CLASSIFICATION OF STABLE MODEL CATEGORIES

PRELIMINARY VERSION

STEFAN SCHWEDE AND BROOKE SHIPLEY

Abstract: A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. We also show that rational stable model categories with a set of generators are equivalent to modules over a differential graded ‘ring with several objects.’

1. Introduction

The recent discovery of highly structured categories of spectra has opened the way for a new wholesale use of algebra in stable homotopy theory. In this paper we use this new algebra of spectra to characterize stable model categories, the settings for doing stable homotopy theory, as categories of highly structured modules. This characterization also leads to a Morita theory for equivalences between categories of highly structured modules.

The motivation and techniques for this paper come from two directions, namely stable homotopy theory and homological algebra. Specifically, stable homotopy theory studies the classical stable homotopy category which is the category of spectra up to homotopy. For our purposes though, the homotopy category is inadequate because too much information is lost, for example the homotopy type of mapping spaces. Instead, we study the model category of spectra which captures the whole stable homotopy theory. More generally we study stable model categories, those model categories which share the main formal property of spectra, namely that the suspension functor is invertible up to homotopy. We list examples of stable model categories in Section 2. Model categories are essential to the formulation of our results. We give specific references throughout; for general background see Quillen’s original article [Qui67], a modern introduction [DS95], or [Hov99a] for a more complete overview.

The algebraic part of the motivation arises as follows. A classical theorem, due to Gabriel [Gab62], characterizes categories of modules as the cocomplete abelian categories with a single small projective generator; the classical Morita theory for equivalences between module categories (see for example [AF92, §21, 22]) follows from this. Later Rickard [Ric89, Ric91] developed a Morita theory for derived categories based on the notion of a tilting complex. In this paper we carry this line of thought one step further. Spectra are the homotopy theoretical generalization of abelian groups and stable model categories are the homotopy theoretic analogue of abelian categories (or rather their categories of chain complexes). Our generalization of Gabriel’s theorem develops a Morita theory for stable model categories. Here the appropriate notion of a model category equivalence is a Quillen equivalence since these equivalences preserve the homotopy theory, not just the homotopy category, see 2.5.

Date: January 24, 2000; 1991 AMS Math. Subj. Class.: 55U35, 55P42.

Research supported by a BASF-Forschungsstipendium der Studienstiftung des deutschen Volkes.

Research partially supported by an NSF Postdoctoral Fellowship.

We have organized our results into four groups, which are not at all independent of each other:

(A) Characterization of module categories. The model category of modules over a ring spectrum has a single compact generator, namely the free module of rank one. But module categories are actually characterized by this property. To every object in a stable model category we associate an endomorphism ring spectrum, see Definition 3.7.5. We show that if there is a single compact generator, then the given stable model category has the same homotopy theory as the modules over the endomorphism ring spectrum of the generator (Theorem 3.1.1). More generally, stable model categories with a set of compact generators are characterized as modules over a ‘ring spectrum with many objects,’ or certain spectrum valued diagram categories (Theorem 3.3.3). See Appendix A.1 for a definition of a ‘ring with many objects’ and its modules. Examples of these characterizations are given in 3.2 and 3.4.

(B) Morita theory for ring spectra. In the classical algebraic context Morita theory describes equivalences between module categories in terms of bimodules, see e.g. [AF92, Thm. 22.1]. In Theorem 4.1.2 we present an analogous result which explains how a chain of Quillen equivalences between module categories over ring spectra can be replaced by a single Quillen equivalence given by smashing with a pair of bimodules.

(C) Generalized tilting theory. In [Ric89, Ric91], Rickard answered the question of when two rings are derived equivalent, i.e., when various derived module categories are equivalent as triangulated categories. Basically, a derived equivalence exists if and only if a so-called tilting complex exists. ¿From our point of view, a tilting complex is a particular compact generator for the derived category of a ring. In Theorem 5.1.1 we obtain a generalized tilting theorem which characterizes stable model categories which are Quillen (or derived) equivalent to the derived category of a ring.

(D) Rational stable model categories. The generalized tilting theorem 5.1.1 shows that under certain conditions a stable model category is algebraic, i.e., Quillen equivalent to the model category of modules over a differential graded ‘ring with many objects.’ We show in Theorem 6.1.3 that the same conclusion always holds in rational stable model categories, i.e., whenever all of the abelian homomorphism groups in the homotopy category are uniquely divisible.

In order to carry out our program it is essential to have available a highly structured model for the category of spectra which admits a symmetric monoidal and homotopically well behaved smash product before passing to the homotopy category. The first examples of such categories were the S-modules of [EKMM] and the symmetric spectra of Jeff Smith [HSS98]; by now several more such categories have been constructed [Lyd98, MMSS]. We work with symmetric spectra because we can replace stable model categories by Quillen equivalent ones which are enriched over symmetric spectra (Section 3.6). Also, symmetric spectra are reasonably easy to define and understand and several other model categories in the literature are already enriched over symmetric spectra. The full strength of our viewpoint comes from combining enriched (over symmetric spectra) category theory with the language of closed model categories. We develop some enriched model category theory in Appendix A.1.

We want to point out the conceptual similarities between the present paper and the work of Keller [Kel94]. Keller uses differential graded categories to give an elegant reformulation (and generalization) of Rickard’s results on derived equivalences for rings. Our approach is similar to Keller’s, but where he considers categories whose hom-objects are chain complexes of abelian groups, our categories have hom-objects which are spectra. Keller does not use the language of model categories, but the ‘P-resolutions’ of [Kel94, 3.1] are basically cofibrant replacements.

Notation and conventions: We use the symbol S_{∗} to denote the category of pointed simplicial sets, and we use Sp^{Σ} for the category of symmetric spectra [HSS98]. The letters C and D usually denote model categories, most of the time assumed to be simplicial and stable. For cofibrant or fibrant approximations of objects in a model category we use superscripts (−)^{c} and (−)^{f}. For an object X in a pointed simplicial model category we use the notation ΣX and ΩX for the simplicial suspension and loop functors (i.e., the pointed tensor and cotensor of an object X with the pointed simplicial circle S^{1} = Δ[1]/∂Δ[1]); one should keep in mind that these objects may have the ‘wrong’ homotopy type if X is not cofibrant or fibrant respectively. Our notation for various kinds of morphism objects is as follows: the set of morphisms in a category C is denoted ‘hom_{C}’; the simplicial set of morphisms in a simplicial category is denoted ‘map’; we use ‘Hom’ for the symmetric function spectrum in a spectral model category (Definition 3.5.1); square brackets ‘[X, Y]^{Ho(C)}’ denote the abelian group of morphisms in the homotopy category of a stable model category C; and for objects X and Y in any triangulated category T we use the notation ‘[X, Y]_{∗}^{T} ’ to denote the graded abelian group of morphisms, i.e., [X, Y]_{n}^{T} = [X[n], Y]^{T} for n ∈ ℤ and where X[n] is the n-fold shift of X.

We want to write the evaluation of a morphism f on an element x as f(x). This determines the following conventions about actions of rings and ring spectra: the endomorphism monoid, ring or ring spectrum End(X) acts on the object X from the left, and it acts on the set (group, spectrum) Hom(X, Y) from the right. A module will always be a right module; this way the left multiplication map establishes an isomorphism between a ring and the endomorphism ring of the free module of rank one. A T -R-bimodule is the same as a (T^{op} ⊗ R)-module (or a (T^{op} ∧ R)-module in the context of ring spectra).

Organization: In Section 2 we recall stable model categories and some of their properties, as well as the notions of compactness and generators, and we give an extensive list of examples. In Section 3 we prove the classification theorems (Theorems 3.1.1 and 3.3.3). In Section 3.6 we introduce the category Sp(C) of symmetric spectra over a simplicial model category C. Under certain technical assumptions we show in Theorem 3.9.2 that it is a stable model category with composable and homotopically well-behaved function symmetric spectra which is Quillen equivalent to the original stable model category C. In Theorem 3.10.3 we then prove Theorem 3.3.3 for spectral model categories (Definition 3.5.1), such as for example Sp(C). This will complete the classification results. In Definition 3.7.5 we associate to an object P of a simplicial stable model category a symmetric endomorphism ring spectrum End(P). In Section 4 we prove the Morita context (Theorem 4.1.2), in Section 5 we prove the tilting theorem (Theorem 5.1.1) and in Section 6 we prove the algebraization of rational stable model categories (Theorem 6.1.3). In Appendix A we give a summary of some enriched model category theory.

Acknowledgments: We would like to thank Greg Arone, Dan Dugger, Bill Dwyer, Mark Hovey, Jeff Smith and Charles Rezk for inspiration and for many helpful discussions on the subjects of this paper.

2. Stable model categories

In this section we recall stable model categories and some of their properties, as well as the notions of compactness and generators, and we give a list of examples.

2.1. Structure on the homotopy category. Recall from [Qui67, I.2] that the homotopy category of a pointed model category supports a suspension functor Σ with a right adjoint loop functor Ω.

Definition 2.1.1. A stable model category is a pointed closed model category for which the functors Ω and Σ on the homotopy category are inverse equivalences.

The homotopy category of a stable model category has a large amount of extra structure, some of which plays a role in this paper. First of all, it is naturally a triangulated category (cf. [Ver96] or [HPS97, A.1]). A complete reference for this fact can be found in [Hov99a, 7.1.6]; we sketch the constructions: by definition of ‘stable’ the suspension functor is a selfequivalence of the homotopy category and it defines the shift functor. Since every object is a two-fold suspension, hence an abelian co-group object, the homotopy category of a stable model category is additive. Furthermore, by [Hov99a, 7.1.11] the cofiber sequences and fiber sequences of [Qui67, 1.3] coincide up to sign in the stable case, and they define the distinguished triangles. Since we required a stable model category to have all limits and colimits, its homotopy category has infinite sums and products. So such a homotopy category behaves like the unbounded derived category of an abelian category. This motivates thinking of a stable model category as a homotopy theoretic analog of an abelian category.

We recall the notions of compactness and generators in the context of triangulated categories:

Definition 2.1.2. Let T be a triangulated category with infinite coproducts. A full triangulated subcategory of T (with shift and triangles induced from T ) is called localizing if it is closed under coproducts in T . A set P of objects of T is called a set of generators if the only localizing subcategory which contains the objects of P is T itself. An object X of T is compact (also called small of finite) if for any family of objects {A_{i}}_{i∈I} the canonical map

⨁_{i∈I} [X, A_{i}]^{T} ⟶ [X, ∐_{i∈I} A_{i}]^{T}

is an isomorphism. Objects of a stable model category are called ‘generators’ or ‘compact’ if they are so when considered as objects of the triangulated homotopy category.

A triangulated category with infinite coproducts and a set of compact generators is often called compactly generated. We avoid this terminology because of the danger of confusing it with the terms ‘cofibrantly generated’ and ‘compactly generated’ in the context of model categories.

2.2. Remarks.

(i) There is a convenient criterion for when a set of compact objects generates a triangulated category. This characterization is well known, but we have been unable to find a reference which proves it in the form we need.

Lemma 2.2.1. Let T be a triangulated category with infinite coproducts and let P be a set of compact objects. Then the following are equivalent:

(i) The set P generates T in the sense of Definition 2.1.2.

(ii) An object X of T is trivial if and only if there are no graded maps from objects of P to X, i.e. [P, X]_{∗} = 0 for all P ∈ P.

Proof. Suppose the set P generates T and let X be an object with the property that [P, X]_{∗} = 0 for all P ∈ P. The full subcategory of T of objects Y satisfying [Y, X]_{∗} = 0 is localizing. Since it contains the set P, it contains all of T . Taking Y = X we see that the identity map of X is trivial, so X is trivial.

The other implication uses the existence of Bousfield localization functors, which in this case is a finite localization first considered by Miller in the context of the stable homotopy category [Mil92]. For every set P of compact objects in a triangulated category with infinite coproducts there exist functors L_{P} (localization) and C_{P} (colocalization) and a natural distinguished triangle

C_{P}X ⟶ X ⟶ L_{P}X ⟶ C_{P}X[1]

such that C_{P}X lies in the localizing subcategory generated by P, and such that [P, L_{P}X]_{∗} = 0 for all P ∈ P and X ∈ T ; one reference for this construction is in the proof of [HPS97, Prop. 2.3.17] see also 3.10.4. So if we assume condition (ii) then for all X the localization L_{P}X is trivial, hence X is isomorphic to the colocalization C_{P}X and thus contained in the localizing subcategory generated by P.

(ii) Our terminology for ‘generators’ is different from the use of the term in category theory; generators in our sense are sometimes called weak generators elsewhere. By Lemma 2.2.1, a set of generators detects if objects are trivial (or equivalently if maps in T are isomorphisms). This notion has to be distinguished from that of a categorical generator which detects if maps are trivial. For example, the sphere spectra are a set of generators (in the sense of Definition 2.1.2) for the stable homotopy category of spectra. Freyd’s generating hypothesis conjectures that the spheres are a set of categorical generators for the stable homotopy category of finite spectra. It is unknown to this day whether the generating hypothesis is true or false.

(iii) An object of a triangulated category is compact if and only if its shifts (suspension and loop objects) are. Any finite coproduct or direct summand of compact objects is again compact. Compact objects are closed under extensions: if two objects in a distinguished triangle are compact, then so is the third one. In other words, the full subcategory of compact objects in a triangulated category is thick. There are non-trivial triangulated categories in which only the zero object is compact. Examples with underlying stable model categories arise for example as suitable Bousfield localizations of the category of spectra, see [HS99, Cor. B.13].

(iv) If a triangulated category has a set of generators, then the coproduct of all of them is a single generator. However, infinite coproducts of compact objects are in general not compact. So the property of having a single compact generator is something special. In fact we see in Theorem 3.1.1 below that this condition characterizes the module categories over ring spectra among the stable model categories. If generators exist, they are far from being unique.

(v) In the following we often consider stable model categories which are cofibrantly generated. Hovey has shown [Hov99a, Thm. 7.3.1] that a cofibrantly generated model category always has a set of generators in the sense of Definition 2.1.2 (the cofibers of any set of generating cofibrations will do). So having generators is not an extra condition in the situation we consider, although these generators may not be compact. See [Hov99a, Cor. 7.4.4] for conditions that guarantee a set of compact generators.

2.3. Examples.

(i) Spectra. As we mentioned in the introduction, one of our main motivating examples is the category of spectra in the sense of stable homotopy theory. The sphere spectrum is a compact generator. Many model categories of spectra have been constructed, for example by Bousfield and Friedlander [BF78]; Robinson [Rob87a, ‘spectral sheaves’]; Jardine [Jar97, ‘n-fold spectra’]; Elmendorf, Kriz, Mandell and May [EKMM, ‘coordinate free spectra’, ‘𝕃spectra’, ‘S-modules’]; Hovey, Shipley and Smith [HSS98, ‘symmetric spectra’]; Lydakis [Lyd98, ‘simplicial functors’]; Mandell, May, Schwede and Shipley [MMSS, ‘orthogonal spectra’, ‘W-spaces’].

(ii) Modules over ring spectra. Modules over an S-algebra [EKMM, VII.1] or modules over a symmetric ring spectrum [HSS98, 5.4.2] form proper, cofibrantly generated, simplicial, stable model categories, see also [MMSS, Sec. 11]. In each case a module is compact if and only if it is weakly equivalent to a retract of a finite cell module. The free module of rank one is a compact generator. More generally there are stable model categories of modules over ‘symmetric ring spectra with several objects’, or spectral categories, see Definition 3.3.1 and Theorem A.1.2.

(iii) Equivariant stable homotopy theory. If G is a compact Lie group, there is a category of G-equivariant coordinate free spectra [LMS86] which is a stable model category. In this case the equivariant suspension spectra of the coset spaces G/H_{+} for all closed subgroups H ⊆ G form a set of compact generators. This equivariant model category is taken up again in Examples 3.4 (i), 5.1.2, and in Remark 6.1.5.

(iv) Presheaves of spectra. For every Grothendieck site Jardine [Jar87] constructs a proper simplicial stable model category of presheaves of Bousfield-Friedlander type spectra; the weak equivalences are the maps which induce isomorphisms of the associated sheaves of stable homotopy groups. For a general site these stable model categories do not seem to have a set of compact generators.

(v) The stabilization of a model category. In principle every model category should give rise to an associated stable model category by ‘inverting’ the suspension functor, i.e., by passage to internal spectra. This has been carried out for certain simplicial model categories in [Sch97] and [Hov98]. The construction of symmetric spectra over a model category (see Section 3.6) is another approach to stabilization.

(vi) Bousfield localization. Following Bousfield [Bou75], localized model structures for modules over an S-algebra are constructed in [EKMM, VIII 1.1]. Hirschhorn [Hir99] shows that under quite general hypotheses the localization of a model category is again a model category. The localization of a stable model category is stable and localization preserves generators. Compactness need not be preserved, see Example 3.2 (iii).

(vii) 𝔸^{1}-local stable homotopy. In [MV98, Voe98] Morel and Voevodsky introduced the 𝔸^{1}- local model category structure for schemes over a base. An associated stable homotopy category of 𝔸^{1}-local T -spectra (where T = 𝔸^{1}/(𝔸^{1}−0) is the ‘Tate-sphere’) is an important tool in Voevodsky’s proof of the Milnor conjecture [Voe96]. This stable homotopy category arises from a stable model category with a set of compact generators, see Example 3.4 (ii) for more details.

2.4. Examples: abelian stable model categories. Some examples of stable model categories are ‘algebraic’, i.e., the model category is also an abelian category. Most of the time the objects consist of chain complexes in some abelian category and depending on the choice of weak equivalences one gets a kind of derived category as the homotopy category. A different kind of example is formed by the stable module categories of Frobenius rings.

For algebraic examples as the ones below, our results are essentially covered by Keller’s paper [Kel94], although Keller does not use the language of model categories. Also there is no need to consider spectra when dealing with abelian model categories: the second author shows [Shi2] that every cofibrantly generated, proper, abelian stable model category is Quillen equivalent to a DG-model category, i.e., a model category enriched over chain complexes of abelian groups.

(i) Complexes of modules. The category of unbounded chain complexes of left modules over a ring supports a model category structure with weak equivalence the quasi-isomorphisms and with fibrations the epimorphisms [Hov99a, Thm. 2.3.11] (this is called the projective model structure). Hence the associated homotopy category is the unbounded derived category of the ring. A chain complex of modules over a ring is compact if and only if it is quasi-isomorphic to a bounded complex of finitely generated projective modules [BN93, Prop. 6.4]. We show in Theorem 6.2.3 that the model category of unbounded chain complexes of A-modules is Quillen equivalent to the category of modules over the symmetric Eilenberg-Mac Lane ring spectrum for A. This example can be generalized in at least two directions: one can consider model categories of chain complexes in an abelian category with enough projectives (see e.g. [Chr99, Thm. 3.4] for a very general construction under mild smallness assumptions). On the other hand one can consider model categories of differential graded modules over a differential graded algebra, or even a ‘DGA with many objects’, see Theorem A.1.2 (i).

(ii) Relative homological algebra. In [Chr99], Christensen introduces model category structures for chain complexes over an abelian category based on a projective class. In the special case where the abelian category is modules over some ring and the projective class consists of all summands of free modules this recovers the (projective) model category structure of the previous example. Another special case of interest is the pure derived category of a ring. Here the projective class consists of all summands of (possibly infinite) sums of finitely generated modules, see also Example 5.1.3.

(iii) Quasi-coherent sheaves. For a nice enough scheme X the derived category of quasicoherent sheaves D(qc/X) arises from a stable model category and has a set of compact generators. More precisely, if X is quasi-compact and quasi-separated, then the so-called injective model structure exists. The objects of the model category are unbounded complexes of quasi-coherent sheaves of O_{X}-modules, the weak equivalences are the quasi-isomorphisms and the cofibrations are the injections [Hov99b, Cor. 2.3 (b)]. If X is separated, then the compact objects of the derived category are precisely the perfect complexes, i.e., the complexes which are locally quasi-isomorphic to a bounded complex of vector bundles [Nee96, 2.3, 2.5]. If X also admits an ample family of line bundles {L_{α}}_{α∈A}, then the set of line bundles {L_{α}^{⊗m} | α ∈ A, m ∈ ℤ}, considered as complexes concentrated in dimension zero, generates the derived category D(qc/X), see [Nee96, 1.11].

