figure: 𝕃𝕃M; μ; 𝕃ξ; 𝕃M; ξ; and; M; =; η; 𝕃M; ξ; 𝕃M; ξ; M; M.]

A map f : M → N of 𝕃-spectra is a map of spectra such that the following diagram commutes:

[figure: 𝕃M; 𝕃f; ξ_{M}; 𝕃N; ξ_{N}; M; f; N.]

We let 𝒮 [𝕃] denote the category of 𝕃-spectra.

There is a dual form of the definition that will occasionally be needed. It is based on the following standard categorical observation.

Lemma 4.3. Let 𝕋 be a monad in a category 𝒞, and suppose that the functor 𝕋 has a right adjoint 𝕋^{#}. Then 𝕋^{#} is a comonad such that the categories of 𝕋-algebras and of 𝕋^{#}-coalgebras are isomorphic.

4. THE CATEGORY OF 𝕃-SPECTRA 19

We shall consistently use the notation 𝕋^{#} for the comonad associated to a monad 𝕋 that has a right adjoint. In particular, by Proposition 2.3, we now have a comonad 𝕃^{#} such that an 𝕃^{#}-coalgebra is the same thing as an 𝕃-spectrum. This implies the following result.

Proposition 4.4. The category of 𝕃-spectra is complete and cocomplete, with both limits and colimits created in the underlying category 𝒮. If X is a based space and M is an 𝕃-spectrum, then M ∧ X and F (X, M) are 𝕃-spectra, and the spectrum level fiber and cofiber of a map of 𝕃-spectra are 𝕃-spectra.

Proof. Since 𝒮 [𝕃] is the category of algebras over the monad 𝕃, the forgetful functor 𝒮 [𝕃] → 𝒮 creates limits [42, VI.2, ex. 2]. Since 𝒮 is complete, this implies the statement about limits. The statement about colimits follows similarly by use of the comonad 𝕃^{#}. The last statement is immediate from the canonical isomorphism

ℒ (1) ⋉ (M ∧ X) ≅ (ℒ (1) ⋉ M) ∧ X

of Proposition 2.2(iv) and its analog [37, VI.1.5]

F [ℒ (1), F (X, M)) ≅ F (X, F [ℒ (1), M)).

Lemma 4.5. The sphere spectrum S is an 𝕃-spectrum. More generally, for based spaces X, Σ^{∞}X ≅ S ∧ X is naturally an 𝕃-spectrum.

Proof. Recall from the proof of Theorem 3.2 that a based space X may be viewed as a spectrum indexed on {0} and that Σ^{∞}X ≅ i_{∗}X, i : {0} ⟶ U. We may rewrite this as Σ^{∞}X = ℒ (0) ⋉ X. Then the structure map is given by

γ ⋉ id : ℒ (1) ⋉ (ℒ (0) ⋉ X) ≅ (ℒ (1) × ℒ (0)) ⋉ X ⟶ ℒ (0) ⋉ X.

In the middle, ℒ (1) × ℒ (0) is regarded as a space over ℒ (0) via γ, and the isomorphism is given by an instance of Proposition 2.2(ii). Of course, γ here is just the unique map from ℒ (1) to the one-point space ℒ (0), and our structure map is just the composite

ℒ (1) ⋉ Σ^{∞}X ≅ Σ^{∞}(ℒ (1)_{+} ∧ X) ⟶ Σ^{∞}(S^{0} ∧ X) ≅ Σ^{∞}X,

where the first isomorphism is given by Proposition 2.1.

Warning 4.6. We issue a technical warning: it is neither necessary nor useful to consider possible 𝕃-spectrum structures on the shift desuspensions Σ_{n}^{∞}X for n > 0. Any spectrum E is isomorphic to the colimit of the shift desuspensions Σ_{n}^{∞}E_{n} of its component spaces [22, I.4.7], and it is easy to construct a fallacious proof that every spectrum is an 𝕃-spectrum, the fallacy being that one cannot give shift desuspensions 𝕃-spectra structures that make the maps of the colimit system maps of 𝕃-spectra.

A homotopy in the category of 𝕃-spectra is a map M ∧ I_{+} ⟶ N. A map of 𝕃-spectra is a weak equivalence if it is a weak equivalence as a map of spectra. The stable homotopy category h̄𝒮 [𝕃] of 𝕃-spectra is constructed from the homotopy category h𝒮 [𝕃] by adjoining formal inverses to the weak equivalences; again, the process is made rigorous by CW approximation. Since the theory of cell and CW 𝕃-spectra is exactly like the theory of cell and CW spectra developed in [37, I§5], we shall not give details. The reader who would like to see an exposition is invited to look ahead to III§2. The theory of cell R-modules to be presented there applies (with minor simplifications) to give what is needed. It is formal that the monad 𝕃 may be viewed as specifying the free functor from spectra to 𝕃-spectra. The sphere 𝕃-spectra that we take as the domains of attaching maps when defining cell 𝕃-spectra are the free 𝕃-spectra 𝕃S^{n} = ℒ (1) ⋉ S^{n}. A weak equivalence of cell 𝕃spectra is a homotopy equivalence, any 𝕃-spectrum is weakly equivalent to a CW 𝕃-spectrum, and h̄𝒮 [𝕃] is equivalent to the homotopy category of CW 𝕃-spectra. We warn the reader that, although S itself is an 𝕃-spectrum, it does not have the homotopy type of a CW 𝕃-spectrum (see Warning 6.8 below). The following comparison between CW spectra and CW 𝕃-spectra establishes an equivalence between h̄𝒮 and h̄𝒮 [𝕃].

Theorem 4.7. The following conclusions hold.

(i) The free functor 𝕃 : 𝒮 ⟶ 𝒮 [𝕃] carries CW spectra to CW 𝕃-spectra.

(ii) The forgetful functor 𝒮 [𝕃] ⟶ 𝒮 carries 𝕃-spectra of the homotopy types

of CW 𝕃-spectra to spectra of the homotopy types of CW spectra. (iii) Every CW 𝕃-spectrum M is homotopy equivalent as an 𝕃-spectrum to 𝕃E

for some CW spectrum E. (iv) The unit η : E ⟶ 𝕃E of the adjunction

𝒮 [𝕃](𝕃E, M) ≅ 𝒮 (E, M)

is a homotopy equivalence if E ∈ 𝒮_{t}, for example if E is a CW spectrum.

(v) The counit ξ : 𝕃M ⟶ M of the adjunction is a homotopy equivalence of spectra if M is tame and is a homotopy equivalence of 𝕃-spectra if M has the homotopy type of a CW 𝕃-spectrum.

The free and forgetful functors establish an adjoint equivalence between the stable homotopy categories h̄𝒮 and h̄𝒮 [𝕃].

Proof. Part (i) is immediate by induction up the sequential filtration (see III.2.1). Part (iv) is immediate from Theorem 2.5 and, applied to sphere spectra, it implies (ii). Since ξ ∘ η = id : M ⟶ M for any M, (iv) and the Whitehead theorem in the category of 𝕃-spectra imply (v). Part (iii) follows from (i) and (v) since there is a CW spectrum E and a homotopy equivalence of spectra E ⟶ M.

5. THE SMASH PRODUCT OF 𝕃-SPECTRA 21

It is a formal consequence of (i) that we have an induced adjunction

h̄𝒮 [𝕃](𝕃E, M) ≅ h̄𝒮 (E, M)

(see [37, I.5.13]), and its unit and counit are natural isomorphisms.

Observe that, dually, we can interpret 𝕃^{#} as specifying the “cofree” functor from spectra to 𝕃-spectra. That is, we have an adjunction

𝒮 [𝕃](M, 𝕃^{#}E) ≅ 𝒮 (M, E). (4.8)

By part (ii) of the theorem and [37, I.5.13], there results an induced adjunction

h̄𝒮 [𝕃](M, 𝕃^{#}E) ≅ h̄𝒮 (M, E).

It is an easy categorical observation that, in any adjoint equivalence of categories, the given left and right adjoints are also right and left adjoint to each other.

Corollary 4.9. The functors 𝕃 : h̄𝒮 ⟶ h̄𝒮 [𝕃] and 𝕃^{#} : h̄𝒮 ⟶ h̄𝒮 [𝕃] are naturally isomorphic.

5. The smash product of 𝕃-spectra

Via instances of the structural maps γ of the operad ℒ, we have a left action of the monoid ℒ (1) and a right action of the monoid ℒ (1) × ℒ (1) on ℒ (2). These actions commute with each other. If M and N are 𝕃-spectra, then ℒ (1) × ℒ (1) acts from the left on the external smash product M ∧ N via the map

[figure: ξ : (ℒ (1) × ℒ (1)) ⋉ (M ∧ N) ≅ (ℒ (1) ⋉ M) ∧ (ℒ (1) ⋉ N); ξ∧ξ; M ∧ N.]

To form the twisted half smash product on the left, we think of ℒ (1) × ℒ (1) as mapping to ℐ (U^{2}, U^{2}) via direct sum of linear isometries. The smash product over ℒ of M and N is simply the balanced product of the two ℒ (1) × ℒ (1)-actions.

Definition 5.1. Let M and N be 𝕃-spectra. Define the operadic smash product M ∧_{ℒ} N to be the coequalizer displayed in the diagram

[figure: (ℒ (2) × ℒ (1) × ℒ (1)) ⋉ (M ∧ N); γ⋉id; id ⋉ξ; ℒ (2) ⋉ (M ∧ N); M ∧_{ℒ} N.]

Here we have implicitly used the isomorphism

(ℒ (2) × ℒ (1) × ℒ (1)) ⋉ (M ∧ N) ≅ ℒ (2) ⋉ [(ℒ (1) × ℒ (1)) ⋉ (M ∧ N)]

given by Proposition 2.2(ii). The left action of ℒ (1) on ℒ (2) induces a left action of ℒ (1) on M ∧_{ℒ} N that gives it a structure of 𝕃-spectrum.

We may mimic tensor product notation and write

M ∧_{ℒ} N = ℒ (2) ⋉_{ℒ (1)×ℒ (1)} (M ∧ N).

We will freely use such notations for coequalizers below. The commutativity of this smash product is immediate.

Proposition 5.2. There is a natural commutativity isomorphism of 𝕃-spectra

τ : M ∧_{ℒ} N ⟶ N ∧_{ℒ} M.

Proof. The permutation σ ∈ Σ_{2} acts on ℒ (2) by fσ = f ∘t, where t : U^{2} → U^{2} is the transposition isomorphism. We may regard σ as a map of spaces over ℒ (2) from id : ℒ (2) ⟶ ℒ (2) to σ : ℒ (2) ⟶ ℒ (2). We have an evident isomorphism ι : t_{∗}(M ∧ N) ≅ N ∧ M on external smash products and, by Proposition 2.2(ii), there results a canonical isomorphism

σ ⋉ ι : ℒ (2) ⋉ M ∧ N ≅ ℒ (2) ⋉ t_{∗}(M ∧ N) ≅ ℒ (2) ⋉ N ∧ M.

There is an analogous isomorphism

(σ×t)⋉ι : (ℒ (2)×ℒ (1)×ℒ (1))⋉(M ∧N) ⟶ (ℒ (2)×ℒ (1)×ℒ (1))⋉(N ∧M).

These maps induce an isomorphism of coequalizer diagrams

[figure: (ℒ (2) × ℒ (1) × ℒ (1)) ⋉ (M ∧ N); (σ×t)⋉ι; γ⋉id; id ⋉ξ; ℒ (2) ⋉ (M ∧ N); σ⋉ι; M ∧_{ℒ} N; τ; (ℒ (2) × ℒ (1) × ℒ (1)) ⋉ (N ∧ M); γ⋉id; id ⋉ξ; ℒ (2) ⋉ (N ∧ M); N ∧_{ℒ} M.]

To show that this smash product is associative, we need some preliminary material on coequalizers. We first recall a standard categorical definition [42, VI.6].

Definition 5.3. Working in an arbitrary category, suppose given a diagram

[figure: A; e; f; B; g; C]

in which ge = gf. The diagram is called a split coequalizer if there are maps

h : C → B and k : B → A

such that gh = id_{C}, fk = id_{B}, and ek = hg. It follows that g is the coequalizer of e and f.

Observe that, while covariant functors need not preserve coequalizers in general, they clearly do preserve split coequalizers. The next observation is crucial; we learned it from Hopkins [31]. Note that, via structural maps γ, ℒ (1) acts from the left on any ℒ (i), hence ℒ (1) × ℒ (1) acts from the left on ℒ (i) × ℒ (j).

Lemma 5.4 (Hopkins). For i ≥ 1 and j ≥ 1, the diagram

[figure: ℒ (2) × ℒ (1) × ℒ (1) × ℒ (i) × ℒ (j); γ×id; id ×γ^{2}; ℒ (2) × ℒ (i) × ℒ (j); γ; ℒ (i + j)]

is a split coequalizer of spaces. Therefore,

ℒ (i + j) ≅ ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (i) × ℒ (j).

Proof. Choose isomorphisms s : U^{i} → U and t : U^{j} → U and define

h(f) = (f ∘ (s ⊕ t)^{−1}, s, t)

and

k(f; g, g′) = (f; g ∘ s^{−1}, g′ ∘ t^{−1}; s, t).

It is trivial to check the identities of Definition 5.3.

Theorem 5.5. There is a natural associativity isomorphism of 𝕃-spectra

(M ∧_{ℒ} N) ∧_{ℒ} P ≅ M ∧_{ℒ} (N ∧_{ℒ} P).

Proof. Note that, for any 𝕃-spectrum N, N ≅ ℒ (1)⋉_{ℒ (1)} N since ℒ (1)⋉N = 𝕃N and, as with any monad [42, p. 148], we have a split coequalizer

\xymatrix{𝕃𝕃N \ar[r] \ar[r] & 𝕃N \ar[r] & N.}

We have the isomorphisms

(M ∧_{ℒ} N) ∧_{ℒ} P ≅ ℒ (2) ⋉_{ℒ (1)^{2}} (ℒ (2) ⋉_{ℒ (1)^{2}} (M ∧ N)) ∧ (ℒ (1) ⋉_{ℒ (1)} P)

≅ (ℒ (2) ×_{ℒ (1)^{2}} ℒ (2) × ℒ (1)) ⋉_{ℒ (1)^{3}} (M ∧ N ∧ P)
≅ ℒ (3) ⋉_{ℒ (1)^{3}} M ∧ N ∧ P.

The symmetric argument shows that this is also isomorphic to M ∧_{ℒ} (N ∧_{ℒ} P).

In view of the generality of Lemma 5.4, the argument iterates to prove the following statement.

Theorem 5.6. For any j-tuple M_{1}, …, M_{j} of 𝕃-spectra, there is a canonical isomorphism of 𝕃-spectra

M_{1} ∧_{ℒ} ⋯ ∧_{ℒ} M_{j} ≅ ℒ (j) ⋉_{ℒ (1)^{j}} (M_{1} ∧ ⋯ ∧ M_{j}),

where the iterated smash product on the left is associated in any fashion.

6. The equivalence of the old and new smash products

We here show that the smash product ∧_{ℒ} does in fact realize the classical smash product of spectra up to homotopy, in the sense that the equivalence between h̄𝒮 and h̄𝒮 [𝕃] preserves smash products.

Fix a linear isometric isomorphism f : U^{2} ⟶ U (not just an isometry) and use it to define the internal smash product of spectra in this section. We begin the comparison of smash products of 𝕃-spectra with smash products of spectra with the following observation.

Proposition 6.1. For spectra X and Y, there are isomorphisms of 𝕃-spectra

𝕃X ∧_{ℒ} 𝕃Y ≅ ℒ (2) ⋉ X ∧ Y ≅ 𝕃f_{∗}(X ∧ Y).

For CW 𝕃-spectra M and N, M ∧_{ℒ} N is a CW 𝕃-spectrum with one (p + q)-cell for each p-cell of M and q-cell of N.

Proof. The first isomorphism is immediate from the definition of ∧_{ℒ} . Regarding f as a point in ℒ (2), we see that γ : ℒ (1)×{f} ⟶ ℒ (2) is a homeomorphism since f is an isomorphism. It follows from Proposition 2.2(ii) that

𝕃f_{∗}(X ∧ Y) = ℒ (1) ⋉ f_{∗}(X ∧ Y) ≅ ℒ (2) ⋉ (X ∧ Y).

When X and Y are sphere spectra, so is f_{∗}(X ∧ Y) [37, II.1.4]. The second statement now follows exactly as for the smash product of CW complexes or CW spectra.

The crux of our comparison of smash products is the following proposition, which implies that 𝕃S is the unit for the smash product in the stable homotopy category h̄𝒮 [𝕃]. We defer the proof to XI§3.

Proposition 6.2. For 𝕃-spectra N, there is a natural weak equivalence of 𝕃spectra ω : 𝕃S ∧_{ℒ} N ⟶ N, and Σ : π_{n}(N) ⟶ π_{n+1}(ΣN) is an isomorphism for all integers n.

If we knew a priori that Σ preserved weak equivalences, we could derive the second clause from the first and the natural isomorphism of 𝕃-spectra

𝕃S ∧_{ℒ} N ≅ Σ(𝕃S^{−1} ∧_{ℒ} N)

by a formal uniqueness of adjoints argument (compare Corollary 3.3). It is a pleasant and surprising technical feature of our theory, immediate from the proposition, that Σ preserves weak equivalences of 𝕃-spectra. That is, the 𝕃 structure somehow has the effect of eliminating point-set pathology. Since Σ on homotopy groups is induced by η : N ⟶ ΩΣN, the proposition also has the following immediate consequence.

Corollary 6.3. For 𝕃-spectra N, the unit η : N ⟶ ΩΣN and counit ε : ΣΩN ⟶ N of the (Σ, Ω)-adjunction are weak equivalences.

Corollary 6.4. Any cofiber sequence N⟶^{f}N′ ⟶ Cf of 𝕃-spectra gives rise to a long exact sequence of homotopy groups

⋯ ⟶ π_{q}(N) ⟶ π_{q}(N′) ⟶ π_{q}(Cf) ⟶ π_{q−1}(N) ⟶ · · · .

Therefore the natural map F f ⟶ ΩCf is a weak equivalence of 𝕃-spectra.

Proof. This follows from Corollary 6.3 via the proof of Corollary 3.4.

Corollary 6.5. Pushouts along cofibrations of 𝕃-spectra preserve weak equivalences.

Proof. Since a cofibration of 𝕃-spectra is a cofibration of spectra, by the retraction of mapping cylinders criterion, this follows from Corollary 6.4 via the proof of Corollary 3.5.

Proposition 6.6. If M is a CW 𝕃-spectrum and ϕ : N ⟶ N′ is a weak equivalence of 𝕃-spectra, then id ∧_{ℒ} ϕ : M∧_{ℒ} N ⟶ M∧_{ℒ} N′ is a weak equivalence of 𝕃-spectra.

Proof. The functor (?) ∧_{ℒ} N preserves cofiber sequences, hence the result for general M follows from Corollary 6.4 and the result for M = 𝕃S^{n}. Here the result for n = 0 follows from Proposition 6.2 and the result for n and −n, n > 0, follows from the result for n = 0 as in the proof of Proposition 3.6.

Thus, for 𝕃-spectra M and N, the smash product M∧_{ℒ} N in the stable homotopy category h̄𝒮 [𝕃] is represented by ΓM ∧_{ℒ} N, where ΓM is a CW 𝕃-spectrum weakly equivalent to M; here we do not need to assume that N is tame. This is analogous to the situation in algebra. When transporting tensor products to algebraic derived categories, we need only apply cell approximation to one of the tensor factors, without any condition on the other [34].

Theorem 6.7. For 𝕃-spectra M and N, there is a natural map of spectra α : f_{∗}(M ∧N) ⟶ M ∧_{ℒ} N, and α is a weak equivalence when M is a CW 𝕃-spectrum and N is a tame spectrum. For any 𝕃-spectrum N, the functor (?) ∧_{ℒ} N from h̄𝒮 [𝕃] to h̄𝒮 computes the derived internal smash product with N.

Proof. Define α to be the composite

f_{∗}(M ∧ N) ⟶ ℒ (2) ⋉ M ∧ N ⟶ M ∧_{ℒ} N

given by the the inclusion of {f} in ℒ (2) and the definition of ∧_{ℒ} . Let M be a CW 𝕃-spectrum throughout the proof. We first show that α is an equivalence when N is also a CW 𝕃-spectrum. In this case, M and N have the homotopy types of CW spectra by Theorem 4.7 and are therefore tame by X.4.3. Thus the first map is a homotopy equivalence by Theorem 2.5. By Theorem 4.7(iii), we may assume without loss of generality that M = ℒ (1) ⋉ X and N = ℒ (1) ⋉ Y for CW spectra X and Y. The second arrow then reduces to the homotopy equivalence

ℒ (2) ⋉ (ℒ (1) ⋉ X) ∧ (ℒ (1) ⋉ Y) ⟶ ℒ (2) ⋉ X ∧ Y

induced by the homotopy equivalence γ : ℒ (2) × ℒ (1) × ℒ (1) ⟶ ℒ (2) via Theorem 2.5. For a general 𝕃-spectrum N, choose a weak equivalence γ : ΓN ⟶ N, where ΓN is a CW 𝕃-spectrum. If N is tame, then Propositions 3.6 and 6.6 imply that the vertical arrows are weak equivalences in the commutative diagram

[figure: f_{∗}(M ∧ ΓN); α; id ∧γ; M ∧_{ℒ} ΓN; id ∧γ; f_{∗}(M ∧ N); α; M ∧_{ℒ} N.]

Thus the bottom arrow α is a weak equivalence since the top one is. For the last statement, simply note that the right-hand composite

(id ∧γ) ∘ α : f_{∗}(M ∧ ΓN) ⟶ M ∧_{ℒ} N

in the diagram is a weak equivalence even when N is not tame.

Warning 6.8. As said before, the sphere 𝕃-spectrum S does not have the homotopy type of a CW 𝕃-spectrum. To see this, assume that it did. Then the action ξ : 𝕃S ⟶ S would be a homotopy equivalence of 𝕃-spectra, by the Whitehead theorem, and the Σ_{2}-equivariant map

ξ ∧_{ℒ} ξ : 𝕃S ∧_{ℒ} 𝕃S ⟶ S ∧_{ℒ} S

would be a homotopy equivalence of 𝕃-spectra and thus of spectra. By Propositions 6.1 and 2.1, 𝕃S ∧_{ℒ} 𝕃S is isomorphic to Σ^{∞}(ℒ (2)_{+}), with Σ_{2}-action induced by that on ℒ (2). By Proposition 8.2 below, S ∧_{ℒ} S is isomorphic to S = Σ^{∞}S^{0} and has trivial action by Σ_{2}. Under these isomorphisms, ξ ∧_{ℒ} ξ coincides with Σ^{∞}π, where π : ℒ (2)_{+} ⟶ S^{0} sends all of ℒ (2) to the non-basepoint. Since ℒ (2)/Σ_{2} ≃ B(Σ_{2}), our assumption implies that we obtain a homotopy equivalence Σ^{∞}B(Σ_{2})_{+} ⟶ Σ^{∞}S^{0} on passage to orbits from ξ ∧_{ℒ} ξ, which is absurd. This argument also shows that our hypothesis that M be a CW 𝕃-spectrum is crucial in the previous two results.

7. FUNCTION 𝕃-SPECTRA 27

7. Function 𝕃-spectra

We here construct a functor F_{ℒ} on 𝕃-spectra that is related to the smash product ∧_{ℒ} by an adjunction of the usual form and consider its homotopical behavior.

Theorem 7.1. Let M, N, and P be 𝕃-spectra. There is a function 𝕃-spectrum functor F_{ℒ} (M, N), contravariant in M and covariant in N, such that

𝒮 [𝕃](M ∧_{ℒ} N, P) ≅ 𝒮 [𝕃](M, F_{ℒ} (N, P)).

Given the adjunction, we can deduce the homotopical behavior of F_{ℒ} from that of ∧_{ℒ} . We run through this before turning to the construction. The following result is a formal consequence of Proposition 6.1; see [37, I.5.13].

Proposition 7.2. If M is a CW 𝕃-spectrum and ϕ : N → N′ is a weak equivalence of 𝕃-spectra, then

F_{ℒ} (id, ϕ) : F_{ℒ} (M, N) ⟶ F_{ℒ} (M, N′)

is a weak equivalence of 𝕃-spectra. There is an induced adjunction

h̄𝒮 [𝕃](M ∧_{ℒ} N, P) ≅ h̄𝒮 [𝕃](M, F_{ℒ} (N, P)).

As in Section 6, we fix a linear isometric isomorphism f : U^{2} ⟶ U and use the isomorphism f_{∗} : 𝒮 U^{2} ⟶ 𝒮 U to define internal smash products f_{∗}(M ∧ N). Recall the external function spectrum F (M,?) and the adjunction displayed for it at the start of Section 2. We use the inverse isomorphism f^{∗} = f_{∗}^{−1} : 𝒮 U ⟶ 𝒮 U^{2} to define internal function spectra F (M, f^{∗}N), as in [37, II.3.11].

Theorem 7.3. For 𝕃-spectra M and N, there is a natural map of spectra

α̃ : F_{ℒ} (M, N) ⟶ F (M, f^{∗}N),

and α̃ is a weak equivalence if M is a CW 𝕃-spectrum. Therefore the equivalence of categories h̄𝒮 [𝕃] ⟶ h̄𝒮 induced by the forgetful functor from 𝕃-spectra to spectra carries the function 𝕃-spectrum functor F_{ℒ} to the internal function spectrum functor F.

Proof. In the category h̄𝒮 [𝕃], F_{ℒ} (M, N) means F_{ℒ} (ΓM, N) where ΓM is a CW 𝕃-spectrum weakly equivalent to M, hence the second statement will follow from the first. The desired map α̃ is the adjoint of the composite

f_{∗}(F_{ℒ} (M, N) ∧ M)⟶^{α}F_{ℒ} (M, N) ∧_{ℒ} M⟶^{ε}N,

where α is given by Theorem 6.7. By that result, if M is a CW 𝕃-spectrum and X is a CW spectrum, then α : f_{∗}(𝕃X ∧ M) ⟶ 𝕃X ∧_{ℒ} M is a weak equivalence of spectra, and it induces a weak equivalence of 𝕃-spectra

𝕃(f_{∗}(𝕃X ∧ M)) ⟶ 𝕃X ∧_{ℒ} M.

Diagram chases show that the map

α̃_{∗} : h̄𝒮 (X, F_{ℒ} (M, N)) ⟶ h̄𝒮 (X, F (M, f^{∗}N))

coincides with the composite of the following chain of natural isomorphisms:

h̄𝒮 (X, F_{ℒ} (M, N)) ≅ h̄𝒮 [𝕃](𝕃X, F_{ℒ} (M, N))
≅ h̄𝒮 [𝕃](𝕃X ∧_{ℒ} M, N) ≅ h̄𝒮 [𝕃](𝕃(f_{∗}(𝕃X ∧ M)), N)
≅ h̄𝒮 (f_{∗}(𝕃X ∧ M), N) ≅ h̄𝒮 (𝕃X ∧ M, f^{∗}N)
≅ h̄𝒮 (𝕃X, F (M, f^{∗}N)) ≅ h̄𝒮 (X, F (M, f^{∗}N)).

Lemma 7.4. The adjoint N ⟶ F_{ℒ} (𝕃S, N) of the unit weak equivalence ω : 𝕃S ∧_{ℒ} N ⟶ N is a weak equivalence.

Proof. This is immediate from the natural isomorphisms

h̄𝒮 [𝕃](M, N) ≅ h̄𝒮 [𝕃](𝕃S ∧_{ℒ} M, N) ≅ h̄𝒮 [𝕃](M, F_{ℒ} (𝕃S, N)).

We must still prove Theorem 7.1. The desired adjunction dictates the definition of F_{ℒ} , and the reader is invited to skip to the next section. It will be simplest to construct F_{ℒ} in two steps. Remember that

M ∧_{ℒ} N = ℒ (2) ⋉_{ℒ (1)×ℒ (1)} M ∧ N.

In the first step we consider general spectra indexed on U^{2} and acted upon by ℒ (1) × ℒ (1), thought of as a space over ℐ (U^{2}, U^{2}) via direct sum of isometries. We call these ℒ (1) × ℒ (1)-spectra and denote the category of such spectra by 𝒮 [ℒ (1) × ℒ (1)]. Of course, the examples we have in mind are of the form M ∧ N. We use the twisted function spectrum construction F [A,?) of Proposition 2.3.

Lemma 7.5. Let N be an 𝕃-spectrum. There is an ℒ (1) × ℒ (1)-spectrum F_{ℒ (1)} [ℒ (2), N) ∈ 𝒮 (U^{2}) such that

𝒮 [𝕃](ℒ (2) ⋉_{ℒ (1)×ℒ (1)} P, N) ≅ 𝒮 [ℒ (1) × ℒ (1)](P, F_{ℒ (1)} [ℒ (2), N))

for ℒ (1) × ℒ (1)-spectra P.

Proof. We construct F_{ℒ (1)} [ℒ (2), N) as the equalizer of two maps

F [ℒ (2), N) ⇉ F [ℒ (1) × ℒ (2), N).

The first is induced by γ : ℒ (1) × ℒ (2) → ℒ (2). The second is the composite

F [ℒ (2), N) ⟶ F [ℒ (1) × ℒ (2), ℒ (1) ⋉ N) ⟶^{F [1,ξ)} F [ℒ (1) × ℒ (2), N);

here the unlabelled arrow is adjoint to

(ℒ (1) × ℒ (2)) ⋉ F [ℒ (2), N) ≅ ℒ (1) ⋉ ℒ (2) ⋉ F [ℒ (2), N) ⟶^{id ⋉ε} ℒ (1) ⋉ N,

where ε is the counit of the adjunction. The left action of ℒ (1) × ℒ (1) on F_{ℒ (1)} [ℒ (2), N) is induced by its right action on ℒ (2).

The second step lands us back in the category of 𝕃-spectra.

Lemma 7.6. Let N be an 𝕃-spectrum and P be an ℒ (1) × ℒ (1)-spectrum. There is an 𝕃-spectrum F̂(N, P) such that

𝒮 [ℒ (1) × ℒ (1)](M ∧ N, P) ≅ 𝒮 [𝕃](M, F̂(N, P))

for 𝕃-spectra M.

Proof. Again, we construct F̂(N, P) as an equalizer, this time of two maps

F (N, P) ⇉ F (𝕃N, P).

The first is induced by the structure map 𝕃N ⟶ N. The second is the composite

F (N, P) ⟶ F (𝕃N, ({1} × ℒ (1)) ⋉ P) ⟶ F (𝕃N, P),

where the second arrow is induced by the structure map of P as an ℒ (1) × ℒ (1)- module and the first arrow is adjoint to

F (N, P) ∧ 𝕃N ≅ ({1} × ℒ (1)) ⋉ F (N, P) ∧ N ⟶^{id ⋉ε} ({1} × ℒ (1)) ⋉ P.

The structure of F̂(N, P) as an 𝕃-spectrum is induced by the action on P of the first factor of ℒ (1) in ℒ (1) × ℒ (1); more precisely, the action 𝕃F (N, P) → F (N, P) is adjoint to the composite

(𝕃F (N, P))∧N ≅ (ℒ (1)×{1})⋉(F (N, P)∧N) ⟶^{id ⋉ε} (ℒ (1)×{1})⋉P ⟶^{ξ} P.

We combine these two functorial constructions to define F_{ℒ} .

Definition 7.7. For 𝕃-spectra M and N, define

F_{ℒ} (M, N) = F̂(M, F_{ℒ (1)} [ℒ (2), N)).

The adjunction of Theorem 7.1 is just the composite of the two adjunctions already obtained.

8. Unital properties of the smash product of 𝕃-spectra

As we have already seen, 𝕃S is a unit for the smash product ∧_{ℒ} on h̄𝒮 [𝕃]. However, for precision in the consideration of algebraic structures, we wish to work in a category of spectra that is actually symmetric monoidal under its smash product, with a point-set level unit isomorphism. The appropriate candidate for a unit object is not 𝕃S but S itself, and at this point another special, and surprising, property of the linear isometries operad comes into play.

Consider the diagram

[figure: ℒ (2) × ℒ (1) × ℒ (1) × ℒ (0) × ℒ (0); γ×id; id ×γ^{2}; ℒ (2) × ℒ (0) × ℒ (0); γ; ℒ (0).]

This is not a split coequalizer, but it turns out to be a coequalizer. The coequalizer of the parallel pair of arrows is the orbit space ℒ (2)/ℒ (1) × ℒ (1).

Lemma 8.1. The orbit space ℒ (2)/ℒ (1) × ℒ (1) consists of a single point.

This is far from obvious, and it is only possible because ℒ (1) is a monoid but not a group. We defer its proof to XI§2. It has the following implication. Recall Lemma 4.5.

Proposition 8.2. There is an isomorphism of 𝕃-spectra λ : S ∧_{ℒ} S ⟶ S such that λτ = λ. For based spaces X and Y, there is a natural isomorphism of 𝕃-spectra

λ : Σ^{∞}X ∧_{ℒ} Σ^{∞}Y ≅ Σ^{∞}(X ∧ Y).

Proof. The second statement follows from the first, or directly: γ induces the isomorphism

ℒ (2) ⋉_{ℒ (1)×ℒ (1)} (ℒ (0) ⋉ X) ∧ (ℒ (0) ⋉ Y) ⟶ ℒ (0) ⋉ X ∧ Y.

The relation λτ = λ : S ∧_{ℒ} S ⟶ S is clear since γτ = γ.

This formalizes our intuition that the smash product should be a stabilized generalization of the smash product of based spaces. It is natural to try to generalize the resulting isomorphism λ : S ∧_{ℒ} Σ^{∞}X ≅ Σ^{∞}X to arbitrary 𝕃-spectra, and the map does generalize.

Proposition 8.3. Let M and N be 𝕃-spectra. There is a natural map of 𝕃spectra λ : S ∧_{ℒ} N ⟶ N. The symmetrically defined map M ∧_{ℒ} S ⟶ M coincides with the composite λτ. Moreover, under the associativity isomorphism,

λτ ∧_{ℒ} id = id ∧_{ℒ} λ : M ∧_{ℒ} S ∧_{ℒ} N ⟶ M ∧_{ℒ} N,

and, under the commutativity isomorphism, these maps also agree with

λ : S ∧_{ℒ} (M ∧_{ℒ} N) ⟶ M ∧_{ℒ} N.

8. UNITAL PROPERTIES OF THE SMASH PRODUCT OF 𝕃-SPECTRA 31

Proof. When N is the free 𝕃-spectrum 𝕃X = ℒ (1) ⋉ X generated by a spectrum X, λ is given by the map

S ∧_{ℒ} 𝕃X = ℒ (2) ⋉_{ℒ (1)×ℒ (1)} (ℒ (0) ⋉ S^{0}) ∧ (ℒ (1) ⋉ X)
≅ (ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (0) × ℒ (1)) ⋉ (S^{0} ∧ X)

⟶^{γ⋉id} ℒ (1) ⋉ X = 𝕃X.

For general N, the map just constructed induces a map of coequalizer diagrams

\xymatrix{S ∧_{ℒ} 𝕃𝕃N \ar[d] \ar[r] \ar[r] & S ∧_{ℒ} 𝕃N \ar[d] \ar[r] & S ∧_{ℒ} N \ar[d] \\ 𝕃𝕃N \ar[r] \ar[r] & 𝕃N \ar[r] & N.}

The symmetry is clear when M is free and follows in general by an easy comparison of coequalizer diagrams. Similarly, suppose that M = 𝕃X and N = 𝕃Y for spectra X and Y. Then, under the associativity isomorphisms of their domains given in the proof of Theorem 5.5, the two unit maps defined on 𝕃X ∧_{ℒ} S ∧_{ℒ} 𝕃Y agree with the map

ℒ (3) ⋉_{ℒ (1)^{3}} ((ℒ (1) ⋉ X) ∧ (ℒ (0) ⋉ S^{0}) ∧ (ℒ (1) ⋉ Y))
≅ (ℒ (3) ×_{ℒ (1)^{3}} ℒ (1) × ℒ (0) × ℒ (1)) ⋉ (X ∧ S^{0} ∧ Y)

⟶^{γ⋉id} ℒ (2) ⋉ (X ∧ Y) ≅ 𝕃X ∧_{ℒ} 𝕃Y.

The conclusion for general M and N follows by another comparison of coequalizer diagrams. The last statement can be proven similarly.

Any attempt to show that S is a strict unit for general 𝕃-spectra founders on the fact that Lemma 5.4 fails if i = 0 or j = 0 and i + j > 0. However, we shall prove the following up to homotopy version of that lemma in XI.2.2.

Lemma 8.4. the space

ℒ̂ (1) ≡ ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (0) × ℒ (1)

is contractible. Therefore γ : ℒ̂ (1) ⟶ ℒ (1) is a homotopy equivalence.

Again, this assertion is far from obvious. It leads us to the following crucial result.

Theorem 8.5. Let M be an 𝕃-spectrum and consider λ : S ∧_{ℒ} M ⟶ M.

(i) If M = 𝕃X for a tame spectrum X, then λ is a homotopy equivalence of spectra and thus a weak equivalence of 𝕃-spectra.

(ii) If M is a CW 𝕃-spectrum, then λ is a homotopy equivalence of 𝕃-spectra. (iii) For any M, λ is a weak equivalence of 𝕃-spectra.

Proof. Since λ = γ ⋉ id on free 𝕃-spectra 𝕃X, Theorem 2.5 and the lemma give (i). By Theorem 4.7(iii), (i) applies to show that λ : S ∧_{ℒ} M ⟶ M is a weak equivalence of 𝕃-spectra when M is a CW 𝕃-spectrum. By the Whitehead theorem for CW 𝕃-spectra, there is a map of 𝕃-spectra ξ : M ⟶ S ∧_{ℒ} M such that λ ∘ ξ ≃ id. To complete the proof of (ii), we must show that ξ ∘ λ ≃ id, and the following commutative diagram identifies this composite with id ∧_{ℒ} (λ ∘ ξ):

[figure: S ∧_{ℒ} M; id ∧ξ; λ; S ∧_{ℒ} S ∧_{ℒ} M; λ; id ∧λ; S ∧_{ℒ} M; M; ξ; S ∧_{ℒ} M.]

The rectangle commutes by the naturality of λ and the triangle commutes by Proposition 8.3. For (iii), let M be arbitrary and consider the diagram

[figure: π_{n}(S ∧_{ℒ} M) ≅ h𝒮 [𝕃](𝕃S^{n}, S ∧_{ℒ} M); λ^{∗}; λ_{∗}; h𝒮 [𝕃](S ∧_{ℒ} 𝕃S^{n}, S ∧_{ℒ} M); λ_{∗}; π_{n}(M) ≅ h𝒮 [𝕃](𝕃S^{n}, M); λ^{∗}; h𝒮 [𝕃](S ∧_{ℒ} 𝕃S^{n}, M).]

By (ii), λ : S ∧_{ℒ} 𝕃S^{n} ⟶ 𝕃S^{n} is a homotopy equivalence of 𝕃-spectra, hence the horizontal arrows are isomorphisms. The right vertical arrow is an isomorphism since, for 𝕃-spectra K,

λ_{∗} : 𝒮 [𝕃](S ∧_{ℒ} K, S ∧_{ℒ} M) ⟶ 𝒮 [𝕃](S ∧_{ℒ} K, M)

is a natural isomorphism; its inverse sends f : S ∧_{ℒ} K ⟶ M to the composite

[figure: S ∧_{ℒ} K; λ^{−1}∧id; S ∧_{ℒ} S ∧_{ℒ} K; id ∧f; S ∧_{ℒ} M.]

(Compare II.1.3 below). Therefore the left vertical arrow is an isomorphism.

Remark 8.6. The weak equivalence ω : 𝕃S ∧_{ℒ} M ⟶ M of Proposition 6.2 is just the composite

[figure: 𝕃S ∧_{ℒ} M; ξ∧id; S ∧_{ℒ} M; λ; M.]

Therefore ξ ∧ id is also a weak equivalence for all 𝕃-spectra M.

Corollary 8.7. For any 𝕃-spectrum M, λ̃ : M ⟶ F_{ℒ} (S, M) is a weak equivalence of 𝕃-spectra.

Proof. For a spectrum X, λ̃_{∗} : 𝒮 (X, M) ⟶ 𝒮 (X, F_{ℒ} (S, M)) can be identified with λ̃_{∗} : 𝒮 [𝕃](𝕃X, M) ⟶ 𝒮 [𝕃](𝕃X, F_{ℒ} (S, M)). In turn, by naturality and adjunction, this can be identified with

λ^{∗} : 𝒮 [𝕃](𝕃X, M) ⟶ 𝒮 [𝕃](S ∧_{ℒ} 𝕃X, M) ≅ 𝒮 [𝕃](𝕃X, F_{ℒ} (S, M)).

If X is a CW spectrum, then λ : S ∧_{ℒ} 𝕃X ⟶ 𝕃X is a homotopy equivalence of 𝕃-spectra, hence the displayed maps all induce isomorphisms on passage to homotopy classes of maps. The conclusion follows by letting X run through the sphere spectra.

34 I. PROLOGUE: THE CATEGORY OF 𝕃-SPECTRA

CHAPTER II Structured ring and module spectra

We can now define and study our basic algebraic objects. We begin with the Smodules, which we think of as analogs of modules over a fixed commutative ring k. Since the category of S-modules is symmetric monoidal under its smash product, we can define S-algebras and commutative S-algebras exactly as we define (associative and unital) k-algebras and commutative k-algebras. Intuitively, S-algebras are as close as one can get to k-algebras in stable homotopy theory, and commutative S-algebras are as close as one can get to commutative k-algebras.

By analyzing free objects, we demonstrate that these new definitions are unital sharpenings of the definitions of A_{∞} and E_{∞} ring spectra that were first given in [47]. This allows us to use [47, 49] to supply examples and is therefore fundamentally important to the theory. We give a parallel analysis of the definitions of modules over S-algebras and commutative S-algebras and over A_{∞} and E_{∞} ring spectra. The new definitions drastically simplify the study of these algebraic structures. For example, in a final categorical section, we prove that our new definitions lead to elementary categorical proofs that the categories of S-algebras and of commutative S-algebras are cocomplete, as was first proven by Hopkins and McClure [31] for the categories of A_{∞} and E_{∞} ring spectra.

1. The category of S-modules

Here, finally, is the promised definition of S-modules.

Definition 1.1. Define an S-module to be an 𝕃-spectrum M which is unital in the sense that λ : S ∧_{ℒ} M ⟶ M is an isomorphism. Let ℳ_{S} denote the full subcategory of 𝒮 [𝕃] whose objects are the S-modules. For S-modules M and N, define

M ∧_{S} N = M ∧_{ℒ} N and F_{S}(M, N) = S ∧_{ℒ} F_{ℒ} (M, N).

The justification for the name “S-module” is given by the commutative diagrams

[figure: S ∧_{S} S ∧_{S} M; λ∧id; id ∧λ; S ∧_{S} M; λ; and; M; λ^{−1}; id; S ∧_{S} M; λ; S ∧_{S} M; λ; M; M.]

For the definition to be useful, we need examples, and I.8.2 and I.8.3 provide many. We consistently retain the notation M ∧_{ℒ} N when the given 𝕃-spectra M and N are not restricted to be S-modules.

Proposition 1.2. For any based space X, Σ^{∞}X is an S-module, and

Σ^{∞}X ∧_{S} Σ^{∞}Y ≅ Σ^{∞}(X ∧ Y).

For any S-module M and any 𝕃-spectrum N, M ∧_{ℒ} N is an S-module. In particular, S ∧_{ℒ} N is an S-module for any 𝕃-spectrum N.

Proof. For the second statement, I.8.3 gives that λ for M ∧_{ℒ} N is determined by λ for M and is therefore an isomorphism.

We have the following categorical relationship between 𝒮 [𝕃] and ℳ_{S}.

Lemma 1.3. The functor S ∧_{ℒ} (?) : 𝒮 [𝕃] ⟶ ℳ_{S} is left adjoint to the functor F_{ℒ} (S,?) : ℳ_{S} ⟶ 𝒮 [𝕃] and right adjoint to the inclusion ℓ : ℳ_{S} ⟶ 𝒮 [𝕃].

Proof. The first adjunction is immediate from I.7.1. For the second, let M be an S-module and N be an 𝕃-spectrum. A map f : M ⟶ S ∧_{ℒ} N of S-modules determines a map λ ∘ f : M ⟶ N of 𝕃-spectra, and a map g : M ⟶ N of 𝕃-spectra determines a map (id ∧g) ∘ λ^{−1} : M ⟶ S ∧_{ℒ} N of S-modules. Using I.8.3, we see that these are inverse bijections.

This implies that to lift right adjoint functors from 𝒮 [𝕃] to ℳ_{S}, we must first forget down to 𝒮 [𝕃], next apply the given functor, and then apply the functor S ∧_{ℒ} (?). For example, limits in ℳ_{S} are created in this fashion.

Proposition 1.4. The category of S-modules is complete and cocomplete. Its colimits are created in 𝒮 [𝕃]. Its limits are created by applying the functor S ∧_{S} (?) to limits in 𝒮 [𝕃]. If X is a based space and M is an S-module, then M ∧ X is an S-module, and the spectrum level cofiber of a map of S-modules is an S-module. For a based space X and S-modules M and N,

ℳ_{S}(M ∧ X, N) ≅ ℳ_{S}(M, S ∧_{ℒ} F (X, N)).

Moreover,

M ∧ X ≅ M ∧_{S} Σ^{∞}X and S ∧_{ℒ} F (X, M) ≅ F_{S}(Σ^{∞}X, M).

1. THE CATEGORY OF S-MODULES 37

Remark 1.5. By the path S-module of an S-module N we must understand S ∧_{ℒ} P N. By the fiber of a map f : M ⟶ N of S-modules, we must understand S ∧_{ℒ} F f. Lemma 1.3 implies that the following square of S-modules is a pullback and that its vertical arrows satisfy the CHP in the category of S-modules.

[figure: S ∧_{ℒ} F f; S ∧_{ℒ} P N; M; f; N.]

The resulting fiber sequences of S-modules behave in exactly the same fashion as fiber sequences of spaces or spectra.

Lemma 1.3 also explains our definition of function S-modules. Its second adjunction and the adjunction of Theorem 7.1 compose to give the adjunction displayed in the following theorem.

Theorem 1.6. The category ℳ_{S} is symmetric monoidal under ∧_{S}, and

ℳ_{S}(M ∧_{S} N, P) ≅ ℳ_{S}(M, F_{S}(N, P))

for S-modules M, N, and P.

A homotopy in the category of S-modules is a map M ∧ I_{+} ⟶ N. A map of S-modules is a weak equivalence if it is a weak equivalence as a map of spectra. The derived category 𝒟_{S} of S-modules is constructed from the homotopy category hℳ_{S} by adjoining formal inverses to the weak equivalences; again, the process is made rigorous by CW approximation. The free 𝕃-spectra 𝕃X are not S-modules, and we define sphere S-modules by

S_{S}^{n} ≡ S ∧_{ℒ} 𝕃S^{n} (1.7)

and use them as the domains of attaching maps when defining cell and CW Smodules. Observe that, by I.8.7 and Lemma 1.3, we have

π_{n}(M) ≡ h𝒮 (S^{n}, M) ≅ h𝒮 [𝕃](𝕃S^{n}, F_{ℒ} (S, M)) ≅ hℳ_{S}(S_{S}^{n}, M) (1.8)

for S-modules M. From here, the theory of cell and CW S-modules is exactly like the theory of cell and CW spectra and is obtained by specialization of the theory of cell R-modules to be presented in Chapter III. A weak equivalence of cell S-modules is a homotopy equivalence, any S-module is weakly equivalent to a CW S-module, and 𝒟_{S} is equivalent to the homotopy category of CW S-modules. Again, as we shall explain in Remark 1.10, the S-module S does not have the homotopy type of a CW S-module. When working homotopically, we replace it with S_{S} ≡ S_{S}^{0}.

The following comparison between CW S-modules and CW 𝕃-spectra establishes an equivalence between 𝒟_{S} and h̄𝒮 [𝕃] and thus between 𝒟_{S} and h̄𝒮.

Theorem 1.9. The following conclusions hold.

(i) The functor S ∧_{ℒ} (?) : 𝒮 [𝕃] ⟶ ℳ_{S} carries CW 𝕃-spectra to CW Smodules.

(ii) The forgetful functor ℳ_{S} ⟶ 𝒮 [𝕃] carries S-modules of the homotopy

types of CW S-modules to 𝕃-spectra of the homotopy types of CW 𝕃-spectra. (iii) Every CW S-module M is homotopy equivalent as an S-module to S ∧_{S} N

for some CW 𝕃-spectrum N. (iv) The unit λ : S ∧_{ℒ} M ⟶ M is a weak equivalence for all 𝕃-spectra M

and is a homotopy equivalence of 𝕃-spectra if M has the homotopy type of a CW 𝕃-spectrum.

The functors S ∧_{ℒ} (?) and the forgetful functor establish an adjoint equivalence between the stable homotopy category h̄𝒮 [𝕃] and the derived category 𝒟_{S}. This equivalence of categories preserves smash products and function spectra.

Proof. Part (i) is immediate by induction up the sequential filtration since the functor S∧_{ℒ} (?) preserves spheres, cones, and colimits. Part (iv) is a recapitulation of I.8.5 and, applied to sphere S-modules, it implies part (ii). Part (iii) follows from (i) and (iv) since there is a CW 𝕃-spectrum M′ and a homotopy equivalence of 𝕃-spectra M′ ⟶ M. The claimed adjoint equivalence of categories is immediate from part (iv). For smash products, the last statement is clear from (ii) and the fact that the smash product M ∧_{S} N of S-modules is their smash product as 𝕃-spectra. The statement for function spectra follows formally.

When doing classical homotopy theory, we can work interchangeably in h̄𝒮, h̄𝒮 [𝕃], or 𝒟_{S}. These three categories are equivalent, and the equivalences preserve all structure in sight. When working on the point set level, we have reached a nearly ideal situation with our construction of ℳ_{S}. We pause to comment on Lewis’s observation [36] that there is no fully ideal situation.

Remark 1.10. Suppose given a symmetric monoidal category of spectra with a suspension spectrum functor Σ^{∞} such that S = Σ^{∞}S^{0} is the unit for the smash product, denoted ∧_{S}, and there is a natural isomorphism

Σ^{∞}X ∧_{S} Σ^{∞}Y ≅ Σ^{∞}(X ∧ Y)

that is suitably compatible with the coherence isomorphisms for the unity, associativity, and commutativity of the respective smash products. Our category of S-modules satisfies all of these properties, and many other desiderata not included among Lewis’s axioms. Suppose further that Σ^{∞} has a right adjoint “Ω^{∞}” and let

2. THE MIRROR IMAGE TO THE CATEGORY OF S-MODULES 39

QX = colim Ω^{n}Σ^{n}X. Then Lewis observes that there cannot be a natural weak equivalence

θ : “Ω^{∞}”Σ^{∞}X ⟶ QX

such that θ ∘ η : X ⟶ QX is the natural inclusion, where η is the unit of the adjunction. In our context, we have the two adjunction homeomorphisms

ℳ_{S}(S ∧_{𝕃} 𝕃Σ^{∞}X, M) ≅ 𝒯 (X, Ω^{∞}F_{𝕃}(S, M))

and

ℳ_{S}(Σ^{∞}X, M) ≅ 𝒯 (X, ℳ_{S}(S, M)),

where 𝒯 is the category of based spaces; see VII§10 for discussion of these topologized Hom sets and of the second of these adjunctions. It is a standard property of any symmetric monoidal category that the self-maps of the unit object form a commutative monoid under composition. In our situation ℳ_{S}(S, S) is therefore a commutative topological monoid. It cannot be weakly equivalent to QS^{0}, and QS^{0} is weakly equivalent to Ω^{∞}F_{𝕃}(S, S). Therefore the weak equivalence S∧_{𝕃}𝕃S ⟶ S cannot be a homotopy equivalence of S-modules and S cannot be of the homotopy type of a CW S-module.

2. The mirror image to the category of S-modules

The categorical picture becomes clearer when we realize that the category of S-modules has a “mirror image” category to which it is naturally equivalent. We find this material quite illuminating, but it will not be used until our discussion of Quillen model categories.

Definition 2.1. Define ℳ^{S} to be the full subcategory of 𝒮 [𝕃] whose objects are those 𝕃-spectra N that are counital, in the sense that λ̃ : N ⟶ F_{ℒ} (S, N) is an isomorphism.

Looking through the mirror at Lemma 1.3 and noting that mirrors interchange left and right, we see the following reflection.

Lemma 2.2. The functor F_{ℒ} (S,?) : 𝒮 [𝕃] ⟶ ℳ^{S} is right adjoint to the functor S ∧_{ℒ} (?) : ℳ^{S} ⟶ 𝒮 [𝕃] and left adjoint to the inclusion r : ℳ^{S} ⟶ 𝒮 [𝕃].

We agree to write

f = F_{ℒ} (S,?) : 𝒮 [𝕃] ⟶ ℳ^{S} and s = S ∧_{ℒ} (?) : 𝒮 [𝕃] ⟶ ℳ_{S} (2.3) in the rest of this section. With this notation, Lemmas 1.3 and 2.2 give the following mirrored pairs of adjunctions, the upper arrow being left adjoint to the lower arrow in each case.

[figure: 𝒮 [𝕃]; s; ℳ_{S}; rfℓ; ℓ; 𝒮 [𝕃]; s; and; 𝒮 [𝕃]; f; ℳ^{S}; ℓsr; r; 𝒮 [𝕃]; f] (2.4)

The display makes new information visible. The composite of the first two left adjoints is just the functor S ∧_{ℒ} (?) and the composite of the second two right adjoints is just the functor F_{ℒ} (S,?). Since these two endo-functors of 𝒮 [𝕃] are left and right adjoint, they must be equivalent to their displayed composite adjoints.

Lemma 2.5. For 𝕃-spectra M, the maps

id ∧_{ℒ} λ̃ : S ∧_{ℒ} M ⟶ S ∧_{ℒ} F_{ℒ} (S, M)

and

F_{ℒ} (id, λ) : F_{ℒ} (S, S ∧_{ℒ} M) ⟶ F_{ℒ} (S, M)

are natural isomorphisms.

We now see that the reflection of a reflection is equivalent to the original.

Proposition 2.6. The functors

fℓ : ℳ_{S} ⟶ ℳ^{S} and sr : ℳ^{S} ⟶ ℳ_{S}

are inverse equivalences of categories. More precisely,

ε : srfℓM = S ∧_{ℒ} F_{ℒ} (S, M) ⟶ M

is an isomorphism for M ∈ ℳ_{S}, and

η : N ⟶ F_{ℒ} (S, S ∧_{ℒ} N) = fℓsrN

is an isomorphism for N ∈ ℳ^{S}, where ε and η are the unit and counit of the (S ∧_{ℒ} (?), F_{ℒ} (S,?)) adjunction.

Proof. The functor sℓ : ℳ_{S} ⟶ ℳ_{S} is an equivalence, and it is left adjoint to the composite srfℓ : ℳ_{S} ⟶ ℳ_{S}. The functor fr : ℳ^{S} ⟶ ℳ^{S} is an equivalence, and it is right adjoint to the composite fℓsr. Therefore these two composites are natural equivalences. A little diagram chase from the previous lemma gives the more precise statement.

Proposition 2.7. The category ℳ^{S}, hence also the category ℳ_{S}, is equivalent to the category of algebras over the monad rf in 𝒮 [𝕃] determined by the adjunction (f, r). The category ℳ_{S}, hence also the category ℳ^{S}, is equivalent to the category of coalgebras over the comonad ℓs in 𝒮 [𝕃] determined by the adjunction (ℓ, s).

3. S-ALGEBRAS AND THEIR MODULES 41

Proof. The unit of the monad rf is λ̃ : M ⟶ F_{ℒ} (S, M) = rfM and its product is the natural isomorphism

μ : rfrfM = F_{ℒ} (S, F_{ℒ} (S, M)) ≅ F_{ℒ} (S, M) = rfM

implied by the isomorphism S ∧_{ℒ} S ≅ S. Clearly, if λ̃ is an isomorphism, then M is an rf-algebra with action λ̃^{−1}. Conversely if ξ : rfM ⟶ M is an action, then ξ ∘ λ̃ = id and the following is a split coequalizer diagram in 𝒮 [𝕃].

[figure: rfrfM; rfξ; μ; rfM; ξ; M.]

Applying f, we obtain a split coequalizer diagram in ℳ^{S}. Since the counit fr → id of the adjunction is an isomorphism, it induces an isomorphism of diagrams

[figure: (frfrfM; frfM); (frfM; fM).]

Applying r, rfM is the (split) coequalizer of the first and M is the (split) coequalizer of the second. The resulting isomorphism rfM → M is just the map ξ, hence is an isomorphism of rf-algebras.

3. S-algebras and their modules

Let 𝒞 be any symmetric monoidal category, with product □ and unit object I. Then a monoid in 𝒞 is an object R together with maps η : I → R and ϕ : R □ R → R such that the evident associativity and unity diagrams commute; R is a commutative monoid if the evident commutativity diagram also commutes. A left R-object over a monoid R is an object M of 𝒞 with a map μ : R□M → M such that the evident unity and associativity diagrams commute, and right R-objects are defined by symmetry. These definitions apply to our symmetric monoidal category ℳ_{S}.

Definition 3.1. An S-algebra is a monoid in ℳ_{S}. A commutative S-algebra is a commutative monoid in ℳ_{S}. For an S-algebra or commutative S-algebra R, a left or right R-module is a left or right R-object in ℳ_{S}. Modules will mean left modules unless otherwise specified, and we let ℳ_{R} denote the category of left R-modules.

Observe that if R is a commutative S-algebra, then an R-module is just a module over R regarded as an S-algebra, as in module theory in algebra. For this reason, even though our main interest is in the much richer commutative context, we work with general S-algebras wherever possible.

We insert the following lemma for later reference. It records specializations of observations that apply to monoids in any symmetric monoidal category.

Lemma 3.2. Let R be an S-algebra and M be an R-module. Then the following diagrams of S-modules are split coequalizers:

[figure: R ∧_{S} R ∧_{S} R; ϕ∧id; id ∧ϕ; R ∧_{S} R; ϕ; R.]

and

[figure: R ∧_{S} R ∧_{S} M; id ∧μ; ϕ∧id; R ∧_{S} M; μ; M.]

While we have given the most conceptual form of the definitions, it is worthwhile to write out the relevant diagrams explicitly. We find that they make perfect sense for 𝕃-spectra that might not be S-modules, and this leads us back to the earlier notions of A_{∞} and E_{∞} ring spectra and their modules.

Definition 3.3. An A_{∞} ring spectrum is an 𝕃-spectrum R with a unit map η : S ⟶ R and a product ϕ : R ∧_{ℒ} R → R such that the following diagrams commute:

[figure: S ∧_{ℒ} R; η∧id; λ; R ∧_{ℒ} R; ϕ; R ∧_{ℒ} S; id ∧η; λτ; R]

and

[figure: R ∧_{ℒ} R ∧_{ℒ} R; ϕ∧id; id ∧ϕ; R ∧_{ℒ} R; ϕ; R ∧_{ℒ} R; ϕ; R;]

R is an E_{∞} ring spectrum if the following diagram also commutes:

[figure: R ∧_{ℒ} R; ϕ; τ; R ∧_{ℒ} R; ϕ; R.]

A module over an A_{∞} or E_{∞} ring spectrum R is an 𝕃-spectrum M with a map μ : R ∧_{ℒ} M → M such that the following diagrams commute:

[figure: S ∧_{ℒ} M; η∧id; λ; R ∧_{ℒ} M; μ; and; R ∧_{ℒ} R ∧_{ℒ} M; ϕ∧id; id ∧μ; R ∧_{ℒ} M; μ; M; R ∧_{ℒ} M; μ; M.]

Lemma 3.4. An S-algebra or commutative S-algebra is an A_{∞} or E_{∞} ring spectrum which is also an S-module. A module over an S-algebra or commutative S-algebra R is a module over R, regarded as an A_{∞} or E_{∞} ring spectrum, which is also an S-module.

In view of Proposition 1.2, this leads to the following observations.

Proposition 3.5. The following statements hold.

(i) S is a commutative S-algebra with unit id and product λ.

(ii) If R and R′ are A_{∞} or E_{∞} ring spectra, then so is R ∧_{ℒ} R′; if either R or

R′ is an S-algebra, then so is R ∧_{ℒ} R′. (iv) If R and R′ are A_{∞} ring spectra, M is an R-module and M′ is an R′-

module, then M ∧_{ℒ} M′ is an R ∧_{ℒ} R′-module.

In particular, we have a functorial way to replace A_{∞} and E_{∞} ring spectra and their modules by S-algebras and commutative S-algebras and their modules.

Corollary 3.6. For an A_{∞} ring spectrum R, S ∧_{ℒ} R is an S-algebra and λ : S ∧_{ℒ} R ⟶ R is a weak equivalence of A_{∞} ring spectra, and similarly in the E_{∞} case. If M is an R-module, then S ∧_{ℒ} M is an S ∧_{ℒ} R-module and λ : S ∧_{ℒ} M ⟶ M is a weak equivalence of R-modules and of modules over S ∧_{ℒ} R regarded as an A_{∞} ring spectrum.

Recall that the tensor product of commutative rings is their coproduct in the category of commutative rings. The proof consists of categorical diagram chases that apply to commutative monoids in any symmetric monoidal category.

Proposition 3.7. If R and R′ are commutative S-algebras, then R ∧_{S} R′ is the coproduct of R and R′ in the category of commutative S-algebras.

We shall construct coproducts in the category of S-algebras in Section 7, where we show more generally that the categories of S-algebras and of commutative Salgebras are cocomplete.

There is a version of the proposition that is true for E_{∞} ring spectra, but this is not obvious. We shall return to this point in Chapter XIII, where we show that the category of 𝕃-spectra under S is symmetric monoidal under a modified smash product ⋆_{S} and that A_{∞} and E_{∞} ring spectra are exactly the monoids and commutative monoids in that symmetric monoidal category. This was the starting point for the earlier version of the present theory announced in [22].

4. Free A_{∞} and E_{∞} ring spectra and comparisons of definitions

We focus on A_{∞} and E_{∞} ring spectra here. It was proven in [47, 49] that various Thom spectra, Eilenberg-Mac Lane spectra, and connective algebraic and topological K-theory spectra are E_{∞} ring spectra. Using the results stated in the previous section, we can convert these E_{∞} ring spectra to weakly equivalent commutative S-algebras. However, on the face of it, the original definitions of A_{∞} and E_{∞} ring spectra appear to be different from those that we have given here. As in algebra, it is important to understand free A_{∞} and E_{∞} ring spectra, and we shall use this understanding to verify that our present definitions agree with the original ones.

There is no difficulty in constructing the relevant monads. In fact, we shall construct two pairs of monads and then relate them. The first is defined on the ground category of spectra and is transparently related to the earlier definitions. The second is defined on the ground category of S-modules and is transparently related to the present definitions. The connection between them will establish the required equivalence of definitions. In effect, our new definition of E_{∞} ring spectra is obtained from the old one simply by factoring the original defining monad ℂ in 𝒮 through a new defining monad ℙ in the more highly structured category 𝒮 [𝕃].

Construction 4.1. Construct monads 𝔹 and ℂ in the category of spectra as follows. Let X be a spectrum and let X^{j} be its j-fold external smash power, with X^{0} = S^{0}. Define

𝔹X ≅ ⋁_{j≥0} ℒ (j) ⋉ X^{j}

and

ℂX ≅ ⋁_{j≥0} ℒ (j) ⋉_{Σ_{j}} X^{j},

where ℒ (j) ⋉_{Σ_{j}} X^{j} is the orbit spectrum (ℒ (j) ⋉ X^{j})/Σ_{j}. The units of these monads are induced by the unit maps X ≅ {1} ⋉ X → ℒ (1) ⋉ X. Their products are induced by wedge sums of maps induced by the structure maps γ of the linear isometries operad ℒ.

The notion of an ℒ -spectrum was defined in [37, VII.2.1]. The definition used permutations, and there is a corresponding notion of a non-Σ ℒ -spectrum. An immediate comparison of definitions gives the following result.

Proposition 4.2. The category of 𝔹-algebras is isomorphic to the category of non-Σ ℒ -spectra. The category of ℂ-algebras is isomorphic to the category of ℒ -spectra.

Actually, 𝒪-spectra were defined in [37, VII.2.1] for any operad 𝒪 that is augmented over ℒ. An E_{∞} operad is one such each 𝒪(j) is Σ_{j}-free and contractible. In earlier work, E_{∞} ring spectra were understood to mean 𝒪-spectra for any

4. FREE A_{∞} AND E_{∞} RING SPECTRA AND COMPARISONS OF DEFINITIONS 45

E_{∞} operad 𝒪 augmented over ℒ. The present theory is based on properties that are special to ℒ. The following result, which will be proven in XII§1, shows that restriction to ℒ results in no loss of generality. There is an analogue for A_{∞} ring spectra that is obtained by forgetting about permutations.

Proposition 4.3. Let 𝒪 be an E_{∞} operad over ℒ. There is a functor V that assigns a weakly equivalent ℒ -spectrum V R to an 𝒪-spectrum R.

Construction 4.4. Construct monads 𝕋 and ℙ in the category of 𝕃-spectra as follows. Let M be an 𝕃-spectrum and let M^{j} be its j-fold power with respect to ∧_{ℒ} , with M^{0} = S. Define

𝕋M ≅ ⋁_{j≥0} M^{j}

and

ℙM ≅ ⋁_{j≥0} M^{j}/Σ_{j}.

Here passage to orbits preserves 𝕃-spectra since it is a finite colimit. The unit is the inclusion of M = M^{1}. The product is induced by the maps

M^{j_{1}} ∧_{ℒ} ⋯ ∧_{ℒ} M^{j_{k}} ⟶ M^{j_{1}+···+j_{k}}

that are given by the evident identifications if each j_{r} ≥ 1 and by use of the unit map λ if any j_{r} = 0. Observe that 𝕋 and ℙ restrict to monads in the category of S-modules.

The letters 𝕋 and ℙ are mnemonic for “tensor algebra” and “polynomial” (or symmetric) algebra. As is clear for S-modules and will be made explicit in Definition 7.1, the definitions fit into a general categorical framework that includes those constructions. The following result is an easy direct consequence of our definitions.

Proposition 4.5. The categories of A_{∞} ring spectra and of S-algebras are isomorphic to the categories of 𝕋-algebras in 𝒮 [𝕃] and of 𝕋-algebras in ℳ_{S}. The categories of E_{∞} ring spectra and of commutative S-algebras are isomorphic to the categories of ℙ-algebras in 𝒮 [𝕃] and of ℙ-algebras in ℳ_{S}.

To relate the monads 𝔹 and ℂ to the monads 𝕋 and ℙ, recall from I.4.2 that the category of 𝕃-spectra is the category of 𝕃-algebras in 𝒮. Together with Propositions 4.2 and 4.5, the following result gives the promised comparison between the old and new definitions of A_{∞} and E_{∞} ring spectra.

Proposition 4.6. The monads 𝔹 and 𝕋𝕃 are isomorphic, hence the categories of non-Σ ℒ -spectra and of A_{∞} ring spectra are isomorphic. The monads ℂ and ℙ𝕃 are isomorphic, hence the categories of ℒ -spectra and of E_{∞} ring spectra are isomorphic.

Proof. The isomorphisms on objects are immediate from I.5.6 applied to 𝕃spectra M_{i} = 𝕃X_{i}. Since these isomorphisms are induced from the structure maps γ of ℒ, the comparison of monad structures is immediate. In both statements, the second clause is a categorical consequence of the first, as we shall show in Lemma 6.1 below.

Remark 4.7. Observe that we have quotient maps of monads 𝔹 ⟶ ℂ and 𝕋 ⟶ ℙ. In Section 6, we shall give categorical definitions that show how to exploit these maps to construct an E_{∞} ring spectrum ℂ ⊗_{𝔹} R (or ℙ ⊗_{𝕋} R) from an A_{∞} ring spectrum R by “passage to quotients”, just as we construct commutative algebras as quotients of associative algebras; see Lemma 6.7 and Corollary 7.3. Formally, ℂ ⊗_{𝔹} R is a coequalizer of a right action of 𝔹 on ℂ and the given action of 𝔹 on R.

Remark 4.8. Passage to orbits and passage to coequalizers are often hard to analyze homotopically. We show how to deal with the first difficulty in III§5, where we show that symmetric powers and extended powers of S-modules (and, more generally, R-modules) are essentially equivalent. One often circumvents the second difficulty by replacing a construction like ℂ ⊗_{𝔹} R with its associated bar construction B(ℂ, 𝔹, R), which we shall introduce in XII§1.

Remark 4.9. There are reduced monads 𝔹̃ and ℂ̃ in the category 𝒮 \S of spectra under S and 𝕋̃ and ℙ̃ in the category 𝒮 [𝕃]\S of 𝕃-spectra under S. They are constructed from the unreduced monads by unit map identifications similar to the basepoint identifications in the James construction or the infinite symmetric product. Observe that 𝒮 \S is the category of algebras over the monad 𝕌 that is specified by 𝕌X = X ∨ S, with product given by the folding map S ∨ S ⟶ S, and similarly for 𝒮 [𝕃]\S. In all four cases, the unreduced monad is the composite of the reduced monad with 𝕌, hence, by Lemma 6.1 below, the reduced and unreduced monads have the same algebras. The difference is that, when considering the reduced monad, one is considering the unit map S → R as preassigned and then ensuring that the unit map created by the monad action coincides with it. It follows that the monad 𝕌 acts from the right on the unreduced monads, and it is easy to write down this action directly. The reduced monad ℂ̃ can then be constructed from ℂ by setting ℂ̃X = ℂ⊗_{𝕌}X for a spectrum X under S, with structure maps induced by passage to coequalizers, and similarly for our other monads. A more explicit description is given in [37, VII§3], where ℂ̃ is denoted by ℂ. While the monad ℂ is more convenient for formal work, the monad ℂ̃ is of far greater homotopical interest.

5. FREE MODULES OVER A_{∞} AND E_{∞} RING SPECTRA 47

5. Free modules over A_{∞} and E_{∞} ring spectra

There is an analogue for modules of the original explicit definition of A_{∞} and E_{∞} ring spectra in terms of twisted half-smash products, and there is an analogous comparison of definitions.

Proposition 5.1. The category of modules over an ℒ -spectrum R is isomorphic to the category of spectra M together with associative, unital, and, in the E_{∞} context, equivariant systems of action maps

ℒ (j) ⋉ (R^{j−1} ∧ M) ⟶ M.

Since we shall not need the details, we shall not write out the relevant diagrams. They make sense for any operad 𝒪 augmented over ℒ, and they are exact analogs of diagrams that are written out in the context of algebraic operads in [34, I.4.1]. Remarkably, with this alternative form of the definition, it is far from obvious that a module over an E_{∞} ring spectrum R is the same thing as a module over R regarded as an A_{∞} ring spectrum. In fact, this appears to be false in the context of modules over an 𝒪-spectrum R for a general E_{∞} operad 𝒪 augmented over ℒ. However, we have the following analogue of Proposition 4.3, which will be proven in XII§1. Again, there is an analogue for A_{∞} ring spectra and modules that is obtained by forgetting about permutations.

