SYMMETRIC SPECTRA

MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH

Contents

Introduction 2 1. Symmetric spectra 4 1.1. Simplicial sets 4 1.2. Symmetric spectra 5 1.3. Simplicial structure on Sp^{Σ} 8 1.4. Symmetric Ω-spectra 10 2. The smash product of symmetric spectra 10 2.1. Symmetric sequences 11 2.2. Symmetric spectra 13 2.3. The ordinary category of spectra 18 3. Stable homotopy theory of symmetric spectra 19 3.1. Stable equivalence 19 3.2. Model categories 25 3.3. Level structure 31 3.4. Stable model category 33 4. Comparison with the Bousfield-Friedlander category 38 4.1. Quillen equivalences 38 4.2. The stable Bousfield-Friedlander category 40 4.3. The Quillen equivalence 41 4.4. Description of V 45 5. Additional properties of symmetric spectra 45 5.1. Level model structure 46 5.2. Stable cofibrations 50 5.3. Pushout smash product 51 5.4. Proper model categories 55 5.5. The monoid axiom 57 5.6. Semistable spectra 58 6. Topological spectra 61 6.1. Topological spectra 61 6.2. Stable model structure 63 6.3. Compactly generated spectra 68 References 69

Date: January 16, 1998.

The first two authors were partially supported by NSF Postdoctoral Fellowships.

The third author was partially supported by an NSF Grant.

Introduction

In this paper we describe a reasonably simple symmetric monoidal category of spectra. The idea of a spectrum is a very simple one: Freudenthal’s suspension theorem implies that the sequence of homotopy classes of maps

[X, Y] ⟶ [ΣX, ΣY] ⟶ … ⟶ [Σ^{n}X, Σ^{n}Y] ⟶ …

is eventually constant for finite-dimensional pointed CW-complexes X and Y, where ΣX = S^{1} ∧ X is the reduced suspension of X. This suggests forming a stable category where the suspension functor is an isomorphism. The standard way to do this is to define a spectrum to be a sequence of pointed topological spaces (or simplicial sets) X_{n} together with structure maps S^{1} ∧ X_{n} ⟶ X_{n+1}. This was first done by Whitehead in [Whi62]. The suspension functor is not an isomorphism in the category of spectra, but becomes an isomorphism when we invert the stable homotopy equivalences. The resulting homotopy category of spectra is often called the stable homotopy category and has been extensively studied, beginning with the work of Boardman [Vog70] and Adams [Ada74] and continuing to this day.

One of the stable homotopy category’s basic features is that it is symmetric monoidal. There is a smash product, built from the smash product of pointed topological spaces and analogous to the tensor product of modules, that is associative, commutative, and unital, up to coherent natural isomorphism. However, the category of spectra itself is not symmetric monoidal. This has been a sticking point for almost forty years now. Indeed, it was long thought that there could be no symmetric monoidal category of spectra; see [Lew91], where it is shown that a symmetric monoidal category of spectra can not have all the properties one would like.

Any good symmetric monoidal category of spectra allows one to perform algebraic constructions on spectra that are impossible without such a category. This is extremely important, for example, in the algebraic K-theory of spectra. In particular, given a good symmetric monoidal category of spectra, it is possible to construct a homotopy category of monoids (ring spectra) and of modules over a given monoid.

In this paper, we describe a symmetric monoidal category of spectra, called the category of symmetric spectra. The ordinary category of spectra as described above is the category of modules over the sphere spectrum. The sphere spectrum is a monoid in the category of sequences of spaces, but it is not a commutative monoid, because the twist map on S^{1}∧S^{1} is not the identity. This explains why the ordinary category of spectra is not symmetric monoidal. To make the sphere spectrum a commutative monoid, we need to keep track of the twist map, and, more generally, of permutations of coordinates. We therefore define a symmetric spectrum to be a sequence of pointed simplicial sets X_{n} together with a pointed action of the permutation group Σ_{n} on X_{n} and equivariant structure maps S^{1} ∧ X_{n} ⟶ X_{n+1}. We must also require that the iterated structure maps S^{p}∧X_{n} ⟶ X_{n+p} be Σ_{p}×Σ_{n}- equivariant. This idea is due to the third author; the first and second authors joined the project later.