This class of examples contains the derived category of a ring as a special case, but it yields a different model structure from the one mentioned in (i). Hovey [Hov99b, Thm. 2.2] generalizes the injective model structure to abelian Grothendieck categories.

(iv) The stable module category of a Frobenius ring. A Frobenius ring is defined by the property that the classes of projective and injective modules coincide. Important examples are finite dimensional Hopf-algebras over a field and in particular group algebras of finite groups. The stable module category is obtained by identifying two module homomorphisms if their difference factors over a projective module. Fortunately the two different meanings of ‘stable’ fit together nicely; the stable module category is the homotopy category associated to an underlying stable model category structure [Hov99a, Sec. 2]. Every finitely generated module is compact when considered as an object of the stable module category. Compare also Example 3.2 (v).

(v) Comodules over a Hopf-algebra. Suppose B is a commutative Hopf-algebra over a field. Hovey, Palmieri and Strickland introduce the category C(B) of chain complexes of injective B-comodules, with morphisms the chain homotopy classes of maps [HPS97, Sec. 9.5]. Compact generators are given by injective resolutions of simple comodules (whose isomorphism classes form a set). In [Hov99a, Thm. 2.5.17], Hovey shows that there is a cofibrantly generated model category structure on the category of all chain complexes of B-comodules whose homotopy category is the category C(B).

2.5. Quillen equivalences. The most highly structured notion to express that two model categories describe the same homotopy theory is that of a Quillen equivalence. An adjoint functor pair between model categories is a Quillen pair if the left adjoint L preserves cofibrations and trivial cofibrations. An equivalent condition is to demand that the right adjoint R should preserve fibrations and trivial fibrations. Under these conditions, the functors pass to an adjoint functor pair on homotopy categories, see [Qui67, I.4 Thm. 3], [DS95, Thm. 9.7 (i)] or [Hov99a, 1.3.10]. A Quillen functor pair is a Quillen equivalence if it induces an equivalence on the homotopy categories. A Quillen pair is a Quillen equivalence if and only if the following criterion holds [Hov99a, 1.3.13]: for every cofibrant object A of the source category of L and for every fibrant object X of the source category of R, a map L(A) ⟶ X is a weak equivalence if and only if its adjoint A ⟶ R(X) is a weak equivalence.

A Quillen equivalence induces an equivalence of homotopy categories, see [Qui67, I.4 Thm. 3], [DS95, Thm. 9.7 (ii)] or [Hov99a, 1.3.13]. As pointed out in [DS95, 9.7 (ii)] and [Qui67, I.4, Thm. 3], Quillen equivalences also preserve the homotopy theory associated to a model category, that is, the higher order structure such as mapping spaces, suspension and loop functors, and cofiber and fiber sequences. Note that the notions of a stable model category, compactness and generators are invariant under a Quillen equivalence of model categories.

For convenience we restrict our attention to simplicial model categories (see [Qui67, II.2]). This is not a big loss of generality; it is shown in [RSS] that every cofibrantly generated, proper, stable model category is in fact Quillen equivalent to a simplicial model category. In [Dug98], Dugger obtains the same conclusion under somewhat different hypotheses. In both cases the candidate is the category of simplicial objects over the given model category endowed with a suitable localization of the Reedy model structure.

3. Classification theorems

3.1. Monogenic stable model categories. Several of the examples of stable model categories mentioned in 2.3 already come as categories of modules over suitable rings or ring spectra. This is no coincidence. In fact, our next classification theorem says that every stable model category with a single compact generator has the same homotopy theory as the modules over a symmetric ring spectrum (see [HSS98, 5.4] for background on symmetric ring spectra). This is analogous to the classical fact [Gab62, V1, p. 405] that module categories are characterized as those cocomplete abelian categories which posses a single small projective generator; the classifying ring is obtained as the endomorphism ring of the generator.

In Definition 3.7.5 we associate to every object P of a simplicial, cofibrantly generated stable model category C a symmetric endomorphism ring spectrum End(P). The ring of homotopy groups π_{∗} End(P) is isomorphic to the ring of graded self maps of P in the homotopy category of C, [P, P]_{∗}^{Ho(C)}.

For the following theorem we have to make two technical assumptions. We need the notion of cofibrantly generated model categories from [DHK] which is reviewed in some detail in [SS00, Sec. 2] and [Hov99a, Sec. 2.1]. We also need properness (see [BF78, Def. 1.2] or [HSS98, Def. 5.5.2]). A model category is left proper if pushouts across cofibrations preserve weak equivalences. A model category is right proper if pullbacks over fibrations preserve weak equivalences. A proper model category is one which is both left and right proper.

Theorem 3.1.1. (Classification of monogenic stable model categories) Let C be a simplicial, cofibrantly generated, proper, stable model category with a compact generator P. Then there exists a chain of simplicial Quillen equivalences between C and the model category of End(P)-modules.

This theorem is a special case of the more general classification result Theorem 3.3.3, which applies to stable model categories with a set of compact generators and which we prove in Section 3.6. Furthermore if in the situation of Theorem 3.1.1, P is a compact object but not necessarily a generator of C, then C still ‘contains’ the homotopy theory of End(P)-modules, see Theorem 3.10.3 (ii) for the precise statement. In the Morita context (Theorem 4.1.2) we also prove a partial converse to Theorem 3.1.1. The properness assumption only enters when we establish the stable model structure for symmetric spectra over C (Theorem 3.9.2). If one is only interested in equivalences of triangulated homotopy categories, as opposed to Quillen equivalences of model categories, then the properness assumption can be dropped, see Section 3.8.

3.2. Examples: stable model categories with a compact generator.

(i) Uniqueness results for stable homotopy theory. The classification theorem above yields a characterization of the model category of spectra: a simplicial, cofibrantly generated, proper, stable model category is simplicially Quillen equivalent to the category of symmetric spectra if and only if it has a compact generator P for which the unit map of ring spectra S ⟶ End(P) is a stable equivalence. The paper [SS99] is devoted to other necessary and sufficient conditions for when a stable model category is Quillen equivalent to spectra – some of them in terms of the homotopy category of C and the natural action of the stable homotopy groups of spheres. By working with spectra in the sense of Bousfield and Friedlander [BF78] (as opposed to the Quillen equivalent symmetric spectra), we can also eliminate the technical conditions ‘cofibrantly generated’ and ‘proper’.

(ii) Chain complexes and Eilenberg-Mac Lane spectra. Let A be a ring. A special case of Theorem 6.2.3 shows that the model category of chain complexes of A-modules is Quillen equivalent to the model category of modules over the symmetric Eilenberg-Mac Lane ring spectrum HA. This can be viewed as an instance of Theorem 3.1.1: the free A-module of rank 1, considered as a complex concentrated in dimension 0, is a compact generator for the unbounded derived category of A. Since the homotopy groups of its endomorphism ring spectrum (as an object of the model category of chain complexes) are concentrated in dimension 0, the endomorphism ring spectrum is stably equivalent to the Eilenberg-Mac Lane ring spectrum for A (see Proposition A.3.1). This also shows that although the model category of chain complexes of A-modules is not simplicial it is Quillen equivalent to a simplicial model category. So although our classification theorems do not apply directly, they do apply indirectly.

(iii) Smashing Bousfield localizations. Let E be a spectrum and consider the E-local model category structure on some model category of spectra (see e.g. [EKMM, VIII 1.1]). This is another stable model category in which the localization of the sphere spectrum L_{E}S^{0} is a generator. This localized sphere is compact if the localization is smashing, i.e., if a certain natural map X ∧ L_{E}S^{0} ⟶ L_{E}X is a stable equivalence for all X. So for a smashing localization the E-local model category of spectra is Quillen equivalent to modules over the ring spectrum L_{E}S^{0} (which is the endomorphism ring spectrum of the localized sphere in the localized model structure).

(iv) K(n)-local spectra. Even if a Bousfield localization is not smashing, Theorem 3.1.1 might be applicable. As an example we consider Bousfield localization with respect to the n-th Morava K-theory K(n) at a fixed prime. The localization of the sphere is still a generator, but for n > 0 it is not compact in the local category, see [HPS97, 3.5.2]. However the localization of any finite type n spectrum F is a compact generator for the K(n)-local category [HS99, 7.3]. Hence the K(n)-local model category is Quillen equivalent to modules over the endomorphism ring End(L_{K(n)}F).

(v) Frobenius rings. As in Example 2.3 (iv) we consider a Frobenius ring and assume that the stable module category has a compact generator. Then we are in the situation of Theorem 3.1.1; however this example is completely algebraic, and there is no need to consider ring spectra to identify the stable module category as the derived category of a suitable ‘ring’. In fact Keller shows [Kel94, 4.3] that in such a situation there exists a differential graded algebra (DGA) and an equivalence between the stable module category and the unbounded derived category of the DGA.

A concrete example of this situation arises for group algebras of p-groups over a field k of characteristic p. In this case the trivial module is the only simple module, and it is a compact generator of the stable module category. More generally a result of Benson [Ben94, Thm. 1.1] says that the trivial module generates the stable module category of the principal block of a group algebra kG if and only if the centralizer of every element of order p is p-nilpotent. So in this situation Keller’s theorem applies and identifies the stable module category as the unbounded derived category of a certain DGA. The homology groups of this DGA are isomorphic (by construction) to the ring of graded self maps of the trivial module in the stable module category, which is just the Tate-cohomology ring Ĥ^{∗}(G; k).

(vi) Stable homotopy of algebraic theories. Another motivation for this paper and an early instance of Theorem 3.1.1 came from the stabilization of the model category of algebras over an algebraic theory [Sch]. For every pointed algebraic theory T, the category of simplicial T -algebras is a simplicial model category so that one has a category Sp(T) of (Bousfield-Friedlander type) spectra of T -algebras, a cofibrantly generated, simplicial stable model category [Sch, 4.3]. The free T -algebra on one generator has an endomorphism ring spectrum which is constructed as a Gamma-ring in [Sch, 4.5] and denoted T^{s}. Then [Sch, Thm. 4.4] provides a Quillen equivalence between the categories of connective spectra of T -algebras and the category of T^{s}-modules (the connectivity condition could be removed by working with symmetric spectra instead of Γ-spaces). This fits with Theorem 3.1.1 because the suspension spectrum of the free T -algebra on one generator is a compact generator for the category Sp(T). See [Sch, Sec. 7] for a list of ring spectra that arise from algebraic theories in this fashion.

Remark 3.2.1. The notion of a compact generator and the homotopy groups of the endomorphism ring spectrum only depend on the homotopy category, and so they are invariant under equivalences of triangulated categories. However, the homotopy type of the endomorphism ring spectrum depends on the model category structure. The following example illustrates this point. Consider the n-th Morava K-theory spectrum K(n) for a fixed prime and some number n > 0. This spectrum admits the structure of an A_{∞}-ring spectrum [Rob89]. Hence it also has a model as an S-algebra or a symmetric ring spectrum and the category of its module spectra is a stable model category. The ring of homotopy groups of K(n) is the graded field 𝔽_{p}[v_{n}, v_{n}^{−1}] with v_{n} of dimension 2p^{n} − 2. Hence the homotopy group functor establishes an equivalence between the homotopy category of K(n)-module spectra and the category of graded 𝔽_{p}[v_{n}, v_{n}^{−1}]-modules.

Similarly the homology functor establishes an equivalence between the derived category of differential graded modules over the graded field 𝔽_{p}[v_{n}, v_{n}^{−1}] and the category of graded 𝔽_{p}[v_{n}, v_{n}^{−1}]- modules. So the two stable model categories of K(n)-module spectra and DG-modules over 𝔽_{p}[v_{n}, v_{n}^{−1}] have equivalent triangulated homotopy categories (including the action of the stable homotopy groups of spheres — all elements in positive dimension act trivially in both cases). But the endomorphism ring spectra of the respective free rank one modules are the Morava Ktheory ring spectrum on the one side and the Eilenberg-Mac Lane ring spectrum for 𝔽_{p}[v_{n}, v_{n}^{−1}] on the other side, which are not stably equivalent. Similarly the two model categories are not Quillen equivalent since for DG-modules all function spaces are products of Eilenberg-Mac Lane spaces, but for K(n)-modules they are not.

3.3. Multiple generators. There is a generalization of Theorem 3.1.1 to the case of a stable model category with a set of compact generators (as opposed to a single compact generator).

Let us recall the algebraic precursors of this result: a ringoid is a category whose hom-sets are abelian groups with bilinear composition. Ringoids are sometimes called pre-additive categories or rings with several objects. Indeed a ring in the traditional sense is the same as a ringoid with one object. A module over a ringoid is defined to be a contravariant additive functor to the category of abelian groups. These more general module categories have been identified as the cocomplete abelian categories which have a set of small projective generators [Fre64, 5.3H, p. 120]. An analogous theory for derived categories of DG categories has been developed by Keller [Kel94].

Our result is very much in the spirit of Freyd’s or Keller’s, with spectra playing the role that is played by abelian groups or chain complexes. A symmetric ring spectrum can be viewed as a category with one object which is enriched over symmetric spectra; the module category then becomes the category of enriched functors to symmetric spectra. So we now look at ‘ring spectra with several objects’ which we call spectral categories. This is analogous to pre-additive, differential graded or simplicial categories which are enriched over abelian groups, chain complexes or simplicial sets respectively.

Definition 3.3.1. A spectral category is a category O which is enriched over the category Sp^{Σ} of symmetric spectra (with respect to smash product, i.e., the monoidal closed structure of [HSS98, 2.2.10]). In other words, for every pair of objects o, o′ in O there is a morphism symmetric spectrum O(o, o′), for every object o of O there is a map from the sphere spectrum S to O(o, o) (the ‘identity element’ of o), and for each triple of objects there is an associative and unital composition map of symmetric spectra

O(o′, o′′) ∧ O(o, o′) ⟶ O(o, o′′) .

An O-module M is a contravariant spectral functor to the category Sp^{Σ} of symmetric spectra, i.e., a symmetric spectrum M(o) for each object of O together with coherently associative and unital maps of symmetric spectra

M(o) ∧ O(o′, o) ⟶ M(o′)

for pairs of objects o, o′ in O. A morphism of O-modules M ⟶ N consists of maps of symmetric spectra M(o) ⟶ N(o) strictly compatible with the action of O. We denote the category of O-modules by mod-O.

Remark 3.3.2. In Definition 3.3.1 we are simply spelling out what it means to do enriched category theory over the symmetric monoidal closed category Sp^{Σ} of symmetric spectra with respect to the smash product and the internal homomorphism spectra. Kelly’s book [Kly82] is an introduction to enriched category theory in general; the spectral categories, modules over these and morphisms of modules as defined above are the Sp^{Σ}-categories, Sp^{Σ}-functors and Sp^{Σ}- natural transformations in the sense of [Kly82, 1.2]. So the precise meaning of the coherence and compatibility conditions in Definition 3.3.1 can be found in [Kly82, 1.2].

We show in Theorem A.1.2 that for any spectral category O the category of O-modules is a model category with objectwise stable equivalences as the weak equivalences. If O has a single object o, then the O-modules are precisely the modules over the symmetric ring spectrum O(o, o), and the model category structure is the one defined in [HSS98, 5.4.2].

In Definition 3.7.5 we associate to every set P of objects of a simplicial, cofibrantly generated stable model category C a spectral endomorphism category E(P) whose objects are the members of the set P and such that there is a natural, associative and unital isomorphism

π_{∗} E(P)(P, P′) ≅ [P, P′]_{∗}^{Ho(C)} .

For a set with a single element this reduces to the notion of the endomorphism ring spectrum.

Theorem 3.3.3. (Classification of stable model categories) Let C be a simplicial, cofibrantly generated, proper, stable model category with a set P of compact generators. Then there exists a chain of simplicial Quillen equivalences between C and the model category of E(P)- modules.

There is an even more general version of Theorem 3.3.3 which also provides information if the set P does not generate the whole homotopy category, see Theorem 3.10.3 (ii). This variant implies that for any set P of compact objects in a proper, cofibrantly generated, simplicial stable model category the homotopy category of E(P)-modules is triangulated equivalent to the localizing subcategory of Ho(C) generated by the set P.

The proof of Theorem 3.3.3 breaks up into two parts. In order to mimic the classical proof for abelian categories we must consider a situation where the hom functor Hom_{C}(P, −) takes values in the category of modules over a suitable endomorphism ring spectrum of P. In Section 3.6 we show how this can be arranged, given the technical conditions that C is cofibrantly generated, proper and simplicial. We introduce the category Sp(C) of symmetric spectra over C and show in Theorem 3.9.2 that it is a stable model category with composable and homotopically wellbehaved function symmetric spectra which is Quillen equivalent to the original stable model category C. On the way we associate to every object P of such a stable model category a symmetric endomorphism ring spectrum End(P), see Definition 3.7.5.

In Theorem 3.10.3 we prove Theorem 3.3.3 under the assumption that C is a spectral model category (Definition 3.5.1), i.e., a model category with composable and homotopically well-behaved function symmetric spectra. Since the model category Sp(C) is spectral and Quillen equivalent to C (given the technical assumptions of Theorem 3.3.3), this will complete the classification results.

In Theorem 3.8.3 we also provide a variant of the classification theorem which only provides an equivalence of homotopy categories, but not a Quillen equivalence of model categories. This does not use the stable model structure on spectra over C, and hence the properness assumption is not needed and the somewhat involved proof for this stable model structure can be avoided. We hope that this more limited, but more easily obtained result may still be of use even though the ultimate goal is to have a Quillen equivalence of model categories.

3.4. Examples: stable model categories with a set of generators.

(i) Equivariant stable homotopy. Let G be a compact Lie group. In [LMS86], the category S_{U}^{G} of G-equivariant spectra based on a universe U is developed. In [GH97], Goerss and Hopkins show that this category is a cofibrantly generated, topological (hence simplicial), proper model category. A closely related category of G-equivariant spectra is developed in [EKMM] (but the monoidal structure plays no role for our present considerations). For every closed subgroup H of G the equivariant suspension spectrum of the homogeneous space G/H_{+} is compact and the set G of these spectra for all closed subgroups H generates the G-equivariant stable homotopy category, see [HPS97, 9.4].

Recall from [LMS86, V §9] that a Mackey functor is a module over the stable homotopy orbit category, i.e., an additive functor from the homotopy orbit category π_{0} E(G) to the category of abelian groups; by [LMS86, V Prop. 9.9] this agrees with the original algebraic definition of Dress [Dre73] in the case of finite groups. The spectral endomorphism category E(G) is a spectrum valued lift of the stable homotopy orbit category. Theorem 3.3.3 shows that the category of G-equivariant spectra is Quillen equivalent to a category of topological Mackey functors, i.e., the category of modules over the stable orbit category E(G). Note that the homotopy type of each morphism spectrum of E(G) depends on the universe U.

After rationalization the Mackey functor analogy becomes even more concrete: we will see in Example 5.1.2 that rational G-equivariant spectral category is in fact Quillen equivalent to the model category of chain complexes of rational Mackey functors for G finite. See also Remark 6.1.5.

(ii) 𝔸^{1}-local stable homotopy of schemes. In [MV98, Voe98] Morel and Voevodsky introduce the 𝔸^{1}-local model category structure for schemes over a base. The objects of their category are sheaves of sets in the Nisnevich topology on smooth schemes of finite type over a fixed base scheme. The weak equivalences are the 𝔸^{1}-local equivalences – roughly speaking they are generated by the projection maps X × 𝔸^{1} ⟶ X for smooth schemes X, where 𝔸^{1} denotes the affine line.

Voevodsky [Voe98, Sec. 5] introduces an associated stable homotopy category by inverting smashing with the ‘Tate-sphere’ T = 𝔸^{1}/(𝔸^{1} − 0). The punch-line is that theories like algebraic K-theory or motivic cohomology are represented by objects in this stable homotopy category [Voe98, Sec. 6], at least when the base scheme is the spectrum of a field.