Proposition 5.2. Let 𝒪 be an E_{∞} operad over ℒ and R be an 𝒪-spectrum. There is a functor V that assigns a weakly equivalent V R-module to an R-module M, where V R is the ℒ -spectrum of Proposition 4.3.

There is a conceptual monadic proof of Proposition 5.1 that is based on analogs of Propositions 4.2, 4.5, and 4.6. To carry out this argument, we need to know that there is a free R-module functor. This is obvious enough when we are considering S-modules: R∧_{S} M is then the free R-module generated by an S-module M. For a general A_{∞} ring spectrum R and an 𝕃-spectrum M, R ∧_{ℒ} M is an R-module but, since M need not be isomorphic to S ∧_{ℒ} M, it is not the free R-module generated by M.

Definition 5.3. For an A_{∞} ring spectrum R and an 𝕃-spectrum M, define an 𝕃-spectrum ℝM and maps of 𝕃-spectra π : R ∧_{ℒ} M ⟶ ℝM and η : M ⟶ ℝM by the pushout diagram

[figure: S ∧_{ℒ} M; η∧id; λ; R ∧_{ℒ} M; π; M; η; ℝM.]

Dually, define an 𝕃-spectrum ℝ^{#}M by the pullback diagram

[figure: ℝ^{#}M; M; λ̃; F_{ℒ} (R, M); F (η,id); F_{ℒ} (S, M).]

These are special cases of general constructions to be studied in Chapter XIII. Such constructions permeated earlier versions of the present theory. Of course, π is an isomorphism if M is an S-module. As will be generalized in XIII.1.4, we deduce the following homotopical property by applying the functor S ∧_{𝕃} (?) to the defining pushout diagram.

Proposition 5.4. The map π : R∧_{ℒ} M ⟶ ℝM is a weak equivalence for any 𝕃-spectrum M.

The unit diagram of an R-module M ensures that its product factors through a map ℝM ⟶ M. More formally, elementary inspections of definitions give the following result.

Proposition 5.5. Let R be an A_{∞} ring spectrum. Then ℝ is a monad in 𝒮 [𝕃] with unit η : M → ℝM and product induced from the product ϕ : R ∧_{ℒ} R ⟶ R. A left R-module is an algebra over the monad ℝ and, for an 𝕃-spectrum M, ℝM is the free R-module generated by M. The functor ℝ^{#} is right adjoint to ℝ and is therefore a comonad in 𝒮 [𝕃] such that an R-module is a coalgebra over ℝ^{#}.

It is logical to denote the category of R-modules by 𝒮 [𝕃][ℝ], reserving the notation ℳ_{R} for the case when R is an S-algebra and R-modules are required to be S-modules. We have freeness and cofreeness adjunctions

𝒮 [𝕃][ℝ](ℝM, N) ≅ 𝒮 [𝕃](M, N)

and

𝒮 [𝕃][ℝ](N, ℝ^{#}M) ≅ 𝒮 [𝕃](N, M)

for 𝕃-spectra M and R-modules N.

Clearly there results a composite adjunction that starts with spectra.

Proposition 5.6. For a spectrum X, define 𝔽X = ℝ𝕃X. Then 𝔽X is the free R-module generated by X. Thus

𝒮 [𝕃][ℝ](𝔽X, N) ≅ 𝒮 (X, N)

for an R-module N. Dually, define 𝔽^{#}X = ℝ^{#}𝕃^{#}X. Then 𝔽^{#}X is the cofree R-module generated by X, so that

𝒮 [𝕃][ℝ](N, 𝔽^{#}X) ≅ 𝒮 (N, X).

In Construction 6.2, we shall show how to combine the monads of the previous section with these free module constructions to obtain monads 𝔹[1] and ℂ[1] in the category of pairs of spectra such that a 𝔹[1]-algebra or ℂ[1]-algebra (R; M) is an A_{∞} or E_{∞} ring spectrum R together with an R-module M in the alternative operad action sense described in Proposition 5.1. The construction will also give monads 𝕋[1] and ℙ[1] in the category of pairs of 𝕃-spectra such that a 𝕋[1]-algebra or ℙ[1]- algebra (R; M) is an A_{∞} or E_{∞} ring spectrum R together with an R-algebra M in the sense of Definition 3.3. The monad 𝔹[1] has the general form

𝔹[1](X; Y) = (𝔹X; 𝔹(X; Y)),

and similarly in the other three cases. Propositions 4.6 and 5.5, together with inspection of the cited construction, directly imply the following analogue of Proposition 4.6. By Lemma 6.1, this in turn implies Proposition 5.1.

Proposition 5.7. The monads 𝔹[1] and 𝕋[1]∘(𝕃, 𝕃) are isomorphic. The monads ℂ[1] and ℙ[1] ∘ (𝕃, 𝕃) are isomorphic. The second coordinates of the four monads are given explicitly as follows. Applied to a pair of spectra (X; Y),

𝔹(X; Y) = ⋁_{j≥1} ℒ (j) ⋉ (X^{j−1} ∧ Y)

and

ℂ(X; Y) = ⋁_{j≥1} ℒ (j) ⋉_{Σ_{j−1}} (X^{j−1} ∧ Y).

Applied to a pair of 𝕃-spectra (M; N),

𝕋(M; N) = ⋁_{j≥1} M^{j−1} ∧_{S} N

and

ℙ(M; N) = ⋁_{j≥1}(M^{j−1}/Σ_{j−1}) ∧_{S} N.

If N is an S-module, then so are 𝕋(M; N) and ℙ(M; N).

Remark 5.8. Construction 6.2 applies equally well to give reduced versions of our four monads, giving monads in the category of pairs (of spectra or 𝕃-spectra), the first coordinate of which lies under S. The monad 𝔹̃[1] has the form

𝔹̃[1](X; Y) = (𝔹̃X; 𝔹̃(X; Y))

and similarly in the other three cases. Inspection of definitions shows that

𝔹̃S = ℂ̃S = S and 𝔹̃(S; Y) = ℂ̃(S; Y) = ℒ (1) ⋉ Y.

This fact dictates our original definition of 𝕃-spectra and is thus the conceptual starting point of our entire theory.

6. Composites of monads and monadic tensor products

In this section and the next, we collect a number of purely categorical observations and constructions that are needed in our work. We shall return to these topics in Chapter VII, but we shall make no further use of this material until then. The reader may prefer to skip these sections on a first reading. We here give the description of algebras over composite monads that was at the heart of our comparisons of definitions and formalize the tensor product construction that appeared briefly in Section 4.

Lemma 6.1. Let 𝕊 be a monad in a category 𝒞 and let 𝕋 be a monad in the category 𝒞 [𝕊] of 𝕊-algebras. Then the category 𝒞 [𝕊][𝕋] of 𝕋-algebras in 𝒞 [𝕊] is isomorphic to the category 𝒞 [𝕋𝕊] of algebras over the composite monad 𝕋𝕊 in 𝒞. Moreover, the unit of 𝕋 defines a map 𝕊 ⟶ 𝕋𝕊 of monads in 𝒞. An analogous assertion holds for comonads.

Proof. Strictly speaking, in constructing 𝕋𝕊, we are regarding 𝕊 as the free 𝕊algebra functor 𝒞 → 𝒞 [𝕊], applying the functor 𝕋, and then applying the forgetful functor back to 𝒞. We continue to neglect notation for forgetful functors and to write 𝕊 and 𝕋 ambiguously for both the given monads and the resulting free functors. The unit of 𝕋𝕊 is given by the composite of unit maps

X ⟶ 𝕊X ⟶ 𝕋𝕊X.

The product of 𝕋𝕊 is given by the composite maps

𝕋𝕊𝕋𝕊X ⟶ 𝕋𝕋𝕊X ⟶ 𝕋𝕊X,

where the second arrow is given by the product of 𝕋 and the first is obtained by application of 𝕋 to the action 𝕊𝕋𝕊X ⟶ 𝕋𝕊X given by the fact that 𝕋 takes 𝕊algebras to 𝕊-algebras. If R is a 𝕋-algebra in 𝒞 [𝕊], with action ξ by 𝕊 and action χ by 𝕋, then it is a 𝕋𝕊-algebra with action the composite

[figure: 𝕋𝕊R; 𝕋ξ; 𝕋R; χ; R.]

If Q is a 𝕋𝕊-algebra with action ω, then Q is a 𝕋-algebra in 𝒞 [𝕊] with actions the composites

[figure: 𝕊Q; η; 𝕋𝕊Q; ω; Q; and; 𝕋Q; 𝕋η; 𝕋𝕊Q; ω; Q.]

These correspondences establish the required isomorphism of categories. Easy diagram chases show that 𝕊 ⟶ 𝕋𝕊 is a map of monads.

When applying this to modules, we used the following construction.

Construction 6.2. For a category 𝒞, let 𝒞 [1] be the category of pairs (X; Y) in 𝒞 and pairs of maps. Let 𝕊 be any of the monads 𝔹, ℂ, 𝕋, or ℙ, and let 𝒞 be its ground category 𝒮, 𝒮 [𝕃], or ℳ_{S}. Construct a monad 𝕊[1] in 𝒞 [1] as follows. On a pair (X; Y), the functor 𝕊[1] is given by

𝕊[1](X; Y) = (𝕊X; 𝕊(X; Y)),

where 𝕊(X; Y) is the free 𝕊X-module generated by Y. This functor factors through the evident category of pairs

(𝕊-algebra; object of 𝒞)

as the composite of (𝕊; id) and (id; free module), where the free module functor is that associated to the algebra in the first variable. Since the identity functor is a monad in a trivial way, each of these functors is a monad. Therefore, by Lemma 6.1, their composite 𝕊[1] is a monad such that an 𝕊[1]-algebra (R; M) is an 𝕊-algebra R together with an R-module M.

We used the following definition in our construction of E_{∞} ring spectra from A_{∞} ring spectra.

Definition 6.3. Let (𝕊, μ, η) be a monad in a cocomplete category 𝒞. A (right) 𝕊-functor in a category 𝒞′ is a functor F : 𝒞 → 𝒞′ together with a natural transformation ν : F 𝕊 → F such that the following diagrams commute:

[figure: F 𝕊; ν; F; F η; id; and; F 𝕊𝕊; F μ; νS; F 𝕊; ν; F; F 𝕊; ν; F.]

Given an 𝕊-algebra (R, ξ), define F ⊗_{𝕊} R to be the coequalizer displayed in the diagram

[figure: F 𝕊R; ν; F ξ; F R; F ⊗_{𝕊} R.]

Given a monad 𝕊′ in 𝒞′ and a left action λ : 𝕊′F → F, we say that F is an (𝕊′, 𝕊)-bifunctor if the following diagram commutes:

[figure: 𝕊′F 𝕊; λ𝕊; 𝕊′ν; F 𝕊; ν; 𝕊′F; λ; F.]

Example 6.4. The functor 𝕊 is an (𝕊, 𝕊)-bifunctor, with both left and right action μ. If π : 𝕊 → 𝕊′ is a map of monads in 𝒞, then 𝕊′ is an (𝕊′, 𝕊)-bifunctor with right action ν = μ′ ∘ 𝕊′π : 𝕊′𝕊 ⟶ 𝕊′. Observe that, for X ∈ 𝒞, 𝕊′ ⊗_{𝕊} 𝕊X ≅ 𝕊′X.

When 𝒞′ in Definition 6.3 has a forgetful functor to the category of spectra, we shall construct a bar construction B(F, 𝕊, R) that will give the appropriate homotopical version of F ⊗_{𝕊} R in XII§1. Assuming that F is an (𝕊′, 𝕊)-bifunctor for one of the monads constructed earlier in this chapter, we will find that B(F, 𝕊, R) is an 𝕊′-algebra. It is natural to ask whether or not F ⊗_{𝕊} R is itself an 𝕊′-algebra. To answer this, we need another categorical definition.

Definition 6.5. In any category 𝒞, a coequalizer diagram

[figure: A; e; f; B; g; C.]

is said to be a reflexive coequalizer if there is a map h : B ⟶ A such that e∘h = id and f ∘ h = id.

The following categorical observation is standard and easy. Although their stated hypotheses are different, the proofs of similar results in [42, p. 147] and [4, pp. 106-108] apply to give the first statement, and the second statement follows.

Lemma 6.6. Let 𝕊 be a monad in 𝒞 such that 𝕊 preserves reflexive coequalizers. If

[figure: A; e; f; B; g; C]

is a reflexive coequalizer in 𝒞 such that A and B are 𝕊-algebras and e and f are maps of 𝕊-algebras, then C has a unique structure of 𝕊-algebra such that g is a map of 𝕊-algebras, and g is the coequalizer of e and f in the category 𝒞 [𝕊]. If, further, 𝕋 is a monad in 𝒞 [𝕊] such that 𝕋 preserves reflexive coequalizers, then 𝕋 ∘ 𝕊 also preserves reflexive coequalizers.

Since the coequalizer diagram used to define F ⊗_{𝕊} R is reflexive, via the map F η : F R ⟶ F 𝕊R, the first statement implies an answer to the question we asked originally.

Lemma 6.7. Let 𝕊 be a monad in 𝒞, 𝕊′ be a monad in 𝒞′, R be an 𝕊-algebra, and F : 𝒞 ⟶ 𝒞′ be an (𝕊′, 𝕊)-bifunctor. If 𝕊′ preserves reflexive coequalizers, then F ⊗_{𝕊} R is an 𝕊′-algebra.

7. Limits and colimits of S-algebras

7. LIMITS AND COLIMITS OF S-ALGEBRAS 53

We here prove that the categories of A_{∞} and E_{∞} ring spectra and of S-algebras and commutative S-algebras are complete and cocomplete. In fact, completeness follows immediately from Proposition 4.5. All four of our categories are categories of algebras over a monad in a complete category, and it follows that they are complete, with their limits created in their respective ground categories [42, VI.2, ex. 2]. The first statement of Lemma 6.6 applies to construct colimits, but to explain this properly we need some preliminary definitions that put our definitions of A_{∞} and E_{∞} ring spectra in perspective.

Definition 7.1. A weak symmetric monoidal category 𝒞 with product □ and unit object I is defined in exactly the same way as a symmetric monoidal category [42, p. 180], except that its unit map λ : I □ X ⟶ X is not required to be an isomorphism; 𝒞 is said to be closed if the functor (?) □ Y has a right adjoint Hom(Y,?) for each Y ∈ 𝒞. Monoids and commutative monoids in 𝒞 are defined in terms of diagrams of the form displayed in Definition 3.3. As in Construction 4.4 and Proposition 4.5, if 𝒞 is cocomplete, then there are monads 𝕋 and ℙ in 𝒞 whose algebras are the monoids and commutative monoids in 𝒞. For X ∈ 𝒞,

𝕋X ≅ ∐_{j≥0} X^{j} and ℙX ≅ ∐_{j≥0} X^{j}/Σ_{j}.

The proof of the following result is abstracted from an argument that Hopkins gave for the monad ℂ [31]. He proceeded by reduction to a proof that the j-fold symmetric powers of based spaces preserve reflexive coequalizers. With our new associative smash products, an abstraction of the latter proof makes the reduction unnecessary.

Proposition 7.2. Let 𝒞 be any cocomplete closed weak symmetric monoidal category. Then the monads 𝕋 and ℙ in 𝒞 preserve reflexive coequalizers.

Proof. For 𝕋, it suffices to prove that the j-fold product X_{1}□⋯□X_{j} preserves reflexive coequalizers. Thus let

[figure: X_{i}; e_{i}; f_{i}; Y_{i}; g_{i}; Z_{i}]

be reflexive coequalizer diagrams in 𝒞, 1 ≤ i ≤ j, and let h_{i} : Y_{i} ⟶ X_{i} satisfy e_{i} ∘ h_{i} = id and f_{i} ∘ h_{i} = id. Let

ε = e_{1} □ ⋯ □ e_{j}, ϕ = f_{1} □ ⋯ □ f_{j}, and γ = g_{1} □ ⋯ □ g_{j}.

Let β : Y_{1} □ ⋯ □ Y_{j} ⟶ Z be the coequalizer of ε and ϕ. Since γε = γϕ, there is a unique map ξ : Z ⟶ Z_{1} □ ⋯ □ Z_{j} such that ξ ∘ β = γ. We claim that ξ is an isomorphism, and we proceed by induction on j. Let

ε_{i} = (id)^{i−1} □ e_{i} □ (id)^{j−i} : Y_{1} □ ⋯ □ Y_{i−1} □ X_{i} □ Y_{i+1} □ ⋯ □ Y_{j} ⟶ Y_{1} □ ⋯ □ Y_{j}

and, similarly, define ϕ_{i} = (id)^{i−1} □ f_{i} □ (id)^{j−i}. We observe first that Z_{1} □ ⋯ □ Z_{j} is the colimit of the diagram given by the j pairs of maps {ε_{i}, ϕ_{i}}. Indeed, for any map α : Y_{1} □ ⋯ □ Y_{j} ⟶ W such that α ∘ ε_{i} = α ∘ ϕ_{i} for 1 ≤ i ≤ j, we obtain unique maps α̂ and α̃ that make the following diagram commute by the induction hypothesis and the fact that the □-product preserves colimits and epimorphisms:

[figure: Y_{1} □ ⋯ □ Y_{j−1} □ X_{j}; g_{1}□···□g_{j−1}□id; id □e_{j}; id □f_{j}; Z_{1} □ ⋯ □ Z_{j−1} □ X_{j}; id □e_{j}; id □f_{j}; Y_{1} □ ⋯ □ Y_{j−1} □ Y_{j}; g_{1}□···□g_{j−1}□id; α; Z_{1} □ ⋯ □ Z_{j−1} □ Y_{j}; α̂; id □g_{j}; W; Z_{1} □ ⋯ □ Z_{j−1} □ Z_{j}.; α̃]

Now let k_{i} = h_{1} □ ⋯ □ h_{i−1} □ id □h_{i+1} □ ⋯ □ h_{j}. Visibly

ε_{i} = ε ∘ k_{i} and ϕ_{i} = ϕ ∘ k_{i}.

Since βε = βϕ, βε_{i} = βϕ_{i} for 1 ≤ i ≤ j and the universal property gives a map ζ : Z_{1} □ ⋯ □ Z_{j} ⟶ Z. It is easy to check from the universal properties that ζ and ξ are inverse isomorphisms. In the symmetric case, we may take our j given coequalizer diagrams to be the same and compose the j-fold power, regarded as a functor to the category of Σ_{j}-objects in 𝒞, with the orbit functor. The latter is constructed as a coequalizer in 𝒞 and is a left adjoint, so preserves coequalizers.

Corollary 7.3. The functors 𝕋 and ℙ on 𝒮 [𝕃], their restrictions to functors 𝕋 and ℙ on ℳ_{S}, and the functors 𝔹 and ℂ on 𝒮 preserve reflexive coequalizers.

Proof. This is immediate since 𝔹 = 𝕋𝕃, ℂ = ℙ𝕃, the functor 𝕃 : 𝒮 ⟶ 𝒮 [𝕃] preserves colimits, and colimits in 𝒮 [𝕃] and in ℳ_{S} are created in 𝒮.

Our claim that the categories of A_{∞} and E_{∞} ring spectra and of S-algebras and commutative S-algebras are cocomplete is now an immediate corollary of the following known result, which we also learned from Hopkins.

Proposition 7.4. Let 𝕊 be a monad in a cocomplete category 𝒞. If 𝕊 preserves reflexive coequalizers, then 𝒞 [𝕊] is cocomplete.

Proof. Consider a diagram {R_{i}} of 𝕊-algebras. Let colim R_{i} be its colimit in 𝒞 and let ι_{i} : R_{i} ⟶ colim R_{i} be the natural maps. Let

α : colim 𝕊R_{i} ⟶ 𝕊 colim R_{i}

be the unique map in 𝒞 whose composite with the natural map 𝕊R_{i} ⟶ colim 𝕊R_{i} is 𝕊ι_{i} for each i. Define colim_{𝕊} R_{i} by the following coequalizer diagram in 𝒞:

[figure: 𝕊(colim 𝕊R_{i}); 𝕊(colim ξ_{i}); μ∘𝕊α; 𝕊(colim R_{i}); colim_{𝕊} R_{i}.]

This is a reflexive coequalizer, via 𝕊(colim η_{i}). Thus, by Lemma 6.6, colim_{𝕊} R_{i} is an 𝕊-algebra such that the displayed diagram is a coequalizer in 𝒞 [𝕊]. It follows easily that colim_{𝕊} R_{i} is the colimit of {R_{i}} in 𝒞 [𝕊].

For later reference, we observe that this result is closely related to the following result of Linton [40] (see also [4, Thm 2, p. 319]).

Theorem 7.5 (Linton). Let 𝕊 be a monad in a cocomplete category 𝒞. If 𝒞 [𝕊] has coequalizers, then 𝒞 [𝕊] is cocomplete.

56 II. STRUCTURED RING AND MODULE SPECTRA

CHAPTER III The homotopy theory of R-modules

We here develop the homotopy theory of modules over an S-algebra R. The classical theory of cell spectra generalizes to give a theory of cell modules over R. We define the smash product over R, ∧_{R}, and the function R-module functor, F_{R}, by direct mimicry of the definitions of tensor product and Hom functors for modules over an algebra. When specialized to commutative S-algebras, our smash product of R-modules is again an R-module, and similarly for F_{R}. Here the category of Rmodules has structure precisely like the category of S-modules, and duality theory works exactly as it does for spectra. We assume familiarity with II§§1,3 and work in the ground category ℳ_{S} of S-modules.

1. The category of R-modules; free and cofree R-modules

Fix an S-algebra R. We understand R-modules to be left R-modules unless otherwise specified. We first observe that the category ℳ_{R} of R-modules is closed under various constructions in the underlying categories of spectra and S-modules. As in algebra, an R-module is the same thing as an algebra over the monad R∧_{S} (?) in ℳ_{S} or, equivalently, a coalgebra over the adjoint comonad F_{S}(R,?) in ℳ_{S}. The functors R ∧_{S} (?) and F_{S}(R,?) from ℳ_{S} to ℳ_{R} are left and right adjoint to the forgetful functor. That is, R ∧_{S} (?) and F_{S}(R,?) are the free and cofree functors from S-modules to R-modules. Together with II.1.4 and formal arguments exactly like those in algebra, this leads to the following result.

Theorem 1.1. The category of R-modules is complete and cocomplete, with both limits and colimits created in the underlying category ℳ_{S}. Let X be a based space, K be an S-module, and M and N be R-modules. Then the following conclusions hold, where the displayed isomorphisms are obtained by restriction of the corresponding isomorphisms for S-modules.

(i) M ∧X is an R-module and the spectrum level cofiber of a map of R-modules is an R-module.

58 III. THE HOMOTOPY THEORY OF R-MODULES

(ii) S ∧_{ℒ} F (X, N) is an R-module and

ℳ_{R}(M ∧ X, N) ≅ ℳ_{R}(M, S ∧_{ℒ} F (X, N)).

(iii) M ∧_{S} K and F_{S}(K, N) are R-modules and

ℳ_{R}(M ∧_{S} K, N) ≅ ℳ_{R}(M, F_{S}(K, N)).

(iv) F_{S}(M, K) is a right R-module.

(v) As R-modules,

M ∧ X ≅ M ∧_{S} Σ^{∞}X and S ∧_{ℒ} F (X, N) ≅ F_{S}(Σ^{∞}X, N).

The cofiber and fiber of a map of R-modules are R-modules, where the fiber is understood to be obtained by application of the functor S ∧_{ℒ} (?) to the fiber constructed in the category of spectra.

Proof. The only point that might need comment is the R-module structure on S ∧_{ℒ} F (X, N). The evaluation map ε : F (X, N)∧X ⟶ N is a map of 𝕃-spectra. The adjoint of R ∧_{ℒ} ε is a map of 𝕃-spectra

ε̃: R ∧_{ℒ} F (X, N) ⟶ F (X, R ∧_{S} N),

and we obtain the desired action upon applying S ∧_{ℒ} ε̃ and using the given action of R on N. This leads to the R-module structure on the specified fiber of a map of R-modules; compare II.1.5.

The free R-module functor on spectra is the starting point of cellular theory.

Definition 1.2. Define the free R-module generated by a spectrum X to be

𝔽_{R}X = R ∧_{S} 𝔽_{S}X,

where 𝔽_{S}X = S ∧_{ℒ} 𝕃X. Equivalently, since R ∧_{S} S ≅ R,

𝔽_{R}X = R ∧_{ℒ} 𝕃X.

We abbreviate 𝔽X = 𝔽_{R}X when R is clear from the context.

The term “free” is technically a misnomer, since 𝔽 is not left adjoint to the forgetful functor. However, it is nearly so.

Proposition 1.3. The functor 𝔽 : 𝒮 ⟶ ℳ_{R} is left adjoint to the functor that sends an R-module M to the spectrum F_{ℒ} (S, M), and there is a natural map of R-modules ξ : 𝔽M ⟶ M whose adjoint M ⟶ F_{ℒ} (S, M) is a weak equivalence of spectra. Therefore

π_{n}(M) ≅ hℳ_{R}(𝔽S^{n}, M).

1. THE CATEGORY OF R-MODULES; FREE AND COFREE R-MODULES 59

Proof. In view of II.1.3, we have the chain of isomorphisms

ℳ_{R}(𝔽_{R}X, N) ≅ ℳ_{S}(𝔽_{S}X, N) ≅ 𝒮 [𝕃](𝕃X, F_{ℒ} (S, N)) ≅ 𝒮 (X, F_{ℒ} (S, N)).

By I.8.7, we have a natural weak equivalence λ̃ : M ⟶ F_{ℒ} (S, M) of 𝒮 [𝕃]-spectra. Thought of as a map of spectra, its adjoint is the required R-map ξ. The statement about the homotopy groups π_{n}(M) = h𝒮 (S^{n}, M) is clear; compare II.1.8.

The following central theorem shows that we have homotopical control on 𝔽X without any hypotheses (such as tameness or CW homotopy type) on R.

Theorem 1.4. In the stable homotopy category h̄𝒮, 𝔽X is naturally isomorphic to the internal smash product R ∧ X. Moreover, the composite

ζ : 𝔽S⟶^{𝔽η}𝔽R⟶^{ξ}R

is a weak equivalence of R-modules.

Proof. The first statement is clear from II.1.9 and I.6.7, but we point out a variant proof that makes clear that the weak equivalence is one of R-module spectra (in the homotopical sense). In X§5, we shall construct a tame A_{∞} ring spectrum KR and a weak equivalence of A_{∞} ring spectra r : KR ⟶ R. Since we are working in the stable homotopy category, we may take X to be a CW spectrum. Then, by I.4.7 and I.6.6,

r ∧_{ℒ} id : KR ∧_{ℒ} 𝕃X ⟶ R ∧_{ℒ} 𝕃X = 𝔽X

is a weak equivalence. By I.4.7 and I.6.7, there are natural weak equivalences

KR ∧ X ⟶ KR ∧ 𝕃X ⟶ KR ∧_{ℒ} 𝕃X.

For the second statement, observe that ζ is the common composite in the diagram

[figure: R ∧_{ℒ} 𝕃S; id ∧ξ; id ∧𝕃η; R ∧_{ℒ} S; id ∧η; ≅; R ∧_{ℒ} 𝕃R; id ∧ξ; R ∧_{ℒ} R; ϕ; R.]

By I.8.6, the top map id ∧ξ is a weak equivalence.

Corollary 1.5. If X is a wedge of sphere spectra, then π_{∗}(𝔽X) is the free π_{∗}(R)-module with one generator of degree n for each wedge summand S^{n}.

We shall need one further fundamental property of free R-modules.

Definition 1.6. A compact spectrum is one of the form Σ_{V}^{∞}X for a compact space X and an indexing space V ⊂ U. A compact R-module is one of the form 𝔽K for a compact spectrum K.

Proposition 1.7. Let L be a finite colimit of compact R-modules and let {M_{i}} be a sequence of R-modules and (spacewise) inclusions M_{i} ⟶ M_{i+1}. Then

ℳ_{R}(L, colim M_{i}) ≅ colim ℳ_{R}(L, M_{i}).

The generalization from compact R-modules to their finite colimits is immediate. The compact case would be elementary if the free functor were left adjoint to the forgetful functor, and we shall show in XI§2 that this is near enough to being true to give the conclusion.

While they play a less central role, we shall also make use of cofree R-modules. Recall from I§4 that 𝕃^{#} : 𝒮 ⟶ 𝒮 is the right adjoint of 𝕃 and gives a comonad whose coalgebras are the 𝕃-spectra. In particular, 𝕃^{#}X is an 𝕃-spectrum for any spectrum X.

Definition 1.8. Define the cofree S-module generated by a spectrum X to be 𝔽_{S}^{#}X = S ∧_{ℒ} 𝕃^{#}X. Then define the cofree R-module generated by X to be

𝔽_{R}^{#}X = F_{S}(R, 𝔽_{S}^{#}X)

with left action of R induced by the right action of R on itself. We abbreviate 𝔽^{#}X = 𝔽_{R}^{#}X when R is clear from the context.

The term “cofree” is not a misnomer, since here we do have the expected adjunction.

Proposition 1.9. The functor 𝔽_{R}^{#} : 𝒮 ⟶ ℳ_{R} is right adjoint to the forgetful functor ℳ_{R} ⟶ 𝒮.

Proof. Let M be an R-module and X be a spectrum. Lemma 5.5(ii) below gives the first of the following isomorphisms, and II.1.3 and I.4.7 give the others:

ℳ_{R}(M, 𝔽_{R}^{#}X) ≅ ℳ_{S}(M, 𝔽_{S}^{#}X) ≅ 𝒮 [𝕃](M, 𝕃^{#}X) ≅ 𝒮 (M, X).

Theorem 1.10. In the stable homotopy category h̄𝒮, 𝔽^{#}X is naturally isomorphic to the internal function spectrum F (R, X).

Proof. This is immediate from II.1.9, I.7.3, and I.4.9.

2. Cell and CW R-modules; the derived category of R-modules

2. CELL AND CW R-MODULES; THE DERIVED CATEGORY OF R-MODULES 61

To develop cell and CW theories for R-modules, we think of the free R-modules S_{R}^{n} ≡ 𝔽S^{n} as “sphere R-modules”. This is consistent with the sphere S-modules of II.1.7. For cells, we note that the cone functor CX = X ∧ I commutes with 𝔽, so that C𝔽S^{n} ≅ 𝔽CS^{n}. Since 𝔽 has a right adjoint, maps out of sphere R-modules and their cones are induced by maps on the spectrum level; the fact that the right adjoint is not the obvious forgetful functor will create no difficulties. In fact, we can simply parrot the cell theory of spectra from [37, I§5], reducing proofs to those given there via adjunction.

Definitions 2.1. We define cell and relative cell R-modules.

(i) A cell R-module M is the union of an expanding sequence of sub R-modules M_{n} such that M_{0} = ∗ and M_{n+1} is the cofiber of a map ϕ_{n} : F_{n} ⟶ M_{n}, where F_{n} is a (possibly empty) wedge of sphere modules S_{R}^{q} (of varying dimensions). The restriction of ϕ_{n} to a wedge summand S_{R}^{q} is called an attaching map. The induced map

CS_{R}^{q} ⟶ M_{n+1} ⊂ M

is called a cell. The sequence {M_{n}} is called the sequential filtration of M. (ii) For an R-module L, a relative cell R-module (M, L) is an R-module M

specified as in (i), but with M_{0} = L. (ii) A map f : M ⟶ N between cell R-modules is sequentially cellular if

f(M_{n}) ⊂ N_{n} for all n. (iii) A submodule L of a cell R-module M is a cell submodule if L is a cell

R-module such that L_{n} ⊂ M_{n} and the composite of each attaching map S_{R}^{q} ⟶ L_{n} of L with the inclusion L_{n} ⟶ M_{n} is an attaching map of M. Thus every cell of L is a cell of M. Observe that (M, L) may be viewed as a relative cell R-module.

(iv) A cell R-module is finite dimensional if it has cells in finitely many dimensions. It is finite if it has finitely many cells.

The sequential filtration is essential for inductive arguments, but it should be regarded as flexible and subject to change whenever convenient. It merely records the order in which cells are attached and, as long as the cells to which new cells are attached are already present, it doesn’t matter in what order cells are attached.

Lemma 2.2. Let f : M ⟶ N be an R-map between cell R-modules. Then M admits a new sequential filtration with respect to which f is sequentially cellular.

Proof. Assume inductively that M_{n} has been given a filtration as a cell Rmodule M_{n} = ∪M′_{q} such that f(M′_{q}) ⊂ N_{q} for all q. Let χ : S_{R}^{r} ⟶ M_{n} be an attaching map for the construction of M_{n+1} from M_{n} and let χ̃ : CS_{R}^{r} ⟶ M_{n+1} be the corresponding cell. By Proposition 1.7, there is a minimal q such that both

Im(χ) ⊂ M′_{q} and Im(f ∘ χ̃) ⊂ N_{q+1}.

Extend the filtration of M_{n} to M_{n+1} by taking χ to be a typical attaching map of a cell of M′_{q+1}.

We shall occasionally need the following two reassuring results. Their proofs are similar to those of their spectrum level analogs [37, pp 494–495] and depend on Proposition 1.7 and its proof.

Lemma 2.3. A map from a compact R-module to a cell R-module has image contained in a finite subcomplex, and a cell R-module is the colimit of its finite subcomplexes.

If K and L are subcomplexes of a cell R-module M, then we understand their intersection and union in the combinatorial sense. That is, K ∩ L is the cell Rmodule constructed from the attaching maps and cells that are in both K and L and K ∪ L is the cell R-module constructed from the attaching maps and cells that are in either K or L. However, we also have their categorical intersection and union, namely the pullback of the inclusions of K and L in M and the pushout of the resulting maps from the categorical intersection to K and to L.

Lemma 2.4. For subcomplexes K and L of a cell R-module M, the canonical map from the combinatorial intersection to the categorical intersection and from the categorical union to the combinatorial union of K and L are isomorphisms of R-modules.

Definition 2.5. A cell R-module M is said to be a CW R-module if each cell is attached only to cells of lower dimension. The n-skeleton M^{n} of a CW R-module is the union of its cells of dimension at most n. A map f : M ⟶ N between CW R-modules is cellular if f(M^{n}) ⊂ N^{n} for all n. We do not require that f also be sequentially cellular but, by Lemma 2.2, that can always be arranged by changing the order in which cells are attached. Relative CW R-modules (M, L) are defined similarly, with each cell attached only to the union of L and the cells of lower dimension.

Proposition 2.6. The collection of cell R-modules enjoys the following closure properties.

(i) A wedge of cell R-modules is a cell R-module.

(ii) The pushout of a map along the inclusion of a cell submodule is a cell

R-module. (iii) The union of a sequence of inclusions of cell submodules is a cell R-module. (iv) The smash product of a cell R-module and a based cell space (with based

attaching maps) is a cell R-module.

(v) The smash product over S of a cell R-module and a cell S-module is a cell

R-module. The same statements hold with “cell” replaced by “CW”, provided that, in (ii), the given map is cellular.

Proof. In (ii), we apply Lemma 2.2 to ensure that the given map is sequentially cellular. Part (v) follows from I.6.1, which implies that the smash product of a sphere R-module and a sphere S-module is a sphere R-module. Otherwise the proofs are the same as for cell and CW spectra [37, I§5].

The following result is the “Homotopy Extension and Lifting Property”.

Theorem 2.7 (HELP). Let (M, L) be a relative cell R-module and let e : N ⟶ P be a weak equivalence of R-modules. Then, given maps f : M ⟶ P, g : L ⟶ N, and h : L ∧ I_{+} ⟶ P such that f|_{L} = hi_{0} and eg = hi_{1} in the following diagram, there are maps g̃ and h̃ that make the entire diagram commute.

[figure: L; i_{0}; L ∧ I_{+}; h; L; i_{1}; g; P; N; e; M; i_{0}; f; M ∧ I_{+}; h̃; M; g̃; i_{1}]

Proof. This is proven for (M, L) = (CS_{R}^{q}, S_{R}^{q}) by reduction to the spectrum level analog. Technically, we use that the fact that our spheres are obtained from sphere spectra by applying a functor that is left adjoint to a functor that preserves weak equivalences (even though it is not the obvious forgetful functor). The general case follows by induction up the sequential filtration, and the inductive step reduces directly to the case of (CS_{R}^{q}, S_{R}^{q}) already handled.

The Whitehead theorem is a formal consequence.

Theorem 2.8 (Whitehead). If M is a cell R-module and e : N ⟶ P is a weak equivalence of R-modules, then e_{∗} : hℳ_{R}(M, N) ⟶ hℳ_{R}(M, P) is an isomorphism. Therefore a weak equivalence between cell R-modules is a homotopy equivalence.

Recall that a spectrum is “connective” if it is (−1)-connected. When R is connective, π_{q}(N/N^{q}) = 0 for any CW R-module and we can prove the following cellular approximation theorem exactly as in [37, I.5.8]. For non-connective R, this result fails and we must content ourselves with cell R-modules. For connective R, there is no significant loss of information if we restrict attention to CW R-modules.

Theorem 2.9 (Cellular approximation). Assume that R is connective and let (M, L) and (M′, L′) be relative CW R-modules. Then any map f : (M, L) ⟶ (M′, L′) is homotopic relative to L to a cellular map. Therefore, for cell R-modules M and M′, any map M ⟶ M′ is homotopic to a cellular map, and any two homotopic cellular maps are cellularly homotopic.

Theorem 2.10 (Approximation by cell modules). For any R-module M, there is a cell R-module ΓM and a weak equivalence γ : ΓM ⟶ M. If R is connective, then ΓM can be chosen to be a CW R-module.

Proof. Choose a wedge of sphere R-modules N_{0} and a map γ_{0} : N_{0} ⟶ M that induces an epimorphism on homotopy groups. Given γ_{n} : N_{n} ⟶ M, we construct N_{n+1} from N_{n} as a homotopy coequalizer of pairs of representative maps for all pairs of unequal elements of any π_{q}(N_{n}) that map to the same element in π_{q}(M). We have homotopies that allow us to extend γ_{n} to γ_{n+1}. We let ΓM be the union of the N_{n}, and the γ_{n} give a map γ : ΓM ⟶ M. We deduce from Proposition 1.7 that γ is a weak equivalence, and we deduce from Proposition 2.6 that ΓM is a cell R-module. If R is connective, we may take our representative maps to be cellular, and ΓN is then a CW R-module.

Construction 2.11. For each R-module M, choose a cell R-module ΓM and a weak equivalence γ : ΓM ⟶ M. By the Whitehead theorem, for a map f : M ⟶ N, there is a map Γf : ΓM ⟶ ΓN, unique up to homotopy, such that the following diagram is homotopy commutative:

[figure: ΓM; Γf; γ; ΓN; γ; M; f; N.]

Thus Γ is a functor hℳ_{R} ⟶ hℳ_{R}, and γ is a natural transformation from Γ to the identity. The derived category 𝒟_{R} can be described as the category whose objects are the R-modules and whose morphisms are specified by

𝒟_{R}(M, N) = hℳ_{R}(ΓM, ΓN),

with the evident composition. When M is a cell R-module,

𝒟_{R}(M, N) ≅ hℳ_{R}(M, N).

Using the identity function on objects and Γ on morphisms, we obtain a functor i : hℳ_{R} ⟶ 𝒟_{R} that sends weak equivalences to isomorphisms and is universal with this property. Let 𝒞_{R} be the full subcategory of ℳ_{R} whose objects are the cell R-modules. Then the functor Γ induces an equivalence of categories 𝒟_{R} ⟶ h𝒞_{R} with inverse the composite of i and the inclusion of h𝒞_{R} in hℳ_{R}.

3. THE SMASH PRODUCT OF R-MODULES 65

Therefore the derived category and the homotopy category of cell R-modules can be used interchangeably. Homotopy-preserving functors on R-modules that do not preserve weak equivalences are transported to the derived category by first applying Γ, then the given functor.

The category 𝒟_{R} has all homotopy limits and colimits. They are created as the corresponding constructions on the underlying diagrams of S-modules; equivalently, homotopy colimits are created on the spectrum level and homotopy limits are created from spectrum level homotopy limits, which are 𝒮 [𝕃]-spectra, by applying the functor S ∧_{ℒ} (?). Explicit functorial constructions will be given in X§3. We have enough information to quote the categorical form of Brown’s representability theorem given in [13]. Adams’ analogue [3] for functors defined only on finite CW spectra also applies in our context, with the same proof.

Theorem 2.12 (Brown). A contravariant functor k : 𝒟_{R} → Sets is representable in the form k(M) ≅ 𝒟_{R}(M, N) for some R-module N if and only if k converts wedges to products and converts homotopy pushouts to weak pullbacks.

Theorem 2.13 (Adams). A contravariant group-valued functor k defined on the homotopy category of finite cell R-modules is representable in the form k(M) ≅ 𝒟_{R}(M, N) for some R-module N if and only if k converts finite wedges to direct products and converts homotopy pushouts to weak pullbacks of underlying sets.

3. The smash product of R-modules

We mimic the definition of tensor products of modules over algebras.

Definition 3.1. Let R be an S-algebra and let M be a right and N be a left R-module. Define M ∧_{R} N to be the coequalizer displayed in the following diagram of S-modules:

[figure: M ∧_{S} R ∧_{S} N; μ∧_{S}id; id ∧_{S}ν; M ∧_{S} N; M ∧_{R} N,]

where μ and ν are the given actions of R on M and N.

When R = S, we are coequalizing the same isomorphism (see I.8.3). Therefore our new M ∧_{S} N coincides with our old M ∧_{S} N.

We shall shortly construct function R-modules satisfying the usual adjunction. It will follow that the functor ∧_{R} preserves colimits in each of its variables. It is clear that smash products with spaces commute with ∧_{R}, in the sense that

(X ∧ M) ∧_{R} N ≅ X ∧ (M ∧_{R} N) ≅ (M ∧_{R} N) ∧ X ≅ M ∧_{R} (N ∧ X).

Therefore the functor ∧_{R} commutes with cofiber sequences in each of its variables. We also have the following adjunction, which complements Theorem 1.1(iv).

Lemma 3.2. For an S-module K,

ℳ_{S}(M ∧_{R} N, K) ≅ ℳ_{R}(N, F_{S}(M, K)).

The commutativity, associativity, and unity properties of the smash product over S and comparisons of coequalizer diagrams give commutativity, associativity, and unity properties of the smash product over R, exactly as in algebra. We state these properties in the generality of their algebraic counterparts.

An S-algebra R with product ϕ : R ∧_{S} R → R has an opposite S-algebra R^{op} with product ϕ ∘ τ, and a left R-module with action μ is a right R^{op}-module with action μ ∘ τ.

Lemma 3.3. For a right R-module M and left R-module N,

M ∧_{R} N ≅ N ∧_{R^{op}} M.

For S-algebras R and R′, we define an (R, R′)-bimodule to be a left R and right R′-module M such that the evident diagram commutes:

\xymatrix{R ∧_{S} M ∧_{S} R′ \ar[r] \ar[d] & M ∧_{S} R′ \ar[d] \\ R ∧_{S} M \ar[r] & M.}

As in algebra, an (R, R′)-bimodule is the same thing as an (R ∧_{S} (R′)^{op})-module.

Proposition 3.4. Let M be an (R, R′)-bimodule, N be an (R′, R′′)-bimodule, and P be an (R′′, R′′′)-bimodule. Then M ∧_{R′} N is an (R, R′′)-bimodule and

(M ∧_{R′} N) ∧_{R′′} P ≅ M ∧_{R′} (N ∧_{R′′} P)

as (R, R′′′)-bimodules.

The unity isomorphism has already been displayed, in the guise of a split coequalizer diagram, in II.3.2. We restate the conclusion.

Lemma 3.5. The action ν : R ∧_{S} N ⟶ N of an R-module N factors through an isomorphism of R-modules λ : R ∧_{R} N ⟶ N.

For an S-module K, R ∧_{S} K ≅ K ∧_{S} R is an (R, R)-bimodule. In particular, this applies to the free left R-module 𝔽_{R}X = R ∧_{S} 𝔽_{S}X generated by a spectrum X, which may be identified with the free right R-module generated by X. The following instances of the isomorphisms above will be used in conjunction with the weak equivalences of I.6.7 and II.1.9. They allow us to deduce homotopical properties of ∧_{R} from corresponding properties of ∧_{S}.

Proposition 3.6. Let K and L be S-modules and let N be an R-module. There is a natural isomorphism of R-modules

(K ∧_{S} R) ∧_{R} N ≅ K ∧_{S} N.

There is also a natural isomorphism of (R, R)-bimodules

(K ∧_{S} R) ∧_{R} (R ∧_{S} L) ≅ R ∧_{S} (K ∧_{S} L).

Using I.6.1, we obtain the following consequence, in which we use an isomorphism of universes f : U ⊕ U → U to define the internal smash product f_{∗}(X ∧ Y).

Corollary 3.7. Let X and Y be spectra and let N be an R-module. There is a natural isomorphism of R-modules

𝔽_{R}X ∧_{R} N ≅ 𝔽_{S}X ∧_{S} N.

There is also a natural isomorphism of (R, R)-bimodules

𝔽_{R}X ∧_{R} 𝔽_{R}Y ≅ 𝔽_{R}f_{∗}(X ∧ Y).

Theorem 3.8. If M is a cell R-module and ϕ : N ⟶ N′ is a weak equivalence of R-modules, then

id ∧_{R}ϕ : M ∧_{R} N ⟶ M ∧_{R} N′

is a weak equivalence of S-modules.

Proof. When M = 𝔽_{R}X for a CW spectrum X, the conclusion is immediate from the corollary and I.6.6. The general case follows from the case of sphere R-modules by induction up the sequential filtration and passage to colimits.

We construct ∧_{R} as a functor

r𝒟_{R} × ℓ𝒟_{R} → 𝒟_{S}

by approximating one of the variables by a cell R-module; here “r” and “ℓ” indicate right and left R-modules. That is, the derived smash product of M and N is represented by ΓM ∧_{R} N.

The following technical sharpening of Corollary 1.5 will be the starting point for our later construction of a spectral sequence for the computation of π_{∗}(M ∧_{R} N).

Proposition 3.9. Let X be a wedge of sphere spectra and let N be a cell Rmodule. Then there is an isomorphism

π_{∗}(𝔽_{R}X ∧_{R} N) ≅ (π_{∗}(R) ⊗ H_{∗}(X)) ⊗_{π_{∗}(R)} π_{∗}(N)

that is natural in the R-modules 𝔽_{R}X and N.

Proof. The point is that naturality on general maps g : 𝔽_{R}X ⟶ 𝔽_{R}X′, with their induced maps π_{∗}(R) ⊗ H_{∗}(X) ⟶ π_{∗}(R) ⊗ H_{∗}(X′), and not just on maps of the form g = 𝔽_{R}f, f : X ⟶ X′, will be essential in the cited application. The diagram

\xymatrix{𝔽_{R}X ∧_{S} R ∧_{S} N \ar[r] \ar[r] & 𝔽_{R}X ∧_{S} N \ar[r] & 𝔽_{R}X ∧_{R} N}

is a split coequalizer in ℳ_{S} and thus in 𝒮, and it is visibly natural in both 𝔽_{R}X and N. It remains a coequalizer on applying π_{∗}, and the required naturality follows.

Finally, we record an analogue of the behavior of tensor products of modules with respect to tensor products of algebras.

Proposition 3.10. Let R and R′ be S-algebras, M and N be right and left R-modules, and M′ and N′ be right and left R′-modules. Then there is a natural isomorphism of S-modules

(M ∧_{S} M′) ∧_{R∧_{S}R′} (N ∧_{S} N′) ≅ (M ∧_{R} N) ∧_{S} (M′ ∧_{R′} N′).

If M is a cell R-module and N′ is a cell R′-module, then M ∧_{S} N′ is a cell R∧_{S} R′- module.

Proof. The first statement is a comparison of coequalizer diagrams. The second statement holds since, on spheres, I.6.1 implies isomorphisms

(𝕃S^{q} ∧_{ℒ} R) ∧_{S} (R′ ∧_{ℒ} 𝕃S^{r}) ≅ (R ∧_{S} R′) ∧_{ℒ} 𝕃S^{q+r}.

4. Change of S-algebras; q-cofibrant S-algebras

In this section, we assume given a map of S-algebras

ϕ : R ⟶ R′,

and we study the relationship between the categories of R-modules and of R′- modules. By pullback along ϕ, we obtain a functor ϕ^{∗} : ℳ_{R′} ⟶ ℳ_{R}. It preserves weak equivalences and thus induces a functor ϕ^{∗} : 𝒟_{R′} ⟶ 𝒟_{R}. It is vital to the theory that this functor is an equivalence of categories when ϕ is a weak equivalence. As we explain, this allows us to replace general S-algebras by better behaved “qcofibrant” ones whenever convenient, without changing the derived category.

Regard R′ as a right R-module via the composite

[figure: R′ ∧_{S} R; id ∧ϕ; R′ ∧_{S} R′; μ′; R′.]

Observe that R′ is an (R′, R)-bimodule with the evident left action by R′ and that, for an R-module M, R′ ∧_{R} M is an R′-module.

4. CHANGE OF S-ALGEBRAS; q-COFIBRANT S-ALGEBRAS 69

Proposition 4.1. Define ϕ_{∗} : ℳ_{R} ⟶ ℳ_{R′} by ϕ_{∗}M = R′ ∧_{R} M. Then ϕ_{∗} is left adjoint to ϕ^{∗}, and the adjunction induces a derived adjunction

𝒟_{R′}(ϕ_{∗}M, M′) ≅ 𝒟_{R}(M, ϕ^{∗}M′).

Moreover, the functor ϕ_{∗} preserves cell modules.

Proof. The required isomorphism

ℳ_{R′}(ϕ_{∗}M, M′) ≅ ℳ_{R}(M, ϕ^{∗}M′)

is proven exactly as in algebra. It sends an R-map M → M′ to the induced composite

R′ ∧_{R} M ⟶ R′ ∧_{R} M′ ⟶ R′ ∧_{R′} M′ ≅ M′,

and it sends an R′-map R′ ∧_{R} M ⟶ M′ to its restriction along the canonical map M ⟶ R′ ∧_{R} M. Since the functor ϕ^{∗} preserves weak equivalences, it is formal that the functor ϕ_{∗} carries R-modules of the homotopy types of cell modules to R′-modules of the homotopy types of cell modules and induces an adjunction on derived categories [37, I.5.13]. Clearly

R′ ∧_{R} (R ∧_{S} L) ≅ R′ ∧_{S} L

for an S-module L. Therefore the functor ϕ_{∗} carries sphere R-modules to sphere R′-modules. Since, as a left adjoint, ϕ_{∗} preserves colimits, this implies that ϕ_{∗} preserves cell modules and not just homotopy types of cell modules.

Theorem 4.2. Let ϕ : R ⟶ R′ be a weak equivalence of S-algebras. Then ϕ_{∗} : 𝒟_{R} ⟶ 𝒟_{R′} and ϕ^{∗} : 𝒟_{R′} ⟶ 𝒟_{R} are inverse adjoint equivalences of categories.

Proof. If M is a cell R-module, then the unit

ϕ ∧_{R} id : M ≅ R ∧_{R} M ⟶ R′ ∧_{R} M

of the adjunction is a weak equivalence by Theorem 3.8. Now let M′ be an R′- module. In the derived category, the composite ϕ_{∗}ϕ^{∗}M′ means R′ ∧_{R} ΓM′, where ΓM′ is a cell R-module for which there is a weak equivalence of R-modules γ : ΓM′ ⟶ ϕ^{∗}M′. The counit of the adjunction is given by

id ∧_{ϕ}γ : R′ ∧_{R} ΓM′ ⟶ R′ ∧_{R′} M′ ≅ M′.

An easy diagram chase shows that the composite map of R-modules

[figure: ΓM′ ≅ R ∧_{R} ΓM′; ϕ∧_{R}id; R′ ∧_{R} ΓM′; id ∧_{ϕ}γ; R′ ∧_{R′} M′ ≅ M′]

coincides with γ. Since ϕ ∧_{R} id is a weak equivalence, so is id ∧_{ϕ}γ.

We shall give the category of S-algebras a Quillen (closed) model category structure in Chapter VII. We will then have the notion of a “q-cofibrant S-algebra”, which is a retract of a “cell S-algebra”. For any S-algebra R, there is a weak equivalence λ : ΛR ⟶ R, where ΛR is a cell S-algebra. By the previous result, λ induces an adjoint equivalence between the categories 𝒟_{R} and 𝒟_{ΛR}. Actually, we will have two quite different model categories, one for S-algebras and another for commutative S-algebras. The comments that we have just made apply in either context. As we shall explain in VII§6, the forgetful functor from R-algebras to R-modules is better behaved homotopically in the non-commutative case than in the commutative case. In fact, VII.6.2 will give the following result.

Theorem 4.3. If R is a q-cofibrant S-algebra, then (R, S) is a relative cell Smodule, the inclusion S ⟶ R being the unit of R. Therefore (R, S) has the homotopy type of a relative CW S-module.

Since we can approximate a commutative S-algebra by a non-commutative cell S-algebra without changing the derived category of modules (up to equivalence), we can use the previous result to obtain homotopical information about the derived categories of commutative S-algebras.

We illustrate the force of these ideas by using them to obtain a complementary adjunction to the case of Proposition 4.1 that is obtained by specializing to the unit η : S ⟶ R of an R-algebra:

𝒟_{R}(R ∧_{S} M, N) ≅ 𝒟_{S}(M, η^{∗}N)

for S-modules M and R-modules N.

Proposition 4.4. The forgetful functor η^{∗} : 𝒟_{R} ⟶ 𝒟_{S} has a right adjoint η^{#} : 𝒟_{S} ⟶ 𝒟_{R}, so that

𝒟_{R}(N, η^{#}M) ≅ 𝒟_{S}(η^{∗}N, M)

for S-modules M and R-modules N.

Proof. On the point set level, we have the adjunction

ℳ_{R}(η^{∗}N, M) ≅ ℳ_{S}(N, F_{S}(R, M)).

Here we regard R as an (S, R)-bimodule, and the right action of R on itself induces a left action of R on F_{S}(R, M) (as with Hom functors in algebra). However, there is no reason to believe that the functor F_{S}(R, M) of M preserves weak equivalences, so that it is not clear how to pass to derived categories. Let λ : ΛR ⟶ R be a weak equivalence of S-algebras, where ΛR is a cell S-algebra. It follows easily from the previous theorem that the functor

F_{S}(ΛR, M) : ℳ_{S} ⟶ ℳ_{ΛR}

5. SYMMETRIC AND EXTENDED POWERS OF R-MODULES 71

of M does preserve weak equivalences. We therefore have an adjunction

𝒟_{S}((η′)^{∗}N′, M) ≅ 𝒟_{ΛR}(N, F_{S}(ΛR, M))

for ΛR-modules N′ and S-modules M, where η′ is the unit of ΛR. Theorem 4.2 implies that we also have an adjunction

𝒟_{ΛR}(λ^{∗}N, N′) ≅ 𝒟_{R}(N, λ_{∗}N′).

Since η = λ ∘ η′ : S ⟶ R and these forgetful functors all preserve weak equivalences, η^{∗} = (η′)^{∗} ∘ λ^{∗} : 𝒟_{R} ⟶ 𝒟_{S}. We define η^{#}(M) = λ_{∗}F_{S}(ΛR, M) and obtain the desired adjunction as the composite of the adjunctions just given.

5. Symmetric and extended powers of R-modules

Let R be an S-algebra and M be an R-module. The jth symmetric power of M is defined to be M^{j}/Σ_{j} and the jth extended power of M is defined to be

D_{j}M = (EΣ_{j})_{+} ∧_{Σ_{j}} M^{j}.

In both notions, M^{j} denotes the j-th power of M with respect to ∧_{R}. One of the most striking features of our smash product of R-modules is that, in the derived category 𝒟_{R}, these are essentially equivalent notions. This fact will give us homotopical control on free R-algebras and will play an important role in our study of Bousfield localizations of R-algebras in Chapter VIII, but it will not be needed before then.

To explain this fact, observe that I.5.6 implies that, for a spectrum K, we have an equivariant isomorphism

(𝕃K)^{j} ≅ ℒ (j) ⋉ K^{j},

where the j-th power is taken with respect to ∧_{ℒ} on the left and with respect to the external smash product on the right. Therefore

(𝕃K)^{j}/Σ_{j} ≅ ℒ (j) ⋉_{Σ_{j}} K^{j}.

Behavior like this propagates through our constructions. However, to retain suffficient homotopical control on our constructions to prove this, we must assume that R is a q-cofibrant S-algebra (in either the non-commutative or the commutative sense) and apply results to be proven in VII§6. Note that S itself is q-cofibrant in both senses.

Theorem 5.1. Let R be a q-cofibrant S-algebra or commutative S-algebra. If M is a cell R-module, then the projection

π : (EΣ_{j})_{+} ∧_{Σ_{j}} M^{j} ⟶ M^{j}/Σ_{j}

is a homotopy equivalence of spectra.

Proof. The conclusion is trivial for j = 1 and we may assume inductively that it holds for i < j.

We first prove the result for any j when M is the free R-module generated by a CW-spectrum X. Expanding definitions and commuting the smash product with (EΣ_{j})_{+} through our constructions, we find that

M^{j} ≅ R ∧_{S} S ∧_{ℒ} (ℒ (j) ⋉ X^{j}),

(EΣ_{j})_{+} ∧ M^{j} ≅ R ∧_{S} S ∧_{ℒ} ((EΣ_{j} × ℒ (j)) ⋉ X^{j}),

and π is induced from the Σ_{j}-equivalence EΣ_{j} × ℒ (j) ⟶ ℒ (j) by passage to orbits. When R = S, π is a homotopy equivalence of spectra by I.2.5 and I.8.5. For general R, VII.6.5 and VII.6.7 imply that the functor R ∧_{S} (?) carries this homotopy equivalence to a weak equivalence. However, the domain and target have the homotopy types of CW spectra, by VII.6.6.

Next, let M be a subcomplex of a cell R-module N and assume that the conclusion holds for M and N/M. As explained for the (external) smash power of spectra in [14, pp 37-38] and works equally well for the (internal) smash power of R-modules, we have a filtration of N^{j} by Σ_{j}-cofibrations of R-modules

M^{j} = F_{j}N^{j} ⊂ F_{j−1}N^{j} ⊂ ⋯ ⊂ F_{1}N^{j} ⊂ F_{0}N^{j} = N^{j}.

Here F_{i}N^{j} is the union of the subcomplexes M_{1} ∧_{R} ⋯ ∧_{R} M_{j}, where each M_{k} is either M or N and i of the M_{k} are M. The subquotients can be identified equivariantly as

F_{i}N^{j}/F_{i+1}N^{j} ≅ Σ_{j} ×_{Σ_{i}×Σ_{j−i}} (M^{i} ∧_{R} (N/M)^{j−i}).

As a (Σ_{i} × Σ_{j−i})-space, EΣ_{j} is homotopy equivalent to EΣ_{i} × EΣ_{j−i}, and there result homotopy equivalences

(EΣ_{j})_{+} ∧_{Σ_{j}} F_{i}N^{j}/F_{i+1}N^{j} ≃ ((EΣ_{i})_{+} ∧_{Σ_{i}} M^{i}) ∧_{R} ((EΣ_{j−i})_{+} ∧_{Σ_{j−i}} (N/M)^{j−i}).

Applying the original induction hypothesis on j and inducting up the filtration, we deduce the conclusion for N.

Finally, turning to the general case, let {M_{n}} be the sequential filtration of M, with M_{0} = ∗. By the first step, the conclusion holds for each M_{n+1}/M_{n}. By the second step, the conclusion for M_{n} implies the conclusion for M_{n+1}. Since M^{j} is the colimit of the sequence of Σ_{j}-cofibrations of R-modules (M_{n})^{j} ⟶ (M_{n+1})^{j}, the conclusion for M follows.

6. FUNCTION R-MODULES 73

6. Function R-modules

Let R be an S-algebra. We have a function R-module functor F_{R} to go with our smash product. Its definition is dictated by the expected adjunction.

Definition 6.1. Let M and N be (left) R-modules. Define F_{R}(M, N) to be the equalizer displayed in the following diagram of S-modules:

[figure: F_{R}(M, N); F_{S}(M, N); μ^{∗}; ω; F_{S}(R ∧_{S} M, N).]

Here μ^{∗} = F_{S}(μ, id) and ω is the adjoint of the composite

[figure: R ∧_{S} (M ∧_{S} F_{S}(M, N)); id ∧ ε; R ∧_{S} N; ν; N.]

When R = S, our new and old function S-modules F_{S}(M, N) are identical. We state the expected adjunction in a general form, but we are most interested in the case R′ = S.

Lemma 6.2. Let M be an (R, R′)-bimodule, N be an R′-module, and P be an R-module. Then

ℳ_{R}(M ∧_{R′} N, P) ≅ ℳ_{R′}(N, F_{R}(M, P)).

Proof. The general case follows from the case R = S of Lemma 3.2 by use of the coequalizer definition of ∧_{R′} and the equalizer definition of F_{R}.

As in algebra, this leads to a function module analogue of Proposition 3.4.

Proposition 6.3. Let M be an (R, R′)-bimodule, N be an (R′, R′′)-bimodule, and P be an (R, R′′′)-bimodule. Then F_{R}(M, P) is an (R′, R′′′)-bimodule, and

F_{R}(M ∧_{R′} N, P) ≅ F_{R′}(N, F_{R}(M, P))

as (R′′, R′′′)-bimodules.

Similarly, the unit isomorphism of Lemma 3.5 implies a counit isomorphism.

Lemma 6.4. The adjoint λ̃ : M → F_{R}(R, M) is an isomorphism.

We also have analogs of Proposition 3.6 and Corollary 3.7. While we are interested primarily in the versions relating F_{R} to the functor ∧_{S}, there are also versions relating F_{R} to the functor F_{S}. The following lemma is needed for the latter versions. Its algebraic analogue is proven by a formal argument that applies equally well in topology.

Lemma 6.5. Let R and R′ be S-algebras.

(i) Let M be an R-module, M′ be an R′-module and P be an R ∧_{S} R′-module. Then there is a natural bijection

ℳ_{R}(M, F_{R′}(M′, P)) ≅ ℳ_{R∧_{S}R′}(M ∧_{S} M′, P).

(ii) Let M be a left R-module, N be a right R-module, and K be an S-module. Then there is a natural bijection

ℳ_{R}(M, F_{S}(N, K)) ≅ ℳ_{S}(N ∧_{R} M, K).

Proof. It suffices to check (i) when M = R ∧_{S} L and M′ = R′ ∧_{S} L′ are the free modules generated by S-modules L and L′. Similarly, it suffices to check (ii) when M = R ∧_{S} L. These cases are easy consequences of our adjunctions.

Proposition 6.6. Let K be an S-module and M be a left R-module. There is a natural isomorphism of left R-modules

F_{R}(K ∧_{S} R, M) ≅ F_{S}(K, M)

and a natural isomorphism of right R-modules

F_{R}(M, F_{S}(R, K)) ≅ F_{S}(M, K).

Proof. The first isomorphism is immediate from the following chain of isomorphisms of represented functors on left R-modules N, which result from Proposition 6.3, Proposition 3.6, and Theorem 1.1(iii), respectively.

ℳ_{R}(N, F_{R}(K ∧_{S} R, M)) ≅ ℳ_{R}((K ∧_{S} R) ∧_{R} N, M)
≅ ℳ_{R}(K ∧_{S} N, M)
≅ ℳ_{R}(N, F_{S}(K, M)).

The second isomorphism results from the following chain of isomorphisms of represented functors on right R-modules N:

ℳ_{R^{op}}(N, F_{R}(M, F_{S}(R, K))) ≅ ℳ_{R∧_{S}R^{op}}(M ∧_{S} N, F_{S}(R, K))
≅ ℳ_{S}(R ∧_{R∧_{S}R^{op}} (M ∧_{S} N), K)
≅ ℳ_{S}(M ∧_{R^{op}} N, K) ≅ ℳ_{R^{op}}(N, F_{S}(M, K)).

The first two isomorphisms are instances of isomorphisms of the lemma. The third follows from the fact that there is a natural isomorphism

R ∧_{R∧_{S}R^{op}} (M ∧_{S} N) ≅ M ∧_{R^{op}} N,

as is easily checked when M and N are free R-modules and follows in general.

Corollary 6.7. Let X be a spectrum and M be an R-module. There is a natural isomorphism of left R-modules

F_{R}(𝔽_{R}X, M) ≅ F_{S}(𝔽_{S}X, M)

and a natural isomorphism of right R-modules

F_{R}(M, 𝔽_{R}^{#}X) ≅ F_{S}(M, 𝔽_{S}^{#}X)).

The functor F_{R}(M, N) converts colimits and cofiber sequences in M to limits and fiber sequences and it preserves limits and fiber sequences in N, as we see formally on the spectrum level (compare [37, III.2.5]) and deduce in order on the levels of 𝕃-spectra, S-modules, and R-modules (compare II.1.5 and Theorem 1.1). Using the previous corollary to deal with sphere R-modules and proceeding by induction up the sequential filtration of M, we obtain the analogue of Theorem 3.8.

Theorem 6.8. If M is a cell R-module and ϕ : N ⟶ N′ is a weak equivalence of R-modules, then

F_{R}(id, ϕ) : F_{R}(M, N) ⟶ F_{R}(M, N′)

is a weak equivalence.

In the derived category 𝒟_{R}, F_{R}(M, N) means F_{R}(ΓM, N), where ΓM is a cell approximation of M. We are entitled to conclude that

𝒟_{R}(M ∧_{S} N, P) ≅ 𝒟_{S}(N, F_{R}(M, P)).

As in Proposition 3.9, we have the following calculational sharpening of Corollary 6.7. It will be the starting point for our later construction of a spectral sequence for the calculation of π_{∗}(F_{R}(M, N)).

Corollary 6.9. Let X be a wedge of sphere spectra and N be an R-module. Then there is an isomorphism

π_{∗}(F_{R}(𝔽X, N)) ≅ Hom_{π_{∗}(R)}(π_{∗}(R) ⊗ H_{∗}(X), π_{∗}(N))

that is natural in the R-modules 𝔽X and N.

We end this section by recording a composition pairing that is a formal implication of Lemma 6.2 and Proposition 6.3. This works exactly as with tensor products and Hom in algebra and, as there, it is convenient for this purpose to use the commutativity of the smash product over S to rewrite our adjunctions and isomorphisms with their variables occurring in the same order on both sides, returning to our original conventions of I§7. Thus, for an S-module L and for R-modules M and N, we have the natural isomorphism of S-modules

F_{R}(L ∧_{S} M, N) ≅ F_{S}(L, F_{R}(M, N)). (6.10)

Let P be another R-module. Using the evaluation R-map

ε : F_{R}(M, N) ∧_{S} M ⟶ N,

we obtain a composite R-map

[figure: F_{R}(N, P) ∧_{S} F_{R}(M, N) ∧_{S} M; id ∧_{S}ε; F_{R}(N, P) ∧_{S} N; ε; P.]

Its adjoint is a composition pairing of S-modules

π : F_{R}(N, P) ∧_{S} F_{R}(M, N) ⟶ F_{R}(M, P). (6.11)

This pairing is unital and associative in the sense that the following diagrams commute; let η : S ⟶ F_{R}(M, M) be the adjoint of λ : S ∧_{S} M ⟶ M:

[figure: F_{R}(N, P) ∧_{S} S; id ∧_{S} η; λτ; F_{R}(N, P) ∧_{S} F_{R}(N, N); π; F_{R}(N, P),]

[figure: S ∧_{S} F_{R}(M, N); η ∧_{S}id; λ; F_{R}(N, N) ∧_{S} F_{R}(M, N); π; F_{R}(M, N),]

and, for another R-module L,

[figure: F_{R}(N, P) ∧_{S} F_{R}(M, N) ∧_{S} F_{R}(L, M); id ∧_{S} π; π ∧_{S}id; F_{R}(N, P) ∧_{S} F_{R}(L, N); π; F_{R}(M, P) ∧_{S} F_{R}(L, M); π; F_{R}(L, P).]

This leads to a host of examples of S-algebras and their modules.

Proposition 6.12. Let R be an S-algebra and let M and N be (left) R-modules. Then F_{R}(N, N) is an S-algebra with product π and unit η. Moreover, F_{R}(M, N) is an (F_{R}(N, N), F_{R}(M, M))-bimodule with left and right actions given by π.

7. COMMUTATIVE S-ALGEBRAS AND DUALITY THEORY 77

7. Commutative S-algebras and duality theory

We assume that R is a commutative S-algebra in this section, and we show that the study of modules over R works in exactly the same way as the study of modules over commutative algebras. If μ : R∧_{S} M → M gives M a left R-module structure, then μ ∘ τ : M ∧_{S} R → M gives M a right R-module structure such that M is an (R, R)-bimodule. As in the study of modules over commutative algebras, this leads to the following important conclusion.

Theorem 7.1. If M and N are R-modules, then M ∧_{R} N and F_{R}(M, N) have canonical R-module structures induced from the R-module structure of M or, equivalently, N. The smash product over R is commutative, associative and unital. There is an adjunction

ℳ_{R}(L ∧_{R} M, N) ≅ ℳ_{R}(L, F_{R}(M, N)). (7.2)

Moreover, the adjunction passes to derived categories.

We have the following consequence of Corollary 3.7.

Proposition 7.3. If M and M′ are cell R-modules, then M ∧_{R} M′ is a cell R-module with one (p + q)-cell for each p-cell of M and q-cell of M′.

For R-modules L, M and N, we have a natural isomorphism of R-modules

F_{R}(L ∧_{R} M, N) ≅ F_{R}(L, F_{R}(M, N)) (7.4)

because both sides represent the same functor. Exactly as in the previous section, but working entirely with R-modules, we obtain a natural associative and R-unital composition pairing

π : F_{R}(M, N) ∧_{R} F_{R}(L, M) ⟶ F_{R}(L, N). (7.5)

The formal duality theory explained in [37, Ch. III] applies to the stable category of R-modules. We define the dual of an R-module M to be D_{R}M = F_{R}(M, R). We have an evaluation map ε : D_{R}M ∧_{R} M ⟶ R and a map η : R → F_{R}(M, M), namely the adjoint of λ : R ∧_{R} M ⟶ M. There is also a natural map

ν : F_{R}(L, M) ∧_{R} N ⟶ F_{R}(L, M ∧_{R} N). (7.6)

By composition with the isomorphism F_{R}(id, λ), ν specializes to a map

ν : D_{R}M ∧_{R} M ⟶ F_{R}(M, M). (7.7)

We say that M is “strongly dualizable”, if it has a coevaluation map η̄ : R ⟶ M ∧_{R} D_{R}M such that the following diagram commutes in 𝒟_{R}:

[figure: R; η̄; η; M ∧_{R} D_{R}M; τ; F_{R}(M, M); D_{R}M ∧_{R} M.; ν] (7.8)

The definition has many purely formal implications. The map ν of (7.6) is an isomorphism in 𝒟_{R} if either L or N is strongly dualizable. The map ν of (7.7) is an isomorphism in 𝒟_{R} if and only if M is strongly dualizable, and the coevaluation map η̄ is then the composite τν^{−1}η in (7.8). The natural map

ρ : M ⟶ D_{R}D_{R}M

is an isomorphism in 𝒟_{R} if M is strongly dualizable. The natural map

∧ : F_{R}(M, N) ∧_{R} F_{R}(M′, N′) ⟶ F_{R}(M ∧_{R} M′, N ∧_{R} N′)

is an isomorphism in 𝒟_{R} if M and M′ are strongly dualizable or if M is strongly dualizable and N = R.