At approximately the same time as the third author discovered symmetric spectra, the team of Elmendorf, Kriz, Mandell, and May [EKMM97] also constructed a symmetric monoidal category of spectra. Their construction appears to be significantly different from symmetric spectra; however, there is work in progress [MMSS2] showing that the two approaches define equivalent stable homotopy categories and equivalent homotopy categories of monoids and modules. Each approach has its own advantages. The category of symmetric spectra is technically much simpler than the S-modules of [EKMM97]; this paper is almost entirely self-contained, depending only on some standard results about simplicial sets and topological spaces. One can form symmetric spectra in very general settings, whereas S-modules appear to be tied to the category of topological spaces. On the other hand, the S-modules of [EKMM97] are very well suited to the varying universes that arise in equivariant stable homotopy theory, whereas we do not yet know how to realize universes in symmetric spectra. Furthermore, at present the theory of S-modules is considerably more developed than the theory of symmetric spectra.

In this paper, we can only begin the study of symmetric spectra. The most significant loose end is the construction of a model category of commutative symmetric ring spectra; such a model category has been constructed by the third author in work in progress. It would also be useful to have a stable fibrant replacement functor, as the usual construction QX does not work in general. A good approximation to such a functor is constructed in [Shi97].

Some generalizations of symmetric spectra appear in [MMSS1]. This paper and [MMSS2] prove that the many different worlds, including S-modules and symmetric spectra, are equivalent in an appropriate sense. Another symmetric monoidal category of spectra sitting between the approaches of [EKMM97] and of this paper is developed in [DS]. We also point out that symmetric spectra are part of a more general theory of localization of model categories [Hir97]; we have not adopted this approach, but both [Hir97] and [DHK] have influenced us considerably.

Symmetric spectra have already proved useful. In [GH97], symmetric spectra are used to extend the definition of topological cyclic homology from rings to schemes. Similarly, in [Shi97], Bökstedt’s approach to topological Hochschild homology [Bök85] is extended to symmetric ring spectra, without connectivity conditions. And in [SS], it is shown that any linear model category is Quillen equivalent to the model category of modules over a symmetric ring spectrum.

The paper is organized as follows. We choose to work in the category of simplicial sets until Section 6, where we discuss topological symmetric spectra. This is a significant technical simplification; while it is possible to develop symmetric spectra in topological spaces from scratch, technical issues arise that are not present when working with simplicial sets. In the first section, we define symmetric spectra, give some examples, and establish some basic properties. In Section 2 we describe the closed symmetric monoidal structure on the category of symmetric spectra, and explain why such a structure can not exist in the ordinary category of spectra. In Section 3 we study the stable homotopy theory of symmetric spectra. This section is where the main subtlety of the theory of symmetric spectra arises: we cannot define stable equivalence by using stable homotopy isomorphisms. Instead, we define a map to be a stable equivalence if it is a cohomology isomorphism for all cohomology theories. The main result of this section is that symmetric spectra, together with stable equivalences and suitably defined classes of stable fibrations and stable cofibrations, form a model category. As expected, the fibrant objects are the Ω-spectra; i.e., symmetric spectra X such that each X_{n} is a Kan complex and the adjoint X_{n} ⟶ X_{n+1}^{S^{1}} of the structure map is a weak equivalence. In Section 4, we prove that the stable homotopy theories of symmetric spectra and ordinary spectra are equivalent. More precisely, we construct a Quillen equivalence of model categories between symmetric spectra and the model category of ordinary spectra described in [BF78].