In [Jar98b], Jardine provides the details of the construction of model categories of T - spectra over the spectrum of a field k. He constructs two Quillen equivalent proper, simplicial model categories of Bousfield-Friedlander type and symmetric 𝔸^{1}-local T -spectra [Jar98b, 2.11, 4.18]. Since T is weakly equivalent to a suspension (of the multiplicative group scheme), this in particular yields a stable model category. A set of compact generators for this homotopy category is given by the T -suspension spectra Σ_{T}^{∞}(SpecR)_{+} when R runs over smooth k-algebras of finite type. So if k is countable then this is a countable set of compact generators, compare [Voe98, Prop. 5.5].

(iii) Algebraic examples. Again the classification theorem 3.3.3 has an algebraic analogue and precursor, namely Keller’s theory of derived equivalences of DG categories [Kel94]. The bottom line is that if an example of a stable model category is algebraic (such as derived or stable module categories in Examples 2.4), then it is not necessary to consider spectra and modules over spectral categories, but one can work with chain complexes and differential graded categories instead. As an example, Theorem 4.3 of [Kel94] identifies the stable module category of a Frobenius ring with the unbounded derived category of a certain differential graded category.

3.5. Prerequisites on spectral model categories. A spectral model category is analogous to a simplicial model category, [Qui67, II.2], but with the category of simplicial sets replaced by symmetric spectra. Roughly speaking, a spectral model category is a pointed model category which is compatibly enriched over the stable model category of symmetric spectra. The compatibility is expressed by the axiom (SP) below which takes the place of [Qui67, II.2 SM7]. For the precise meaning of ‘tensors’ and ‘cotensors’ over symmetric spectra see e.g. [Kly82, 3.7]. A spectral model category is the same as a ‘Sp^{Σ}-model category’ in the sense of [Hov99a, Def. 4.2.18], where the category of symmetric spectra is endowed with the stable model structure of [HSS98, 3.4.4]. Condition two of [Hov99a, 4.2.18] is automatic since the unit S for the smash product of symmetric spectra is cofibrant. Examples of spectral model categories are module categories over a symmetric ring spectrum, module categories over a spectral category (Theorem A.1.2) and the category of symmetric spectra over a suitable simplicial model category (Theorem 3.9.2).

Definition 3.5.1. A spectral model category is a model category C which is tensored, cotensored and enriched (denoted Hom_{C}) over the category of symmetric spectra with the closed monoidal structure of [HSS98, 2.2.10] such that the following compatibility axiom (SP) holds:

(SP) For every cofibration A ⟶ B and every fibration X ⟶ Y in C the induced map

Hom_{C}(B, X) ⟶ Hom_{C}(A, X) ×_{Hom_{C}(A,Y)} Hom_{C}(B, Y)

is a stable fibration of symmetric spectra. If in addition one of the maps A ⟶ B or X ⟶ Y is a weak equivalence, then the resulting map of symmetric spectra is also a stable equivalence. We use the notation K ∧ X and X^{K} to denote the tensors and cotensors for X ∈ C and K a symmetric spectrum.

In analogy with [Qui67, II.2 Prop. 3] the compatibility axiom (SP) in Definition 3.5.1 of a spectral model category can be cast into two adjoint forms, one of which will be of use for us. Given a categorical enrichment of a model category C over the category of symmetric spectra, then axiom (SP) is equivalent to (SPb) below. The equivalence of conditions (SP) and (SPb) is a consequence of the adjointness properties of the tensor and cotensor functors, see [Hov99a, Lemma 4.2.2] for the details.

(SPb) For every cofibration A ⟶ B in C and every stable cofibration K ⟶ L of symmetric spectra, the canonical map (pushout product map)

L ∧ A ∪_{K∧A} K ∧ B ⟶ L ∧ B

is a cofibration; the pushout product map is a weak equivalence if in addition A ⟶ B is a weak equivalence in C or K ⟶ L is a stable equivalence of symmetric spectra.

In a spectral model category the levels of the symmetric function spectra Hom_{C}(X, Y) can be rewritten as follows. The adjunctions give an isomorphism of simplicial sets

Hom_{C}(X, Y)_{n} ≅ map_{Sp}(F_{n}S^{0}, Hom_{C}(X, Y)) ≅ map_{C}(F_{n}S^{0} ∧ X, Y) ≅ map_{C}(X, Y^{F_{n}S^{0}})

where F_{n}S^{0} is the free symmetric spectrum generated at level n by the 0-sphere (see [HSS98, 2.2.5] or Definition 3.6.5).

Lemma 3.5.2. A spectral model category is in particular a simplicial and stable model category. For X a cofibrant and Y a fibrant object of a spectral model category C there is a natural isomorphism of graded abelian groups π_{∗}^{s} Hom_{C}(X, Y) ≅ [X, Y]_{∗}^{Ho(C)}.

Proof. The tensor and cotensors of an object of C with a pointed simplicial set K is defined by applying the tensor and cotensor with the symmetric suspension spectrum Σ^{∞}K. The homomorphism simplicial set between two objects of C is the 0-th level of the homomorphism symmetric spectrum. The necessary adjunction formulas and the compatibility axiom [Qui67, II.2 SM7] hold because the suspension spectrum functor Σ^{∞}: S_{∗} ⟶ Sp from the category of pointed simplicial sets to symmetric spectra is the left adjoint of a Quillen adjoint functor pair and preserves the smash product (i.e., it is strong symmetric monoidal). In order to see that C is stable we recall that the shift functor in the homotopy category of C is the suspension functor. For cofibrant objects suspension is represented on the model category level by the smash product with the one-dimensional sphere spectrum Σ^{∞}S^{1}. This sphere spectrum is invertible, up to stable equivalence of symmetric spectra, with inverse the (-1)-dimensional sphere spectrum (modeled as a symmetric spectrum by F_{1}S^{0}). Since the action of symmetric spectra on C is associative up to coherent isomorphism this implies that suspension is a self-equivalence of the homotopy category of C. This in turn implies that the right adjoint loop functor has to be an inverse equivalence. If X is cofibrant and Y fibrant in C, then by the compatibility axiom (SP) the symmetric spectrum Hom_{C}(X, Y) is stably fibrant, i.e., an Ω-spectrum. So for n ≥ 0 we have isomorphisms

π_{n} Hom_{C}(X, Y) ≅ π_{n} map_{C}(X, Y) ≅ π_{0} map_{C}(X, Ω^{n}Y) ≅ [X, Ω^{n}Y]^{Ho(C)} ;

and for n ≤ 0 we have the isomorphisms

π_{n} Hom_{C}(X, Y) ≅ π_{0} Hom_{C}(X, Y)_{n} ≅ π_{0} map_{C}(F_{n}S^{0} ∧ X, Y) ≅ [Ω^{n}X, Y]^{Ho(C)} .

3.6. Symmetric spectra over a category. Throughout this section we assume that C is a cocomplete category which is tensored and cotensored over the category S_{∗} of pointed simplicial sets, with this action denoted by ⊗ and morphism simplicial sets denoted map_{C}. We let S^{1} = Δ[1]/∂Δ[1] be our model for the simplicial circle and we set S^{n} = (S^{1})^{∧n} for n > 1; the symmetric group on n letters acts on S^{n} by permuting the coordinates.

Definition 3.6.1. Let C be a category which is tensored over the category of pointed simplicial sets. A symmetric sequence over C is a sequence of objects X = {X_{n}}_{n≥0} in C together with a left action of the symmetric group Σ_{n} on X_{n} for all n ≥ 0. A symmetric spectrum over C is a symmetric sequence in C with coherently associative Σ_{p} × Σ_{q}-equivariant morphisms

S^{p} ⊗ X_{q} ⟶ X_{p+q}

(for all p, q ≥ 0). A morphism of symmetric sequences or symmetric spectra X ⟶ Y consists of a sequence of Σ_{n}-equivariant morphisms X_{n} ⟶ Y_{n} which commute with the structure maps. We denote the category of symmetric sequences by C^{Σ} and the category of symmetric spectra by Sp(C).

Since C is simplicial, the action of S_{∗} on C extends to an action of S_{∗}^{Σ}, the category of symmetric sequences over S_{∗}, on C^{Σ}:

Definition 3.6.2. Given X a symmetric sequence over C, and K a symmetric sequence over S_{∗}, we define their tensor product, K ⊗ X, by the formula

(K ⊗ X)_{n} = ⋁_{p+q=n} Σ_{n}^{+} ⊗_{Σ_{p}×Σ_{q}} (K_{p} ⊗ X_{q}).

The symmetric sequence S = (S^{0}, S^{1}, ⋯, S^{n}, ⋯) of simplicial sets is a commutative monoid in the symmetric monoidal category (S_{∗}^{Σ}, ⊗). The unit map here is the identity in the first spot and the base point elsewhere, η: (S^{0}, ∗, ∗, ⋯) = u ⟶ S. In this language a symmetric spectrum over C can be redefined as a left S-module in the category of symmetric sequences over C.

Definition 3.6.3. Let X be an object in Sp(C) and K a symmetric spectrum in Sp^{Σ}. Define their smash product, K ∧ X, as the coequalizer of the two maps

K ⊗ S ⊗ X \overrightarrow{⟶} K ⊗ X

induced by the action of S on X and K respectively. So Sp(C) is tensored over the symmetric monoidal category of symmetric spectra. Dually, we define a symmetric spectrum valued hom object Hom_{Sp(C)}(X, Y) ∈ Sp^{Σ} for X, Y ∈ Sp(C). As a preliminary step, define a shifting down functor, sh_{n}: Sp(C) ⟶ Sp(C), by (sh_{n} X)_{m} = X_{n+m} where Σ_{m} acts via the standard inclusion into Σ_{n+m}. Note there is a leftover action of Σ_{n} on sh_{n} X. Define Hom_{Σ}(X, Y) ∈ S_{∗}^{Σ} for X, Y objects in C^{Σ} by Hom_{Σ}(X, Y)_{n} = map(X, sh_{n} Y), the simplicial mapping space given by the simplicial structure on C^{Σ} with the Σ_{n} action given by the leftover action of Σ_{n} on sh_{n} Y as mentioned above. Then Hom_{Sp(C)}(X, Y) ∈ Sp^{Σ} is the equalizer of the two maps Hom_{Σ}(X, Y) ⟶ Hom_{Σ}(S ⊗ X, Y).

Using this spectrum valued hom functor, for K in Sp^{Σ} and Y in Sp(C), we define Y^{K} ∈ Sp(C) as the adjoint of the functor K ⊗ −. That is, for any X ∈ Sp(C) define Y^{K} such that (3.6.4.) Hom_{Sp(C)}(K ⊗ X, Y) ≅ Hom_{Sp(C)}(X, Y^{K}) ≅ Hom_{Sp^{Σ}}(K, Hom_{Sp(C)}(X, Y)).

Definition 3.6.5. The nth evaluation functor Ev_{n} : Sp(C) ⟶ C is given by Ev_{n}(X) = X_{n}, ignoring the action of the symmetric group. The functor Ev_{n} has a left adjoint F_{n}: C ⟶ Sp(C) which has the form (F_{n}X)_{m} = Σ_{m}^{+} ⊗_{Σ_{m−n}} S^{m−n} ⊗ X where S^{n} = ∗ for n < 0. We use Σ^{∞} as another name for F_{0} and call it the suspension spectrum.

3.7. The level model structure on Sp(C). There are two model category structures on symmetric spectra over C which we consider; the level model category which we discuss in this section, and the stable model category (see 3.9). The level model category is a stepping stone for defining the stable model category, but it also allows us to define endomorphism ring spectra (Definition 3.7.5) and to construct equivalences of triangulated homotopy categories (see 3.8).

Definition 3.7.1. Let f : X ⟶ Y be a map in Sp(C). The map f is a level equivalence if each f_{n} : X_{n} ⟶ Y_{n} is a weak equivalence in C, ignoring the Σ_{n} action. It is a level fibration if each f_{n} is a fibration in C. It is a cofibration if it has the left lifting property with respect to all level trivial fibrations.

Proposition 3.7.2. For any simplicial, cofibrantly generated model category C, Sp(C) with the level equivalences, level fibrations, and cofibrations described above forms a cofibrantly generated model category referred to as the level model category, and denoted by Sp(C)^{lv}. Furthermore the following level analogue of the spectral axiom (SP) holds: (SP^{lv}) for every cofibration A ⟶ B and every level fibration X ⟶ Y in Sp(C) the induced map

Hom_{Sp(C)}(B, X) ⟶ Hom_{Sp(C)}(A, X) ×_{Hom_{Sp(C)}(A,Y)} Hom_{Sp(C)}(B, Y)

is a level fibration of symmetric spectra. If in addition one of the maps A ⟶ B or X ⟶ Y is a level equivalence, then the resulting map of symmetric spectra is also a level equivalence.

There are various theorems in the model category literature which are useful in establishing model category structures. These theorems separate formal considerations that tend to show up routinely from the properties which require special arguments in each specific case. Since we will construct model category structures several times in this paper, we recall one such result that we apply in our cases. We work with the concept of cofibrantly generated model categories, introduced by Dwyer, Hirschhorn and Kan [DHK]; see also Section 2.1 of Hovey’s book [Hov99a] for a detailed treatment of this concept.

We use the same terminology as [Hov99a, Sec. 2.1]. Let I be a set of maps in a category. A map is a relative I-cell complex if it is a (possibly transfinite) composition of cobase changes of maps in I. An I-injective map is a map with the right lifting property with respect to every map in I. An I-cofibration is a map with the left lifting property with respect to I-injective maps. For the definition of smallness relative to a set of maps see [Hov99a, 2.1.3].

One of the main properties of cofibrantly generated model categories is that they admit an abstract version of Quillen’s small object argument [Qui67, II 3.4].

Lemma 3.7.3. [DHK], [Hov99a, 2.1.14, 2.1.15] Let C be a cocomplete category and I a set of maps in C whose domains are small relative to the relative I-cell complexes. Then

• there is a functorial factorization of any map f in C as f = qi with q an I-injective map and i a relative I-cell complex, and thus

• every I-cofibration is a retract of a relative I-cell complex.

Theorem 3.7.4. [DHK], [Hov99a, Thm. 2.1.19] Let C be a complete and cocomplete category and I and J two sets of maps of C such that the domains of the maps in I and J are small with respect to the relative I-cell complexes and the relative J-cell complexes respectively. Suppose also that a subcategory of C is specified whose morphisms are called ‘weak equivalences’.

Then there is a cofibrantly generated model structure on C with the given class of weak equivalences, with I a set of generating cofibrations, and with J a set of generating trivial cofibrations if the following conditions hold:

(1) if f and g are composable morphisms such that two of the three maps f, g and gf are weak equivalences, then the third is also a weak equivalence.

(2) every relative J-cell complex is an I-cofibration and a weak equivalence.

(3) the I-injectives are precisely the maps which are both J-injective and weak equivalences.

Proof of Proposition 3.7.2. Let I_{C} and J_{C} be sets of generators for the cofibrations and trivial cofibrations of C. We define sets of generators for the level model category by F I_{C} = {F_{n}I_{C}}_{n≥0} and F J_{C} = {F_{n}J_{C}}_{n≥0}, i.e., F_{n} applied to the generators of C for each n. Then the F I_{C}- injectives are precisely the levelwise trivial fibrations and the F J_{C}-injectives are precisely the level fibrations. We claim that every relative F I_{C}-cell complex is levelwise a cofibration in C, and similarly every relative F J_{C}-cell complex is levelwise a trivial cofibration in C. We show this for relative F J_{C}-cell complexes; the argument for F I_{C} is the same. Since level evaluation preserves colimits it suffices to check the claim for the generating cofibrations, F_{n}A ⟶ F_{n}B for A ⟶ B ∈ J_{C}. But the mth level of this map is a coproduct of m!/(m − n)! copies of the map S^{m−n} ∧ A ⟶ S^{m−n} ∧ B. By the simplicial compatibility axiom [Qui67, II.2 SM7], smashing with a simplicial sphere preserves trivial cofibrations, so we are done.

Now we apply Theorem 3.7.4 to the sets F I_{C} and F J_{C} with the level equivalences as weak equivalences. Checking that the maps in F I_{C} are small with respect to relative F I_{C}-cell complexes comes down (by adjointness) to checking that the domains of the maps in I_{C} are small with respect to the levels of relative F I_{C}-cell complexes, i.e. the cofibrations in C. By [Hir99, 14.2.14], since the domains in I_{C} are small with respect to the relative I_{C}-cell complexes they are also small with respect to all cofibrations. The argument for F J_{C} is the same. Conditions (1) and (3) of Theorem 3.7.4 hold and condition (2) follows from the above claim. So we indeed have a cofibrantly generated level model structure.

To prove the property (SP^{lv}) it suffices to check its adjoint pushout product form, i.e., the level analogue of condition (SPb) of 3.5; it is enough to show that the pushout product of two generating cofibrations is a cofibration, and similarly when one of the maps is a trivial cofibration (see [SS00, 2.3 (1)] or [Hov99a, Cor. 4.2.5]). So let i ∈ I_{S_{∗}} and let j ∈ I_{C}. Then the product F_{n}i ∧ F_{m}j is isomorphic to F_{n+m}(i ∧ j) and the result follows since the free functors preserve cofibrations and trivial cofibrations.

Now that we defined the category of symmetric spectra over a simplicial model category and discussed the level model category, we can introduce endomorphism ring spectra and endomorphism categories.

Definition 3.7.5. Let C be a simplicial and cofibrantly generated model category and P a set of cofibrant objects. For every object P ∈ P let Σ_{f}^{∞}P be a fibrant replacement of the symmetric suspension spectrum of P in the level model structure on Sp(C) of Proposition 3.7.2. We define the endomorphism category E(P) as the full spectral subcategory of Sp(C) with objects Σ_{f}^{∞}P for P ∈ P. To simplify notation, we usually denote objects of E(P) by P instead of Σ_{f}^{∞}P. If P has a single object P we also refer to the symmetric ring spectrum E(P)(P, P) = Hom_{Sp(C)}(Σ_{f}^{∞}P, Σ_{f}^{∞}P) as the endomorphism ring spectrum of the object P.

Up to stable equivalence, the definition of the endomorphism category does not depend on the choices of fibrant replacements.

Lemma 3.7.6. Let C be a simplicial and cofibrantly generated model category and P a set of cofibrant objects. Suppose \{Σ_{f}^{∞}P \}_{P ∈P} and \{\overline{Σ_{f}^{∞}P} \}_{P ∈P} are two sets of level fibrant replacements of the symmetric suspension spectra. Then the two full spectral subcategories of Sp(C) with objects \{Σ_{f}^{∞}P \}_{P ∈P} and \{\overline{Σ_{f}^{∞}P} \}_{P ∈P} respectively are stably equivalent.

Proof. The proof uses the notion of quasi-equivalence, see Definition A.2.1. For every P ∈ P we choose a level equivalence ϕ_{P} : \overline{Σ_{f}^{∞}P} ⟶ Σ_{f}^{∞}P. We define a E(P)-\overline{E(P)}-bimodule M by the rule

M(P, P′) = Hom_{Sp(C)}(\overline{Σ_{f}^{∞}P} , Σ_{f}^{∞}P) .

Because of the property (SP^{lv}) of the homomorphism spectra in Sp(C) the bimodule M is a quasi-equivalence with respect to the maps ϕ_{P}, and the result follows from Lemma A.2.3.

3.8. An equivalence of homotopy categories. We have now assembled enough tools to show that a stable model category with a set of compact generators has the same homotopy category as the modules over the endomorphism ring spectrum of the generators. The ultimate goal is to refine this equivalence of homotopy categories to a chain of Quillen equivalences as stated in Theorem 3.3.3; that step requires additional technical assumptions and extra work, and we complete it in Section 3.9 and 3.10. The extra assumption we do not yet need is properness, and the additional work goes into proving the existence of the stable model structure on Sp(C). The properness assumption is of a technical nature, and many model categories satisfy it. However having to assume properness is a nuisance and we know of no general approach designed to arrange for properness.

Suppose C is a simplicial, cofibrantly generated model category. Then the level model structure of Proposition 3.7.2 exists on the category Sp(C) of symmetric spectra over C.

Definition 3.8.1. Let λ : F_{1}S^{1} ⟶ F_{0}S^{0} ≅ S be the stable equivalence of symmetric spectra which is adjoint to the identity map on the first level. A spectrum Z in Sp(C) is an Ω-spectrum if Z is fibrant on each level and the map Z ≅ Z^{F_{0}S^{0}} ⟶ Z^{F_{1}S^{1}} induced by λ is a level equivalence.