Say that a cell R-module N is a wedge summand up to homotopy of a cell Rmodule M if there is a homotopy equivalence of R-modules between M and N ∨N′ for some cell R-module N′. In contrast with the usual stable homotopy category, if M is finite it does not follow that N must have the homotopy type of a finite cell R-module. Via Eilenberg-Mac Lane spectra, finitely generated projective modules that are not free give rise to explicit counterexamples. Define a semi-finite Rmodule to be an R-module that is a wedge summand up to homotopy of a finite cell R-module, and note for use in Chapter VI that this notion makes sense even when R is not commutative.

Theorem 7.9. A cell R-module is strongly dualizable if and only if it is semifinite.

Proof. Observe first that S_{R}^{q} is strongly dualizable with dual S_{R}^{−q}, hence any finite wedge of sphere R-modules is strongly dualizable. Observe next that the cofiber of a map between strongly dualizable R-modules is strongly dualizable. In fact, the evaluation map ε induces a natural map

ε_{#} : 𝒟_{R}(L, N ∧_{R} D_{R}M) → 𝒟_{R}(L ∧_{R} M, N),

and M is strongly dualizable if and only if ε_{#} is an isomorphism for all L and N [37, III.1.6]. Since both sides convert cofiber sequences in the variable M into long exact sequences, the five lemma gives the observation. We conclude by induction on the number of cells that a finite cell R-module is strongly dualizable. It is formal that a wedge summand in 𝒟_{R} of a strongly dualizable cell R-module is strongly dualizable. For the converse, let N be a cell R-module that is strongly dualizable with coevaluation map η̄ : R → N ∧_{R} D_{R}N. Since η̄ is determined by its restriction to S and S is compact, η̄ factors through M ∧_{R} D_{R}N for some finite cell subcomplex M of N. By [37, III.1.2], the bottom composite in the following commutative diagram is the identity (in 𝒟_{R}):

[figure: M ∧_{R} D_{R}N ∧_{R} N; id ∧ε; M ∧_{R} R; ≃; M; N ≅ R ∧_{R} N; η̄∧id; N ∧_{R} D_{R}N ∧_{R} N; id ∧ε; N ∧_{R} R; ≃; N.]

Therefore N is a retract up to homotopy and thus, by a comparison of exact triangles, a wedge summand up to homotopy of M: retractions split in triangulated categories.

80 III. THE HOMOTOPY THEORY OF R-MODULES

CHAPTER IV The algebraic theory of R-modules

We define generalized Tor and Ext groups as the homotopy groups of derived smash product and function modules, and we interpret these groups in terms of generalized homology and cohomology theories on R-modules. Specializing to Eilenberg-Mac Lane spectra, these groups give the classical Tor and Ext groups, and we show how to topologically realize classical algebraic derived categories of complexes of modules over a ring. Starting with a connective S-algebra R, rather than an Eilenberg-Mac Lane spectrum, the discussion generalizes to give ordinary homology and cohomology theories on R-modules, together with Atiyah-Hirzebruch spectral sequences for the computation of generalized homology and cohomology theories on R-modules.

In Sections 4 and 5, we construct “hyperhomology” spectral sequences for the calculation of our generalized Tor and Ext groups in terms of ordinary Tor and Ext groups, and we show that these specialize to give universal coefficient and Künneth spectral sequences for homology and cohomology theories defined on spectra. In Sections 6 and 7, we generalize to Eilenberg-Moore spectral sequences for the computation of E_{∗}(M ∧_{R} N) under varying hypotheses on R and E. In particular, we give a bar construction approximation to M ∧_{R} N that allows us to view the classical space level Eilenberg-Moore-Rothenberg-Steenrod spectral sequence as a special case.

Except that his theory was intrinsically restricted to the A_{∞} context, Robinson’s series of papers [56, 57, 58, 59, 60] gave earlier versions of many of the results of this chapter. Of course, with the earlier technology, the proofs were substantially more difficult.

As usual, for a spectrum E, we shall often abbreviate notations by setting

E_{n} = π_{n}(E) = E^{−n}.

82 IV. THE ALGEBRAIC THEORY OF R-MODULES

1. Tor and Ext; homology and cohomology; duality

Definition 1.1. Let R be an S-algebra. For a right R-module M and a left R-module N, define

Tor_{n}^{R}(M, N) = π_{n}(M ∧_{R} N).

For left R-modules M and N, define

Ext_{R}^{n}(M, N) = π_{−n}(F_{R}(M, N)).

Here the smash product and function modules are understood to be taken in the derived category 𝒟_{R}. For Tor, this means that M or N must be replaced by a weakly equivalent cell R-module before applying the module level functor ∧_{R}. For Ext, this means that M must be approximated by a cell R-module before applying F_{R}. At this point in our work, however, we act as traditional topologists, taking it for granted that all spectra and modules are to be approximated as cell modules, without change of notation, whenever necessary. We will point out explicitly any places where this gives rise to mathematical issues.

Clearly Tor_{∗}^{R}(M, N) and Ext_{R}^{∗}(M, N) are R_{∗}-modules when R is commutative. Various properties reminiscent of those of the classical Tor and Ext functors follow directly from the definition and the results of the previous chapters. The intuition is that the definition gives an analogue of the differential Tor and Ext functors (alias hyperhomology and cohomology functors) in the context of differential graded modules over differential graded algebras. In particular, the grading should not be thought of as the resolution grading of the classical torsion product, but rather as a total grading that sums a resolution degree and an internal degree; this idea will be made precise by the grading of the spectral sequences that we shall construct for the calculation of these functors.

Proposition 1.2. Tor_{∗}^{R}(M, N) satisfies the following properties.

(i) If R, M, and N are connective, then Tor_{n}^{R}(M, N) = 0 for n < 0.

(ii) A cofiber sequence N′ → N → N′′ gives rise to a long exact sequence

⋯ → Tor_{n}^{R}(M, N′) → Tor_{n}^{R}(M, N) → Tor_{n}^{R}(M, N′′) → Tor_{n−1}^{R}(M, N′) → · · · .

(iii) Tor_{∗}^{R}(M, R) ≅ π_{∗}(M) and, for a spectrum X,

Tor_{∗}^{R}(M, 𝔽X) ≅ π_{∗}(M ∧ X).

(iv) The functor Tor_{∗}^{R}(M,?) carries wedges to direct sums.

Proof. In (i), M and N can be approximated by CW R-modules with cells of non-negative dimension, hence it suffices to check the conclusion for N = S_{R}^{r}, r ≥ 0, in which case it is immediate from (iii). Part (iii) follows from III.1.4 and III.3.7.

The commutativity and associativity relations for the smash product imply various further properties. We content ourselves with the following specialization.

Proposition 1.3. If R is commutative, then

Tor_{∗}^{R}(M, N) ≅ Tor_{∗}^{R}(N, M)

and

Tor_{∗}^{R}(M ∧_{R} N, P) ≅ Tor_{∗}^{R}(M, N ∧_{R} P).

Say that a spectrum N is coconnective if π_{q}N = 0 for q > 0.

Proposition 1.4. Ext_{R}^{∗}(M, N) satisfies the following properties.

(i) If R and M are connective and N is coconnective, then Ext_{R}^{n}(M, N) = 0 for n < 0.

(ii) Fiber sequences N′ → N → N′′ and cofiber sequences M′ → M → M′′ give rise to long exact sequences

⋯ → Ext_{R}^{n}(M, N′) → Ext_{R}^{n}(M, N) → Ext_{R}^{n}(M, N′′) → Ext_{R}^{n+1}(M, N′) → ⋯

and

⋯ → Ext_{R}^{n}(M′′, N) → Ext_{R}^{n}(M, N) → Ext_{R}^{n}(M′, N) → Ext_{R}^{n+1}(M′′, N) → · · · .

(iii) Ext_{R}^{∗}(R, N) ≅ π_{−∗}(N) and, for a spectrum X,

Ext_{R}^{∗}(𝔽X, N) ≅ π_{−∗}(F (X, N))

and

Ext_{R}^{∗}(M, 𝔽^{#}X) ≅ π_{−∗}(F (M, X)).

(iv) The functor Ext_{R}^{∗}(?, N) carries wedges to products and the functor Ext_{R}^{∗}(M,?) carries products to products.

Proof. It suffices to check (i) for M = S_{R}^{r}, r ≥ 0, in which case the conclusion is immediate from (iii). Part (iii) follows from III.1.4, III.1.10, and III.6.7.

Passing to homotopy groups from the pairings of III.6.11 and III.7.5, we obtain the following further property.

Proposition 1.5. There is a natural, associative, and unital system of pairings

π : Ext_{R}^{∗}(M, N) ⊗_{π^{∗}(S)} Ext_{R}^{∗}(L, M) ⟶ Ext_{R}^{∗}(L, N).

If R is commutative, then these are pairings of R_{∗}-modules, and the tensor product may be taken over R_{∗}.

Proof. The first statement is clear. The second uses the the fact that π_{q}(M) = 𝒟_{R}(S_{R}^{q}, M) and the associative and unital system of equivalences of R-modules

S_{R}^{q} ∧_{R} S_{R}^{r} ≅ S_{R}^{q+r}

given by III.3.7.

The formal duality theory of III§6 implies the following result, together with various other such isomorphisms.

Proposition 1.6. Let R be commutative. For a finite cell R-module M and any R-module N,

Tor_{n}^{R}(D_{R}M, N) ≅ Ext_{R}^{−n}(M, N).

We think of the derived category 𝒟_{R} as a stable homotopy category. Changing notations, we may reinterpret the functors Tor and Ext as prescribing homology and cohomology theories in this category.

Definition 1.7. Let E′ be a right and E a left R-module. For left R-modules M and N, define

E′_{n}^{R}(M) = π_{n}(E′ ∧_{R} M) and E_{R}^{n}(M) = π_{−n}(F_{R}(M, E)).

The properties of Tor and Ext translate directly to statements about homology and cohomology. All of the standard homotopical machinery is available to us, and the previous result now takes the form of Spanier-Whitehead duality.

Corollary 1.8. Let R be commutative. For a finite cell R-module M and any R-module E,

E_{n}^{R}(D_{R}M) ≅ E_{R}^{−n}(M).

Since the equivalence between the classical stable homotopy category and the derived category of S-modules preserves smash products and function spectra, we obtain versions of all of the usual homology and cohomology theories on spectra by taking R = S. Moreover the following reinterpretation of Propositions 1.2(iii) and 1.4(iii) shows that the specializations to R-modules of all of the usual homology and cohomology theories on spectra are given by instances of our new homology and cohomology theories on R-modules.

Corollary 1.9. For a spectrum E and a (left) R-module M,

E_{∗}(M) ≅ (𝔽E)_{∗}^{R}(M) and E^{∗}(M) ≅ (𝔽^{#}E)_{R}^{∗}(M).

2. Eilenberg-Mac Lane spectra and derived categories

In this section, we change notation and let R denote a discrete ring. Applying multiplicative infinite loop space theory [49] to obtain an A_{∞} ring spectrum and then applying the functor S ∧_{ℒ} (?), we obtain an Eilenberg-Mac Lane spectrum HR = K(R, 0) that is an S-algebra and is a commutative S-algebra if R is commutative. An elaboration of multiplicative infinite loop space theory, followed by application of the functor S ∧_{ℒ} (?), can be used to realize passage to Eilenberg-Mac Lane spectra as a point-set level functor H from R-modules in the sense of algebra to HR-modules. We shall shortly use the present theory to give two different homotopical constructions of such Eilenberg-Mac Lane HR-modules. Granting this for the moment, we have the following result.

Theorem 2.1. For a ring R and R-modules M and N,

Tor_{∗}^{R}(M, N) ≅ Tor_{∗}^{HR}(HM, HN)

and

Ext_{R}^{∗}(M, N) ≅ Ext_{HR}^{∗}(HM, HN).

If R is commutative, then these are isomorphisms of R-modules. Under the second isomorphism, the topologically defined pairing

Ext_{HR}^{∗}(HM, HN) ⊗ Ext_{HR}^{∗}(HL, HM) ⟶ Ext_{HR}^{∗}(HL, HN)

coincides with the algebraic Yoneda product.

Proof. If 0 → N′ → N → N′′ → 0 is a short exact sequence of R-modules, then HN′ → HN → HN′′ is equivalent to a cofiber sequence. The conclusion is now immediate from Propositions 1.2 and 1.4, together with the axioms for algebraic Tor and Ext functors. It should be noted that right exactness and proper behavior on free modules together imply algebraically that

Tor_{0}^{R}(M, N) ≅ M ⊗_{R} N and Ext_{R}^{0}(M, N) ≅ Hom_{R}(M, N).

It is important to remember that the axioms for Ext require verifications about free or injective modules, but not both. The last statement follows from Yoneda’s axiomatization [70], which only requires proper behavior in degree zero and proper behavior relating connecting homomorphisms to products. The last follows topologically from commutation with cofiber sequences, which is easily derived from the adjoint construction of our pairings in III§5.

We can elaborate this result to an equivalence of derived categories. We shall restrict attention to morphisms of degree zero since the extension to graded morphisms is formal. Recall from [66] or [34, Ch.III] that the derived category 𝒟_{R} is obtained from the homotopy category of chain complexes over R by localizing at the quasi-isomorphisms, exactly as we obtained the category 𝒟_{HR} from the homotopy category of HR-modules by localizing at the weak equivalences. The algebraic theory of cell and CW chain complexes over R in [34, Ch.III] makes the analogy especially close. The proof of the equivalence is quite simple. The category 𝒟_{HR} is equivalent to the homotopy category of CW HR-modules and cellular maps. We will see that CW HR-modules have associated chain complexes. This gives a functor 𝒟_{HR} ⟶ 𝒟_{R}, and we will obtain an inverse functor directly from Brown’s representability theorem.

Definition 2.2. Let M be a CW HR-module. Define the associated chain complex C_{∗}(M) of R-modules by letting C_{n}(M) = π_{n}(M^{n}, M^{n−1}) and letting the differential d_{n} : C_{n}(M) ⟶ C_{n−1}(M) be the connecting homorphism of the triple (M^{n}, M^{n−1}, M^{n−2}). Observe that a cellular map of HR-modules induces a map of chain complexes and that a cellular homotopy induces a chain homotopy. Observe too that, since M^{n}/M^{n−1} is a wedge of free modules S_{HR}^{n} ≃ HR ∧ S^{n}, C_{n}(M) is a free R-module.

Lemma 2.3. For CW HR-modules M, the homology groups H_{∗}(C_{∗}(M)) are naturally isomorphic to the homotopy groups of M.

Proof. Since HR is connective, the inclusion M^{n} ⟶ M induces a bijection on π_{q} for q < n and a surjection on π_{n}. By induction up the sequential filtration of M^{n−1}, π_{q}(M^{n−1}) = 0 for q ≥ n. Therefore the quotient map M ⟶ M/M^{n−1} induces a monomorphism on π_{n}. The conclusion follows by a simple diagram chase.

Theorem 2.4. The cellular chain functor C_{∗} on HR-modules induces an equivalence of categories 𝒟_{HR} ⟶ 𝒟_{R}. The inverse equivalence Φ satisfies

H_{∗}(X) ≅ π_{∗}(Φ(X)).

Proof. The functor C_{∗} carries wedges to direct sums and carries homotopy colimits of cellular diagrams to chain level homotopy colimits. For a fixed chain complex X, the functor k on 𝒟_{HR} specified by k(M) = 𝒟_{R}(C_{∗}(M), X) therefore satisfies the wedge and Mayer-Vietoris axioms. By Brown’s representability theorem, III.2.12, k is represented by an HR-module spectrum Φ(X). By the functoriality of the representation, this gives a functor Φ : 𝒟_{R} ⟶ 𝒟_{HR} and an adjunction

𝒟_{R}(C_{∗}(M), X) ≅ 𝒟_{HR}(M, Φ(X)).

Since H_{n}(X) ≅ 𝒟_{R}(Σ^{n}R, X), where Σ^{n}R is the free R-module on one generator of degree n and C_{∗}(S_{HR}^{n}) = Σ^{n}R, this implies that H_{∗}(X) ≅ π_{∗}(Φ(X)). We claim that the unit η : M ⟶ Φ(C_{∗}(M)) and counit ε : C_{∗}(Φ(X)) ⟶ X of the adjunction are natural isomorphisms. On hom sets, the functor C_{∗} coincides with

η_{∗} : 𝒟_{HR}(L, M) ⟶ 𝒟_{HR}(L, Φ(C_{∗}(M))) ≅ 𝒟_{R}(C_{∗}(L), C_{∗}(M)).

As L runs through the S_{HR}^{n}, η_{∗} runs through the isomorphisms

π_{n}(M) ⟶ H_{n}(C_{∗}(M))

of the previous lemma. Therefore η is an isomorphism in 𝒟_{HR} for all M. Since the composite

ΦX⟶^{η}ΦC_{∗}(ΦX)⟶^{Φε}ΦX

is the identity, it follows that Φε is an isomorphism in 𝒟_{HR} for all X. The following natural diagram commutes:

[figure: 𝒟_{HR}(L, ΦC_{∗}(ΦX)) ≅ 𝒟_{R}(C_{∗}(L), C_{∗}(ΦX)); (Φε)_{∗}; ε_{∗}; 𝒟_{HR}(L, ΦX) ≅ 𝒟_{R}(C_{∗}(L), X).]

As L runs through the sphere modules S_{HR}^{n}, the resulting isomorphisms ε_{∗} show that ε induces an isomorphism on all homology groups and is therefore an isomorphism in 𝒟_{R}.

In the commutative case, we have the following important addendum to the theorem. See [34, III] for a discussion of tensor product and Hom functors in the derived category 𝒟_{R}. As in topology, they are constructed by first applying CW approximation of R-modules and then taking the point-set level functor.

Proposition 2.5. Assume that R is commutative. If M and N are CW HRmodules, then M ∧_{HR} N is a CW HR-module such that

C_{∗}(M ∧_{HR} N) ≅ C_{∗}(M) ⊗_{R} C_{∗}(N).

Therefore such an isomorphism holds in the derived category 𝒟_{R} for general HRmodules M and N. Moreover, in 𝒟_{R},

C_{∗}(F_{HR}(M, N)) ≅ Hom_{R}(C_{∗}(M), C_{∗}(N)).

If X and Y are chain complexes, then

Φ(X ⊗_{R} Y) ≅ ΦX ∧_{HR} ΦY

and

Φ Hom_{R}(X, Y) ≅ F_{HR}(ΦX, ΦY)

in 𝒟_{HR}.

Proof. The first statement is implied by III.7.3, and the last three derived category level isomorphisms are all formal consequences of the first.

Regarding R-modules as chain complexes concentrated in degree zero, we see that the functor Φ restricts to a functor H that assigns an Eilenberg-Mac Lane HRmodule spectrum HM to an R-module M. We give a more explicit construction.

Construction 2.6. (i) For an R-module M, we construct HM = K(M, 0) as a CW module L with sequential filtration {L_{n}} and skeletal filtration {L^{n}} related by L^{n−1} = L_{n}. Choose a free resolution

⋯ ⟶ F_{n}⟶^{d_{n}}F_{n−1} ⟶ ⋯ ⟶ F_{0}⟶^{ε}M ⟶ 0

of M. Let K_{0} be a wedge of 0-spheres, with one sphere for each basis element of F_{0}. For n ≥ 1, let K_{n} be a wedge of (n − 1)-spheres, with one sphere for each basis element of F_{n}. Define L_{1} = 𝔽K_{0}. For n ≥ 2, L_{n} will have two non-vanishing homotopy groups, namely π_{0}(L_{n}) = M and π_{n−1}(L_{n}) = Im d_{n}, and the inclusion i_{n} : L_{n} ⟶ L_{n+1} will induce an isomorphism on π_{0}. By freeness, we can realize d_{1} by a map of HR-modules 𝔽K_{1} ⟶ 𝔽K_{0}. Let L_{2} be its cofiber. Then the resulting map 𝔽K_{0} ⟶ L_{2} realizes ε on π_{0} and the resulting map L_{2} ⟶ Σ𝔽K_{1} realizes the inclusion Im d_{2} ⊂ F_{1} on π_{1}. Inductively, given L_{n}, we can realize d_{n} : F_{n} ⟶ Im d_{n} on the (n − 1)st homotopy group by a map of HR-modules 𝔽K_{n} ⟶ L_{n}. We let L_{n+1} be its cofiber. The claimed properties follow immediately. The union L = ∪L_{n} is the desired CW HR-module HM. (ii) Given a map f : M ⟶ M′ of R-modules, we construct a cellular map Hf : HM ⟶ HM′ of CW HR-modules that realizes f on π_{0}. Construct L′ = HM′ as above, writing F′_{n}, etc. As usual, we can construct a sequence of R-maps f_{n} : F_{n} ⟶ F′_{n} that give a map of resolutions. We can realize f_{n} on homotopy groups by an HR-map 𝔽K_{n} ⟶ 𝔽K′_{n}. Starting with L_{1} = 𝔽K_{0} and proceeding inductively, we can use a standard cofibration sequence argument, carried out in the category of HR-modules, to construct HR-maps L_{n} ⟶ L′_{n} such that the middle squares commute and the left and right squares commute up to homotopy in the following diagrams of HR-modules:

\xymatrix{𝔽K_{n+1} \ar[r] \ar[d] & L_{n} \ar[r] \ar[d] & L_{n+1} \ar[r] \ar[d] & Σ𝔽K_{n+1} \ar[d] \\ 𝔽K′_{n+1} \ar[r] & L′_{n} \ar[r] & L′_{n+1} \ar[r] & Σ𝔽K′_{n+1}.}

On passage to unions, we obtain the desired cellular map Hf : HM ⟶ HM′. A similar argument works to show that if we choose another map g_{∗} : F_{∗} ⟶ F′_{∗} of resolutions over f and repeat the construction, then the resulting HR-maps are homotopic.

Remark 2.7. Since they are Hℤ-module spectra, the underlying spectra of the HR-module spectra studied in this section all have the homotopy types of Eilenberg-Mac Lane spectra.

3. The Atiyah-Hirzebruch spectral sequence

We assume given a connective S-algebra R in this section, and we let k = π_{0}(R). Since R is connective, its derived category 𝒟_{R} is equivalent to the homotopy category h𝒞 𝒲_{R} of CW R-modules and cellular maps. We shall see that the Eilenberg-Mac Lane spectrum Hk is an R-module that plays a role in the study of R-modules analogous to the role played by Hℤ in the category of spectra. We use this insight to construct Atiyah-Hirzebruch spectral sequences and prove a Hurewicz theorem in the category of R-modules. Although we shall not assume this, the theory is most useful when R is commutative; of course, k may well be commutative even when R is not. Remember that modules mean left modules unless otherwise specified.

Proposition 3.1. There is a map of S-algebras π : R ⟶ Hk that realizes the identity homomorphism on π_{0}(R) = k.

Proof. We sketch two proofs. The first is an application of multiplicative infinite loop space theory. By [47, VII.2.4], the zeroth space R_{0} of R is an A_{∞} ring space. Modulo some point-set care to ensure continuity (e.g, we could replace R by a weakly equivalent “q-cofibrant” S-algebra, which is of the homotopy type of a CW spectrum by VII.6.5 and VII.6.6), we obtain a discretization map δ : R_{0} ⟶ k, and it is immediate from the definitions that it is an A_{∞} ring map. By [47, VII§4], there is a functor E from A_{∞} ring spaces to A_{∞} ring spectra, hence there results a map of A_{∞} ring spectra ER_{0} ⟶ Ek. By [47, VII.3.2 and 4.3] and the connectivity of R, there is a natural weak equivalence of A_{∞} ring spectra between ER_{0} and R, and the homotopical properties of E immediately imply that Ek is an Hk. Now apply the functor S ∧_{ℒ} (?) to replace A_{∞} ring spectra by S-algebras, and replace R by the weakly equivalent S-algebra S ∧_{ℒ} ER_{0}.

The second proof makes more serious use of the Quillen model category structure on the category of S-algebras that we construct in VII§§4,5. Using it, we can mimic the classical space level argument of killing higher homotopy groups, successively attaching cell S-algebras to kill the higher homotopy groups of R.

It follows that Hk is an (R, R)-bimodule. If R and therefore also Hk are commutative S-algebras, then Hk is a commutative R-algebra in the sense of VII§1 below. If j is a k-module, then the Hk-module Hj is an R-module by pullback along π. We consider the homology and cohomology theories represented by the Hj as ordinary homology and cohomology theories defined on the derived category of R-modules: they clearly satisfy the analogs of the Eilenberg–Steenrod axioms for ordinary homology and cohomology theories. We agree to alter the notations of Definition 1.7 by setting

H_{∗}^{R}(M; j′) = (Hj′)_{∗}^{R}(M) and H_{R}^{∗}(M; j) = (Hj)_{R}^{∗}(M) (3.2)

for a left R-module M, a right k-module j′ and a left k-module j. We have symmetric definitions with left and right reversed.

These theories can be calculated as the homology and cohomology of the cellular chain complexes of CW R-modules M. In fact, the definition of the associated chain complex of a CW R-module M is formally identical to Definition 2.2.

Definition 3.3. Let M be a CW R-module. Define the associated chain complex C_{∗}^{R}(M) of k-modules by letting C_{n}^{R}(M) = π_{n}(M^{n}, M^{n−1}) and letting the differential d_{n} : C_{n}^{R}(M) ⟶ C_{n−1}^{R}(M) be the connecting homorphism of the triple (M^{n}, M^{n−1}, M^{n−2}). Observe that a cellular map of R-modules induces a map of chain complexes and that a cellular homotopy induces a chain homotopy. Observe too that, since M^{n}/M^{n−1} is a wedge of free modules S_{R}^{n} ≃ R ∧ S^{n}, C_{n}^{R}(M) is a free k-module. For right and left k-modules j′ and j, define chain and cochain complexes of abelian groups

C_{∗}^{R}(M; j′) = j′ ⊗_{k} C_{∗}^{R}(M) and C_{R}^{∗}(M; j) = Hom_{k}(C_{∗}^{R}(M), j).

Clearly these chain and cochain functors induce covariant and contravariant functors from the derived category 𝒟_{R} to the derived category 𝒟_{ℤ} of chain complexes over ℤ, interpreted as homologically or cohomologically graded, respectively. When k is commutative, these functors take values in 𝒟_{k}. We have the following analogue of Proposition 2.5.

Proposition 3.4. If R is a commutative S-algebra and M and N are CW Rmodules, then M ∧_{R} N is a CW R-module such that

C_{∗}^{R}(M ∧_{R} N) ≅ C_{∗}^{R}(M) ⊗_{k} C_{∗}^{R}(N).

Therefore such an isomorphism holds in the derived category 𝒟_{k} for general Rmodules M and N. Moreover, in 𝒟_{k}, there is a natural map

ε̃: C_{∗}^{R}(F_{R}(M, N)) ⟶ Hom_{k}(C_{∗}^{R}(M), C_{∗}^{R}(N)),

and ε̃ is an isomorphism if M is a finite CW R-module.

Proof. The first statement is implied by III.7.3. For the second, the evaluation map F_{R}(M, N) ∧_{R} M ⟶ N induces a map

C_{∗}^{R}(F_{R}(M, N)) ⊗_{k} C_{∗}^{R}(M) ≅ C_{∗}^{R}(F_{R}(M, N) ∧_{R} M) ⟶ C_{∗}(N)

in 𝒟_{k}, and its adjoint is the required map ε̃. Clearly ε̃ is an isomorphism when M is a sphere R-module. It is therefore an isomorphism for all finite CW R-modules since the functors F_{R} and Hom_{k} both convert cofibration sequences in the first variable to fibration sequences.

We cannot expect the derived chain complex functor to preserve function objects in general, as the case R = S makes clear.

By checking axioms, as in the classical case, we reach the following conclusion.

Theorem 3.5. For R-modules M and right and left k-modules j′ and j, there are natural isomorphisms

H_{∗}^{R}(M; j′) = H(C_{∗}^{R}(M; j′)) and H_{R}^{∗}(M; j) = H(C_{R}^{∗}(M; j)).

The map π ∧ id : M ≅ R ∧_{R} M ⟶ Hk ∧_{R} M induces the Hurewicz homomorphism h : π_{∗}(M) ⟶ H_{∗}^{R}(M; k), and the proof of the Hurewicz theorem is exactly the same induction over skeleta as in the classical case.

Theorem 3.6 (Hurewicz). Let M be an (n − 1)-connected R-module. Then H_{i}^{R}(M; k) = 0 for i < n and h : π_{n}(M) ⟶ H_{n}^{R}(M; k) is an isomorphism.

Applying a generalized homology or cohomology theory to the skeletal filtration of a CW R-module M, we obtain an exact couple and thus a spectral sequence that generalizes the chain and cochain description of the ordinary represented homology and cohomology of M.

Theorem 3.7 (Atiyah-Hirzebruch Spectral Sequence). For a homology theory E_{∗}^{R} and a cohomology theory E_{R}^{∗} on R-modules, there are natural spectral sequences of the form

E_{p,q}^{2} = H_{p}^{R}(M; E_{q}^{R}) ⟹ E_{p+q}^{R}(M)

and

E_{2}^{p,q} = H_{R}^{p}(M; E_{R}^{q}) ⟹ E_{R}^{p+q}(M).

Convergence is as in the classical case, and we refer the reader to Boardman [7, §14] (see also [24, App B]) for discussion. If M is bounded below, then the homology spectral sequence converges strongly to E_{∗}^{R}(M) and the cohomology spectral sequence converges conditionally to E_{R}^{∗}(M). If, further, for each fixed (p, q) there are only finitely many r such that d^{r} is non-zero on E_{p,q}^{r}, then the cohomology spectral sequence converges strongly.

The multiplicative properties of the spectral sequences are as one would expect from Proposition 3.4.

4. Universal coefficient and Künneth spectral sequences

There are spectral sequences for the calculation of our Tor and Ext groups that are analogous to the Eilenberg-Moore (or hyperhomology) spectral sequences in differential homological algebra. Compare [18, 28, 34]. They specialize to give universal coefficient and Künneth spectral sequences in the homology and cohomology theory of spectra. We state our results in this section and give the construction in the next. Fix an S-algebra R.

Theorem 4.1. For right R-modules M and left R-modules N, there is a natural spectral sequence of the form

E_{p,q}^{2} = Tor_{p,q}^{R_{∗}}(M_{∗}, N_{∗}) ⟹ Tor_{p+q}^{R}(M, N). (4.2)

For left R-modules M and N, there is a natural spectral sequence of the form

E_{2}^{p,q} = Ext_{R^{∗}}^{p,q}(M^{∗}, N^{∗}) ⟹ Ext_{R}^{p+q}(M, N). (4.3)

If R is commutative, then these are spectral sequences of differential R_{∗}-modules.

The Tor spectral sequence is of standard homological type, with

d_{p,q}^{r} : E_{p,q}^{r} ⟶ E_{p−r,q+r−1}^{r}.

It lies in the right half-plane and converges strongly. The Ext spectral sequence is of standard cohomological type, with

d_{r} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1}.

It lies in the right half plane and converges conditionally. We have the following addendum.

Proposition 4.4. The pairing F_{R}(M, N) ∧_{S} F_{R}(L, M) → F_{R}(L, N) induces a pairing of spectral sequences that coincides with the algebraic Yoneda pairing

Ext_{R^{∗}}^{∗,∗}(M^{∗}, N^{∗}) ⊗_{S^{∗}} Ext_{R^{∗}}^{∗,∗}(L^{∗}, M^{∗}) ⟶ Ext_{R^{∗}}^{∗,∗}(L^{∗}, N^{∗})

on the E_{2}-level and that converges to the induced pairing of Ext groups.

The rest of the results of this section are corollaries of the results already stated. With the specializations of variables that we cite, the conclusions are immediate from the properties of our free and cofree functors and properties of Tor and Ext recorded in Section 1. Recalling Definition 1.7, we see that our spectral sequences can be viewed as universal coefficient spectral sequences for the computation of homology and cohomology theories on R-modules. Via Corollary 1.9, they specialize to give universal coefficient spectral sequences for the computation of homology and cohomology theories on spectra. Thus, setting M = 𝔽X in the two spectral sequences of Theorem 4.1, we obtain the following result. We have written the stars to indicate the way the grading is usually thought of in cohomology.

Theorem 4.5 (Universal coefficient). For an R-module N and any spectrum X, there are spectral sequences of the form

Tor_{∗,∗}^{R_{∗}}(R_{∗}(X), N_{∗}) ⟹ N_{∗}(X)

and

Ext_{R^{∗}}^{∗,∗}(R_{−∗}(X), N^{∗}) ⟹ N^{∗}(X).

Of course, replacing R and N by Eilenberg-Mac Lane spectra HR and HN for a ring R and R-module N, we obtain the classical universal coefficient theorem. Here we are thinking of the module N as defining theories acting on general spectra. By instead taking N = 𝔽E and N = 𝔽^{#}E in the two spectral sequences of Theorem 4.1, we obtain spectral sequences that are suitable for calculating the E-homology and cohomology of M.

Theorem 4.6. For an R-module M and any spectrum E, there are spectral sequences of the form

Tor_{∗,∗}^{R_{∗}}(M_{∗}, E_{∗}(R)) ⟹ E_{∗}(M)

and

Ext_{R^{∗}}^{∗,∗}(M^{∗}, E^{∗}(R)) ⟹ E^{∗}(M).

When E is also an R-module, we can take M = E and so obtain spectral sequences that converge to the E-Steenrod algebra E^{∗}(E) and its dual E_{∗}(E). For example, when R = S and M = E = Hℤ_{p}, the cohomology spectral sequence is a backwards Adams spectral sequence that converges from Ext_{S^{∗}}^{∗,∗}(ℤ_{p}, ℤ_{p}) to the mod p Steenrod algebra A. Such a spectral sequence was first studied in [39].

Replacing N by 𝔽Y and by F_{R}(𝔽Y, R) in the two universal coefficient spectral sequences, we arrive at Künneth spectral sequences.

Theorem 4.7 (Künneth). For any spectra X and Y, there are spectral sequences of the form

Tor_{∗,∗}^{R_{∗}}(R_{∗}(X), R_{∗}(Y)) ⟹ R_{∗}(X ∧ Y)

and

Ext_{R^{∗}}^{∗,∗}(R_{−∗}(X), R^{∗}(Y)) ⟹ R^{∗}(X ∧ Y).

A reference to Adams [1] is mandatory. He was the first to observe that one can derive Künneth spectral sequences from universal coefficient spectral sequences, and he observed that, by duality, the four spectral sequences of Theorems 4.5 and 4.7 imply two more universal coefficient and two more Künneth spectral sequences. He derived spectral sequences of this sort under the hypothesis that his given ring spectrum E is the colimit of finite subspectra E_{α} such that H^{∗}(E_{α}; E^{∗}) is E^{∗}-projective and the Atiyah-Hirzebruch spectral sequence converging from H^{∗}(E_{α}; E^{∗}) to E^{∗}(E_{α}) satisfies E_{2} = E_{∞}. Of course, this is an ad hoc calculational hypothesis that requires case-by-case verification. It covers some cases that are not covered by our results, and conversely.

5. The construction of the spectral sequences

The construction is similar to the construction of Eilenberg-Mac Lane spectra at the end of Section 2. For a right R-module M, we choose an R_{∗}-free resolution

⋯ ⟶ F_{p}⟶^{d_{p}}F_{p−1} ⟶ ⋯ ⟶ F_{0}⟶^{ε}M_{∗} ⟶ 0. (5.1)

Let Q_{0} = ker ε and Q_{p} = ker d_{p} for p ≥ 1, so that d_{p} defines an epimorphism F_{p} → Q_{p−1}. For p ≥ 0, let K_{p} be the wedge of one (p + s)-sphere for each basis element of F_{p} of degree s and let M_{0} = M. Proceeding inductively, we can use freeness to construct cofiber sequences of right R-modules

𝔽K_{p}⟶^{k_{p}}M_{p}⟶^{i_{p}}M_{p+1}⟶^{j_{p+1}}Σ𝔽K_{p} (5.2)

for p ≥ 0 that satisfy the following properties:

(i) k_{0} realizes ε on π_{∗}.

(ii) π_{∗}(M_{p}) = Σ^{p}Q_{p−1} for p ≥ 1.

(iii) k_{p} realizes Σ^{p}d_{p} : Σ^{p}F_{p} ⟶ Σ^{p}Q_{p−1} on π_{∗} for p ≥ 1.

(iv) i_{p} induces the zero homomorphism on π_{∗} for p ≥ 0.

(v) j_{p+1} realizes the inclusion Σ^{p+1}Q_{p} ⟶ Σ^{p+1}F_{p} on π_{∗} for p ≥ 0.

Observe that (iii) implies the case p + 1 of (ii) together with (iv) and (v).

To obtain the spectral sequence for Tor, we define

D_{p,q}^{1} = π_{p+q+1}(M_{p+1} ∧_{R} N) and E_{p,q}^{1} = π_{p+q}(𝔽K_{p} ∧_{R} N). (5.3)

The maps displayed in (5.2) give maps

i ≡ (i_{p})_{∗} : D_{p−1,q+1}^{1} ⟶ D_{p,q}^{1}
j ≡ (j_{p+1})_{∗} : D_{p,q}^{1} ⟶ E_{p,q}^{1}
k ≡ (k_{p})_{∗} : E_{p,q}^{1} ⟶ D_{p−1,q}^{1}.

These display an exact couple in standard homological form. We see from III.3.9 that E_{p,q}^{1} ≅ (F_{p} ⊗_{R_{∗}} N_{∗})_{q} and that d_{1} agrees under the isomorphism with d ⊗ 1. This proves that

E_{p,q}^{2} = Tor_{p,q}^{R_{∗}}(M_{∗}, N_{∗}).

Observe that k : E_{0,q}^{1} ⟶ D_{−1,q}^{1} can and must be interpreted as

π_{q}(𝔽K_{0} ∧_{R} N) ⟶ π_{q}(M ∧_{R} N).

On passage to E^{2}, it induces the edge homomorphism

E_{0,q}^{2} = M_{∗} ⊗_{R_{∗}} N_{∗} ⟶ π_{∗}(M ∧_{R} N). (5.4)

The convergence is standard, although it appears a bit differently than in most spectral sequences in current use. Write i_{0,p} for both the evident composite map M ⟶ M_{p} and its smash product with N. We filter π_{∗}(M ∧_{R} N) by letting F_{p}π_{∗}(M ∧_{R} N) be the kernel of

(i_{0,p+1})_{∗} : π_{∗}(M ∧_{R} N) ⟶ π_{∗}(M_{p+1} ∧_{R} N).

By (iv) above, we see that the telescope tel M_{p} is trivial. Since the functor (?)∧_{R} N commutes with telescopes, tel(M_{p} ∧_{R} N) is also trivial. This implies that the filtration is exhaustive. Consider the (p, q)th term of the associated bigraded group of the filtration. It is defined as usual by

E_{p,q}^{0}π_{∗}(M ∧_{R} N) = F_{p}π_{p+q}(M ∧_{R} N)/F_{p−1}π_{p+q}(M ∧_{R} N),

and the definition of the filtration immediately implies that this group is isomorphic to the image of

(i_{0,p})_{∗} : π_{p+q}(M ∧_{R} N) ⟶ π_{p+q}(M_{p} ∧_{R} N).

The target of (i_{0,p})_{∗} is D_{p−1,q}^{1}, and of course E_{p,q}^{1} = π_{p+q}(𝔽K_{p} ∧_{R} N) also maps into D_{p−1,q}^{1}, via k. It is a routine exercise in the definition of a spectral sequence to check that k induces an isomorphism

E_{p,q}^{∞} ⟶ Im(i_{0,p})_{∗}.

(We know of no published source, but this verification is given in [7, §6].)

To see the functoriality of the spectral sequence, suppose given a map f : M → M′ of R-modules and apply the constructions above to M′, writing F′_{p}, etc. Construct a sequence of maps of R_{∗}-modules f_{p} : F_{p} → F′_{p} that give a map of resolutions. We can realize the maps f_{p} on homotopy groups by R-module maps 𝔽K_{p} → 𝔽K′_{p}. Starting with f = f_{0} and proceeding inductively, a standard cofiber sequence argument allows us to construct a map M_{p+1} → M′_{p+1} such that the following diagram of R-modules commutes up to homotopy:

\xymatrix{F K_{p} \ar[r] \ar[d] & M_{p} \ar[r] \ar[d] & M_{p+1} \ar[r] \ar[d] & Σ𝔽K_{p} \ar[d] \\ F K′_{p} \ar[r] & M′_{p} \ar[r] & M′_{p+1} \ar[r] & Σ𝔽K′_{p}.}

There results a map of spectral sequences that realizes the induced map

Tor_{∗,∗}^{R_{∗}}(M_{∗}, N_{∗}) ⟶ Tor_{∗,∗}^{R_{∗}}(M′_{∗}, N_{∗})

on E^{2} and converges to (f ∧_{R} id)_{∗}. Functoriality in N is obvious.

To obtain the analogous spectral sequence for Ext, we switch from right to left modules in our resolution (5.1) of M_{∗} and its realization by R-modules. We define

D_{1}^{p,q} = π_{−p−q}(F_{R}(M_{p}, N)) and E_{1}^{p,q} = π_{−p−q}(F_{R}(𝔽K_{p}, N)). (5.5)

The maps displayed in (5.1) give maps

i ≡ (i_{p})^{∗} : D_{1}^{p+1,q−1} ⟶ D_{1}^{p,q}
j ≡ (k_{p})^{∗} : D_{1}^{p,q} ⟶ E_{1}^{p,q}
k ≡ (j_{p+1})^{∗} : E_{1}^{p,q} ⟶ D_{1}^{p+1,q}.

These display an exact couple in standard cohomological form. We see by III.6.9 that E_{1}^{p,q} ≅ Hom_{R^{∗}}^{q}(F_{p}, N^{∗}), where F_{p} is regraded cohomologically, and that d_{1} agrees with Hom(d, 1) under the isomorphism. This proves that

E_{2}^{p,q} = Ext_{R^{∗}}^{p,q}(M^{∗}, N^{∗}).

Observe that j : D_{1}^{0,q} → E_{1}^{0,q} can and must be interpreted as

π_{−q}(F_{R}(M, N)) ⟶ π_{−q}(F_{R}(𝔽K_{0}, N)).

On passage to E_{2}, it induces the edge homomorphism

π_{−q}(F_{R}(M, N)) ⟶ Hom_{R^{∗}}^{q}(M^{∗}, N^{∗}) = E_{2}^{0,q}. (5.6)

To see the convergence, let

ι^{0,p} : F_{R}(M_{p}, N) ⟶ F_{R}(M, N)

be the map induced by the evident iterate M → M_{p} and filter π_{∗}(F_{R}(M, N)) by letting F^{p}π_{∗}(F_{R}(M, N)) be the image of

(ι^{0,p})_{∗} : π_{∗}(F_{R}(M_{p}, N)) ⟶ π_{∗}(F_{R}(M, N)).

The (p, q)th term of the associated bigraded group of the filtration is

E_{0}^{p,q}π_{∗}(F_{R}(M, N)) = F^{p}π_{−p−q}(F_{R}(M, N))/F^{p+1}π_{−p−q}(F_{R}(M, N)).

The group E_{∞}^{p,q} is defined as the subquotient Z_{∞}^{p,q}/B_{∞}^{p,q} of E_{1}^{p,q}, where

B_{∞}^{p,q} = j(ker(ι^{0,p})_{∗}),

and a routine exercise in the definition of a spectral sequence shows that the additive relation (ι^{0,p})_{∗} ∘ j^{−1} induces an isomorphism

E_{∞}^{p,q} ≅ E_{0}^{p,q}π_{∗}(F_{R}(M, N)).

Since tel M_{p} is trivial, so is the homotopy limit, or “microscope”,

mic F_{R}(M_{p}, N) ≅ F_{R}(tel M_{p}, N).

By the lim^{1} exact sequence for the computation of π_{∗}(mic F_{R}(M_{p}, N)), we conclude that

lim π_{∗}(F_{R}(M_{p}, N)) = 0 and lim^{1}π_{∗}(F_{R}(M_{p}, N)) = 0.

This means that the spectral sequence {E_{r}^{p,q}} is conditionally convergent. The functoriality of the spectral sequence is clear from the argument for torsion products already given.

Finally, turning to the proof of Proposition 4.4, consider the pairing

F_{R}(M, N) ∧_{R} F_{R}(L, M) → F_{R}(L, N).

Construct a sequence {L_{p}} as in (5.2). Then the maps M → M_{p} induce a compatible system of pairings

\xymatrix{F_{R}(M_{p}, N) ∧_{R} F_{R}(L_{p′}, M) \ar[r] & F_{R}(M, N) ∧_{R} F_{R}(L_{p′}, M) \ar[r] & F_{R}(L_{p′}, N).}

These induce the required pairing of spectral sequences. The convergence is clear, and the behavior on E_{2} terms is correct by comparison with the axioms or by comparison with the usual construction of Yoneda products.

6. Eilenberg-Moore type spectral sequences

Let R be an S-algebra and let M be a right and N a left R-module. Let E be an associative ring spectrum in the sense of classical stable homotopy theory. By I.6.7 and II.1.9, we may assume without loss of generality that E is an associative Sring spectrum (in the sense to be defined formally in V§2). Under several different further hypotheses, we shall construct a spectral sequence of the form

Tor_{p,q}^{E_{∗}(R)}(E_{∗}(M), E_{∗}(N)) ⟹ E_{p+q}(M ∧_{R} N). (6.1)

The simplest version of this spectral sequence is the following one.

Theorem 6.2. A spectral sequence of the form (6.1) exists if E_{∗}(R) is a flat right R_{∗}-module.

Proof. By a standard comparison of homology theories argument, the flatness hypothesis implies that, for left R-modules N, the natural map

E_{∗}(R) ⊗_{R_{∗}} N_{∗} ⟶ π_{∗}((E ∧_{S} R) ∧_{R} N) ≅ π_{∗}(E ∧_{S} N) = E_{∗}(N)

is an isomorphism. It also ensures that the functor E_{∗}(R) ⊗_{R_{∗}} (?) carries the exact sequence (5.1) to an exact sequence of E_{∗}(R)-modules. We now apply the functor E_{∗}(?) = π_{∗}(E∧_{S}?), rather than the functor π_{∗}, to the sequence of cofibrations obtained from (5.2) by smashing over R with N and find that the rest of the proof of Theorem 4.1 carries over verbatim. In fact, if R is commutative, then the spectral sequence (6.1) results from the first spectral sequence of Theorem 4.1 by applying the exact functor E_{∗}(R) ⊗_{R_{∗}} (?).

This flatness hypothesis is generally unrealistic. By assuming that E is also an S-algebra and exploiting the S-algebra E ∧_{S} R, we can obtain a theorem like this without flatness hypotheses. We need a lemma.

Lemma 6.3. Let R be an S-algebra such that (R, S) has the homotopy type of a relative CW S-module and let M and N be right and left cell R-modules. Then M, N, and M ∧_{R} N have the homotopy types of cell S-modules.

Proof. Up to homotopy, S ⟶ R is a cofibration of S-modules and R/S is a CW S-module. Since 𝔽_{R}X = R ∧_{S} 𝔽_{S}X, it follows from the cofiber sequence

𝔽_{S}X ⟶ R ∧_{S} 𝔽_{S}X ⟶ R/S ∧_{S} 𝔽_{S}X

that 𝔽_{R}X has the homotopy type of a CW S-module if X has the homotopy type of a CW spectrum. Therefore sphere R-modules and, by III.3.7, their smash products are of the homotopy types of CW S-modules. The conclusion follows.

Theorem 6.4. Let E and R be S-algebras and assume that (R, S) is of the homotopy type of a relative CW S-module. Let M be a right and N a left Rmodule. Then there is a spectral sequence of the form

Tor_{p,q}^{E_{∗}(R)}(E_{∗}(M), E_{∗}(N)) ⟹ E_{p+q}(M ∧_{R} N).

Proof. Replace the triple (R; M, N) in Theorem 4.1 by the triple

(E ∧_{S} R; E ∧_{S} M, E ∧_{S} N).

The E^{2}-term of the resulting spectral sequence is

Tor_{∗,∗}^{(E∧_{S}R)_{∗}}((E ∧_{S} M)_{∗}, (E ∧_{S} N)_{∗}).

It converges to π_{∗}((E ∧_{S} M) ∧_{E∧_{S}R} (E ∧_{S} N)) and, by III.3.10, we have

(E ∧_{S} M) ∧_{E∧_{S}R} (E ∧_{S} N) ≅ E ∧_{S} (M ∧_{R} N).

7. THE BAR CONSTRUCTIONS B(M, R, N) AND B(X, G, Y) 99

Since we are working in derived categories, we may assume that M and N are cell R-modules. Then M, N, and M ∧_{R} N are of the homotopy types of CW S-modules, and I.6.7 and II.1.9 imply that their smash products over S with E are isomorphic in h̄𝒮 to the corresponding internal smash products. This is also true for R/S, and of course E ∧_{S} S ≅ E ≃ E ∧ S. We conclude that

(E ∧_{S} R)_{∗} ≅ E_{∗}(R), (E ∧_{S} M)_{∗} ≅ E_{∗}(M) and (E ∧_{S} N)_{∗} ≅ E_{∗}(N),

so that the E^{2} term of the spectral sequence is as stated, and

π_{∗}((E ∧_{S} M) ∧_{E∧_{S}R} (E ∧_{S} N)) ≅ E_{∗}(M ∧_{R} N),

so that the target of the spectral sequence is also as stated.

The hypothesis that (R, S) is of the homotopy type of a relative CW S-module results in no loss of generality since, as discussed in III§4, the model category theory of Chapter VII implies that, for any S-algebra R, there is a q-cofibrant S-algebra ΛR and a weak equivalence λ : ΛR ⟶ R. The map λ induces an equivalence of categories 𝒟_{R} ≈ 𝒟_{ΛR}, and (ΛR, S) is of the homotopy type of a relative CW S-module.

Remark 6.5. To deal with multiplicative structures, it is important to work with commutative S-algebras. As we shall see in Chapter VII, the category of commutative S-algebas also admits a model category structure. However, we do not believe that its q-cofibrant objects are of the homotopy types of relative CW S-modules. This is a significant technical difference between the theories of Salgebras and of commutative S-algebras. One way of getting around this difficulty is to approximate commutative S-algebras by q-cofibrant non-commutative S-algebras. We shall find a more satisfactory solution in VII§6, where we examine the homotopical properties of q-cofibrant commutative S-algebras. The results there show that the proofs of Theorem 6.4 and of Theorem 7.7 below work in the commutative context provided that we assume that our given commutative S-algebras are q-cofibrant.

7. The bar constructions B(M, R, N) and B(X, G, Y)

The spectral sequence (6.1) is reminiscent of the Rothenberg-Steenrod-Eilen-berg-Moore spectral sequence

Tor_{∗,∗}^{E_{∗}(G)}(E_{∗}(X), E_{∗}(Y)) ⟹ E_{∗}B(X, G, Y), (7.1)

where G is a topological monoid, X is a right G-space, Y is a left G-space, and B(X, G, Y) is the two-sided bar construction [50]. We here describe a spectrum level two-sided bar construction B(M, R, N) that explains the analogy. We will use the bar construction to derive a version of (6.1) for general commutative ring spectra E that applies under different, and more realistic, flatness hypotheses than those of Theorem 6.2, and we will show that the classical spectral sequence (7.1) is a special case.

Definition 7.2. For an S-algebra (R, ϕ, η), a right R-module (M, μ), and a left R-module (N, ν), define a simplicial S-module B_{∗}(M, R, N) by setting

B_{p}(M, R, N) = M ∧_{S} R^{p} ∧_{S} N,

where R^{p} is the p-fold ∧_{S}-power, interpreted as S if p = 0. The face and degeneracy operators on B_{p}(M, R, N) are

d_{i} = \begin{cases}μ ∧ (id_{R})^{p−1} ∧ id_{N} & if i = 0 \\ id_{M} ∧(id_{R})^{i−1} ∧ ϕ ∧ (id_{R})^{p−i−1} ∧ id_{N} & if 0 < i < p \\ id_{M} ∧(id_{R})^{p−1} ∧ ν & if i = p \end{cases}

and s_{i} = id_{M} ∧(id_{R})^{i} ∧ η ∧ (id_{R})^{p−i} ∧ id_{N} if 0 ≤ i ≤ p. If M is an (R′, R)-bimodule, then B_{∗}(M, R, N) is a simplicial R′-module.

We will discuss the geometric realization of simplicial spectra in X§§1-2, and we agree to write

B(M, R, N) = |B_{∗}(M, R, N)|.

By X.1.5, geometric realization carries simplicial R-modules to R-modules. By XII.1.2 and X.1.2, there is a natural map

ψ : B(R, R, N) ⟶ N (7.3)

of R-modules that is a homotopy equivalence of S-modules. More generally, by use of the product on R and its action on the given modules, we obtain a natural map of S-modules

ψ : B(M, R, N) ⟶ M ∧_{R} N. (7.4)

Proposition 7.5. For a cell R-module M and any R-module N,

ψ : B(M, R, N) ⟶ M ∧_{R} N

is a weak equivalence of S-modules.

Proof. If \underline{M} is the constant simplicial R-module at M, then, by X.1.3 and the isomorphism M ∧_{R} R ≅ M, we have

M ∧_{R} B(R, R, N) ≅ |\underline{M} ∧_{R} B_{∗}(R, R, N)| ≅ B(M, R, N).

Moreover, under this isomorphism, id ∧_{R}ψ agrees with ψ. Since ψ of (7.3) is a weak equivalence of R-modules, the conclusion follows from III.3.8.

For the bar construction to be useful calculationally, the simplicial spectrum B_{∗}(M, R, N) must be proper, in the sense of X.2.1 and X.2.2. By the following result, which is part of IX.2.7, we lose no generality by assuming this.

Proposition 7.6. If R is a q-cofibrant S-algebra, then B_{∗}(M, R, N) is a proper simplicial S-module.

By X.2.9, when B_{∗}(M, R, N) is proper, we can use the simplicial filtration of B(M, R, N) to construct a well-behaved spectral sequence that converges to E_{∗}B(M, R, N) for any spectrum E. When E is a commutative ring spectrum, we can use flatness hypotheses to identify the E_{2}-term. Recall that, in algebra, if A is an algebra over a commutative ring k, then there is a notion of a relatively flat A-module F, for which the functor (?) ⊗_{A} F is exact when applied to k-split exact sequences. The obvious examples are the relatively free A-modules A ⊗_{k} L for k-modules L. There is a concomitant relative torsion product Tor_{∗}^{(A,k)}(M, N), and similarly for graded algebras over commutative graded rings. When k is a field, these reduce to ordinary absolute torsion products.

Theorem 7.7. Let E be a commutative ring spectrum. Let R be an S-algebra such that (R, S) is of the homotopy type of a relative CW S-module. Let M be a right and N a left cell R-module such that B_{∗}(M, R, N) is proper. If E_{∗}(R) and either E_{∗}(M) or E_{∗}(N) is E_{∗}-flat, then the bar construction spectral sequence converging to

E_{∗}B(M, R, N) ≅ E_{∗}(M ∧_{R} N)

satisfies

E_{p,q}^{2} = Tor_{p,q}^{(E_{∗}(R),E_{∗})}(E_{∗}(M), E_{∗}(N))

Proof. Our hypotheses ensure that we can use smash products over S and internal smash products interchangeably when computing homology and homotopy groups. Our flatness hypotheses ensure that

E_{∗}(M ∧_{S} R^{p} ∧_{S} N) ≅ E_{∗}(M) ⊗_{E_{∗}} E_{∗}(R)^{⊗p} ⊗_{E_{∗}} E_{∗}(N),

where the p-fold tensor power is taken over E_{∗}. This determines the E-homology of the spectrum of p-simplices of B_{∗}(M, R, N). Since B_{∗}(M, R, N) is proper, it follows that the complex that computes E^{2} (see X.2.9) is the standard bar complex for the computation of the relative torsion product.

Intuitively, interchanging the roles of M and N in the proof of Proposition 7.5, we see that the filtration quotients

F_{p}B(M, R, R)/F_{p−1}B(M, R, R)

play a role similar to that played by the terms 𝔽K_{p} in the construction of the spectral sequence of Theorem 6.2.

As promised, we have the following result, which shows that the spectral sequence of (7.2) is a special case.

Theorem 7.8. Let G be a topological monoid, X a right G-space, and Y a left G-space. Then Σ^{∞}G_{+} is an S-algebra, Σ^{∞}X_{+} is a right Σ^{∞}G_{+}-module, and Σ^{∞}Y_{+} is a left Σ^{∞}G_{+}-module. Moreover, there is a natural isomorphism of S-modules

Σ^{∞}B(X, G, Y)_{+} ≅ B(Σ^{∞}X_{+}, Σ^{∞}G_{+}, Σ^{∞}Y_{+}),

and B_{∗}(Σ^{∞}X_{+}, Σ^{∞}G_{+}, Σ^{∞}Y_{+}) is proper if G is nondegenerately based.

Proof. The first statement is immediate from I.8.2 and II.1.2, together with the obvious identification

X_{+} ∧ Y_{+} ≅ (X × Y)_{+}

for spaces X and Y. The product on Σ^{∞}G_{+} is induced from the product on G,

Σ^{∞}G_{+} ∧_{S} Σ^{∞}G_{+} ≅ Σ^{∞}(G × G)_{+} ⟶ Σ^{∞}G_{+},

and similarly for the actions on Σ^{∞}X_{+} and Σ^{∞}Y_{+}. The second statement follows from the fact that the functors Σ^{∞} and geometric realization commute, by X.1.3, and that Σ^{∞} preserves properness; see X.2.1. We obtain an identification of simplicial spectra

Σ^{∞}B_{∗}(X, G, Y)_{+} ≅ B_{∗}(Σ^{∞}X_{+}, Σ^{∞}G_{+}, Σ^{∞}Y_{+})

by applying Σ^{∞} to the spaces

(X × G^{p} × Y)_{+} ≅ X_{+} ∧ (G_{+})^{p} ∧ Y,

where (G_{+})^{p} is the p-fold smash power. If G is non-degenerately based, then B_{∗}(X, G, Y)_{+} is a proper simplicial based space.

CHAPTER V R-ring spectra and the specialization to MU

In this chapter, we think of the derived category of R-modules as an analog of the stable homotopy category. From this point of view, we have the notion of an R-ring spectrum, which is just like the classical notion of a ring spectrum in the stable homotopy category. We shall study such homotopical structures in this chapter.

We first show how to construct quotients M/IM and localizations M[X^{−1}] of modules over a commutative S-algebra R. We then study when these constructions inherit a structure of R-ring spectrum from an R-ring spectrum structure on M.

When specialized to MU, our results give more highly structured versions of spectra that in the past have been constructed by means of the Baas-Sullivan theory of manifolds with singularities or the Landweber exact functor theorem. At least at odd primes, we obtain an entirely satisfactory, and surprisingly simple, treatment of MU-ring structures on the resulting MU-modules.

1. Quotients by ideals and localizations

Let R be a commutative S-algebra. We assume that all given R-modules M are of the homotopy types of cell R-modules, but we must keep in mind that R itself will not be of the homotopy type of a cell R-module. By III.1.4, we have a canonical weak equivalence of R-modules ζ : S_{R} ⟶ R, where the sphere R-module S_{R} = 𝔽_{R}S is the free R-module generated by S, and we implicitly replace R by S_{R} when performing constructions on R regarded as an R-module. Concomitantly, we must sometimes replace the unit isomorphism R ∧_{R} M ≅ M by its composite with the weak equivalence ζ ∧ id. This is consistent with the standard practice of replacing spectra by weakly equivalent CW spectra without change of notation.

104 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU

We shall work throughout in the derived category 𝒟_{R} of R-modules. To deduce S-module or spectrum level conclusions from our R-module level arguments, we must apply the forgetful functors 𝒟_{R} ⟶ 𝒟_{S} and 𝒟_{S} ⟶ h̄𝒮. The process is routine, but it does entail reapproximating cell R-modules by CW S-modules or CW spectra since, in general, cell R-modules need not be of the homotopy types of CW S-modules or of CW spectra.

We are interested in homotopy groups, and we make use of the isomorphisms

π_{n}(M) = h𝒮 (S^{n}, M) ≅ hℳ_{S}(S_{S}^{n}, M) ≅ hℳ_{R}(S_{R}^{n}, M) (1.1)

to represent elements as maps of R-modules, where, as usual, S_{R}^{n} = 𝔽_{R}S^{n}. We write M_{∗} = π_{∗}(M), and we do not distinguish notationally between a representative map of spectra S^{n} ⟶ M and a representative map of R-modules S_{R}^{n} ⟶ M.

By III.3.7 and standard properties of spectrum level spheres ([37, pp 386-389]), we have a system of equivalences of R-modules

S_{R}^{q} ∧_{R} S_{R}^{r} ≃ S_{R}^{q+r} (1.2)

that is associative, commutative, and unital up to a coherent system of homotopy equivalences and is compatible with suspension as q and r vary. For a pairing of R-modules L ∧_{R} M ⟶ N, we therefore obtain a pairing of homotopy groups

L_{∗} ⊗_{R_{∗}} M_{∗} ⟶ N_{∗}.

Of course, L ∧_{R} M is an R-module since R is commutative.

For x ∈ R_{n}, thought of as an R-map S_{R}^{n} ⟶ R, we have the R-map

S_{R}^{n} ∧_{R} M⟶^{x∧id}R ∧_{R} M ≅ M. (1.3)

This map of R-modules realizes multiplication by x on M_{∗}. We agree to write Σ^{n}M for S_{R}^{n} ∧_{R} M and to write x : Σ^{n}M ⟶ M for the map (1.3) throughout this chapter. By III.3.7, S_{R}^{n} ∧_{R} M is isomorphic as an R-module to S_{S}^{n} ∧_{S} M and, by I.6.7 and II.1.9, S_{S}^{n} ∧_{S} M is weakly equivalent as a spectrum to S^{n} ∧M. Therefore, on passage to h̄𝒮, the R-module Σ^{n}M is a model for the spectrum level suspension of M.

Definition 1.4. Define M/xM to be the cofiber of the map (1.3) and let ρ : M ⟶ M/xM be the canonical map. Inductively, for a finite sequence {x_{1}, …, x_{n}} of elements of R_{∗}, define

M/(x_{1}, …, x_{n})M = N/x_{n}N, where N = M/(x_{1}, …, x_{n−1})M.

For a (countably) infinite sequence X = {x_{i}}, define M/XM to be the telescope of the M/(x_{1}, …, x_{n})M, where the telescope is taken with respect to the successive canonical maps ρ.

We have the following analogue of the universal property of quotients by principal ideals in algebra.

Lemma 1.5. Let N be an R-module such that x : Σ^{n}N ⟶ N is zero and let α : M ⟶ N be a map of R-modules. Then there is a map of R-modules α̃ : M/xM ⟶ N such that α̃ ∘ ρ = α, and α̃ is unique if [Σ^{n+1}M, N]_{R} = 0.

Proof. This is obvious from the diagram

[figure: Σ^{n}M; x; α; M; α; ρ; M/xM; Σ^{n+1}M; Σ^{n}N; x; N,]

in which the row is the cofiber sequence that defines M/xM.

Clearly we have a long exact sequence

⋯ ⟶ π_{q−n}(M)⟶^{x}π_{q}(M)⟶^{ρ_{∗}}π_{q}(M/xM) ⟶ π_{q−n−1}(M) ⟶ · · · . (1.6)

If x is not a zero divisor for π_{∗}(M), then ρ_{∗} induces an isomorphism of R_{∗}-modules

π_{∗}(M)/xπ_{∗}(M) ≅ π_{∗}(M/xM). (1.7)

If {x_{1}, …, x_{n}} is a regular sequence for π_{∗}(M), in the sense that x_{i} is not a zero divisor for π_{∗}(M)/(x_{1}, …, x_{i−1})π_{∗}(M) for 1 ≤ i ≤ n, then

π_{∗}(M)/(x_{1}, …, x_{n})π_{∗}(M) ≅ π_{∗}(M/(x_{1}, …, x_{n})M), (1.8)

and similarly for a possibly infinite regular sequence X = {x_{i}}. We shall see in a moment that M/XM is independent of the ordering of the elements of the set X. If I denotes the ideal generated by a regular sequence X, then, by Corollary 2.10 below, M/XM is independent of the choice of regular sequence (under reasonable hypotheses) and it is reasonable to define

M/IM = M/XM. (1.9)

However, this notation must be used with caution since, if we fail to restrict attention to regular sequences X, the homotopy type of M/XM will depend on the set X and not just on the ideal it generates. For example, quite different modules are obtained if we repeat a generator x_{i} of I in our construction.

As in algebra, we can describe the construction on general R-modules M as the smash product of M with the construction on R regarded as an R-module. We write R/X or R/I instead of R/XR or R/IR.

Lemma 1.10. For a sequence X of elements of R_{∗}, there is a natural isomorphism in 𝒟_{R}

(R/X) ∧_{R} M ⟶ M/XM.

In particular, for a finite sequence X = {x_{1}, …, x_{n}},

R/(x_{1}, …, x_{n}) ≃ (R/x_{1}) ∧_{R} ⋯ ∧_{R} (R/x_{n}),

and R/X is therefore independent of the ordering of the elements of X.

Proof. Working on the point-set level, we have an isomorphism of cofiber sequences :

[figure: S_{R}^{n} ∧_{R} R ∧_{R} M; x∧id; id ∧λ; R ∧_{R} M; ρ∧id; λ; (R/xR) ∧_{R} M; S_{R}^{n} ∧_{R} M; x; M; ρ; M/xM.]

We only claim an isomorphism in 𝒟_{R} since, working homotopically, we should replace R by S_{R} and use the weak equivalence ζ∧id : S_{R}∧_{R}M ⟶ R∧_{R}M to obtain a composite weak equivalence (S_{R}/xS_{R}) ∧_{R} M ⟶ (R/xR) ∧_{R} M ⟶ M/xM. The rest follows by iteration and use of the commutativity of ∧_{R}.

We turn next to localizations of R-modules at subsets X = {x_{i}} of R_{∗}. We restrict attention to countable sets for notational convenience, but this restriction can easily be removed. Let {y_{i}} be any cofinal sequence of products of the x_{i}, such as that specified inductively by y_{1} = x_{1} and y_{i} = x_{1} ⋯ x_{i}y_{i−1}. If y_{i} ∈ R_{n_{i}}, we may represent y_{i} by an R-map S_{R}^{0} ⟶ S_{R}^{−n_{i}}, which we also denote by y_{i}. Let q_{0} = 0 and, inductively, q_{i} = q_{i−1} + n_{i}. The map of R-modules

S_{R}^{0} ∧_{R} M ⟶^{y_{i}∧id}S_{R}^{−n_{i}} ∧_{R} M

represents y_{i}. Smashing over R with S_{R}^{−q_{i−1}} and using equivalences (1.2), we obtain a sequence of maps of R-modules

S_{R}^{−q_{i−1}} ∧_{R} M ⟶ S_{R}^{−q_{i}} ∧_{R} M. (1.11)

Definition 1.12. Define the localization of M at X, denoted M[X^{−1}], to be the telescope of the sequence of maps (1.11). Since M ≃ S_{R}^{0} ∧_{R} M in 𝒟_{R}, we may regard the inclusion of the initial stage S_{R}^{0} ∧_{R} M of the telescope as a natural map λ : M ⟶ M[X^{−1}].

Again, we have an analogue of the standard universal property of localization in algebra.

Lemma 1.13. Let N be an R-module such that x_{i} : Σ^{k_{i}}N ⟶ N, deg x_{i} = k_{i}, is an equivalence for each i and let α : M ⟶ N be a map of R-modules. Then there is a unique map of R-modules α̃ : M[X^{−1}] ⟶ N such that α̃ ∘ λ = α.

2. LOCALIZATIONS AND QUOTIENTS OF R-RING SPECTRA 107

Proof. Passage to telescopes gives α̃ : M[X^{−1}] ⟶ N[X^{−1}] ≃ N. The lim^{1} term is zero in the exact sequence

0 ⟶ lim^{1} [Σ^{1−q_{i}}M, N]_{R} ⟶ [M[X^{−1}], N]_{R} ⟶ lim [Σ^{−q_{i}}M, N]_{R} ⟶ 0

since the maps of the inverse system are isomorphisms. Therefore α̃ is unique.

Since homotopy groups commute with localization, by III.1.7, we see immediately that λ induces an isomorphism of R_{∗}-modules

π_{∗}(M[X^{−1}]) ≅ π_{∗}(M)[X^{−1}]. (1.14)

Arguing as in Lemma 1.10, we see that the localization of M is the smash product of M with the localization of R.

Lemma 1.15. For a set X of elements of R_{∗}, there is a natural weak equivalence

R[X^{−1}] ∧_{R} M ⟶ M[X^{−1}].

Moreover, R[X^{−1}] is independent of the ordering of the elements of X. For sets X and Y, R[(X ∪ Y)^{−1}] is equivalent to the composite localization R[X^{−1}][Y^{−1}].

Proof. The independence of ordering is shown by use of the union of any two given cofinal sequences. The last statement is shown by use of the usual Fubini type theorem for iterated homotopy colimits.

2. Localizations and quotients of R-ring spectra

Again, fix a commutative S-algebra R. Since 𝒟_{R} is a symmetric monoidal category under ∧_{R} with unit R, we have the notion of a monoid or a commutative monoid in 𝒟_{R}. These are the analogs of associative or of associative and commutative ring spectra in classical stable homotopy theory. As there, we must allow weaker structures.

Definition 2.1. An R-ring spectrum A is an R-module A with unit η : R ⟶ A and product ϕ : A ∧_{R} A ⟶ A in 𝒟_{R} such that the following left and right unit diagram commutes in 𝒟_{R}:

[figure: R ∧_{R} A; η∧id; λ; A ∧_{R} A; ϕ; A ∧_{R} R; id ∧η; λτ; A.]

Of course, by neglect of structure, an R-ring spectrum A is a ring spectrum in the sense of classical stable homotopy theory; its unit is the composite of the unit of R and the unit of A and its product is the composite of the product of A and the canonical map

A ∧ A ≃ A ∧_{S} A ⟶ A ∧_{R} A.

Similarly, for an R-ring spectrum A, we have the evident homotopical notion of an A-module spectrum M. Here, in conformity with the definition just given, we only require that the action μ : A∧_{R} M ⟶ M satisfy the unit condition μ∘(η ∧id) = id in 𝒟_{R}. When A is associative, it is conventional to insist that M also satisfy the evident associativity condition. These structures play a role in the study of our new derived categories of R-modules that is analogous to the role played by ring spectra and their module spectra in classical stable homotopy theory. When R = S, I.6.7 and II.1.9 imply that S-ring spectra and their module spectra are equivalent to classical ring spectra and their module spectra.