In Section 5 we discuss some of the properties of symmetric spectra. In particular, in Section 5.1, we tie up a loose end from Section 3 by establishing two different model categories of symmetric spectra where the weak equivalences are the level equivalences. We characterize the stable cofibrations of symmetric spectra in Section 5.2. In Section 5.3, we show that the smash product of symmetric spectra interacts with the model structure in the expected way. This section is crucial for the applications of symmetric spectra, and, in particular, is necessary to be sure that the smash product of symmetric spectra does define a symmetric monoidal structure on the stable homotopy category. We establish that symmetric spectra are a proper model category in Section 5.4, and use this to verify the monoid axiom in Section 5.5. The monoid axiom is required to construct model categories of monoids and of modules over a given monoid, see [SS97]. In Section 5.6, we define semistable spectra, which are helpful for understanding the difference between stable equivalences and stable homotopy equivalences. Finally, we conclude the paper by considering topological symmetric spectra in Section 6.

The authors would like to thank Dan Christensen, Bill Dwyer, Phil Hirschhorn, Dan Kan, Haynes Miller, John Palmieri, Charles Rezk, and Stefan Schwede for many helpful conversations about symmetric spectra.

We now establish some notation we will use throughout the paper. Many of the categories in this paper have an enriched Hom as well as a set valued Hom. To distinguish them: in a category 𝒞, the set of maps from X to Y is denoted 𝒞(X, Y); in a simplicial category 𝒞, the simplicial set of maps from X to Y is denoted Map_{𝒞} (X, Y) or Map(X, Y); in a category 𝒞 with an internal Hom, the object in 𝒞 of maps from X to Y is denoted Hom_{𝒞} (X, Y) or Hom(X, Y). In case 𝒞 is the category of modules over a commutative monoid S, we also use Hom_{S}(X, Y) for the internal Hom.

1. Symmetric spectra

In this section we construct the category of symmetric spectra over simplicial sets. We discuss the related category of topological symmetric spectra in Section 6. We begin this section by recalling the basic facts about simplicial sets in Section 1.1, then we define symmetric spectra in Section 1.2. We describe the simplicial structure on the category of symmetric spectra in Section 1.3. The homotopy category of symmetric Ω-spectra is described in Section 1.4.

1.1. Simplicial sets. With the exception of Section 6 dealing with topological spectra, this paper is written using simplicial sets. We recall the basics. Consult [May67] or [Cur71] for more details.

The category Δ has the ordered sets [n] = {0, 1, …, n} for n ≥ 0 as its objects and the order preserving functions [n] → [m] as its maps. The category of simplicial sets, denoted 𝒮, is the category of functors from Δ^{op} to the category of sets. The set of n-simplices of the simplicial set X, denoted X_{n}, is the value of the functor X at [n]. The standard n-simplex Δ[n] is the contravariant functor Δ(−, [n]). Varying n gives a covariant functor Δ[−]: Δ → 𝒮. By the Yoneda lemma, 𝒮(Δ[n], X) = X_{n} and the contravariant functor 𝒮(Δ[−], X) is naturally isomorphic to X.

Let G be a discrete group. The category of G-simplicial sets is the category 𝒮^{G} of functors from G to 𝒮, where G is regarded as a category with one object. A G-simplicial set is therefore a simplicial set X with a left simplicial G-action, i.e., a homomorphism G → 𝒮(X, X).

A basepoint of a simplicial set X is a distinguished 0-simplex ∗ ∈ X_{0}. The category of pointed simplicial sets and basepoint preserving maps is denoted 𝒮_{∗}. The simplicial set Δ[0] = Δ(−, [0]) has a single simplex in each degree and is the terminal object in 𝒮. A basepoint of X is the same as a map Δ[0] → X. The disjoint union X_{+} = X ⨿ Δ[0] adds a disjoint basepoint to the simplicial set X. For example, the 0-sphere is S^{0} = Δ[0]_{+}. A basepoint of a G-simplicial set X is a G-invariant 0-simplex of X. The category of pointed G-simplicial sets is denoted 𝒮_{∗}^{G}.