The above definition of Ω-spectrum is just a rewrite of the usual one since the n-th level of the spectrum Z^{F_{1}S^{1}} is isomorphic to ΩZ_{n+1}. However this form is more convenient here.

Lemma 3.8.2. Let K be a cofibrant symmetric spectrum, A a cofibrant spectrum in Sp(C) and Z an Ω-spectrum in Sp(C). Then the symmetric function spectrum Hom_{Sp(C)}(A, Z) is a symmetric Ω-spectrum and the function spectrum Z^{K} is an Ω-spectrum in Sp(C).

Proof. By the adjunctions between smash products and function spectra (see 3.6.4) we can rewrite the symmetric function spectrum Hom_{Sp(C)}(A, Z)^{F_{1}S^{1}} as Hom_{Sp(C)}(A, Z^{F_{1}S^{1}}) in such a way that the map Hom_{Sp(C)}(A, Z)^{λ} is isomorphic to the map Hom_{Sp(C)}(A, Z^{λ}); since Z^{λ} : Z ≅ Z^{F_{0}S^{0}} ⟶ Z^{F_{1}S^{1}} is a level equivalence between level fibrant objects, the first claim follows from property (SP^{lv}) of Proposition 3.7.2.

Similarly we can rewrite the spectrum (Z^{K})^{F_{1}S^{1}} as (Z^{F_{1}S^{1}})^{K} in such a way that the map (Z^{K})^{λ} is isomorphic to the map (Z^{λ})^{K}; since Z^{λ} is a level equivalence between level fibrant objects, the second claim follows from the adjoint form (SP^{lv}(a)) of property (SP^{lv}) of Proposition 3.7.2, see [Qui67, II.2 SM7(a)].

If C is also stable, then for every cofibrant object A, the functorial level fibrant replacement of the suspension spectrum Σ_{f}^{∞}A is an Ω-spectrum. Let P be a set of cofibrant objects of C. We define a functor

Hom(P, −) : C ⟶ mod-E(P)

by the formula

Hom(P, A)(P) = Hom_{Sp(C)}(Σ_{f}^{∞}P, Σ_{f}^{∞}A) .

The endomorphism category E(P) of P (see Definition 3.7.5) acts on Hom(P, A) from the right by composition.

The functor Σ_{f}^{∞} : C ⟶ Sp(C) takes weak equivalences between cofibrant objects to level equivalences between level fibrant spectra. So by the property (SP^{lv}) of Proposition 3.7.2 the composite functor Hom(P, −) takes weak equivalences between cofibrant objects to objectwise equivalences of E(P)-modules. Hence there is a left derived functor

L Hom(P, −) : Ho(C) ⟶ Ho(mod-E(P))

on the level of homotopy categories.

Theorem 3.8.3. Let C be a cofibrantly generated, simplicial stable model category and P a set of cofibrant objects. Then the derived functor

L Hom(P, −) : Ho(C) ⟶ Ho (mod-E(P))

is naturally a triangulated functor. If all objects in P are compact, then the functor L Hom(P, −) restricts to a triangulated equivalence between the localizing subcategory of Ho(C) generated by P and the homotopy category of E(P)-modules.

Proof. A priori there is no functor going in the opposite direction, so we first show that the functor L Hom(P, −) is full and faithful: fix an object X of Ho(C) and denote by T_{X} the full subcategory of those objects A ∈ Ho(C) for which the map

L Hom(P, −) : [A, X]^{Ho(C)} ⟶ [L Hom(P, A), L Hom(P, X)]^{Ho(mod-E(P))}

is an isomorphism. If P is one of the objects of P, then Hom(P, P) is isomorphic to the free module F_{P}^{E(P)}, see section A.1, and so the right hand side is isomorphic to

[F_{P}^{E(P)}, L Hom(P, X)]^{Ho(mod-E(P))} ≅ π_{0} Hom(P, X)(P) ≅ π_{0} Hom_{Sp(C)}(Σ_{f}^{∞}P, Σ_{f}^{∞}X)

which in turn is isomorphic to the group of maps from P to X in the homotopy category of C. The functor L Hom(P, −) preserves the triangulated structure, so the subcategory T_{X} is thick. Since the objects in P are compact, one can pass to infinite coproducts as well, and conclude that T_{X} is in fact a localizing subcategory. Since the objects in P belong to the subcategory T_{X}, it contains the entire localizing subcategory generated by P. In other words: the restriction of L Hom(P, −) to the localizing subcategory generated by P is full and faithful. To see that L Hom(P, −) is also surjective on homotopy types we note that the generating E(P)-modules F_{P}^{E(P)}, see Theorem A.1.2, are in the image and that every E(P)-module is built from these. Since L Hom(P, −) is full and faithful, a presentation of an arbitrary module M in terms of the generating modules can be lifted to the homotopy category of C to yield a pre-image of M.

3.9. The stable model structure on Sp(C). In this section we provide the details of the stable model category structure for symmetric spectra over C; the result is summarized as Theorem 3.9.2. We use the level model category to define the stable model category structures on Sp(C). The stable model category is more difficult to establish than the level model category, and we need to assume that C is a simplicial, cofibrantly generated, proper, stable model category. The proof of the stable model structure for Sp(C) is similar to the proof of the stable model structure for Sp^{Σ} in [HSS98, 3.4], except for one point in the proof of Proposition 3.9.7 where we use the stability of C instead of the fact that fiber sequences and cofiber sequences of spaces are stably equivalent.

Categories of symmetric spectrum objects over a model category have been considered more generally by Hovey in [Hov98]. Hovey relies on the general localization machinery of [Hir99]. Theorem 3.9.2 below should be compared to [Hov98, Thms. 10.11 and 11.1] which are more general but have slightly different technical assumptions.

Definition 3.9.1. A map g : A ⟶ B in Sp(C) is a stable equivalence if the induced map

Hom_{Sp(C)}(cg, Z) : Hom_{Sp(C)}(cA, Z) ⟶ Hom_{Sp(C)}(cB, Z)

is a level equivalence of symmetric spectra for any Ω-spectrum Z, see Definition 3.8.1; here c(−) denotes a cofibrant replacement functor in the level model category structure. A map is a stable cofibration if it has the left lifting property with respect to each level trivial fibration, i.e., if it is a cofibration in the level model category structure. A map is a stable fibration if it has the right lifting property with respect to each map which is both a stable cofibration and a stable equivalence.

Lemma 3.9.6 combined with the fact that the stable fibrations are the J-injective maps shows that the stably fibrant objects are precisely the Ω-spectra.

Theorem 3.9.2. Let C be a simplicial, cofibrantly generated, proper, stable model category. Then Sp(C) supports the structure of a spectral model category – referred to as the stable model structure – such that the adjoint functors Σ^{∞} and evaluation Ev_{0}, are a Quillen equivalence between C and Sp(C) with the stable model structure.

Above we have defined the stable model structure as a localization of the level model structure with respect to the Ω-spectra. We start the proof by giving a characterization for a cofibration to be a stable equivalence.

Lemma 3.9.3. Let C be a simplicial, cofibrantly generated and left proper model category. Then a cofibration A ⟶ B is a stable equivalence if and only if for every Ω-spectrum Z the symmetric function spectrum Hom_{Sp(C)}(B/A, Z) is level contractible.

Proof. Choose a factorization of the functorial level cofibrant replacement cA ⟶ cB of the given cofibration as a cofibration i : cA ⟶ B̄ followed by a level equivalence q : B̄ ⟶ cB. Then q is a level equivalence between cofibrant objects, so for every Ω-spectrum Z, the induced map Hom_{Sp(C)}(q, Z) is a level equivalence. Hence f is a stable equivalence if and only if

Hom_{Sp(C)}(i, Z) : Hom_{Sp(C)}(cA, Z) ⟶ Hom_{Sp(C)}(B̄, Z)

is a level equivalence for every Ω-spectrum Z.

The symmetric spectrum Hom_{Sp(C)}(B̄/cA, Z) is the fiber of the level fibration Hom_{Sp(C)}(i, Z) (by (SP^{lv})) between symmetric Ω-spectra (by Lemma 3.8.2). Hence the given map is a stable equivalence if and only if Hom_{Sp(C)}(B̄/cA, Z) is level contractible.

Since cA ⟶ A and B̄ ⟶ B are level equivalences and C is left proper, the induced map on the cofibers B/cA ⟶ B/A is a level equivalence between level cofibrant objects. So for every Ωspectrum Z the induced map Hom_{Sp(C)}(B̄/A, Z) ⟶ Hom_{Sp(C)}(B̄/cA, Z) is a level equivalence (by (SP^{lv})) between symmetric Ω-spectra (by Lemma 3.8.2). Hence the given map is a stable equivalence if and only if Hom_{Sp(C)}(B/A, Z) is level contractible, which proves the lemma.

We now show that Sp(C) satisfies (SPb) of Section 3.5, an adjoint form of (SP) from Definition 3.5.1. This shows that Sp(C) is a spectral model category as soon as the stable model structure on Sp(C) is established.

Proposition 3.9.4. Let C be a simplicial, cofibrantly generated and left proper model category. Let i : A ⟶ B be a cofibration in Sp(C) and j : K ⟶ L a stable cofibration of symmetric spectra. Then the pushout product map

j □ i : L ∧ A ∪_{K∧A} K ∧ B ⟶ L ∧ B

is a cofibration in Sp(C); the pushout product map is a stable equivalence if in addition i is a stable equivalence in Sp(C) or j is a stable equivalence of symmetric spectra.

Proof. Since the cofibrations coincide in the level and the stable model structures for Sp(C) and for symmetric spectra, we know by property (SP^{lv}) of Proposition 3.7.2 that j □ i is again a cofibration in Sp(C). Now suppose that one of the maps is in addition a stable equivalence. The pushout product map j □ i is a cofibration with cofiber isomorphic to (L/K) ∧ (B/A). So by Lemma 3.9.3 it suffices to show that Hom_{Sp(C)}((L/K) ∧ (B/A), Z) is level contractible for every Ω-spectrum Z. If i is a stable acyclic cofibration, then we can rewrite this function spectrum as

Hom_{Sp(C)}((L/K) ∧ (B/A), Z) ≅ Hom_{Sp(C)}(B/A, Z^{(L/K)}) ;

the latter spectrum is level contractible by Lemma 3.9.3 since Z^{(L/K)} is an Ω-spectrum by Lemma 3.8.2 and since i is a cofibration and stable equivalence. If j is a stable acyclic cofibration, then we similarly rewrite the spectrum as

Hom_{Sp(C)}((L/K) ∧ (B/A), Z) ≅ Hom_{Sp^{Σ}}(L/K, Hom_{Sp(C)}(B/A, Z)) ;

the latter spectrum is level contractible by [HSS98, 5.3.9] since Hom_{Sp(C)}(B/A, Z) is a symmetric Ω-spectrum by Lemma 3.8.2 and since L/K is stably contractible.

We use Theorem 3.7.4 to verify the stable model category structure on Sp(C). We first define two sets I and J of maps in Sp(C) which will be generating sets for the cofibrations and stable trivial cofibrations. Since the stable cofibrations are the same class of maps as the cofibrations in the level model structure we let I be the generating set F I_{C} which was used in Proposition 3.7.2 to construct the level model structure. With this choice the I-injectives are precisely the level trivial fibrations.

The generating set for the stable trivial cofibrations is the union J = F J_{C} ∪ K, where F J_{C} is the generating set of trivial cofibrations for the level model category (see the proof of Proposition 3.7.2) and K is defined as follows. In the category of symmetric spectra over simplicial sets there is a map λ : F_{1}S^{1} ⟶ F_{0}S^{0} = S which is adjoint to the identity map on the first level. Let Mλ be the mapping cylinder of this map, formed by taking the mapping cylinder of simplicial sets on each level. So λ = rκ with κ : F_{1}S^{1} ⟶ Mλ a stable equivalence and stable cofibration and r : Mλ ⟶ S a simplicial homotopy equivalence, see [HSS98, 3.4.9]. Then K is the set of maps

K = {κ □ F I_{C}} = {κ □ F_{n}i|i ∈ I_{C}} ,

where for i : A ⟶ B,

κ □ F_{n}i : (F_{1}S^{1} ∧ F_{n}B) ∪_{F_{1}S^{1}∧F_{n}A} (Mλ ∧ F_{n}A) ⟶ Mλ ∧ F_{n}B .

Here we only use the pushout product, □, as a convenient way of naming these maps, see also [HSS98, 5.3]. Now we can verify condition (2) of Theorem 3.7.4.

Proposition 3.9.5. Let C be a simplicial, cofibrantly generated and left proper model category. Then every relative J-cell complex is an I-cofibration and a stable equivalence.

Proof. All maps in J are cofibrations in the level model structure on Sp(C) of Proposition 3.7.2, hence the relative J-cell complexes are contained in the I-cofibrations.

We claim that for every J-cofibration A ⟶ B and every Ω-spectrum Z, the map

Hom_{Sp(C)}(B, Z) ⟶ Hom_{Sp(C)}(A, Z)

is a level trivial fibration of symmetric spectra. Hence the fiber, the symmetric spectrum Hom_{Sp(C)}(B/A, Z), is level contractible and A ⟶ B is a stable equivalence by Lemma 3.9.3.

The property of inducing a trivial fibration after applying Hom_{Sp(C)}(−, Z) is closed under pushout, transfinite composition and retract, so by the small object argument 3.7.3 it suffices to check this for the generating maps in J = F J_{C} ∪ K. The generating cofibrations in F J_{C} are level trivial cofibrations, so for these the claim holds by the compatibility axiom (SP^{lv}). A map in the set K is of the form κ □ F_{n}i where κ : F_{1}S^{1} ⟶ Mλ is stable trivial cofibration of symmetric spectra and F_{n}i is a cofibration in Sp(C); hence the map κ □ F_{n}i is a stable trivial cofibration between cofibrant objects by Proposition 3.9.4. So the induced map of symmetric spectra Hom_{Sp(C)}(κ □ F_{n}i, Z) is a level fibration (by (SP^{lv})) between Ω-spectra by Lemma 3.8.2. In addition the fiber of the map Hom_{Sp(C)}(κ □ F_{n}i, Z) is level contractible by Lemma 3.9.3, so the map is indeed a level trivial fibration.

Before turning to property (3) of Theorem 3.7.4, we need the following lemma.

Lemma 3.9.6. Let C be a simplicial, cofibrantly generated model category and X a symmetric spectrum over C. Then the map X ⟶ ∗ is J-injective if and only if X is an Ω-spectrum.

Proof. The maps in F J_{C} generate the trivial cofibrations in the level model structure of Proposition 3.7.2, so X ⟶ ∗ is F J_{C}-injective if an only if X is levelwise fibrant. So we assume that X is levelwise fibrant and show that then the map X ⟶ ∗ is K-injective if and only if X is an Ω-spectrum. By adjointness X ⟶ ∗ is K-injective if and only if the map X^{κ} : X^{Mλ} ⟶ X^{F_{1}S^{1}} is a level trivial fibration. The projection r : Mλ ⟶ S is a simplicial homotopy equivalence, so it induces a level equivalence X ⟶ X^{Mλ}. So X ⟶ ∗ is K-injective if and only if the map X^{λ} is a level equivalence, which precisely means that X is an Ω-spectrum.

Proposition 3.9.7. Let C be a simplicial, cofibrantly generated, right proper and stable model category. Then a map is J-injective and a stable equivalence if and only if it is a level trivial fibration.

Proof. Every level equivalence is a stable equivalence, and the level trivial fibrations are precisely the I-injectives; since the J-cofibrations are contained in the I-cofibrations, the I-injectives are also J-injective. The converse is more difficult to prove.

Since J in particular contains maps of the form F_{n}A ⟶ F_{n}B where n runs over the natural numbers and A ⟶ B runs over a set of generating trivial cofibrations for C, J-injective maps are level fibrations. So we show that a J-injective stable equivalence, E ⟶ B, is a level equivalence. Let F denote the fiber and choose a cofibrant replacement F^{c} ⟶ F in the level model category structure. Then choose a factorization in the level model category structure

F^{c}  E^{c}  E v  w ^{lv ∼} w

of the composite map F^{c} ⟶ E as a cofibration followed by a level equivalence. Since C is right proper, each level of F ⟶ E ⟶ B is a homotopy fibration sequence in C. Each level of F^{c} ⟶ E^{c} ⟶ E^{c}/F^{c} is a homotopy cofibration sequence in C, left properness is not needed here since each object is cofibrant. So since C is stable, we see that E^{c}/F^{c} ⟶ B is a level equivalence. Thus E^{c} ⟶ E^{c}/F^{c} is a stable equivalence. For any Ω-spectrum Z there is a fiber sequence of symmetric Ω-spectra

Hom_{C}(E^{c}/F^{c}, Z) ⟶ Hom_{C}(E^{c}, Z) ⟶ Hom_{C}(F^{c}, Z)

in which the left map is a level equivalence and the right map is a level fibration. Hence the symmetric spectrum Hom_{C}(F^{c}, Z) is level contractible which means that F is stably contractible.

Since F ⟶ ∗ is the pull back of the map E ⟶ B, it is a J-injective map. So F is an Ω-spectrum by Proposition 3.9.6. Since F is both stably contractible and an Ω-spectrum, the spectrum Hom_{C}(F^{c}, F) is level contractible, so F is level equivalent to a point. But this means that E^{c} ⟶ E^{c}/F^{c}, and thus also E ⟶ B is a level equivalence.

Proof of Theorem 3.9.2. We apply Theorem 3.7.4 to show that the I-cofibrations, stable equivalences, and J-injectives form a cofibrantly generated model category on Sp(C). Since the I-cofibrations are exactly the stable cofibrations this implies that the J-injectives are the maps with the right lifting property with respect to the stable trivial cofibrations, i.e., the stable fibrations as defined before the statement of the theorem. The 2-out-of-3 condition, part (1) in Theorem 3.7.4, is clear from the definition of stable equivalences. Condition (2) is verified in Proposition 3.9.5 and condition (3) is verified in Proposition 3.9.7 (since the I-injectives are precisely the levelwise trivial fibrations). So to conclude that the sets I and J generate a model structure with the stable equivalences as weak equivalence it is enough to verify that the domains of the generators are small with respect to the level cofibrations. This has already been checked in the proof of Proposition 3.7.2 for the generators in F I_{C} and F J_{C}. So only the generators in K remain. Since a pushout of objects which are small is small, we only need to check that F_{k}A ∧ Mλ and F_{k}A ∧ F_{1}S^{1} are small with respect to the level cofibrations. Here A is small with respect to relative I_{C}-cell complexes and hence also the cofibrations by [Hir99, 14.2.14]. F_{k}A is small with respect to relative I-cell complexes in Sp(C) by adjointness and F_{1}S^{1} is small with respect to all of Sp^{Σ}. So by various adjunctions F_{1}S^{1} ∧ F_{k}A is small with respect to level cofibrations. Since Mλ is the pushout of small objects, similar arguments show that F_{k}A ∧ Mλ is also small with respect to the level cofibrations in Sp(C).

The spectral compatibility axiom is verified in Proposition 3.9.4 in its adjoint form (SPb). So, it remains to show that the adjoint functors Σ^{∞} and Ev_{0} are a Quillen equivalence. The suspension spectrum functor Σ^{∞} takes (trivial) cofibrations to (trivial) cofibrations in the level model structure. Hence Σ^{∞} also preserves (trivial) cofibrations with respect to the stable model structure. So the adjoint functors Ev_{0} and Σ^{∞} are a Quillen pair between C and Sp(C).

To show that the functors are a Quillen equivalence it suffices to show (see [Hov99a, Cor. 1.3.16]) that Ev_{0} reflects stable equivalences between stably fibrant objects and that for every cofibrant object A of C the map A ⟶ Ev_{0} R(Σ^{∞}A) is a weak equivalence where R denotes any stably fibrant replacement in Sp(C). So suppose that f : X ⟶ Y is a map between Ω-spectra with the property that f_{0} : X_{0} ⟶ Y_{0} is a weak equivalence in C. Since X is an Ω-spectrum, X_{0} ⟶ Ω^{n}X_{n} is a weak equivalence, and similarly for Y. Hence the map Ω^{n}f_{n} : Ω^{n}X_{n} ⟶ Ω^{n}Y_{n} is a weak equivalence in C. Since C is stable, the loop functor is a self-Quillen equivalence, so it reflects weak equivalences between fibrant objects, and so f_{n} : X_{n} ⟶ Y_{n} is a weak equivalence in C. Hence f is a level, and thus a stable equivalence of spectra over C.