Lemma 2.2. If A and B are R-ring spectra, then so is A ∧_{R} B. If A and B are associative or commutative, then so is A ∧_{R} B.

We ask the behavior of quotients and localizations with respect to R-ring structures. For localizations, the answer is immediate, and we shall give a sharper point-set level analogue in VIII§3.

Proposition 2.3. Let X be a set of elements of R_{∗}. If A is an R-ring spectrum, then A[X^{−1}] is an R-ring spectrum such that λ : A ⟶ A[X^{−1}] is a map of R-ring spectra. If A is associative or commutative, then so is A[X^{−1}].

Proof. Since A[X^{−1}] ≃ R[X^{−1}] ∧_{R} A, it suffices to prove that R[X^{−1}] is an associative and commutative R-ring spectrum with unit λ. Lemma 1.15 gives an equivalence

R[X^{−1}] ∧_{R} R[X^{−1}] ≃ R[X^{−1}][X^{−1}] ≃ R[X^{−1}]

under R, and this equivalence is the required product.

This doesn’t work for quotients since (R/X)/X is not equivalent to R/X. However, we can analyze the problem by analyzing the deviation, and, by Lemma 1.10, we may as well work one element at a time. We have a necessary condition for R/x to be an R-ring spectrum that will be familiar from classical stable homotopy theory. We generally write η and ϕ for the units and products of R-ring spectra; as stated before, we write Σ^{n} for the module theoretic suspension functor S_{R}^{n} ∧_{R} (?).

Lemma 2.4. Let A be an R-ring spectrum. If A/xA admits an R-ring spectrum structure such that ρ : A ⟶ A/xA is a map of R-ring spectra, then x : A/xA ⟶ A/xA is null homotopic as a map of R-modules.

Proof. We have the following commutative diagram (where we omit suspension coordinates from the labels of maps):

[figure: Σ^{n}R ∧_{R} (A/xA); λ; η∧id; Σ^{n}(A/xA) ∧_{R} (A/xA); x∧id; ϕ; (A/xA) ∧_{R} (A/xA); ϕ; Σ^{n}(A/xA); x; A/xA.]

In view of the following commutative diagram, its top composite is null homotopic:

[figure: Σ^{n}A; ρ; x; A; ρ; Σ^{n}R; η; η; Σ^{n}(A/xA); x; A/xA.]

Thus, for example, the Moore spectrum S/2 is not an S-algebra since the map 2 : S/2 ⟶ S/2 is not null homotopic.

We need a lemma in order to obtain an R-ring spectrum structure on R/x in appropriate generality.

Lemma 2.5. Let ρ : R ⟶ M be any map of R-modules. Then

(ρ ∧ id) ∘ ρ = (id ∧ρ) ∘ ρ : R ⟶ M ∧_{R} M.

Proof. Since λ = λ ∘ τ : R ∧_{R} R ⟶ R, the following diagram commutes:

[figure: R; ρ; ρ; M; M; R ∧_{R} R; λ; id ∧ρ; ρ∧id; R ∧_{R} M; λ; ρ∧id; M ∧_{R} R; id ∧ρ; λ∘τ; M ∧_{R} M.]

Theorem 2.6. Let x ∈ R_{n}, where π_{n+1}(R/x) = 0 and π_{2n+1}(R/x) = 0. Then R/x admits a structure of R-ring spectrum with unit ρ : R ⟶ R/x. Therefore A/XA admits a structure of R-ring spectrum such that ρ : A ⟶ A/XA is a map of R-ring spectra for every R-ring spectrum A and every sequence X of elements of R_{∗} such that π_{n+1}(R/x) = 0 and π_{2n+1}(R/x) = 0 for all x ∈ X, where deg x = n.

Proof. Consider the following diagram in the derived category 𝒟_{R}:

(2.7)[figure: Σ^{2n+1}R; x; Σ^{n+1}R; ρ; ν; x; ΣR; ρ; Σ^{n}(R/x); x; R/x; ρ∧id; (R/x) ∧_{R} (R/x); π; ϕ; Σ^{n+1}(R/x); x; π′; σ; Σ(R/x); Σ^{2n+2}R.]

The map x is that specified by (1.3). The bottom row is the cofiber sequence that results from the equivalence

(R/x) ∧_{R} (R/x) ≃ (R/x)/x

of Lemma 1.10, and the column is also a cofiber sequence. The composite x ∘ ρ is null homotopic since ρ ∘ x is null homotopic and the square commutes. Therefore there is a map ν such that π ∘ ν = ρ, and ν is unique since π_{n+1}(R/x) = 0. Since π ∘ ν ∘ x = ρ ∘ x = 0, ν ∘ x factors through a map Σ^{2n+1}R ⟶ R/x. Since π_{2n+1}(R/x) = 0, such maps are null homotopic. Thus ν ∘ x is null homotopic. Therefore there is a map σ such that σ ∘ ρ = ν. Now π ∘ σ ∘ ρ = π ∘ ν = ρ, hence (π∘σ−id)ρ = 0. Therefore π∘σ−id factors through a map Σ^{2n+2}R ⟶ Σ^{n+1}(R/x). Again, such maps are null homotopic. Therefore π∘σ = id. Thus the bottom cofiber sequence splits (proving in passing that x : Σ^{n}(R/x) ⟶ R/x is null homotopic, as it must be). A choice ϕ of a splitting gives a product on R/x. The unit condition ϕ ∘ (ρ ∧ id) = id is automatic. To see that ϕ ∘ (id ∧ρ) = id, we observe that, by the lemma,

(ϕ ∘ (id ∧ρ) − id) ∘ ρ = ϕ ∘ (id ∧ρ − ρ ∧ id) ∘ ρ = 0.

Therefore ϕ ∘ (id ∧ρ) − id factors through a map Σ^{n+1}R ⟶ R/x. Again, such maps are null homotopic, hence ϕ ∘ (id ∧ρ) = id. This completes the proof that R/x is an R-ring spectrum with unit ρ. The rest follows from Lemmas 1.10 and 2.2.

3. THE ASSOCIATIVITY AND COMMUTATIVITY OF R-RING SPECTRA 111

The product on R/x can be described a little more concretely. The wedge sum

(ρ ∧ id) ∨ σ : (R/x) ∨ Σ^{n+1}(R/x) ⟶ (R/x) ∧_{R} (R/x) (2.8)

is an equivalence. The product ϕ restricts to the identity on the first wedge summand and to the trivial map on the second wedge summand. Thus the product is determined by the choice of σ, and two choices of σ differ by a composite

[figure: Σ^{n+1}(R/x); π′; Σ^{2n+2}R; (R/x) ∧_{R} (R/x).] (2.9)

By the splitting (2.8) and the assumption that π_{n+1}(R/x) = 0, we can view the second map as an element of π_{2n+2}(R/x). If x is not a zero divisor, then π′_{∗} = 0 on homotopy groups and any two products have the same effect on homotopy groups.

Before continuing our discussion of these R-ring spectra, we insert the following easy consequence of the mere existence of the R-ring structure.

Corollary 2.10. Assume that R_{i} = 0 if i is odd. Let X and Y be regular sequences in R_{∗} that generate the same ideal. Then there is an equivalence of R-modules ξ : R/X ⟶ R/Y under R.

Proof. It suffices to construct a map ξ under R since it will automatically induce an isomorphism on homotopy groups. Each x_{i} is an R_{∗}-linear combination of the y_{j} and each y_{j} : R/Y ⟶ R/Y is zero. By Lemma 1.5, we obtain a unique map ξ_{i} : R/x_{i} ⟶ R/Y under R. Since R/Y is an R-ring spectrum, we may first take the smash product of these maps and then use the product (associated conveniently) on R/Y and passage to telescopes (if X is infinite) to obtain ξ.

3. The associativity and commutativity of R-ring spectra

We assume given an R-ring spectrum A. For x ∈ R_{n} as in Theorem 2.6, we give A/xA ≃ (R/x) ∧_{R} A the product induced by one of our constructed products on R/x and the given product on A. We refer to any such product as a “canonical” product on A/xA. Of course, we do not claim that every product on A/xA is canonical. Observe that, by first using the product on A, the product on A/xA can be factored through

ϕ ∧_{R} id : (R/x) ∧_{R} (R/x) ∧_{R} A ⟶ (R/x) ∧_{R} A.

This allows us to smash any diagram giving information about the product on R/x with A and so obtain information about the product on A/xA. Obviously any diagram so constructed is a diagram of right A-module spectra via the product action of A on itself. This smashing with A can kill obstructions. Clearly, a map of A-modules Σ^{q}A ⟶ M is determined by its restriction S^{q} ⟶ M along the unit of A regarded as a map of spectra (or S-modules), which is just an element of π_{q}(M). These considerations lead to the following result.

Theorem 3.1. Let x ∈ R_{n}, where π_{n+1}(R/x) = 0 and π_{2n+1}(R/x) = 0. Let A be an R-ring spectrum and assume that π_{2n+2}(A/xA) = 0. Then there is a unique canonical product on A/xA. If A is commutative, then A/xA is commutative. If A is associative and π_{3n+3}(A/xA) = 0, then A/xA is associative.

Proof. The second arrow of (2.9) becomes zero after smashing with A since it is then given by an element of π_{2n+2}(A/xA) = 0. This proves the uniqueness statement. The commutativity statement follows since if ϕ is a canonical product on A/xA, then so is ϕτ. However, it may be worth displaying the obstruction that lies in π_{2n+2}(A/xA). Looking at the splitting (2.8), we see that ϕ is commutative on the wedge summand R/x by the unit property. For the summand Σ^{n+1}R, consider the diagram

[figure: Σ^{n+1}R; ρ; Σ^{n+1}(R/x); π′; (ϕ−ϕ∘τ)∘σ; R/x.; Σ^{2n+2}R; γ]

The horizontal composite is null homotopic since π_{n+1}(R/x) = 0. Thus there exists γ such that the triangle commutes. It is the obstruction to the commutativity of R/x, and smashing with A gives the obstruction to the commutativity of A/xA.

For the associativity, consider the splitting displayed in the following diagram:

[figure: (R/x) ∨ Σ^{n+1}(R/x) ∨ Σ^{n+1}(R/x) ∨ Σ^{2n+2}(R/x); ≃; [(R/x) ∨ Σ^{n+1}(R/x)] ∨ Σ^{n+1}[(R/x) ∨ Σ^{n+1}(R/x)]; [(ρ∧id)∨σ]∨Σ^{n+1}[(ρ∧id)∨σ]; [(R/x) ∧_{R} (R/x)] ∨ Σ^{n+1}[(R/x) ∧_{R} (R/x)]; ≃; (R/x) ∧_{R} [(R/x) ∨ Σ^{n+1}(R/x)]; id ∧[(ρ∧id)∨σ]; (R/x) ∧_{R} (R/x) ∧_{R} (R/x).]

The question of associativity can be considered separately on the restrictions of the iterated product to the four wedge summands. Via easy diagram chases, we see that, under the splitting (2.8) and unit isomorphisms, the natural maps

ρ ∧ id ∧ id : R ∧_{R} (R/x) ∧_{R} (R/x) ⟶ (R/x) ∧_{R} (R/x) ∧_{R} (R/x)

and

id ∧ρ ∧ id : (R/x) ∧_{R} R ∧_{R} (R/x) ⟶ (R/x) ∧_{R} (R/x) ∧_{R} (R/x)

correspond to the inclusions of the first and third and first and second wedge summands, respectively. Therefore the unital property of ϕ and the unital and associativity properties of ∧_{R} imply that the restriction of ϕ to the first three wedge summands is associative. Let ω denote the displayed inclusion of the fourth wedge summand and consider the diagram

[figure: Σ^{2n+2}R; ρ; Σ^{2n+2}(R/x); π′; [ϕ∘(ϕ∧id)−ϕ∘(id ∧ϕ)]∘ω; R/x; Σ^{3n+3}R]

Call the horizontal composite α. If α is nullhomotopic then the deviation from associativity [ϕ ∘ (ϕ ∧ id) − ϕ ∘ (id ∧ϕ)] ∘ ω factors through a map Σ^{3n+3}R ⟶ R/x. Thus if π_{3n+3}(R/x) = 0, then the element α ∈ π_{2n+2}(R/x) is the obstruction to the associativity of R/x. If both relevant homotopy groups become zero after smashing with A, we can conclude that A/xA is associative if A is associative.

We can iterate the argument to arrive at the following fundamental conclusion.

Theorem 3.2. Assume that R_{i} = 0 if i is odd and let X be a sequence of non zero divisors in R_{∗} such that π_{∗}(R/X) is concentrated in degrees congruent to zero mod 4. Then R/X has a unique canonical structure of R-ring spectrum, and it is commutative and associative.

Proof. We first observe that for an element x ∈ π_{q}(R), an R-module M, and an R-module N such that x : Σ^{q}N ⟶ N is null homotopic, the map ρ : M ⟶ M/xM induces an epimorphism

ρ^{∗} : [M/xM, N]_{R} ⟶ [M, N]_{R}

since the action x^{∗} : [M, N]_{R} ⟶ [Σ^{q}M, N]_{R} can be computed from the action on N and is therefore zero. Let X_{n} be the subsequence consisting of the first n elements of the sequence X. Then R/X is defined to be the telescope of the R/X_{n}, and Lemma 2.4 implies that multiplication by x_{n} is null homotopic on R/X for each n. Since R/X ∧_{R} R/X is equivalent to the telescope of the R/X_{n} ∧_{R} R/X_{n}, we obtain a product on R/X from a canonical product on the R/X_{n} by passage to telescopes. Moreover, by the Mittag-Leffler criterion, our first observation implies that all relevant lim^{1} terms are zero. Thus it suffices to show that any two products on X_{n} become equal and the commutativity and associativity diagrams for R/X_{n} become commutative upon composition with the map R/X_{n} ⟶ R/X, and we may proceed by induction on n. The conclusion follows since the obstructions to uniqueness, commutativity, and associativity of each R/x_{n} become trivial when we map to R/X.

4. The specialization to MU-modules and algebras

It was observed in [47] that MU can be constructed as an E_{∞} ring spectrum, and we apply S ∧_{ℒ} (?) to convert it to a commutative S-algebra. Of course, its homotopy groups are concentrated in even degrees, and every non-zero element is a non zero divisor. Thus Proposition 2.3, Theorem 2.6, and Theorem 3.2 combine to give the following result.

Theorem 4.1. Let X be a regular sequence in MU_{∗}, let I be the ideal generated by X, and let Y be any sequence in MU_{∗}. Then there is an MU-ring spectrum (MU/X)[Y^{−1}] and a natural map of MU-ring spectra (the unit map)

η : MU ⟶ (MU/X)[Y^{−1}]

such that

η_{∗} : MU_{∗} ⟶ π_{∗}((MU/X)[Y^{−1}])

realizes the natural homomorphism of MU_{∗}-algebras

MU_{∗} ⟶ (MU_{∗}/I)[Y^{−1}].

If MU_{∗}/I is concentrated in degrees congruent to zero mod 4, then there is a unique canonical product on (MU/X)[Y^{−1}], and this product is commutative and associative.

In comparison with constructions of this sort based on the Baas-Sullivan theory of manifolds with singularities or on Landweber’s exact functor theorem (where it applies), we have obtained a simpler proof of a substantially stronger result. We emphasize that an MU-ring spectrum is a much richer structure than just a ring spectrum and that commutativity and associativity in the MU-ring spectrum sense are much more stringent conditions than mere commutativity and associativity of the underlying ring spectrum. Observe that the assumption that X is regular is used only to obtain the calculational description of η_{∗}.

We illustrate by explaining how BP appears in this context. Fix a prime p and write (?)_{p} for localization at p. Let BP be the Brown-Peterson spectrum at p. We are thinking of Quillen’s idempotent construction, and we have the splitting maps i : BP ⟶ MU_{p} and e : MU_{p} ⟶ BP. These are maps of commutative and associative ring spectra such that e ∘ i = id. Let I be the kernel of the composite

MU_{∗} ⟶ MU_{p∗} ⟶ BP_{∗}.

4. THE SPECIALIZATION TO MU-MODULES AND ALGEBRAS 115

Then I is generated by a regular sequence X, and our MU/X is a canonical integral version of BP. For the moment, let BP′ = (MU/X)_{p}. Let ξ : BP ⟶ BP′ be the composite

[figure: BP; i; MU_{p}; ζ_{p}; BP′.]

It is immediate that ξ is an equivalence. In effect, since we have arranged that ζ_{p} has the same effect on homotopy groups as e, ξ induces the identity map of (MU_{∗}/I)_{p} on homotopy groups. By the splitting of MU_{p} and the fact that selfmaps of MU_{p} are determined by their effect on homotopy groups [2, II.9.3], maps MU_{p} ⟶ BP are determined by their effect on homotopy groups. This implies that ξ ∘ e = ζ_{p} : MU_{p} ⟶ BP′. The product on BP is the composite

[figure: BP ∧ BP; i∧i; MU_{p} ∧ MU_{p}; ϕ; MU_{p}; e; BP.]

Since ζ_{p} is a map of MU-ring spectra and thus of ring spectra, a trivial diagram chase now shows that the equivalence ξ : BP ⟶ BP′ is a map of ring spectra.

We conclude that our BP′ is a model for BP that is an MU-ring spectrum, commutative and associative if p > 2. The situation for p = 2 is interesting. We conclude from the equivalence that BP′ is commutative and associative as a ring spectrum, although we do not know that it is commutative or associative as an MU-algebra.

Recall that π_{∗}(BP) = ℤ_{(p)}[v_{i}|deg(v_{i}) = 2(p^{i} − 1)], where the generators v_{i} come from π_{∗}(MU) (provided that we use the Hazewinkel generators). We list a few of the spectra derived from BP, with their coefficient rings. Let 𝔽_{p} denote the field with p elements.

\begin{matrix}BP ⟨n⟩ & ℤ_{(p)}[v_{1}, …, v_{n}] & E(n) & ℤ_{(p)}[v_{1}, …, v_{n}, v_{n}^{−1}] \\ P (n) & 𝔽_{p}[v_{n}, v_{n+1}, …] & B(n) & 𝔽_{p}[v_{n}^{−1}, v_{n}, v_{n+1}, …] \\ k(n) & 𝔽_{p}[v_{n}] & K(n) & 𝔽_{p}[v_{n}, v_{n}^{−1}] \end{matrix}

By the method just illustrated, we can construct canonical integral versions of the BP ⟨n⟩ and E(n). All of these spectra fit into the context of Theorem 4.1. If p > 2, they all have unique canonical commutative and associative MU-ring spectra structures. Further study is needed when p = 2, but we leave that to the interested reader. In any case, our theory makes it unnecessary to appeal to Baas-Sullivan theory or to Landweber’s exact functor theorem for the construction and analysis of spectra such as these.

We have started with MU because it appears in nature with a canonical structure as a commutative S-algebra. However, it is also possible to start with BP, using the second author’s result that BP admits a commutative S-algebra structure; in fact, it admits uncountably many different ones [33].

116 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU

CHAPTER VI Algebraic K-theory of S-algebras

In this chapter we apply the basic constructions of algebraic K-theory to the new categories of module over S-algebras. We show how to construct a K-theory spectrum KR for each S-algebra R in such a way that K becomes a functor from the category of S-algebras to the stable category. When R is a connective commutative S-algebra, so is KR. We prove that weakly equivalent S-algebras have equivalent K-theory, and we prove a Morita invariance result. When R is connective we are able to give an alternate description of this K-theory in terms of Quillen’s plus construction, a “plus equals S_{•}” theorem. When R = Hk is an Eilenberg-Mac Lane S-algebra, this K-theory is essentially Quillen’s algebraic K-theory of the ring k. When R = Σ^{∞}|GSX|_{+} is the suspension spectrum of a a disjoint basepoint plus the geometric realization of the loop group of the singular complex of a topological space X, this K-theory is Waldhausen’s algebraic K-theory of the space X.

1. Waldhausen categories and algebraic K-theory

We first review the basic definitions of Waldhausen [68] that we shall use.

Definition 1.1. A category with w-cofibrations 𝒞 is a (small) category with preferred zero object “∗”, together with a chosen subcategory co(𝒞) that satisfies the following three axioms:

(i) Any isomorphism in 𝒞 is a morphism in co(𝒞); in particular, co(𝒞) contains all the objects of 𝒞.

(ii) For every object A in 𝒞, the unique map ∗ ⟶ A is in co(𝒞).

(iii) If A ⟶ B is a map in co(𝒞), and A ⟶ C is a map in 𝒞, then the pushout B ⨿_{A} C exists in 𝒞 and the canonical map C ⟶ B ⨿_{A} C is in co(𝒞); in particular, 𝒞 has finite coproducts.

118 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS

We call the morphisms in co(𝒞) w-cofibrations, and often use the feathered arrow “↣” to denote them in diagrams. Although Waldhausen called these arrows “cofibrations” we will consistently use the term “w-cofibration” so that there will be no confusion with our standard use of the word “cofibration” to mean those maps that satisfy the homotopy extension property (HEP).

Definition 1.2. A Waldhausen category (in [68], “a category with cofibrations and weak equivalences”) is a category with w-cofibrations 𝒞 and a chosen subcategory w(𝒞) of 𝒞 that satisfies the following axioms:

(i) Any isomorphism in 𝒞 is a morphism in w(𝒞); in particular, w(𝒞) contains all the objects of 𝒞.

(ii) Given any commutative diagram in 𝒞

[figure: C; A; B; C′; A′; B′]

in which the vertical maps are in w(𝒞) and the feathered arrows are in co(𝒞), the induced map B ⨿_{A} C ⟶ B′ ⨿_{A′} C′ is in w(𝒞).

We call the morphisms in w(𝒞) weak equivalences, and often use the arrows “⟶^{∼}” to denote them. We say that the weak equivalences are saturated or that 𝒞 is a saturated Waldhausen category if whenever f and g are composable arrows in 𝒞 and any two of f, g, and gf are weak equivalences then so is the third.

Definition 1.3. A functor between Waldhausen categories is exact if it preserves all of the above structure; i.e. it must send w-cofibrations to w-cofibrations, weak equivalences to weak equivalences, the preferred zero object to the preferred zero object, and it must preserve the pushouts along a w-cofibration.

We now have all the necessary ingredients to describe Waldhausen’s S_{•} construction [68]. Let 𝒞 be a Waldhausen category. For each n ≥ 0, define a category S_{n}𝒞 as follows. An object of S_{n}𝒞 consists of n + 1 composible arrows in co(𝒞) starting from the preferred zero object ∗,

[figure: ∗ = A_{0}; α_{0}; A_{1}; α_{1}; ⋯; α_{n}; A_{n},]

together with objects A_{i,j} for 0 ≤ i ≤ j ≤ n and maps a_{i,j}: A_{j} ⟶ A_{i,j} such that A_{i,i} = ∗, A_{j} = A_{0,j} with a_{0,j} the identity map, and the diagrams

[figure: A_{i}; α_{j−1}∘···∘α_{i}; A_{j}; a_{i,j}; ∗; A_{i,j}]

are pushouts for 0 ≤ i < j ≤ n. A morphism of S_{n}𝒞 from {A_{j}, A_{i,j}, α_{j}, a_{i,j}} to {A′_{j}, A′_{i,j}, α′_{j}, a′_{i,j}} is a sequence of maps f_{j}: A_{j} ⟶ A′_{j} such that the diagram

[figure: A_{0}; α_{0}; f_{0}; A_{1}; α_{1}; f_{1}; ⋯; α_{n}; A_{n}; f_{n}; A′_{0}; α′_{0}; A′_{1}; α′_{1}; ⋯; α′_{n}; A′_{n}]

commutes. Observe that by the universal property of pushouts, we have induced maps A_{i,j} ⟶ A′_{i,j} making all the appropriate diagrams commute. We give S_{n}𝒞 the structure of a Waldhausen category by defining a map {f_{0}, …, f_{n}} to be a w-cofibration (resp. weak equivalence) if each f_{j} is a w-cofibration (resp. weak equivalence) of 𝒞. Observe that when {f_{0}, …, f_{n}} is a w-cofibration (resp. weak equivalence) all the induced maps A_{i,j} ⟶ A′_{i,j} are w-cofibrations (resp. weak equivalences). Notice that S_{0}𝒞 is the trivial category and that S_{1}𝒞 is isomorphic to 𝒞.

For 0 ≤ i ≤ n, define d_{k}: S_{n}𝒞 ⟶ S_{n−1}𝒞 to be the functor that drops the k-th row and k-th column from the matrix {A_{i,j}}. More precisely, d_{0} sends the object {A_{j}, A_{i,j}, α_{j}, a_{i,j}} of S_{n}𝒞 to the object {B_{j}, B_{i,j}, β_{j}, b_{i,j}} of S_{n−1}𝒞 where B_{j} = A_{1,j+1}, B_{i,j} = A_{i+1,j+1}, and the maps β_{j} and b_{i,j} are the maps induced from α_{j+1} and a_{i+1,j+1} by the universal property of the pushout. For k > 0, the functor d_{k} is defined similarly. For 0 ≤ k ≤ n, define s_{k}: S_{n}𝒞 ⟶ S_{n+1}𝒞 to be the functor that repeats the k-th row and k-th column in the matrix {A_{i,j}}. More precisely, s_{k} sends the object {A_{j}, A_{i,j}, α_{j}, a_{i,j}} of S_{n}𝒞 to the object {B_{j}, B_{i,j}, β_{j}, b_{i,j}} of S_{n+1}𝒞 where

B_{j} = \begin{cases}A_{j} & if j ≤ k \\ A_{j−1} & if j > k \end{cases}

B_{i,j} = \begin{cases}A_{i,j} & if j ≤ k \\ A_{i,j−1} & if j > k and i ≤ k \\ A_{i−1,j−1} & if i > k \end{cases}

β_{j} = \begin{cases}α_{j} & if j < k \\ id & if j = k \\ α_{j−1} & if j > k \end{cases}

b_{i,j} = \begin{cases}a_{i,j} & if j ≤ k \\ a_{i,j−1} & if j > k and i ≤ k \\ a_{i−1,j−1} if i > k. \end{cases}

Observe that the functors d_{k} and s_{k} satisfy the simplicial identities and the collection {S_{n}𝒞 } assembles into a simplicial category, which we denote S_{•}𝒞. Furthermore, the functors d_{k} and s_{k} are exact and S_{•} has the structure of a simplicial Waldhausen category. In particular, we can iterate this construction to form the bisimplicial Waldhausen category S_{•}^{(2)}𝒞 = S_{•}S_{•}𝒞, and the polysimplicial Waldhausen categories S_{•}^{(n)}𝒞 = S_{•} ⋯ S_{•}𝒞. We abbreviate the notation for the category of weak equivalences in S_{•}^{(n)}𝒞 to wS_{•}^{(n)}𝒞. We are interested in the classifying spaces of these categories, the spaces |wS_{•}^{(n)}𝒞 |. Since S_{0}𝒞 is the trivial category, we see that |wS_{•}𝒞 | is connected, and it is not much harder to see that in general |wS_{•}^{(n)}𝒞 | is (n − 1)-connected. Observe that the identifications of 𝒞 with S_{1}𝒞 and more generally S_{•}^{(n)}𝒞 with S_{1}S_{•}^{(n)}𝒞 induce maps Σ|w𝒞 | ⟶ |wS_{•}𝒞 | and Σ|wS_{•}^{(n)}𝒞 | ⟶ |wS_{•}^{(n+1)}𝒞 | which are inclusions of subcomplexes. It is a fundamental observation of [68, §1.3, 1.5.3] that in the sequence

|w𝒞 | ⟶ Ω|wS_{•}𝒞 | ⟶ Ω^{2}|wS_{•}^{(2)}𝒞 | ⟶ ⋯

all maps beyond the first are homotopy equivalences. This motivates the following definition.

Definition 1.4. The algebraic K-theory of the Waldhausen category 𝒞 is the spectrification of the Σ-cofibrant prespectrum {|w𝒞 |, |wS_{•}𝒞 |, |wS_{•}^{(2)}𝒞 |, … }. We denote this spectrum by the symbol K𝒞. The algebraic K-groups of 𝒞 are the homotopy groups of this spectrum, K_{n}𝒞 = π_{n}K𝒞 = π_{n+1}|wS_{•}𝒞 |. In particular, K_{n}𝒞 = 0 for n < 0.

Waldhausen observes that in the special case when 𝒞 is an exact category (where the w-cofibrations are the admissible monos, and the weak equivalences are the isomorphisms) the algebraic K-groups defined above agree with those defined by Quillen [55]. In fact, the basic properties of the Q-construction are all easily provable in terms of the S_{•} construction [64] (see also [51]).

Observe that an exact functor 𝒞 ⟶ 𝒟 induces an exact functor S_{•}𝒞 ⟶ S_{•}𝒟 and hence exact functors S_{•}^{(n)}𝒞 ⟶ S_{•}^{(n)}𝒟. This induces a map of prespectra {wS_{•}^{(n)}𝒞 } ⟶ {wS_{•}^{(n)}𝒟} and hence a map of spectra K𝒞 ⟶ K𝒟. If the map |wS_{•}𝒞 | ⟶ |wS_{•}𝒟 | is a weak equivalence, then the maps |wS_{•}^{(n)}𝒞 | ⟶ |wS_{•}^{(n)}𝒟 | are weak equivalences and therefore homotopy equivalences. Since these prespectra are Σ-cofibrant, maps between them that are spacewise homotopy equivalences induce homotopy equivalences of their spectrifications. In other words, an exact functor that induces a weak equivalence on |wS_{•}−| induces a homotopy equivalence of K-theory spectra. For this reason, although Waldhausen defines the algebraic K-theory of a Waldhausen category 𝒞 to be the space Ω|wS_{•}𝒞 |, all the results we use from [68] apply to the K-theory spectra, even when they are stated only for the K-theory spaces, and we will use them this way without further comment; moreover, whenever we shall assert a result about K-theory, we shall mean the result about the K-theory spectra unless otherwise noted.

2. Cylinders, homotopies, and approximation theorems

Let MOR 𝒞 be the category whose objects are the morphisms of 𝒞 and whose morphisms are the commutative diagrams. For A, A′, B, B′ objects of 𝒞 and a: A ⟶ A′ and b: B ⟶ B′ maps of 𝒞, a and b are objects of MOR 𝒞. If f: A ⟶ B and f′: A′ ⟶ B′ are maps in 𝒞 that make the diagram

[figure: A; a; f; B; b; A′; f′; B′]

commute, then (f, f′) is a map in MOR 𝒞. Whenever 𝒞 is a Waldhausen category, we can give MOR 𝒞 the structure of a Waldhausen category by saying that a map (f, f′) is a w-cofibration (resp. weak equivalence) of MOR 𝒞 if both f and f′ are w-cofibrations (resp weak equivalences) of 𝒞.

Definition 2.1. [68, 1.6] Let 𝒞 be a Waldhausen category. A cylinder functor is a functor T: MOR 𝒞 ⟶ 𝒞 together with natural transformations i_{1}, i_{2}, and p that make the following diagram commute for a morphism f: A ⟶ B in 𝒞:

[figure: A; i_{1}; f; T f; p; B; i_{2}; B]

and that satisfies the following properties:

(i) i_{1} ⨿ i_{2} : A ⨿ B ↣ T f is in co(𝒞).

(ii) The functor (A ⟶ B) ↦ (A ⨿ B ⟶^{i_{1}⨿i_{2}} T f) is an exact functor MOR 𝒞 ⟶

MOR 𝒞. (iii) T (∗ ⟶ B) = B, with p and i_{2} the identity map. We say that the cylinder functor satisfies the cylinder axiom if in addition

p: T f ⟶ B is in w(𝒞) for all morphisms f. We will often refer to i_{1} and i_{2} as face maps and to p as the collapse map.

Theorem 2.2. (Waldhausen, [68, 1.6.7], “The Approximation Theorem”) Let 𝒜 and ℬ be saturated Waldhausen categories, where 𝒜 has a cylinder functor that satisfies the cylinder axiom. Let F: 𝒜 ⟶ ℬ be an exact functor such that

(i) If f is a morphism in 𝒜 such that F (f) is in w(ℬ), then f is in w(𝒜). (ii) For any object A ∈ 𝒜 and any map f: F A ⟶ B, there exists a map

g: A ⟶ A′ in 𝒜 and a weak equivalence h: F A′ ⟶ B in w(ℬ) such

that f = h ∘ F (g):
[figure: F (A); F (g); f; B.; F (A′); h]

Then F induces a homotopy equivalence K𝒜 ⟶ Kℬ.

Remark 2.3. In [68] it is required that g be a w-cofibration, but [65] points out that this requirement is unnecesary since we can use cylinders to replace an arbitrary map with a w-cofibration.

Often, we have the situation where it is easy to make the diagram in 2.2(ii) commute up to some kind of homotopy. By integrating the idea of homotopy into Waldhausen’s language of K-theory, we can prove two easy but extremely useful corollaries of the approximation theorem. To this end, we offer the following definitions.

Definitions 2.4. Let 𝒞 be a Waldhausen category with cylinder functor T. Observe that T gives an exact functor I: 𝒞 ⟶ 𝒞 by restriction along the exact functor 1: 𝒞 ⟶ MOR 𝒞 that sends an object to its identity morphism. We call (W, j_{1}, j_{2}, q) a cylinder object of the object X if W = IX, q = p (the collapse map) and either j_{1} = i_{1} and j_{2} = i_{2}, or j_{1}=i_{2} and j_{2} = i_{1}. We say that (W, j_{1}, j_{2}, q) is a generalized cylinder object of the object X, if W is the pushout over alternate face maps of a sequence of cylinder objects, j_{1}, j_{2} : X ↣ W are the two unused face maps, and q: W ⟶ X is the gluing of the collapse maps; in particular observe that q ∘ j_{i} = 1_{X} for i = 1, 2. We call two maps f_{1}, f_{2} : X ⟶ Y homotopic if there exists a generalized cylinder object W of X and a map ϕ: W ⟶ Y such that ϕ ∘ j_{i} = f_{i} for i = 1, 2. It is easy to see that this specifies an equivalence relation.

Let us say that an exact functor F: 𝒞 ⟶ 𝒟 between Waldhausen categories with cylinder functors preserves cylinder objects if there is a natural isomorphism α: F I_{𝒞} ≅ I_{𝒟} F such that α ∘ F (i_{k}) = i_{k} and p ∘ α = F (p):

[figure: F IA; α; F A; F (i_{1}); i_{1}; F A; F (i_{2}); i_{2}; IF A; F IA; α; F (p); F A.; IF A; p]

Observe that a functor that preserves cylinder objects also preserves generalized cylinder objects, in the sense that α gives an isomorphism of F W to a generalized cylinder object W′ with α ∘ F (j_{k}) = j′_{k} and q′ ∘ α = F (q). It is easy to see that when F preserves cylinder objects, F also preserves the relation of homotopy of morphisms.

Theorem 2.5. (Homotopy Approximation Theorem) Let 𝒜 and ℬ be small saturated Waldhausen categories with cylinder functors satisfying the cylinder axiom. Let F: 𝒜 ⟶ ℬ be an exact functor that preserves cylinder objects and such that

(i) If f is a morphism in 𝒜 such that F (f) is in w(ℬ), then f is in w(𝒜). (ii) For any object A ∈ 𝒜 and any map f: F A ⟶ B, there exists a map

a: A ⟶ X in 𝒜 and a weak equivalence e: F X ⟶ B in w(ℬ) such that f is homotopic to e ∘ F (a).

Then F induces a homotopy equivalence K𝒜 ⟶ Kℬ.

Proof. We produce an object A′ and maps g, h that satisfy condition (ii) of the Approximation Theorem.

We have assumed that f is homotopic to e ∘ F (a), so there exists a generalized cylinder object (W′, j′_{1}, j′_{2}, q′) of F A and ψ: W′ ⟶ B with ψ ∘ j′_{1} = f, ψ ∘ j′_{2} = e ∘ F a. We can construct in 𝒜 the generalized cylinder object W with the same gluings of faces; then we have α: F W ≅ W′ with α ∘ F (j_{i}) = j′_{i}, since F preserves cylinder objects.

Let A′ = W ⨿_{a} X, and let g be the evident map A ⟶ A′ induced by j_{1} : A ⟶ W. Then α induces an isomorphism F A′ ⟶ W′ ⨿_{F (a)} F X, which we will denote by α̃. Consider the map h̃ = ψ ⨿_{F (a)} e: W′ ⨿_{F (a)} F X ⟶ B. The inclusion F X ⟶ W′ ⨿_{F (a)} F X is a weak equivalence by property 1.2.(ii), since it is the pushout of the following weak equivalence of diagrams

[figure: F X; ∼; id; F A; F (a); ∼; id; id; F A; ∼; j′_{1}; F X; F A; F (a); j′_{1}; W′.]

The composite of this inclusion with h̃ is the weak equivalence e, so we concluse that h̃ is a weak equivalence since ℬ is saturated by assumption. Choosing h to be the weak equivalence h̃ ∘ α̃ makes diagram 2.2(ii) commute.

The next corollary of Waldhausen’s Approximation Theorem requires some preliminary definitions.

Definitions 2.6. We say that a map f: X ⟶ Y is a homotopy equivalence if there exists a morphism g: Y ⟶ X so that f ∘ g and g ∘ f are homotopic to the respective identity morphisms. In this case, we call g a homotopy inverse to f. We say that 𝒞 is a category with w-cofibrations and homotopy equivalences or a Waldhausen homotopy category (WH category for short) if 𝒞 is a saturated Waldhausen category with cylinder functor satisfying the cylinder axiom such that the weak equivalences are the homotopy equivalences.

We need to make an observation about the “derived” category of a WH category, the category formed by inverting the homotopy equivalences. First note that if f is a homotopy equivalence and g a homotopy inverse to f, then g ∘ f and f ∘ g are both identity morphisms in the derived category. From this, it is easy to see that every map in the derived category of 𝒞 is represented by a map in 𝒞.

One can ask when two maps give the same map in the derived category. First observe that for homotopic maps f_{1}, f_{2} and homotopic maps g_{1}, g_{2}, the compositions f_{1} ∘g_{1} and f_{2} ∘g_{2} are homotopic; so we can form the homotopy category, h𝒞 whose objects are the objects of 𝒞 and whose maps are homotopy classes of maps. It is straightforward to verify that homotopic maps represent the same map in the derived category, and that if two maps represent the same map in the derived category then they are homotopic. We conclude that the natural map from 𝒞 to its derived category factors through h𝒞, and that the map from h𝒞 to the derived category is actually an isomorphism (not merely equivalence) of categories.

The next result is now an immediate corollary to Theorem 2.5.

Corollary 2.7. Let 𝒜 and ℬ be WH categories. Suppose F: 𝒜 ⟶ ℬ is an exact functor that preserves cylinder objects and that passes to an equivalence on the derived categories. Then F induces a homotopy equivalence K𝒜 ⟶ Kℬ.

Proof. We reduce to Theorem 2.5: Condition (i) is clear. For condition (ii), choose X ∈ 𝒜 so that F X is isomorphic to B in the derived category of ℬ. Since every map in the derived category of ℬ is represented by an actual map in ℬ, we can choose e: F X ⟶ B that represents this isomorphism. Then e is a homotopy equivalence; let e′ be a homotopy inverse. Now there exists a map a: A ⟶ X so that F a: F A ⟶ F X represents the same map as e′ ∘ f in the derived category of ℬ. We conclude that f is homotopic to e ∘ F a.

3. Application to categories of R-modules

3. APPLICATION TO CATEGORIES OF R-MODULES 125

For an S-algebra R, let 𝒞_{R} be the full subcategory ℳ_{R} consisting of the cell R-modules, and 𝒞 𝒲 _{R} the category of CW R-modules and cellular maps. We denote by f𝒞_{R} the full subcategory of 𝒞_{R} of finite cell R-modules and f𝒞 𝒲 _{R} the full subcategory of 𝒞 𝒲 _{R} of finite CW R-modules; more precisely, we must choose small full subcategories containing at least one object of each isomorphism class, but the fact that the category of spectra has canonical colimits allows a strict interpretation of the definition of cell and CW R-modules under which the categories f𝒞_{R} and f𝒞 𝒲 _{R} are already small. When 𝒞 is one of the categories f𝒞_{R} or f𝒞 𝒲 _{R}, we can give 𝒞 the structure of a WH category as follows. We define the category of w-cofibrations, co(𝒞), to consist of those maps which are isomorphic (in MOR 𝒞) to the inclusion of a subcomplex, and the category of weak equivalences w(𝒞) to consist of those maps in 𝒞 which are homotopy equivalences. We take as our cylinder functor the ordinary mapping cylinder.

Proposition 3.1. These definitions specify the structure of WH-categories on f𝒞_{R} and f𝒞 𝒲 _{R} and the inclusion f𝒞 𝒲 _{R} ⟶ f𝒞_{R} is an exact functor which preserves cylinder objects. Furthermore, when R is connective, this inclusion induces a homotopy equivalence of K-theory spectra.

Proof. We check the definitions directly. Let 𝒞 be either f𝒞_{R} or f𝒞 𝒲_{R} with co(𝒞) and w(𝒞) as above.

First we check that co(𝒞) is a category. Let f: A ⟶ B and g: B ⟶ C be arrows in 𝒞. These are isomorphic to inclusions of subcomplexes, and without loss of generality we can assume that f is the inclusion of a subcomplex and that g is isomorphic to an inclusion g′: B′ ⟶ C with the isomorphism on the codomain C the identity:

[figure: B; b; g; B′; g′; C; C.]

By choosing different sequential filtrations if necessary we can assume that the map B ⟶ C is sequentially cellular and therefore so is the map B ⟶ B′ (we adjust the sequential filtration on A as well if necessary so that it remains a subcomplex of B). Since C is built from B′ by attaching cells, we can form an isomorphic complex D by attaching the same cells to B via b^{−1}. In the CW case, D is CW and the isomorphisms to and from C are cellular because we have assumed that the isomorphisms b and b^{−1} are cellular. Now A is a subcomplex of B, which is a subcomplex of D and the map A ⟶ D is isomorphic to the map A ⟶ C.

Properties 1.1(i) and (ii) and 1.2(i) are clear as will be 1.2(ii) once we show 1.1(iii). Given a diagram in 𝒞,

[figure: A; f; g; B; C]

with f an arrow in co(𝒞), we show that we can find a pushout in 𝒞 so that the map from C is the inclusion of a subcomplex. We can assume without loss of generality that f is the inclusion of a subcomplex and that the map g is sequentially cellular. Since B is built from A by attaching cells, we can form a cellular complex D by attaching the cells to C via g: A ⟶ C. In the CW case, D will be a CW complex since we have assumed that g is cellular. For categorical reasons, D must be a pushout of the above diagram, and by construction the map from C is the inclusion of a subcomplex.

If R is connective, we can approximate any finite cellular R-module by a finite CW R-module, and the last statement follows by Theorem 2.5.

Definition 3.2. We define the algebraic K-theory of the S-algebra R to be the spectrum Kf𝒞_{R}, and we denote it by KR. We define the algebraic K-groups of R to be the homotopy groups of this spectrum, K_{n}R = π_{n}KR = K_{n}f𝒞_{R}.

Although the categories f𝒞_{R} and f𝒞 𝒲 _{R} seem the most natural choices for Ktheory, there are many other possibilities. Indeed, since pushouts along cofibrations in ℳ_{R} preserve weak equivalences, it is easy to see that any subcategory of ℳ_{R} that is a category with w-cofibrations such that all of the w-cofibrations are cofibrations becomes a Waldhausen category by taking the weak equivalences to be the ordinary weak equivalences. In particular, when 𝒳 is small full subcategory of ℳ_{R} that contains the trivial R-module, and is closed under pushouts along cofibrations, then 𝒳 is a category with w-cofibrations the set of all cofibrations in 𝒳 and in this way becomes a Waldhausen category. We shall call the resulting Waldhausen category structure on 𝒳 the standard Waldhausen structure. If 𝒞 is a full subcategory of 𝒞_{R} that is small, contains the trivial R-module and is closed under pushouts along maps isomorphic to inclusions of subcomplexes, then 𝒞 is a category with wcofibrations the set of maps in 𝒞 isomorphic to the inclusion of a subcomplex and in this way becomes a Waldhausen category. We shall call the resulting Waldhausen category structure on 𝒞 the standard cellular Waldhausen structure. If 𝒳 or 𝒞 is closed under smashing with I_{+}, then the mapping cylinder gives a cylinder functor satisfying the cylinder axiom, which we shall call the standard cylinder functor. Since a standard cellular Waldhausen category with the the standard cylinder functor is a WH category, we shall call such a category a standard WH category. It might at first appear that the standard Waldhausen structures are somewhat rare, but the following remark demonstrates that they are actually quite common.

Remark 3.3. (Smallest Standard Waldhausen Categories) Given a set of objects 𝒜 ⊂ ℳ_{R} that is not necessarily closed under pushouts along cofibrations, we can form a small category ℬ containing 𝒜 that is. We let ℬ be the union of an expanding sequence of small categories 𝒜_{0} ⟶ 𝒜_{1} ⟶ 𝒜_{2} ⟶ ⋯, where Ob(𝒜_{0}) = 𝒜 and 𝒜_{n+1} is the full subcategory of ℳ_{R} of objects that are pushouts of diagrams (with one leg a cofibration) in 𝒜_{n} (one choice of object for each such diagram). Since the set of all maps in 𝒜_{n} is small (by induction), 𝒜_{n+1} is a small category. It is easy to see that ℬ has a kind of universal property: whenever a standard Waldhausen category contains a full subcategory equivalent to 𝒜 (regarded as a full subcategory of ℳ_{R}), it must contain a full subcategory compatibly equivalent to ℬ. For this reason, we will refer to ℬ as the smallest standard Waldhausen category containing 𝒜. Observe that when all the objects of 𝒜 have the weak homotopy types of finite cell complexes then so do all the objects of ℬ (by Corollary I.6.5).

Often we will want our standard Waldhausen categories to have the standard cylinder functor. In forming 𝒜_{n+1} from 𝒜_{n} above we could also include X ∧ I_{+} for each X ∈ 𝒜_{n}. In this case, 𝒜_{n+1} will still be small, but now ℬ will be closed under smashing with I_{+}, and hence have the standard cylinder functor. It is easy to see that ℬ will now have a similar universal property with respect to standard Waldhausen categories with the standard cylinder functor. For this reason we will refer to the category constructed in this way as the smallest standard Waldhausen category with standard cylinder functor containing 𝒜. Again, when all the objects of 𝒜 have the weak homotopy types of finite cell complexes then so do all the objects of ℬ.

If 𝒜 ⊂ 𝒞_{R}, we can do a similar construction but using maps isomorphic to inclusions of subcomplexes in place of cofibrations. Then the resulting category ℬ ⊂ 𝒞_{R} is a standard cellular Waldhausen category (a WH category if we include smashes with I_{+}) and has a similar universal property with respect to standard cellular Waldhausen categories. We shall not actually use this construction, but we could call ℬ in this case the smallest standard cellular Waldhausen category containing 𝒜 (or if we include smashes with I_{+}, smallest standard WH category containing 𝒜). Furthermore, observe that if all the objects of 𝒜 have the homotopy types of finite cell complexes, then so do all the objects of ℬ.

One advantage of the standard Waldhausen structures is that inclusions of subcategories are exact functors.

Proposition 3.4. Suppose 𝒳 is a subcategory of 𝒴 . If 𝒳 and 𝒴 are both standard Waldhausen categories or both standard cellular Waldhausen categories, then the inclusion 𝒳 ⟶ 𝒴 is an exact functor. If 𝒳 is a standard cellular Waldhausen category and 𝒴 is a standard Waldhausen category, then the inclusion 𝒳 ⟶ 𝒴 is an exact functor.

Many standard Waldhausen categories have K-theory equivalent to f𝒞_{R}. The following proposition follows directly from Theorem 2.5 (and the Whitehead Theorem) and will often apply to the smallest standard categories constructed above.

Proposition 3.5. Let 𝒳 be a standard Waldhausen category with standard cylinder functor or a standard WH category. If 𝒳 contains f𝒞_{R}, and if furthermore each object of 𝒳 is weakly equivalent to a finite cell complex, then the induced map of K-theory spectra is a homotopy equivalence.

The K-theory we consider, the K-theory of f𝒞_{R}, is best thought of as analogous to the K-theory of finitely generated free modules: indeed, since all objects are constructed from the sphere R-modules by a finite number of extensions by spheres, it follows immediately that the obvious homomorphism K_{0}^{f}π_{0}R ⟶ K_{0}R (induced by [n] ↦ ∨_{i=1}^{n}S_{R}) is surjective, where “K_{0}^{f}π_{0}R” denotes K_{0} of the finitely generated free modules of the ring π_{0}R. When R is connective this homomorphism is an isomorphism whose inverse is given by the Euler characteristic of a CW object X, the alternating sum of the classes of C_{n}(X), where C_{∗} is the chain functor of IV.3. For this reason, categories that could be reasonable alternatives to the categories f𝒞_{R} and f𝒞 𝒲 _{R} would be those small categories of semi-finite cell R-modules that are standard WH categories. When such a category contains f𝒞_{R}, it follows from [65, 1.10.1], [68, 1.5.9] and the argument of [23, §1] (as observed in [65, 1.10.2]) that the inclusion will induce an isomorphism of homotopy groups of K-theory spectra except in dimension zero. Intuitively, whereas the K-theory of f𝒞_{R} or f𝒞 𝒲 _{R} is like the K-theory of the finitely generated free modules, we might think of the Ktheory of semi-finite objects as analogous to the K-theory of the finitely generated projective modules.

We conclude this section by remarking that when R is an A_{∞} ring spectrum but not an S-algebra, we can make analogous observations about the K-theory of categories of its modules. However the functor S ∧_{ℒ} (?) is an exact functor that converts such a category to the corresponding category of S ∧_{ℒ} R-modules and induces a homotopy equivalence of K-theory spectra by Theorem 2.5. Thus, results about the K-theory of A_{∞} ring spectra follow from results about the Ktheory of S-algebras.

4. Homotopy invariance and Quillen’s algebraic K-theory of rings

In this section we prove some properties of the K-theory of the category f𝒞_{R} and compare with the K-theory of (discrete) rings.

We observe that K-theory as defined above gives a functor from the category of S-algebras to the stable category which has nice homotopical properties.

Proposition 4.1. If ϕ: A ⟶ B is a map of S-algebras, then the functor B ∧_{A} (−): f𝒞_{A} ⟶ f𝒞_{B} (or f𝒞 𝒲 _{A} ⟶ f𝒞 𝒲 _{B}) is exact and preserves cylinder objects. This association makes K into a functor from the category of S-algebras to the stable category.

Proof. The first statement follows from III.4.1, the second from the isomorphisms C ∧_{B} (B ∧_{A} (−)) ≅ C ∧_{A} (−) and A ∧_{A} (−) ≅ id.

Proposition 4.2. If ϕ: A ⟶ B is a map of S-algebras that is a weak equivalence, then Kϕ is a homotopy equivalence.

4. HOMOTOPY INVARIANCE AND QUILLEN’S ALGEBRAIC K-THEORY 129

Proof. From III.4.2, B ∧_{A} (−) induces an equivalence of derived categories 𝒟_{A} ⟶ 𝒟_{B}, which restricts to an equivalence of the derived categories of finite cell complexes by an easy application of the Whitehead theorem. The result follows from Corollary 2.7.

We compare this K-theory with Quillen’s algebraic K-theory. Let k be a ring, and let Hk denote the Eilenberg–Mac Lane S-algebra of k. We shall use the symbol K^{f}k for the algebraic K-theory of the finitely generated free modules of k, a covering spectrum of Kk.

Theorem 4.3. KHk is homotopy equivalent to K^{f}k, naturally in k.

Proof. We can identify K^{f}k with the K-theory of the WH category of finite free k-chain complexes with w-cofibrations the split monics, weak equivalences the quasi-isomorphisms, and the cylinder functor given by the usual mapping cylinder. (see, for example, [65, 1.11.7].) We will denote this WH category as f𝒞 𝒲 _{k}.

The functor C_{∗}: f𝒞 𝒲 _{Hk} ⟶ f𝒞 𝒲 _{k} of IV.2 is exact and preserves cylinder objects. By the Hurewicz theorem IV.3.6, a map between finite CW modules is a weak equivalence if and only if its image under C_{∗} is a quasi-isomorphism, hence the theorem will follow from Theorem 2.2 if we can show that condition (ii) holds.

Given a finite free chain complex M_{∗}, we can actually construct a CW Hkmodule X whose cellular chain complex C_{∗}(X) is isomorphic to M_{∗}. We proceed by induction. Since M_{∗} is finite, M_{i} is zero below some m, and we take the iskeleton of X, X^{i} to be the trivial Hk-module for i < m. Now assume that we have constructed X^{n} and an isomorphism C_{∗}(X^{n}) ⟶ M_{∗}^{≤n}, where M_{∗}^{≤n} denotes the brutal n-truncation of M_{∗}, i.e. M_{i}^{≤n} = M_{i} for i ≤ n and M_{i}^{≤n} = 0 for i > n. By IV.2.3, π_{n}(X^{n}) ≅ H_{n}(M_{∗}^{≤n}), which is the kernel of the differential d_{n−1}, i.e. the cycles of M_{n}. Via this isomorphism and a choice of basis for M_{n+1}, the differential d_{n} specifies a homotopy class of maps from a wedge of S_{Hk}^{n} to X^{n}. Choose a representative of this homotopy class, and let X^{n+1} be the CW complex formed by attaching (n + 1)-cells along this map. By construction C_{n+1}(X^{n+1}) ≅ M_{n+1}, compatibly with the differentials.

Given an Hk-module A and a map f: C_{∗}(A) ⟶ M_{∗}, we show that we can find a map a: A ⟶ X such that C_{∗}(a) agrees with f via the isomorphism contructed above. Assume we have constructed this map as far as the n-skeleton of A, i.e. we have a_{n} : A^{n} ⟶ X such that C_{∗}(a_{n}) = f^{≤n}. Now A^{n+1} is formed from A^{n} by attaching a finite wedge ∨CS_{Hk}^{n} along a map α: ∨ S_{Hk}^{n} ⟶ A^{n}. The map f_{n+1} : C_{n+1}(A) ⟶ M_{n+1} specifies a homotopy class of maps (∨CS_{Hk}^{n}, ∨S_{Hk}^{n}) ⟶ (X^{n+1}, X^{n}) ⟶ (X, X^{n}), whose class on ∨S_{Hk}^{n} agrees with [a_{n} ∘ α], since π_{n}(X_{n}) coincides with the cycles of M_{n}. We choose a representative in the homotopy class whose restriction to S_{Hk}^{n} is a_{n}∘α. This extends to a map a_{n+1} : A^{n+1} ⟶ X^{n+1} ⟶ X, and by construction C_{∗}(a_{n+1}) agrees with f^{≤n+1}.

Remark 4.4. For R a connective S-algebra, the functor C_{∗} of IV.3 is an exact functor f𝒞 𝒲 _{R} ⟶ f𝒞 𝒲 _{k}, for k = π_{0}R. The induced map of K-theory can be thought of as “discretization” and factors as KR ⟶ KHk ⟶ K^{f}k.

Remark 4.5. Another question one may ask is how the K-theory of k compares with the K-theory of k regarded as an A_{∞} ring, i.e. the K-theory of the category of finite cell A_{∞} k-modules (as constructed in [34]). In fact, Propositions 4.1 and 4.2 have exact analogs in the theory of discrete A_{∞} rings (with close analogs of the proofs). In particular, it follows from Corollary 2.7 that the natural quasiisomorphism of the ring k with its A_{∞} enveloping algebra induces a homotopy equivalence from the free K-theory of k to its A_{∞}-K-theory.

5. Morita equivalence

Next, we discuss Morita equivalence, the relationship of the category of Rmodules to the category of modules over (the analogue of) a matrix ring of R. We introduce the shorthand notation ∨_{n}X for ∨_{i=1}^{n}X, and we (temporarily) define M_{n}R = F_{R}(∨_{n}R, ∨_{n}R), M_{n1} = F_{R}(R, ∨_{n}R) ≅ ∨_{n}R, and M_{1n} = F_{R}(∨_{n}R, R) ≅ ∏_{n} R. By III.6.12, we see that M_{n}R is an S-algebra, M_{n1} an (M_{n}R, R)-bimodule, and M_{1n} an (R, M_{n}R)-bimodule. Classical Morita equivalence is the theorem that for a (discrete) ring R, tensoring with these two bimodules gives an equivalence between the category of R-modules and the category of M_{n}R-modules. The observation that this restricts to an equivalence between the categories of finitely generated projective modules proves that Quillen’s algebraic K-theory is Morita invariant.

In the case we consider, it is unreasonable to hope for an equivalence between ℳ_{R} and ℳ_{M_{n}R} since products and coproducts are not isomorphic, but we can ask for an equivalence of 𝒟_{R} and 𝒟_{M_{n}R}. Furthermore, because our K-theory is really the K-theory of free modules, we cannot expect the induced map of K-theory to give an isomorphism on K_{0} in general (since for a (non-zero) discrete ring, the image of the free module of rank one is a projective but not free module for n > 1), but we can ask for an isomorphism of the higher K-groups. In this section we find affirmative answers to each of these questions in the following theorems.

Theorem 5.1. (Morita Equivalence) The derived functors of M_{n1} ∧_{R} (?) and M_{1n} ∧_{M_{n}R} (?) give an equivalence of categories 𝒟_{R} ≃ 𝒟_{M_{n}R}, which restricts to an equivalence of the derived categories of semi-finite objects.

Theorem 5.2. (Morita Invariance of K-Theory) The functor M_{1n} ∧_{M_{n}R} (?) induces a map of K-theory KM_{n}R ⟶ KR, which on homotopy groups (K-groups) sends a generator in dimension zero to n times a generator, and gives an isomorphism on the higher groups.

We prove Theorem 5.1 by imitating as much as possible the proof of classical Morita equivalence. The following lemma gives a good start in this direction.

Lemma 5.3. The (R, R)-bimodules R and M_{1n} ∧_{M_{n}R} M_{n1} are isomorphic.

Proof. By a comparison of colimits, using the map of S-algebras R ⟶ M_{n}R, it is not hard to see that the following diagram is a coequalizer (cf. VII.1.9)

\xymatrix{M_{1n} ∧_{R} M_{n}R ∧_{R} M_{n1} \ar[r] \ar[r] & M_{1n} ∧_{R} M_{n1} \ar[r] & M_{1n} ∧_{M_{n}R} M_{n1}.}

The evaluation map M_{1n} ∧_{R} M_{n1} ⟶ R coequalizes this diagram, so induces a map M_{1n} ∧_{M_{n}R} M_{n1} ⟶ R, which is evidently an (R, R)-bimodule map. We show that this is an isomorphism by observing that

\xymatrix{M_{1n} ∧_{R} M_{n}R ∧_{R} M_{n1} \ar[r] \ar[r] & M_{1n} ∧_{R} M_{n1} \ar[r] & R}

is a split coequalizer of S-modules. The splitting is given by maps analogous to those in the discrete case: The map R ≅ R ∧_{R} R ⟶ M_{1n} ∧_{R} M_{n1} is the smash over R of the map R ⟶ ∨_{n}R that includes it as the first wedge-summand with the map R ⟶ F_{R}(∨_{n}R, R) induced by the map ∨_{n}R ⟶ R which collapses onto the first summand. The map

M_{1n} ∧_{R} M_{n1} ≅ M_{1n} ∧_{R} M_{n1} ∧_{R} R ⟶ M_{1n} ∧_{R} M_{n}R ∧_{R} M_{n1}

is induced by the identity on M_{1n} smashed over R with the map M_{n1} ⟶ M_{n}R (F_{R}(R, ∨_{n}R) ⟶ F_{R}(∨_{n}R, ∨_{n}R)) induced by the map ∨_{n}R ⟶ R collapsing onto the first summand in the first variable smashed over R with the inclusion of R as the first summand in M_{n1}. It is straight-forward to verify that the composites are as required to split the diagram.

Proof of Theorem 5.1. We verify that the composite 𝒟_{R} ⟶ 𝒟_{M_{n}R} ⟶ 𝒟_{R} is naturally isomorphic to the identity. Let X be a cellular R-module, and let Y be a cellular M_{n}R-module approximation to M_{n1} ∧_{R} X; we must show that the map

M_{1n} ∧_{M_{n}R} Y ⟶ M_{1n} ∧_{M_{n}R} M_{n1} ∧_{R} X ≅ X

is a weak equivalence. Observe that the obvious map ∨_{n}M_{n1} ⟶ M_{n}R is a weak equivalence and a map of (M_{n}R, R)-bimodules. Since X is a cellular R-module, the map ∨_{n}M_{n1} ∧_{R} X ⟶ M_{n}R ∧_{R} X is a weak equivalence and the composite map ∨_{n}Y ⟶ M_{n}R ∧_{R} X is a homotopy equivalence. Now we conclude that M_{1n} ∧_{M_{n}R} ∨_{n}Y ⟶ ∨_{n}X must be a weak equivalence, since the map ∨_{n}X ⟶ (∏_{n} R) ∧_{R} X ≅ M_{1n} ∧_{R} X is, but the induced map on homotopy groups is just the direct sum of n copies of the map we are interested in, so this map must also be a weak equivalence.

The reverse composite 𝒟_{M_{n}R} ⟶ 𝒟_{R} ⟶ 𝒟_{M_{n}R} is simpler. Let X be a cellular M_{n}R-module. Since M_{n1} ∧_{R} (?) preserves weak equivalences, the composite functor can be represented by X ↦ M_{n1} ∧_{R} M_{1n} ∧_{M_{n}R} X. Observe that the evaluation map M_{n1} ∧_{R} M_{1n} ⟶ M_{n}R is a weak equivalence of (M_{n}R, M_{n}R)-bimodules. On the underlying R-modules it is a map ∨_{n} ∏_{n} R ⟶ ∏_{n} ∨_{n}R inducing an isomorphsim ⊕ ∏ R_{∗} ⟶ ∏ ⊕R_{∗}. This induces the natural isomorphism (in 𝒟_{M_{n}R}) to the identity.

Since the derived categories of semi-finite objects are full subcategories of the derived categories 𝒟_{R} and 𝒟_{M_{n}R}, we see that this equivalence restricts, since both functors send wedge summands of finite objects to wedge summands of finite objects.

Let 𝒞 be the smallest standard Waldhausen category with standard cylinder functor containing f𝒞_{R} and the image of 𝒞_{M_{n}R}. By Proposition 3.5, the K-theory of 𝒞 is homotopy equivalent to KR. Let ℐ be the full subcategory of 𝒞 of objects weakly equivalent to objects in the image of f𝒞_{M_{n}R}. Since pushouts along cofibrations are homotopy equivalent to homotopy pushouts, which M_{1n} ∧_{M_{n}R} (?) preserves, it is easy to check that ℐ is closed under pushouts along cofibrations and therefore is a standard Waldhausen category with standard cylinder functor; moreover, the functor M_{1n} ∧_{M_{n}R} (?): f𝒞_{M_{n}R} ⟶ ℐ is exact.

Lemma 5.4. The exact functor M_{1n} ∧_{M_{n}R} (?): f𝒞_{M_{n}R} ⟶ ℐ induces a homotopy equivalence of K-theory.

Proof. We apply Theorem 2.5: for A ∈ f𝒞_{M_{n}R}, B ∈ ℐ, and f: M_{1n} ∧_{M_{n}R} A ⟶ B, we find X ∈ f𝒞_{M_{n}R}, a weak equivalence e: M_{1n} ∧_{M_{n}R} X ⟶ B and a: A ⟶ X, such that e∘M_{1n}∧_{M_{n}R}a is homotopic to f. By assumption, B is weakly equivalent to the image of some X ∈ f𝒞_{M_{n}R}, so M_{n1}∧_{R}B is an M_{n}R-module weakly equivalent to M_{n1} ∧_{R} M_{1n} ∧_{M_{n}R} X, which in turn is weakly equivalent to X. Thus, by the Whitehead Theorem, there exists a weak equivalence ϵ: X ⟶ M_{n1} ∧_{R} B. Since the natural map i: M_{n1} ∧_{R} M_{1n} ∧_{M_{n}R} A ⟶ M_{n}R ∧_{M_{n}R} A ≅ A is a weak equivalence, it has a homotopy retraction r, and there exists a map a: A ⟶ X such that ϵ ∘ a is homotopic to (M_{n1} ∧_{R} f) ∘ r, again by the Whitehead theorem. Thus the solid line part of the following diagram commutes up to homotopy.

[figure: M_{n1} ∧_{R} M_{1n} ∧_{M_{n}R} A; i; M_{n1}∧f; A; r; a; M_{n1} ∧_{R} B.; X; ≃; ϵ]

We apply the functor M_{1n} ∧_{M_{n}R} (?). The isomorphism constructed in Lemma 5.3 induces a natural transformation μ: id ⟶ M_{1n} ∧_{M_{n}R} M_{n1} ∧_{R} (?), from which we get the diagram

[figure: M_{1n} ∧_{M_{n}R} A; (M_{1n}∧i)∘μ; f; M_{1n} ∧_{M_{n}R} A; M_{1n}∧r; M_{1n}∧a; B.; M_{1n} ∧_{M_{n}R} X; ≃; μ^{−1}∘(M_{1n}∧ϵ)]

By the associativity of the multiplication pairing, the diagram

[figure: M_{1n} ∧_{M_{n}R} M_{n1} ∧_{R} M_{1n}; M_{1n}∧i; ≅; R ∧_{R} M_{1n}; ≅; M_{1n} ∧_{M_{n}R} M_{n}R; ≅; M_{1n}]

must commute, and we conclude that (M_{1n} ∧_{M_{n}R} i) ∘ μ is the identity. Now, letting e = μ^{−1} ∘M_{1n} ∧_{M_{n}R} ϵ, we see that e∘M_{1n} ∧_{M_{n}R} a is homotopic to f as required.

Lemma 5.5. ℐ is closed under extensions in 𝒞.

Proof. We need to show that for a cofibration sequence A ↣ B ↠ C in 𝒞, if A and C are in ℐ, then B is also in ℐ. It suffices to consider the case when A, B, and C are celluar R-modules, since any cofibration sequence can be replaced by a weakly equivalent one of this form. Using the proof of the last lemma, the map ∗ ⟶ C allows us to find X in f𝒞_{M_{n}R} and a weak equivalence e: M_{1n} ∧_{M_{n}R} X ⟶ C. Composing with the map c: C ⟶ ΣA implied by the cofibration sequence, and applying once again the proof of the last lemma, we find an object Y in f𝒞_{M_{n}R}, a map a: X ⟶ Y, and a weak equivalence f: M_{1n} ∧_{M_{n}R} Y ⟶ ΣA that make the following square homotopy commute:

[figure: M_{1n} ∧_{M_{n}R} X; M_{1n}∧a; e; ≃; M_{1n} ∧_{M_{n}R} Y; f; ≃; C; c; ΣA.]

We conclude that the induced map on cofibers C(M_{1n} ∧_{M_{n}R} a) ⟶ C(c) is a weak equivalence. The functor M_{1n} ∧_{M_{n}R} (?) commutes with smashing over S on the right with S_{S}^{−1} and smashing on the right with the space S^{1}, both composites of which are homotopic to the identity in f𝒞_{M_{n}R}. We conclude that the map C(M_{1n} ∧_{M_{n}R} a) ∧_{S} S_{S}^{−1} ⟶ C(c) ∧_{S} S_{S}^{−1} is a weak equivalence. But C(c) ∧_{S} S_{S}^{−1} is homotopy equivalent to B and C(M_{1n} ∧_{M_{n}R} a) ∧_{S} S_{S}^{−1} ≅ M_{1n} ∧_{M_{n}R} (C(a) ∧_{S} S_{S}^{−1}) is in the image of f𝒞_{M_{n}R}, hence B is in ℐ.

Proof of Theorem 5.2. The inclusion ℐ ⟶ 𝒞 is an exact functor. It is easy to see that on K_{0} it sends a generator (S_{M_{n}R}) to n times a generator (M_{1n} ∧_{M_{n}R} M_{n}R ∧_{S} S_{S} ≅ M_{1n} ∧_{S} S_{S} ≃ ∨_{n}S_{R}). We want to see that it induces an isomorphism of the higher K-groups. Let 𝒥 be the full subcategory of 𝒞 of all objects whose class in K_{0}𝒞 is in the image of K_{0}ℐ (so in particular ℐ ⊂ 𝒥). It follows from the relations that define K_{0} that 𝒥 inherits the structure of a standard Waldhausen category with standard cylinder functor. By Proposition 3.4, the inclusions ℐ ⟶ 𝒥 and 𝒥 ⟶ 𝒞 are exact functors.

We use the argument of [23, §1] to show that ℐ is strictly cofinal in 𝒥 (in the sense of [68, 1.5.9]). We define an equivalence relation on the objects of 𝒞 by letting A and A′ be equivalent if there exists some X ∈ ℐ such that A∨X is weakly equivalent to A′ ∨ X. Let G be the set of equivalence classes under this relation. Then G is a group under the operation “∨” with the inverse of A represented by ∨_{n−1}A. We have an obvious homomorphism G ⟶ K_{0}𝒞 /K_{0}ℐ; we construct an inverse to this homomorphism. If A ↣ B ↠ C is a cofibration sequence in 𝒞, then ∨_{n}A ↣ B ∨ (∨_{n−1}A) ∨ (∨_{n−1}C) ↠ ∨_{n}C is a cofibration sequence. But ∨_{n}A and ∨_{n}C are in ℐ, so B ∨ (∨_{n−1}A) ∨ (∨_{n−1}C) is in ℐ since ℐ is closed under extensions in 𝒞; therefore, B ∨(∨_{n−1}A)∨(∨_{n−1}C) represents the identity in G and hence B represents the same element as A ∨ C in G. If A is weakly equivalent to A′ then they represent the same element in G. We see that G satisfies the universal relations that define K_{0}𝒞, and so specifies a map K_{0}𝒞 ⟶ G. This map clearly factors through a map K_{0}𝒞 /K_{0}ℐ ⟶ G that is evidently inverse to the map above. Now we see that 𝒥 consists of the objects whose class in G is the identity, so we conclude that for any X ∈ 𝒥, there exists Y ∈ ℐ so that X ∨ Y is weakly equivalent to an object of ℐ and hence X ∨ Y is an object of ℐ.

Now by [68, 1.5.9], ℐ ⟶ 𝒥 induces a homotopy equivalence of K-theory, but by [65, 1.10.1], K_{i}𝒥 ⟶ K_{i}𝒞 is an isomorphism for i > 0.

6. Multiplicative structure in the commutative case

In this section, we prove the following theorem (cf. [62]).

Theorem 6.1. If R is a connective commutative S-algebra then KR is homotopy equivalent to an E_{∞} ring spectrum and therefore weakly equivalent to a commutative S-algebra.

This result and the results of the next section depend on the following technical lemma, which the reader may recognize as a simple application of the theory of [68, §1.6–1.8] to our new categories. Although we believe that Theorem 6.1 may generalize to non-connective commutative S-algebras, this lemma is peculiar to the connective case and relies on the existence of the ordinary homology theories of IV.3. For this lemma, we need R to be a connective but not necessarily commutative S-algebra. Let 𝒞 be a standard Waldhausen category of R-modules that contains f𝒞_{R} and that only contains objects of the weak homotopy type of finite cell R-modules. We see by Proposition 3.5 that K𝒞 is homotopy equivalent to KR. We denote by 𝒞^{m} the full subcategory of objects weakly equivalent to a finite wedge of S_{R}^{m}. Observe that for each m, the category of weak equivalences of 𝒞^{m} is a symmetric monoidal category under the operation of wedge, and denote the associated spectrum as k𝒞^{m}. Suspension induces a system of maps of spectra k𝒞^{m} ⟶ k𝒞^{m+1}.