The smash product X ∧ Y of the pointed simplicial sets X and Y is the quotient (X × Y)/(X ∨ Y) that collapses the simplicial subset X ∨ Y = X × ∗ ∪ ∗ × Y to a point. For pointed G-simplicial sets X and Y, let X ∧_{G} Y be the quotient of X ∧ Y by the diagonal action of G. For pointed simplicial sets X, Y, and Z, there are natural isomorphisms (X ∧ Y) ∧ Z ≅ X ∧ (Y ∧ Z), X ∧ Y ≅ Y ∧ X and X ∧ S^{0} ≅ X. In the language of monoidal categories, the smash product is a symmetric monoidal product on the category of pointed simplicial sets. We recall the definition of symmetric monoidal product, but for more details see [ML71, VII] or [Bor94, 6.1].

Definition 1.1.1. A symmetric monoidal product on a category 𝒞 is: a bifunctor ⊗: 𝒞 × 𝒞 → 𝒞; a unit U ∈ 𝒞; and coherent natural isomorphisms (X ⊗ Y) ⊗ Z ≅ X ⊗ (Y ⊗ Z) (the associativity isomorphism), X ⊗ Y ≅ Y ⊗ X (the twist isomorphism), and U ⊗ X ≅ X (the unit isomorphism). The product is closed if the functor X ⊗ (−) has a right adjoint Hom(X, −) for every X ∈ 𝒞. A (closed) symmetric monoidal category is a category 𝒞 with a (closed) symmetric monoidal product.

Coherence of the natural isomorphisms means that all reasonable diagrams built from the natural isomorphisms also commute [ML71]. When the product is closed, the pairing Hom(X, Y): 𝒞^{op} × 𝒞 → 𝒞 is an internal Hom. For example, the smash product on the category 𝒮_{∗} of pointed simplicial sets is closed. For X, Y ∈ 𝒮_{∗}, the pointed simplicial set of maps from X to Y is Map_{𝒮_{∗}} (X, Y) = 𝒮_{∗}(X∧Δ[−]_{+}, Y). For pointed G-simplicial sets X and Y, the simplicial subset of G-equivariant pointed maps is Map_{G}(X, Y) = 𝒮_{∗}^{G}(X ∧ Δ[−]_{+}, Y).

1.2. Symmetric spectra. Let S^{1} be the simplicial circle Δ[1]/∂Δ[1], obtained by identifying the two vertices of Δ[1].

Definition 1.2.1. A spectrum is 1. a sequence X_{0}, X_{1}, …, X_{n}, … of pointed simplicial sets; and

2. a pointed map σ: S^{1} ∧ X_{n} → X_{n+1} for each n ≥ 0. The maps σ are the structure maps of the spectrum. A map of spectra f: X → Y is a sequence of pointed maps f_{n}: X_{n} → Y_{n} such that the diagram

S^{1} ∧ X_{n} ⟶^{σ} X_{n+1}
S^{1}∧f_{n}↓  ^{f_{n+1}}↓
S^{1} ∧ Y_{n} ⟶^{σ} Y_{n+1}

is commutative for each n ≥ 0. Let Sp^{ℕ} denote the category of spectra.

Replacing the sequence of pointed simplicial sets by a sequence of pointed topological spaces in 1.2.1 gives the original definition of a spectrum (due to Whitehead and Lima). The categories of simplicial spectra and of topological spectra are discussed in the work of Bousfield and Friedlander [BF78].

A symmetric spectrum is a spectrum to which symmetric group actions have been added. Let Σ_{p} be the group of permutations of the set \overline{p} = \{1, 2, …, p\}, with \overline{0} = ∅. As usual, embed Σ_{p} ×Σ_{q} as the subgroup of Σ_{p+q} with Σ_{p} acting on the first p elements of \overline{p + q} and Σ_{q} acting on the last q elements of \overline{p + q}. Let S^{p} = (S^{1})^{∧p} be the p-fold smash power of the simplicial circle with the left permutation action of Σ_{p}.