Since C is stable the spectrum Σ_{f}^{∞}A (the fibrant replacement of the suspension spectrum in the level model structure) is an Ω-spectrum, and thus stably fibrant. Hence we may take Σ_{f}^{∞}A as the stably fibrant replacement R(Σ^{∞}A), which proves that A ⟶ Ev_{0} R(Σ^{∞}A) is a weak equivalence in C.

3.10. The Quillen equivalence. In this section we prove Theorem 3.3.3, i.e., we show that a suitable model category with a set of compact generators is Quillen equivalent to the modules over the spectral endomorphism category of the generators.

In Theorem 3.10.3 we first formulate the result for spectral model categories; this gives a more general result since the conditions about cofibrant generation and properness in C are not needed. We then combine this with the fact that every suitable stable model category is Quillen equivalent to a spectral model category to prove our main classification theorem.

Definition 3.10.1. Let G be a set of objects in a spectral model category D. We denote by E(G) the full spectral subcategory of D with objects G, i.e., E(G)(G, G′) = Hom_{D}(G, G′). We let

Hom(G, −) : D ⟶ mod-E(G)

denote the tautological functor given by Hom(G, Y)(G) = Hom_{D}(G, Y).

We want to stress the reassuring fact that the stable equivalence type of the spectral endomorphism category E(G) only depends on the weak equivalence types of the objects in the set G, as long as these are all fibrant and cofibrant, see Corollary A.2.4. This is not completely obvious since taking endomorphisms is not a functor.

The earlier Definition 3.7.5 of the endomorphism ring spectrum and endomorphism category of objects in a simplicial stable model category C is a special case of Definition 3.10.1 with D = Sp(C) and G the level fibrant replacements of the suspension spectra of the chosen objects in C. Again, if G = {G} has a single element then E(G) is determined by the single symmetric ring spectrum, End_{D}(G) = Hom_{D}(G, G).

Definition 3.10.2. Let C and D be spectral model categories. A spectral Quillen pair is a Quillen adjoint functor pair L : C ⟶ D and R : D ⟶ C together with a natural isomorphism of symmetric homomorphism spectra

Hom_{C}(A, RX) ≅ Hom_{D}(LA, X)

which on the vertices of the 0-th level reduces to the adjunction isomorphism. A spectral Quillen pair is a spectral Quillen equivalence if the underlying Quillen functor pair is an ordinary Quillen equivalence.

In the terminology of [Hov99a, Def. 4.2.18] a spectral Quillen pair would be called a ‘Sp^{Σ}- Quillen functor’ for the monoidal model category of symmetric spectra.

Theorem 3.10.3. Let D be a spectral model category and G a set of of cofibrant and fibrant objects.

(i) The tautological functor

Hom(G, −) : D ⟶ mod-E(G)

is the right adjoint of a spectral Quillen functor pair. The left adjoint is denoted − ∧_{E(G)} G. (ii) If all objects in G are compact, then the total derived functors of Hom(G, −) and − ∧_{E(G)} G restrict to a triangulated equivalence between the homotopy category of E(G)-modules and the localizing subcategory of Ho(D) generated by G. (iii) If G is a set of compact generators for D, then the adjoint functor pair Hom(G, −) and − ∧_{E(G)} G form a spectral Quillen equivalence.

Proof. (i) For an E(G)-module M the object M ∧_{E(G)} G is given by an enriched coend [Kly82, 3.10]. This means that M ∧_{E(G)} G is the coequalizer of the two maps

⋁_{G,G′∈G} M(G′) ∧ E(G)(G, G′) ∧ G \overrightarrow{⟶} ⋁_{G∈G} M(G) ∧ G .

One map in the diagram is induced by the evaluation map E(G)(G, G′)∧G ⟶ G′ and the other is induced by the action map M(G′)∧E(G)(G, G′) ⟶ M(G). The tautological functor Hom(G, −) preserves fibrations and trivial fibrations by the compatibility axiom (SP) of Definition 3.5.1, since all objects of G are cofibrant. So together with its left adjoint it forms a spectral Quillen pair.

(ii) Since the functors Hom(G, −) and − ∧_{E(G)} G are a Quillen pair, they have adjoint total derived functors on the level of homotopy categories [Qui67, I.4]; we denote these derived functors by RHom(G, −) and − ∧_{E(G)}^{L} G respectively. The functor − ∧_{E(G)}^{L} G commutes with suspension and preserves cofiber sequences, and the functor RHom(G, −) commutes with taking loops and preserves fiber sequences [Qui67, I.4 Prop. 2]. In the homotopy category of a stable model category, the cofiber and fiber sequences coincide up to sign and they constitute the distinguished triangles. So both total derived functors preserve shifts and triangles, i.e., they are exact functors of triangulated categories.

For every G ∈ G the E(G)-module Hom(G, G) is isomorphic to the free module F_{G}^{E(G)} (see A.1) by inspection and F_{G}^{E(G)} ∧_{E(G)} G is isomorphic to G since they represent the same functor on D. As a left adjoint, the functor − ∧_{E(G)}^{L} G preserves coproducts. We claim that the right adjoint

RHom(G, −) also preserves coproducts. Since the free modules F_{G}^{E(G)} form a set of compact generators for the category of E(G)-modules (see Theorem A.1.2), it suffices to show that for all G ∈ G and for every family {A_{i}}_{i∈I} of objects of D the natural map

⨁_{i∈I} [F_{G}^{E(G)}, RHom(G, A_{i})]_{∗}^{Ho(mod-E(G))} ≅ [F_{G}^{E(G)}, ∐_{i∈I} RHom(G, A_{i})]_{∗}^{Ho(mod-E(G))} ⟶

[F_{G}^{E(G)}, RHom(G, ∐_{i∈I} A_{i})]_{∗}^{Ho(mod-E(G))}

is an isomorphism. By the adjunctions and the identification F_{G}^{E(G)} ∧_{E(G)}^{L} G ≅ G this map is isomorphic to the natural map

⨁_{i∈I} [G, A_{i}]_{∗}^{Ho(D)} ⟶ [G, ∐_{i∈I} A_{i}]_{∗}^{Ho(D)} .

But this last map is an isomorphism since G was assumed to be compact.

Both derived functors preserve shifts, triangles and coproducts; since they match up the free E(G)-modules F_{G}^{E(G)} with the objects of G, they restrict to adjoint functors between the localizing subcategories generated by the free modules on the one side and the objects of G on the other side. We consider the full subcategories of those M ∈ Ho(mod-E(G)) and X ∈ Ho(D) respectively for which the unit of the adjunction

η : M ⟶ RHom(G, M ∧_{E(G)}^{L} G)

or the counit of the adjunction

ν : RHom(G, X) ∧_{E(G)}^{L} X ⟶ X

are isomorphisms. Since both derived functors are exact and preserve coproducts, these are localizing subcategories. Since F_{G}^{E(G)} ∧_{E(G)}^{L} G ≅ G and RHom(G, G) ≅ F_{G}^{E(G)}, the map η is an isomorphism for every free module, and the map ν is an isomorphism for every object of G. Since the free modules F_{G}^{E(G)} generate the homotopy category of E(G)-modules, the claim follows.

(iii) Now the localizing subcategory generated by G is the entire homotopy category of D, so part (ii) of the theorem implies that the total derived functors of Hom(G, −) and − ∧_{E(G)} G are inverse equivalences of homotopy categories. But a Quillen adjoint functor pair is a Quillen equivalence if and only if the total derived functors are inverse equivalences of homotopy categories [Hov99a, Prop. 1.3.13].

Now we can finally give the

Proof of Theorem 3.3.3: We can combine Theorem 3.9.2 and Theorem 3.10.3 (iii) to obtain a diagram of model categories and Quillen equivalences

C \begin{aligned}& Σ^{∞} \\ & & w \\ u \\ Ev & 0 \end{aligned} Sp(C) \begin{matrix}−∧_{E(G)}G \\ u \\ w \\ Hom(G,−) \end{matrix} mod-E(G)

(the left adjoints are on top). First, since compact generators are preserved by weak equivalence we cofibrantly replace each element in P. Since the left Quillen functor pair above induces an equivalence of homotopy categories, the suspension spectra of the new set P of compact cofibrant generators for C form a set of compact cofibrant generators for Sp(C).

We denote by G the set of level fibrant replacements Σ_{f}^{∞}P of the given generators in C. The spectral endomorphism category E(G) in the sense of Definition 3.10.1 is equal to the endomorphism categories associated to P by Definition 3.7.5. Since C is stable, the spectra Σ_{f}^{∞}P are Ω-spectra, hence they are both fibrant by Lemma 3.9.6 and cofibrant in the stable model structure on Sp(C). So we can apply Theorem 3.10.3 (iii) to get the second Quillen equivalence.

Remark 3.10.4. Finite localization and E(P)-modules. Suppose P is a set of compact objects of a triangulated category T with infinite coproducts. Then there always exists an idempotent localization functor L_{P} on T whose acyclics are precisely the objects of the localizing subcategory generated by P (compare [Mil92] or the proofs of Lemma 2.2.1 or [HPS97, Prop. 2.3.17]). These localizations are often referred to as finite Bousfield localizations away from P.

Theorem 3.10.3 gives a lift of finite localization to the model category level. Suppose C is a stable model category with a set P of compact objects, and let L_{P} denote the associated localization functor on the homotopy category of C. By Theorem 5.3.2 (ii) the acyclics for L_{P} are equivalent to the homotopy category of E(P)-modules, the equivalence arising from a Quillen adjoint functor pair. Furthermore the counit of the derived adjunction

Hom(P, X) ∧_{E(P)}^{L} P ⟶ X

is the acyclicization map and its cofiber is a model for the localization L_{P}X.

4. Morita context

In the classical algebraic context there is a characterization of equivalences of module categories in terms of bimodules, see for example [AF92, §22]. We provide an analogous result for module categories over ring spectra. As usual, here instead of actual equivalences of module categories one obtains Quillen equivalences of model categories. We state the Morita context for symmetric ring spectra and spectral Quillen equivalences (see Definition 3.10.2).

Definition 4.1.1. If R is a symmetric ring spectrum and C a spectral model category, then an R-C-bimodule is an object X of C on which R acts through C-morphisms, i.e., a homomorphism of symmetric ring spectra from R to the endomorphism ring spectrum of X.

If C is the category of modules over another symmetric ring spectrum T, then this notion of bimodule specializes to the usual one, i.e., an R-(T -mod)-bimodule is the same as an R^{op} ∧ T - module. In the following Morita theorem, the implication (2) ⟹ (3) is a special case of the classification of monogenic stable model categories (Theorem 3.1.1); hence the implication (3) ⟹ (2) is a partial converse to that classification result. The implication (3) ⟹ (1) says that certain chains of Quillen equivalences can be rectified into a single Quillen equivalence whose left adjoint is given by smashing with a bimodule.

Theorem 4.1.2. (Morita context) Consider the following statements for a symmetric ring spectrum R and a spectral model category C.

(1) There exists an R-C-bimodule M such that smashing with M over R is the left adjoint of a Quillen equivalence between the category of R-modules and C.

(2) The category C has a compact, cofibrant and fibrant generator M such that R is stably equivalent to the endomorphism ring spectrum of M.

(3) There exists a chain of spectral Quillen equivalences through spectral model categories between C and the category of R-modules.

Then conditions (2) and (3) are equivalent, and condition (1) implies both conditions (2) and (3). If R is cofibrant as a symmetric ring spectrum, then all three conditions are equivalent.

Furthermore, if C is the category of modules over another symmetric ring spectrum T which is cofibrant as a symmetric spectrum, then condition (1) is equivalent to the condition

(4) There exists an R-T -bimodule M and a T -R-bimodule N which are cofibrant as right modules such that

• M ∧_{T} N is stably equivalent to R as an R-bimodule and

• N ∧_{R} M is stably equivalent to T as a T -bimodule.

Remark 4.1.3. The cofibrancy conditions in the Morita theorem can always be arranged since every symmetric ring spectrum has a stably equivalent cofibrant replacement in the model category of symmetric ring spectra [HSS98, 5.4.3]; furthermore the underlying symmetric spectrum of a cofibrant ring spectrum is always cofibrant ([SS00, 4.1] or [HSS98, 5.4.3]). It should not be surprising that cofibrancy conditions have to be imposed in the Morita theorem. In the algebraic context the analogous conditions show up in Rickard’s paper [Ric91]: when trying to realize derived equivalences between k-algebras by derived tensor product with bimodule complexes, he has to assume that the algebras are flat over the ground ring k, see [Ric91, Sec. 3].

Proof of the Morita theorem. We construct several stable equivalences with the use of quasiequivalences which are introduced in Definition A.2.1.

(1) ⟹ (2): We let φ : M ⟶ M^{f} be a trivial cofibration in C such that M^{f} is fibrant. Since M ≅ R ∧_{R} M is the image of the cofibrant R-module R under a left Quillen functor, the object M, and hence also M^{f}, is cofibrant in C.

We claim that the R-End_{C}(M^{f})-bimodule Hom_{C}(M, M^{f}) is a quasi-equivalence with respect to φ, so that Lemma A.2.3 yields a chain of stable equivalences between R and the endomorphism ring spectrum of M^{f}. Indeed the left multiplication map ·φ : End_{C}(M^{f}) ⟶ Hom_{C}(M, M^{f}) is precomposition with the trivial cofibration φ, and is thus a stable equivalence of symmetric spectra since M^{f} is fibrant. The right multiplication map φ· : R ⟶ Hom_{C}(M, M^{f}) is adjoint to the equivalence φ : R ∧_{R} M ≅ M ⟶ M^{f}, hence is a stable equivalence of R-modules since R is cofibrant, M^{f} is fibrant and − ∧_{R} M is a left Quillen equivalence. Since M^{f} is the image of the free R-module of rank one under the derived equivalence − ∧_{R}^{L} M of homotopy categories, it is a compact generator for C and it satisfies condition (2).

(2) ⟹ (3): This is essentially a special case of Theorem 3.10.3 (iii). More precisely, that theorem constructs a spectral Quillen equivalence between C and the category of modules over the endomorphism ring spectrum of the generator M. Furthermore, restriction and extension of scalars are spectral Quillen equivalences for two stably equivalent ring spectra [HSS98, Thm. 5.4.5], see also [MMSS, Thm. 11.1], which establishes condition (3).

(3) ⟹ (2): This implication follows from the homotopy invariance of endomorphism ring spectra under spectral Quillen equivalences, see Corollary A.2.4. We choose a chain of spectral Quillen equivalences through spectral model categories. Then we choose a trivial cofibration R ⟶ R^{f} of R-modules such that R^{f} is fibrant; then R is stably equivalent to the endomorphism ring spectrum of R^{f}. We define an object M of C by iteratively applying the functors in the chain of Quillen equivalences, starting with R^{f}. In addition we take a fibrant or cofibrant replacement after each application depending on whether we use a left or right Quillen functor. By a repeated application of Corollary A.2.4 the endomorphism ring spectra of these objects, including the final one, M, are all stably equivalent to R. By construction the object M is isomorphic in the homotopy category of C to the image of the free R-module of rank one under the equivalence of homotopy categories induced by the Quillen equivalences. Hence M is also a compact generator for C and satisfies condition (2).

(2) ⟹ (1), provided R is cofibrant as a symmetric ring spectrum: Since M is fibrant and cofibrant the endomorphism ring spectrum End_{C}(M) is fibrant. Since R is cofibrant as a symmetric ring spectrum any isomorphism between R and End_{C}(M) in the homotopy category of symmetric ring spectra can be lifted to a stable equivalence η : R ⟶ End_{C}(M). In particular the stable equivalence η makes M into an R-C-bimodule. The functor X ↦ X ∧_{R} M is left adjoint to the functor Y ↦ Hom_{C}(M, Y) from C to the category of R-modules. To show that these adjoint functors form a Quillen equivalence we note that they factor as the composite of two adjoint functor pairs

C −∧_{End_{C} (M)}M ; u ; w ; Hom_{C}(M,−) mod-End_{C}(M) \begin{aligned}& η_{∗} \\ u \\ & & w \\ & η^{∗} \end{aligned} mod-R

(the left adjoints are on top). Since M is a cofibrant and fibrant compact generator for C, the left pair is a Quillen equivalence by Theorem 3.10.3 (iii). The other adjoint functor pair is restriction and extension of scalars along the stable equivalence of ring spectra η : R ⟶ End_{C}(M), which is a Quillen equivalence by [HSS98, Thm. 5.4.5] or [MMSS, Thm. 11.1].

For the rest of the proof we assume that C is the category of modules over a symmetric ring spectrum T which is cofibrant as a symmetric spectrum.

(1) ⟹ (4): We start by noting that since smashing with M over R is a left Quillen functor and since the free R-module of rank one is cofibrant, M ≅ R ∧_{R} M is cofibrant as a right T -module. We choose a fibrant replacement T ⟶ T^{f} of T in the category of T -bimodules and we let N be a cofibrant replacement of the T -R-bimodule Hom_{T} (M, T^{f}). The forgetful functor from T -R-bimodules to right R-modules preserves cofibrations since its right adjoint

Hom_{Sp^{Σ}}(T, −) : mod-R ⟶ T -mod-R

preserves trivial fibrations (because T is cofibrant as a symmetric spectrum). In particular the bimodule N is cofibrant as a right R-module. We will exhibit two chains of stable equivalences of bimodules

N ∧_{R} M ⟶^{∼} T^{f} ⟵^{∼} T and M ∧_{T} N ⟶^{∼} Hom_{T} (M, (M ∧_{T} T^{f})^{f}) ⟵^{∼} R

where M ∧_{T} T^{f} ⟶ (M ∧_{T} T^{f})^{f} is a fibrant approximation in the category of R-T -bimodules. This will establish condition (4).

Since − ∧_{R} M was assumed to be a left Quillen equivalence and the approximation map N ⟶ Hom_{T} (M, T^{f}) is a weak equivalence, so is its adjoint N ∧_{R} M ⟶ T^{f}; but this adjoint is even a map of T -bimodules. The equivalence T ⟶ T^{f} was chosen in the beginning. For the next equivalence we smash the T -bimodule equivalence N ∧_{R} M ⟶ T^{f} with the right-cofibrant bimodule M to get a stable equivalence of R-T -bimodules

M ∧_{T} N ∧_{R} M ⟶^{∼} M ∧_{T} T^{f} .

We then compose with the approximation map and obtain a stable equivalence of R-T -bimodules M ∧_{T} N ∧_{R} M ⟶ (M ∧_{T} T^{f})^{f}. Since M and N are cofibrant as right modules, the R-bimodule M ∧_{T} N is cofibrant as a right R-module. Since −∧_{R} M is a left Quillen equivalence, the adjoint M ∧_{T} N ⟶ Hom_{T} (M, (M ∧_{T} T^{f})^{f}) is thus a stable equivalence of R-bimodules. For the same reason the adjoint of the composite stable equivalence

R ∧_{R} M ≅ M ∧_{T} T ⟶^{∼} M ∧_{T} T^{f} ⟶^{∼} (M ∧_{T} T^{f})^{f}

gives the final stable equivalence of R-bimodules R ⟶ Hom_{T} (M, (M ∧_{T} T^{f})^{f}).

(4) ⟹ (1): Let M and N be bimodules which satisfy the conditions of (4). The functor X ↦ X ∧_{R} M is left adjoint to the functor Y ↦ Hom_{T} (M, Y) from the category of T -modules to the category of R-modules. Since M is cofibrant as a right T -module, this right adjoint preserves fibrations and trivial fibrations by the spectral axiom (SP). So − ∧_{R} M and Hom_{T} (M, −) form a Quillen functor pair. By condition (4) the left derived functor of − ∧_{R} M is an equivalence of derived categories (with inverse the left derived functor of smashing with the bimodule N). This implies, by [Hov99a, 1.3.13], that the left Quillen functor − ∧_{R} M is indeed a Quillen equivalence.