Lemma 6.2. The homotopy colimit of the system {k𝒞^{m}} is homotopy equivalent to K𝒞.

Proof. The Hurewicz theorem IV.3.6 allows us to identify 𝒞^{m} with the full subcategory of 𝒞 of objects whose ordinary homology H_{∗}^{R} is zero in all dimensions except m, and in dimension m is a finitely generated free module. Let 𝒞̃^{m} be the full subcategory of 𝒞 of objects whose ordinary homology H_{∗}^{R} is zero in all dimensions except m and in dimension m is a finitely generated stably free module, i.e. is isomorphic to the kernel of a surjective map of finitely generated free modules. Let 𝒞^{≥n} be the full subcategory of 𝒞 of those objects which are (n − 1)-connected. By the Hurewicz theorem IV.3.6 these are exactly the objects whose homology is zero in dimensions less than n. We give the categories 𝒞^{m} and 𝒞̃^{m} Waldhausen structures by defining the w-cofibrations to be the w-cofibrations of 𝒞 whose quotients lie in the subcategory in question. The categories 𝒞^{≥n} have the structure of standard Waldhausen categories with the standard cylinder functor.

Suspension is an exact functor 𝒞̃^{m} ⟶ 𝒞̃^{m+1} and 𝒞^{≥n} ⟶ 𝒞^{≥n}. The inclusion of 𝒞̃^{m} in 𝒞^{≥n} is an exact functor, and induces a map

hocolim_{m→∞} |wS_{•}𝒞̃^{m} | ⟶ hocolim_{Σ} |wS_{•}𝒞^{≥n} |

for each n. Next observe that ordinary homology H_{∗}^{R} restricted to 𝒞^{≥n} is a homology theory in the sense of [68, §1.7] (at least after shifting the indexing), and that the categories 𝒞̃^{m} form categories of “spherical objects” for 𝒞^{≥n} for the class of finitely generated stably free modules. Since this theory satisfies the “Hypothesis” of [68, 1.7.1], we conclude that the map above is a weak equivalence. On the other hand the inclusions 𝒞^{≥n} ⟶ 𝒞^{≥n+1} are exact functors which induce cofibrations |wS_{•}𝒞^{≥n} | ⟶ |wS_{•}𝒞^{≥n+1} |, whose colimit is |wS_{•}𝒞 |. Taking the colimit (over n) of the homotopy equivalence above, we get a homotopy equivalence

hocolim_{m→∞} |wS_{•}𝒞̃^{m} | ⟶ hocolim_{Σ} |wS_{•}𝒞 |.

The maps on the right are all homotopy equivalences (by [68, 1.6.2]), so we conclude that there exists a homotopy equivalence hocolim K𝒞̃^{m} ⟶ K𝒞.

We apply the Strict Cofinality Theorem [68, 1.5.9] to conclude that K𝒞̃^{m} is homotopy equivalent to K𝒞^{m}. Now we are reduced to comparing K𝒞^{m} with k𝒞^{m}. According to [68, 1.8.1], it suffices to observe that cofibrations in 𝒞^{m} are “splittable up to weak equivalence”. Given a cofibration A ↣ B, we can find a basis of the free module H_{m}(B) that represents the union of bases for H_{m}A and H_{m}(B/A). The Hurewicz theorem IV.3.6 now specifies a homotopy class of weak equivalence from the wedge of A and a wedge of spheres to both B and A ∨ B/A, relative to the maps from A.

Proof of Theorem 6.1 Let 𝒞 be the smallest standard Waldhausen category with standard cylinder functor containing R, S_{R}^{n} (all n) and all finite smash products over R of these. It is easy to check that the bifunctor (?)∧_{R}(?) restricts to 𝒞 (up to equivalence), so 𝒞 is a symmetric bimonoidal category under coproduct and smash product over R. Let 𝒞^{0} be as in the lemma above. Then 𝒞 is the full subcategory of 𝒞 of objects weakly equivalent to a finite wedge of S_{R}. Since smash product over R with R and with S_{R}^{n} preserve weak equivalences, so do smash products over R with any object of 𝒞, and the smash product over R of objects in 𝒞^{0} is weakly equivalent to a finite wedge of S_{R} and therefore is an object of 𝒞^{0}. Thus the smash product over R restricts to a bifunctor on 𝒞^{0} that makes 𝒞^{0} a symmetric bimonoidal category. By the work of [49], we can construct k𝒞^{0} functorially as an E_{∞} ring spectrum.

Next observe that suspension and S_{R}^{−1} ∧_{R} (?) give functors 𝒞^{m} ⟶ 𝒞^{m+1} and 𝒞^{m+1} ⟶ 𝒞^{m} for which both composites are weakly equivalent to the identity. We conclude that suspension gives a homotopy equivalence k𝒞^{m} ⟶ k𝒞^{m+1}, and that k𝒞^{0} is homotopy equivalent to K𝒞 by the previous lemma.

7. The plus construction description of KR

We have observed that the category f𝒞_{R} gives a K-theory Kf𝒞_{R} that has some right to be called the algebraic K-theory of R. This section is devoted to a comparison with another possible definition, based on Quillen’s plus construction. In what follows, R is a fixed connective S-algebra, and k = π_{0}R. We shall make use of classifying spaces of the topological monoids ℳ_{R}(X, X). Unfortunately even when X = S_{R}, we cannot guarantee that the inclusion of the identity element is a cofibration. There are well-known ways of overcoming this difficulty, i.e. whiskering the monoids [44] or using thickened realizations [63]. In this and the next section, we shall take advantage of such techniques implicitly wherever necessary without further comment.

7. THE PLUS CONSTRUCTION DESCRIPTION OF KR 137

Let M̃_{n}R be the topological space f𝒞_{R}(⋁_{n} S_{R}, ⋁_{n} S_{R}); then π_{0}M̃_{n}R ≅ M_{n}(k), the ordinary matrix ring of the ring k. Let G̃L_{n}R be the space consisting of those connected components of M̃_{n}R whose image in M_{n}(k) is invertible. Then G̃L_{n}R is a topological monoid; indeed, it is the monoid of homotopy equivalences in M̃_{n}R. We can consider its classifying space BG̃L_{n}R. We have the inclusion i_{n}: G̃L_{n}R ⟶ G̃L_{n+1}R obtained by sending the last wedge summand to the last wedge summand via the identity map, and it induces Bi_{n}: BG̃L_{n}R ⟶ BG̃L_{n+1}R. Let BG̃LR be the telescope of these maps.

Now π_{1}BG̃LR ≅ GL(k) has a perfect normal subgroup, so we can form BG̃LR^{+} (Quillen’s plus construction). We shall see in a moment that K_{0}^{f}k × BG̃LR^{+} is an infinite loop space. Define K^{+}R to be the connective spectrum obtained by delooping K_{0}^{f}k × BG̃LR^{+}. We prove the following “plus equals S_{•}” theorem.

Theorem 7.1. K^{+}R is weakly equivalent to KR.

First we need to specify the infinite loop space structure on K_{0}^{f}k ×BG̃LR^{+}. For this, we observe that K_{0}^{f}k×BG̃LR^{+} is the group completion of the classifying space of the topological category 𝒲 whose (discrete) set of objects is the finite wedges of S_{R} and whose space of morphisms is the set of homotopy equivalences topologized as a subspace of the space of morphisms of ℳ_{R}. Call this group completion B. In the case when K_{0}^{f}k is the integers, the classifying space of 𝒲 is the disjoint union of the BG̃L_{n}R and we may apply the remarks of [63, §4] to conclude that we have a homology isomorphism to B from the telescope of maps ∐ BG̃L_{n}R to itself induced by the maps Bi_{n}: BG̃L_{n}R ⟶ BG̃L_{n+1}R. This telescope is easily seen to be K_{0}^{f}k × hocolim_{n} BG̃L_{n}R. We conclude that B ≃ K_{0}^{f}k × (hocolim_{n} BG̃L_{n}R)^{+}.

In the pathological case when K_{0}^{f}k is not the integers, i.e. when there exists a homotopy equivalence ∨_{j}S_{R} ≃ ∨_{k}S_{R} for j ≠ k, we still have a homology isomorphism to the group completion B from the telescope T of maps from B𝒲 to itself induced by addition of an identity map on the wedges of sphere R-modules. Proposition 7.2 below allows us to see that B𝒲 is homotopy equivalent to a disjoint union of of some of the BG̃L_{n}R, one choice for each isomorphism class of finitely generated free π_{0}R-modules. Now we see that the telescope T is homotopy equivalent to K_{0}^{f}k × hocolim_{n} BG̃L_{n}R, and we conclude that B ≃ K_{0}^{f}k × (hocolim_{n} BG̃L_{n}R)^{+}.

To identify the homotopy type of B𝒲 in the pathological case above, we need the following proposition. We will need a similar result again later, and we have written this proposition in the minimal possible generality necessary to handle both cases. The proposition says essentially that if the morphisms in a category are all homotopy equivalences (in a certain sense), then the classifying space of the monoid of endomorphisms of any object is homotopy equivalent to its connected component in the classifying space of the category. Because this proposition has obvious generalizations with more general scope than its use in this section, we break our rule of not mentioning the necessary cofibration assumptions. As always the reader has the choice of deleting the cofibration assumption by using a whiskering technique or employing the thickened realization.

Proposition 7.2. (cf. [68, 2.2.7]) Let 𝒞 be a topological category with discrete set of objects such that the identity morphism (from objects to morphisms) is a cofibration. Let X be an object of 𝒞 and denote by 𝒞_{X} the full subcategory of 𝒞 containing X. Suppose that for each morphism f: Y ⟶ Z in 𝒞, there is some f′: Z ⟶ Y so that f′ ∘ f and f ∘ f′ each lie in the same path component of 𝒞 (Y, Y) and 𝒞 (Z, Z) as the respective identity elements. Then the inclusion 𝒞_{X} ⟶ 𝒞 induces a homotopy equivalence of the classifying space of 𝒞_{X} with the connected component of its image in the classifying space of the category 𝒞.

Proof. First observe that Quillen’s “Theorem A” [55] holds with essentially the same proof for continuous functors between topological categories with discrete object sets whose identity map (objects to morphisms) is a cofibration.

Since the connected component of the image of 𝒞_{X} in the classifying space of 𝒞 is the classifying space of the connected component (as a graph) of 𝒞 that contains X, we can reduce to this smaller category and assume without loss of generality that 𝒞 is connected (as a graph). Applying Quillen’s Theorem A (dual formulation), we are reduced to showing that for every Y in 𝒞, the topological category 𝒞_{X}/Y is contractible. But if f: Y ⟶ Z is morphism in 𝒞, then we have f′: Z ⟶ Y and paths γ: f′ ∘f ⇝ 1_{Y} and γ′: f ∘f′ ⇝ 1_{Z}. We can interpret the morphisms f and f′ as continuous functors 𝒞_{X}/Y ⟶ 𝒞_{X}/Z, 𝒞_{X}/Z ⟶ 𝒞_{X}/Y, and the paths γ and γ′ as continuous functors 𝒞_{X}/Y × I ⟶ 𝒞_{X}/Y, 𝒞_{X}/Z × I ⟶ 𝒞_{X}/Z. Passing to the classifying spaces we see that the paths represent homotopies B(𝒞_{X}/Y) × I ⟶ B(𝒞_{X}/Y) and B(𝒞_{X}/Z) × I ⟶ B(𝒞_{X}/Z) from the compositions Bf′ ∘ Bf and Bf ∘ Bf′ to the repective identities. In short, 𝒞_{X}/Y and 𝒞_{X}/Z are homotopy equivalent. Since we have reduced to the case when 𝒞 is connected (as a graph), we see that 𝒞_{X}/Y is homotopy equivalent to 𝒞_{X}/X. The lemma is established by the observation that 𝒞_{X}/X has a final object and therefore is contractible.

We begin to compare K^{+}R to Kf𝒞_{R}. One obvious obstacle is that we have defined K^{+}R in terms of a topological category and Kf𝒞_{R} in terms of a discrete one. Let w𝒞^{0} denote the (discrete) full subcategory of w(f𝒞_{R}) whose objects are homotopy equivalent to wedges of S_{R}; the set of morphisms is the set of homotopy equivalences. Using arguments similar to [68, 2.2], we relate w𝒞^{0} to both 𝒲 and wS_{•}f𝒞_{R}.

Lemma 7.3. (cf. [68, 2.2.5]) There is a chain of weak equivalences relating the classifying spaces of the categories 𝒲 and w𝒞^{0}. Each map in the chain is a map of E_{∞} spaces.

Proof. For each k, let 𝒲 Δ_{k} be the (discrete) category whose objects are the objects of w𝒞^{0} and whose morphisms 𝒲 Δ_{k}(X, Y) consist of the set of continuous maps Δ[k] ⟶ f𝒞_{R}(X, Y) whose image lands in the component of a weak equivalence, where Δ[k] denotes the standard topological k-simplex. In light of the adjunction 𝒯 (Δ[k]_{+}, f𝒞_{R}(X, Y)) ≅ f𝒞_{R}(X ∧ Δ[k]_{+}, Y), we see that this is the same as the set of weak equivalences X ∧ Δ[k]_{+} ⟶ Y. This is a simplicial category. Let N_{j,k} be the nerve of this category.

If we realize N_{j,k} in the k direction, we obtain a simplicial set that is the nerve of a topological category with a discrete set of objects. We denote this category as |𝒲 Δ |. In particular, the objects of |𝒲 Δ | are the objects of w𝒞^{0} and the morphism space |𝒲 Δ |(X, Y) is the geometric realization of the total singular complex of the subspace of f𝒞_{R}(X, Y) consisting of those components which contain homotopy equivalences. For each X ∈ 𝒲 , let |𝒲 Δ |_{X} be the full subcategory of |𝒲 Δ | consisting of the single object X. By the previous proposition, the inclusion |𝒲 Δ |_{X} ⟶ |𝒲 Δ | induces a homotopy equivalence from the classifying space of |𝒲 Δ |_{X} to its connected component in the classifying space of |𝒲 Δ |. On the other hand we have a natural weak equivalence of monoids |𝒲 Δ |(X, X) ⟶ 𝒲 (X, X), giving a weak equivalence of their classifying spaces. Let |𝒲 ΔS_{R} | be the full subcategory of |𝒲 Δ | consisting of the finite wedges of S_{R}. Then we have weak

equivalences ∥N_{j,k}∥ ⟵^{∼} ∥𝒲 ΔS_{R} ∥ ⟶^{∼} |𝒲 |.

Next we produce a weak equivalence between w𝒞^{0} and 𝒲 Δ_{∗}. The map Δ[k] ⟶ S^{0} induces a functor F: w𝒞^{0} ⟶ 𝒲 Δ_{k} that is the identity on objects. Let G: 𝒲 Δ_{k} ⟶ w𝒞^{0} be the functor induced by the map S^{0} ⟶ Δ[k]_{+} that sends the non-basepoint to the zeroth vertex of Δ[k]. Then GF is the identity functor on w𝒞^{0}. We show that F G is homotopic to the identity. Let H: 𝒲 Δ_{k} ⟶ 𝒲 Δ_{k} be the functor that takes X to X ∧ I_{+} and that on morphisms is induced by a map I × Δ[k] ⟶ Δ[k] that is the identity on the bottom face and sends the whole top face to the zeroth vertex. There are obvious natural transformations id ⟶ H and F G ⟶ H given by the inclusion of bottom face and the inclusion of top face, from which we conclude that F G is homotopic to the identity. We may regard w𝒞^{0} as a simplicial category constant in the k direction. The functors F are compatible with the faces and degeneracies (in k), and therefore assemble to a simplicial functor w𝒞^{0} ⟶ 𝒲 Δ_{∗} that induces a homotopy equivalence upon passage to classifying spaces.

It is easy to see that the simplicial maps above realize to maps of E_{∞} spaces as they are induced by functors that preserve wedges.

Proof of Theorem 7.1 If we let 𝒞 be the category f𝒞_{R}, then w𝒞^{0} is exactly the subcategory of weak equivalences of the category 𝒞^{0} defined above Lemma 6.2, the associated spectrum of which we denoted k𝒞^{0}. Again suspension and S_{R}^{−1} ∧_{R} (?) give functors 𝒞^{m} ⟶ 𝒞^{m+1} and 𝒞^{m+1} ⟶ 𝒞^{m} whose composites are weakly equivalent to the identity. We conclude that the maps in the homotopy colimit are homotopy equivalences and that k𝒞^{0} is homotopy equivalent to KR. On the other hand, the previous proposition shows that K^{+}R is weakly equivalent to k𝒞^{0}.

Remark 7.4. Note that we only needed the connectivity hypothesis to show the relationship between k𝒞^{0} and Kf𝒞_{R}. More generally we do have a homotopy equivalence k𝒞^{0} ≃ K^{+}R (the spectrum whose zeroth space is K_{0}^{f}k × BG̃LR^{+}), but there is no reason to expect that the map k𝒞^{0} ⟶ Kf𝒞_{R} will be a homotopy equivalence. In particular k𝒞^{0} cannot see any relationships between spheres of different dimensions. For example, if K_{0}^{f}k = Z, but S_{R} ≃ S_{R}^{1}, then |wS_{•}f𝒞_{R} | is contractible but |wN_{•}𝒞^{0} | is not.

Remark 7.5. We should also observe that this allows another interpretation of the discretization map: π_{0} applied to the simplicial space N𝒲 gives an E_{∞} map KR(0) ≃ K_{0}^{f}k ×BG̃LR^{+} ⟶ K_{0}^{f}k ×BGLk^{+} ≃ K^{f}k(0), which evidently coincides with the discretization map and is a weak equivalence in the case when R = Hk.

Remark 7.6. (Monomial Matrices) Let 𝒱 be the subcategory of 𝒲 of those maps ∨_{n}S_{R} ⟶ ∨_{n}S_{R} that are wedges of n maps S_{R} ⟶ S_{R} in any order. Thinking of 𝒲 (∨_{n}S_{R}, ∨_{n}S_{R}) as analogous to GL_{n}R, then 𝒱 (∨_{n}S_{R}, ∨_{n}S_{R}) is analogous to the subgroup of monomial matrices, those matrices with a single non-zero entry in each row and column. Let R^{×} denote the monoid 𝒱 (S_{R}, S_{R}) = 𝒲 (S_{R}, S_{R}). Then 𝒱 (∨_{n}S_{R}, ∨_{n}S_{R}) is isomorphic to the monoid Σ_{n} ∫ R^{×} and the classifying space of 𝒱 is isomorphic to the disjoint union of the classifying spaces of these monoids; moreover, under this isomorphism the E_{∞} space structure induced by wedge sums becomes the E_{∞} space structure induced by block sums. We conclude that the group completion of the classifying space of 𝒱 is homotopic to QBR^{×}_{+}, and that 𝒱 ⟶ 𝒲 induces a map of spectra Σ^{∞}BR^{×}_{+} ⟶ KR.

Remark 7.7. (Naturality) Let A ⟶ B be a map of S-algebras. We saw in Propostion 4.1 that the functor B∧_{A}(?) induces a map of K-theory spectra KA ⟶ KB. This also restricts to a continuous functor of topological categories 𝒲_{A} ⟶ 𝒲_{B} that induces a map of the plus contruction spectra above. We conclude that these two maps represent the same map in the stable category, since this functor commutes up to natural isomorphism with the functors used in comparing K^{+} with K.

8. Comparison with Waldhausen’s K-theory of spaces

Now we compare the new algebraic K-theory with Waldhausen’s algebraic Ktheory of spaces. For this, let X be a connected pointed topological space, and let G = |GSX|, the geometric realization of the Kan loop group of the based singular complex of X. This is a topological group with non-degenerate identity. We let R = Σ^{∞}(G_{+}) (where the plus subscript is union with a disjoint basepoint) and we note that R is an S-algebra (IV.7.8) with k = π_{0}R = Z[π_{0}G].

Definition 8.1. Let H_{n}^{m} denote the topological monoid of pointed G-equivari-ant homotopy equivalences of ⋁_{n} Σ^{m}G_{+} with itself, and let BH_{n}^{m} denote its classifying space. We have monoid maps H_{n}^{m} ⟶ H_{n}^{m+1}, and H_{n}^{m} ⟶ H_{n+1}^{m} which are induced by suspension and by addition of an identity map on the last wedge summand and which are cofibrations. The algebraic K-theory of the space X is defined to be the space A(X) = K_{0}^{f}Z[π_{0}G] × (colim_{m,n} BH_{n}^{m})^{+}. This is obviously equivalent to Waldhausen’s definition [68, 2.2.1]. We shall also use the symbol A(X) to denote the spectrum associated to its delooping, and under this interpretation we will prove the following result.

Theorem 8.2. The spectra KΣ^{∞}G_{+} and A(X) are homotopy equivalent, naturally in X.

Observe that the functors Σ_{m}^{∞} give maps of topological monoids

H_{n}^{m} ⟶ 𝒮 (⋁_{n} Σ^{∞}G_{+}, ⋁_{n} Σ^{∞}G_{+})

which are easily seen to be compatible with suspension and addition of an identity map. Composing with the functors 𝕃 and S ∧_{ℒ} (?), we obtain maps of topological monoids

H_{n}^{m} ⟶ ℳ_{S}(⋁_{n} S ∧_{ℒ} 𝕃Σ^{∞}G_{+}, ⋁_{n} S ∧_{ℒ} 𝕃Σ^{∞}G_{+}).

We denote this composite functor by L_{n}^{m}. The observation that the functor G_{+}∧(?) is naturally isomorphic to the functor R ∧_{S} (?) immediately implies that L_{n}^{m} sends G-equivariant maps to R-module maps; therefore, we can interpret L_{n}^{m} as a map of topological monoids H_{n}^{m} ⟶ M̃_{n}R. Since for m ≥ 2, H_{n}^{m} consists of the subspace of those connected components of Map_{G}(⋁_{n} Σ^{m}G_{+}, ⋁_{n} Σ^{m}G_{+}) which π_{0} maps to GL_{n}(R), we see that L_{n}^{m} restricts to a map of monoids H_{n}^{m} ⟶ G̃L_{n}R. We will show that in the colimit this map is a homotopy equivalence.

Proposition 8.3. The map of topological monoids

L_{n}: colim_{m} H_{n}^{m} ⟶ G̃L_{n}R

is a homotopy equivalence.

Proof. We have defined L_{n}^{m} via a composition of functors so that it would be easy to see that it is a map of monoids; we rewrite this composition to make it easier to analyze homotopically.

Consider the map of spaces

f_{m}: 𝒯 (⋁_{n} S^{m}, ⋁_{n} Σ^{m}G_{+}) ⟶ ℳ_{S}(⋁_{n} S_{S}, ⋁_{n} S_{R})

(for fixed n) induced by the composite of the functors Σ_{m}^{∞}, 𝕃, and S ∧_{S} (?). The colimit of the f_{m} is the composite of the maps

colim_{m} 𝒯 (⋁_{n} S^{m}, ⋁_{n} Σ^{m}G_{+}) ⟶ 𝒮 (⋁_{n} S, ⋁_{n} Σ^{∞}G_{+})
⟶ 𝒮 [𝕃](⋁_{n} 𝕃S, ⋁_{n} 𝕃Σ^{∞}G_{+})
⟶ ℳ_{S}(⋁_{n} S_{S}, ⋁_{n} S_{R}),

each of which is a homotopy equivalence. Via the obvious isomorphisms, the map L_{n} agrees with the restriction of this map to the connected components that consist of weak equivalences, and so it is also a homotopy equivalence.

Since the inclusion of the identity in G is a cofibration, we see that induced map colim_{m} BH_{n}^{m} ⟶ BG̃L_{n}R is a homotopy equivalence, and hence the induced map on the plus constructions of the telescopes is a homotopy equivalence.

Proof of Theorem 8.1. We need to show that we have a map of spectra. But the infinite loop space structure on A(X) comes from the operation wedge on the colimit of the topological categories whose objects are finite wedges of Σ^{m}G_{+} (for each m) and whose maps are the H_{n}^{m}. The functors L_{n}^{m} assemble to a continuous functor from this colimit to the category 𝒲 which commutes with wedges and which coincides with the above homotopy equivalence on the plus constructions. We conclude that the map constructed above

K_{0}^{f}Z[π_{0}G] × (colim_{m,n} BH_{n}^{m})^{+} ⟶ K_{0}^{f}Z[π_{0}G] × BG̃LR^{+}

is a map of E_{∞} spaces.

Since G is a CW space, Σ^{∞}G_{+} is a CW spectrum, so M̃_{n}R, G̃L_{n}R, BG̃L_{n}R, BG̃LR, and BG̃LR^{+} have the homotopy type of CW spaces; therefore, the plus construction of the previous section produces a spectrum homotopy equivalent to KR. We conclude that the spectrum A(X) is homotopy equivalent to KR.

Remark 8.4. (Linearization) The map R ⟶ HZ ∧_{S} R is a map of S-algebras and a rational equivalence. The map

HZ ∧_{S} (?): 𝒲_{R}(∨_{n}S_{R}, ∨_{m}S_{R}) ⟶ 𝒲_{HZ∧_{S}R}(∨_{n}S_{HZ∧_{S}R}, ∨_{m}S_{HZ∧_{S}R})

induces an equivalence on rational homology. We conclude that the induced map KR ⟶ K(HZ ∧_{S} R) is a rational equivalence. A comparison of the categories of modules for the S-algebra HZ ∧_{S} R and the simplicial ring Z[GSX] would then

8. COMPARISON WITH WALDHAUSEN’S K-THEORY OF SPACES 143

give a linearization result. We save this and other observations along these lines for a future paper.

144 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS

CHAPTER VII R-algebras and topological model categories

In Chapter II, we set up the ground category of S-modules, and we developed the theory of S-algebras and their modules by exploiting the good formal properties of that category. In Chapter III, we set up a ground category of modules over a commutative S-algebra R that enjoys the same formal properties as the category of S-modules, and the previous three chapters gave applications of that theory. As we discuss in Section 1, we can go on to define R-algebras and their modules simply by changing ground categories from ℳ_{S} to ℳ_{R}.

At this point, we face a homotopical problem. We want to use point-set level constructions, such as bar constructions and constructions of topological Hochschild homology, that involve taking smash powers of a commutative R-algebra A. To make homotopical use of these constructions, we need to know that the underlying R-modules of these smash powers represent their smash powers in the derived category of R-modules. However, A need not have the homotopy type of a cell R-module, so we must approximate it by a weakly equivalent R-algebra with better properties. We first attacked this problem by use of the bar construction of Chapter XII, but we shall here deal with it by use of Quillen model categories.

Thus we shall prove that all of our various categories of A_{∞} and E_{∞} ring spectra, R-algebras, commutative R-algebras, and modules over any of these are complete and cocomplete, tensored and cotensored, topologically enriched categories that admit canonical (closed) model structures in the sense of Quillen [54]. Since cofibrations and fibrations in the classical sense are important in our theory, we shall use the terms q-cofibration and q-fibration for the model category concepts.

The proofs that our categories are so richly structured are almost entirely formal, and these formal structures do not solve or even address the motivating homotopical problem since forgetful functors need not preserve q-cofibrant homotopy types. However, we shall see that the problem can be solved by combining the formal theory with the homotopical analysis of the linear isometries operad.

146 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES

Much of the formal theory in this chapter is based on ideas and results originally due to Hopkins and McClure (in part in [31], but we have also benefited from many profitable conversations) or to McClure, Schwänzl and Vogt [52].

1. R-algebras and their modules

We fix a commutative S-algebra R and work in the symmetric monoidal category ℳ_{R} of R-modules.

Definition 1.1. An R-algebra is a monoid in ℳ_{R}. A commutative R-algebra is a commutative monoid in ℳ_{R}.

As in algebra, we obtain free R-algebras by “extension of scalars” from S to R. To show this, we use an alternative description of R-algebras and commutative R-algebras, which again is the same as in algebra. Say that a map η : R ⟶ A of R-algebras is central if the following diagram commutes:

[figure: R ∧_{S} A; τ; η∧id; A ∧_{S} R; id ∧η; A ∧_{S} A; ϕ; A; A ∧_{S} A; ϕ]

We learned the following interpretation of this definition from McClure.

Remark 1.2. The center of an associative k-algebra A with product ϕ can be written as the equalizer displayed in the diagram

[figure: C(A); A; ϕ̃; _{ϕτ}; Hom_{k}(A, A);]

here ϕ̃(a)(b) = ab and ϕ̃τ(a)(b) = ba. This suggests that the center C(A) of an S-algebra A should be defined as the equalizer displayed in the diagram

[figure: C(A); A; ϕ̃; _{ϕτ}; F_{S}(A, A).]

The definition of a central map η : R ⟶ A then says precisely that η factors through C(A).

Lemma 1.3. An R-algebra A is an S-algebra with a central map R ⟶ A of S-algebras. A commutative R-algebra A is a commutative S-algebra with a map R ⟶ A of S-algebras.

1. R-ALGEBRAS AND THEIR MODULES 147

Proof. Trivially, if A is an R-algebra, then its unit η : R ⟶ A is a central map of R-algebras. Conversely, if A is an S-algebra and η : R ⟶ A is a map of S-algebras, then A is a left R-module via the composite

[figure: R ∧_{S} A; η∧id; A ∧_{S} A; ϕ; A.]

There is a symmetrically defined right action of R on A that makes A an (R, R)- bimodule. Centrality ensures that the left and right actions agree under the commutativity isomorphism of their domains. The product of A therefore factors through A ∧_{R} A to give the required R-algebra structure.

We leave the proofs of the next few results as exercises; as in the proofs above, one first writes down the proof of the algebraic analogue and then replaces tensor products with smash products.

Proposition 1.4. If Q is an S-algebra, then R ∧_{S} Q is the free R-algebra generated by Q, hence R ∧_{S} 𝕋M is the free R-algebra generated by an S-module M. If Q is a commutative S-algebra, then R ∧_{S} Q is the free commutative R-algebra generated by Q, hence R ∧_{S} ℙM is the free commutative R-algebra generated by M.

Remark 1.5. We may think of R ∧_{S} (S ∧_{ℒ} 𝔹X) as the “free” R-algebra generated by a spectrum X and R ∧_{S} (S ∧_{ℒ} ℂX) as the “free” commutative R-algebra generated by a spectrum X. However, in view of II.1.3 (see also III§1), this is a misnomer since the right adjoints of these functors from the category of spectra to the category of R-algebras or commutative R-algebras are weakly equivalent rather than equal to the obvious forgetful functors.

Proposition 1.6. Let f : R ⟶ R′ and g : R ⟶ R′′ be maps of commutative S-algebras. Then R′ ∧_{R} R′′ is both the coproduct of R′ and R′′ in the category of commutative R-algebras and the pushout of f and g in the category of commutative S-algebras. More generally, let f : A ⟶ A′ and g : A ⟶ A′′ be maps of commutative R-algebras. Then A′ ∧_{A} A′′ is the pushout of f and g in the category of commutative R-algebras.

As in algebra, we can define the notion of a module over an R-algebra A, but it turns out to be equivalent to the notion of a module over A regarded just as an S-algebra. Recall III.3.1.

Definition 1.7. Let A be an R-algebra. A left or right A-module is a left or right A-object in ℳ_{R}.

The free A-module generated by an S-module M is

A ∧_{R} (R ∧_{S} M) ≅ A ∧_{S} M.

This gives an isomorphism of monads that implies the following result.

Lemma 1.8. Let A be an R-algebra. A module over A regarded as an S-algebra is the same thing as a module over A regarded as an R-algebra. That is, an action A ∧_{S} M ⟶ M necessarily factors through an action A ∧_{R} M ⟶ M.

Similarly, if M and N are A-modules, then M ∧_{A} N is the same whether defined using a coequalizer diagram in the category of R-modules or in the category of S-modules.

Lemma 1.9. Let A be an R-algebra, and let M be a right and N a left A-module. Then M ∧_{A} N can be identified with the coequalizer M ∧_{(A,R)} N displayed in the diagram

[figure: M ∧_{R} A ∧_{R} N; μ∧_{R}id; id ∧_{R}ν; M ∧_{R} N; M ∧_{(A,R)} N,]

The analogous result holds for function A-modules.

Proof. The proof is a formal categorical chase of the following schematic diagram:

\xymatrix{ & M ∧_{S} R ∧_{S} N \ar[dl] \ar[d] \ar[d] \\ M ∧_{S} A ∧_{S} N \ar[r] \ar[r] \ar[d] & M ∧_{S} N \ar[r] \ar[d] & M ∧_{A} N \ar[d] \\ M ∧_{R} A ∧_{R} N \ar[r] \ar[r] & M ∧_{R} N \ar[r] \ar@{-->}[ur] & M ∧_{(A,R)} N. \ar@{-->}[u]}

Here the left vertical arrow is an epimorphism, and this implies that the diagonal dotted arrow factors through the dotted right vertical arrow.

Although we have only one notion of an A-module, it is helpful to think of its study as divided into an “absolute theory”, in which we take the ground ring to be S, and a “relative theory”, in which we take the ground ring to be R. The absolute theory is a special case of the study of modules over algebras that we developed in Chapter III. In particular, III.1.4 shows that 𝔽_{A}X is weakly equivalent to A ∧ X for a CW spectrum X. Here the free functor 𝔽_{A} is isomorphic to the composite functor A ∧_{R} (R ∧_{S} 𝔽_{S}) from spectra to A-modules. Again, the term free is a misnomer since the right adjoint of 𝔽_{S} is only weakly equivalent to the forgetful functor. The theory of cell and CW A-modules and the definition of the derived category of A-modules are part of the absolute theory.

The previous lemma shows that the absolute smash product ∧_{A} and function module functors F_{A} are isomorphic to the relative functors, so that M ∧_{A} N and F_{A}(M, N) are R-modules. Of course, if A is a commutative R-algebra, then these are A-modules and duality theory applies. In the relative theory, if we replace (R, S) by (A, R), with concomitant changes of notations for various functors, then all of the statements in Chapter III which make sense remain true. Note, for example, that we have relative versions of III.3.10 and of the pairings discussed in III§5. The results on pairings give the following generalization of III.6.12.

Proposition 1.10. Let R be a commutative S-algebra, A be an R-algebra, and M and N be A-modules. Then F_{A}(M, M) is an R-algebra and F_{A}(M, N) is an (F_{A}(N, N), F_{A}(M, M))-bimodule.

Of course, the case R = A is of particular interest.

2. Tensored and cotensored categories of structured spectra

As in I§1, consider the categories 𝒫 and 𝒮 of prespectra and spectra indexed on a universe U. It was proven in [37, p.17-18] that these categories are topologically enriched, in the sense that their Hom sets are based topological spaces such that composition is continuous. For prespectra D and D′, 𝒫(D, D′) is topologized as a subspace of the product over indexing spaces V of the function spaces F (DV, D′V). Since maps between spectra are just maps between their underlying prespectra, this fixes the topology on 𝒮 (E, E′). It was also observed in [37, p.18] that all of the functors introduced in that volume are continuous and all of the adjunctions proven in it are given by homeomorphisms of Hom sets.

For example, by [37, I.3.3], there are natural homeomorphisms

𝒮 (E ∧ X, E′) ≅ 𝒯 (X, 𝒮 (E, E′)) ≅ 𝒮 (E, F (X, E′)) (2.1)

for spaces X and spectra E and E′, where 𝒯 denotes the category of based spaces. In categorical language [32, §3.7], (2.1) states that 𝒮 is tensored with tensors E∧X and cotensored with cotensors F (X, E). Adjoining disjoint basepoints to unbased spaces X, we obtain similar homeomorphisms involving the category 𝒰 of unbased spaces. We give a formal definition in the unbased context.

Definition 2.2. Let ℰ be a category enriched over the category 𝒰 of unbased spaces. Then ℰ is tensored if there is a functor ⊗_{ℰ} : ℰ × 𝒰 ⟶ ℰ, continuous in both variables, together with a natural homeomorphism

ℰ (E ⊗_{ℰ} X, E′) ≅ 𝒰 (X, ℰ (E, E′))

for spaces X and objects E and E′ of ℰ. We write ⊗ for ⊗_{ℰ} when ℰ is clear from the context. Dually, ℰ is cotensored if there is a functor F_{ℰ} : 𝒰^{op} × ℰ ⟶ ℰ, continuous in both variables, together with a natural homeomorphism

𝒰 (X, ℰ (E, E′)) ≅ ℰ (E, F_{ℰ} (X, E′)).

As in the motivating example (2.1), F_{ℰ} will always admit an explicit description. The tensors are more interesting and less familiar. We will give a way of describing them for many spaces X in the next section.

Again, by the argument illustrated in [37, p.18-19], colimits and limits of spectra are continuous in the sense that the isomorphisms

(2.3) 𝒮 (colim E_{i}, F) ≅ lim 𝒮 (E_{i}, F)

and

𝒮 (F, lim E_{i}) ≅ lim 𝒮 (F, E_{i}) (2.4)

are homeomorphisms.

The continuity can also be deduced categorically. There are valuable general notions of indexed colimits and limits in enriched categories, which are defined and discussed in Kelly [32, §3.1]. Indexed colimits include tensors with spaces and continuous colimits as special cases, and dually for limits. We shall not repeat the general definition, since we shall not have occasion to use it, and we shall rely on the following result of Kelly [32, 3.69-3.73] to deduce the existence of indexed colimits and limits.

Definition 2.5. A category ℰ enriched over 𝒰 is topologically cocomplete if it has all indexed colimits and topologically complete if it has all indexed limits.

Theorem 2.6 (Kelly). Let ℰ be a category enriched over the category of based or unbased spaces. Then ℰ is topologically cocomplete if it is cocomplete and admits tensor products and is topologically complete if it is complete and admits cotensor products. In particular, the given colimits and limits are continuous.

Our various categories of structured ring, module, and algebra spectra inherit subspace topologies on their Hom sets. Thus they are all topologically enriched. All of the functors and adjunctions that we have constructed in this paper are continuous, by the cited arguments of [37, p.18-19]. We claim that our various categories of rings, modules, and algebras are topologically cocomplete and complete.

For modules, this is immediate from II.1.4, III.1.1, and inspection. If R is an S-algebra, M is an R-module, and X is a based space, then

ℳ_{R}(M ∧ X, M′) ≅ 𝒯 (X, ℳ_{R}(M, M′)) ≅ ℳ_{R}(M, S ∧_{ℒ} F (X, M′)). (2.7)

We deduce the first isomorphism from the first isomorphism of (2.1) by first writing ℳ_{S}(M, M′) as the equalizer of a pair of maps 𝒮 (M, M′) ⟶ 𝒮 (𝕃M, M′) and then writing ℳ_{R}(M, M′) as the equalizer of a pair of maps ℳ_{S}(M, M′) ⟶ ℳ_{S}(R ∧_{S} M, M′). We deduce the second isomorphism from the first by use of the isomorphisms

M ∧ X ≅ M ∧_{S} Σ^{∞}X and S ∧_{ℒ} F (X, M) ≅ F_{S}(Σ^{∞}X, M).

Proposition 2.8. For any S-algebra R, ℳ_{R} is topologically cocomplete and complete. Its tensors M ∧ X and all other indexed colimits are created in ℳ_{S} or, equivalently, in 𝒮. Its cotensors F_{S}(Σ^{∞}X, M) and all other indexed limits are created in ℳ_{S} or, equivalently, by applying the functor S ∧_{ℒ} (?) to indexed limits created in 𝒮.

Now consider the categories of R-algebras and of commutative R-algebras. We agree to denote these categories by 𝒜_{R} and 𝒞 𝒜_{R}, respectively. We must enrich these categories over 𝒰 , since there are no “trivial maps” to take as basepoints of Hom sets. We have already observed in II§7 that the categories 𝒜_{R} and 𝒞 𝒜_{R} are complete and cocomplete. Continuing that discussion, we obtain the following result. The proof works equally well in the categories of A_{∞} and E_{∞} ring spectra, where the result is due to Hopkins and McClure [31] and, in the E_{∞} case, is the main technical result of McClure, Schwänzl, and Vogt [52, Thm A].

Theorem 2.9. For any commutative S-algebra R, the categories 𝒜_{R} of Ralgebras and 𝒞 𝒜_{R} of commutative R-algebras are topologically cocomplete and complete. Their cotensors and all other indexed limits are created in ℳ_{S} or, equivalently, by applying the functor S ∧_{ℒ} (?) to indexed limits created in 𝒮.

Proof. By II.7.1 (compare II.4.5), we have monads 𝕋 and ℙ in the category of R-modules whose algebras are the R-algebras and commutative R-algebras, and these monads are continuous (e.g., by inspection when R = S and use of Proposition 1.4). Now II.7.2 and II.7.4 apply to show that 𝒜_{R} and 𝒞 𝒜_{R} are cocomplete. In the commutative case, the construction of colimits is quite simple since Proposition 1.6 gives coproducts and pushouts, and it is trivial to construct coequalizers from them. Moreover, by an easy bootstrap argument from the continuity of colimits in the ground category of spectra, coequalizers in 𝒜_{R} and 𝒞 𝒜_{R} are continuous. Now the following categorical result completes the proof.

Proposition 2.10. Let 𝕋 : 𝒞 ⟶ 𝒞 be a continuous monad defined on a topologically enriched category 𝒞 and let 𝒞 [𝕋] be the category of algebras over 𝕋. Assume that 𝒞 is topologically cocomplete and complete.

(i) The forgetful functor 𝒞 [𝕋] ⟶ 𝒞 creates all indexed limits.

(ii) If 𝒞 [𝕋] has continuous coequalizers, then 𝒞 [𝕋] has all indexed colimits.

Proof. The first part is the enriched version of [42, VI.2, Ex 2]. Our proof of the second part is due to Hopkins [31]. (The later argument of [52, 2.7] is slightly flawed.) Let (C, ξ) be a 𝕋-algebra and X be a space. We must construct their tensor 𝕋-algebra. Let C ⊗ X denote their tensor in 𝒞. Define ν : 𝕋C ⊗ X ⟶ 𝕋(C ⊗ X) to be the adjoint of the composite map of spaces

[figure: X; 𝒞 (C, C ⊗ X); 𝕋; 𝒞 (𝕋C, 𝕋(C ⊗ X)),]

where the first arrow is adjoint to the identity map C ⊗ X ⟶ C ⊗ X. Let 𝕋 also denote the free functor from 𝒞 to 𝒞 [𝕋]. Define C ⊗_{𝒞 [𝕋]} X to be the coequalizer in 𝒞 [𝕋] of the pair of maps

[figure: 𝕋(𝕋C ⊗ X); 𝕋(ξ⊗id); μ∘𝕋ν; 𝕋(C ⊗ X).]

We easily check the required adjunction

𝒞 [𝕋](C ⊗_{𝒞 [𝕋]} X, C′) ≅ 𝒰 (X, 𝒞 [𝕋](C, C′))

by using the fact that 𝒞 [𝕋](C, C′) is the equalizer of

[figure: 𝒞 (C, C′); 𝒞 (ξ,id); 𝒞 (𝕋C, C′)]

and

[figure: 𝒞 (C, C′); 𝕋; 𝒞 (𝕋C, 𝕋C′); 𝒞 (id,ξ′); 𝒞 (𝕋C, C′).]

By Linton’s theorem (II.7.5), 𝒞 [𝕋] is cocomplete since it has coequalizers. By Kelly’s theorem (Theorem 2.4), 𝒞 [𝕋] has all indexed colimits.

In particular, F_{S}(Σ^{∞}X_{+}, A) is the cotensor of a space X and an R-algebra or commutative R-algebra A. The diagonal on X and the product on A induce the product on F_{S}(Σ^{∞}X_{+}, A). The following instance of a general categorical observation explains the relationship between the smash product A ∧ X_{+} in the category of R-modules and the tensor A ⊗ X in the category of R-algebras or commutative R-algebras.

Proposition 2.11. For R-algebras A and spaces X there is a natural map of R-modules

ω : A ∧ X_{+} ⟶ A ⊗ X

such that ω = id if X = {∗} and the following transitivity diagrams commute:

[figure: (A ∧ X_{+}) ∧ Y_{+}; ω∧id; ≅; (A ⊗ X) ∧ Y_{+}; ω; (A ⊗ X) ⊗ Y; ≅; A ∧ (X × Y)_{+}; ω; A ⊗ (X × Y).]

For x ∈ X, let i_{x} : A ⟶ A∧X_{+} be the map induced by the inclusion {x}_{+} ⟶ X_{+}. A map f : A ∧ X_{+} ⟶ B of spectra into an R-algebra B such that each composite f ∘ i_{x} : A ⟶ B is a map of R-algebras uniquely determines a map of R-algebras f̃: A ⊗ X ⟶ B such that f = f̃∘ ω. The same statement holds for commutative R-algebras.

Proof. We have a natural map

𝒜_{R}(A ⊗ X, B) ≅ 𝒰 (X, 𝒜_{R}(A, B)) ⟶ 𝒰 (X, ℳ_{R}(A, B)) ≅ ℳ_{R}(A ∧ X_{+}, B),

and ω is the image of the identity map of A ⊗ X. The rest is easy diagram chasing, using the natural map ℳ_{R}(A∧X_{+}, B) ⟶ 𝒮 (A∧X_{+}, B) for the last statement.

Remark 2.12. For R-algebras A and B, the previous result says that a map A ⊗ X ⟶ B of R-algebras determines and is determined by a map A ∧ X_{+} ⟶ B of spectra that is pointwise a map of R-algebras. A similar construction and result apply whenever one has a tensored category ℰ with a continuous forgetful functor to spectra. For objects A and B in ℰ, we define a homotopy to be a map h : A ⊗ I ⟶ B. Then h is induced by a homotopy A ∧ I_{+} ⟶ B through maps in ℰ.

3. Geometric realization and calculations of tensors

To prepare for our construction of model structures and our study of thh, we explain how to calculate tensors E ⊗ X for certain spaces X, and we use this calculation to study pushouts and cofibrations in the context of R-algebras. Our main tool is geometric realization, and the reader is urged to read the first two sections of Chapter X, which give a down to earth study of the geometric realization of simplicial spectra, before reading this section.

Fix a topologically complete and cocomplete category ℰ with a continuous forgetful functor to spectra. We have the notion of a simplicial object E_{∗} in ℰ. There are two notions of the geometric realization of such an object. We can first forget down to the category of simplicial spectra and take the geometric realization |E_{∗}| there, or we can rework the definition and carry out the construction entirely in ℰ, obtaining the internal geometric realization |E_{∗}|_{ℰ} . Explicitly, |E_{∗}|_{ℰ} is the coend

|E_{∗}|_{ℰ} = ∫ ^{Δ} E_{q} ⊗_{ℰ} Δ_{q}. (3.1)

The following relationships between these two kinds of geometric realization generalize and clarify observations of McClure, Schwänzl, and Vogt [52, 4.3, 4.4] about the category of E_{∞} ring spectra. We defer the proofs to the end of the section.

Proposition 3.2. Let X_{∗} be a simplicial space and let A ∈ ℰ. Then there is a natural isomorphism

A ⊗_{ℰ} |X_{∗}| ≅ |A ⊗_{ℰ} X_{∗}|_{ℰ} ,

of objects of ℰ.

The realization of underlying simplicial spectra is more amenable to homotopical analysis than the internal realization. In favorable cases, the realization |E_{∗}| will again be an object of ℰ, but this is not formal. We shall prove in X§1 that this holds for all of the categories of interest to us. In such cases, the two geometric realizations are isomorphic. In particular, the following result holds.

Proposition 3.3. Let R be any commutative S-algebra, such as R = S. For simplicial R-algebras A_{∗}, there is a natural isomorphism of R-algebras

|A_{∗}| ≅ |A_{∗}|_{𝒜_{R}} ,

and similarly for simplicial commutative R-algebras.

Corollary 3.4. For R-algebras A and simplicial spaces X_{∗}, there is a natural isomorphism of R-algebras

A ⊗_{𝒜_{R}} |X_{∗}| ≅ |A ⊗_{𝒜_{R}} X_{∗}|,

and similarly for commutative R-algebras.

In the following discussion, we let ℰ denote either 𝒜_{R} or 𝒞 𝒜_{R} and write ⊗ for ⊗_{ℰ} . We use the term R-algebra in either case. The computation of A ⊗ |X_{∗}| just given applies particularly effectively to simplicial sets X_{∗}, regarded as discrete simplicial spaces. We have a categorical coproduct ⨿ in ℰ. This is ∧_{R} in the commutative case, but it is the “free product” in the non-commutative case. In the commutative case, the codiagonal map ▽ : A ⨿ A ⟶ A is the product on A. In both cases, the unit η : R ⟶ A is the unique map from the initial object. Since a discrete set \underline{n} with n points is the coproduct of its elements and the functor A⊗(?) preserves coproducts, A⊗\underline{n} is the coproduct of n copies of A. To calculate A⊗|X_{∗}|, we need only identify the induced face and degeneracy operators on coproducts of copies of A in terms of the structure maps ▽ and η.

In order to understand homotopy theory in ℰ, we need to understand A ⊗ I. We shall describe it in terms of a bar construction that is defined on R-algebras. Recall that we defined the bar construction B(M, R, N) for a commutative Salgebra R and R-modules M and N in IV.7.2. We shall later use the evident generalization in which we replace R and its modules by a commutative R-algebra A and its modules. We here introduce a variant that applies equally well to either commutative or non-commutative R-algebras. In the commutative case, it is just the specialization of the cited generalization in which the given A-modules are restricted to be commutative A-algebras.

Definition 3.5. Let A be an R-algebra, and let f : A ⟶ A′ and g : A ⟶ A′′ be maps of R-algebras. These maps and the identity maps of A′ and A′′ determine maps of R-algebras

μ : A′ ⨿ A ⟶ A′ and ν : A ⨿ A′′ ⟶ A′′

Define a simplicial R-algebra β_{∗}(A′, A, A′′) by replacing ∧_{S} and ϕ by ⨿ and ▽ in IV.7.2. Then define an R-algebra β^{R}(A′, A, A′′) by

β^{R}(A′, A, A′′) = |β_{∗}^{R}(A′, A, A′′)|.

There is an evident natural map of R-algebras

ψ : β^{R}(A′, A, A′′) ⟶ A′ ⨿_{A} A′′

from the bar construction to the displayed pushout.

Define the double mapping cylinder R-algebra M(A′, A, A′′) by

M(A′, A, A′′) = A′ ⨿_{A} (A ⊗ I) ⨿_{A} A′′ (3.6)

and observe that the map I ⟶ {pt} induces a collapse map

ψ : M(A′, A, A′′) ⟶ A′ ⨿_{A} A′′.

We have the following identification of these two constructions.

Proposition 3.7. Let A be an R-algebra with given maps to R-algebras A′ and A′′. Then there is a natural isomorphism

β^{R}(A, A, A) ≅ A ⊗ I

of R-algebras over A and under A ⨿ A, and there is a natural isomorphism

β^{R}(A′, A, A′′) ≅ M(A′, A, A′′)

of R-algebras over A′ ⨿_{A} A′′ and under A′ ⨿ A′′.

Proof. Let I_{∗} be the standard simplicial 1-simplex with realization I. It has p + 2 p-simplices, and a simple comparison of its face and degeneracy operations (e.g., [43, p.14]) with those of the bar construction shows that we have a natural identification of simplicial R-algebras

β_{∗}^{R}(A, A, A) ≅ A ⊗ I_{∗}.

The rest follows.

In fact, one can see this quite directly, since the only non-degenerate simplices of I_{∗} are a 1-simplex Δ_{1} and its faces, and similarly for β_{∗}^{R}(A, A, A).

We use this to obtain a result about cofibrations that will be at the heart of our construction of model structures on ℰ. Let 𝕋 : ℳ_{R} ⟶ ℰ be the free Ralgebra functor; thus 𝕋 must be interpreted as ℙ in the commutative case. Since 𝕋 preserves tensors and pushouts and since R = 𝕋(∗), we have

𝕋CM ≅ R ⨿_{𝕋M} (𝕋M ⊗ I).

Proposition 3.8. For any R-module M, and any map of R-algebras 𝕋M ⟶ A, the natural map of R-algebras

ψ : M(𝕋CM, 𝕋M, A) ⟶ 𝕋CM ⨿_{𝕋M} A

is homotopic rel A to an isomorphism.

Proof. For a based space X, it is trivial to see that the map

CX ∪_{X} (X ∧ I_{+}) ⟶ CX

that retracts the cylinder onto the base of the cone is homotopic to a homeomorphism. Working in the category of R-modules, the same argument works with X replaced by M. Applying the functor 𝕋, the cited map then becomes the map

ρ : R ⨿_{𝕋M} (𝕋M ⊗ I) ⨿_{𝕋M} (𝕋M ⊗ I) ⟶ R ⨿_{𝕋M} (𝕋M ⊗ I)

that retracts the second copy of 𝕋M ⊗ I onto the base of the first. We have

M(𝕋CM, 𝕋M, A) ≅ R ⨿_{𝕋M} (𝕋M ⊗ I) ⨿_{𝕋M} (𝕋M ⊗ I) ⨿_{𝕋M} A,

and ψ is obtained by applying the functor (?) ⨿_{𝕋M} A to ρ. The conclusion follows.

We shall prove in XII.2.3 that the functor 𝕋 preserves cofibrations of R-modules, and this makes the following result plausible. We shall have more to say about this in Section 6.

Proposition 3.9. For any pushout diagram of R-algebras

[figure: 𝕋M; A; i; 𝕋CM; B,]

the map i is a cofibration of R-modules and therefore of spectra.

Proof. The essential point is just that the unit map η : R ⟶ 𝕋M is the inclusion of a wedge summand of R-modules and a retract of R-algebras. From this, we find that the induced map A ⟶ 𝕋M ⨿ A of R-algebras is also the inclusion of a wedge summand of R-modules and a retract of R-algebras. By the previous lemma and proposition, the pushout is isomorphic under A to the bar construction β^{R}(𝕋CM, 𝕋M, A). All of the degeneracy operators of β_{∗}^{R}(𝕋CM, 𝕋M, A) are inclusion of wedge summands of R-modules, and it follows that β_{∗}^{R}(𝕋CM, 𝕋M, A) is proper in the sense of X.1.2. This implies that the map from the zero skeleton 𝕋CM ⨿ A into β^{R}(𝕋CM, 𝕋M, A) is a cofibration, and the conclusion follows.

We shall also need the following elementary complement.

Lemma 3.10. Let {A_{i}} be a sequence of maps of R-algebras that are cofibrations of spectra. Then the underlying spectrum of the colimit of the sequence computed in the category of R-algebras is the colimit of the sequence computed in the category of spectra.

Proof. The colimit in the category of spectra computes the colimit in the category of R-modules and satisfies

(colim A_{i}) ∧_{R} (colim A_{i}) ≅ colim (A_{i} ∧_{R} A_{i}).

Therefore the spectrum level colimit inherits an R-algebra structure from the A_{i}, and the universal property in the category of R-algebras follows from the universal property in the category of R-modules.

We must still prove Propositions 3.2 and 3.3. Let s𝒞 denote the category of simplicial objects in a category 𝒞.

Proof of Proposition 3.2. For a space Y, let 𝒰 (Δ_{∗}, Y) be the evident simplicial space with q-simplices 𝒰 (Δ_{q}, Y). This functor of Y is right adjoint to geometric realization,

𝒰 (|X_{∗}|, Y) ≅ s𝒰 (X_{∗}, 𝒰 (Δ_{∗}, Y)). (3.11)

Similarly, for an object F of ℰ, let F_{ℰ} (Δ_{∗}, F) be the evident simplicial object of ℰ with q-simplices F_{ℰ} (Δ_{q}, F). This functor of F is right adjoint to the internal geometric realization,

ℰ (|E_{∗}|_{ℰ} , F) ≅ sℰ (E_{∗}, F_{ℰ} (Δ_{∗}, F)). (3.12)

These adjunctions, together with tensor and cotensor adjunctions, give the chain of natural isomorphisms

ℰ (E ⊗_{ℰ} |X_{∗}|, F) ≅ 𝒰 (|X_{∗}|, ℰ (E, F))
≅ s𝒰 (X_{∗}, 𝒰 (Δ_{∗}, ℰ (E, F))
≅ s𝒰 (X_{∗}, ℰ (E, F_{ℰ} (Δ_{∗}, F))
≅ sℰ (E ⊗_{ℰ} X_{∗}, F_{ℰ} (Δ_{∗}, F))
≅ ℰ (|E ⊗_{ℰ} X_{∗}|_{ℰ} , F).

The conclusion follows.

Proof of Proposition 3.3. Our interest is in the examples ℰ = 𝒜_{R} and ℰ = 𝒞 𝒜_{R}, but the argument is general. In all cases where realizations |E_{∗}| inherit structure present in ℰ, the induced structure “arises pointwise”. To explain what this means, note that we have an adjunction like those of (3.11) and (3.12) for simplicial spectra K_{∗} and spectra L, namely

𝒮 (|K_{∗}|, L) ≅ s𝒮 (K_{∗}, F ((Δ_{∗})_{+}, L)), (3.13)

where F ((Δ_{∗})_{+}, L) has q-simplices F ((Δ_{q})_{+}, L). Now let E_{∗} be a simplicial object of ℰ and F be an object of ℰ. When |E_{∗}| is again an object of ℰ, we have the subspace

ℰ (|E_{∗}|, F) ⊂ 𝒮 (|E_{∗}|, F)

of maps in ℰ. This subspace coincides under the adjunction (3.13) with the subspace of s𝒮 (E_{∗}, F ((Δ_{∗})_{+}, F)) consisting of those points f = {f_{q}} such that the adjoint f̃_{q} : E_{q} ∧ (Δ_{q})_{+} ⟶ F of f_{q} : E_{q} ⟶ F ((Δ_{q})_{+}, F)) restricts to a map E_{q} ⟶ F in ℰ on the copy of E_{q} in E_{q} ∧ (Δ_{q})_{+} determined by each point of Δ_{q}. By Proposition 2.11 and Remark 2.12, such a map f̃_{q} extends uniquely to a map g̃_{q} : E_{q} ⊗_{ℰ} Δ_{q} ⟶ F in ℰ. In turn, under the tensor-cotensor adjunction, g̃_{q} corresponds to a map g_{q} : E_{q} ⟶ F_{ℰ} (Δ_{q}, F) in ℰ. The function {f_{q}} ⟶ {g_{q}} determines an adjunction

ℰ (|E_{∗}|, F) ≅ sℰ (E_{∗}, F_{ℰ} (Δ_{∗}, F)). (3.14)

Comparison of (3.12) and (3.14) gives the conclusion. An alternative argument based on the properties of the monads 𝕋 and ℙ is also possible. The adjunctions above can be used to check that

𝕋|A_{∗}| ≅ |𝕋(A_{∗})|_{𝒜_{R}} .

The functor 𝕋 commutes with |?| on simplicial R-modules, the functor |?|_{𝒜_{R}} preserves coequalizers, and a comparison of coequalizer diagrams gives the result.

4. MODEL CATEGORIES OF RING, MODULE, AND ALGEBRA SPECTRA 159

4. Model categories of ring, module, and algebra spectra

We shall prove that our various categories of structured spectra admit model structures. A more general, axiomatic, framework is possible; compare Blanc [5]. We assume familiarity with the language of model categories, by which we understand closed model categories in Quillen’s original sense [54]. A good exposition is given in [17]. We explain our results in this section and prove them in the next.

In this paper, cofibrations and fibrations in any of our categories mean maps that satisfy the homotopy extension property (HEP) or covering homotopy property (CHP) in that category. Cofibrations in this sense will play a central role in the work of the next section. It is a pity that the language of model categories has, in the literature, been superimposed on the classical language, with resulting ambiguity. We shall use q-cofibrations and q-fibrations for the model theoretic terms.

In all of our model categories, the weak equivalences in the model sense will be those maps in the category which are weak equivalences of underlying spectra. We say that the weak equivalences are created in 𝒮. Observe that a retract of a weak equivalence is a weak equivalence. Recall that a q-fibration or q-cofibration in a model category is said to be acyclic if it is a weak equivalence.

Implicitly or explicitly, we must constantly think in terms of diagrams

[figure: E; α; i; X; p; F; β; g; Y,]

where the square is given to be commutative and we seek a lift g that makes both triangles commute. We say that i has the left lifting property (LLP) with respect to a class of morphisms 𝒫 if there exists such a lift g for any square in which p ∈ 𝒫. We say that p satisfies the right lifting property (RLP) with respect to a class of morphisms ℐ if there exists such a lift g for any square in which i ∈ ℐ.

For example, a Serre fibration of spectra is a map that satisfies the CHP with repect to the set of “cone spectra”

{Σ_{q}^{∞}CS^{n} | q ≥ 0 and n ≥ 0}.

This means that it is a map that satisfies the RLP with respect to the set of inclusions

i_{0} : Σ_{q}^{∞}CS^{n} ⟶ Σ_{q}^{∞}CS^{n} ∧ I_{+}.

Again, a retract of a Serre fibration is a Serre fibration. The q-fibrations in 𝒮 will be the Serre fibrations.

The following definition will allow us to give succinct statements of our results.

Definition 4.1. Let 𝒞 be a model category with a forgetful functor to 𝒮 that creates weak equivalences and let ℰ be a category with a forgetful functor to 𝒞. We say that 𝒞 creates a model structure in ℰ if ℰ is a model category whose weak equivalences are created in 𝒮 and whose q-fibrations are created in 𝒞. That is, a map in ℰ is a q-fibration if it is a q-fibration when regarded as a map in 𝒞. The q-cofibrations in ℰ must then be those maps which satisfy the left lifting property with respect to the acyclic q-fibrations.

Our categories are enriched, and our model structures will reflect this. Quillen defined the notion of a simplicial model category in [54, II§2], and the appropriate topological analogue of his definition reads as follows.

Definition 4.2. A model category ℰ is topological if it is topologically complete and cocomplete and if, for any q-cofibration i : E ⟶ F and q-fibration p : X ⟶ Y, the induced map

(i^{∗}, p_{∗}) : ℰ (F, X) ⟶ ℰ (E, X) ×_{ℰ (E,Y)} ℰ (F, Y) (4.3)

is a Serre fibration of spaces which is acyclic if either i or p is acyclic.

Theorem 4.4. The category 𝒮 is a topological model category with respect to the weak equivalences and Serre fibrations. If 𝕋 : 𝒮 ⟶ 𝒮 is a continuous monad such that the category 𝒮 [𝕋] of 𝕋-algebras has continuous coequalizers and satisfies the “Cofibration Hypothesis”, then 𝒮 creates a topological model structure in 𝒮 [𝕋].

We think of the first statement as the specialization to the identity monad of the second. We shall specify the “Cofibration Hypothesis” shortly. It will obviously be satisfied by the identity monad and by the monad 𝕃, and arguments like those of the previous section verify it for the monads 𝕋𝕃 and ℙ𝕃 that define A_{∞} and E_{∞} ring spectra.

Corollary 4.5. The categories of 𝕃-spectra and of A_{∞} and E_{∞} ring spectra are topological model categories.

Of course, we are far more interested in our categories of modules and algebras. The crux of the proof of Theorem 4.4 is the adjunction

𝒮 [𝕋](𝕋X, A) ≅ 𝒮 (X, A)

for spectra X and 𝕋-algebras A. For S-modules, this must be replaced by the adjunction

ℳ_{S}(S ∧_{ℒ} 𝕃X, M) ≅ 𝒮 (X, F_{ℒ} (S, M))

for S-modules M that we obtain by composition of the first adjunction of II.2.2 with the freeness adjunction for the monad 𝕃. Thus we must change our forgetful functor from the obvious one to the functor F_{ℒ} (S,?). Since F_{ℒ} (S,?) is naturally weakly equivalent to M, by I.8.7, the weak equivalences are unchanged. However, the q-fibrations are changed. Although the functor 𝕋X = S ∧_{ℒ} 𝕃X from spectra to S-modules is not a monad, the proof of Theorem 4.4 will apply verbatim to it to give the following result.

Theorem 4.6. The category ℳ_{S} is a topological model category with weak equivalences created in 𝒮. Its q-fibrations are those maps f : M ⟶ N of S-modules such that

F (id, f) : F_{ℒ} (S, M) ⟶ F_{ℒ} (S, N)

is a Serre fibration of spectra.

To understand this, it is useful to think in terms of the “mirror image category” ℳ^{S} of counital 𝕃-spectra specified in II.2.1. By II.2.7 and composition (see II.6.1), we have a continuous monad F_{ℒ} (S, 𝕃(?)) on 𝒮 whose algebras are the counital 𝕃spectra. We have a topological equivalence of categories ℳ^{S} ⟶ ℳ_{S} that carries N to S ∧_{ℒ} N. By II.2.5, S ∧_{ℒ} F_{ℒ} (S, M) is naturally isomorphic to S ∧_{ℒ} M for any 𝕃-spectrum M. Thus the monad that defines counital 𝕃-spectra is transported under the equivalence to the functor 𝕋 relevant to the construction of the model structure on ℳ_{S}. The equivalence has the effect of changing the forgetful functor.

The proof of Theorem 4.4 will apply equally well if we change our ground category to ℳ_{S}.

Theorem 4.7. If 𝕋 : ℳ_{S} ⟶ ℳ_{S} is a continuous monad such that the category ℳ_{S}[𝕋] of 𝕋-algebras has continuous coequalizers and satisfies the “Cofibration Hypothesis”, then ℳ_{S} creates a topological model structure in ℳ_{S}[𝕋].

Of course, the description of the q-fibrations as maps f such that F_{ℒ} (S, f) is a Serre fibration persists. Again, the Cofibration Hypothesis will be specified shortly and holds in our examples.

Corollary 4.8. The categories of S-algebras, commutative S-algebras, and modules over an S-algebra R are topological model categories.

Now that we have a model structure on ℳ_{R}, we can generalize Theorem 4.7 by changing its ground category to ℳ_{R}.

Theorem 4.9. Let R be a commutative S-algebra. If 𝕋 : ℳ_{R} ⟶ ℳ_{R} is a continuous monad such that the category ℳ_{R}[𝕋] of 𝕋-algebras has continuous coequalizers and satisfies the “Cofibration Hypothesis”, then ℳ_{R} creates a topological model structure in ℳ_{R}[𝕋].

Corollary 4.10. The categories of algebras and commutative algebras over a commutative S-algebra R are topological model categories.

In fact, Theorems 4.7 and 4.9 both apply, and they give the same model structures since they give the same q-fibrations and weak equivalences. We prefer to think of the model structure as created in ℳ_{R}, since that makes visible more information about the q-cofibrations. While the model category theory dictates what the qcofibrations must be, the proofs of the theorems will lead to explicit descriptions.

Definition 4.11. Let 𝕋 be a monad in 𝒮 as in Theorem 4.7. A relative cell 𝕋-algebra Y under a 𝕋-algebra X is a 𝕋-algebra Y = colim Y_{n}, where Y_{0} = X and Y_{n+1} is obtained from Y_{n} as the pushout of a sum of attaching maps 𝕋S^{q} ⟶ Y_{n} along the coproduct of the natural maps 𝕋S^{q} ⟶ 𝕋CS^{q}. When X is an initial 𝕋-algebra, we say that Y is a cell 𝕋-algebra. Relative and absolute cell 𝕋-algebras are defined in precisely the same way for a monad 𝕋 in ℳ_{R} as in Theorem 4.9, except that the sphere spectra S^{q} are replaced by the sphere R-modules S_{R}^{q}.

Remark 4.12. The functor 𝕋 : 𝒮 ⟶ 𝒮 [𝕋], being a left adjoint, preserves coproducts. Thus, when attaching a coproduct of cells 𝕋CS^{q} to Y_{n} to obtain Y_{n+1}, we are considering a pushout in 𝒮 [𝕋] of the general form

[figure: 𝕋E; A; i; 𝕋CE; B,] (4.13)

where E is a wedge of spheres, and similarly when the ground category is ℳ_{S} or ℳ_{R}.

The Cofibration Hypothesis is just the minimum condition necessary to obtain homotopical control over these pushout diagrams and their colimits. It holds in our examples by Proposition 3.9 and Lemma 3.10.

Cofibration Hypothesis. The map i in any pushout of the form (4.13) is a cofibration of spectra (for Theorem 4.4) or of S-modules (for Theorem 4.7) or of R-modules (for Theorem 4.9). The underlying spectrum of the 𝕋-algebra colimit of a sequence of cofibrations of 𝕋-algebras is their colimit as a sequence of maps of spectra.

Actually, for the model structure in Theorems 4.7 and 4.9, we only need the maps i to be cofibrations of spectra. However, the stronger R-module cofibration condition holds in practice and is important in the applications.

Theorem 4.14. Under the hypotheses of Theorems 4.4, 4.7, and 4.9, a map of 𝕋-algebras is a q-cofibration if and only if it is a retract of a relative cell 𝕋-algebra. Moreover, any q-cofibration is a cofibration of underlying spectra (in Theorem 4.4) or of underlying S-modules (in Theorem 4.7) or of underlying R-modules (in Theorem 4.9).

By the Cofibration Hypothesis, the second statement will follow from the first. In all of our categories of 𝕋-algebras, the trivial spectrum is a terminal object and every 𝕋-algebra is q-fibrant. By the previous result, a 𝕋-algebra is q-cofibrant if and only if it is a retract of a cell 𝕋-algebra. Note in particular that the unit R ⟶ A of a q-cofibrant R-algebra or commutative R-algebra is a cofibration of R-modules.

As in our discussion of Theorem 4.6, the proof of the previous theorem will apply to give the following expected conclusion.

Theorem 4.15. For an S-algebra R, such as R = S, a map of R-modules is a q-cofibration if and only if it is a retract of a relative cell R-module.

Thus, in the case of R-modules, model category theory just brings us back to the cell theory that we took as our starting point. We can turn this around. We certainly want the weak equivalences and q-cofibrations in ℳ_{R} to be the weak equivalences of underlying spectra and the retracts of relative cell R-modules. Since the weak equivalences and q-cofibrations determine the q-fibrations, we see that the q-fibrations specified in Theorem 4.6 are in fact forced on us by the cell theory that we began with.

5. The proofs of the model structure theorems

We must prove Theorems 4.4, 4.6, 4.7, 4.9, 4.14, and 4.15. For uniformity of treatment, let 𝒞 be either 𝒮 or ℳ_{R} for a commutative S-algebra R. Logically, of course, we should treat the case R = S before going on to the general case. Let 𝕋 be a continuous monad in 𝒞 such that 𝒞 [𝕋] has continuous coequalizers. By Proposition 2.10, we already know that 𝒞 [𝕋] is complete, cocomplete, tensored and cotensored, and indeed has all indexed limits and colimits. It is clear that if g ∘f is defined and two of f, g, and g ∘ f are weak equivalences, then so is the third. It is also clear that the collections of q-fibrations, q-cofibrations, and weak equivalences are closed under composition and retracts and contain all isomorphisms. It remains to prove that arbitrary maps factor appropriately and that the q-fibrations satisfy the right lifting property (RLP) with respect to the acyclic q-cofibrations. The essential point is that Quillen’s “small object argument” applies to construct the required factorizations. A general version of Quillen’s original argument is given in [17, §6], and we shall give a modified version of that argument.