Definition 1.2.2. A symmetric spectrum is 1. a sequence X_{0}, X_{1}, …, X_{n}, … of pointed simplicial sets;

2. a pointed map σ: S^{1} ∧ X_{n} → X_{n+1} for each n ≥ 0; and

3. a basepoint preserving left action of Σ_{n} on X_{n} such that the composition

σ^{p} = σ ∘ (S^{1} ∧ σ) ∘ ⋯ ∘ (S^{p−1} ∧ σ): S^{p} ∧ X_{n} → X_{n+p},

of the maps S^{i} ∧ S^{1} ∧ X_{n+p−i−1} ⟶^{S^{i}∧σ} S^{i} ∧ X_{n+p−i} is Σ_{p} × Σ_{n}-equivariant

for p ≥ 1 and n ≥ 0. A map of symmetric spectra f: X → Y is a sequence of pointed maps f_{n}: X_{n} → Y_{n} such that f_{n} is Σ_{n}-equivariant and the diagram

S^{1} ∧ X_{n} ⟶^{σ} X_{n+1}
S^{1}∧f_{n}↓  ^{f_{n+1}}↓
S^{1} ∧ Y_{n} ⟶^{σ} Y_{n+1}

is commutative for each n ≥ 0. Let Sp^{Σ} denote the category of symmetric spectra.

Remark 1.2.3. In part three of Definition 1.2.2, one need only assume that the maps σ: S^{1} ∧ X_{n} → X_{n+1} and σ^{2} : S^{2} ∧ X_{n} → X_{n+2} are equivariant; since the symmetric groups Σ_{p} are generated by transpositions (i, i + 1), if σ and σ^{2} are equivariant then all the maps σ^{p} are equivariant.

Example 1.2.4. The symmetric suspension spectrum Σ^{∞}K of the pointed simplicial set K is the sequence of pointed simplicial sets S^{n} ∧ K with the natural isomorphisms σ: S^{1} ∧ S^{n} ∧ K → S^{n+1} ∧ K as the structure maps and the diagonal action of Σ_{n} on S^{n} ∧ K coming from the left permutation action on S^{n} and the trivial action on K. The composition σ^{p} is the natural isomorphism which is Σ_{p}×Σ_{n}-equivariant. The symmetric sphere spectrum S is the symmetric suspension spectrum of the 0-sphere; S is the sequence of spheres S^{0}, S^{1}, S^{2}, … with the natural isomorphisms S^{1} ∧ S^{n} → S^{n+1} as the structure maps and the left permutation action of Σ_{n} on S^{n}.

Example 1.2.5. The Eilenberg-Mac Lane spectrum Hℤ is the sequence of simplicial abelian groups ℤ ⊗ S^{n}, where (ℤ ⊗ S^{n})_{k} is the free abelian group on the non-basepoint k-simplices of S^{n}. We identify the basepoint with 0. The symmetric group Σ_{n} acts by permuting the generators, and one can easily verify that the evident structure maps are equivariant. One could replace ℤ by any ring.

Remark 1.2.6. Bordism is most easily defined as a topological symmetric spectrum, see Example 6.1.3. As explained in [GH97, Section 6], many other examples of symmetric spectra arise as the K-theory of a category with cofibrations and weak equivalences as defined by Waldhausen [Wal85, p.330].

A symmetric spectrum with values in a simplicial category 𝒞 is obtained by replacing the sequence of pointed simplicial sets by a sequence of pointed objects in 𝒞. In particular, a topological symmetric spectrum is a symmetric spectrum with values in the simplicial category of topological spaces; see Section 6.

By ignoring group actions, a symmetric spectrum is a spectrum and a map of symmetric spectra is a map of spectra. When no confusion can arise, the adjective “symmetric” may be dropped.