5. A generalized tilting theorem

In this section we state and prove a generalization of Rickard’s “Morita theory for derived categories”, [Ric89]. Rickard studies the question of when two rings are derived equivalent, i.e., when there exists a triangulated equivalence between various derived categories of the module categories. He shows [Ric89, Thm. 6.4] that a necessary and sufficient condition for such a derived equivalence is the existence of a tilting complex. A tilting complex for a pair of rings Γ and Λ is a bounded complex X of finitely generated projective Γ-modules which generates the derived category and whose graded ring of self extension groups [X, X]_{∗}^{D(Γ)} is isomorphic to Λ, concentrated in dimension zero.

We generalize the result in two directions. First, we allow the input to be a stable model category (which generalizes categories of chain complexes of modules). Second, we allow for a set of special generators, as opposed to a single tilting complex. The compact objects in the unbounded derived category of a ring are precisely the perfect complexes, i.e., those chain complexes which are quasi-isomorphic to a bounded complex of finitely generated projective modules [BN93, Prop. 6.4]. In our context we thus define a set of tiltors in a stable model category to be a set 𝕋 of compact generators such that for any two objects T, T′ ∈ 𝕋 the graded homomorphism group [T, T′]_{∗}^{Ho(C)} in the homotopy category is concentrated in dimension zero. Then Theorem 5.1.1 shows that the existence of a set of tiltors is necessary and sufficient for a stable model category to be Quillen equivalent or derived equivalent to the category of chain complexes over a ringoid (ring with several objects).

Of course an interesting special case is that of a stable model category with a single tiltor, i.e., a single compact generator whose graded endomorphism ring in the homotopy category is concentrated in dimension zero. Then ringoids simplify to rings, and the Eilenberg-Mac Lane spectral category becomes an Eilenberg-Mac Lane ring spectrum. In particular when C is the model category of chain complexes of Γ-modules for some ring Γ, then a single tiltor is the same (up to quasi-isomorphism) as a tilting complex, and the equivalence of conditions (2’) and (3) below recovers Rickard‘s ‘Morita theory for derived categories’ [Ric89, Thm. 6.4]. If one is only interested in in the equivalence of conditions (2), (2’) and (3) in the following tilting theorem (disregarding the question of Quillen equivalence), then the properness assumption on C can be dropped, see Section 3.8.

Theorem 5.1.1. (Tilting theorem) Let C be a simplicial, cofibrantly generated, proper, stable model category and \underline{A} a ringoid. Then the following conditions are equivalent:

(1) There is a chain of Quillen equivalences between C and the model category of chain complexes of \underline{A}-modules.

(1’) There is a chain of Quillen equivalences between C and the model category of module spectra over the Eilenberg-Mac Lane spectral category H\underline{A}.

(2) The homotopy category of C is triangulated equivalent to D(\underline{A}), the unbounded derived category of the ringoid \underline{A}.

(2’) C has a set of compact generators and the full subcategory of compact objects in Ho(C) is triangulated equivalent to K^{b}(proj-\underline{A}), the homotopy category of bounded chain complexes of finitely generated projective \underline{A}-modules.

(3) The model category C has a set of tiltors whose endomorphism ringoid in the homotopy category of C is isomorphic to \underline{A}.

Example 5.1.2. Let G be a finite group. As in Example 3.4 (i) the category S_{U}^{G} of G-equivariant spectra based on a universe U is a simplicial, stable model category, and the equivariant suspension spectra of the homogeneous spaces G/H^{+} form a set of compact generators as H runs through the subgroups of G. Rationalization is a smashing Bousfield localization so the rationalized suspension spectra form a set of compact generators of the rational G-equivariant stable homotopy category. In S_{U}^{G} the homotopy groups of the function spectra between the various generators are torsion in dimensions different from 0 [GM95, Prop. A.3], so the rationalized suspension spectra form a set of tiltors. Modules over the associated ringoid are nothing but rational Mackey functors, so the tilting theorem 5.1.1 shows that the rational G-equivariant stable homotopy category is equivalent to the derived category of rational Mackey functors. In turn, since these rational Mackey functors are all projective and injective the derived category is equivalent to the graded category. So this recovers the Theorem A.1 in [GM95].

Example 5.1.3. Let A be a ring and consider the pure projective model category structure – in the sense of Christensen [Chr99, Sec. 4] – on the category of chain complexes of A-modules (see also Example 2.3 (xiii)). A map X ⟶ Y of complexes is a weak equivalence if and only if for every finitely generated A-module M the induced map of mapping complexes Hom_{A}(M, X) ⟶ Hom_{A}(M, Y) is a quasi-isomorphism. Let G be a set of representatives of the isomorphism classes of indecomposable finitely generated A-modules. Then G forms a set of compact generators for the pure derived category D_{P}(A). Since furthermore every finitely generated module is pure projective, maps in the pure derived category between modules in G are concentrated in dimension 0. In other words, the indecomposable finitely generated A-modules form a set of tiltors. So Theorem 5.1.1 implies that the pure projective model category of A is Quillen equivalent to the modules over the ringoid given by the full subcategory of A-modules with objects G.

Remark 5.1.4. We want to emphasize one special feature of the tilting situation. For general stable model categories the notion of Quillen equivalence is considerably stronger than triangulated equivalence of homotopy categories (see Remark 3.2.1 for an example). Hence it is somewhat remarkable that for chain complexes of modules over ringoids the two notions are in fact equivalent. In general the homotopy category determines the homotopy groups of the spectral endomorphism category, but not its homotopy type. The real reason behind the equivalences of conditions (1) and (2) above is the fact that in contrast to arbitrary ring spectra or spectral categories, Eilenberg-Mac Lane objects are determined by their homotopy groups, see Proposition A.3.1.

Proof of Theorem 5.1.1. The equivalence of conditions (1) and (1’) follows from the fact that the model category of chain complexes over a ringoid and the category of modules over the Eilenberg-Mac Lane ring spectral category are Quillen equivalent (this is a special case of Theorem 6.2.3). Every Quillen equivalence of stable model categories induces an equivalence of triangulated homotopy categories, so condition (1) implies condition (2). Any triangulated equivalence restricts to an equivalence between the respective subcategories of compact objects. By the same argument as [BN93, Prop. 6.4] (which deals with the special case of complexes of modules over a ring), K^{b}(proj-\underline{A}) is equivalent to the full subcategory of compact objects in D(\underline{A}). Since the derived category of a ringoid has a set of compact generators, so does any equivalent triangulated category. Hence condition (2) implies condition (2’).

Now we assume condition (2’) and choose a triangulated equivalence K^{b}(proj-\underline{A}) ≅ Ho^{compact}(C). For a ∈ \underline{A} we let T_{a} be a representative in C of the image of the representable module F_{a} (see A.1), viewed as a complex concentrated in dimension 0. Since the collection of complexes F_{a}[0] is a set of tiltors for the derived category, the set \{T_{a}\}_{a∈\underline{A}} has all the properties of a set of tiltors, except that it may not generate the full homotopy category of C. However the localizing subcategory generated by the T_{a}’s coincides with the localizing subcategory generated by all compact objects since on the other side of the equivalence the complexes F_{a}[0] generate the category K^{b}(proj-\underline{A}). In general the compact objects might not generate all of Ho(C) (see [HS99, Cor. B.13] for some extreme cases where the zero object is the only compact object), but here this is part of the assumptions in (2’). So the T_{a}’s generate C, hence they are a set of tiltors, and so condition (3) holds.

If on the other hand C has a set of tiltors 𝕋, then 𝕋 is in particular a set of compact generators, so by Theorem 3.1.1, C is Quillen equivalent to the category of modules over the endomorphism category End(𝕋). In this special case the homotopy type of the spectral category End(𝕋) is determined by its homotopy groups: since the homotopy groups of End(𝕋) are concentrated in dimension 0, End(𝕋) is stably equivalent to the Eilenberg-Mac Lane spectral category of its component ringoid \underline{A} by Proposition A.3.1. Thus the categories of End(𝕋)-modules and H\underline{A}-modules are Quillen equivalent by Theorem A.1.2 and so condition (1’) holds.

6. Rational stable model categories

In this section we prove that rational stable model categories are algebraic, i.e., Quillen equivalent to the modules over a ‘differential graded algebra with many objects’ or a DG-category. There is a well-known special case saying that the stable homotopy category of rational spectra is equivalent to the category of graded ℚ-vector spaces. This equivalence comes from a Quillen equivalence between the rational model category of spectra and the model category of unbounded chain complexes of ℚ-vector spaces. We show here that a similar Quillen equivalence can be constructed whenever a stable model category has a set of compact generators and the morphism groups in the homotopy category are ℚ-vector spaces. In particular, this applies to rational equivariant spectra, see Remark 6.1.5 below.

In Section 6.2 below we construct the Eilenberg-Mac Lane functor H from chain complexes to symmetric spectra. We show that this functor induces the right adjoint of a Quillen equivalence between modules over a differential graded category, A, and modules over the associated Eilenberg-Mac Lane spectral category HA. This is also used in section 5. The proof of the algebraization of rational stable model categories is finished in Section 6.3.

6.1. Algebraization of rational stable model categories. We recall differential graded categories and their modules:

Definition 6.1.1. Let k be a commutative ring. A differential graded k-category, or simply DG-category if the ring k is understood, is a small category A enriched over chain complexes of k-modules. In other words: for every pair of objects a, b ∈ A there is a chain complex of k-modules Hom(a, b), for every object a ∈ A there is an identity element 1_{a} ∈ Hom(a, a)_{0}, and there are associative and unital composition maps of complexes

Hom(b, c) ⊗_{k} Hom(a, b) ⟶ Hom(a, c) .

By passing to the homology of the morphism complexes a DG-category A gives rise to a graded k-category H_{∗}A with the same set of objects, called the homology category. A module over a DG-category A is an enriched contravariant functor from A to the category of complexes of k-modules; we denote by mod-A the category of A-modules.

A more down-to-earth definition of DG-categories and their modules can be found in [Kel94, 1.2]. A DG-category with a single object corresponds to a differential graded k-algebra, and the modules in the sense of 6.1.1 coincide with the DG-modules in the ordinary sense.

We show in Theorem A.1.2 (i) that the module category over a DG-category supports a model category structure with weak equivalences the module maps which are objectwise quasiisomorphisms. Hence the associated homotopy category is the derived category of the DGcategory as discussed by Keller in [Kel94, 4.1].

Definition 6.1.2. A stable model category C is rational if for every pair of objects X and Y in C the abelian homomorphism group [X, Y]^{Ho(C)} in the homotopy category is uniquely divisible, i.e. a ℚ-vector space.

Theorem 6.1.3. Let C be a rational spectral model category with a set P of compact generators. Then there exists a differential graded ℚ-category A and a chain of Quillen equivalences between C and the category of A-modules. Furthermore the objects of A correspond to the objects in P and there is an isomorphism of graded ℚ-categories between the homology category H_{∗}A and the full graded subcategory of Ho(C) with objects P.

Suppose C is a rational stable model category which is also cofibrantly generated, simplicial and proper. By Theorem 3.9.2, C is then Quillen equivalent to the spectral model category Sp(C) of symmetric spectra over C. Combining this with the previous theorem gives the

Corollary 6.1.4 (Algebraization of rational stable model categories). Let C be a cofibrantly generated, simplicial, proper, stable model category. If C is rational and has a set of compact generators, then C is Quillen equivalent to the category of modules over a differential graded ℚ-category.

Remark 6.1.5. This corollary applies to the category of rational G-equivariant spectra for any compact Lie group G, see also Examples 2.3 (iii), 3.4 (i), and 5.1.2. The homotopy of the function spectra between generators is no longer concentrated in degree zero though for non-finite compact Lie groups G. Hence in this case it is more difficult to explicitly describe the associated differential graded ℚ-category. This algebraization is used in [GS] to show that the explicit algebraic model given in [Gr] is Quillen equivalent to the category of rational T^{n}-equivariant spectra for T^{n} the n-dimensional torus. In particular, this lifts the homotopy category equivalence for S^{1} produced in [Gr99].

Another application of this corollary is to the study of rational 𝔸^{1}-local stable homotopy theory of schemes, see Examples 2.3 (vii) and 3.4 (ii). Again, it is difficult to explicitly describe the associated differential graded ℚ-category.

The main new input for the proof of Theorem 6.1.3 is the fact that rational spectral categories are always algebraic, i.e., equivalent to Eilenberg-Mac Lane spectral categories. This fact is stated in Proposition 6.3.2 after the discussion of the Eilenberg-Mac Lane functor in Section 6.2.

6.2. Eilenberg-Mac Lane spectra and chain complexes. We now provide some general facts about the Eilenberg-Mac Lane spectral categories associated to differential graded categories. In particular we show that for every DG-category A, the model category of HA-modules is Quillen equivalent to the model category of differential graded A-modules (Theorem 6.2.3). This property is not unexpected, and variations have been proved for the special case of ring spectra in different frameworks. Indeed the Quillen equivalence of Theorem 6.2.3 is a generalization and strengthening of the fact first proved in [Rob87b] that the unbounded derived category of R-modules is equivalent to the homotopy category of HR-modules, see also [EKMM, IV Thm. 2.4] in the context of S-algebras.

Construction 6.2.1. We define an Eilenberg-Mac Lane functor H from the category of chain complexes of abelian groups to the category of symmetric spectra which is lax monoidal with respect to the tensor product of chain complexes and the smash product of symmetric spectra. Thus, when we apply the Eilenberg-Mac Lane functor pointwise to a differential graded category, it yields a spectral category. Furthermore for every DG-category A the Eilenberg-Mac Lane functor turns out to be the right adjoint of a Quillen equivalence between the category of differential graded A-modules and the category of HA-module spectra.

For any pointed simplicial set K we denote by NK the normalized chain complex of the reduced free simplicial abelian group generated by K. So NK is a non-negative dimensional chain complex which in dimension n is isomorphic to the free abelian group on the non-degenerate non-basepoint n-simplices of K. A functor W from the category of chain complexes Ch_{ℤ} to the category of simplicial abelian groups is defined by

(W C)_{k} = hom_{Ch_{ℤ}}(N(Δ[k]_{+}), C) .

For non-negative dimensional complexes, W is just the inverse to the normalized chain functor in the Dold-Kan equivalence between simplicial abelian groups and non-negative dimensional chain complexes [Do58, 1.9]. For an arbitrary complex C there is a natural chain map NW C ⟶ C which is an isomorphism in positive dimensions and which expresses NW C as the (−1)-connected cover of C.

For a complex C we define the symmetric Eilenberg-Mac Lane spectrum HC by the formula

(HC)_{n} = W (C[n])

where C[n] denotes the n-fold shift suspension of the complex C. We let the symmetric group Σ_{n} act on the simplicial abelian group (HC)_{n} by the sign representation. To define the symmetric structure maps we use the Alexander-Whitney map, see [EM54, 2.9] or [May67, 29.7]. This map is a natural, associative and unital transformation (but not an isomorphism) of simplicial abelian groups

AW : W (C) ⊗ W (D) ⟶ W (C ⊗ D) .

Here the left tensor product is the dimensionwise tensor product of simplicial abelian groups, whereas the right one is the tensor product of chain complexes. By our conventions the psphere S^{p} is the p-fold smash product of the simplicial circle S^{1} = Δ[1]/∂Δ[1]. So the reduced free abelian group generated by S^{p} is the p-fold tensor product of the simplicial abelian group ℤ̃[S^{1}] = W (ℤ[1]) (where ℤ[1] is the chain complex which contains a single copy of the group ℤ in dimension 1). So for every chain complex D the Alexander-Whitney map gives a map

ℤ̃[S^{p}] ⊗ W D ≅ \underbrace{W (ℤ[1]) ⊗ ⋯ ⊗ W (ℤ[1])}_{p} ⊗W D ⟶^{AW} W (\underbrace{ℤ[1] ⊗ ⋯ ⊗ ℤ[1]}_{p} ⊗D) ≅ W (D[p]) .

This map is Σ_{p}-equivariant for the permutation action on the source and the sign action on the target. In particular for D = C[q] we get a Σ_{p} × Σ_{q}-equivariant map

ℤ̃[S^{p}] ⊗ (HC)_{q} = ℤ̃[S^{p}] ⊗ W (C[q]) ⟶ W (C[p + q]) = (HC)_{p+q} .

This map makes HC into a symmetric spectrum of simplicial abelian groups, hence by neglect of structure into a symmetric spectrum. The symmetric spectrum HC is always an Ω-spectrum and the stable homotopy groups of HC are naturally isomorphic to the homology groups of the chain complex C.

Now we claim that the Eilenberg-Mac Lane functor H is lax monoidal, i.e., that there are natural associative and unital maps of symmetric spectra

S ⟶ Hℤ  and  HC ∧ HD ⟶H(C ⊗ D)

(for C and D chain complexes). The unit map is given at level n by the composite

S^{n} ⟶^{generators} ℤ̃[S^{n}] ⟶^{AW} W (ℤ[n]) .

The multiplication map is induced by the universal property of the smash product by the maps

(HC)_{n} ∧ (HD)_{m} ⟶ W (C[n]) ⊗ W (D[m]) ⟶^{AW} W (C[n] ⊗ D[m]) ≅ (H(C ⊗ D))_{n+m} .

Remark 6.2.2. For an abelian group A considered as a complex concentrated in dimension 0, the symmetric spectrum HA is different from, but level equivalent to the model given in [HSS98, 1.2.5]. The simplicial abelian group (HA)_{n} = W (A[n]) is the minimal model of an Eilenberg-Mac Lane space of type K(A, n) and it is determined (up to preferred isomorphism) by the property that its normalized chain complex consists only of one copy of A in dimension n. The Alexander-Whitney map takes the form of a weak equivalence of simplicial abelian groups Z̃[S^{n}] ⊗ A ⟶ W (A[n]), equivariant with respect to the permutation and the sign action of Σ_{n} respectively. So these Alexander-Whitney maps assemble into a level equivalence of symmetric spectra from the Eilenberg-Mac Lane model of [HSS98, 1.2.5] to HA.

The Eilenberg-Mac Lane functor H is lax monoidal, so applying it pointwise to a differential graded category yields a spectral category with the same set of objects. Similarly for a fixed DG-category A, the functor H induces a functor from the category of A-modules to the category of HA-modules.

Theorem 6.2.3. Let A be a differential graded category. Then the Eilenberg-Mac Lane functor H is the right adjoint of a Quillen equivalence between the model category of A-modules and the model category of modules over the Eilenberg-Mac Lane spectral category HA.

Proof. The Eilenberg-Mac Lane functor commutes with limits, so by Lemma A.1.5 it has a left adjoint λ : mod-HA ⟶ mod-A. The Eilenberg-Mac Lane functor takes values in the category of Ω-spectra and it takes fibrations of chain complexes (i.e., epimorphisms) to level fibrations. So H takes objectwise quasi-isomorphisms of A-modules to objectwise level equivalences of HAmodules; similarly H takes objectwise fibrations to objectwise level fibrations. Thus H and λ form a Quillen adjoint functor pair. Moreover, the right adjoint H reflects weak equivalences, so it suffices to show [Hov99a, 1.3.16] that the unit of the derived adjunction η_{M} : M ⟶ HλM on the level of homotopy categories is an isomorphism for every HA-module M.

We first verify this for the representable HA-modules F_{a}^{HA}. We claim that for every object a ∈ A the A-module λF_{a}^{HA} is isomorphic (after passage to the homotopy category) to the representable module F_{a}^{A}. Indeed, by the adjunction isomorphisms the representable functor [λF_{a}^{HA}, −]^{Ho(mod-A)} is isomorphic to taking the 0-th homology of an A-module at the object a, which is in turn naturally isomorphic to [F_{a}^{A}, −]^{Ho(mod-A)}. Hence in the diagram of HA-modules

F_{a}^{HA} ⟶^{η} HλF_{a}^{HA} ⟶^{∼} HF_{a}^{A}

the second map is a stable equivalence. Since the composite map is an isomorphism (even before passage to the homotopy category), the unit of the adjunction is indeed an isomorphism for every representable module.