Definition 5.1. For the purposes of this section, define a finite pair of spectra to be a pair of the form (Σ_{q}^{∞}B, Σ_{q}^{∞}A), where B is a finite based CW complex, A is a subcomplex, and q ≥ 0. Define a finite pair of 𝕃-spectra to be a pair obtained by applying 𝕃 to a finite pair of spectra. Define a finite pair of R-modules to be a pair obtained by applying 𝔽_{R} to a finite pair of spectra.

As a matter of esoterica, we actually only need A, not B, to be finite in our arguments.

Lemma 5.2. Let ℱ be a set of maps in 𝒞 [𝕋], each of which is of the form 𝕋E ⟶ 𝕋F for some finite pair (F, E) in 𝒞. Then any map f : X ⟶ Y in 𝒞 [𝕋] factors as a composite

[figure: X; i; X′; p; Y,]

where p satisfies the RLP with respect to each map in ℱ and i satisfies the LLP with respect to any map that satisfies the RLP with respect to each map in ℱ.

Proof. Let X = X_{0}. We construct a commutative diagram

[figure: X_{0}; f=p_{0}; i_{0}; X_{1}; p_{1}; ⋯; X_{n}; i_{n}; p_{n}; X_{n+1}; p_{n+1}; ⋯; Y; id; Y; ⋯; Y; id; Y; ⋯] (5.3)

as follows. Suppose inductively that we have constructed p_{n}. Consider all maps from a map in ℱ to p_{n}. Each such map is a commutative diagram of the form

[figure: 𝕋E; α; X_{n}; p_{n}; 𝕋F; β; Y.] (5.4)

Summing over such diagrams, we construct a pushout diagram of the form

[figure: ∐ 𝕋E; ∑ α; X_{n}; i_{n}; ∐ 𝕋F; X_{n+1}.]

The maps β induce a map p_{n+1} : X_{n+1} ⟶ Y such that p_{n+1} ∘ i_{n} = p_{n}. Let X′ = colim X_{n}, let i : X ⟶ X′ be the canonical map, and let p : X′ ⟶ Y be obtained by passage to colimits from the p_{n}. Constructing lifts by passage to coproducts, pushouts, and colimits in 𝒞 [𝕋], we see that each i_{n} and therefore also i satisfies the LLP with respect to maps that satisfy the RLP with respect to maps in ℱ. Assume given a commutative square

[figure: 𝕋E; α′; i; X′; p; 𝕋F; β; g; Y,]

where i is in ℱ. To verify that p satisfies the RLP with respect to i, we must construct a map g that makes the diagram commute. The Cofibration Hypothesis implies that X′ is constructed as the colimit of a sequence of cofibrations of spectra. By [37, App.3.9], a cofibration of spectra is a spacewise closed inclusion. Therefore, using that 𝕋 is the free functor from 𝒞 to 𝒞 [𝕋], we see by III.1.7 that the natural map

colim 𝒞 [𝕋](𝕋E, X_{n}) ⟶ 𝒞 [𝕋](𝕋E, X′) (5.5)

is a bijection. This ensures that α′ : 𝕋E ⟶ X′ factors through some X_{n}, giving us one of the commutative squares (5.4) used in the construction of X_{n+1}. By construction, there is a map 𝕋F ⟶ X_{n+1} whose composite with the natural map to X′ gives a map g as required.

Lemma 5.6. Any map f : X ⟶ Y in 𝒞 [𝕋] factors as p ∘ i, where i is an acyclic q-cofibration that satisfies the LLP with respect to any q-fibration and p is a q-fibration.

Proof. Let ℱ be the set of pairs obtained by letting (B, A) in Definition 5.1 run through all pairs of spaces (CS^{n} ∧I_{+}, CS^{n} ∧{0}_{+}), n ≥ 0. By chasing through adjunctions and using the definition of q-fibrations and q-cofibrations, we see that a map is a q-fibration if and only if it satisfies the RLP with respect to every map in ℱ and that every map in ℱ is a q-cofibration, of course an acyclic one. Note the relevance of the first adjunction of II.2.2 when 𝒞 = ℳ_{S}: this is where the definition of q-fibrations in Theorem 4.6 is forced on us. Now use Lemma 5.2 to factor f. Then that lemma says that p is a q-fibration and that i satisfies the LLP with respect to all q-fibrations. In particular, i is a q-cofibration. We use the cylinders (?) ⊗ I to define homotopies in the category 𝒞 [𝕋], as discussed in Remark 2.12. Then the free functor 𝕋 and the adjoint forgetful functor preserve homotopies. A formal argument shows that each i_{n} is the inclusion of a deformation retraction of 𝕋-algebras, and it follows that i is also a deformation retraction. Therefore i is an acyclic q-cofibration.

Lemma 5.7. q-fibrations satisfy the RLP with respect to acyclic q-cofibrations.

Proof. This is formal. Let f : E ⟶ F be any acyclic q-cofibration. We must show that f satisfies the LLP with respect to q-fibrations. By the previous lemma, we may factor f as f = p ∘ i, where i : E ⟶ E′ is an acyclic q-cofibration that does satisfy the LLP with respect to q-fibrations and p : E′ ⟶ F is a q-fibration. Since f and i are weak equivalences, so is p. Since f satisfies the LLP with respect to acyclic q-fibrations, there exists g : F ⟶ E′ such that g ∘ f = i and p ∘ g = id_{F}. Clearly p and g, together with the identity map on E, express f as a retract of i. Since i satisfies the LLP with respect to q-fibrations, so does f.

Lemma 5.8. Any map f : X ⟶ Y in 𝒮 [𝕋] factors as p ∘ i, where i is a q-cofibration and p is an acyclic q-fibration.

Proof. This is another application of Lemma 5.2. Let 𝒜 ℱ be the set of pairs obtained by letting (B, A) in Definition 5.1 run through all pairs of spaces (CS^{n}, S^{n}), n ≥ 0. By tracing through adjunctions again, we see that a map of 𝕋-algebras is an acyclic q-fibration if and only if it satisfies the RLP with respect to all maps in 𝒜 ℱ and that each map in 𝒜 ℱ is thus a q-cofibration. In the factorization f = p ∘ i that we now obtain from Lemma 5.2, that lemma says that p is an acyclic q-fibration and i is a q-cofibration.

This completes the proofs of Theorems 4.4, 4.6, 4.7, and 4.9, and we turn to Theorem 4.14. As in the proof of Lemmas 5.2 and 5.8, a relative cell 𝕋-algebra E ⟶ E′ satisfies the LLP with respect to the acyclic q-fibrations and is thus a q-cofibration. Let f : E ⟶ F be a q-cofibration. The proof of Lemma 5.8 gives a factorization of f as the composite of a relative cell 𝕋-algebra i : E ⟶ E′ and an acyclic q-fibration p : E′ ⟶ F. As in the proof of Lemma 5.7, there exists g : F ⟶ E′ such that g ∘ f = i and p ∘ g = id_{F}, and p and g express f as a retract of i.

We must still prove that 𝒞 [𝕋] is topological, in the sense of Definition 4.2. As in [54, SM7(a), p.II.2.3], the description of q-cofibrations as retracts of relative cell 𝕋-algebras implies that we need only check that the map (4.3) is a Serre fibration when i : E ⟶ F is in the set ℱ defined in the proof of Lemma 5.6 and an acyclic Serre fibration when i : E ⟶ F is in the set 𝒜 ℱ defined in the proof of Lemma 5.8. The freeness adjunction for the monad 𝕋 reduces this to the case of spectra or of R-modules, and further adjunctions then reduce it to its known space level analog.

The reader should be convinced that the construction of model structures is a nearly formal consequence of the monadic descriptions of our various notions of structured ring, module, and algebra spectra.

6. THE UNDERLYING R-MODULES OF q-COFIBRANT R-ALGEBRAS 167

Remark 5.9. In [34], categories of A_{∞} and E_{∞} k-algebras and their modules were defined, and derived categories of modules were constructed, using a cell theory based on “sphere and cone modules”. Replacing the ground category 𝒮 with the ground category ℳ_{k} of differential graded k-modules, the arguments of this section apply to give model structures to the analogous algebraic categories. Unlike the treatments in [54, 5, 17], with this approach there is not the slightest reason to restrict attention to bounded below k-modules.

6. The underlying R-modules of q-cofibrant R-algebras

Let R be a fixed q-cofibrant commutative S-algebra. We study the underlying Rmodules of q-cofibrant R-algebras and commutative R-algebras. The main point is to prove that the point-set level iterated smash products of q-cofibrant R-algebras represent their smash product in the derived category 𝒟_{R}, but we also prove a key technical lemma on cofibrations.

We begin with the simpler non-commutative case, and we do not need R to be q-cofibrant in the following two results. Recall from III.7.3 that smash products of cell R-modules are cell R-modules.

Proposition 6.1. Let A and B be R-algebras that are cell R-modules relative to R. Then their coproduct A⨿B is a cell R-module relative to R. In more detail, A⨿B is the colimit of an expanding sequence of relative cell R-modules {C_{n}} such that C_{0} = R and, for n ≥ 1, C_{n}/C_{n−1} is the wedge of the two monomial word modules of length n in A/R and B/R.

Proof. The monomial word modules in R-modules M and N are the smash products

M ∧_{R} N ∧_{R} M ∧_{R} ⋯ and N ∧_{R} M ∧_{R} N ∧_{R} · · · .

By II.7.4, we see that A ⨿ B is constructed via a coequalizer diagram in ℳ_{R}

\xymatrix{𝕋(𝕋A ∨ 𝕋B) \ar[r] \ar[r] & 𝕋(A ∨ B) \ar[r] & A ⨿ B.}

Writing out the source and target of the pair of parallel arrows as wedges of smash products and restricting to those wedge summands with at most n smash factors, we define C_{n} to be the coequalizer of the resulting restricted parallel pair of arrows. Clearly there result compatible maps C_{n} ⟶ C_{n+1} and C_{n} ⟶ A ⨿ B such that A ⨿ B is the colimit of the C_{n}. In view of the use of the action maps 𝕋A ⟶ A and 𝕋B ⟶ B in II.7.4, we see that the wedge of the monomial words in A and B of length at most n maps onto C_{n}. That is, elements of word monomials involving A∧_{R}A or B∧_{R}B are identified in the coequalizer with elements of word monomials of lower length.

Let U_{n} be the coproduct in the category of R-modules under R of the two monomial words in A and B of length n, so that the copies of R in these R-modules under R are identified. Then U_{n} is a relative cell R-module. Let V_{n} be the union of the subcomplexes of U_{n} that are obtained by replacing any one A or B in either of the monomial words by its submodule R. The isomorphisms R ∧_{R} A ⟶ A and R ∧_{R} B ⟶ B induce a map V_{n} ⟶ C_{n}. Inspection of the restricted coequalizer diagrams (and comparison with algebra for intuition) shows that there result pushout diagrams of R-modules

\xymatrix{V_{n} \ar[r] \ar[d] & C_{n} \ar[d] \\ U_{n+1} \ar[r] & C_{n+1}.}

Inductively, the C_{n} and A ⨿ B are cell R-modules relative to R.

Theorem 6.2. If A is a q-cofibrant R-algebra, then A is a retract of a cell R-module relative to R. Thus the unit R ⟶ A is a q-cofibration of R-modules.

Proof. If M is a cell R-module, then M^{j} is a cell R-module for j ≥ 1 and (𝕋M, R) is a relative cell R-module. Moreover, since M ⟶ CM is cellular, 𝕋M ⟶ 𝕋CM is the inclusion of a subcomplex in a relative cell R-module. Now suppose that (A, R) is a relative cell R-module and that we have a pushout diagram of R-algebras

\xymatrix{𝕋M \ar[r] \ar[d] & A \ar[d] \\ 𝕋CM \ar[r] & B.}

As in the proof of Proposition 3.9, B is isomorphic to the geometric realization of a simplicial R-module that is proper because its degeneracies are given by inclusions of wedge summands. The previous proposition implies that its R-module of psimplices is a cell R-module relative to R. Moreover, the face and degeneracy maps are sequentially cellular. Therefore, by X.2.7, (B, R) is isomorphic to a relative cell R-module, and A is a subcomplex. By passage to colimits, any cell R-algebra is a relative cell R-module. The conclusion follows from Theorem 4.14.

In the commutative case, the argument fails because we must pass to orbits over actions of symmetric groups. Tracing the proof of III.7.3 back to that of I.6.1, we see that it depends on the homeomorphism ℒ (j) ≅ ℒ (1) induced by a linear isomorphism f : U^{j} ⟶ U. Since this homeomorphism is not Σ_{j}-equivariant, we cannot deduce that symmetric powers of CW 𝕃-spectra are, or even have the homotopy types of, CW 𝕃-spectra, although they do have the homotopy types of CW spectra. For this reason, we cannot conclude that the symmetric power M^{j}/Σ_{j} of a cell R-module M has the homotopy type of a cell R-module; we refer the reader ahead to VIII.2.7 for an analysis of the homotopy type of its underlying spectrum. We get around this problem by use of the following result, which gives canonical CW S-module approximations of smash products of “extended powers” of CW spectra. Here the jth extended power of X is defined to be

D_{j}X = (𝕃X)^{j}/Σ_{j} ≅ ℒ (j) ⋉_{Σ_{j}} X^{j}.

We adopt the convention that D_{0}X = S.

Theorem 6.3. Let {X_{1}, …, X_{n}} be CW spectra, let j_{i} ≥ 0, and consider the following commutative diagram of 𝕃-spectra:

[figure: ⋀_{ℒ} (S ∧_{ℒ} 𝕃D_{j_{i}}X_{i}); ∧_{ℒ} (id ∧_{ℒ} ξ); ∧_{ℒ} λ; ⋀_{ℒ} 𝕃D_{j_{i}}X_{i}; ∧_{ℒ} ξ; ⋀_{ℒ} (S ∧_{ℒ} D_{j_{i}}X_{i}); ∧_{ℒ} λ; ⋀_{ℒ} D_{j_{i}}X_{i}.]

All spectra in the diagram have the homotopy types of CW-spectra, all maps in the diagram are homotopy equivalences of spectra, and ∧_{S}(S ∧_{ℒ} 𝕃D_{j_{i}}X_{i}) has the homotopy type of a CW S-module.

Proof. By [37, VI.5.2 or VIII.2.4], X^{j} has the homotopy type of a Σ_{j}-CW spectrum indexed on U^{j}. By XI.1.7, ℒ (j) has the homotopy type of a Σ_{j}-CW complex. Therefore, by the equivariant form of I.2.6, ℒ (j)⋉X^{j} has the homotopy type of a Σ_{j}-CW spectrum indexed on U. Thus, by [37, I.5.6], D_{j}X has the homotopy type of a CW-spectrum. By I.4.7 and II.1.9, 𝕃D_{j}X has the homotopy type of a CW 𝕃-spectrum and S ∧_{ℒ} 𝕃D_{j}X has the homotopy type of a CW Smodule. These conclusions pass to smash products by I.6.1 and III.7.3. The top horizontal arrow is a homotopy equivalence of 𝕃-spectra by I.4.7 and I.8.5, and the bottom horizontal arrow is a homotopy equivalence of spectra by XI.2.4. We claim that the right vertical arrow and therefore the left vertical arrow are also homotopy equivalences of spectra. Indeed, if all j_{i} ≥ 1, then use of I.5.4 and I.5.6 shows that the right vertical arrow is isomorphic to

[figure: (ℒ (n) × ℒ (j_{1}) × ⋯ × ℒ (j_{n})) ⋉_{Σ_{j_{1}}×···×Σ_{jn}} (X_{1}^{j_{1}} ∧ ⋯ ∧ X_{n}^{j_{n}}); γ⋉id; ℒ (j_{1} + ⋯ + j_{n}) ⋉_{Σ_{j_{1}}×···×Σ_{jn}} (X_{1}^{j_{1}} ∧ ⋯ ∧ X_{n}^{j_{n}}).]

Since γ is a (Σ_{j_{1}} × ⋯ × Σ_{j_{n}})-equivariant homotopy equivalence, the map before passage to orbits is an equivariant homotopy equivalence by the equivariant version of I.2.5. If any j_{i} = 0, then use of I.6.1 reduces us to the case when a single j_{i} = 0, and in this case the conclusion follows from I.8.6.

Now return to consideration of our given q-cofibrant commutative S-algebra R.

Definition 6.4. Define ℰ_{R} to be the collection of R-modules of the form

R ∧_{S} (S ∧_{ℒ} D_{j}X),

where X is any spectrum of the homotopy type of a CW-spectrum and j ≥ 0. Define ℰ̄_{R} to be the closure of ℰ_{R} under finite ∧_{R}-products, wedges, pushouts along cofibrations, colimits of countable sequences of cofibrations, and homotopy equivalences, where all of these operations are taken in the category of R-modules. That is, if {M_{1}, …, M_{n}} ⊂ ℰ̄_{R}, then M_{1} ∧_{R} ⋯ ∧_{R} M_{n} ∈ ℰ̄_{R}, and so forth.

Observe that ℰ̄_{R} contains all R-modules of the homotopy types of cell R-modules, that being the collection that would be obtained if we only allowed j = 1 in our initial class. One point of the definition is the following observation. Its proof is just like that of Theorem 6.2, and we shall say more about the commutative case shortly.

Theorem 6.5. The underlying R-module of a q-cofibrant R-algebra or commutative R-algebra A is in ℰ̄_{R}.

Another point is the following reassuring consequence of Theorem 6.3 and the definition.

Proposition 6.6. The underlying spectrum of an R-module in ℰ̄_{R} has the homotopy type of a CW-spectrum.

These lead to the main point, which is that we have control of the behavior of derived smash product of R-modules that are in ℰ̄_{R}.

Theorem 6.7. Let R be a q-cofibrant commutative S-algebra. Choose a cell R-module ΓM and a weak equivalence of cell R-modules γ : ΓM ⟶ M for each M ∈ ℰ̄_{R}. Then, for any finite subset {M_{1}, …, M_{n}} of ℰ̄_{R},

γ ∧_{R} ⋯ ∧_{R} γ : ΓM_{1} ∧_{R} ⋯ ∧_{R} ΓM_{n} ⟶ M_{1} ∧_{R} ⋯ ∧_{R} M_{n}

is a weak equivalence of R-modules. That is, the derived smash product of the M_{i} in the category 𝒟_{R} is represented by their point-set level smash product.

Proof. When R = S and each M_{i} is in ℰ_{S}, Theorem 6.3 gives the conclusion. The conclusion for general M_{i} follows by standard commutation formulas relating smash products to the chosen operations. For general R and M_{i} = R ∧_{S} N_{i}, where N_{i} ∈ ℰ_{S} has CW S-approximation ΓN_{i}, R ∧_{S} ΓN_{i} is a CW R-approximation of M_{i}. Here we have the identification

(R ∧_{S} N_{1}) ∧_{R} ⋯ ∧_{R} (R ∧_{S} N_{n}) ≅ R ∧_{S} (N_{1} ∧_{S} ⋯ ∧_{S} N_{n}),

and similarly and compatibly for the ΓN_{i}. By Theorem 6.5, R is in ℰ̄_{S}, hence the result for S implies the result for these M_{i}. The result for general M_{i} follows as in the case R = S.

Observe that the γ and their smash products are necessarily homotopy equivalences of underlying spectra, since these are CW homotopy types.

We conclude with the following lemma on cofibrations. It will imply that the simplicial R-modules used in the following chapters are proper, in the sense of X.2.2. As explained at the start of X§2, we abuse language by writing about unions, images, and inclusions when we should be writing more precisely about maps from a suitable coend to A^{q}. The abuse is justified by the conclusion, since a cofibration of spectra is a spacewise closed inclusion [37, I.8.1].

Lemma 6.8. Let A be a q-cofibrant R-algebra or a q-cofibrant commutative Ralgebra. Let sA^{q} ⊂ A^{q} be the “union of the images” of the maps

s_{i} = (id)^{i} ∧ η ∧ (id)^{q−i} : A^{q−1} ⟶ A^{q}.

Then the “inclusion” sA^{q} ⊂ A^{q} is a cofibration of R-modules. In particular, the unit η : R ⟶ A is a cofibration of R-modules.

Proof. In the non-commutative case, (A, R) is a relative cell R-module. Its qth smash power inherits such a structure, by III.7.3, and sA^{q} is a subcomplex. In the commutative case, we can apply the same brief argument, once we observe that (A, R) is a suitably general kind of relative cell R-module. Thus we consider generalized relative cell R-modules that are constructed with (cell, sphere) pairs replaced by pairs of the form (N ∧ B_{+}^{q}, N ∧ S_{+}^{q−1}), where N runs through all finite smash products over R of R-modules of the form (S_{R}^{n})^{j}/Σ_{j} or (CS_{R}^{n})^{j}/Σ_{j} for integers n and for j ≥ 1. Here the (B^{q}, S^{q−1}) are ordinary space level (cell, sphere) pairs. Observe that these R-modules are finite colimits of compact R-modules, so that III.1.7 applies to them. Let ℙ be the monad in the category of R-modules that defines commutative R-algebras. Obviously ℙM and ℙCM are relative cell R-modules in this generalized sense when M is a wedge of sphere R-modules. Equally obviously, the smash product over R of two such generalized relative cell R-modules is another such. Suppose that A is such a commutative R-algebra and consider a pushout diagram

\xymatrix{ℙM \ar[r] \ar[d] & A \ar[d] \\ ℙCM \ar[r] & B.}

As explained in 3.5–3.9, B is isomorphic to the geometric realization of the proper simplicial R-module β_{∗}^{R}(ℙCM, ℙM, A). Remember that, here in the commutative case, the coproduct used in VII§3 is the smash product over R. We may construct the geometric realization by first using degeneracy identifications, which serve simply to eliminate redundant wedge summands from the relevant coend, and then face identifications; compare X.2.6. The latter identifications can be expressed by pushout diagrams of R-modules

[figure: β_{q}^{R}(ℙCM, ℙM, A) ∧ ∂Δ_{q+}; g; F_{q−1}β^{R}(ℙCM, ℙM, A); β_{q}^{R}(ℙCM, ℙM, A) ∧ Δ_{q+}; F_{q}β^{R}(ℙCM, ℙM, A).]

Of course, we think of (Δ_{q}, ∂Δ_{q}) as a model for (B^{q}, S^{q−1}). Proceeding inductively and using III.1.7 and the proof of III.2.2, we can make g a sequentially cellular map and deduce that the qth filtration is a generalized relative cell R-module. By passage to colimits, so is any commutative cell R-algebra. The same holds for smash powers. The inclusion of a subcomplex is a cofibration, by reduction to the obvious case of cell pairs, and the conclusion follows.

Remark 6.9. By their characterization in terms of the LLP, coproducts of qcofibrations are q-cofibrations In any model category. This observation may help clarify the previous result, since it ensures that the smash product over R of two q-cofibrant commutative R-algebras is again a q-cofibrant commutative R-algebra.

CHAPTER VIII Bousfield localizations of R-modules and algebras

We study Bousfield localizations in this chapter. For any S-algebra R and cell R-module E, we show that ℳ_{R} admits a new model category structure in which the weak equivalences are the E-equivalences. With this model structure, a factorization of the trivial map M ⟶ ∗ as the composite of an E-acyclic E-cofibration and an E-fibration constructs the localization of M at E.

Restricting to a q-cofibrant commutative S-algebra R, we combine formal constructions with the homotopical analysis of the previous chapter to prove that localizations at E of cell R-algebras can be constructed as cell R-algebras, and similarly for commutative cell R-algebras. Of course, this applies quite generally since any R-algebra is weakly equivalent to a cell R-algebra. That is, we can conclude that Bousfield localizations of R-algebras and commutative R-algebras are again such. In the case R = S, Hopkins and McClure had an unpublished argument, sketched in e-mails to us, that Bousfield localizations of E_{∞} ring spectra are E_{∞} ring spectra.

We deduce that the derived category of E-local R-modules is equivalent to the full subcategory of the derived category of R_{E}-modules whose objects are those R_{E}- modules that are E-local as R-modules. In particular our new derived categories of S_{E}-modules are intrinsically important to a complete understanding of the classical Bousfield localizations of spectra.

As a simple direct application, we deduce that KO and KU are commutative ko and ku-algebras since they are Bousfield localizations of ko and ku obtained by inverting the Bott elements. By neglect of structure, they are therefore commutative S-algebras. This solves a problem that was first studied in McClure’s 1978 PhD thesis.

We refer the reader to [25] for a discussion of the special cases of Bousfield localization that give localizations and completions of R-modules at ideals of the coefficient ring R_{∗}.

174 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS

1. Bousfield localizations of R-modules

Let R be an S-algebra, such as S itself, and suppose given a cell R-module E. We shall construct Bousfield localizations of R-modules at E. The treatment is based on Bousfield’s papers [11, 12], but in the latter he was handicapped by working in a primitive category of spectra that did not admit a model category structure.

A map f : M ⟶ N of R-modules is said to be an E-equivalence if the induced map

id ∧_{R}f : E ∧_{R} M ⟶ E ∧_{R} N

is a weak equivalence. Homologically, we should call such maps E_{∗}^{R}-equivalences, and we shall often refer to them as E-acyclic maps. An R-module W is said to be E-acyclic if E ∧_{R} W ≃ ∗, and a map f is E-acyclic if and only if its cofiber is Eacyclic. We say that an R-module L is E-local if f^{∗} : 𝒟_{R}(N, L) ⟶ 𝒟_{R}(M, L) is an isomorphism for any E-equivalence f or, equivalently, if 𝒟_{R}(W, L) = 0 for any Eacyclic R-module W. Since this is a derived category criterion, it suffices to test it when W is a cell R-module. A localization of M at E is a map λ : M ⟶ M_{E} such that λ is an E-equivalence and M_{E} is E-local. Of course, the formal properties of such localizations discussed in [11, 12] carry over verbatim to the present context. We shall construct a model structure on ℳ_{R} that implies the existence of Elocalizations of R-modules.

Theorem 1.1. The category ℳ_{R} admits a new structure as a topological model category in which the weak equivalences are the E-equivalences and the cofibrations are the q-cofibrations in the model structure already constructed. The fibrations in the new model structure are the maps that satisfy the right lifting property with respect to the E-acyclic q-cofibrations.

Although the theorem gives the best way to think about the new model structure, it will be convenient to construct it in a way that parallels the proofs in VII§5. To that end, we give apparently different definitions of E-fibrations and E-cofibrations.

Definition 1.2. A map f : M ⟶ N is an E-fibration if it has the right lifting property with respect to the E-acyclic inclusions of subcomplexes in cell R-modules. A map f : M ⟶ N is an E-cofibration if it satisfies the left lifting property with respect to the E-acyclic E-fibrations.

The following comparisons will emerge during our proof of Theorem 1.1.

Lemma 1.3. A map is an E-cofibration if and only if it is a q-cofibration.

Lemma 1.4. A map is an E-fibration if and only if it satisfies the RLP with respect to the E-acyclic q-cofibrations.

1. BOUSFIELD LOCALIZATIONS OF R-MODULES 175

An R-module L is said to be E-fibrant if the unique map L ⟶ ∗ is an Efibration.

Proposition 1.5. An R-module is E-fibrant if and only if it is E-local.

Proof. The argument is the same as that of [11, 3.5]. As in [11, 3.6], one checks that the class of E-equivalences in the derived category 𝒟_{R} admits a calculus of left fractions. This implies [11, 2.5] that an R-module L is E-local if and only if f^{∗} : 𝒟_{R}(N, L) ⟶ 𝒟_{R}(M, L) is a surjection for any E-equivalence f : M ⟶ N. Since we are working in derived categories, there is no loss of generality to assume that f is the inclusion of a subcomplex in a cell R-module. If L is E-fibrant, the RLP already gives surjectivity on the point-set level, hence on the level of homotopy classes. If L is E-local, we have surjectivity on the level of homotopy classes and deduce it on the point-set level by use of HEP.

Theorem 1.6. Every R-module M admits a localization λ : M ⟶ M_{E}.

Proof. We may factor the trivial map M ⟶ ∗ as the composite of an Eacyclic E-cofibration λ : M ⟶ M_{E} and an E-fibration M_{E} ⟶ ∗.

Localizations of the underlying spectra of R-modules at spectra can be recovered as special cases of our new localizations of R-modules at R-modules. Therefore, up to equivalence, the localization of an R-module at a spectrum can be constructed as a map of R-modules.

Proposition 1.7. Let K be a CW-spectrum and let E be the R-module 𝔽_{R}K. Regarded as a map of spectra, a localization λ : M ⟶ M_{E} of an R-module M at E is a localization of M at K.

Proof. By IV.1.9, we have K_{∗}(M) ≅ E_{∗}^{R}(M) for R-modules M. Therefore an E-equivalence of R-modules is a K-equivalence of spectra. If W is a K-acyclic spectrum, then 𝔽_{R}W is an E-acyclic R-module since E ∧_{R} 𝔽_{R}W is equivalent to 𝔽_{R}(K ∧W). Therefore, if N is an E-local R-module, then [W, N] ≅ [𝔽_{R}W, N]_{R} = 0 and N is a K-local spectrum. The conclusion follows.

The argument generalizes to show that, for an R-algebra A, the localization of an A-module at an R-module E can be constructed as a map of A-modules.

Proposition 1.8. Let A be a q-cofibrant commutative R-algebra, let E be a cell R-module and let F be the A-module A ∧_{R} E. Regarded as a map of R-modules, a localization λ : M ⟶ M_{F} of an A-module M at F is a localization of M at E.

We prove Theorem 1.1 in the rest of the section. Of course, ℳ_{R} is topologically complete and cocomplete. It is clear that retracts of E-equivalences, E-cofibrations, and E-fibrations are again such (and similarly with cofibrations and fibrations as in the statement of the theorem). The following result motivates our definition of Efibrations in terms of inclusions of subcomplexes rather than general q-cofibrations. Let #X denote the cardinality of the set of cells of a cell R-module X and let c be a fixed infinite cardinal that is at least the cardinality of E_{∗}^{R}(S_{R}). Long exact sequences and the commutation of homology with colimits imply that if X ⟶ Y is the inclusion of a subcomplex in a cell R-module Y such that #Y ≤ c, then the cardinality of E_{∗}^{R}(Y/X) is at most c. Let 𝒯 be the set of E-acyclic inclusions of subcomplexes in cell R-modules Y such that #Y ≤ c. Then 𝒯 is a test set for E-fibrations.

Lemma 1.9. A map f : M ⟶ N is an E-fibration if and only p has the RLP with respect to maps in 𝒯 .

Proof. Arguing as in [11, 11.2,11.3], we see that for any proper inclusion X ⟶ Y of a subcomplex in a cell R-module Y, there is a subcomplex X′ ⊂ Y such that #X′ ≤ c, X′ is not contained in X, and X′ ∩ X ⟶ X′ is E-acyclic. We construct X′ as the union of a sequence of subcomplexes X_{n} of Y such that #X_{n} ≤ c, X_{n} is not contained in X, and the map

E_{∗}^{R}(X_{n}/X_{n} ∩ X) ⟶ E_{∗}^{R}(X_{n+1}/X_{n+1} ∩ X)

is zero. The fact that homology commutes with colimits and that E_{∗}^{R}(Y/X) = 0 allows us to perform the inductive step by adjoining finite subcomplexes of Y to X_{n} to kill elements of E_{∗}^{R}(X_{n}/X_{n} ∩ X). We conclude that if f has the RLP with respect to maps in 𝒯 , then it has the RLP with respect to X ⟶ X′ ∪ X since it has the RLP with respect to X′ ∩ X ⟶ X′. By transfinite induction, it follows that f has the RLP with respect to X ⟶ Y.

Lemma 1.10. Any map f : M ⟶ N factors as a composite

M⟶^{i}M′⟶^{p}N,

where p is an E-fibration and i is an E-acyclic q-cofibration that satisfies the LLP with respect to E-fibrations.

Proof. The construction is exactly like that in the proof of VII.5.2, with 𝒯 here playing the role of ℱ there. However, since we do not have compactness, we must perform the construction transfinitely. We carry the construction through to the least ordinal of cardinality greater than c. We can then use set theory rather than (VII.5.5) to ensure the requisite factorization α′ in the cited proof. The resulting map p is certainly an E-fibration. The construction by successive pushouts of wedges of maps in 𝒯 , for ordinals n with successors, and passage to colimits, for limit ordinals, shows that the constructed map i is E-acyclic and satisfies the LLP with respect to the E-fibrations. To see that i, despite its transfinite construction, is a q-cofibration, we must specify a sequential filtration inductively. Given the sequential filtration on M_{n}, where n has a successor, we obtain a sequential filtration on M_{n+1} by using III.2.3 to arrange that the pushout that constructs M_{n+1} is a diagram of sequentially cellular maps. If n is a limit ordinal and we have compatible sequential filtrations on the M_{m} for m < n, then each cell of M_{n} has a preassigned filtration and we take the qth filtration of M_{n} to be the union of the qth filtrations of the M_{m} for m < n.

Remark 1.11. The previous proof uses that if i : X ⟶ Y is an E-acyclic qcofibration and f : X ⟶ M is any map, then the pushout j : M ⟶ M ∪_{X} Y is Eacyclic. This holds because i and j are cofibrations of R-modules with isomorphic quotients M/X ≅ (M ∪_{X} Y)/M.

Lemma 1.12. The following conditions on a map f : M ⟶ N are equivalent.

(i) f is an E-acyclic E-fibration.

(ii) f is an E-acyclic map that satisfies the RLP with respect to all q-cofibrations. (iii) f is an acyclic q-fibration.

Proof. Obviously (ii) implies (i) and (iii) implies (ii). Assume (i). Clearly f is a q-fibration, and we must prove that it is a weak equivalence. By factoring a weak equivalence from a cell R-module to M as the composite of an acyclic q-cofibration and a q-fibration, we can construct an acyclic q-fibration p : M′ ⟶ M, where M′ is a cell R-module. By VII.5.8, we may factor f ∘ p as the composite of a q-cofibration i : M′ ⟶ N′ and an acyclic q-fibration p′ : N′ ⟶ N. By III.2.3 and the proof of VII.5.2, we can arrange that N′ is a cell R-module that contains M′ as a subcomplex. Since f ∘ p is E-acyclic, so is i. Summarizing, we have the diagram

[figure: M′; i; M′; f∘p; p; M; f; N′; r; p′; N; N.]

Since f ∘ p is an E-fibration and i is an E-acyclic inclusion of a subcomplex in a cell R-module, there exists a lift r. This expresses f ∘ p as a retract of the weak equivalence p′. Therefore f ∘ p and f are weak equivalences.

Observe that Lemma 1.3 is an immediate consequence of Lemma 1.12.

Proof of Lemma 1.4. It suffices to show that an E-fibration satisfies the RLP with respect to all E-acyclic q-cofibrations i : X ⟶ Y. Factor i as in Lemma 1.8 and consider the resulting diagram

[figure: X; j; i; X′; p; Y; ι; Y,]

where p is an E-fibration and j is an E-acyclic q-cofibration that satisfies the LLP with respect to the E-fibrations. Clearly p is E-acyclic, hence Lemma 1.10 implies that it satisfies the RLP with respect to the cofibration i. There results a lift ι, and ι and p show that i is a retract of j. Since j satisfies the LLP with respect to all E-fibrations, so does i.

Proof of Theorem 1.1. We have proven one of the factorization axioms in Lemma 1.10, and the remaining axioms for a model structure are now direct consequences of the corresponding axioms for the original model structure on ℳ_{R}.

2. Bousfield localizations of R-algebras

In this section, we restrict R to be a q-cofibrant commutative S-algebra and let E be a cell R-module. We shall prove the following pair of theorems.

Theorem 2.1. For a cell R-algebra A, the localization λ : A ⟶ A_{E} can be constructed as the inclusion of a subcomplex in a cell R-algebra A_{E}. Moreover, if f : A ⟶ B is a map of R-algebras into an E-local R-algebra B, then f lifts to a map of R-algebras f̃: A_{E} ⟶ B such that f̃∘ λ = f; if f is an E-equivalence, then f̃ is a weak equivalence.

Theorem 2.2. For a commutative cell R-algebra A, the localization λ : A ⟶ A_{E} can be constructed as the inclusion of a subcomplex in a commutative cell Ralgebra A_{E}. Moreover, if f : A ⟶ B is a map of R-algebras into an E-local commutative R-algebra, then f lifts to a map of R-algebras f̃ : A_{E} ⟶ B such that f̃∘ λ = f; if f is an E-equivalence, then f̃ is a weak equivalence.

Proofs. The idea is to replace the category ℳ_{R} by either the category 𝒜_{R} or 𝒞 𝒜_{R} in the work of the previous section. Most arguments go through with little change, the crucial exception being the part of the proof of Lemma 1.10 that is singled out in Remark 1.11. The problem there is that, to prove the analogue of the cited lemma in full generality, we would have to allow A to be an arbitrary R-algebra or commutative R-algebra. However, to keep homotopical control, we need A to be a cell R-algebra. This is enough to prove our theorems, although we don’t actually obtain new model structures on 𝒜_{R} and 𝒞 𝒜_{R}.

2. BOUSFIELD LOCALIZATIONS OF R-ALGEBRAS 179

For definiteness, consider the non-commutative case. Proceeding as in the proofs of VII.5.2 and Lemma 1.10, we let A_{0} = A and construct a transfinite sequence

[figure: A_{0}; i_{0}; A_{1}; ⋯; A_{n}; i_{n}; A_{n+1}; ⋯] (2.3)

as follows. Suppose inductively that we have constructed A_{n} and that n has a successor. Consider all diagrams of R-modules

[figure: Y; X; i; α; A_{n},] (2.4)

where i is in 𝒯 . Using the free R-algebra functor 𝕋 on R-modules, we take the sum over such diagrams of the adjoint maps α̃ : 𝕋X ⟶ A_{n} and construct the pushout diagram of R-algebras

[figure: ∐ 𝕋X; ^{∑ α̃}; A_{n}; i_{n}; ∐ 𝕋Y; A_{n+1}.] (2.5)

If n is a limit ordinal and A_{m} has been constructed for m < n, we let A_{n} be the colimit of the A_{m}. We take A_{E} to be the colimit up to the least ordinal of cardinality greater than c. Then any map of R-modules from a cell R-module X with #X ≤ c into A_{E} factors through some A_{n}, and we let λ : A ⟶ A_{E} be the canonical map. Regarded as an R-module, A_{E} is E-fibrant and therefore E-local. To see this, consider a diagram of R-modules

[figure: X; α′; i; A_{E},; Y; g]

where i is in 𝒯 . We must construct a map g that makes the diagram commute. Since α′ factors through some A_{n}, we have one of the diagrams (2.4) used in the construction of A_{n+1}. By construction, there is a map 𝕋Y ⟶ A_{n+1} the adjoint of whose composite with the natural map to A_{E} is a map g as required. By arguments like those in Lemmas 1.10 and 1.12, i : A ⟶ A_{E} is the inclusion of a subcomplex in a cell R-algebra.

We must prove that i is an E-equivalence. By the commutation of homology with directed colimits, only the pushout maps A_{n} ⟶ A_{n+1} are at issue. Observe that ∐ 𝕋X ≅ 𝕋(∨X). Lemma 2.6 below shows that the left vertical arrow in (2.5) is an E-equivalence. In the commutative case, we must replace 𝕋 by ℙ, and here Lemma 2.7 shows that the left vertical arrow in the commutative analogue of (2.5) is an E-equivalence. Finally, in both cases, Lemma 2.9 shows that the right vertical arrow A_{n} ⟶ A_{n+1} is an E-equivalence.

We prove the lifting statement for a map f : A ⟶ B by inductively lifting f to maps f_{n} : A_{n} ⟶ B. We proceed by passage to colimits when n is a limit ordinal. For the inductive step when f_{n} : A_{n} ⟶ B is given and n has a successor, we apply the fact that B is E-fibrant to lift the evident composites

X ⟶ 𝕋X ⟶ A_{n} ⟶ B

to maps of R-modules Y ⟶ B and then apply freeness to obtain maps of Ralgebras 𝕋Y ⟶ B that lift the maps of R-algebras 𝕋X ⟶ B. Passage to pushouts then gives the required map f_{n+1} : A_{n+1} ⟶ B. It is clear that f̃ must be an E-equivalence and therefore a weak equivalence if f is an E-equivalence.

Lemma 2.6. If f : M ⟶ N is an E-equivalence of cell R-modules, then 𝕋f : 𝕋M ⟶ 𝕋N is an E-equivalence of R-modules.

Proof. If f : M ⟶ N and f′ : M′ ⟶ N′ are E-equivalences, then, factoring

id ∧_{R}f ∧_{R} f′ : E ∧_{R} M ∧_{R} M′ ⟶ E ∧_{R} N ∧_{R} N′

as the composite (id ∧_{R}f ∧_{R} id)(id ∧_{R} id ∧_{R}f′) and using the commutativity and associativity of ∧_{R} and the fact that all R-modules in sight are cell R-modules, we see that id ∧_{R}f ∧_{R} f′ is an equivalence and thus that f ∧_{R} f′ is an E-equivalence. Inductively, f^{j} : M^{j} ⟶ N^{j} is an E-equivalence for all j ≥ 0.

Lemma 2.7. If f : M ⟶ N is an E-equivalence of cell R-modules, then ℙf : ℙM ⟶ ℙN is an E-equivalence of R-modules.

Proof. We must show that f^{j}/Σ_{j} : M^{j}/Σ_{j} ⟶ N^{j}/Σ_{j} is an E-equivalence for all j ≥ 0. By III.6.1, this will hold if

id ∧f^{j} : (EΣ_{j})_{+} ∧_{Σ_{j}} M^{j} ⟶ (EΣ_{j})_{+} ∧_{Σ_{j}} N^{j}

is an E-equivalence. By the previous proof, we have an E-equivalence before passage to orbits. Using the skeletal filtration of EΣ_{j}, we may set up a natural spectral sequence

H_{∗}(Σ_{j}; E_{∗}^{R}(M^{j})) ⟹ E_{∗}^{R}((EΣ_{j})_{+} ∧_{Σ_{j}} M^{j})

and so deduce the conclusion.

Lemma 2.8. Suppose given a pushout diagram of R-algebras

[figure: A; i; f; C; j; B; g; D,]

where i is an E-acyclic inclusion of a subcomplex in a cell R-algebra and C is a cell R-algebra. Then j is E-acyclic. The same conclusion holds in the case of commutative R-algebras.

Proof. Recall the bar construction β^{R}(B, A, C) of VII.3.5. By definition or by VII.3.7, we may interpret β^{R}(B, A, C) as the homotopy pushout of i and f. Since i is a cofibration of R-algebras, the natural map β^{R}(B, A, C) ⟶ D is a homotopy equivalence of R-algebras under C. Moreover, the map C ⟶ β^{R}(B, A, C) factors as the composite of a map η : C ⟶ β^{R}(A, A, C) and the map

β^{R}(i, id, id) : β^{R}(A, A, C) ⟶ β^{R}(B, A, C).

Here η is a homotopy equivalence of R-modules by XII.1.2 and X.1.2. The map β^{R}(i, id, id) is the geometric realization of a map of proper simplicial R-modules, where properness is defined in X.2.2. Properness holds in the case of R-algebras since, by VII.6.2, the inclusions of degeneracy sub R-modules are inclusions of subcomplexes in relative cell R-modules. It holds in the case of commutative Ralgebras by VII.6.8. The smash product over R with E commutes with geometric realization by X.1.4. Since i is an E-equivalence, we find in the R-algebra case that the map E ∧_{R} β_{q}^{R}(i, id, id) on q-simplices is a homotopy equivalence for each q because it is a weak equivalence between relative cell R-modules. In the commutative case, this map is the smash product over R of the weak equivalence E ∧_{R} A ⟶ E ∧_{R} B with the identity map on A^{q} ∧_{R} C and is therefore a weak equivalence by VII.6.7. In either case, we conclude from X.2.4 that β^{R}(i, id, id) is an E-equivalence.

3. Categories of local modules

Again, let R be a q-cofibrant commutative S-algebra and E be a cell R-module. let R_{E} be a q-cofibrant commutative R-algebra whose unit is a localization of R at E. The fact that Bousfield localization preserves R-algebras and commutative R-algebras gives a powerful tool for the construction of new R-algebras and opens up a new approach to the study of Bousfield localizations.

To see the latter, let us compare the derived category 𝒟_{R_{E}} to the stable homotopy category 𝒟_{R}[E^{−1}] associated to the model structure on ℳ_{R} determined by E. Here 𝒟_{R}[E^{−1}] is obtained from 𝒟_{R} by inverting the E-equivalences and is equivalent to the full subcategory of 𝒟_{R} whose objects are the E-local R-modules. Observe that, for a cell R-module M, we have the canonical E-equivalence

ξ = η ∧ id : M ≅ R ∧_{R} M ⟶ R_{E} ∧_{R} M.

The following observation is the same as in the classical case.

Lemma 3.1. If M is a finite cell R-module, then R_{E} ∧_{R} M is E-local and therefore ξ is the localization of M at E.

Proof. If W is an E-acyclic R-module, then

𝒟_{R}(W, R_{E} ∧_{R} M) ≅ 𝒟_{R}(W ∧_{R} D_{R}M, R_{E}) = 0

since W ∧_{R} D_{R}M is E-acyclic and R_{E} is E-local.

We say that localization at E is smashing if, for all cell R-modules M, R_{E} ∧_{R} M is E-local and therefore ζ is the localization of M at E. The following observation is due to Wolbert [69].

Proposition 3.2 (Wolbert). If localization at E is smashing, then the categories 𝒟_{R}[E^{−1}] and 𝒟_{R_{E}} are equivalent.

These categories are closely related even when localization at E is not smashing, as the following elaboration of Wolbert’s result shows.

Theorem 3.3. The following three categories are equivalent.

(i) The category 𝒟_{R}[E^{−1}] of E-local R-modules.

(ii) The full subcategory 𝒟_{R_{E}}[E^{−1}] of 𝒟_{R_{E}} whose objects are the R_{E}-modules

that are E-local as R-modules. (iii) The category 𝒟_{R_{E}}[(R_{E} ∧_{R} E)^{−1}] of (R_{E} ∧_{R} E)-local R_{E}-modules.

This implies that the question of whether or not localization at E is smashing is a question about the category of R_{E}-modules, and it leads to the following factorization of the localization functor. In the classical case R = S, this shows that our new commutative S-algebras S_{E} and their categories of modules are intrinsic to the theory of Bousfield localization.

Theorem 3.4. The E-localization functor 𝒟_{R} ⟶ 𝒟_{R}[E^{−1}] is equivalent to the composite of the extension of scalars functor

R_{E} ∧_{R} (?) : 𝒟_{R} ⟶ 𝒟_{R_{E}}

and the (R_{E} ∧_{R} E)-localization functor

(?)_{R_{E}∧_{R}E} : 𝒟_{R_{E}} ⟶ 𝒟_{R_{E}}[(R_{E} ∧_{R} E)^{−1}].

Corollary 3.5. Localization at E is smashing if and only if all R_{E}-modules are E-local as R-modules, so that

𝒟_{R}[E^{−1}] ≈ 𝒟_{R_{E}} ≈ 𝒟_{R_{E}}[(R_{E} ∧_{R} E)^{−1}].

The proofs of the results above are based on the following generalization of III.4.4 and a special case of III.4.1 from the ground category S to the ground category R. The proofs are the same as those of the cited results.

Proposition 3.6. Let A be an R-algebra with unit η : R ⟶ A. The forgetful functor η^{∗} : 𝒟_{A} ⟶ 𝒟_{R} has the functor A ∧_{R} (?) : 𝒟_{R} ⟶ 𝒟_{A} as left adjoint, and it also has a right adjoint η^{#} : 𝒟_{R} ⟶ 𝒟_{A}.

Proof of Theorem 3.3. We apply the previous result to η : R ⟶ R_{E}, obtaining

𝒟_{R_{E}}(N, η^{#}M) ≅ 𝒟_{R}(η^{∗}N, M)

and

𝒟_{R_{E}}(R_{E} ∧_{R} M, N) ≅ 𝒟_{R}(M, η^{∗}N)

for R-modules M and R_{E}-modules N. We claim that the functors η^{∗} and η^{#} of the first adjunction restrict to give inverse adjoint equivalences between 𝒟_{R}[E^{−1}] and 𝒟_{R_{E}}[E^{−1}], and we also claim that an R_{E}-module N is (R_{E} ∧_{R} E)-acyclic if and only if η^{∗}N is E-acyclic. These claims will give the conclusion.

If W is an E-acyclic R-module, then R_{E}∧_{R}W is an (R_{E}∧_{R}E)-acyclic R_{E}-module since

(R_{E} ∧_{R} W) ∧_{R_{E}} (R_{E} ∧_{R} E) ≅ R_{E} ∧_{R} (W ∧_{R} E) ≃ ∗.

Therefore, by the second adjunction, if N is an (R_{E} ∧_{R} E)-local R_{E}-module, then η^{∗}N is an E-local R-module.

If V is an (R_{E} ∧_{R} E)-acyclic R_{E}-module, then η^{∗}V is an E-acyclic R-module since

(η^{∗}V) ∧_{R} E ≅ η^{∗}(V ∧_{R_{E}} (R_{E} ∧_{R} E)) ≃ ∗.

Therefore, by the first adjunction, if M is an E-local R-module, then η^{#}M is an (R_{E} ∧_{R} E)-local R_{E}-module and thus η^{∗}η^{#}M is again an E-local R-module.

We claim that if if M is E-local, then the counit ε : η^{∗}η^{#}M ⟶ M of the first adjunction is a weak equivalence of R-modules. To see this, consider the commutative diagrams:

[figure: 𝒟_{R_{E}}(R_{E} ∧_{R} S^{q}, η^{#}M); η^{∗}; ≅; 𝒟_{R}(η^{∗}(R_{E} ∧_{R} S^{q}), η^{∗}η^{#}M); ξ^{∗}; ε_{∗}; 𝒟_{R}(S^{q}, η^{∗}η^{#}M); ε_{∗}; 𝒟_{R}(η^{∗}(R_{E} ∧_{R} S^{q}), M); ξ^{∗}; 𝒟_{R}(S^{q}, M).]

The left vertical composite ε_{∗} ∘ η^{∗} is an instance of the first adjunction. The right diagonal is an instance of the second adjunction. The horizontal arrows are induced by ξ : S^{q} ⟶ R_{E} ∧_{R} S^{q} and are isomorphisms since ξ is localization at E and M and η^{∗}η^{#}M are E-local. Therefore the maps ε_{∗} in the diagram are isomorphisms and ε is a weak equivaleance.

If N is an R_{E}-module such that η^{∗}N is E-local, then the unit ζ : N ⟶ η^{#}η^{∗}N of the first adjunction is a weak equivalence since the composite

η^{∗}N⟶^{η^{∗}ζ}η^{∗}η^{#}η^{∗}N⟶^{ε}η^{∗}N

is the identity and ε is a weak equivalence. Since η^{#}η^{∗}N is (R_{E} ∧_{R} E)-local, this also implies that N is (R_{E} ∧_{R} E)-local and so completes the proof.

Proof of Theorem 3.4. Any E-local R-module is isomorphic in 𝒟_{R} to one of the form η^{∗}N, where N is an R_{E}-module that is E-local as an R-module. Therefore the E-localization of any R-module M is given by a map λ : M ⟶ M_{E}, where M_{E} is an R_{E}-module. Such a map λ factors uniquely through a map λ̃ : R_{E} ∧_{R} M ⟶ M_{E} in 𝒟_{R_{E}}. Clearly λ̃ is an E-equivalence of R-modules and therefore an (R_{E} ∧_{R} E)-equivalence of R_{E}-modules. This proves the claimed factorization of localization at E.

Proof of Corollary 3.5. Localization at E is smashing if and only if all R-modules of the form R_{E} ∧_{R} M for an R-module M are E-local. In this case, if M is an R_{E}-module, then, as an R-module, M is a retract of R_{E} ∧_{R} M and is therefore also E-local.

4. Periodicity and K-theory

We illustrate the constructive power of our results on R-algebras by showing that the algebraic localizations of R considered in Chapter V take R to commutative R-algebras on the point set level and not just on the homotopical level studied in V.2.3. Thus let X be a set of elements of R_{∗} and consider R[X^{−1}]. We saw in V.2.3

4. PERIODICITY AND K-THEORY 185

that R[X^{−1}] is a commutative and associative R-ring spectrum whose product is an equivalence under R. For an R-module M, we have the canonical map

λ = λ ∧_{R} id : M ≅ R ∧_{R} M ⟶ R[X^{−1}] ∧_{R} M.

Proposition 4.1. For any R-module M, λ is the Bousfield localization of M at R[X^{−1}].

Proof. Upon smashing with R[X^{−1}], λ becomes an equivalence with inverse given by the product on R[X^{−1}]. Thus λ is an R[X^{−1}]-equivalence. By a standard argument, R[X^{−1}]∧_{R}M is R[X^{−1}]-local since it is an R[X^{−1}]-module spectrum.

So far we have been working homotopically, in the derived category 𝒟_{R}. Theorem 2.1 allows us to translate to the point-set level: R is a cell R-algebra, so its localization at E = R[X^{−1}] can be constructed as a map of R-algebras. We are entitled to the following conclusion.

Theorem 4.2. The localization R ⟶ R[X^{−1}] can be constructed as the unit of a cell R-algebra.

By multiplicative infinite loop space theory [49] and our model category structure on the category of S-algebras, the spectra ko and ku that represent real and complex connective K-theory can be taken to be q-cofibrant commutative S-algebras. It is standard (see e.g. [47, II§3]) that the spectra that represent periodic K-theory can be reconstructed up to homotopy by inverting the Bott element β_{O} ∈ π_{8}(ko) or β_{U} ∈ π_{2}(ku). That is,

KO ≃ ko[β_{O}^{−1}] and KU ≃ ku[β_{U}^{−1}].

We are entitled to the following result as a special case of the previous one.

Theorem 4.3. The spectra KO and KU can be constructed as commutative ko and ku-algebras.

Restricting the unit maps ko ⟶ KO and ku ⟶ KU along the unit maps S ⟶ ko and S ⟶ ku, we see that KO and KU are commutative S-algebras. McClure studied the problem of obtaining such a structure in his thesis. He proved that KO and KU are H_{∞} ring spectra, this being a weakened up-to-homotopy analogue of the notion of an E_{∞} ring spectrum, with some additional structure; see [14, VII§7]. More recently, in unpublished work, he returned to the problem and proved that the completion of KU at a prime p is an E_{∞} ring spectrum. Of course, this also follows from our work since completion at p is another example of a Bousfield localization.

Wolbert [69] is studying the algebraic structure of the derived categories of modules over the connective and periodic versions of the real and complex K-theory S-algebras.

Remark 4.4. It is often important in algebraic K-theory to invert a “Bott element” β ∈ K_{2}(R), where R is a suitable discrete ring. Since multiplicative infinite loop space theory implies that the algebraic K-theory spectrum KR can be taken to be a q-cofibrant commutative S-algebra, our arguments construct KR[β^{−1}] as a commutative KR-algebra.

Remark 4.5. For finite groups G, Theorem 4.3 applies with the same proof to construct the periodic spectra KO_{G} and KU_{G} of equivariant K-theory as commutative ko_{G} and ku_{G}-algebras. As explained in [26], this leads to an elegant proof of the Atiyah-Segal completion theorem in equivariant K-cohomology and of its analogue for equivariant K-homology.

CHAPTER IX Topological Hochschild homology and cohomology

As another application of our theory, we construct the topological Hochschild homology of R-algebras with coefficients in bimodules. The relevance to THH of a theory such as ours has long been known. McClure and Staffeldt gave a good introduction of ideas in [53, §3], and in fact our paper provides the foundations that were optimistically assumed in theirs (with reference to a four author paper in preparation that will never exist). Our analogue of Bökstedt’s topological Hochschild homology [8] is under active investigation by a number of people, and we shall just lay the foundations.

Actually, in Sections 1 and 2, we give two different constructions. For a qcofibrant commutative S-algebra R, a q-cofibrant R-algebra A, and an (A, A)- bimodule M, we first define T HH^{R}(A; M) to be the derived smash product M ∧_{A^{e}} A, exactly as in algebra (for flat algebras over rings). With this definition, we prove that algebraic Hochschild homology can be realized as the homotopy groups of the topological Hochschild homology of suitable Eilenberg-Mac Lane spectra, and we construct spectral sequences for the calculation of the homotopy and homology groups of T HH^{R}(A; M) in general.

We then define thh^{R}(A; M) by mimicking the standard complex for the computation of algebraic Hochschild homology and we prove that, when M is a cell A^{e}-module, thh^{R}(A; M) and T HH^{R}(A; M) are equivalent. When M = A, the resulting construction thh^{R}(A) has exceptionally nice formal properties. For example, it is immediate from the construction that thh^{R}(A) is a commutative Ralgebra when A is. While A is not equivalent to a cell A^{e}-module, we shall use our standing hypotheses that R and A are q-cofibrant to prove that T HH^{R}(A) is equivalent to thh^{R}(A).

Further formal properties of thh^{R}(A) are explained in the brief Section 3, which contains the results of the recent paper [52] of McClure, Schwänzl, and Vogt. They exploit the tensor structure of the category of commutative S-algebras to give a conceptual description of thh^{R}(A) as A ⊗ S^{1} when A is a commutative R-algebra. Their paper was based on the now obsolete definitions of a preliminary draft of our paper, it had some errors of detail, and we have since found simpler ways to carry out their intriguing application of our theory.

1. Topological Hochschild homology: first definition

We assume given an algebra A over a commutative S-algebra R and an (A, A)- bimodule M. We here define the topological Hochschild homology and cohomology spectra T HH^{R}(A; M) and T HH_{R}(A; M). The former presumably generalizes Bökstedt’s original definition [8] (see also [10]), although a precise comparison has not been established. There is no precursor of the latter in the literature. We mimic the conceptual definition of Hochschild homology and cohomology given by Cartan-Eilenberg [15, IX§§3-4]. In the next section, we give an alternative construction that mimics Hochschild’s original definition in terms of the standard complex [30] and compare definitions.

We are only interested in relative (A, A)-bimodules, that is, those for which the induced left and right actions of R agree under transposition of M and R, and we define the enveloping R-algebra of A by

A^{e} = A ∧_{R} A^{op}.

Then an (A, A)-bimodule M can be viewed as either a left or a right A^{e}-module. We usually view A itself as a left A^{e}-module and our general bimodule M as a right or a left A^{e}-module, whichever is convenient. If A is commutative, then A^{e} = A ∧_{R} A, the product A^{e} ⟶ A is a map of R-algebras, and A can be viewed as an (A^{e}, A)-bimodule.

We assume once and for all that our given commutative S-algebra R is q-cofibrant in the model category of commutative S-algebras and that A is q-cofibrant in the model category of R-algebras or of commutative R-algebras. There is no loss of generality in these assumptions since we could replace any given pair (A, R) by a weakly equivalent pair that satisfies our assumptions. By VII.6.7, these assumptions ensure that if γ : ΓA ⟶ A is a weak equivalence of R-modules, where ΓA is a cell R-module, then

γ ∧ γ : ΓA ∧_{R} ΓA ⟶ A ∧_{R} A

is a weak equivalence of R-modules. Thus the underlying R-module of A^{e} represents A ∧_{R} A in the derived category 𝒟_{R}.

Definition 1.1. Working in derived categories, define topological Hochschild homology and cohomology with values in 𝒟_{R} by

T HH^{R}(A; M) = M ∧_{A^{e}} A and T HH_{R}(A; M) = F_{A^{e}}(A, M).

If A is commutative, then these functors take values in the derived category 𝒟_{A^{e}}. On passage to homotopy groups, define

T HH_{∗}^{R}(A; M) = Tor_{∗}^{A^{e}}(M, A) and T HH_{R}^{∗}(A; M) = Ext_{A^{e}}^{∗}(A, M).

When M = A, we delete it from the notations.

Since we are working in derived categories, we are implicitly taking M to be a cell A^{e}-module in the definition of T HH^{R}(A; M) and approximating A by a weakly equivalent cell A^{e}-module in the definition of T HH_{R}(A; M). When A is commutative, we have the following observation, which will be amplified in the next section.

Proposition 1.2. If A is a commutative R-algebra, then T HH^{R}(A) is isomorphic in 𝒟_{A^{e}} to a commutative A^{e}-algebra.

Proof. By VII.6.9, A^{e} is a q-cofibrant commutative R-algebra since A is assumed to be one, and A is clearly a commutative A^{e}-algebra. Let ΨA ⟶ A be a weak equivalence of A^{e}-algebras from a q-cofibrant commutative A^{e}-algebra ΨA to A. Then, by VII.6.5 and VII.6.7, the commutative A^{e}-algebra ΨA ∧_{A^{e}} ΨA is isomorphic in 𝒟_{A^{e}} to T HH^{R}(A).

The module structures on T HH^{R}(A; M) have the following standard implication.

Proposition 1.3. If either R or A is the Eilenberg-Mac Lane spectrum of a commutative ring, then T HH^{R}(A; M) is a product of Eilenberg-Mac Lane spectra.

There is no analogue in the literature of our T HH^{R}(A; M) except in the absolute case R = S, and there is no analogue of our T HH_{R}(A; M) even then. However, the relationship between algebraic and topological Hochschild homology becomes far more transparent when one works in full generality. To describe this relationship, we must first fix notations for algebraic Hochschild homology and cohomology. For a commutative graded ring R_{∗}, a graded R_{∗}-algebra A_{∗} that is flat as an R_{∗}-module, and a graded (A_{∗}, A_{∗})-bimodule M_{∗}, we write

HH_{p,q}^{R_{∗}}(A_{∗}; M_{∗}) = Tor_{p,q}^{(A_{∗})^{e}}(M_{∗}, A_{∗}) and HH_{R^{∗}}^{p,q}(A^{∗}; M^{∗}) = Ext_{(A^{∗})^{e}}^{p,q}(A^{∗}, M^{∗}),

where p is the homological degree and q is the internal degree. When M_{∗} = A_{∗}, we delete it from the notation.

Observe that there is an evident epimorphism

ι : M_{∗} ⟶ HH_{0,∗}^{R_{∗}}(A_{∗}; M_{∗}) (1.4)

and that ι is an isomorphism if the left and right action of A on M are related by ξ_{ℓ} = ξ_{r} ∘ τ. If A_{∗} is commutative, then HH_{∗,∗}^{R_{∗}}(A_{∗}) is a graded A_{∗}-algebra and ι is a ring homomorphism; see [15, XI§6].

There is also a map

σ : A_{∗} ⟶ HH_{1,∗}^{R_{∗}}(A_{∗}) (1.5)

that sends an element a to the 1-cycle 1 ⊗ a in the standard complex, and σ is a derivation if A is commutative.

Specialization of IV.4.1 gives the following result. Observe that (A^{op})_{∗} = (A_{∗})^{op}.

Theorem 1.6. There are spectral sequences of the form

E_{p,q}^{2} = Tor_{p,q}^{R_{∗}}(A_{∗}, A_{∗}^{op}) ⟹ (A^{e})_{p+q},
E_{p,q}^{2} = Tor_{p,q}^{(A^{e})_{∗}}(M_{∗}, A_{∗}) ⟹ T HH_{p+q}^{R}(A; M),

and

E_{2}^{p,q} = Ext_{(A^{e})^{∗}}^{p,q}(A^{∗}, M^{∗}) ⟹ T HH_{R}^{p+q}(A; M).

If A_{∗} is a flat R_{∗}-module, so that the first spectral sequence collapses, then the initial terms of the second and third spectral sequences are, respectively,

HH_{∗,∗}^{R_{∗}}(A_{∗}; M_{∗}) and HH_{R^{∗}}^{∗,∗}(A^{∗}; M^{∗}).

This is of negligible use in the absolute case R = S, where the flatness hypothesis is unrealistic. However, in the relative case, it implies that algebraic Hochschild homology and cohomology are special cases of topological Hochschild homology and cohomology.

Theorem 1.7. Let R be a (discrete, ungraded) commutative ring, let A be an R-flat R-algebra, and let M be an (A, A)-bimodule. Then

HH_{∗}^{R}(A; M) ≅ T HH_{∗}^{HR}(HA; HM)

and

HH_{R}^{∗}(A; M) ≅ T HH_{HR}^{∗}(HA; HM).

If A is commutative, then HH_{∗}^{R}(A) ≅ T HH_{∗}^{HR}(HA) as A-algebras.

Proof. By VII.1.3 and the naturality of multiplicative infinite loop space theory [49], we can construct HA as an HR-algebra, commutative if A is so. The results of IV§2 construct HM as an (HA, HA)-bimodule. Thus the statement makes sense. The spectral sequences collapse since their internal gradings are concentrated in degree zero. We will prove the last statement below.

We concentrate on homology henceforward. In the absolute case R = S, it is natural to approach T HH_{∗}^{S}(A; M) by first determining the ordinary homology of T HH^{S}(A; M), using the case E = H𝔽_{p} of the following spectral sequence, and then using the Adams spectral sequence. A spectral sequence like the following one was first obtained by Bökstedt [9]. An interesting case, essentially T HH(ku), was worked out by McClure and Staffeldt [53], who assumed without proof that the second spectral sequence in the following theorem could be constructed. The flatness hypotheses required when E is only a commutative ring spectrum are still unrealistic in the absolute case, but the situation is saved by the lack of need for such hypotheses when E is a commutative S-algebra, such as H𝔽_{p}.

Observe that there is a natural map

ζ = id ∧ϕ : M ≅ M ∧_{A^{e}} A^{e} ⟶ M ∧_{A^{e}} A = T HH_{R}(A; M). (1.8)

Theorem 1.9. Let E be a commutative ring spectrum. If E_{∗}(R) is a flat R_{∗}- module, or if E is a commutative S-algebra, there is a spectral sequence of differential E_{∗}(R)-modules of the form

E_{p,q}^{2} = Tor_{p,q}^{E_{∗}R}(E_{∗}A, E_{∗}(A^{op})) ⟹ E_{p+q}(A^{e}).

If E_{∗}(A^{e}) is a flat (A^{e})_{∗}-module, or if E is a commutative S-algebra, there is a spectral sequence of differential E_{∗}(R)-modules of the form

E_{p,q}^{2} = Tor_{p,q}^{E_{∗}(A^{e})}(E_{∗}(M), E_{∗}(A)) ⟹ E_{p+q}(T HH^{R}(A; M)).

In either case, if E_{∗}(A) is E_{∗}(R)-flat, so that

E_{p,q}^{2} = HH_{∗,∗}^{E_{∗}(R)}(E_{∗}(A); E_{∗}(M))

in the second spectral sequence, then the composite

[figure: E_{∗}(M); ι; E_{0,∗}^{2}; E_{0,∗}^{∞}; ⊂; E_{∗}(T HH^{R}(A; M))]

coincides with ζ_{∗} : E_{∗}(M) ⟶ E_{∗}(T HH^{R}(A; M)).

Proof. When E is just a commutative ring spectrum, both spectral sequences are immediate from IV.6.2. When E is a commutative S-algebra, both spectral sequences are immediate from IV.6.4 (see also IV.6.6). The statement about ζ is clear from the discussion of the edge homomorphism in IV§5.

Applied to Eilenberg-Mac Lane spectra, the following complement implies the last statement of Theorem 1.7. Clearly Proposition 1.2 implies that if A is a commutative R-algebra, then T HH^{R}(A) is a commutative R-ring spectrum. Moreover, by Corollary 3.8 below, there is then a natural map (in 𝒟_{R})

ω : A ∧ S_{+}^{1} ⟶ T HH^{R}(A). (1.10)

Proposition 1.11. Let A be a commutative R-algebra and assume sufficient hypotheses that Theorem 1.9 gives a spectral sequence

E_{p,q}^{2} = HH_{p,q}^{E_{∗}(R)}(E_{∗}(A)) ⟹ E_{p+q}(T HH^{R}(A)).

Then this is a spectral sequence of differential E_{∗}(A)-algebras such that E_{2} has the standard product in Hochshild homology. Moreover, the composite

[figure: E_{∗}(A); σ; E_{1,∗}^{2}; E_{1,∗}^{∞}; ⊂; E_{∗}(T HH^{R}(A))/ im ζ_{∗}]

coincides with the restriction of

(ω)_{∗} : E_{∗}(A ∧ S_{+}^{1}) ⟶ E_{∗}(T HH^{R}(A))/ im ζ_{∗}

to the wedge summand ΣA.

Proof. We may use the standard complex for the computation of algebraic Hochschild homology in the construction of the spectral sequences in IV§5. We can then construct a pairing of the resolutions constructed there that realizes the standard product and so deduce a pairing of spectral sequences. We omit details, since the result will be more transparent with the alternative construction of the spectral sequence (albeit under different hypotheses) in the next section. For the last statement, under the usual stable splitting of S_{+}^{1} as S^{0} ∨ S^{1}, the restriction of ω to the wedge summand A coincides with ζ. Under the equivalence with the standard complex description of T HH in the next section, ω lands in simplicial filtration one. Thinking of ΣA as Σ(𝔽S ∧_{A^{e}} A), where 𝔽 is the free A^{e}-module functor, we find that the restriction of ω to ΣA provides a factorization through A∧_{A^{e}} A of the first stage of the inductive construction of the spectral sequence given in IV§5. Again, this will be more transparent with the alternative construction of the spectral sequence.

Remark 1.12. We have given Theorem 1.9 in the form appropriate to classical stable homotopy theory. It is perhaps more natural to give a version that makes sense from the point of view of the multiplicative homology theories E_{∗}^{R} on Rmodules of IV.1.7, where E is a commutative R-algebra. The ground ring in this context is E_{∗} = E_{∗}^{R}(R). We leave the details to the reader. The essential point is the relative case of III.3.10.

2. Topological Hochschild homology: second definition

We again assume given a q-cofibrant commutative S-algebra R, a q-cofibrant R-algebra or q-cofibrant commutative R-algebra A, and an (A, A)-bimodule M. Write A^{p} for the p-fold ∧_{R}-power of A, and let

ϕ : A ∧_{R} A ⟶ A and η : R ⟶ A

be the product and unit of A. Let

ξ_{ℓ} : A ∧_{R} M ⟶ M and ξ_{r} : M ∧_{R} A ⟶ M

be the left and right action of A on M. We have canonical cyclic permutation isomorphisms

τ : M ∧_{R} A^{p} ∧_{R} A ⟶ A ∧_{R} M ∧_{R} A^{p}.

The following definition precisely mimics the definition of the standard complex for the computation of Hochschild homology, as given in [15, p. 175]. The topological analogue of passage from a simplicial k-module to a chain complex of k-modules is passage from a simplicial spectrum E_{∗} to its geometric realization |E_{∗}|.

Definition 2.1. Define a simplicial R-module thh^{R}(A; M)_{∗} as follows. Its Rmodule of p-simplices is M ∧_{R} A^{p}. Its face and degeneracy operators are

d_{i} = \begin{cases}ξ_{r} ∧ (id)^{p−1} & if i = 0 \\ id ∧(id)^{i−1} ∧ ϕ ∧ (id)^{p−i−1} & if 1 ≤ i < p \\ (ξ_{ℓ} ∧ (id)^{p−1}) ∘ τ & if i = p \end{cases}

and s_{i} = id ∧(id)^{i} ∧ η ∧ (id)^{p−i}. Define

thh^{R}(A; M) = |thh^{R}(A; M)_{∗}|.