Definition 1.2.7. Let X be a symmetric spectrum. The underlying spectrum UX is the sequence of pointed simplicial sets (UX)_{n} = X_{n} with the same structure maps σ: S^{1} ∧ (UX)_{n} → (UX)_{n+1} as X but ignoring the symmetric group actions. This gives a faithful functor U: Sp^{Σ} → Sp^{ℕ}.

Since the action of Σ_{n} on S^{n} is non-trivial for n ≥ 2, it is usually impossible to obtain a symmetric spectrum from a spectrum by letting Σ_{n} act trivially on X_{n}. However, many of the usual functors to the category of spectra lift to the category of symmetric spectra. For example, the suspension spectrum of a pointed simplicial set K is the underlying spectrum of the symmetric suspension spectrum of K.

Many examples of symmetric spectra and of functors on the category of symmetric spectra are constructed by prolongation of simplicial functors.

Definition 1.2.8. A pointed simplicial functor or 𝒮_{∗}-functor is a pointed functor R: 𝒮_{∗} → 𝒮_{∗} and a natural transformation h: RX ∧ K → R(X ∧ K) of bifunctors such that the composition RX ∧ S^{0} → R(X ∧ S^{0}) → R(X) is the unit isomorphism and the diagram of natural transformations

[figure: (RX ∧ K) ∧ L; h∧L; R(X ∧ K) ∧ L; h; RX ∧ (K ∧ L); h; R(X ∧ K ∧ L)]

is commutative. A pointed simplicial natural transformation, or 𝒮_{∗}-natural transformation, from the 𝒮_{∗}-functors R to the 𝒮_{∗}-functor R′ is a natural transformation τ: R → R′ such that τh = h′(τ ∧ K).

Definition 1.2.9. The prolongation of a 𝒮_{∗}-functor R: 𝒮_{∗} → 𝒮_{∗} is the functor R: Sp^{Σ} → Sp^{Σ} defined as follows. For X a symmetric spectrum, RX is the sequence of pointed simplicial sets RX_{n} with the composition σ: S^{1} ∧ R(X_{n}) → R(S^{1} ∧ X_{n}) ⟶^{Rσ} R(X_{n+1}) as the structure maps and the action of Σ_{n} on R(X_{n}) obtained by applying the functor R to the action of Σ_{n} on X_{n}. Since R is a 𝒮_{∗}- functor, each map σ^{p} is equivariant and so RX is a symmetric spectrum. For f a map of symmetric spectra, Rf is the sequence of pointed maps Rf_{n}. Since R is an 𝒮_{∗}-functor, Rf is a map of spectra. Similarly, we can prolong an 𝒮_{∗}-natural transformation to a natural transformation of functors on Sp^{Σ}.

Proposition 1.2.10. The category of symmetric spectra is bicomplete (every small diagram has a limit and a colimit).

Proof. For any small category I, the limit and colimit functors 𝒮_{∗}^{I} → 𝒮_{∗} are pointed simplicial functors; for K ∈ 𝒮_{∗} and D ∈ Set^{I} there is a natural isomorphism

K ∧ colim D ≅ colim(K ∧ D)

and a natural map

K ∧ lim D → lim(K ∧ D).

A slight generalization of prolongation gives the limit and the colimit of a diagram of symmetric spectra.

In particular, the underlying sequence of the limit is (lim D)_{n} = lim D_{n} and the underlying sequence of the colimit is (colim D)_{n} = colim D_{n}.

1.3. Simplicial structure on Sp^{Σ}. For a pointed simplicial set K and a symmetric spectrum X, prolongation of the 𝒮_{∗}-functor (−) ∧ K: 𝒮_{∗} → 𝒮_{∗} defines the smash product X ∧ K and prolongation of the 𝒮_{∗}-functor (−)^{K}: 𝒮_{∗} → 𝒮_{∗} defines the power spectrum X^{K}. For symmetric spectra X and Y, the pointed simplicial set of maps from X to Y is Map_{Sp^{Σ}}(X, Y) = Sp^{Σ}(X ∧ Δ[−]_{+}, Y).