Since the composite derived functor Hλ is exact and commutes with coproducts (since HF_{a}^{A} is isomorphic to F_{a}^{HA}, and thus a compact object in the homotopy category of HA-modules), the full subcategory of those HA-modules for which the unit is an isomorphism is a localizing subcategory. Since it also contains the generating representable modules, it coincides with the full homotopy category of HA-modules.

6.3. Proof of the algebraization theorem. There are two different kinds of adjoints to the Eilenberg-Mac Lane functor H, depending on whether we apply it to categories or to graphs (see Section A.1 for a discussion of how enriched categories can be viewed as monoids in the category of enriched graphs). If I is any set, we can apply H pointwise to I-graphs of chain complexes and view it as a functor

H : Ch_{k}-Gr_{I} ⟶ Sp^{Σ}-Gr_{I} .

Then H preserves limits, so it has a left adjoint (Lemma A.1.5) which we denote λ_{k}.

The Eilenberg-Mac Lane functor H is lax monoidal, so pointwise application to a differential graded category yields a spectral category with the same set of objects. The restriction of H to the category of k-linear DG-categories

H : Ch_{k}-Cat_{I} ⟶ Sp^{Σ}-Cat_{I}

also has a left adjoint which we denote L_{k}.

If O is an I-indexed spectral category, then there is an adjunction unit map of spectral categories

O ⟶ HL_{k}O

If we forget the multiplication and view this as a map of I-graphs, we can form the adjoint

λ_{k}O ⟶ L_{k}O

Since the left adjoint λ_{k} is not monoidal, this map of I-graphs of complexes is not an isomorphism. The map is not even a map of DG-categories since the left hand side is not a DG-category; the functor L_{k} takes the composition in the spectral category O into account, but the functor λ_{k} only depends on the underlying I-graph of O.

However the map is a pointwise equivalence for cofibrant objects as defined in section A.1.

Theorem 6.3.1. For every cofibrant I-indexed spectral category O the map

λ_{k}O ⟶ L_{k}O

is a pointwise quasi-isomorphism of I-graphs in Ch_{k}.

The proof of this statement appears at the end of this section. This theorem is used to prove the following properties of the rational adjunctions. We call a symmetric spectrum rational if the homotopy groups of some (hence any) stably fibrant replacement are uniquely divisible. A graph of spectra (see Definition A.1.3) or a spectral category is called rational if it is pointwise rational.

Proposition 6.3.2. (i) If X is a cofibrant rational symmetric spectrum, then the adjunction

unit X ⟶ Hλ_{ℚ}X is a stable equivalence. (ii) If I is a set and Y is a cofibrant rational I-indexed graph of symmetric spectra, then the

adjunction unit Y ⟶ Hλ_{ℚ}Y is a pointwise stable equivalence. (iii) If I is a set and O a cofibrant rational I-indexed spectral category, then the adjunction unit

O ⟶ HL_{ℚ}O is a pointwise stable equivalence.

With these statements in hand, the algebraization of rational stable model categories follows.

Proof of Theorem 6.1.3. By Theorem 3.10.3, C is Quillen equivalent to the category of modules over the spectral endomorphism category of the generators, E(P). Because of the isomorphism

π_{∗} E(P)(P, P′) ≅ [P, P′]_{∗}^{Ho(C)}

the homomorphism spectra E(P)(P, P′) have uniquely divisible homotopy groups, and so E(P) is rational. We let E(P)^{c} be a cofibrant replacement of E(P) given by Theorem A.1.4 (iv). By Proposition 6.3.2 (iii), E(P)^{c} is stably equivalent to the Eilenberg-Mac Lane spectral category HL_{ℚ}E(P)^{c}. So if we set A = L_{ℚ}E(P)^{c}, then extension and restriction of scalars provide a chain of Quillen equivalences between the category of E(P)-modules and the category of HA-modules by Theorem A.1.2 (ii). The hypotheses of Theorem A.1.2 (ii) hold over symmetric spectra by arguments analogous to [MMSS, 12.7]. Theorem 6.2.3 provides another Quillen equivalence between the category of HA-modules and the category of DG-modules over A.

Proof of 6.3.2. (i) By the proof of 6.2.3, H is a right Quillen functor, hence λ_{ℚ} is a left Quillen functor and it preserves weak equivalences between cofibrant objects. The functor H preserves all weak equivalences, so the composite functor Hλ_{ℚ} has a total left derived functor, which is an exact endofunctor of the homotopy category of symmetric spectra. Furthermore the adjunction unit descends to a natural transformation from the identity functor to the total left derived functor of Hλ_{ℚ}.

It suffices to show that for every object in the homotopy category of symmetric spectra which has rational homotopy groups, this map is an isomorphism. Since the derived functor of Hλ_{ℚ} is exact and commutes with infinite coproducts (it takes homology groups of complexes to homotopy groups of spectra), the full subcategory of the homotopy category of spectra for which the map is an isomorphism is a localizing subcategory. Since the rational Eilenberg-Mac Lane spectrum generates the category of rational spectra and Theorem 6.2.3 shows that this localizing subcategory contains (a cofibrant replacment of) Hℚ, the statement follows.

(ii) The functors H and λ_{ℚ} for graphs are obtained from the corresponding functors for symmetric spectra by pointwise applications. Furthermore a graph is cofibrant if and only if it is pointwise cofibrant as a symmetric spectrum, hence the claim follows from part (i).

(iii) It suffices to show that the adjunction unit is a pointwise equivalence of underlying I-graphs of symmetric spectra. In the category Sp^{Σ}-Gr_{I}, the map factors as a composite

O ⟶ Hλ_{ℚ}O ⟶ HL_{ℚ}O .

Cofibrant spectral categories are also cofibrant as I-graphs (Theorem A.1.4), so the first map is a pointwise equivalence by part (ii).

The second map Hλ_{ℚ}O ⟶ HL_{ℚ}O is a pointwise stable equivalence by Theorem 6.3.1 since H takes quasi-isomorphisms to stable equivalences.

The proof of Theorem 6.3.1 occupies the rest of this section. We also denote by λ_{k} : Sp^{Σ} ⟶ Ch_{k} the left adjoint of the Eilenberg-Mac Lane functor H restricted to the category of complexes of k-modules (this is the special case where I is a one element set). This should not cause problems since λ_{k} as defined above is just pointwise application of the second λ_{k}. The functor λ_{k} is not monoidal so λ_{k} does not induce a functor on the categories of monoids. If λ_{k} were monoidal, then the map of Theorem 6.3.1 would be an isomorphism. Nevertheless, λ_{k} is ‘lax comonoidal’ and the comparison map is a quasi-isomorphism in good cases. In more detail: in Construction 6.2.1 we showed that the functor H is lax monoidal, i.e., we produced associative and unital maps of symmetric spectra

S ⟶ Hk  and  HC ∧ HD ⟶ H(C ⊗_{k} D) .

Taking adjoints we obtain maps

λ_{k}S ⟶ k  and  λ_{k}(HC ∧ HD) ⟶ C ⊗_{k} D .

The left map is actually an isomorphism. Specializing the second map to C = λ_{k}A and D = λ_{k}B for symmetric spectra A and B and composing with unit of the adjunction we get a map

λ_{k}(A ∧ B) ⟶ λ_{k}(Hλ_{k}A ∧ Hλ_{k}B) ⟶ λ_{k}A ⊗_{k} λ_{k}B .

Lemma 6.3.3. If A and B are cofibrant symmetric spectra, then the map

λ_{k}(A ∧ B) ⟶ λ_{k}A ⊗_{k} λ_{k}B

is a quasi-isomorphism.

Proof. When A is the sphere spectrum the map in question is an isomorphism. If we fix a cofibrant symmetric spectrum B, then both sides of the map are left Quillen functors in A. Hence both sides have total left derived functors from the stable homotopy category to the derived category of k which are exact functors and commute with infinite wedges. So the full subcategory of the stable homotopy category of spectra for which the derived map is an equivalence is a localizing subcategory. Since it also contains the sphere spectrum, it contains all spectra.

Proof of Theorem 6.3.1. We verify the statement by a cell induction in the category of I-indexed spectral categories. The theorem is true for the initial spectral category in Sp^{Σ}-Cat_{I}. Suppose the claim has already been verified for some cofibrant I-indexed category X, and suppose P is obtained from X as a pushout, along a free extension

T (A)  T (B) X  P w u  u w

where A ⟶ B is a pointwise cofibration of I-graphs over Sp^{Σ} and T is the free monoid functor from Sp^{Σ}-Gr_{I} to Sp^{Σ}-Cat_{I}

Such pushouts are analyzed in the proof of [SS00, Lem. 6.2], and we make use of this description. Note that the part of the analysis which we use here does not depend on the assumption in [SS00] that the monoidal product is symmetric. The underlying I-graph of the spectral category P can be written as a colimit of a sequence of cofibrations of I-graphs P_{n−1} ⟶ P_{n} each of which is a pushout, in the category of I-graphs, of a cofibration of the form Q_{n} ⟶ (X∧B)^{∧n}∧X.

Since L_{k} is a left adjoint we also have a pushout diagram of I-indexed DG-categories

L_{k}T (A)  L_{k}T (B) L_{k}X  L_{k}P w u  u w

Moreover L_{k}T (A) is T λ_{k}(A) the free I-indexed DG-category on the I-graph of k-chain complexes λ_{k}(A). Hence, applying [SS00, Lem. 6.2] to this pushout out, L_{k}P is the colimit, in the category Ch_{k}-Gr_{I}, of a sequence of cofibrations R_{n−1} ⟶ R_{n}. The map from λ_{k}P to L_{k}P takes the n-th filtration λ_{k}P_{n} of the source to R_{n}; we now show by induction that for all n the map λ_{k}P_{n} ⟶ R_{n} is a pointwise quasi-isomorphism in Ch_{k}-Gr_{I}.

We are led to consider a commutative diagram

λ_{k}P_{n−1}  λ_{k}P_{n}  λ_{k} [(X ∧ B/A)^{∧n} ∧ X] R_{n−1}  R_{n}  (L_{k}(X) ⊗ λ_{k}(B/A))^{⊗n} ⊗ (L_{k}X) u  w u  w  u w  w

in which both rows are pointwise short exact sequences of chain complexes. By induction the left vertical map is a pointwise quasi-isomorphism. The right vertical map factors as a composite

λ_{k} [(X ∧ B/A)^{∧n} ∧ X] ⟶ (λ_{k}(X) ⊗ λ_{k}(B/A))^{⊗n} ⊗ (λ_{k}X) ⟶ (L_{k}(X) ⊗ λ_{k}(B/A))^{⊗n} ⊗ (L_{k}X) .

Since X is also pointwise cofibrant as an I-graph of symmetric spectra, by Theorem A.1.4, Lemma 6.3.3 shows that the left map is a pointwise quasi-isomorphism. We assumed that λ_{k}X ⟶ L_{k}X is also a pointwise quasi-isomorphism. So since all objects in sight are pointwise cofibrant, the right map and hence the composite are pointwise quasi-isomorphisms. Thus, the map λ_{k}P_{n} ⟶ R_{n} is a pointwise quasi-isomorphism.

Since homology of chain complexes commutes with filtered colimits we conclude that the map λ_{k}P ⟶ L_{k}P is a pointwise quasi-isomorphism. Hence this map is a pointwise quasiisomorphism for all cofibrant spectral categories which can be built from the initial one by finitely many pushouts of the form considered. This also holds for (possibly transfinite) composites and retracts of such spectral categories. Since all cofibrant spectral categories can be obtained this way by Theorem A.1.4, this finishes the proof of the theorem.

Appendix A. Some enriched model category theory

In this appendix we develop some general theory of modules over ‘rings with several objects’ based on a monoidal model category in the sense of [SS00]. In this paper we are really interested in two special cases, namely modules over a spectral category and modules over a differential graded category. However the arguments work in perfect generality. Most of this is a relatively straightforward generalization from the case of ‘ring objects’ (which in the special cases are symmetric ring spectra or differential graded algebras) to ‘rings with several objects’.

A.1. Model category structures for enriched modules. Throughout this section, V is a monoidal model category satisfying the monoid axiom in the sense of [SS00], see Definition A.1.1. The two cases of particular interest are when V is

• the category of symmetric spectra under smash product and with the stable model structure of [HSS98]; the general theory then specializes to modules over spectral categories in the sense of Definition 3.3.1;

• the category of ℤ-graded chain complexes of abelian groups under tensor product of complexes with the projective model structure of [Hov99a]; the general theory then specializes to DG-modules over a differential graded category (compare Definition 6.1.1 or [Kel94, 1.2]).

In Section 5 of [SS00] we list further examples of monoidal model categories to which the general theory applies.

If V is a category with a monoidal product ∧ and I is a class of maps in V, we denote by I ∧ V the class of maps of the form

A ∧ Z ⟶ B ∧ Z

for A ⟶ B a map in I and Z an object of V.

Definition A.1.1. A model category V is a monoidal model category if it is endowed with a closed symmetric monoidal structure and satisfies the following pushout product axiom. We denote the symmetric monoidal product by ∧ and the unit object by 𝕀.

Pushout product axiom. Let A ⟶ B and K ⟶ L be cofibrations in V. Then the map

A ∧ L ∪_{A∧K} B ∧ K ⟶ B ∧ L

is also a cofibration. If in addition one of the former maps is a weak equivalence, so is the latter map.

If K is a class of morphisms in a category, then ‘K-cell’ denotes the class of relative K-cell complexes [Hov99a, 2.1.9] (called regular K-cofibrations in [SS00]), that is the maps which can be obtained from maps in K by pushouts and (possibly transfinite) composition. A monoidal model category V satisfies the monoid axiom if every map in

({acyc. cofibrations} ∧ V)-cell

is a weak equivalence.

Let O be a small V-category [Kly82, 1.2] (a category enriched over V). One may think of O as a ‘ring/monoid with many objects’, enriched in V. Indeed if O has one object ∗, then it is completely determined by the endomorphism monoid O(∗, ∗), which is a monoid in V. Moreover the O-modules as defined below coincide with the O(∗, ∗)-modules in the ordinary sense. If V is the category of symmetric spectra, then a V-category is a spectral category as defined in Definition 3.3.1; if V is the category of chain complexes of abelian groups, then a V-category is the same as a differential graded category as defined in Definition 6.1.1 or [Kel94, 1.2].

An O-module is a contravariant V-functor from O to the category V. A map of O-modules is a V-natural transformation. A map of O-modules f: X ⟶ Y is an objectwise equivalence (resp. objectwise fibration) if f(o): X(o) ⟶ Y (o) is a weak equivalence (resp. fibration) in V for each object o in O. For every object o ∈ O, there is a free or representable O-module which we denote F_{o}^{O} and define by F_{o}^{O}(o′) = O(o′, o). As the name suggests, homomorphisms from F_{o}^{O} into a module M are in bijective correspondence with elements of M(o) by the enriched Yoneda Lemma [Kly82]. The evaluation functor at o, Ev_{o}: mod-O ⟶ V has a left adjoint ‘free’ functor which sends an object X of V to the module F_{o}^{O}X = X ∧ F_{o}^{O}, where X ∧ − is defined by smashing objectwise with X.

If R and O are V-categories, their smash product R∧O is the V-category whose set of objects is the cartesian product of the objects of R and O and whose morphism objects are defined by the rule

R ∧ O((r, o), (r′, o′)) = R(r, r′) ∧ O(o, o′) .

An R-O-bimodule is by definition an R^{op} ∧ O-module. Since modules for us are always contravariant functors, an R-O-bimodule translates to a covariant V-functor from O^{op} ∧ R to V.

Unless R has a single object, there is no preferred forgetful functor from R-O-bimodules to O-modules. In contrast, for every object r ∈ R and every R-O-bimodule M we get an O-module by restricting the action along the map

O^{op} ≅ O^{op} ∧ 𝕀 ⟶^{id∧r} O^{op} ∧ R .

Here 𝕀 is the V-category on the same objects as R with 𝕀(r, r) = 𝕀 and 𝕀(r, r′) = ∅, the initial object, for r ≠ r′. We denote this O-module by M(−, r). Similarly restriction to an object o ∈ O yields a left R-module which we denote M(o, −).

A morphism Ψ : O ⟶ R of V-categories is simply a V-functor. The restriction of scalars

Ψ^{∗} : mod-R ⟶ mod-O ; M ⟼ M ∘ Ψ

has a left adjoint functor Ψ_{∗}, also denoted − ∧_{O} R, which we refer to as extension of scalars. As usual it is given by an enriched coend, i.e., for an O-module N the R-module Ψ_{∗}N = N ∧_{O} R is given by the coequalizer of the two R-module homomorphisms

⋁_{o,p∈O} N(p) ∧ O(o, p) ∧ F_{Ψ(o)}^{R} \overrightarrow{⟶} ⋁_{o∈O} N(o) ∧ F_{Ψ(o)}^{R} .

We call Ψ : O ⟶ R a weak equivalence of V-categories if it is a bijection on objects and if for all objects o, o′ in O the map

Ψ_{o,o′} : O(o, o′) ⟶ R(Ψ(o), Ψ(o′))

is a weak equivalence in V. Next we establish the model category structure for O-modules and show its invariance under restriction of scalars along a weak equivalence of V-categories.

Theorem A.1.2. Let V be a cofibrantly generated monoidal model category which satisfies the monoid axiom and such that every object of V is small with respect to the whole category.

(i) Let O be a small V-category. Then the category of O-modules with the objectwise equivalences, objectwise fibrations, and cofibrations is a cofibrantly generated model category.

(ii) Assume Ψ : O ⟶ R is a weak equivalence of V-categories. Suppose that for every cofibrant right O-module N, the induced map N ∧_{O} O ⟶ N ∧_{O} R is an objectwise weak equivalence. Then restriction and extension of scalars along Ψ form a V-Quillen equivalence of the module categories.

Proof. We use [SS00, 2.3] to establish the model category for O-modules. The unit morphism 𝕀 ⟶ O induces adjoint functors of restriction and extension of scalors between the module categories. This produces a triple − ∧_{𝕀} O on 𝕀-modules with the algebras over this triple the O-modules. An 𝕀-module is just a diagram in V indexed on the objects of O. Hence it has a cofibrantly generated model category inherited from V in which the cofibrations, fibrations and weak equivalences are objectwise. Here the generating trivial cofibrations are maps between modules concentrated at one object, i.e. of the form A_{o} with A_{o}(o) = A and A_{o}(o′) = ∅ if o ≠ o′. Then the generating trivial cofibrations for O-modules are maps between modules of the form A_{o} ∧_{𝕀} O = A ∧ O(−, o). Hence the monoid axiom applies to show that the new generating trivial cofibrations and their relative cell morphisms are weak equivalences. Thus, since all objects in V are assumed to be small, the model category structure follows by criterion (1) of [SS00, 2.3]. Note, we used [SS00, 2.3] instead of [SS00, 4.1] because the V-categories have underlying graphs, see Definition A.1.3 below, indexed by the cartesian product of the objects with themselves whereas the diagrams underlying the modules are only indexed by the objects.

The proof of (ii) follows as in [SS00, 4.3]. The restriction functor Ψ^{∗} preserves objectwise fibrations and objectwise equivalences, so restriction and extension of scalars form a Quillen adjoint pair. By assumption, for N a cofibrant right O-module

N ≅ N ∧_{O} O ⟶ N ∧_{O} R

is a weak equivalence. Thus if M is a fibrant right R-module, an O-module map N ⟶ Ψ^{∗}M is a weak equivalence if and only if the adjoint R-module map Ψ_{∗}N = N ∧_{O} R ⟶ M is a weak equivalence.

In [Mac, II.7] it is explained how a small category with object set I can be viewed as a monoid in the category of I-indexed graphs. We need an enriched version of this giving V-categories as the monoids with respect to a monoidal product on the category of V-graphs, so that we can apply some of the general theory of [SS00]. Note that here the product on V-graphs is not symmetric monoidal, so we must take care in applying [SS00].

Definition A.1.3. Let V be a symmetric monoidal closed category and I a set. Denote the unit object of V by 𝕀 and suppose V has an initial object ∅. The category of I-graphs in V, denoted V-Gr_{I} is simply the product category of copies of V indexed by the set I × I. If G and H are two I-graphs, then their smash product is defined by the formula

(G ∧ H)(i, j) = ⋁_{k∈I} G(k, j) ∧ H(i, k) .