When M = A, we delete it from the notation, writing thh^{R}(A)_{∗} and |thh^{R}(A)_{∗}|.

Since geometric realization converts simplicial R-modules to R-modules, by X.1.5, thh^{R}(A; M) and thh^{R}(A) are R-modules. Observe that the maps

ζ_{p} = id ∧η^{p} : M ≅ M ∧_{R} R^{p} ⟶ M ∧_{R} A^{p}

specify a map of simplicial R-modules from the constant simplicial R-module \underline{M} to thh^{R}(A; M)_{∗}; it induces a natural map of R-modules

ζ = |ζ_{∗}| : M ⟶ thh^{R}(A; M).

Inspection of the simplicial structure shows that, if A is commutative, there is a natural map of R-modules

ω : A ∧ S_{+}^{1} ⟶ thh^{R}(A)

with image in the simplicial 1-skeleton; see (3.2) and Corollary 3.8 below. Moreover, we then have the following observation.

Proposition 2.2. Let A be a commutative R-algebra. Then thh^{R}(A) is naturally a commutative A-algebra with unit ζ : A ⟶ thh^{R}(A), and thh^{R}(A; M) is a thh^{R}(A)-module. By neglect of structure, thh^{R}(A) is a commutative R-algebra.

Proof. Clearly thh^{R}(A)_{∗} is a simplicial commutative R-algebra, thh^{R}(A; M)_{∗} is a simplicial thh^{R}(A)_{∗}-module, and ζ_{∗} : \underline{A} ⟶ thh^{R}(A)_{∗} is a map of simplicial commutative R-algebras. Since all structure in sight is preserved by geometric realization, by X.1.5, this implies the result.

The definition of T HH^{R}(A; M) was homotopical and led directly to spectral sequences for its calculational study. The definition of thh^{R}(A; M) is formal and algebraic. We must establish a connection between these two definitions.

The starting point is the relative two-sided bar construction B^{R}(M, A, N), which is defined for a commutative S-algebra R, an R-algebra A, and right and left Amodules M and N. The definition is the same as that of B(M, R, N) = B^{S}(M, R, N) in IV.7.2, except that smash products over S are replaced by smash products over R. By XII.1.2 and X.1.2, there is a natural map

ψ : B^{R}(A, A, N) ⟶ N

of A-modules that is a homotopy equivalence of R-modules. More generally, by use of the product on A and its action on the given modules, we obtain a natural map of R-modules

ψ : B^{R}(M, A, N) ⟶ M ∧_{A} N.

The proof of the following result is the same as that of its special case IV.7.5.

Proposition 2.3. For a cell A-module M and an A-module N,

ψ : B(M, A, N) ⟶ M ∧_{A} N

is a weak equivalence of R-modules.

The relevance of the bar construction to thh is shown by the following observation, which is the same as in algebra. We agree to write

B^{R}(A) = B^{R}(A, A, A).

Observe that B^{R}(A) is an (A, A)-bimodule; on the simplicial level, B_{0}^{R}(A) = A^{e}.

Lemma 2.4. For (A, A)-bimodules M, there is a natural isomorphism

thh^{R}(A; M) ≅ M ∧_{A^{e}} B^{R}(A).

Proof. If \underline{M} is the constant simplicial (A, A)-bimodule at M, then

M ∧_{A^{e}} B^{R}(A) ≅ |\underline{M} ∧_{A^{e}} B_{∗}^{R}(A, A, A)|.

We have canonical isomorphisms

M ∧_{R} A^{p} ≅ M ∧_{A^{e}} (A^{e} ∧_{R} A^{p}) ≅ M ∧_{A^{e}} (A ∧_{R} A^{p} ∧_{R} A)

given by the permutation of A^{op} = A past A^{p}. Inspection shows that these commute with the face and degeneracy operations and so induce the stated isomorphism.

Since the natural map ψ : B^{R}(A) ⟶ A of (A, A)-bimodules is a homotopy equivalence of R-modules, this has the following immediate consequence, by III.3.8.

Proposition 2.5. For cell A^{e}-modules M, the natural map

thh^{R}(A; M) ≅ M ∧_{A^{e}} B^{R}(A)⟶^{id ∧ψ}M ∧_{A^{e}} A = T HH^{R}(A; M)

is a weak equivalences of R-modules, or of A^{e}-modules if A is commutative.

While we are perfectly happy, indeed forced, to assume that M is a cell A^{e}- module in our derived category level definition of T HH, we are mainly interested in the case M = A of our point-set level construction thh, and A is not of the A^{e}-homotopy type of a cell A^{e}-module except in trivial cases. However, we have the following result.

Theorem 2.6. Let γ : M ⟶ A be a weak equivalence of A^{e}-modules, where M is a cell A^{e}-module. Then the map

thh^{R}(id; γ) : thh^{R}(A; M) ⟶ thh^{R}(A; A) = thh^{R}(A)

is a weak equivalence of R-modules, or of A^{e}-modules if A is commutative. Therefore T HH^{R}(A; M) is weakly equivalent to thh^{R}(A).

Proof. With the notation of VII.6.4, it is clear from VII.6.5 that M and A are both in ℰ̄_{R}, and it follows from VII.6.7 that the map thh_{p}^{R}(id; γ) of p-simplices is a weak equivalence for each p. The following result gives that the simplicial R-modules B_{∗}^{R}(A) and thh_{∗}^{R}(A) are proper, in the sense of X.2.2, and X.2.4 gives the conclusion.

Proposition 2.7. For right and left A-modules M and N, B_{∗}^{R}(M, A, N) is a proper simplicial R-module. For an (A, A)-bimodule M, thh_{∗}^{R}(A; M) is a proper simplicial R-module.

Proof. The condition of being proper involves only the degeneracy and not the face operators of a simplicial R-module. In our cases, the degeneracies are obtained from the unit η : R ⟶ A and, since smashing over R with M and N in the first statement and with M in the second preserves cofibrations of R-modules, the result in both cases is an immediate consequence of VII.6.8.

Returning to the study of spectral sequences in the previous section, we find that use of the standard complex gives us spectral sequences under different flatness hypotheses, just as in IV§7. We consider the spectral sequences derived in X.2.9 from the simplicial filtration of thh^{R}(A; M). For simplicity, we restrict attention to the absolute case R = S.

Theorem 2.8. Let E be a commutative ring spectrum, let A be an S-algebra, and let M be a cell A^{e}-module. If E_{∗}(A) is E_{∗}-flat, then there is a spectral sequence of the form

E_{p,q}^{2} = HH_{p,q}^{E_{∗}}(E_{∗}(A); E_{∗}(M)) ⟹ E_{p+q}(thh^{S}(A; M)).

The composite

[figure: E_{∗}(M); ι; E_{0,∗}^{2}; E_{0,∗}^{∞}; ⊂; E_{∗}(thh^{S}(A; M))]

coincides with ζ_{∗} : E_{∗}(M) ⟶ E_{∗}(thh^{S}(A; M)). If A is a commutative S-algebra then the spectral sequence

E_{p,q}^{2} = HH_{p,q}^{E_{∗}}(E_{∗}(A)) ⟹ E_{p+q}(thh^{S}(A))

is a spectral sequence of differential E_{∗}(A)-algebras, and the composite

[figure: E_{∗}(A); σ; E_{1,∗}^{2}; E_{1,∗}^{∞}; ⊂; E_{∗}(thh^{S}(A))/ im ζ_{∗}]

coincides with the restriction of

(ω)_{∗} : E_{∗}(A ∧ S_{+}^{1}) ⟶ E_{∗}(thh^{S}(A))/ im ζ_{∗}

to the wedge summand ΣA.

Proof. Using VII.6.2 or VII.6.7 and our standing q-cofibrancy hypothesis to see that our point-set level constructions can be used to compute derived smash products, we see that the E^{1}-terms are exactly the standard complexes for the computation of the algebraic Hochschild homology groups in the E^{2}-term. The standard product on the standard complex is realized on E^{1}, and the rest is clear.

3. The isomorphism between thh^{R}(A) and A ⊗ S^{1}

We here explain a reinterpretation of the definition of thh^{R}(A) that was observed by McClure, Schwänzl, and Vogt [52]. Recall that the category 𝒞 𝒜_{R} of commutative R-algebras 𝒞 𝒜_{R} is tensored and cotensored over the category 𝒰 of unbased topological spaces, so that we have adjunction homeomorphisms

𝒞 𝒜_{R}(A ⊗ X, B) ≅ 𝒰 (X, 𝒞 𝒜_{R}(A, B)) ≅ 𝒞 𝒜_{R}(A, F (X_{+}, B)). (3.1)

As in VII.3.7, we easily obtain an identification of simplicial commutative Ralgebras

thh^{R}(A)_{∗} ≅ A ⊗ S_{∗}^{1}. (3.2)

by writing out the standard simplicial set S_{∗}^{1} whose realization is the circle and comparing faces and degeneracies. We give a slightly different proof of the following result. We think of S^{1} as the unit complex numbers.

3. THE ISOMORPHISM BETWEEN thh^{R}(A) AND A ⊗ S^{1} 197

Theorem 3.3 (McClure, Schwänzl, Vogt). For commutative R-algebras A, there is a natural isomorphism of commutative R-algebras

thh^{R}(A) ≅ A ⊗ S^{1}.

The product of thh^{R}(A) is induced by the codiagonal S^{1} ∐ S^{1} ⟶ S^{1}. The unit ζ : A ⟶ thh^{R}(A) is induced by the inclusion {1} → S^{1}.

Proof. We may identify S^{1} with the pushout in the diagram

\xymatrix{∂I \ar[r] \ar[d] & I \ar[d] \\ \{pt\} \ar[r] & S^{1}.}

We arrange our identification to map {pt} to {1}. Applying the functor A ⊗ (?), we obtain a pushout diagram

\xymatrix{A ⊗ ∂I \ar[r] \ar[d] & A ⊗ I \ar[d] \\ A ⊗ \{pt\} \ar[r] & A ⊗ S^{1}.}

By VII.3.7, we have B^{R}(A) ≅ A ⊗ I. By Lemma 2.4, thh^{R}(A) ≅ A ∧_{A^{e}} B^{R}(A). This gives the isomorphism thh^{R}(A) ≅ A ⊗ S^{1} by a comparison of pushouts. The statement about the product follows from the isomorphism of coproducts

(A ⊗_{R} S^{1}) ∧_{R} (A ⊗_{R} S^{1}) ≅ A ⊗ (S^{1}∐S^{1}).

Since ζ is determined by {pt} ⟶ S^{1}, the last statement is clear.

Corollary 3.4. The pinch map S^{1} ⟶ S^{1} ∨ S^{1} and trivial map S^{1} ⟶ ∗ induce a (homotopy) coassociative and counital coproduct and counit

ψ : thh^{R}(A) ⟶ thh^{R}(A) ∧_{A} thh^{R}(A) and ε : thh^{R}(A) ⟶ A

that make thh^{R}(A) a homotopical Hopf A-algebra.

Proof. Since S^{1} ∨S^{1} is the pushout of S^{1} ⟵ ∗ ⟶ S^{1} and the functor A⊗(?) preserves pushouts, we see from VII.1.6 that

thh^{R}(A) ∧_{A} thh^{R}(A) ≅ thh^{R}(A) ⊗ (S^{1} ∨ S^{1}).

The rest is clear.

The next few corollaries are based on the case X = S^{1} of the adjunctions (3.1). Of course, left adjoints preserve colimits.

Corollary 3.5. The functor thh^{R}(A) preserves colimits in A.

The word “naive” in the following corollary refers to the fact that we only consider spectra indexed on universes with trivial S^{1}-actions here; genuine S^{1}-spectra must be indexed on universes that contain all representations of S^{1}. In the naive sense, we define commutative S^{1}-S-algebras, and so on, simply by requiring S^{1} to act compatibly with all structure in sight. We think of S^{1} as acting trivially on R and A.

Corollary 3.6. thh^{R}(A) is a naive commutative S^{1}-R-algebra. If B is a naive commutative S^{1}-R-algebra and f : A ⟶ B is a map of commutative R-algebras, then there is a unique map of naive commutative S^{1}-R-algebras f̃: thh^{R}(A) ⟶ B such that f̃∘ ζ = f.

Proof. The product on S^{1} gives a map

α : (A ⊗ S^{1}) ⊗ S^{1} ≅ A ⊗ (S^{1} × S^{1}) ⟶ A ⊗ S^{1}.

Its adjoint S^{1} ⟶ 𝒞 𝒜_{R}(thh^{R}(A), thh^{R}(A)) gives actions by elements of S^{1} via maps of commutative R-algebras, with the requisite continuity, and the adjunction immediately implies the universal property.

For an integer r, define ϕ^{r} : S^{1} ⟶ S^{1} by

ϕ^{r}(e^{2πit}) = e^{2πirt}.

It is immediate that these induce power operations Φ^{r} on thh^{R}(A).

Corollary 3.7. There are natural maps of commutative R-algebras

Φ^{r} : thh^{R}(A) ⟶ thh^{R}(A)

such that

Φ^{0} = ζε, Φ^{1} = id, Φ^{r} ∘ Φ^{s} = Φ^{rs},

and the following diagrams commute:

[figure: thh^{R}(A) ⊗ S^{1}; α; Φ^{r}⊗ϕ^{s}; thh^{R}(A); Φ^{r+s}; thh^{R}(A) ⊗ S^{1}; α; thh^{R}(A).]

Corollary 3.8. There is a natural S^{1}-equivariant map of R-modules

ω : A ∧ S_{+}^{1} ⟶ thh^{R}(A)

such that if B is a commutative R-algebra and f : A ∧ S_{+}^{1} ⟶ B is a map of spectra such that the composite f ∘ (id ∧i_{x}) : A ⟶ B, i_{x} : {x}_{+} ⊂ S_{+}^{1}, is a map of R-algebras for each x ∈ S^{1}, then f uniquely determines a map of R-algebras f̃ : thh^{R}(A) ⟶ B such that f = f̃ ∘ ω. Moreover, ω is obtained by passage to geometric realization from the natural map of simplicial spectra

ω_{∗} : A ∧ (S_{∗}^{1})_{+} ⟶ A ⊗ S_{∗}^{1}.

Proof. This is immediate from VII.2.11. Its transitivity diagram and a naturality diagram imply the S^{1}-equivariance.

The image of ω lies in the simplicial 1-skeleton. The intuition is that the rest of thh^{R}(A) freely builds up the R-algebra structure.

Remark 3.9. When A is an R-algebra, inspection of the simplicial structure shows that thh^{R}(A)_{∗} is a cyclic spectrum, and it follows exactly as for cyclic spaces [16] that thh^{R}(A) is a naive S^{1}-R-module. We believe that thh^{R}(A) can in fact be constructed as a genuine S^{1}-spectrum that is cyclotomic in the sense of Hesselholt and Madsen [29, 1.2], and we intend to return to this elsewhere.

200 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY

CHAPTER X Some basic constructions on spectra

We have used geometric realization of simplicial spectra in several places, and we shall make further use of it to prove some of our earlier claims. We study it in the first two sections, concentrating on formal properties in Section 1 and on homotopical properties in Section 2. We then use geometric realization to define homotopy colimits in Section 3. All of the basic definitions and much of the work in the first three sections carries over to any of our model categories of module, ring, and algebra categories, as we have already indicated in VII§3. We prefer to be more concrete in this service chapter. It will be evident that geometric realization in the category of 𝕃-spectra or the category of R-modules for an S-algebra R is given by geometric realization in the underlying category of spectra.

After discussing various special kinds of prespectra in Section 4, we use homotopy colimits to construct the “cylinder functor” in Section 5. This functor converts spectra to weakly equivalent Σ-cofibrant spectra while preserving ring, module, and algebra structures.

Much of the material of this chapter has long been known to the authors, and to others, but little if any of it has appeared in the literature.

1. The geometric realization of simplicial spectra

We first recall from [42] the definition of a coend, or tensor product of functors. Let Λ be any small category and let 𝒞 be any category that has all (small) colimits. Write ∨ for the categorical coproduct in 𝒞. Suppose given a functor

F : Λ^{op} × Λ ⟶ 𝒞 .

Define the coend of F, denoted

∫ ^{Λ} F (n, n)

to be the coequalizer of the pair of maps

[figure: ⋁_{ϕ:m→n} F (n, m); e; f; ⋁_{n} F (n, n)]

where the restriction of e to the ϕth summand is F (ϕ, id) and the restriction of f to the ϕth summand is F (id, ϕ). Using equalizers, we obtain the dual notion of the end of F, denoted ∫_{Λ} F (n, n).

Now recall that a simplicial object in any category 𝒞 is a contravariant functor from the simplicial category Δ to 𝒞. We have the classical geometric realization functor, denoted |X_{∗}|, from simplicial spaces to spaces, and we need to extend it to the level of spectra. We shall begin with a spectrum level definition, and we shall then show how to interpret it in terms of the space level definition. This will allow us to deduce many properties of the spectrum level functor simply by quoting standard properties of the space level functor. Recall that, using the usual face and degeneracy maps, we obtain a covariant functor from Δ to spaces that sends q to the standard topological q-simplex Δ_{q}.

Definition 1.1. Let K_{∗} be a simplicial spectrum. Define its geometric realization to be the coend

|K_{∗}| = ∫ ^{Δ} K_{q} ∧ (Δ_{q})_{+}.

Of course, the functor Δ^{op} × Δ ⟶ 𝒮 that is implicit in the definition sends (p, q) to K_{p} ∧ (Δ_{q})_{+}. The geometric realization of a simplicial space X_{∗} is defined similarly:

|X_{∗}| = ∫ ^{Δ} X_{q} × Δ_{q}.

If X_{∗} is a simplicial based space, so that all its face and degeneracy maps are basepoint-preserving, then all points of each subspace ∗ × Δ_{q} are identified to the point (∗, 1) ∈ X_{0} × Δ_{0} in the construction of |X_{∗}|, hence

|X_{∗}| = ∫ ^{Δ} X_{q} ∧ (Δ_{q})_{+}.

This places the two definitions in the same form. Actually, as with any categorical colimit, the geometric realization of a simplicial spectrum is obtained by applying the spectrification functor L to the spacewise geometric realization of its underlying simplicial prespectrum. In more detail, for a simplicial prespectrum K_{∗}, we have simplicial based spaces K_{∗}(V) for indexing spaces V. Their geometric realizations form a prespectrum with structural maps induced from those of K_{∗} as the composites

Σ^{W −V} |K_{∗}(V)| ≅ |Σ^{W −V} K_{∗}(V)| ⟶ |K_{∗}(W)|.

If K_{∗} is a simplicial spectrum, then |K_{∗}| is obtained by applying L to this prespectrum.

As we explain in the proof, the following result has two different senses, and it is correct in both senses.

Proposition 1.2. The geometric realization functor from simplicial spectra to spectra preserves homotopies.

Proof. As on the space level [44, §11], we have two kinds of homotopy between simplicial maps, and geometric realization carries both to homotopies of the usual sort. One kind is just a simplicial map with domain of the form K_{∗} ∧ I_{+}, and geometric realization preserves this kind of homotopy by part (ii) of the next proposition applied to the constant simplicial space at I_{+}. The other is the combinatorial kind of homotopy that makes sense for maps between simplicial objects in any category [44, 9.1]. It can be viewed as a simplicial map with domain of the form K_{∗} ∧ Δ_{∗}[1]_{+}, where Δ_{∗}[1] is the standard simplicial 1-simplex viewed as a discrete simplicial space, hence part (ii) of the next proposition also applies to show that geometric realization preserves this kind of homotopy.

On the level of based spaces, it is standard that geometric realization commutes with wedges and products and therefore with smash products. This easily implies a direct proof of (ii) and (iv) of the following result, and (i) can be viewed as a special case of (ii). Recall that functors on 𝒞 are extended termwise to simplicial objects in 𝒞; for example (J_{∗} ∧ K_{∗})_{q} = J_{q} ∧ K_{q} for simplicial spectra J_{∗} and K_{∗}.

Proposition 1.3. Geometric realization enjoys the following properties.

(i) For simplicial based spaces X_{∗}, there is a natural isomorphism

Σ^{∞}|X_{∗}| ≅ |Σ^{∞}X_{∗}|.

(ii) For simplicial based spaces X_{∗} and simplicial spectra K_{∗}, there is a natural

isomorphism

|K_{∗} ∧ X_{∗}| ≅ |K_{∗}| ∧ |X_{∗}|.

(iii) For simplicial spectra K_{∗} indexed on U and spaces A over ℐ (U, U′), there is a natural isomorphism

|A ⋉ K_{∗}| ≅ A ⋉ |K_{∗}|.

(iv) For simplicial spectra |J_{∗}| and |K_{∗}|, there is a natural isomorphism

|J_{∗} ∧ K_{∗}| ≅ |J_{∗}| ∧ |K_{∗}|,

where external smash products are understood.

(v) For simplicial spectra K_{∗}, there are natural isomorphisms

|𝔹K_{∗}| ≅ 𝔹|K_{∗}| and |ℂK_{∗}| ≅ ℂ|K_{∗}|,

and similarly for the monads 𝔹[1] and ℂ[1] and for the corresponding reduced monads that were defined in II§§4–5.

Proof. Part (i-iii) hold since left adjoints commute with colimits. Parts (iv) and (v) follow from parts (i)-(iii) and a Fubini theorem for iterated coends.

Proposition 1.4. For simplicial 𝕃-spectra K_{∗} and L_{∗}, there is a natural isomorphism

|K_{∗}| ∧_{ℒ} |L_{∗}| ≅ |K_{∗} ∧_{ℒ} L_{∗}|.

For an S-algebra R, such as R = S, and simplicial R-modules M_{∗} and N_{∗}, there is a natural isomorphism

|M_{∗}| ∧_{R} |N_{∗}| ≅ |M_{∗} ∧_{R} N_{∗}|.

Proof. The first isomorphism is immediate from the previous proposition, and it directly implies the second when R = S. In view of the coequalizer description of smash products over R, the case of general R follows by the commutation of coequalizers with geometric realization.

Proposition 1.5. The geometric realization of a simplicial A_{∞} or E_{∞} ring spectrum is an A_{∞} or E_{∞} ring spectrum. For a commutative S-algebra R, such as R = S, the geometric realization of a simplicial (commutative) R-algebra is a (commutative) R-algebra. The analogous preservation properties hold for modules.

2. Homotopical and homological properties of realization

To discuss the behavior of geometric realization with respect to equivalences and CW homotopy types, and to obtain useful spectral sequences from its canonical filtration, we need the following technical definition.

Definition 2.1. Let K_{∗} be a simplicial spectrum and let sK_{q} ⊂ K_{q} be the “union” of the subspectra s_{j}K_{n−1}, 0 ≤ j < q. Say that K_{∗} is proper if the “inclusion” sK_{q} ⟶ K_{q} is a cofibration for each q ≥ 0.

2. HOMOTOPICAL AND HOMOLOGICAL PROPERTIES OF REALIZATION 205

The term “union” must be interpreted in terms of appropriate pushout diagrams. The corresponding “inclusions” must be interpreted, a priori, in terms of associated maps. However, a cofibration of spectra is a spacewise closed inclusion [37, I.8.1]. Rigorous notation would make this section unreadable, so we shall use notations as if we were dealing with simplicial spaces, leaving the translation to rigorous categorical language to the reader. One way to be precise about the degeneracy subspectrum sK_{q} and its associated map to K_{q} is to interpret the latter as the following map of coends

∫ ^{D_{q−1}} K_{p} ∧ D(\underline{q}, \underline{p})_{+} ⟶ ∫ ^{D_{q}} K_{p} ∧ D(\underline{q}, \underline{p})_{+} ≅ K_{q}.

Here D is the subcategory of Δ consisting of the monotonic surjections (which index the degeneracy and identity maps), and D_{q} is its full subcategory of objects \underline{i} with 0 ≤ i ≤ q. The isomorphism is an application of Yoneda’s lemma. With this interpretation, we can generalize the context to 𝕃-spectra or to R-modules for a fixed S-algebra R. Recall that colimits in the categories of 𝕃-spectra and of R-modules are created in the category of spectra.

Definition 2.2. A simplicial 𝕃-spectrum is proper if the canonical map of 𝕃spectra sK_{q} ⟶ K_{q} is a cofibration for each q ≥ 0. A simplicial R-module is proper if the canonical map of R-modules sK_{q} ⟶ K_{q} is a cofibration for each q ≥ 0.

Since mapping cylinders of R-modules are created in 𝒮, a cofibration of 𝕃spectra or of R-modules is a cofibration of spectra, but not conversely. Note that the inclusion M ⟶ N of a relative cell R-module (N, M) is a cofibration, by HELP, and that VII.4.14 gives that q-cofibrations of R-algebras and of commutative R-algebras are cofibrations of R-modules.

Lemma 2.3. Let i : A ⟶ X be a cofibration of spectra, 𝕃-spectra, or Rmodules. Then

j = i ∧ id : (A ∧ Δ_{q+}) ∪ (X ∧ ∂Δ_{q+}) ⟶ X ∧ Δ_{q+}

is a cofibration of spectra, 𝕃-spectra, or R-modules. Therefore, if K_{∗} is a proper simplicial spectrum, 𝕃-spectrum, or R-module, then the inclusion

(sK_{q} ∧ Δ_{q+}) ∪ (K_{q} ∧ ∂Δ_{q+}) ⟶ K_{q} ∧ Δ_{q+}

is a cofibration for each q ≥ 1.

Proof. With the usual conventions on products and smash products of pairs, we are given that

(X, A) ∧ (I_{+}, {0}_{+})

is a retract pair, and we must deduce that

(X, A) ∧ (Δ_{q+}, ∂Δ_{q+}) ∧ (I_{+}, {0}_{+})

is a retract pair. There is a homeomorphism of pairs

(Δ_{q}, ∂Δ_{q}) × (I, {0}) ≅ (Δ_{q} × I, Δ_{q} × {0}).

In based notation, this clearly implies a homeomorphism of pairs

(Δ_{q+}, ∂Δ_{q+}) ∧ (I_{+}, {0}_{+}) ≅ (I_{+}, {0}_{+}) ∧ Δ_{q+}.

The conclusion follows upon smashing with (X, A) and using the given retraction.

Similarly, the theorems to follow are valid with essentially identical proofs in the contexts of spectra, 𝕃-spectra, and R-modules.

Theorem 2.4. Let f_{∗} : K_{∗} ⟶ K′_{∗} be a map of proper simplicial spectra, 𝕃spectra, or simplicial R-modules.

(i) If each f_{q} : K_{q} ⟶ K′_{q} is a homotopy equivalence, then so is |f_{∗}| : |K_{∗}| ⟶ |K′_{∗}|.

(ii) In the 𝕃-spectrum case, and therefore also in the R-module case, if each f_{q} : K_{q} ⟶ K′_{q} is a weak equivalence, then so is |f_{∗}| : |K_{∗}| ⟶ |K′_{∗}|; in the spectrum case, this holds if all given and constructed spectra are tame.

Proof. The proof is precisely parallel to the proof of the space level analogs [45, A.4]. The essential point is just the gluing theorem to the effect that a pushout of (weak) equivalences is a (weak) equivalence when corresponding legs of the given pushout diagrams are cofibrations. For tame spectra, the weak version of this statement is a consequence of I.3.4. For 𝕃-spectra, the weak version is a consequence of I.6.5.

For 0 ≤ k < q, let s^{k}K_{q} be the union over 0 ≤ j ≤ k of the subspectra s_{j}K_{q−1} of K_{q}. We claim first that the inclusion s^{k−1}K_{q} ⟶ s^{k}K_{q} is a cofibration for 0 < k < q. This holds vacuously if q = 1. We assume it for q − 1 and deduce it for q. Since a composite of cofibrations is a cofibration and K_{∗} is proper, the left vertical inclusion in the following commutative diagram is a cofibration:

\xymatrix{s^{k−1}K_{q−1} \ar[r] \ar[d] & s^{k−1}K_{q} ∩ s_{k}K_{q−1} \ar[r] \ar[d] & s^{k−1}K_{q} \ar[d] \\ K_{q−1} \ar[r] & s_{k}K_{q−1} \ar[r] & s^{k}K_{q}.} (2.5)

The left horizontal arrows are induced by s_{k} and are isomorphisms with inverses induced by d_{k+1}, and the right square is a pushout. Therefore the middle and right vertical arrows are also cofibrations. This proves our claim.

Since s_{0} : K_{q−1} ⟶ s^{0}K_{q} is an isomorphism, we find by induction on q and, for fixed q, by induction on k that f_{q} : s^{k}K_{q} ⟶ s^{k}K′_{q} is a (weak) equivalence for each k and q. In particular, f_{q} : sK_{q} ⟶ sK′_{q} is a (weak) equivalence for each q. As usual, |K_{∗}| is filtered, and we have successive pushouts

[figure: sK_{q} ∧ ∂Δ_{q+}; K_{q} ∧ ∂Δ_{q+}; sK_{q} ∧ Δ_{q+}; (sK_{q} ∧ Δ_{q+}) ∪ (K_{q} ∧ ∂Δ_{q+}); g; F_{q−1}|K_{∗}|; K_{q} ∧ Δ_{q+}; F_{q}|K_{∗}|.] (2.6)

Here the restrictions of the map g to sK_{q} ∧ Δ_{q+} and K_{q} ∧ ∂Δ_{q+} are dictated by the coequalizer description of |K_{∗}|. The vertical arrows are cofibrations, the bottom middle one by Lemma 2.3. Therefore the restrictions |f_{∗}| : F_{q}|K_{∗}| ⟶ F_{q}|K′_{∗}| are (weak) equivalences by successive applications of the gluing theorem, and |f_{∗}| is a (weak) equivalence by passage to colimits over q, using III.1.7 in the weak case.

Theorem 2.7. Let K_{∗} be a simplicial spectrum, 𝕃-spectrum, or R-module.

(i) If each K_{q} is a cell object, each degeneracy operator is the inclusion of a subcomplex, and each face operator is sequentially cellular, then |K_{∗}| is a cell object, and similarly for CW objects.

(ii) If K_{∗} is proper and if each K_{q} has the homotopy type of a cell object, then so does |K_{∗}|, and similarly for CW objects provided that, in the R-module case, R is connective.

Proof. The proofs follow the same outline as in the previous theorem, and part (i) is clear from the second pushout in (2.6). For the CW case of (ii), If each K_{q} has the homotopy type of a CW object, then, by induction on q and, for fixed q, by induction on k, (2.5) shows that each s^{k}K_{q} has the homotopy type of a CW spectrum. By induction on q, (2.6) then shows that each F_{q}|K_{∗}| has the homotopy type of a CW spectrum. Therefore |K_{∗}| has the homotopy type of a CW spectrum.

The essential point is that if J, K, and L are CW homotopy types, then a pushout diagram obtained from a cofibration J → K and a map J → L is equivalent to a pushout diagram obtained from the inclusion of a subcomplex J′ in a CW object K′ and a cellular map from J′ to a CW object L′. Since we are applying the cellular approximation theorem, we must assume that R is connective in the Rmodule case. By the gluing theorem, the pushout K ∪_{J} L is therefore homotopy equivalent to the CW object K′ ∪_{J′} L′. Since colimits are constructed from wedges and coequalizers and thus from wedges and pushouts, it follows that the colimit of a sequence of cofibrations of objects of the homotopy types of CW objects has the homotopy type of a CW object. The proof of the statement about cell objects is similar; here III.2.2 substitutes for the cellular approximation theorem.

Remark 2.8. A result similar to Theorem 2.4(i) was proven for simplicial LEC spectra (see §4) in [20]; in that context the argument proceeds by an immediate reduction to the space level analog. However, that proof does not work to give Theorem 2.7(ii) and does not apply to the R-module setting.

The following is the spectrum level analogue of a frequently used space level spectral sequence. We have used it in our discussion of bar constructions and of topological Hochschild homology. For a spectrum E, we can apply E_{q} to the simplicial spectrum K_{∗} to obtain a simplicial abelian group E_{q}(K_{∗}). Taking the homology of its normalized chain complex, we obtain groups H_{p}(E_{q}(K_{∗})), and we obtain the same groups if we take the homology of its unnormalized chain complex [43, 22.3]. If E is an R-module and K_{∗} is a simplicial R-module, then, with the notation of IV.1.7, we obtain groups H_{p}(E_{q}^{R}(K_{∗})) the same way. If R is commutative, then each H_{p}(E_{∗}^{R}(K_{∗})) is an R_{∗}-module. If, further, E is a commutative R-ring spectrum, then H_{p}(E_{∗}^{R}(K_{∗})) is an E_{∗}-module.

Theorem 2.9. Let K_{∗} be a proper simplicial spectrum and let E be any spectrum. There is a natural homological spectral sequence {E_{p,q}^{r}K_{∗}} such that

E_{p,q}^{2}K_{∗} = H_{p}(E_{q}(K_{∗}))

and {E_{p,q}^{r}K_{∗}} converges strongly to E_{∗}(|K_{∗}|). With E_{∗} replaced by E_{∗}^{R}, the same statement holds for a proper simplicial R-module K_{∗} and an R-module E. Here, if R is commutative, then the spectral sequence is one of differential R_{∗}-modules, and if E is a commutative R-ring spectrum, then the spectral sequence is one of differential E_{∗}-modules.

Proof. Let F_{p}|K_{∗}| be the image in |K_{∗}| of the wedge over 0 ≤ q ≤ p of the spectra K_{q} ∧ (Δ_{q+}). Then the inclusions F_{p−1}|K_{∗}| ⊂ F_{p}|K_{∗}| are cofibrations, and we have isomorphisms

F_{p}|K_{∗}|/F_{p−1}|K_{∗}| ≅ (K_{p}/sK_{p}) ∧ (Δ_{p}/∂Δ_{p}) ≅ Σ^{p}(K_{p}/sK_{p}).

We apply E_{∗} to obtain an exact couple, and thus a spectral sequence, with

E_{p+q}(F_{p}|K_{∗}|/F_{p−1}|K_{∗}|) = E_{p,q}^{1}K_{∗}.

We now see that E_{p,q}^{1}K_{∗} is isomorphic to the p-chain group of the normalized chain complex of E_{q}(K_{∗}), and a diagram chase just like that of [44, p. 111] shows that d^{1} agrees with the differential of this chain complex. This identifies E^{2}, and the convergence is standard.

3. Homotopy colimits and limits

Homotopy colimits and limits of spectra do not appear explicitly in the literature. However, in view of our results on geometric realization, these constructions present no more difficulty in the category 𝒮 of spectra than they do in the category 𝒯 of based spaces. Of course, we are concerned with precise point-set level versions rather than with the cruder homotopical versions that are present in any Quillen model category. We record the definitions in this section. The same definitions apply in the category of R-modules for any S-algebra R.

Let 𝒟 be any small category. Let B_{q}(𝒟) be the set of q-tuples \underline{f} = (f_{1}, …, f_{q}) of composable arrows of 𝒟, depicted

[figure: d_{0}; d_{1}; f_{1}; ⋯; f_{2}; d_{q},; f_{q}] (3.1)

and let S(\underline{f}) = d_{q} and T (\underline{f}) = d_{0} be the source and target of \underline{f}. We understand B_{0}(𝒟) to be the set 𝒪 of objects of 𝒟. With the usual faces and degeneracies, B_{∗}(𝒟) is a simplicial set whose geometric realization is the classifying space B(𝒟).

We first specify homotopy colimits. A 𝒟-shaped diagram of spectra is a covariant functor D : 𝒟 ⟶ 𝒮. For any such D, there is a simplicial spectrum B_{∗}(∗, 𝒟, D), the space level analogue of which was specified in [46, §12]. (The left variable ∗ is a place holder.) The spectrum of q-simplices is the wedge over all \underline{f} ∈ B_{q}(𝒟) of the spectra D(S(\underline{f})). The faces and degeneracies are the standard ones of the twosided bar construction [46, §7]. Applied to \underline{f} as in (3.1), the last face on B_{q}(𝒟) forgets f_{q}; the last face on the \underline{f}th wedge summand of B_{q}(∗, 𝒟, D) is the map D(f_{q}) : D(d_{q}) ⟶ D(d_{q−1}). By definition, hocolim D is the geometric realization of this simplicial spectrum. Using the abbreviation B(?) = |B_{∗}(?)|, we may write this as

hocolim D = B(∗, 𝒟, D). (3.2)

For example, if \underline{K} : 𝒟 ⟶ 𝒮 is the constant functor at a spectrum K, then we see by inspection of definitions that

hocolim \underline{K} ≅ B(𝒟)_{+} ∧ K. (3.3)

A map f : D ⟶ D′ of diagrams is a natural transformation of functors, and, since B_{∗}(∗, 𝒟, D) is clearly proper, Theorem 2.4 has the following immediate implication.

Proposition 3.4. If f : D ⟶ D′ is a map of diagrams such that each f(d) is a homotopy equivalence, then hocolim f : hocolim D ⟶ hocolim D′ is a homotopy equivalence. If all given and constructed spectra are tame, the same holds for weak equivalences.

We shall need the following lemma on cofibrations. While we shall only use it in the context of based spaces, it works equally well in the context of spectra.

Lemma 3.5. Let 𝒟′ be a subcategory of 𝒟 such that any morphism of 𝒟 with target in 𝒟′ is a morphism of 𝒟′. Let D be a functor from 𝒟 to the category of spaces, based spaces, or spectra and let D′ be the restriction of D to 𝒟′. Then the induced map hocolim D′ ⟶ hocolim D is a cofibration.

Proof. We work with based spaces for definiteness, but the argument is the same in the other two cases. It suffices to construct a retraction

r : (hocolim D) ∧ I_{+} ⟶ (hocolim D) ∪ ((hocolim D′) ∧ I_{+})

from the reduced cylinder to the reduced mapping cylinder of the inclusion. A point z = |(\underline{f}, x), u| ∧ s of the domain is given by a composable tuple \underline{f} of maps as in (3.1), a point x ∈ D(d_{q}), a point u ∈ Δ_{q}, and a point s ∈ I. There is an i ≥ 0 such that the maps f_{1}, ⋯, f_{i} are in 𝒟 − 𝒟′ and the maps f_{i+1}, ⋯, f_{q} are in 𝒟′. If i = 0, define r(z) = z. If i = q, define r(z) by replacing s by 0; that is, r is here just the retraction to the base of the cylinder. If 0 < i < q, write u = (tv, (1−t)w), where v ∈ Δ_{i−1}, w ∈ Δ_{q−i}, and t ∈ I. Then define

r(z) = \begin{cases}|(f, x), (0, w)| ∧ (1 − 2t)s & if 0 ≤ t ≤ 1/2 \\ |(f, x), ((2t − 1)v, (2 − 2t)w)| ∧ 0 & if 1/2 ≤ t ≤ 1. \end{cases}

It is straightforward to check that r is a well-defined retraction.

Although we shall not need it here, we take the opportunity to record our preferred definition of homotopy limits, which is precisely dual to the definition of homotopy colimits. We suppose given a contravariant functor E : 𝒟 ⟶ 𝒮. We obtain a cosimplicial spectrum C_{∗}(E, 𝒟, ∗), a two-sided cobar construction. Its spectrum of q-cosimplices is the product over all \underline{f} ∈ B_{q}(𝒟) of the spectra E(T (\underline{f})). The cofaces and codegeneracies with target C_{q}(E, 𝒟, ∗) have \underline{f}th coordinate obtained by projecting onto the coordinate of the source that is indexed by the corresponding face or degeneracy applied to \underline{f}, and, for the zeroth coface, composing with the map E(f_{1}) : E(d_{0}) ⟶ E(d_{1}).

We define the geometric realization or totalization “Tot K_{∗}” of a cosimplicial spectrum K_{∗} to be the end

Tot K_{∗} = ∫_{Δ} F ((Δ_{q})_{+}, K_{q}). (3.6)

Here we are using the evident functor Δ^{op}×Δ ⟶ 𝒮 that sends (p, q) to F ((Δ_{p})_{+}, K_{q}).

We define the homotopy limit of a contravariant functor E : 𝒟 ⟶ 𝒮 to be the totalization

(3.7) holim E = Tot C_{∗}(E, 𝒟, ∗).

For example, if \underline{K} : 𝒟 ⟶ 𝒮 is the constant functor at a spectrum K, then we see by use of adjunctions and inspection of definitions that

holim \underline{K} ≅ F (B(𝒟)_{+}, K). (3.8)

The essential point is that the definition makes perfect sense with the precise pointset level definitions of product and function spectra given in [37, pp. 13, 17].

4. Σ-cofibrant, LEC, and CW prespectra

We here discuss several special types of prespectra that play an important technical role in point-set level studies in stable homotopy theory. We first put the notion of a Σ-cofibrant spectrum (from I.2.4) into perspective by recalling the following definitions from [37, I§8] and [20]. A space X is said to be LEC (locally equiconnected) if the inclusion of its diagonal subspace is an unbased cofibration; see e.g. Lewis [35] for a discussion of such spaces.

Definition 4.1. A prespectrum D is said to be

(i) Σ-cofibrant if its structure maps

σ : Σ^{W −V} DV → DW

are based cofibrations. (ii) an inclusion prespectrum if its adjoint structure maps

σ̃ : DV → Ω^{W −V} DW

are inclusions. (iii) cofibrant if its adjoint structure maps σ̃ are based cofibrations. (iv) LEC if it is Σ-cofibrant and each space DV is LEC.

(v) CW if it is LEC and each DV has the homotopy type of a CW complex. A spectrum E is said to be Σ-cofibrant or LEC if it is isomorphic to LD for some Σ-cofibrant or LEC prespectrum D.

If E is a spectrum, then the maps σ̃ are homeomorphisms. Therefore, as a prespectrum, E is cofibrant, but it is not Σ-cofibrant (unless it is trivial). Although we have concentrated on Σ-cofibrant prespectra and spectra, the following result of Lewis [35] gives one reason for interest in the LEC notion.

Lemma 4.2. A Σ-cofibrant prespectrum is an inclusion prespectrum. An LEC prespectrum is cofibrant.

Our CW prespectra must not be confused with CW spectra; the latter are defined in terms of spectrum-level spheres and attaching maps. Since we are interested in notions that are appropriate for serious point-set level work and that admit usable equivariant generalizations, we have no interest in the old-fashioned and, to our minds, obsolete, notion of a CW prespectrum that requires CW complexes D_{n} and cellular structure maps ΣD_{n} ⟶ D_{n+1}. We have the following relations between CW prespectra and CW spectra [37, I.8.12-14].

Theorem 4.3. If D is a CW prespectrum, then LD has the homotopy type of a CW spectrum. If E is a CW spectrum, then each space EV has the homotopy type of a CW complex and E is homotopy equivalent to LD for some CW prespectrum D. Thus a spectrum has the homotopy type of a CW spectrum if and only if it has the homotopy type of LD for some CW prespectrum D.

The first statement is an immediate consequence of the following description of spectra in terms of shift desuspensions of spaces [37, I.4.7]. The second is based on use of the cylinder construction defined in the next section.

Proposition 4.4. If D is an inclusion prespectrum, then

LD ≅ colim Σ_{V}^{∞}DV,

where the colimit is computed as the prespectrum level colimit of the maps

Σ_{W}^{∞}σ : Σ_{V}^{∞}DV ≅ Σ_{W}^{∞}Σ^{W −V} DV ⟶ Σ_{W}^{∞}DW.

That is, the prespectrum level colimit is a spectrum that is isomorphic to LD.

In particular, if D is Σ-cofibrant, then LD is the colimit of shift desuspensions of space level based cofibrations. This makes the point-set level analysis of such spectra particularly convenient. The condition of being Σ-cofibrant is quite weak. It is clear from Theorem 4.3 that tame spectra, that is, spectra of the homotopy types of Σ-cofibrant spectra, are considerably more general than spectra of the homotopy types of CW spectra. The output spectra of the standard infinite loop space machines are Σ-cofibrant no matter what their input. The following closure properties of the category of Σ-cofibrant spectra are more directly relevant to us.

Lemma 4.5. The suspension and shift desuspension spectra of based spaces are Σ-cofibrant.

Proof. The prespectrum level structure maps of shift desuspensions are identity maps or the inclusions of basepoints (which are always based cofibrations) [37, I.4.1]. Explicitly, Σ_{V}^{∞}X = LΠ_{V}^{∞}X for an indexing space V ⊂ U, where

(Π_{V}^{∞}X)(W) = Σ^{W −V} X if W ⊃ V and (Π_{V}^{∞}X)(W) = {∗} otherwise.

Proposition 4.6. Assume given compact spaces A_{i}, based cofibrations A_{i} ⟶ B_{i}, and indexing spaces V_{i}, where i runs through any indexing set. If

\xymatrix{⋁_{i} Σ_{V_{i}}^{∞}A_{i} \ar[r] \ar[d] & E \ar[d] \\ ⋁_{i} Σ_{V_{i}}^{∞}B_{i} \ar[r] & F}

is a pushout of spectra and E is Σ-cofibrant, then F is Σ-cofibrant. If

\xymatrix{⋁_{i} 𝕃Σ_{V_{i}}^{∞}A_{i} \ar[r] \ar[d] & L \ar[d] \\ ⋁_{i} 𝕃Σ_{V_{i}}^{∞}B_{i} \ar[r] & M.}

is a pushout of 𝕃-spectra and L is tame, then M is tame.

Proof. Let E = LD, where D is a Σ-cofibrant prespectrum. Since E is a prespectrum level colimit, A_{i} is compact, and Σ_{V_{i}}^{∞} is adjoint to the V_{i}th space functor, we find that the given map Σ_{V_{i}}^{∞}A_{i} ⟶ E is induced by a map A_{i} ⟶ Ω^{W_{i}−V_{i}}DW_{i} for some W_{i} ⊃ V_{i}. There results a map of prespectra Π_{W_{i}}^{∞}Σ^{W_{i}−V_{i}}A_{i} ⟶ D that induces the given map of spectra under the isomorphism Σ_{V_{i}}^{∞}A_{i} ≅ Σ_{W_{i}}^{∞}Σ^{W_{i}−V_{i}}A_{i}. This allows us to construct the pushout on the prespectrum level, where an inspection from the fact that the structure maps of the Π_{W_{i}}^{∞}Σ^{W_{i}−V_{i}}A_{i} and Π_{W_{i}}^{∞}Σ^{W_{i}−V_{i}}B_{i} are wedges of identity maps or inclusions of basepoints shows that the pushout is Σ-cofibrant. This proves the first statement. Since there is no claim about the action of 𝕃, the second statement is an easy consequence, by comparisons of pushout diagrams and use of the fact that η : Σ_{V}^{∞}A ⟶ 𝕃Σ_{V}^{∞}A is a homotopy equivalence for any A and V.

Proposition 4.7. The external smash product of two Σ-cofibrant spectra is Σcofibrant. The j-fold external smash power of a Σ-cofibrant spectrum is Σ-cofibrant as a Σ_{j}-spectrum.

Proof. The smash product f ∧ g of based cofibrations is a based cofibration since it is the composite of based cofibrations f ∧ id and id ∧g. Indexing smash products on inner product spaces V ⊕ V′, as we may, we see immediately that the smash product of Σ-cofibrant prespectra is a Σ-cofibrant prespectrum. Similarly, for j-fold smash powers, we may index on j-fold sums V^{j} and use the fact that the jth smash power of a based cofibration is a based Σ_{j}-cofibration to see that the j-fold smash power of a Σ-cofibrant prespectrum is a Σ-cofibrant Σ_{j}-prespectrum. By the following lemma, these prespectrum level observations imply the desired spectrum level conclusions.

Lemma 4.8. For prespectra D and D′, the units D ⟶ ℓLD and D′ ⟶ ℓLD′ of the spectrification adjunction induce an isomorphism of spectra

L(D ∧ D′) ⟶ L(ℓLD ∧ ℓLD′).

Proof. Factoring the map in question through L(D ∧ℓLD′) in the evident way, we see that it suffices to prove that L(D ∧D′) ⟶ L(D ∧ℓLD′) is an isomorphism. We have adjoint function prespectra and spectra [37, II.3.3] such that if E is a spectrum and D′ is a prespectrum, then F (D′, ℓE) is a spectrum. Moreover, a glance at the cited definition shows that

F (D′, ℓE) ≅ ℓF (LD′, E).

This isomorphism of right adjoints implies the desired isomorphism of left adjoints.

Although the fact will not be used in our work, results like those above also apply to LEC spectra [20]: the suspension and shift desuspensions of LEC spaces are LEC spectra, and the smash product of LEC spectra is LEC.

For twisted half-smash products, we only have an up to homotopy version of the previous proposition.

Proposition 4.9. If E ∈ 𝒮 U is Σ-cofibrant and A is a compact space over ℐ (U, U′), then A ⋉ E is Σ-cofibrant. If E ∈ 𝒮 U is tame and A is a space over ℐ (U, U′) that has the homotopy type of a colimit of a sequence of cofibrations between compact spaces, then A ⋉ E is tame.

Proof. The first statement is immediate from the prespectrum level construction of twisted half-smash products [37, VI.2.7]. For the second statement, we may assume that E = LD, where D is a Σ-cofibrant prespectrum and, by I.2.5, we may also assume that A = colim A_{i} for a sequence of cofibrations A_{i} ⟶ A_{i+1} between compact spaces A_{i}. Then [37, VI.2.5 and VI.2.18] give a concrete description of A ⋉ E as L(A ⋉ D), where the prespectrum A ⋉ D is obviously Σ-cofibrant.

For example, the second statement applies when A has the homotopy type of a CW complex with finite skeleta.

5. The cylinder construction

We show here that we can functorially replace an A_{∞} or E_{∞} ring spectrum R by a weakly equivalent Σ-cofibrant A_{∞} or E_{∞} ring spectrum KR, and similarly for modules. This replacement already works on the prespectrum level. We have used it in several technical proofs, and we shall use it again later. An iterated mapping cylinder functor K that sends prespectra to weakly equivalent Σ-cofibrant prespectra was constructed in [37, I.6.8]. We shall use the language of homotopy colimits to give a more conceptual version of the construction that allows us to prove that it preserves structured ring and module spectra.

Construction 5.1. Let D be a prespectrum indexed on U. Define KD as follows. For an indexing space V, let \underline{V} be the category of subspaces V′ ⊂ V and inclusions. Define a functor D_{V} from \underline{V} to the category of based spaces by letting D_{V} (V′) = Σ^{V −V′}DV′. For an inclusion V′′ ⟶ V′,

V − V′′ = (V − V′) ⊕ (V′ − V′′),

and σ : Σ^{V′−V′′}DV′′ ⟶ DV′ induces D_{V} (V′′) ⟶ D_{V} (V′). Define

(KD)(V) = hocolim D_{V} .

An inclusion i : V ⟶ W induces a functor \underline{i} : \underline{V} ⟶ \underline{W}, the functor Σ^{W −V} commutes with homotopy colimits, and there is an evident isomorphism Σ^{W −V} D_{V} ≅ D_{W} \underline{i} of functors \underline{V} ⟶ \underline{W}. Therefore \underline{i} induces a map

σ : Σ^{W −V} hocolim D_{V} ≅ hocolim Σ^{W −V} D_{V} ≅ hocolim D_{W} \underline{i} ⟶ hocolim D_{W} ,

and this map is a cofibration by Lemma 3.5. Thus, with these structural maps, KD is a Σ-cofibrant prespectrum. The structural maps σ : D_{V} V′ ⟶ DV specify a natural transformation to the constant functor at DV and so induce a map r : (KD)(V) ⟶ DV, and these maps r specify a map of prespectra. Regarding the object V as a trivial subcategory of \underline{V}, we obtain j : DV ⟶ (KD)(V). Clearly rj = id, and jr ≃ id via a canonical homotopy since V is a terminal object of \underline{V}. The maps j do not specify a map of prespectra, but they do specify a weak map, in the sense that jσ ≃ σΣ^{W −V} j : Σ^{W −V} DV ⟶ (KD)(W), via a canonical homotopy. Clearly K is functorial and homotopy-preserving, and r is natural.

The following example may be illuminating.

Example 5.2. Let X be a based space and let D be the suspension prespectrum of X, so that DV = Σ^{V} X and the structure maps σ : Σ^{W −V} Σ^{V} X ⟶ Σ^{W} X are the evident identifications. Via these identifications, the functor D_{V} is isomorphic to the constant functor at Σ^{V} X, hence

(KD)(V) ≅ B(\underline{V})_{+} ∧ Σ^{V} X.

The structure maps of KD are induced by the cited identifications and the maps B(\underline{i}). In this case, we can use the initial objects \{0\} of the \underline{V} rather than the terminal objects V to obtain maps Σ^{V} X ⟶ (KD)(V). Because the functors \underline{i} preserve initial points, this gives a map of prespectra ν : D ⟶ KD such that rν = id. We have simply fattened up the Σ^{V} X via the compatible system of contractible spaces B(\underline{V}).

Construction 5.1 is a conceptual version of [37, I.6.8], and the discussion of “preternaturality” given in [37, I.7.5-I.7.7] applies to it. As usual, we extend the construction to spectra by setting KE = LKℓE, and we then have a natural weak equivalence r = Lrℓ : KE ⟶ E. The following result implies the second statement of Theorem 4.3.

Proposition 5.3. (i) If each space DV has the homotopy type of a CW

complex, then LKD has the homotopy type of a CW spectrum. (ii) If E has the homotopy type of a CW spectrum, then KE has the homotopy

type of a CW spectrum, hence r : KE ⟶ E is a homotopy equivalence.

Proof. By Proposition 4.4, LKD ≅ colim Σ_{n}^{∞}(KD)_{n}, (KD)_{n} = (KD)(ℝ^{n}), where the colimit is taken over the cofibrations

Σ_{n}^{∞}(KD)_{n} ≅ Σ_{n+1}^{∞}Σ(KD)_{n} ⟶ Σ_{n+1}^{∞}(KD)_{n+1}.

The conclusion of (i) follows since the colimit of a sequence of cofibrations of spectra of the homotopy types of CW spectra has the homotopy type of a CW spectrum. By [37, I.8.14], each space EV of a CW spectrum E has the homotopy type of a CW complex. Thus (ii) follows from (i) and the Whitehead theorem.

We must still discuss the behavior of the functor K with respect to smash products and twisted half-smash products.

Proposition 5.4. Let D and D′ be prespectra indexed on U and U′. Then there is a natural unital, associative, and commutative system of isomorphisms

ω : KD ∧ KD′ ⟶ K(D ∧ D′)

over D ∧ D′, where external smash products are understood.

Proof. Recall that the prespectrum level external smash product D ∧ D′ is naturally indexed on direct sums V ⊕ V′ of indexing spaces V in U and V′ in U′. Clearly, with our restricted set of indexing spaces, the product category \underline{V} × \underline{V′} is isomorphic to \underline{V ⊕ V′}. By definition,

(D ∧ D′)(V ⊕ V′) = DV ∧ D′V′,

with the evident structural maps. Since homotopy colimits are two-sided bar constructions and geometric realization and simplicial bar constructions commute suitably with products, we obtain isomorphisms

(hocolim D_{V}) ∧ (hocolim D′_{V′}) ≅ hocolim(D_{V} ∧ D′_{V′}) ≅ hocolim(D ∧ D′)_{V ⊕V′}

that are evidently compatible with the retractions to DV ∧ D′V′. The coherence statements are easily verified. For the unital condition, we allow U = {0}, in which case K is the identity functor; the space S^{0} is the unit for the external smash product.

Clearly this extends to j-fold external smash products, with all possible associativity and equivariance. We next consider changes of universe, preparatory to considering twisted half-smash products.

Lemma 5.5. Let f : U ⟶ U′ be a linear isometry. For prespectra D′ indexed on U′, Kf^{∗}D′ is isomorphic over f^{∗}D′ to f^{∗}KD′. For spectra E indexed on U, there is a natural map ω : f_{∗}KE ⟶ Kf_{∗}E over f_{∗}E.

Proof. For an indexing space V in U, f induces an isomorphism of categories \underline{V} ⟶ \underline{f(V)}. By definition, (f^{∗}D′)(V) = D′f(V), with the evident structural maps. By inspection,

(Kf^{∗}D′)(V) = hocolim(f^{∗}D′)_{V} ≅ hocolim D′_{f(V)} = (f^{∗}KD′)(V),

and these isomorphisms are compatible with the retractions to (f^{∗}D′)(V). The functor f_{∗} is left adjoint to f^{∗} [37, p.58]. For a prespectrum D indexed on U, the unit D ⟶ f^{∗}f_{∗}D of the adjunction induces a natural map

KD ⟶ Kf^{∗}f_{∗}D ≅ f^{∗}Kf_{∗}D.

The adjoint of this map is a natural map ϕ : f_{∗}KD ⟶ Kf_{∗}D over f_{∗}D. The spectrum level left adjoint to f^{∗} is f_{∗}E = Lf_{∗}ℓE. The unit D ⟶ ℓLD of the (L, ℓ) adjunction induces natural maps

Lf_{∗}D ⟶ Lf_{∗}ℓLD = f_{∗}LD and LKD ⟶ LKℓLD = KLD.

By [37, pp. 19, 58], the first of these is an isomorphism of spectra since f^{∗}ℓE = ℓf^{∗}E. Therefore ϕ specializes to give the required map

f_{∗}KE = Lf_{∗}ℓLKℓE ≅ Lf_{∗}KℓE ⟶ LKf_{∗}ℓE ⟶ LKℓLf_{∗}ℓE = Kf_{∗}E.

Lemma 5.6. For based spaces X, there is a natural map ω : Σ^{∞}X ⟶ KΣ^{∞}X such that r ∘ ω = id.

Proof. We can obtain ω by applying the previous lemma to i : {0} ⟶ U since, as noted in the proof of I.3.2, i_{∗}X = Σ^{∞}X. The map ω so obtained is the same as the map of spectra induced by the map ν of prespectra described in Example 5.2.

Proposition 5.7. Let α : A ⟶ ℐ (U, U′) be a space over ℐ (U, U′). For spectra E ∈ 𝒮 U, there is a natural map

ω : A ⋉ KE ⟶ K(A ⋉ E)

over A ⋉ E.

Proof. According to [37, II.2.17], a map of spectra A ⋉ E ⟶ E′ determines and is determined by maps of spectra α(a)_{∗}E ⟶ E′ for a ∈ A that satisfy a certain continuity condition. In particular, the identity map of A ⋉ E is determined by the evident maps ι(a) : α(a)_{∗}E ⟶ A ⋉ E. Composing maps ω from Lemma 5.5 with maps Kι(a), we obtain maps

α(a)_{∗}KE ⟶ K(α(a)_{∗}E) ⟶ K(A ⋉ E).

It is not hard to trace through the definitions to check the required continuity condition, and it is clear by pointwise inspection that the resulting map ω covers the retractions to A ⋉ E, r ∘ ω = id ⋉ω.

There are coherence diagrams that relate the maps ω of the proposition to the isomorphisms recorded in I.2.2. Putting these results together, using the definitions of 𝕃-spectra and their smash product and its unit map (I.4.2, I.5.1, I.8.3) and the definition of the 𝕃-spectrum structure on Σ^{∞}X (I.4.5), we arrive at the following conclusions.

Theorem 5.8. If N is an 𝕃-spectrum with action ξ, then KN is an 𝕃-spectrum with action the composite

[figure: ℒ (1) ⋉ KN; ω; K(ℒ (1) ⋉ N); Kξ; KN.]

Moreover, r : KN ⟶ N is a map of 𝕃-spectra. If X a based space, then ω : Σ^{∞}X ⟶ KΣ^{∞}X is a map of 𝕃-spectra over Σ^{∞}X. For 𝕃-spectra M and N, there is a natural map of 𝕃-spectra

ω : KM ∧_{ℒ} KN ⟶ K(M ∧_{ℒ} N)

over M ∧_{ℒ} N such that the following unit, associativity, and commutativity diagrams commute:

[figure: S ∧_{ℒ} KN; ω∧id; λ; KS ∧_{ℒ} KN; ω; KN; K(S ∧_{ℒ} N),; Kλ]

[figure: KL ∧_{ℒ} KM ∧_{ℒ} KN; ω∧id; id ∧ω; K(L ∧_{ℒ} M) ∧_{ℒ} KN; ω; KL ∧_{ℒ} K(M ∧_{ℒ} N); ω; K(L ∧_{ℒ} M ∧_{ℒ} N),]

and

[figure: KM ∧_{ℒ} KN; τ; ω; KN ∧_{ℒ} KM; ω; K(M ∧_{ℒ} N); Kτ; K(M ∧_{ℒ} N).]

Theorem 5.9. Let R be an A_{∞} ring spectrum with unit η and product ϕ. Then KR is an A_{∞} ring spectrum with unit and product the composites

[figure: S; ω; KS; Kη; KR; and; KR ∧_{ℒ} KR; ω; K(R ∧_{ℒ} R); Kϕ; KR.]

Moreover, r : KR ⟶ R is a map of A_{∞} ring spectra. If R is an E_{∞} ring spectrum, then so is KR. If M is an R-module (in the sense of II.3.3), then KM is a KR-module such that r : KM ⟶ M is a map of KR-modules.

220 X. SOME BASIC CONSTRUCTIONS ON SPECTRA

CHAPTER XI Spaces of linear isometries and technical theorems

This chapter contains a number of deferred proofs concerning the structure of the linear isometries operad and the behavior of twisted half-smash products with respect to equivalences and cofibrations. As we said in the introduction, these results are at the technical heart of our work. We emphasize that these results were used to build the foundations of Chapter I and were not referred to later until VII§6. Logically, they precede the formal introduction of S-modules.

1. Spaces of linear isometries

Many of our results depend on understanding the point-set topological and homotopical properties of spaces of linear isometries. We collect together the results that we need in this section and the next. However, we begin with a result on limits of cofibrations of unbased spaces. To prove it, we need the following generalization of the standard fact that a cofibration which is a homotopy equivalence is the inclusion of a strong deformation retract; it applies when the given map is also the map of total spaces of a pair of fibrations.

Lemma 1.1. Assume given a commutative diagram of spaces

[figure: A; p; i; X; q; B; j; Y]

in which p and q are fibrations and i and j are cofibrations and homotopy equivalences. Assume given a map r̄ : X ⟶ B such that r̄ ∘ i = p together with a homotopy h̄ : q ≃ j ∘ r̄ rel A. Then there is a map r : X ⟶ A such that r ∘ i = id and p ∘ r = r̄ together with a homotopy h : id ≃ ir rel A such that q ∘ h = h̄.

Proof. Both (X, A) and (X, A) × (I, ∂I) = (X × I, X × ∂I ∪ A × I) are DRpairs. The standard lifting property for fibrations and trivial cofibrations gives r and h via the diagrams:

[figure: A; i; id; A; p; and; X × ∂I ∪ A × I; k; X; q; X; r̄; r; B; X × I; h̄; h; Y,]

where k(x, 0, t) = x, k(x, 1, t) = ir(x), and k(a, s) = i(a).

Proposition 1.2. For n ≥ 1, assume given a commutative diagram of spaces

[figure: A_{n}; p_{n}; e_{n}; X_{n}; q_{n}; A_{n−1}; e_{n−1}; X_{n−1}]

in which the p_{n} and q_{n} are fibrations and the e_{n} are cofibrations and homotopy equivalences. Then the induced map

e : A ≡ lim A_{i} ⟶ lim X_{i} ≡ X

is the inclusion of a strong deformation retract and a cofibration.

Proof. Proceeding inductively, we use the lemma to construct retractions r_{n} : X_{n} ⟶ A_{n} and homotopies h_{n} : id ≃ e_{n} ∘ r_{n} rel A_{n} that are compatible with the given fibrations. The roles of r̄ and h̄ in the lemma are played by r_{n−1} ∘ q_{n} and h_{n−1} ∘ (q_{n} × id). We obtain the retraction r : X ⟶ A and homotopy h : id ≃ e∘r rel A by passage to limits. By the standard (N)DR-pair criterion, to show that e is a cofibration, we need only construct a map u : X ⟶ I such that u^{−1}(0) = A; of course, this is given by Urysohn’s lemma if X is normal (e.g., metric). Since each (X_{n}, A_{n}) is a DR-pair, there are maps v_{n} : X_{n} ⟶ I such that v_{n}^{−1}(0) = A_{n}. Let u_{n} = v_{n} ∘ π_{n}, where π_{n} : X → X_{n} is the projection, and define

u(x) = ∑_{n=1}^{∞} \frac{1}{2^{n}} u_{n}(x).

Then u is continuous and u(x) = 0 if and only π_{n}(x) ∈ A_{n} for each n.

Remark 1.3. The preceding may appear to be a model category result, but it depends on properties peculiar to the classical cofibrations of topological spaces.

Now Let U and U′ be universes and write U and U′ as the unions of expanding sequences of finite dimensional subspaces {V_{n}} and {V′_{n}}, with the topology of the union. Thus a subset N of U is open if it intersects each V_{n} in an open subset. This topology is finer than the evident metric topology. If we identify U with ℝ^{∞} and think of ℝ^{∞} as a subset of the product of countably many copies of ℝ, then the intersection of ℝ^{∞} with the product of the intervals (−1/q, 1/q) is an open neighborhood of zero which is not an open set in the metric topology.

For finite dimensional inner product spaces V and V′, the space ℐ (V, V′) of linear isometries from V to V′ is a smooth compact manifold. For a finite dimensional V, ℐ (V, U′) is the union of the ℐ (V, V′_{n}). Since a regular topological space that is the union of a sequence of compact spaces is paracompact, ℐ (V, U′) is paracompact. As a union of smooth compact manifolds, ℐ (V, U′) can be triangulated as a CW complex (and this also implies paracompactness).

The space ℐ (U, U′) is the inverse limit of the ℐ (V_{n}, U′). As such, it is a subspace of the product of the ℐ (V_{n}, U′), and it is therefore compactly generated. Each projection ℐ (V_{i+1}, V′_{j}) ⟶ ℐ (V_{i}, V′_{j}) is a bundle. By checking that the trivializations extend as j increases, one can deduce that each projection ℐ (V_{i+1}, U′) ⟶ ℐ (V_{i}, U′) is also a bundle.

Recall that a space X is LEC if the diagonal map X ⟶ X × X is a cofibration. It is standard that the inclusion {x} ⟶ X is then a cofibration for all x ∈ X; that is, every point is a nondegenerate basepoint. In fact, more generally, the inclusion of a retract in an LEC space is a cofibration [35, 3.1].

Proposition 1.4. The space ℐ (U, U′) is LEC.

Proof. Any CW complex is LEC [35, 2.4], hence each ℐ (V_{i}, U′) is LEC. Since ℐ (V_{i}, U′) is also contractible [37, II.1.5] (or see the following lemma), its diagonal map is a cofibration and a homotopy equivalence. The diagonal map of ℐ (U, U′) is the inverse limit of the diagonal maps of the ℐ (V_{i}, U′). Now the conclusion is immediate from Proposition 1.2.

Breaking with our rule of ignoring equivariant considerations, we prove the following result in full equivariant generality. As we have already used, ℐ (U, U′) is contractible. Thus, trivially, ℐ (U, U′) has the homotopy type of a CW complex. We record equivariant generalizations of these facts. We assume that some compact Lie group G acts on U and U′. Then G acts on ℐ (U, U′) by conjugation.

Lemma 1.5. For a G-space X, any two G-maps f, g : X ⟶ ℐ (U, U′) are homotopic.

Proof. Write U′ = U′_{1} ⊕U′_{2}, where U′_{1} and U′_{2} are G-universes isomorphic to U′. Deformations of the identity on U′ to isometries U′ ⟶ U′_{1} and U′ ⟶ U′_{2} show that f and g are homotopic to maps f′ : X ⟶ ℐ (U, U′_{1}) and g′ : X ⟶ ℐ (U, U′_{2}). Orthogonalization of the linear homotopy, (1 − t)f′ + tg′ shows that f′ ≃ g′.

Lemma 1.6. Assume that there is a finite dimensional representation V ⊂ U such that the projection π : ℐ (U, U′) ⟶ ℐ (V, U′) is a weak G-equivalence. Then π is a G-homotopy equivalence, hence ℐ (U, U′) has the homotopy type of a G-CW complex.

Proof. As a union of smooth G-manifolds, ℐ (V, U′) is triangulable as a G-CW complex. Therefore, by the G-Whitehead theorem, there is a G-map ϕ : ℐ (V, U′) ⟶ ℐ (U, U′) such that π ∘ ϕ ≃ id. The previous lemma gives that ϕ ∘ π ≃ id, completing the proof.

In practice, ℐ (U, U′) and ℐ (V, U′) have the appropriate behavior on fixed point spaces to be universal ℱ -spaces for some family ℱ of subgroups of G [37, II.2.11], and this allows one to verify the hypothesis on π. We will shortly use the following example.

Lemma 1.7. The space ℒ (j) = ℐ (U^{j}, U), U ≅ ℝ^{∞}, has the homotopy type of a Σ_{j}-CW complex.

Proof. The previous lemma applies with V replaced by V^{j} for any non-zero finite dimensional V ⊂ U.

2. Fine structure of the linear isometries operad

We here prove two deferred technical results about the linear isometries operad that were used in I§8, namely I.8.1 and I.8.4. Actually, we will prove a generalization of I.8.4 and of its consequence I.8.5, and we shall also prove III.1.8. The essential point is to give the analogs of I.5.4 in the cases i = 0 and j = 0 that it excludes.

Proof of I.8.1. We must prove that ℒ (2)/ℒ (1) × ℒ (1) consists of a single point. Consider points f and g in ℒ (2). Let V_{1} and V_{2} be the images of the restrictions of f to the two copies of U in U ⊕ U, and let W_{1} and W_{2} be the images of the restrictions of g to these copies of U. Clearly the point f is specified by isomorphisms U → V_{1} and U → V_{2}. We can find pairwise orthogonal infinite dimensional sub inner product spaces V′_{1}, V′_{2}, W′_{1}, and W′_{2} of V_{1}, V_{2}, W_{1}, and W_{2}, respectively. If f′ is specified by isomorphisms U → V′_{1} ⊂ V_{1} and U → V′_{2} ⊂ V_{2}, then f′ = fk for some k ∈ ℒ (1) × ℒ (1), namely k = f^{−1}(i_{1} ⊕ i_{2})f′, where i_{1} : V′_{1} → V_{1} and i_{2} : V′_{2} → V_{2} are the inclusions. Via the right action of ℒ (1) × ℒ (1), f is equivalent to such an f′, which in turn is equivalent to a point specified by isomorphisms U → V′_{1} ⊕ W′_{1} and U → V′_{2} ⊕ W′_{2}. By symmetry, the same is true for g, hence f is equivalent to g.

To explain the analogue of I.5.4 for the case i = 0 or j = 0, let ℒ^{0}(j) denote the subspace of ℒ (j) that consists of those linear isometries whose images have infinite dimensional orthogonal complements. Note that, for f ∈ ℒ^{0}(j), im(f) ⊕ im(f)^{⊥} is contained in but not necessarily equal to U. Define

ℒ̂ (j) = ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (0) × ℒ (j). (2.1)

Lemma 2.2. The map

γ : ℒ (2) × ℒ (0) × ℒ (j) ⟶ ℒ (j)

induces a continuous bijection

γ̂ : ℒ̂ (j) ⟶ ℒ^{0}(j).