In the language of enriched category theory, the following proposition says that the smash product X ∧ K is a closed action of 𝒮_{∗} on Sp^{Σ}. We leave the straightforward proof to the reader.

Proposition 1.3.1. Let X be a symmetric spectrum. Let K and L be pointed simplicial sets.

1. There are coherent natural isomorphisms X ∧ (K ∧ L) ≅ (X ∧ K) ∧ L and X ∧ S^{0} ≅ X.

2. (−) ∧ K: Sp^{Σ} → Sp^{Σ} is the left adjoint of the functor (−)^{K}: Sp^{Σ} → Sp^{Σ}.

3. X ∧ (−): 𝒮_{∗} → Sp^{Σ} is the left adjoint of the functor Map_{Sp^{Σ}}(X, −): Sp^{Σ} → 𝒮_{∗}.

The evaluation map X ∧ Map_{Sp^{Σ}}(X, Y) → Y is the adjoint of the identity map on Map_{Sp^{Σ}}(X, Y). The composition pairing

Map_{Sp^{Σ}}(X, Y) ∧ Map_{Sp^{Σ}}(Y, Z) → Map_{Sp^{Σ}}(X, Z)

is the adjoint of the composition

X ∧ Map_{Sp^{Σ}}(X, Y) ∧ Map_{Sp^{Σ}}(Y, Z) → Y ∧ Map_{Sp^{Σ}}(Y, Z) → Z

of two evaluation maps. In the language of enriched category theory, a category with a closed action of 𝒮_{∗} is the same as a tensored and cotensored 𝒮_{∗}-category. The following proposition, whose proof we also leave to the reader, expresses this fact.

Proposition 1.3.2. Let X, Y, and Z be symmetric spectra and let K be a pointed simplicial set.

1. The composition pairing Map_{Sp^{Σ}}(X, Y) ∧ Map_{Sp^{Σ}}(Y, Z) → Map_{Sp^{Σ}}(X, Z) is associative.

2. The adjoint S^{0} → Map_{Sp^{Σ}}(X, X) of the isomorphism X ∧ S^{0} → X is a left and a right unit of the composition pairing.

3. There are natural isomorphisms

Map_{Sp^{Σ}}(X ∧ K, Y) ≅ Map_{Sp^{Σ}}(X, Y^{K}) ≅ Map_{Sp^{Σ}}(X, Y)^{K}.

Proposition 1.3.1 says that certain functors are adjoints, whereas Proposition 1.3.2 says more; they are simplicial adjoints.

The category of symmetric spectra satisfies Quillen’s axiom SM7 for simplicial model categories.

Definition 1.3.3. Let f: U → V and g: X → Y be maps of pointed simplicial sets. The pushout smash product f □ g is the natural map on the pushout

f □ g: V ∧ X ⨿_{U∧X} U ∧ Y → V ∧ Y.

induced by the commutative square

U ∧ X ⟶^{f∧X} V ∧ X
U∧g↓  ↓^{V ∧g}
U ∧ Y ⟶_{f∧Y} V ∧ Y.

Let f be a map of symmetric spectra and g be a map of pointed simplicial sets. The pushout smash product f □ g is defined by prolongation, (f □ g)_{n} = f_{n} □ g.

Recall that a map of simplicial sets is a weak equivalence if its geometric realization is a homotopy equivalence of CW-complexes. One of the basic properties of simplicial sets, proved in [Qui67, II.3], is:

Proposition 1.3.4. Let f and g be monomorphisms of pointed simplicial sets. Then f □ g is a monomorphism, which is a weak equivalence if either f or g is a weak equivalence.

Prolongation gives a corollary for symmetric spectra. A map f of symmetric spectra is a monomorphism if f_{n} is a monomorphism of simplicial sets for each n ≥ 0.

Definition 1.3.5. A map f of symmetric spectra is a level equivalence if f_{n} is a weak equivalence of simplicial sets for each n ≥ 0.