The I-graph 𝕀_{I} is defined by

𝕀_{I}(i, j) = \{ \begin{matrix}𝕀 & if i = j \\ ∅ & if i ≠ j \end{matrix}

If G is an I-graph in V, then one can think of G(i, j) as the V-object of arrows pointing from the vertex i to the vertex j. The smash product makes the category of I-graphs in V into a monoidal category with unit object 𝕀_{I}. If V is a model category, then the category V-Gr_{I} inherits a model structure in which cofibrations, fibrations and weak equivalences are defined pointwise, i.e., for each (i, j). Furthermore if the original category V satisfies the pushout product axiom (Definition A.1.1), then so does the smash product of I-graphs (since the coproduct of any number of (trivial) cofibrations is another (trivial) cofibration). If the monoid axiom (Definition A.1.1) holds for V then it also holds for I-graphs, by [Hov99a, 2.1.13] which states that coproducts of relative cofibrations are relative cofibrations. We denote by V-Cat_{I} the category of all V-categories whose set of objects is equal to I and with morphisms the V-functors which are the identity on objects. This is precisely the category of monoids in V-Gr_{I} with respect to the smash product of I-graphs in V.

Since V-Gr_{I} is not a symmetric monoidal category, we cannot apply [SS00] to produce a model category on the monoids, V-Cat_{I}. There is still a useful notion of cofibration and of cofibrant replacement though. A cofibration in V-Cat_{I} is a map which has the left lifting property with respect to any pointwise trivial fibration.

Let V be a cofibrantly generated monoidal model category. Then V-Gr_{I} is also a cofibrantly generated model category. The generating cofibrations are of the form A_{i,j} ⟶ B_{i,j} where A ⟶ B is a generating cofibration in V and A_{i,j} is the V-Gr_{I} with value A concentrated at (i, j). Define a set K_{T} to be the image of the free monoid functor T : V-Gr_{I} ⟶ V-Cat_{I} on the set of generating cofibrations in V-Gr_{I}. A cofibrant replacement O^{c} of an object O in V-Cat_{I} has a factorization of the unit map of O as 𝕀 ⟶ O^{c} ⟶ O, a cofibration followed by a pointwise trivial fibration.

Theorem A.1.4. Let V be a cofibrantly generated monoidal model category such that every object of V is small with respect to the whole category.

(i) The pointwise trivial fibrations in V-Cat_{I} are the K_{T} -injectives. (ii) Any cofibration in V-Cat_{I} is a retract of a map in K_{T} -cell. (iii) There is a functorial factorization of any map f in V-Cat_{I} as f = qi with q in K_{T} -injectives

and i in K_{T} -cell. (iv) Any object in V-Cat_{I} has a cofibrant replacement.

(v) If the unit object 𝕀 is cofibrant in V then every V-category which is cofibrant in V-Cat_{I} is also pointwise cofibrant in V-Gr_{I}.

Proof. By adjointness and the fact that K_{T} is the image of the generating cofibrations in V-Gr_{I} a K_{T} -injective is an underlying trivial fibration in V-Gr_{I}. The cofibrations in V-Cat_{I} are the K_{T} -cofibrations, hence by [SS00, 2.1] (ii) and (iii) hold. The factorization of the unit produces a cofibrant repalcement for any object. The last item follows from the analysis of the pushout in V-Cat_{I} in the last paragraph of [SS00]. This analysis does not require a symmetric monoidal product.

The following lemma constructs the left adjoints to the Eilenberg-Mac Lane functor H from the category of chain complexes to the category of symmetric spectra constructed in 6.2.1. Assuming that the left adjoints existed, we showed that H is the right adjoint in several Quillen equivalences of model categories without needing to know the explicit form of the left adjoints. So we use Freyd’s Special Adjoint Functor Theorem to construct the left adjoints. For us a ‘category’ by definition has small hom-sets. A category is well-powered if every object has only a set of subobjects (i.e., equivalence classes of monomorphisms into the object). An object q is called a cogenerator of the category if for every pair of distinct morphisms h, h′ : a ⟶ b there exists a morphism : b ⟶ q such that fh ≠ fh′. One form of the Special Adjoint Functor Theorem (see [Mac, V.8 Corollary]) says that a functor from a complete, well-powered category with a cogenerator has a left adjoint if and only if it commutes with limits. We want to apply this criterion to categories made of chain complexes, so we verify the hypothesis in the following lemma:

Lemma A.1.5. Let A a differential graded category. Then the category of A-modules is complete, well-powered and has a cogenerator. Hence every limit-preserving functor out of the category of A-modules has a left adjoint.

Proof. Module categories have limits since these are created objectwise on underlying chain complexes. The monomorphisms are the injective maps, hence every subobject is represented by the inclusion of a subobject in the classical sense, of which there are only a set’s worth. So module categories are well-powered.

For every integer n and object a of A the contravariant functor from mod−A to the category of abelian groups which sends M to Hom_{ℤ}(M(a)_{n}, ℚ/ℤ) is co-representable. In other words, there exists an A-module I(a, n) and a natural isomorphism

Hom_{mod-A}(M, I(a, n)) ≅ Hom_{ℤ}(M(a)_{n}, ℚ/ℤ) .

Suppose h, h′ : M ⟶ N are two distinct morphisms of A-modules. Then h and h′ differ at some object a ∈ A in some dimension n. Since ℚ/ℤ is a cogenerator for the category of abelian groups, there exists a homomorphism N(a)_{n} ⟶ ℚ/ℤ which separates the components of h and h′. Hence there is an adjoint map of A-modules f : N ⟶ I(a, n) satisfying fh ≠ fh′. So the product of the modules I(a, n), for a ∈ Ob A and n ∈ ℤ is a cogenerator for the category of A-modules.

A.2. Quasi-equivalences. This section introduces quasi-equivalences which are a bookkeeping device for producing stable equivalences between symmetric ring spectra or spectral categories, see Lemma A.2.3 below. The name is taken from [Kel94, Summary, p. 64], where this notion is discussed in the context of differential graded algebras. Every (stable) equivalence of ring spectra gives rise to a quasi-equivalence; conversely the proof of Lemma A.2.3 shows that a single quasi-equivalence encodes a zig-zag of four stable equivalences relating two ring spectra or spectral categories. One place where quasi-equivalences arise ‘in nature’ is the proof that weakly equivalent objects in a model category have weakly equivalent endomorphism monoids, see Corollary A.2.4. The arguments involved are entirely based on enriched model category theory, and they work in any monoidal model category in the sense of Definition A.1.1, hence we present them in this generality.

Definition A.2.1. Let V be a monoidal model category and let R and O be two V-categories with the same set I of objects. Then a quasi-equivalence is an R-O-bimodule M together with a collection of ‘elements’ φ_{i} ∈ M(i, i) (i.e., V-morphisms 𝕀 ⟶ M(i, i)) for all i ∈ I such that the following holds: for all pairs i and j of objects the right multiplication with φ_{i} and the left multiplication with φ_{j},

R(i, j) ⟶^{·φ_{i}} M(i, j) ⟵^{φ_{j}·} O(i, j)

are weak equivalences in V.

Remark A.2.2. In the important special case of spectral categories with a single object, i.e., for two symmetric ring spectra R and T, a quasi-equivalence is an R-T -bimodule M together with an element φ ∈ M (i.e., a vertex of the 0-th level of M or equivalently a map S^{0} ⟶ M of symmetric spectra) such that the left and right multiplication maps with φ,

R ⟶^{·φ} M ⟵^{φ·} T

are stable equivalences of symmetric spectra.

If Ψ : O ⟶ R is a weak equivalence of V-categories, then the target R becomes an R-O-bimodule if O acts on the right via Ψ. Furthermore the identity elements in R(i, i) for all objects i of R make R into a quasi-equivalence. The following lemma shows that conversely that quasi-equivalent V-categories are related by a chain of weak equivalences:

Lemma A.2.3. Let R and O be two small V-categories with the same set of objects. If there exists a quasi-equivalence between R and O, then there also is a chain of pointwise equivalences between R and O.

Proof. i) Special case: suppose that there exists a quasi-equivalence (M, {φ_{i}}_{i∈I}) for which all of the right multiplication maps ·φ_{i} : R(i, j) ⟶ M(i, j) are trivial fibrations. In this case we define a new V-category E(M, φ) with objects I as the pullback of R and O over M. More precisely for every pair i, j ∈ I the homomorphism object E(M, φ)(i, j) is defined as the pullback in V

O(i, j) R(i, j)  M(i, j)_{u_{φ_{j}·}} w_{·φ_{i}}

Using the universal property of the pullback there is a unique way to define composition and identity morphisms in E(M, φ) in such a way that the maps E(M, φ) ⟶ O and E(M, φ) ⟶ R are homomorphisms of V-categories.

Since M is a quasi-equivalence, all the maps in the defining pullback diagrams are weak equivalences. By assumption the horizontal ones are even trivial fibrations, so both base change maps End(M, φ) ⟶ O and End(M, φ) ⟶ R are pointwise equivalences of V-categories. The same argument works if instead of the right multiplication maps all the left multiplication maps φ_{j}· : O(i, j) ⟶ M(i, j) are trivial fibrations.

ii) General case: taking fibrant replacement if necessary we can assume that the bimodule M is objectwise fibrant. The ‘element’ φ_{j} of M(j, j) corresponds to a map F_{j}^{O} ⟶ M(−, j) from the representable O-module; the map is left multiplication by φ_{j} and is thus an objectwise equivalence since M is a quasi-equivalence. We factor this O-module equivalence as a trivial cofibration α_{j} : F_{j}^{O} ⟶ (F_{j}^{O})^{f} followed by a trivial fibration ψ_{j} : (F_{j}^{O})^{f} ⟶ M(−, j). Now we appeal twice to the special case that we already proved.

We let E((F^{O})^{f}) denote the endomorphism V-category of the fibrant replacements, i.e., the full V-subcategory of the category of O-modules with objects (F_{j}^{O})^{f} for j ∈ I. We define a E((F^{O})^{f})-O-bimodule W by the rule

W (i, j) = Hom_{mod-O}(F_{i}^{O}, (F_{j}^{O})^{f}) ≅ (F_{j}^{O})^{f}(i) .

The bimodule W is a quasi-equivalence with respect to the maps α_{j}. Moreover the right multiplication map ·α_{i} is the restriction map

α_{i}^{∗} : Hom_{mod-O}((F_{i}^{O})^{f}, (F_{j}^{O})^{f}) ⟶ Hom_{mod-O}(F_{i}^{O}, (F_{j}^{O})^{f})

But α_{i}^{∗} is a trivial fibration since α_{i} is a trivial cofibration of O-modules and (F_{j}^{O})^{f} is a fibrant module. So case i) above provides a chain of pointwise equivalences between O and the endomorphism category of the cofibrant-fibrant O-modules (F_{j}^{O})^{f}.

Now we define an R-E((F^{O})^{f})-bimodule V by the rule

V (i, j) = Hom_{mod-O}((F_{i}^{O})^{f}, M(−, j)).

The bimodule V is a quasi-equivalence with respect to the maps ψ_{j}. The left multiplication map ψ_{j}· is the composition

(ψ_{j})_{∗} : Hom_{mod-O}((F_{i}^{O})^{f}, (F_{j}^{O})^{f}) ⟶ Hom_{mod-O}((F_{i}^{O})^{f}, M(−, j))

This time (ψ_{j})_{∗} is a trivial fibration since ψ_{j} is a trivial fibration of O-modules and (F_{i}^{O})^{f} is a cofibrant module. Furthermore the right multiplication map

·ψ_{i} : R(i, j) ⟶ Hom_{mod-O}((F_{i}^{O})^{f}, M(−, j))

is an equivalence because its composite with the map

α_{i}^{∗} : Hom_{mod-O}((F_{i}^{O})^{f}, M(−, j)) ⟶ Hom_{mod-O}(F_{i}^{O}, M(−, j)) ≅ M(i, j)

is right multiplication by ψ_{i}, an equivalence by assumption. So case i) gives a chain of pointwise equivalences between R and the endomorphism V-category of the O-modules (F_{j}^{O})^{f}; combining this with the previous paragraph shows that R and O are pointwise equivalent.

As a corollary we obtain the homotopy invariance of endomorphism categories under V-Quillen equivalences.

Corollary A.2.4. Let V be a monoidal model category and suppose C and D are V-model categories and L : C ⟶ D is the left adjoint of a V-Quillen equivalence. Suppose I is a set, {P_{i}}_{i∈I} and {Q_{i}}_{i∈I} are sets of cofibrant and fibrant objects of C and D respectively, and that for all i ∈ I, LP_{i} is weakly equivalent to Q_{i} in D. Then the C-endomorphism categories of {P_{i}}_{i∈I} and {Q_{i}}_{i∈I} are pointwise equivalent. In particular the V-endomorphism category of {P_{i}}_{i∈I} depends up to pointwise equivalence only on the weak equivalence type of the objects P_{i}.

Proof. Since the object LP_{i} is cofibrant and weakly equivalent to the fibrant object Q_{i}, we can choose a weak equivalence φ_{i} : LP_{i} ⟶ Q_{i} for every i ∈ I. We claim that the collection of homomorphism objects Hom_{D}(LP_{i}, Q_{j}) forms a quasi-equivalence for the endomorphism Vcategories of {P_{i}}_{i∈I} and {Q_{i}}_{i∈I} with respect to the equivalences φ_{i}. Indeed the endomorphism category of {Q_{i}}_{i∈I} acts on the left by composition; also right multiplication by φ_{j} is a stable equivalence since Q_{j} is fibrant and φ_{j} is a weak equivalence between cofibrant objects. If R denotes the right adjoint of L, then Hom_{D}(LP_{i}, Q_{j}) is isomorphic to Hom_{C}(P_{i}, RQ_{j}), so the endomorphism category of {P_{i}}_{i∈I} acts on the right by composition. Since R and L form a V-Quillen equivalence, the adjoints φ̂_{i} : P_{i} ⟶ RQ_{i} are weak equivalences between fibrant objects; so left multiplication by φ_{i} is a stable equivalence since P_{i} is cofibrant. The last statement is the special case where D = C and L is the identity functor.

A.3. Characterization of Eilenberg-Mac Lane spectra. In this section we show that Eilenberg-Mac Lane spectral categories with homotopy concentrated in dimension 0 are determined up to stable equivalence by their homotopy groups (Proposition A.3.1).

Recall that a ringoid is a small category whose hom-sets carry an abelian group structure for which composition is bilinear. A module over a ringoid \underline{A} is a contravariant additive functor to the category of abelian groups. From our point of view a ringoid is merely a special kind of differential graded category, namely one in which all homomorphism complexes are concentrated in dimension 0. Then complexes of \underline{A}-modules correspond to modules over \underline{A} when viewed as a DG-category.

Proposition A.3.1. Let R be a spectral category all of whose morphism spectra are stably fibrant and have homotopy groups concentrated in dimension 0. Then there exists a natural chain of stable equivalences of spectral categories between R and H\underline{π_{0}R}.

The proposition is a special case of the following

Lemma A.3.2. There are functors M and E from the category of spectral categories to itself and natural transformations

Id ⟶^{α} M ⟵^{β} E ⟶^{γ} H\underline{π_{0}}

with the following properties: for every spectral category R with connective stably fibrant morphism spectra the maps α_{R} and β_{R} are stable equivalences and the map γ_{R} induces the canonical isomorphism on component ringoids.

Proof. The strategy of proof is to transfer the corresponding statement from the category of Gamma-rings (where it is easy to prove) to the category of symmetric ring spectra and extend it to the ‘multiple object case’. We use in a crucial way Bökstedt’s hocolim_{I} construction [Bök]. The functors M, E as well as an intermediate functor D all arise as lax monoidal functors from the category of symmetric spectra to itself, and the natural maps between them are monoidal transformations. This implies that when we apply them to the morphism spectra of a spectral category, then the outcome is again a spectral category in a natural way, and the transformations assemble into spectral functors.

The two functors M and D from the category of symmetric spectra to itself are defined in [Shi, Sec. 3]. The n-th space of the symmetric spectrum MX is defined as the homotopy colimit

(MX)_{n} = hocolim_{k∈I} Ω^{k}Sing|X_{k+n}| .

Here I is a skeleton of the category of finite sets and injections with objects k = {0, 1, …, k}; for the precise definition and the structure maps making this a symmetric spectrum see [Shi, Sec. 3]. The map α : X ⟶ MX is induced by the inclusion of X into the colimit diagram at k = 0. In the proof of [Shi, Prop. 3.1.9] it is shown that the map α is a stable equivalence (even a level equivalence) for every stably fibrant symmetric spectrum X.

The n-th level of the functor D (for ‘detection’ - it detects stable equivalences of symmetric spectra) is defined as

(DX)_{n} = hocolim_{k∈I} Ω^{k}Sing|X_{k} ∧ S^{n}| ,

see [Shi, Def. 3.1.1]. Also in the proof of [Shi, Prop. 3.1.9] a natural map DX ⟶ MX is constructed which we denote β_{X}^{1} and which is a stable equivalence (even a level equivalence) for every stably fibrant spectrum X.

The symmetric spectrum DX in fact arises from a simplicial functor QX. The value of QX at a pointed simplicial set K is given by

QX(K) = hocolim_{k∈I} Ω^{k}Sing|X_{k} ∧ K| .

A simplicial functor F can be evaluated on the simplicial spheres to give a symmetric spectrum, which we denote F (𝕊). In the situation at hand we thus have DX = QX(𝕊). If we restrict the simplicial functor QX to the category Γ^{op} of finite pointed sets we obtain a Γ-space [Seg74, BF78] denoted ρQX. Every Γ-space can be prolonged to a simplicial functor defined on the category of pointed simplicial sets [BF78, §4]. Prolongation is left adjoint to the restriction functor ρ, and we denote it by P. We then set EX = (P ρQX)(𝕊). The unit P ρQX ⟶ QX of the adjunction between restriction and prolongation, evaluated at the spheres, gives a map of symmetric spectra

β_{X}^{2} : EX = (P ρQX)(𝕊) ⟶QX(𝕊) = DX .

If X is a stably fibrant symmetric spectrum, the simplicial functor QX takes values in infinite loop spaces (see [MS99]) and the map QX(K ∨ L) ⟶ QX(K) × QX(L) is a weak equivalence for every pair of pointed simplicial sets K and L. In particular the restricted Γ-space ρQX is very special [BF78, p. 97], and hence by [BF78, Thm. 4.2], EX = (P ρQX)(𝕊) is a connective Ω-spectrum.

The map β_{X} : EX ⟶ MX is defined as the composite of the maps β_{X}^{2} : EX ⟶ DX and β_{X}^{1} : DX ⟶ MX. Since both EX and MX are connective Ω-spectra and the map β_{X} is an isomorphism at level 0, β_{X} is in fact a level equivalence, hence a stable equivalence of symmetric spectra.

Every Γ-space Y has a natural monoidal map Y ⟶ Hπ_{0}Y to the Eilenberg-Mac Lane Γ-space ([Seg74, §0], [Sch99, Sec. 1]) of its component group which induces the canonical isomorphism on π_{0}, see [Sch99, Lemma 1.2]. (This map is in fact the unit of another monoidal adjunction, namely, between the Eilenberg-Mac Lane Γ-space functor and the π_{0}-functor.) In particular there is such a map of Γ-spaces ρQX ⟶ Hπ_{0}(ρQX). From this we get the map

γ_{X} : EX = (P ρQX)(𝕊) ⟶ (P Hπ_{0}(ρQX))(𝕊) = Hπ_{0}X

by prolongation and evaluation of the adjunction unit on spheres. Whenever X is a stably fibrant, the component groups π_{0}X and π_{0}(ρQX) are isomorphic. The symmetric spectrum associated to the Eilenberg-Mac Lane Γ-space by prolongation and then restriction to spheres is the Eilenberg-Mac Lane model given in [HSS98, 1.2.5]. As we remarked in 6.2.2, this is level equivalent to the Eilenberg-Mac Lane spectral category defined by the functor H in Construction 6.2.1.

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Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany

   E-mail address: schwede@mathematik.uni-bielefeld.de

Department of Mathematics, Purdue University, W. Lafayette, IN 47907, USA

   E-mail address: bshipley@math.purdue.edu