Both ℒ̂ (j) and ℒ^{0}(j) are Σ_{j}-free and contractible and have the homotopy types of Σ_{j}-CW complexes, hence γ and γ̂ are Σ_{j}-equivariant homotopy equivalences.

Before giving the proof, we explain some consequences. We begin with the proof of III.1.7, and we need the following observation. As in I§1, we let 𝒮 U denote the category of spectra indexed on U when U is not clear from context.

Lemma 2.3. Let A be a space over ℐ (U, U′) and let ν : B ⟶ A be a surjective map. Then the induced map

𝒮 U′(A ⋉ E, E′) ⟶ 𝒮 U′(B ⋉ E, E′)

is injective for spectra E ∈ 𝒰 and E′ ∈ 𝒰′. Moreover, if α : E′ ⟶ E′′ is a spacewise inclusion of spectra, then the following diagram is a pullback of spaces:

[figure: 𝒮 U′(A ⋉ E, E′); α_{∗}; ν^{∗}; 𝒮 U′(A ⋉ E, E′′); ν^{∗}; 𝒮 U′(B ⋉ E, E′); α_{∗}; 𝒮 U′(B ⋉ E, E′′).]

Proof. The adjunction in Theorem 4.1 below reduces this to easily verified space level assertions.

Proof of III.1.7. Given a compact R-module L and a sequence of spacewise inclusions of R-modules M_{i} ⟶ M_{i+1}, we must prove that the natural map

colim ℳ_{R}(L, M_{i}) ⟶ ℳ_{R}(L, colim M_{i})

is a bijection. Let L = 𝔽_{R}K for a compact spectrum K. Since colimits of Rmodules are computed on the spectrum level and

ℳ_{R}(𝔽_{R}K, M) ≅ ℳ_{S}(S ∧_{ℒ} 𝕃K, M) = 𝒮 [𝕃](S ∧_{ℒ} 𝕃K, M),

it suffices to show that

colim 𝒮 [𝕃](S ∧_{ℒ} 𝕃K, M_{i}) ⟶ 𝒮 [𝕃](S ∧_{ℒ} 𝕃K, colim M_{i})

is a bijection for a sequence M_{i} ⟶ M_{i+1} of spacewise inclusions of 𝕃-spectra. We have S ∧_{ℒ} 𝕃K ≅ ℒ̂ (1) ⋉ K. Fix f ∈ ℒ^{0}(1) such that im(f) ⊕ im(f)^{⊥} = U and let g be its preimage in ℒ̂ (1). Clearly any other point f′ ∈ ℒ^{0}(1) has the form e ∘ f for some e ∈ ℒ (1). That is, ℒ^{0}(1) consists of a single orbit under the action of ℒ (1). Obviously this property is inherited by ℒ̂ (1). Thus the map ν : ℒ (1) ⟶ ℒ̂ (1) defined by ν(e) = e · g is a surjection. Using I.2.2, we see that ν induces a map of 𝕃-spectra

ν ⋉ id : 𝕃f_{∗}K ⟶ S ∧_{ℒ} 𝕃K.

Consider the following commutative diagram:

[figure: colim 𝒮 [𝕃](S ∧_{ℒ} 𝕃K, M_{i}); ν^{∗}; 𝒮 [𝕃](S ∧_{ℒ} 𝕃K, colim M_{i}); ν^{∗}; colim 𝒮 [𝕃](𝕃f_{∗}K, M_{i}); ≅; 𝒮 [𝕃](𝕃f_{∗}K, colim M_{i}); ≅; colim 𝒮 (K, f^{∗}M_{i}); 𝒮 (K, f^{∗} colim M_{i}).]

Since ν is a surjection, we can deduce from the preceding lemma and the evident description of 𝒮 [𝕃](M, N) as an equalizer that the top two vertical maps are injections and that, for each i, the ith top square, before passage to colimits on the left, is a pullback. We have f^{∗}(M)(V) = M(f(V)), hence f^{∗} colim M_{i} is the colimit of the sequence of inclusions f^{∗}M_{i} ⟶ f^{∗}M_{i+1}. The spectrum level analogue [37, I.4.8] gives that the bottom horizontal arrow is a bijection. It follows immediately that the top horizontal arrow is a bijection, the cited pullback squares implying the surjectivity.

The following result generalizes the first statement of I.8.5 from j = 1 to j ≥ 1. It was used in the proof of VII.6.3.

Theorem 2.4. Let j_{i} ≥ 1 and let Y_{i}, 1 ≤ i ≤ n, be a tame Σ_{j_{i}}-spectrum indexed on U^{j_{i}}, such as Y_{i} = (X_{i})^{j_{i}} for a tame spectrum X indexed on U. Then

⋀_{i=1}^{n} λ : ⋀_{ℒ} S ∧_{ℒ} (ℒ (j_{i}) ⋉ Y_{i}) ⟶ ⋀_{ℒ} ℒ (j_{i}) ⋉ Y_{i}

is a (Σ_{j_{1}} × ⋯ × Σ_{j_{n}})-equivariant homotopy equivalence of spectra. If the Y_{i} have the homotopy types of CW Σ_{j_{i}}-spectra, then ⋀_{ℒ} ℒ (j_{i}) ⋉ Y_{i} has the homotopy type of a CW (Σ_{j_{1}} × ⋯ × Σ_{j_{n}})-spectrum, and its orbit spectrum has the homotopy type of a CW spectrum. In particular, if X is a tame spectrum indexed on U, then

λ : S ∧_{ℒ} 𝔹X ⟶ 𝔹X and λ : S ∧_{ℒ} ℂX ⟶ ℂX

are homotopy equivalences of spectra, and similarly for smash products over ℒ of such maps.

Proof. By the associativity and commutativity of ∧_{ℒ} and the isomorphism S ∧_{ℒ} S ≅ S,

⋀_{ℒ} S ∧_{ℒ} (ℒ (j_{i}) ⋉ Y_{i}) ≅ S ∧_{ℒ} (⋀_{ℒ} ℒ (j_{i}) ⋉ Y_{i})).

Using I.5.4 and I.5.6, we find that

⋀_{ℒ} ℒ (j_{i}) ⋉ Y_{i} ≅ ℒ (j_{1} + ⋯ + j_{n}) ⋉ (Y_{1} ∧ ⋯ ∧ Y_{n}).

Let j = j_{1} + ⋯ + j_{n} and Z = Y_{1} ∧ ⋯ ∧ Y_{n}. By definition and inspection (see I.5.1 and the proofs of I.3.2 and I.8.3),

S ∧_{ℒ} (ℒ (j) ⋉ Z) = ℒ (2) ⋉_{ℒ (1)×ℒ (1)} (ℒ (0) ⋉ S^{0}) ∧ (ℒ (j) ⋉ Z) ≅ ℒ̂ (j) ⋉ Z

and, under our isomorphisms, the smash product of maps λ coincides with

λ = γ ⋉ id : ℒ̂ (j) ⋉ Z ⟶ ℒ (j) ⋉ Z.

The claimed homotopy equivalence follows from the equivariant form of I.2.5. The statements about CW homotopy type follow from [37, II.3.8, VI.5.2, and I.5.6], together with the equivariant form of I.2.6.

Proof of Lemma 2.2. We obtain a homeomorphism ℒ (j) ⟶ ℒ (1) of left ℒ (1)-spaces by composing on the right with an isomorphism U^{j} ⟶ U. It follows that, except for the equivariance statements, the result will be true in general if it is true when j = 1. It is clear that both spaces are Σ_{j}-free since ℒ (j) is, and an elaboration of Lemma 1.6 gives the assertion about Σ_{j}-CW homotopy types. Thus we assume that j = 1 in the rest of the proof.

Let i : {0} ⟶ U and i_{2} = i ⊕ id : U ⟶ U ⊕ U be the obvious isometries. Then, for f ∈ ℒ (1) and g ∈ ℒ (2),

γ(g; i, f) = g ∘ i_{2} ∘ f.

To see the surjectivity of γ̂, let h ∈ ℒ^{0}(1), let V be the orthogonal complement of the image of h, and choose an isomorphism j : U ⊕ V ⟶ V. Then h is the composite

U⟶^{i_{2}}U ⊕ U ⟶^{id ⊕h}U ⊕ h(U) ⊂ U ⊕ V ⊕ h(U)⟶^{j⊕id}V ⊕ h(U) ⊂ U.

Let g = (j ⊕ id) ∘ (id ⊕h). Then h = g ∘ i_{2} = γ(g; i, id).

To see the injectivity, consider (g, f) and (g′, f′) in ℒ (2) × ℒ (1). Let “∼” be the equivalence relation generated by (g, f) ∼ (g′, f′) if

g′ = g ∘ (j_{1} ⊕ j_{2}) and j_{2} ∘ f′ = f (2.5)

for points j_{1} and j_{2} in ℒ (1). It suffices to show that

(g′, f′) ∼ (g, f) ⟺ g′ ∘ i_{2} ∘ f′ = g ∘ i_{2} ∘ f,

and the forward implication is clear. The isometry g is given by the orthogonal pair of subspaces V_{1} = (g ∘i_{1})(U) and V_{2} = (g ∘i_{2})(U) of U together with isomorphisms U → V_{1} and U → V_{2}, and we let h = g ∘i_{2} ∘f : U ⟶ V_{2}. Thus (g, f) determines a triple (V_{1}, V_{2}, h) consisting of a pair of orthogonal infinite dimensional subspaces of U ⊕ U and a linear isometry h : U → V_{2}. Moreover, every such triple comes from some (g, f), as we see by choosing a linear isometry g such that V_{1} = (g ∘ i_{1})(U) and V_{2} = (g ∘ i_{2})(U) and setting f = (g ∘ i_{2})^{−1} ∘ h. Let “∼” be the equivalence relation on such triples generated by (V_{1}, V_{2}, h) ∼ (V′_{1}, V′_{2}, h′) if V′_{1} ⊂ V_{1}, V′_{2} ⊂ V_{2}, and h = h′ as maps U → U. If these triples arise from (g, f) and (g′, f′) and we set

j_{1} = (g ∘ i_{1})^{−1} ∘ g′ ∘ i_{1} and j_{2} = (g ∘ i_{2})^{−1} ∘ g′ ∘ i_{2},

then we find that (2.5) holds and can conclude that (g′, f′) ∼ (g, f). Thus the injectivity will follow if we can show that (V_{1}, V_{2}, h) ∼ (V′_{1}, V′_{2}, h′) for any two triples such that h = h′. Choose infinite dimensional subspaces W_{1} of V_{1} and W′_{1} of V′_{1} such that W_{1}, W′_{1}, and h(U) are mutually orthogonal. Then

(V_{1}, V_{2}, h) ∼ (W_{1}, h(U), h) ∼ (W_{1} + W′_{1}, h(U), h) ∼ (W′_{1}, h′(U), h′) ∼ (V′_{1}, V′_{2}, h′).

This proves the injectivity and thus the bijectivity of γ̂.

The contractibility of ℒ^{0}(1) is clear since it is closed under the homotopies described in Lemma 1.5. To see the contractibility of ℒ̂ (1), write U = U_{1} ⊕ U_{2}, where U_{1} and U_{2} are isomorphic to U, and define

𝒦 (2) = {g|g({0} ⊕ U) ⊂ U_{2}} ⊂ ℒ (2),
𝒦̂ (1) = 𝒦 (2) ×_{ℒ (1)×ℒ (1)} ℒ (0) × ℒ (1),

and

𝒦 ^{0}(1) = {f|f(U) ⊂ U_{2}} ⊂ ℒ^{0}(1).

Since 𝒦 ^{0}(1) ≅ ℐ (U, U_{2}), it is contractible. Since ℒ^{0}(1) is also contractible, the inclusion 𝒦 ^{0}(1) ⟶ ℒ^{0}(1) is obviously a homotopy equivalence. We have the following commutative diagram, in which the vertical arrows are inclusions:

[figure: 𝒦̂ (1); β̂; 𝒦 ^{0}(1); ℒ̂ (1); γ̂; ℒ^{0}(1).]

Modifying the proof that γ̂ is a bijection by restricting V_{2} to be contained in U_{2}, we see that β̂ is also a bijection. Choose linear isometric isomorphisms k_{1} : U ⟶ U_{1} and k_{2} : U ⟶ U_{2} and define

σ : 𝒦 ^{0}(1) ⟶ 𝒦 (2) × ℒ (0) × ℒ (1)

by σ(f) = (k_{1} ⊕ k_{2}, i, k_{2}^{−1} ∘ f). Then σ is a continuous section of

γ : 𝒦 (2) × ℒ (0) × ℒ (1) ⟶ 𝒦 ^{0}(1).

It follows that β̂ is a homeomorphism and in particular that 𝒦̂ (1) is contractible. We claim that the inclusion ι : 𝒦 (2) ⟶ ℒ (2) is a homotopy equivalence of right ℒ (1) × ℒ (1)-spaces. It will follow that the inclusion 𝒦̂ (1) ⟶ ℒ̂ (1) is a homotopy equivalence, proving the contractibility of ℒ̂ (1).

Define ρ : ℒ (2) ⟶ 𝒦 (2) by ρ(g) = i_{2} ∘k_{2} ∘g. To prove our claim, it suffices to find a homotopy h : i_{2} ∘ k_{2} ≃ id such that h_{t}(U_{2}) ⊂ U_{2} for all t, for then g ∈ 𝒦 (2) will imply h_{t} ∘ g ∈ 𝒦 (2) and, via right composition with maps g, h will induce the required homotopies id ≃ ι ∘ ρ and id ≃ ρ ∘ ι. Trivially, we have the following commutative diagram:

[figure: U_{1} ⊕ U_{2}; i_{2}⊕id; k_{2}; U_{1} ⊕ U_{1} ⊕ U_{2}; id ⊕k_{2}; U_{2}; i_{2}; U_{1} ⊕ U_{2}.]

We can homotope i_{2} ⊕ id to i_{1} ⊕ id by homotoping i_{2} to i_{1}, after which the right composite becomes id ⊕(k_{2}|_{U_{2}}). We can then homotope k_{2}|_{U_{2}} to the identity, after which the right composite becomes the identity. It is clear that U_{2} is carried into U_{2} by these homotopies.

Remark 2.6. It seems unlikely to us that γ̂ is actually a homeomorphism.

3. The unit equivalence for the smash product of 𝕃-spectra

We here prove I.6.2, and we restate each of its clauses as a lemma.

Lemma 3.1. For 𝕃-spectra N, there is a natural weak equivalence of 𝕃-spectra

ω : 𝕃S ∧_{ℒ} N ⟶ N.

Proof. Let ℒ (1) act from the right on ℒ (2) by setting fe = f ∘ (1 ⊕ e) for f ∈ ℒ (2) and e ∈ ℒ (1). Regard ℒ (2) as a space over ℒ (1) via the map σ_{2} : ℒ (2) ⟶ ℒ (1) specified by σ_{2}(f) = f ∘ i_{2}, where i_{2} : U ⟶ U^{2} is the inclusion of the second summand. Then

𝕃S ∧_{ℒ} N ≅ ℒ (2) ⋉_{ℒ (1)} N, (3.2)

and σ_{2} induces the required natural map

ω = σ_{2} ⋉ id : ℒ (2) ⋉_{ℒ (1)} N ⟶ ℒ (1) ⋉_{ℒ (1)} N ≅ N.

Observe that, by I.2.5, ω is a homotopy equivalence of spectra when N = 𝕃S^{n}.

We must prove that ω induces an isomorphism on homotopy groups. By adjunction, we may identify π_{n}(N) with h𝒮 [𝕃](𝕃S^{n}, N). First, to prove surjectivity, suppose given a map of 𝕃-spectra α : 𝕃S^{n} ⟶ N. Write

α̂ = id ∧_{ℒ} α : 𝕃S ∧_{ℒ} 𝕃S^{n} ⟶ 𝕃S ∧_{ℒ} N.

The following diagram commutes:

[figure: 𝕃S ∧_{ℒ} 𝕃S^{n}; ω; α̂; 𝕃S ∧_{ℒ} N; ω; 𝕃S^{n}; α; N.]

Since ω on the left is an equivalence, α ∈ Im(ω_{∗}).

To prove injectivity, suppose given an 𝕃-map β : 𝕃S^{n} → 𝕃S ∧_{ℒ} N such that α ≡ ω ∘ β ≃ 0. Define α̂ as above. Since α ≃ 0 and 𝕃S ∧_{ℒ} (?) : 𝒮 [𝕃] ⟶ 𝒮 [𝕃] is a homotopy preserving functor, α̂ ≃ 0. Define

β̂ = β ∘ ω : 𝕃S ∧_{ℒ} 𝕃S^{n} ⟶ 𝕃S ∧_{ℒ} N.

Since ω is an equivalence, to prove that β ≃ 0, it suffices to prove that α̂ ≃ β̂. Here, by the naturality of ω, β̂ coincides with the composite

𝕃S ∧_{ℒ} 𝕃S^{n}⟶^{id ∧β}𝕃S ∧_{ℒ} (𝕃S ∧_{ℒ} N)⟶^{ω}𝕃S ∧_{ℒ} N,

while α̂ coincides with the composite

𝕃S ∧_{ℒ} 𝕃S^{n}⟶^{id ∧β}𝕃S ∧_{ℒ} (𝕃S ∧_{ℒ} N)⟶^{id ∧ω}𝕃S ∧_{ℒ} N.

3. THE UNIT EQUIVALENCE FOR THE SMASH PRODUCT OF 𝕃-SPECTRA 231

Thus it suffices to show that ω ≃ id ∧ω. Let ℒ (1) act from the right on ℒ (3) by setting ge = g ∘ (1 ⊕ 1 ⊕ e) for g ∈ ℒ (3) and e ∈ ℒ (1). Regard ℒ (3) as a space over ℒ (1) via the map σ_{3} : ℒ (3) ⟶ ℒ (1) specified by σ_{3}(f) = f ∘ i_{3}, where i_{3} : U ⟶ U^{3} is the inclusion of the third summand. By the proof of the associativity isomorphism I.5.5, we have

𝕃S ∧_{ℒ} (𝕃S ∧_{ℒ} N) ≅ ℒ (3) ⋉_{ℒ (1)} N. (3.3)

Under the identifications (3.2) and (3.3), the maps ω and id ∧ω in our factorizations of β̂ and α̂ coincide with the maps

σ_{2,3} ⋉_{ℒ (1)} id : ℒ (3) ⋉_{ℒ (1)} N ⟶ ℒ (2) ⋉_{ℒ (1)} N

and

σ_{1,3} ⋉_{ℒ (1)} id : ℒ (3) ⋉_{ℒ (1)} N ⟶ ℒ (2) ⋉_{ℒ (1)} N,

where

σ_{2,3} : ℒ (3) ⟶ ℒ (2) and σ_{1,3} : ℒ (3) ⟶ ℒ (2)

are the maps that restrict g ∈ ℒ (3) to the second and third and first and third coordinates, respectively. Thus, it suffices to show that σ_{2,3} and σ_{1,3} are homotopic as maps of right ℒ (1)-spaces over ℒ (1). Since the images under σ_{2,3}(g) and σ_{1,3}(g) of the first copy of U in U^{2} are orthogonal and the right action of ℒ (1) is on the second copy, to which we restrict when mapping to ℒ (1), we obtain the required homotopy by normalizing the evident linear homotopy

h(g, t)(u_{1}, u_{2}) = tg(u_{1}, 0, u_{2}) + (1 − t)g(0, u_{1}, u_{2}).

Lemma 3.4. The suspension homomorphism Σ : π_{n}(N) ⟶ π_{n+1}(ΣN) is an isomorphism for any 𝕃-spectrum N and integer n.

Proof. We shall construct an explicit inverse isomorphism

Σ^{−1} : π_{n+1}(ΣN) ⟶ π_{n}(N).

We again think of π_{n}(N) as h𝒮 [𝕃](𝕃S^{n}, N). Since the functors Σ and 𝕃 commute, we may identify 𝕃S^{n+1} with Σ𝕃S^{n}. Similarly, we have a natural isomorphism

ι : 𝕃S^{−1} ∧_{ℒ} ΣN ≅ 𝕃S ∧_{ℒ} N.

Since 𝕃S ∧_{ℒ} 𝕃S^{n} and 𝕃S^{n} are CW 𝕃-spectra, the weak equivalence

ω : 𝕃S ∧_{ℒ} 𝕃S^{n} ⟶ 𝕃S^{n}

is a homotopy equivalence of 𝕃-spectra, and we choose a homotopy inverse ν. Suppose given an 𝕃-map β : Σ𝕃S^{n} ⟶ ΣN. We define Σ^{−1}β to be the composite

𝕃S^{n}⟶^{ν}𝕃S ∧_{ℒ} 𝕃S^{n}⟶^{ι^{−1}}𝕃S^{−1} ∧_{ℒ} Σ𝕃S^{n}⟶^{id ∧β}𝕃S^{−1} ∧_{ℒ} ΣN⟶^{ι}𝕃S ∧_{ℒ} N⟶^{ω}N.

If β = Σα, then the naturality of ι and ω imply that Σ^{−1}β = α ∘ ω ∘ ν ≃ α. Thus Σ^{−1} ∘ Σ = id. To evaluate Σ ∘ Σ^{−1}, consider the following diagram:

[figure: Σ(𝕃S ∧_{ℒ} 𝕃S^{n}); Σι^{−1}; Σ(𝕃S^{−1} ∧_{ℒ} Σ𝕃S^{n}); ι; Σ(id ∧β); Σ(𝕃S^{−1} ∧_{ℒ} ΣN); Σι; ι; Σ(𝕃S ∧_{ℒ} N); Σω; 𝕃S ∧_{ℒ} Σ𝕃S^{n}; id ∧β; ω; 𝕃S ∧_{ℒ} ΣN; ω; Σ𝕃S^{n}; Σν; Σ𝕃S^{n}; β; ΣN; ΣN.]

The left hand dotted arrow is a homotopy equivalence. The maps Σι and ι appearing at the right differ by an interchange of circle coordinates, hence we obtain a dotted homotopy equivalence making the right rectangle homotopy commute by using a map of degree minus one on the circle coordinate. This implies that the composite Σ ∘ Σ^{−1} is an isomorphism, and it follows formally that it must be the identity.

4. Twisted half-smash products and shift desuspension

We prove I.2.1 here. Our first proof was lengthy and unilluminating. The present conceptual argument is based on the first author’s work in [19], and we must explain its relevant features. In this section and the next, we write 𝒮 U for the category of spectra indexed on U, where U is a finite or countably infinite dimensional real inner product space.

There is an enlarged category of spectra, which we shall denote by 𝒮 in this section and the next, that contains all of the 𝒮 U. An object of 𝒮 is just a spectrum in 𝒮 U for some U. A map E ⟶ E′ in 𝒮 between spectra E ∈ 𝒮 U and E′ ∈ 𝒮 U′ is a linear isometry f : U ⟶ U′ together with a map g : E ⟶ f^{∗}E′ in 𝒮 U; we write g for such a map, letting f be understood. The full definition of 𝒮 exploits the topology of Grassmannian manifolds to topologize the set 𝒮 (E, E′) of maps E ⟶ E′ in such a way that the function

ε : 𝒮 (E, E′) ⟶ ℐ (U, U′)

that sends a map g to its underlying linear isometry f is continuous. The twisted half-smash product and twisted function spectra are implicitly built into the topology in view of the following result [19, Thm 0.5]. Let 𝒯 and 𝒰 denote the categories of based and unbased spaces, respectively, and let 𝒰 /_{ℐ (U,U′)} denote the category of spaces over ℐ (U, U′).

Theorem 4.1. There are natural homeomorphisms

𝒮 U′(A ⋉ E, E′) ≅ 𝒰 /_{ℐ (U,U′)} (A, 𝒮 (E, E′)) ≅ 𝒮 U(E, F [A, E′))

for spaces A over ℐ (U, U′) and spectra E ∈ 𝒮 U and E′ ∈ 𝒮 U′.

The proof of I.2.1 is based on an understanding of the topology of 𝒮 (E, E′) when E is a shift desuspension. To describe the topology, we first recall a general construction. Let B be any space and let 𝒯 /_{B} be the category of sectioned spaces over B. A based space X determines an object X × B of 𝒯 /_{B}. By [19, 1.3], there is an internal hom functor F_{B} in 𝒯 /_{B}. It is characterized by the adjunction

𝒯 /_{B}(W ∧_{B} X, Y) ≅ 𝒯 /_{B}(W, F_{B}(X, Y)),

where ∧_{B} is the fiberwise smash product. The fiber of F_{B}(X, Y) over b ∈ B is the function space F (X_{b}, Y_{b}) of based maps between the fibers of X and Y over b.

Let G_{n}(U) be the Grassmannian manifold of n-planes in the universe U. Let E_{n} be the disjoint union of the sets EV, where V runs over the points of G_{n}(U). Then E_{n} admits a natural topology such that the evident projection E_{n} ⟶ G_{n}(U) is a bundle [19, 2.15], and E_{n} is sectioned via the basepoints of the EV.

Now let V be a given n-dimensional subspace of U, and observe that there is a bundle ℐ (V, U) ⟶ G_{n}(U) that sends an isometry f to its image plane f(V). A spectrum E ∈ 𝒮 V is determined by the space X = EV and has zeroth space Ω^{V} X. We agree to write X(V) for the spectrum indexed on V that is determined by a based space X. The space 𝒮 (X(V), E′) can be identified with the pullback displayed in the following diagram.

[figure: 𝒮 (X(V), E′); ε; F_{G_{n}(U′)}(X × G_{n}(U′), E′_{n}); ℐ (V, U′); G_{n}(U′)] (4.2)

In fact, the identification of sets is easy from the definitions, and the bijection dictates the topology. Since E′_{n} and X × G_{n}(U′) are bundles over G_{n}(U′), so is the right vertical arrow. Therefore the left vertical arrow ε is also a bundle.

Let i : V ⟶ U be the inclusion and let j : ℐ (U, U′) ⟶ ℐ (V, U′) be induced by i. Recall that Σ_{V}^{∞} : 𝒯 ⟶ 𝒮 U is the left adjoint of the V th space functor 𝒮 U ⟶ 𝒯 . The functor that sends X to i_{∗}X(V) is also left adjoint to the V th space functor, hence

Σ_{V}^{∞}X ≅ i_{∗}X(V).

Define a space j^{∗}𝒮 (X(V), E′) by the pullback diagram

[figure: j^{∗}𝒮 (X(V), E′); 𝒮 (X(V), E′); ε; ℐ (U, U′); j; ℐ (V, U′)] (4.3)

By [19, proof of 4.9], the (i_{∗}, i^{∗}) adjunction leads to a homeomorphism

𝒮 (Σ_{V}^{∞}X, E′) ≅ j^{∗}𝒮 (X(V), E′) (4.4)

over ℐ (U, U′). Again, the identification of sets is evident from the definitions, and the bijection dictates the topology. We conclude that

ε : 𝒮 (Σ_{V}^{∞}X, E′) ⟶ ℐ (U, U′)

is a bundle. Since it is a pullback of a bundle over the paracompact contractible space ℐ (V, U′), it is a trivial bundle. This implies the following result.

Proposition 4.5. Let X be a based space, let V be a finite dimensional subspace of U, let V′ be a subspace of U′ isomorphic to V, and let E′ be a spectrum in 𝒮 U′. Then there is a homeomorphism

𝒮 (Σ_{V}^{∞}X, E′) ≅ ℐ (U, U′) × 𝒯 (X, E′V′)

of spaces over ℐ (U, U′) that is natural in X and E′.

In turn, this implies the following sharpening of [37, VI.1.6] (which deals with the case when A is compact); I.2.1 is a special case.

Proposition 4.6. Let α : A ⟶ ℐ (U, U′) be a map and X be a based space. Let V ⊂ U and V′ ⊂ U′ be isomorphic. Then there is an isomorphism of spectra

A ⋉ Σ_{V}^{∞}X ≅ A_{+} ∧ Σ_{V′}^{∞}X

that is natural in both A, as a space over ℐ (U, U′), and X.

Proof. This is an immediate consequence of the following chain of natural homeomorphisms.

𝒮 U′(A ⋉ Σ_{V}^{∞}X, E′) ≅ 𝒰 /_{ℐ (U,U′)} (A, 𝒮 (Σ_{V}^{∞}X, E′))
≅ 𝒰 /_{ℐ (U,U′)} (A, ℐ (U, U′) × 𝒯 (X, E′V′))
≅ 𝒰 (A, 𝒯 (X, E′V′))
≅ 𝒯 (A_{+}, 𝒯 (X, E′V′))
≅ 𝒯 (A_{+} ∧ X, E′V′)
≅ 𝒮 U′(Σ_{V′}^{∞}(A_{+} ∧ X), E′).

5. Twisted half-smash products and cofibrations

We here prove an analogue of I.2.5 concerning the behavior of twisted half-smash products with respect to cofibrations. It depends on the following sharpening of [19, 4.1].

Theorem 5.1. Let E ∈ 𝒮 U be a CW spectrum or a Σ-cofibrant spectrum and let E′ ∈ 𝒮 U′. Then ε : 𝒮 (E, E′) ⟶ ℐ (U, U′) is a fibration.

Proof. When E is a CW spectrum, this is [19, 4.1], except that the cited result only claimed a fibration over compact subspaces of ℐ (U, U′) because the elementary fact that G_{n}(U) is paracompact was overlooked there. The proof depends on three basic facts: E is the colimit of its finite subcomplexes, a finite CW spectrum is a shift desuspension Σ_{n}^{∞}X of a space X, and the inclusion of one finite subcomplex in another is a cofibration. As we now indicate, the proof of [19, 4.1] applies with only minor changes to give the conclusion when E is Σ-cofibrant, so that E = LD is a colimit as in X.4.4. The space 𝒮 (E, E′) is the limit of the spaces 𝒮 (Σ_{n}^{∞}D_{n}, E′), by [19, 4.7], and ε : 𝒮 (∗, E′) ⟶ ℐ (U, U′) is clearly a homeomorphism. Since the limit of a sequence of fibrations is a fibration, it suffices to show that each map

𝒮 (Σ_{n}^{∞}D_{n}, E′) ⟶ 𝒮 (Σ_{n−1}^{∞}D_{n−1}, E′) (5.2)

is a fibration. This map is induced by application of Σ_{n}^{∞} to the based cofibration σ : ΣD_{n−1} ⟶ D_{n}. Choose a numerable open cover {N} of G_{n}(U′) by coordinate neighborhoods for the bundle E′_{n} ⟶ G_{n}(U′). By (4.2)–(4.4) and a comparison of pullback diagrams (compare [19, p. 76]), the map (5.2) will be a fibration if the map

F_{G_{n}(U′)}(D_{n} × G_{n}(U′), E′_{n}) ⟶ F_{G_{n}(U′)}(ΣD_{n−1} × G_{n}(U′), E′_{n}) (5.3)

is a fibration. For a neighborhood N and point V′ ∈ N, the restriction over N of the map (5.3) is the map

F (D_{n}, E′V′) × N ⟶ F (ΣD_{n−1}, E′V′) × N (5.4)

induced by σ. Since σ is a based cofibration, it is formal that the induced map of function spaces

F (D_{n}, E′V′) ⟶ F (ΣD_{n−1}, E′V′)

is a based fibration. However, by application of the based CHP to test spaces Y_{+}, the based CHP implies the unbased CHP. Thus the map (5.4) is a fibration. Since the inverse images of the sets N give a numerable cover of F_{G_{n}(U′)}(ΣD_{n−1} × G_{n}(U′), E′_{n}), it follows that the map (5.3) is a fibration.

Roughly, the following result says about cofibrations what I.2.5 says about homotopy equivalences.

Proposition 5.5. Let A be a space over ℐ (U, U′) and let ϕ : A′ ⟶ A be a cofibration. If E ∈ 𝒮 U is either Σ-cofibrant or a CW spectrum, then the map ϕ ⋉ id : A′ ⋉ E ⟶ A ⋉ E is a cofibration of spectra in 𝒮 U′.

Corollary 5.6. Let A be a space over ℐ (U, U′) and let ϕ : A′ ⟶ A and ψ : B′ ⟶ B be cofibrations. If E ∈ 𝒮 U is either Σ-cofibrant or a CW-spectrum, then the map

((A′ ⋉ E) ∧ B_{+}) ∪ ((A ⋉ E) ∧ B′_{+}) ⟶ (A ⋉ E) ∧ B_{+}

induced by ϕ and ψ is a cofibration of spectra in 𝒮 U′.

Proof. By [37, VI.1.7], if we regard A × B as a space over ℐ (U, U′) via the composite of the projection to A and the given map A ⟶ ℐ (U, U′), then

(A ⋉ E) ∧ B_{+} ≅ (A × B) ⋉ E. (5.7)

Under such isomorphisms, the map in question agrees with the evident map

(A′ × B ∪ A × B′) ⋉ E ⟶ (A × B) ⋉ E.

By Lillig’s theorem [38], this reduces the corollary to a special case of the proposition.

Remark 5.8. These results, like our others, apply equally well to G-spectra for any compact Lie group G. Since passage to orbits commutes with pushouts and with smash products with spaces with trivial G-action, it carries equivariant mapping cylinders to nonequivariant ones. Therefore, by the standard characterization of cofibrations in terms of retractions to mapping cylinders, passage to orbits carries cofibrations of G-spectra to cofibrations of spectra.

Proof of Proposition 5.5. Consider a typical test diagram in 𝒮 U′ for the homotopy extension property:

[figure: A ⋉ E; (A ⋉ E) ∧ I_{+}; h; E′; X ⋉ E; f; (X ⋉ E) ∧ I_{+}; h̄]

Applying (5.7) (with B = I) and using the first homeomorphism of Theorem 4.1, we see that this is equivalent to the following test diagram of spaces:

[figure: A; A × I; h̃; 𝒮 (E, E′); ε; ℐ (U, U′); X; f̃; X × I; ε∘π; H]

Note the elaboration necesssary to take into account the fact that we are dealing with the category of spaces over ℐ (U, U′). The classical covering homotopy extension property (CHEP) is concerned with precisely such diagrams and asserts that there is a map H that makes the diagram commute. Its adjoint gives a map h̄ that makes the original diagram commute.

238 XI. SPACES OF LINEAR ISOMETRIES AND TECHNICAL THEOREMS

CHAPTER XII The monadic bar construction

The monadic bar construction was a central tool in earlier drafts of this paper, but it plays a very minor role in this version. It is nevertheless an important construction. We shall say just enough about it to prove the two deferred results that depend on it and to allow rigorous use of it in later work. The essential point is to prove certain lemmas on cofibrations, one of which played a role in our construction of model structures on the categories of R-algebras and commutative R-algebras.

1. The bar construction and two deferred proofs

Recall the definitions of an action of a monad on a functor and of a monadic bifunctor from II.6.3.

Definition 1.1. For a triple (F, 𝕊, R) consisting of a monad (𝕊, μ, η) in a category 𝒞, an 𝕊-algebra (R, ξ), and an 𝕊-functor (F, ν) in 𝒞′, define a simplicial object B_{∗}(F, 𝕊, R) in 𝒞′ by letting the q-simplices B_{q}(F, 𝕊, R) be F 𝕊^{q}R (where 𝕊^{q} denotes 𝕊 composed with itself q times); the faces and degeneracies are given by

d_{i} = \begin{cases}ν𝕊^{q−1} & if i = 0 \\ F 𝕊^{i−1}μ𝕊^{q−i−1} & if 1 ≤ i < q \\ F 𝕊^{q−1}ξ & if i = q \end{cases}

and s_{i} = F 𝕊^{i}η𝕊^{q−i}. If 𝕊′ is a monad in 𝒞′ and F is an (𝕊′, 𝕊)-bifunctor, then B_{∗}(F, 𝕊, R) is a simplicial 𝕊′-algebra.

When F takes values in a category with a forgetful functor to 𝒮, we write

B(F, 𝕊, R) = |B_{∗}(F, 𝕊, R)|.

We use a similar notation for pairs when F takes pairs of spectra as values. All of the bar constructions used earlier, such as B^{R}(M, A, N), can be interpreted as instances of this general construction. In the context of II.6.4, we have the following standard example.

Example 1.2. We have a simplicial 𝕊-algebra B_{∗}(𝕊, 𝕊, R) associated to an 𝕊algebra R. Let \underline{R} denote R regarded as a constant simplicial object, \underline{R}_{q} = R for all q, with each face and degeneracy the identity map. Iterates of μ and ξ give a map ε_{∗} : B_{∗}(𝕊, 𝕊, R) ⟶ \underline{R} of simplicial 𝕊-algebras in 𝒞. Similarly, iterates of η give a map η_{∗} : \underline{R} ⟶ B_{∗}(𝕊, 𝕊, R) of simplicial objects in 𝒞 such that ε_{∗}η_{∗} = id. Moreover, there is a simplicial homotopy η_{∗}ε_{∗} ≃ id [44, 9.8].

By II.4.1, we have monads 𝔹 and ℂ in 𝒮 whose algebras are the A_{∞} and E_{∞} ring spectra. We shall work in the ground category of spectra, rather than that of 𝕃spectra, for definiteness and because we envision more applications in that setting. Recall that geometric realization carries simplicial A_{∞} and E_{∞} rings, modules, and algebras to A_{∞} and E_{∞} rings, modules, and algebras, by X.1.5. We assume that all given spectra are Σ-cofibrant. In the contrary case, we first apply the cylinder construction K to make them so. By the results of X§4, this implies that the spectra of q-simplices in all of our constructions are tame. As we shall explain in the next section, it also implies that our simplicial spectra are proper, so that our homotopical results on geometric realization apply.

Definition 1.3. For an A_{∞} ring spectrum R, define an A_{∞} ring spectrum UR by

UR = B(𝔹, 𝔹, R).

For an E_{∞} ring spectrum R, define an E_{∞} ring spectrum UR by

UR = B(ℂ, ℂ, R).

The following result is immediate from Example 1.2 and X.1.2.

Lemma 1.4. For E_{∞} ring spectra R there is a natural map of E_{∞} ring spectra ε : UR ⟶ R that is a homotopy equivalence of spectra, and similarly for A_{∞} ring spectra.

We shall prove the following addendum in the next section.

Lemma 1.5. The unit η : S ⟶ UR is a cofibration of 𝕃-spectra.

Remark 1.6. The A_{∞} and E_{∞} versions of the lemmas are compatible. If R is an E_{∞} ring spectrum, then the natural map

B(𝔹, 𝔹, R) ⟶ B(ℂ, ℂ, R)

of A_{∞} ring spectra is a map under S and over R and is therefore a homotopy equivalence of spectra.

We now prove our change of operads result II.4.3. The proof is virtually the same as that given on the space level in [44], and we should admit that we know of no applications that cannot be handled by the space level result. Operads other than ℒ are essential to infinite loop space theory, but there we can convert an 𝒪-space to an ℒ -space before passage to spectra.

Proof of II.4.3. We are given an E_{∞} operad 𝒪 over ℒ and an 𝒪-spectrum R. Technically, we must assume that the unit element of 𝒪(1) is a nondegenerate basepoint in order to ensure that the simplicial spectra we use are proper. As in II.4.1, we have a monad 𝕆 whose algebras are the 𝒪-spectra together with a map 𝕆 ⟶ ℂ of monads. We define V R to be the bar construction B(ℂ, 𝕆, R). By I.2.5 and the results of X§4, 𝕆X ⟶ ℂX is a homotopy equivalence of spectra for any tame spectrum X. By X.2.4, there result maps of 𝒪-spectra

R ⟵ B(𝕆, 𝕆, R) ⟶ B(ℂ, 𝕆, R) = V R

that are homotopy equivalences of spectra.

We have precisely analogous constructions for modules. By II.6.2, we have a monad ℂ[1] in the category of pairs of spectra whose algebras are pairs consisting of an E_{∞} ring spectrum and a module over it. Its second coordinate is made explicit in II.5.7. We have a similar monad 𝔹[1] in the A_{∞} case.

Definition 1.7. For an E_{∞} ring spectrum R and an R-module M, define a UR-module UM by

(UR; UM) = B(ℂ[1], ℂ[1], (R; M))

Replacing ℂ by 𝔹, we obtain an analogous functor U on modules over an A_{∞} ring spectrum.

Lemma 1.8. There is a natural map of UR-modules ε : UM ⟶ M that is a homotopy equivalence of spectra.

Remark 1.9. The A_{∞} and E_{∞} interpretations of the lemma are compatible. If M is a module over an E_{∞} ring spectrum R, then the natural map

B(𝔹[1], 𝔹[1], (R; M)) ⟶ B(ℂ[1], ℂ[1], (R; M))

is a map over (R; M) and is thus a pair of homotopy equivalences of spectra.

Proof of II.5.2. The argument is much the same as the proof of II.4.3. As in II.5.7, we have a monad 𝕆[1] such that a 𝕆[1]-algebra is a pair consisting of an 𝒪-spectrum and a module over it, and we have a map of monads 𝕆[1] ⟶ ℂ[1]. For an 𝒪-spectrum R and an R-module M, we define V M to be the V R-module given by the second coordinate of B(ℂ[1], 𝕆[1], (R; M)). The second coordinate of the weak equivalence

(R; M) ⟵ B(𝕆[1], 𝕆[1], (R; M)) ⟶ B(ℂ[1], 𝕆[1], (R; M)) = (V R; V M)

is the required weak equivalence between the R-module M and the V R-module V M; it is a homotopy equivalence of spectra.

2. Cofibrations and the bar construction

We must prove that our simplicial bar constructions are proper and prove Lemma 1.5. Recall the definition of a proper simplicial 𝕃-spectrum |K_{∗}| from X.2.2, and remember that the simplicial filtration of |K_{∗}| is then given by a sequence of cofibrations of 𝕃-spectra; it follows that the inclusion K_{0} ⟶ |K_{∗}| is a cofibration of 𝕃-spectra.

Proposition 2.1. The simplicial bar constructions used to construct the various functors U and V in the previous section are all proper simplicial 𝕃-spectra.

Proof of Lemma 1.5. The unit S ⟶ UR is the composite of the inclusion of S as a wedge summand of ℂR and the inclusion ℂR ⟶ UR.

The following two lemmas directly imply Proposition 2.1 in the case of UR. The proofs in the remaining cases are similar. Looking back at B_{∗}(F, 𝕊, R), we see that all of its degeneracy operators are maps of the form F (?), so that the inclusion

sB_{q}(F, 𝕊, R) ⊂ B_{q}(F, 𝕊, R)

is obtained by applying the functor F to an inclusion that we may write

s𝕊^{q}R ⊂ 𝕊^{q}R. (2.2)

As explained after X.2.1, pedantic care with pushouts and coends is needed to be precise about this. The following lemma shows that the functor ℂ converts cofibrations of spectra to cofibrations of 𝕃-spectra. Here we are thinking of ℂ as playing the role of F in our bar construction. The second part of the lemma is more interesting. It was essential to make sense of the Cofibration Hypothesis of VII§4.

Lemma 2.3. The following statements hold.

(i) The monads 𝕋 and ℙ in 𝒮 [𝕃] that define A_{∞} and E_{∞} ring spectra preserve cofibrations of 𝕃-spectra. Therefore the monads 𝔹 and ℂ in 𝒮 convert cofibrations of spectra to cofibrations of 𝕃-spectra.

(ii) For any commutative S-algebra R, the monads 𝕋 and ℙ in ℳ_{R} that define R-algebras and commutative R-algebras preserve cofibrations of R-modules.

Proof. We prove the second statement. The first is similar; its second statement holds because 𝔹 = 𝕋𝕃, ℂ = ℙ𝕃, and 𝕃 carries cofibrations of spectra to cofibrations of 𝕃-spectra. Let f_{i} : M_{i} ⟶ N_{i} be cofibrations of R-modules, 1 ≤ i ≤ j. We claim that f_{1} ∧_{R} ⋯ ∧_{R} f_{j} is a cofibration of R-modules. There are retractions of R-modules r_{i} : N_{i} ∧ I_{+} ⟶ Mf_{i}, where Mf_{i} = N_{i} ∪ (M_{i} ∧ I_{+}) is the mapping cylinder of f_{i}. The diagonal map Δ : I ⟶ I^{j} is a deformation retraction of spaces, with retraction ρ given by the averaging map (t_{1}, ⋯, t_{j}) ⟶ ∑ t_{i}/j. The following composite is a retraction of R-modules, proving our claim:

[figure: (N_{1} ∧_{R} ⋯ ∧_{R} N_{j}) ∧ I_{+}; id ∧Δ_{+}; (N_{1} ∧_{R} ⋯ ∧_{R} N_{j}) ∧ (I^{j})_{+} ≅ (N_{1} ∧ I_{+}) ∧_{R} ⋯ ∧_{R} (N_{j} ∧ I_{+}); r_{1}∧···∧r_{j}; (Mf_{1}) ∧_{R} ⋯ ∧_{R} (Mf_{j}) ≅ (N_{1} ∧_{R} ⋯ ∧_{R} N_{j}) ∪ ((M_{1} ∧_{R} ⋯ ∧_{R} M_{j}) ∧ (I^{j})_{+}); id ∪(id ∧ρ_{+}); (N_{1} ∧_{R} ⋯ ∧_{R} N_{j}) ∪ ((M_{1} ∧_{R} ⋯ ∧_{R} M_{j}) ∧ I_{+}) = M(f_{1} ∧ ⋯ ∧ f_{j}).]

Now let f_{1} = ⋯ = f_{j} = f, say. Then the j-fold ∧_{R}-power f^{j} is a cofibration. Moreover, since Δ and ρ are Σ_{j}-equivariant, f^{j} is a Σ_{j}-cofibration. Since passage to orbits carries Σ_{j}-cofibrations to cofibrations, f^{j}/Σ_{j} is also a cofibration of Rmodules. By II.7.1, our monads are given by wedges of j-fold smash powers or j-fold symmetric smash powers, and the conclusion follows.

Lemma 2.4. For each q ≥ 1, the inclusion sℂ^{q}R ⊂ ℂ^{q}R is a cofibration of spectra.

Proof. Recall that ℂR = ⋁_{j≥0} ℒ (j) ⋉_{Σ_{j}} R^{j}. Our standing assumption that R is Σ-cofibrant implies that each R^{j} is Σ_{j}-equivariantly Σ-cofibrant, by X.4.7. This allows us to apply XI.5.5 and its equivariant version to show that maps between twisted half-smash products are cofibrations. The unit map η : R ⟶ ℂR is the composite

η ⋉ id : R = {id} ⋉ R ⟶ ℒ (1) ⋉ R ⊂ ℂR.

Here the first map is induced by the inclusion {1} ⟶ ℒ (1) and is a cofibration by XI.1.4 and XI.5.5, and the second map is the inclusion of a wedge summand. We shall prove that the inclusion sℂ^{q}R ⊂ ℂ^{q}R is a wedge of inclusions of the general form A′ ⋉_{G} R^{j} ⟶ A ⋉_{G} R^{j}, where G is a subgroup of Σ_{j} and A′ ⟶ A is a Gcofibration. By XI.5.5, this will imply the conclusion. Although the combinatorics of the proof are a bit messy, it is easy to see what is going on by writing out the first few cases explicitly. The subspectrum sℂ^{q} is the union of the images of the inclusions ℂ^{i}ηℂ^{q−i−1} : ℂ^{q−1}R ⟶ ℂ^{q}R, 0 ≤ i ≤ q − 1. We first show by induction on q that ℂ^{q}R is a wedge of spectra of the form A ⋉_{G} R^{j}. This is obvious if q = 1 and we assume it for q − 1. We first consider ℒ (j) ⋉_{Σ_{j}} Q^{j}, where Q is a wedge of spectra Q_{v} indexed on a totally ordered set 𝒱 . (See [14, II§2] for more details of this analysis of extended powers of wedges.) Let V run through the set of ordered j-tuples of elements of 𝒱 ; these V can be viewed as canonical elements in the distinct orbits of 𝒱 ^{j} under the permutation action of Σ_{j}. For such a V, let (v(1), ⋯, v(n)) be the distinct elements of 𝒱 appearing in V, let v(i) appear j(i) times, so that j = ∑ j(i), and let Σ_{V} ⊂ Σ_{j} be the image of Σ_{j(1)} × ⋯ × Σ_{j(n)} under the block sum homomorphism. Then

ℒ (j) ⋉_{Σ_{j}} Q^{j} ≅ ⋁_{V} ℒ (j) ⋉_{Σ_{V}} (Q_{v(1)}^{j(1)} ∧ ⋯ ∧ Q_{v(n)}^{j(n)}).

Now suppose that each spectrum Q_{v} is of the form X(v) ⋉_{G(v)} R^{k(v)} for some subgroup G(v) of Σ_{k(v)} and some G(v)-space X(v) over ℒ (k(v)). Then canonical isomorphisms in I.2.2 imply that

ℒ (j) ⋉_{Σ_{V}} (Q_{v(1)}^{j(1)} ∧ ⋯ ∧ Q_{v(n)}^{j(n)}) ≅ A(V) ⋉_{G(V)} R^{k(V)},

where k(V) = ∑ j(i)k(v(i)), G(V) ⊂ Σ_{k(V)} is the image under the canonical homomorphism of the product of wreath products

Σ_{j(1)} ∫ G_{v(1)} × ⋯ × Σ_{j(n)} ∫ G_{v(n)},

and

A(V) = ℒ (j) × X_{v(1)}^{j(1)} × ⋯ × X_{v(n)}^{j(n)}

with its structural map γ to ℒ (k(V)). This implies that ℂ^{q}R is a wedge of spectra of the form A ⋉_{G} R^{j}, and it implicitly gives a complete inductive description of the relevant spaces A and groups G. The wedge summands are indexed by certain directed trees T. Each vertex of T has a prescribed level i, 1 ≤ i ≤ q. There is a unique vertex of level q, there are directed edges from vertices of level i to vertices of level i − 1, and each vertex of level less than q is the target of exactly one edge:

[figure: •; •; •; •; •; •; •; •; •]

Each vertex is labelled with some ℒ (j), where, if the vertex has level greater than 1, then the vertex is the source of j edges. (We allow j = 0, when the vertex is the source of no edges.) For such a tree T, the space A(T) that is used to construct the corresponding wedge summand is the product of the labelling spaces ℒ (j), ordered as prescribed by the tree and the inductive specification of the wedge summands given above. The degeneracy subspace lies in those wedge summands whose corresponding trees have all of their labels ℒ (1) at one or more levels, and it is obtained by replacing ℒ (1) by the point {1} in the labels of vertices at those levels. This proves the lemma.

246 XII. THE MONADIC BAR CONSTRUCTION

CHAPTER XIII Epilogue: The category of 𝕃-spectra under S

In a previous draft of this paper, certain variants of smash products that are defined between 𝕃-spectra under S played a central role. In the present version, the only vestige remaining is the definition of free modules over A_{∞} ring spectra given in II.5.3.

However, the parallel algebraic theory of [34] still requires such variant tensor products. We imagine that there is a functor, like the singular chain complex functor, from topological A_{∞} and E_{∞} rings and modules to algebraic ones. Such a construction would require the old definitions. The point is that, in algebra, it seems that one cannot hope to have an analogue of the isomorphism S ∧_{ℒ} S ≅ S. The theory of [34] is based on the algebraic operad 𝒞 = C_{∗}(ℒ), where C_{∗} is the singular chain complex functor. Hopkins’ lemma, I.5.4, carries over since C_{∗} preserves split coequalizers. However, the relation ℒ (2)/(ℒ (1) × ℒ (1)) = {∗} does not carry over, and in fact one cannot have the relation 𝒞 (2)⊗_{𝒞 (1)⊗𝒞 (1)} ℤ = ℤ in any E_{∞} operad of (connected) chain complexes. Thus the topological theory is intrinsically better behaved algebraically than the parallel algebraic theory.

We explain just enough of the old definitions to give the idea and to explain how the new theory gives homotopical information about the old definitions.

1. The modified smash products ◁_{L}, ▷_{L}, and ⋆_{L}

We return to the prologue and work in the category of 𝕃-spectra in this section. We shall leave all proofs as exercises for the reader. They are easy consequences of results in Chapter I. Let 𝒮 [𝕃]\S denote the category of 𝕃-spectra under S. We write η generically for the given map S ⟶ M.

248 XIII. EPILOGUE: THE CATEGORY OF 𝕃-SPECTRA UNDER S

Definition 1.1. Let M be an 𝕃-spectrum under S and let N be an 𝕃-spectrum. Define the mixed smash product M ◁_{ℒ} N to be the pushout displayed in the following diagram of 𝕃-spectra:

[figure: S ∧_{ℒ} N; λ; η∧id; M ∧_{ℒ} N; N; M ◁_{ℒ} N.] (1.2)

Define N ▷_{ℒ} M by symmetry.

If we apply the functor S ∧_{ℒ} (?) to the diagram (1.2), we obtain a weakly equivalent pushout diagram whose left arrow is an isomorphism and whose right arrow is therefore also an isomorphism. This implies the basic relation

S ∧_{ℒ} (M ◁_{ℒ} N) ≅ (S ∧_{ℒ} M) ∧_{S} (S ∧_{ℒ} N), (1.3)

which allows us to deduce homotopical properties of ◁_{ℒ} from homotopical properties of ∧_{S}. It also implies the following result.

Proposition 1.4. For any 𝕃-spectrum N, the canonical map

N ⟶ S ◁_{ℒ} N

is an isomorphism of 𝕃-spectra and the canonical map

M ∧_{ℒ} N ⟶ M ◁_{ℒ} N

is a weak equivalence of 𝕃-spectra.

For any 𝕃-spectrum N under S, the canonical map

S ∧_{ℒ} N ⟶ S ▷_{ℒ} N

is an isomorphism because λ : S ∧_{ℒ} S ⟶ S is an isomorphism. Composing the inverse of this isomorphism with the unit weak equivalence S ∧_{ℒ} N ⟶ N, we obtain the following result.

Proposition 1.5. For 𝕃-spectra N under S, there is a natural weak equivalence of 𝕃-spectra λ : S ▷_{ℒ} N ⟶ N.

Lemma 1.6. Let M and N be 𝕃-spectra. Then

(M ∨ S) ◁_{ℒ} N ≅ (M ∧_{ℒ} N) ∨ N.

The commutativity and associativity of ∧_{ℒ} imply the following commutativity and associativity isomorphisms relating ∧_{ℒ} and ◁_{ℒ} ; these isomorphisms imply various others. The monad on 𝒮 [𝕃] whose algebras are the 𝕃-spectra under S sends M to M ∨ S, hence the 𝕃-spectra M ∨ S are the free 𝕃-spectra under S. Results like the following one can be proven by first checking them on the M ∨ S and then deducing them in general.

1. THE MODIFIED SMASH PRODUCTS ◁_{L}, ▷_{L}, AND ⋆_{L} 249

Lemma 1.7. Let M and M′ be 𝕃-spectra under S and let N and N′ be 𝕃-spectra. Then there are natural isomorphisms

M ◁_{ℒ} N ≅ N ▷_{ℒ} M,
M ◁_{ℒ} (N ∧_{ℒ} N′) ≅ (M ◁_{ℒ} N) ∧_{ℒ} N′,

and

M ◁_{ℒ} (N ▷_{ℒ} M′) ≅ (M ◁_{ℒ} N) ▷_{ℒ} M′.

With a view towards generalization to arbitrary ground E_{∞} ring spectra, for which R ∧_{ℒ ,R} R will not be isomorphic to R, we give the following definition in a form that does not rely on the isomorphism S ∧_{ℒ} S ≅ S.

Definition 1.8. Let M and N be 𝕃-spectra under S. The coproduct of M and N in 𝒮 [𝕃]\S is the pushout M ∪_{S} N. There is an analogous pushout

(M ∧_{ℒ} S) ∪_{S∧_{ℒ} S} (S ∧_{ℒ} N),

and the unit maps λ determine a natural map of 𝕃-spectra

λ : (M ∧_{ℒ} S) ∪_{S∧_{ℒ} S} (S ∧_{ℒ} N) ⟶ M ∪_{S} N.

The restrictions to S ∧_{ℒ} S of the maps

id ∧_{ℒ} η : M ∧_{ℒ} S ⟶ M ∧_{ℒ} N and η ∧_{ℒ} id : S ∧_{ℒ} N ⟶ M ∧_{ℒ} N

coincide, hence these maps determine a map

θ : (M ∧_{ℒ} S) ∪_{S∧_{ℒ} S} (S ∧_{ℒ} N) ⟶ M ∧_{ℒ} N.

Define the unital operadic smash product M ⋆_{ℒ} N to be the pushout displayed in the following diagram of 𝕃-spectra:

[figure: (M ∧_{ℒ} S) ∪_{S∧_{ℒ} S} (S ∧_{ℒ} N); θ; λ; M ∪_{S} N; M ∧_{ℒ} N; M ⋆_{ℒ} N.]

Then M ⋆_{ℒ} N is an 𝕃-spectrum under S with unit the composite of the unit S ⟶ M ∪_{S} N and the displayed canonical map M ∪_{S} N → M ⋆_{ℒ} N.

The essential, obvious, point is that S is a strict unit for the product ⋆_{ℒ} .

Remark 1.9. Because S ∧_{ℒ} S ≅ S, ⋆_{ℒ} can be defined less conceptually but more succinctly as the pushout in the diagram

[figure: (M ∧_{ℒ} S) ∨ (S ∧_{ℒ} N); θ; λ; M ∨ N; M ∧_{ℒ} N; M ⋆_{ℒ} N.]

An immediate comparison of pushout diagrams gives a natural map

M ▷_{ℒ} N ⟶ M ⋆_{ℒ} N,

and a diagram chase shows that the product ⋆_{ℒ} can be constructed in terms of the product ▷_{ℒ} .

Lemma 1.10. If M and N are 𝕃-spectra under S, then the following diagram is a pushout:

[figure: S ▷_{ℒ} N; λ; η▷id; M ▷_{ℒ} N; N; M ⋆_{ℒ} N.] (1.11)

If M is an 𝕃-spectrum and N is an 𝕃-spectrum under S, then

(M ∨ S) ⋆_{ℒ} N ≅ (M ▷_{ℒ} N) ∨ N.

Applying the functor S ∧_{ℒ} (?) to the diagram (1.11) and using (1.3), we find that

S ∧_{ℒ} (M ⋆_{ℒ} N) ≅ (S ∧_{ℒ} M) ∧_{S} (S ∧_{ℒ} N). (1.12)

Again, homotopical properties of ⋆_{ℒ} can be deduced from homotopical properties of ∧_{S}, and we have the following result.

Proposition 1.13. The canonical map

M ▷_{ℒ} N ⟶ M ⋆_{ℒ} N

is a weak equivalence of 𝕃-spectra.

Lemma 1.14. Let M and N be 𝕃-spectra. Then

(M ∨ S) ⋆_{ℒ} (N ∨ S) ≅ (M ∧_{ℒ} N) ∨ M ∨ N ∨ S.

Lemma 1.15. The following associativity relation holds, where M and M′ are 𝕃-spectra under S and N is an 𝕃-spectrum:

(M ⋆_{ℒ} M′) ◁_{ℒ} N ≅ M ◁_{ℒ} (M′ ◁_{ℒ} N).

2. THE MODIFIED SMASH PRODUCTS ◁_{R}, ▷_{R}, AND ⋆_{R} 251

Theorem 1.16. The category 𝒮 [𝕃]\S is symmetric monoidal under ⋆_{ℒ} . The categories of monoids and commutative monoids in 𝒮 [𝕃]\S are isomorphic to the categories of A_{∞} ring spectra and E_{∞} ring spectra.

We have a mixed function 𝕃-spectrum (but not a unital operadic one).

Definition 1.17. Let M be an 𝕃-spectrum under S and N be an 𝕃-spectrum. Define F_{ℒ}^{▷}(M, N) to be the 𝕃-spectrum displayed in the following pullback diagram:

[figure: F_{ℒ}^{▷}(M, N); F_{ℒ} (M, N); η^{∗}; N; F_{ℒ} (S, N);]

here the bottom arrow is adjoint to λτ : N ∧_{ℒ} S ≅ S ∧_{ℒ} N ⟶ N.

Proposition 1.18. Let M be an 𝕃-spectrum under S and L and N be 𝕃-spectra. Then

𝒮 [𝕃](M ◁_{ℒ} L, N) ≅ 𝒮 [𝕃](L ▷_{ℒ} M, N) ≅ 𝒮 [𝕃](L, F_{ℒ}^{▷}(M, N)).
2. The modified smash products ◁_{R}, ▷_{R}, and ⋆_{R}

We assume that R is an A_{∞} ring spectrum in this section, and we understand modules in the sense of II.3.3. Thus the unit weak equivalences are not required to be isomorphisms. The cell and CW theory of such R-modules is developed in the same fashion as for modules over S-algebras. The appropriate definition of a smash product over R in this context reads as follows.

Definition 2.1. Let R be an A_{∞} ring spectrum and let M be a right and N be a left R-module. Define M ∧_{ℒ ,R} N to be the coequalizer displayed in the following diagram of ℒ -spectra:

[figure: (M ▷_{ℒ} R) ∧_{ℒ} N ≅ M ∧_{ℒ} (R ◁_{ℒ} N); μ∧id; id ∧ν; M ∧_{ℒ} N; M ∧_{ℒ ,R} N,]

where μ and ν are the given actions of R on M and N; the canonical isomorphism of the terms on the left is implied by Lemma 1.7.

When R = S, M ▷_{ℒ} S ≅ M, S ◁_{ℒ} N ≅ N, and we are coequalizing the same isomorphism. Therefore our new M ∧_{ℒ ,S} N coincides with our old M ∧_{ℒ} N. We have used the notation ∧_{ℒ ,R} to emphasize the conceptual point that we are here generalizing ∧_{ℒ} rather than ∧_{S}.

Remark 2.2. We have given the definition in the form most convenient for proofs, because the displayed coequalizer is split. However it is equivalent to define M ∧_{ℒ ,R} N more intuitively as the coequalizer displayed in the diagram

[figure: M ∧_{ℒ} R ∧_{ℒ} N; μ∧id; id ∧ν; M ∧_{ℒ} N; M ∧_{ℒ ,R} N.]

We can define modified smash products ◁_{R}, ▷_{R}, and ⋆_{R} when one or both of the variables comes with a given map of R-modules η : R → M, copying Definitions 1.1 and 1.8 with S replaced by R.

Definition 2.3. Let M be a right R-module under R and let N be a left Rmodule. Define the mixed smash product M ◁_{R} N to be the pushout displayed in the following diagram of 𝕃-spectra:

[figure: R ∧_{ℒ ,R} N; λ; η∧id; M ∧_{ℒ ,R} N; N; M ◁_{R} N.]

Define ▷_{R} by symmetry. Observe that the displayed pushout is a diagram of R-modules when R is an E_{∞} ring spectrum.

Definition 2.4. Let M be a right and N a left R-module under R. Define the unital smash product M ⋆_{R} N to be the pushout displayed in the following diagram of 𝕃-spectra:

[figure: (M ∧_{ℒ ,R} R) ∪_{R∧_{ℒ ,R} R} (R ∧_{ℒ ,R} N); θ; λ; M ∪_{R} N; M ∧_{ℒ ,R} N; M ⋆_{R} N.]

Here, as in Definition 1.8, λ is induced by the unit maps λ of M and N and θ is induced by the structure maps η of M and N. Observe that the displayed pushout is a diagram of R-modules when R is an E_{∞} ring spectrum.

All of the results of the previous section apply verbatim in the context of Rmodules, except for those that depend on the isomorphism S ∧_{ℒ} S ≅ S.

Lemma 2.5. Let M be a right and N be a left R-module. Then

(M ∨ R) ◁_{R} N ≅ (M ∧_{ℒ ,R} N) ∨ N

and

(M ∨ R) ⋆_{R} (N ∨ R) ≅ (M ∧_{ℒ ,R} N) ∨ M ∨ N ∨ R.

If N is an R-module under R, then

(M ∨ R) ⋆_{R} N ≅ (M ▷_{R} N) ∨ N.

If M and N are R-modules under R, then the following diagram is a pushout, where the unit map λ : R ▷_{R} N ⟶ N is constructed by a comparison of pushout diagrams:

[figure: R ▷_{R} N; λ; η▷id; M ▷_{R} N; N; M ⋆_{R} N.]

Using this, we can deduce alternative expressions for these products in terms of coequalizer diagrams like that which defines ∧_{ℒ ,R} .

Lemma 2.6. For a right R-module M under R and a left R-module N, M ◁_{R} N can be identified with the coequalizer displayed in the diagram

[figure: (M ⋆_{ℒ} R) ◁_{ℒ} N ≅ M ◁_{ℒ} (R ◁_{ℒ} N); μ◁id; id◁ν; M ◁_{ℒ} N; M ◁_{R} N.]

For a right R-module M under R and a left R-module N under R, M ⋆_{R} N can be identified with the coequalizer displayed in the diagram

[figure: (M ⋆_{ℒ} R) ⋆_{ℒ} N ≅ M ⋆_{ℒ} (R ⋆_{ℒ} N); μ⋆id; id ⋆ν; M ⋆_{ℒ} N; M ⋆_{R} N.]

Proposition 2.7. (i) For any R-module N, the canonical map of R-modules

N ⟶ R ◁_{R} N

is an isomorphism. (ii) For any R-module M under R, the canonical map of R-modules

M ∧_{ℒ ,R} N ⟶ M ◁_{R} N

is a weak equivalence. (iii) For any R-module N under R, the canonical map of R-modules

λ : R ▷_{R} N ⟶ N

is a weak equivalence. (iv) For any R-modules M and N under R, the canonical map of R-modules

M ▷_{R} N ⟶ M ⋆_{R} N

is a weak equivalence.

We have various commutativity and associativity isomorphisms that involve several A_{∞} rings, as in III.3.4. Note that an (R, R′)-bimodule is the same thing as an (R ⋆_{ℒ} R′^{op})-module, and an (R ⋆_{ℒ} R′^{op})-module under (R ⋆_{ℒ} R′^{op}) is both a right R-module under R and a left R′-module under R′.

Proposition 2.8. Let M be an (R, R′)-bimodule, N be an (R′, R′′)-bimodule, and P be an (R′′, R′′′)-bimodule.

(i) If M is an R′-module under R′, then

M ◁_{R′} N ≅ N ▷_{R′}^{op} M

as R ⋆_{ℒ} R′′^{op}-modules and

M ◁_{R′} (N ∧_{R′′} P) ≅ (M ◁_{R′} N) ∧_{R′′} P

as (R, R′′′)-bimodules. (ii) If M is an R′-module under R′ and P is an R′′-module under R′′, then

M ◁_{R′} (N ▷_{R′′} P) ≅ (M ◁_{R′} N) ▷_{R′′} P

as (R, R′′′)-bimodules. (iii) If M is an R′-module under R′ and N is an (R′ ⋆_{ℒ} R′′^{op})-module under (R′ ⋆_{ℒ} R′′^{op}), then

(M ⋆_{R′} N) ◁_{R′′} P ≅ M ◁_{R′} (N ◁_{R′′} P)

as (R, R′′′)-bimodules. (iv) If M is an R′-module under R′, N is an (R′ ⋆_{ℒ} R′′^{op})-module under (R′ ⋆_{ℒ} R′′^{op}), and P is an R′′-module under R′′, then

(M ⋆_{R′} N) ⋆_{R′′} P ≅ M ⋆_{R′} (N ⋆_{R′′} P)

as (R, R′′′)-bimodules.

Proof. This is a formal exercise in the commutation of coequalizers with coequalizers, starting with the analogous isomorphisms for our various smash products over S. One writes down three-by-three diagrams of coequalizers and uses that the coequalizer of coequalizers is a coequalizer vertically and horizontally to deduce the stated isomorphisms. The top left-hand corner of the diagram needed for the isomorphism of (iii), for example, is

(((M ⋆_{ℒ} R′) ⋆_{ℒ} N) ⋆_{ℒ} R′′) ◁_{ℒ} P ≅ M ◁_{ℒ} (R′ ◁_{ℒ} (N ◁_{ℒ} (R′′ ◁_{ℒ} P))).

Similarly, all other results of III§§3-4 carry over directly to the present context, and we reach the following conclusion in the commutative case.

Theorem 2.9. The category of R-modules under R is symmetric monoidal under ⋆_{R}.

We can define R-algebras and commutative R-algebras to be monoids and commutative monoids in this symmetric monoidal category. Of course, if we apply the functor S ∧_{ℒ} (?) to such algebras we obtain weakly equivalent algebras over the S-algebra S ∧_{ℒ} R.

Again, we have a mixed function R-spectrum giving an adjunction like Proposition 1.18.

Definition 2.10. Let M be an R-module under R and let N be an R-spectrum. Define F_{R}^{▷}(M, N) to be the 𝕃-spectrum displayed in the following pullback diagram:

[figure: F_{R}^{▷}(M, N); F_{R}(M, N); η^{∗}; N; F_{R}(R, N);]

here the bottom arrow is adjoint to λτ : N ∧_{R} R ≅ R ∧_{R} N ⟶ N. If R is an E_{∞} ring spectrum, then F_{R}^{▷}(M, N) is an R-module.

Remark 2.11. Geometric realization behaves as expected. If L_{∗} and L′_{∗} are simplicial R-modules under R and K_{∗} is a simplicial R-module, then

|L_{∗}| ◁_{ℒ} |K_{∗}| ≅ |L_{∗} ◁_{ℒ} K_{∗}|

and

|L_{∗}| ⋆_{ℒ} |L′_{∗}| ≅ |L_{∗} ⋆_{ℒ} L′_{∗}|.

Similarly our work on enriched model categories carries over to the present framework. There is an analogue of VII.2.8 for the category of R-modules under R. Here we must enrich over the category of unbased spaces. Recall that a colimit is said to be connected if it is indexed on a diagram whose domain category is connected.

Proposition 2.12. The category of R-modules under R is topologically cocomplete and complete. The cotensors F (X_{+}, E) and all other indexed limits are created in 𝒮; ordinary connected colimits are also created in 𝒮. For an R-module N, the functor M ◁_{R} N on R-modules M under R preserves connected colimits. For an R-module N under R, the functor M ⋆_{R} N on R-modules M under R preserves connected colimits.

Proof. The colimit, colim_{relR} D of a diagram D of R-modules under R is computed from its colimit as a diagram of R-modules via the pushout

\xymatrix{colim \underline{R} \ar[d] \ar[r] & R \ar[d] \\ colim D \ar[r] & colim_{relR} D;}

the right vertical arrow gives the unit. Here \underline{R} is the constant diagram at R. The top horizontal arrow is an isomorphism if the domain category of D is connected, and the relative and ordinary colimits colimits then agree. Since this holds for coequalizers, VII.2.6 and 2.8 imply the first statement. The remaining statements follow since the products ◁ and ⋆ are defined in terms of (?) ∧_{ℒ ,R} N and pushouts, which preserve colimits of R-modules.

Theorem 2.13. For any E_{∞} ring spectrum R the categories of R-algebras and of commutative R-algebras are topologically cocomplete and complete. Their cotensors and all other indexed limits are created in 𝒮.

Similarly, all categories in sight admit model structures.

Theorem 2.14. The categories of modules over an A_{∞} ring spectrum R and of algebras and commutative algebras over an E_{∞} ring spectrum R are topological model categories. In all cases, the weak equivalences and q-fibrations are the maps which are weak equivalences or Serre fibrations of underlying spectra.

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