Corollary 1.3.6. Let f be a monomorphism of symmetric spectra and let g be a monomorphism of pointed simplicial sets. Then f □ g is a monomorphism, which is a level equivalence if either f is a level equivalence or g is a weak equivalence.

By definition, a 0-simplex of Map_{Sp^{Σ}}(X, Y) is a map X ∧ Δ[0]_{+} → Y, but X ∧Δ[0]_{+} ≅ X and so a 0-simplex of Map_{Sp^{Σ}}(X, Y) is a map X → Y. A 1-simplex of Map_{Sp^{Σ}}(X, Y) is a simplicial homotopy H: X ∧ Δ[1]_{+} → Y from H ∘ (X ∧ i_{0}) to H ∘ (X ∧ i_{1}) where i_{0} and i_{1} are the two inclusions Δ[0] → Δ[1]. Simplicial homotopy generates an equivalence relation on Sp^{Σ}(X, Y) and the quotient is π_{0} Map_{Sp^{Σ}}(X, Y). A map f: X → Y is a simplicial homotopy equivalence if it has a simplicial homotopy inverse, i.e., a map g: Y → X such that gf is simplicially homotopic to the identity map on X and fg is simplicially homotopic to the identity map on Y. If f is a simplicial homotopy equivalence of symmetric spectra, then each of the maps f_{n} is a simplicial homotopy equivalence, and so each of the maps f_{n} is a weak equivalence. Every simplicial homotopy equivalence is therefore a level equivalence. The converse is false; a map can be a level equivalence and NOT a simplicial homotopy equivalence.

1.4. Symmetric Ω-spectra. The stable homotopy category can be defined using Ω-spectra and level equivalences.

Definition 1.4.1. A Kan complex (see Example 3.2.6) is a simplicial set that satisfies the Kan extension condition. An Ω-spectrum is a spectrum X such that for each n ≥ 0 the simplicial set X_{n} is a Kan complex and the adjoint X_{n} → Map_{𝒮_{∗}} (S^{1}, X_{n+1}) of the structure map S^{1} ∧ X_{n} → X_{n+1} is a weak equivalence of simplicial sets.

Let ΩSp^{ℕ} ⊆ Sp^{ℕ} be the full subcategory of Ω-spectra. The homotopy category Ho(ΩSp^{ℕ}) is obtained from ΩSp^{ℕ} by formally inverting the level equivalences. By the results in [BF78], the category Ho(ΩSp^{ℕ}) is naturally equivalent to Boardman’s stable homotopy category (or any other). Likewise, let ΩSp^{Σ} ⊆ Sp^{Σ} be the full subcategory of symmetric Ω-spectra (i.e., symmetric spectra X for which UX is an Ω-spectrum). The homotopy category Ho(ΩSp^{Σ}) is obtained from ΩSp^{Σ} by formally inverting the level equivalences. Since the forgetful functor U: Sp^{Σ} → Sp^{ℕ} preserves Ω-spectra and level equivalences, it induces a functor Ho(U): Ho(ΩSp^{Σ}) → Ho(ΩSp^{ℕ}). As a corollary of Theorem 4.3.2, the functor Ho(U) is a natural equivalence of categories. Thus the category Ho(ΩSp^{Σ}) is naturally equivalent to Boardman’s stable homotopy category. To describe an inverse of Ho(U), let Ω^{∞}: Sp^{ℕ} → 𝒮_{∗} be the functor that takes a spectrum to the 0-space of its associated Ω-spectrum. For any spectrum E ∈ Sp^{ℕ}, the symmetric spectrum V E = Ω^{∞}(E∧S) is the value of the prolongation of the 𝒮_{∗}-functor Ω^{∞}(E∧−) at the symmetric sphere spectrum S; the underlying sequence is V E_{n} = Ω^{∞}(E ∧S^{n}). The functor V preserves Ω-spectra, preserves level equivalences, and induces a functor Ho(V): Ho(ΩSp^{ℕ}) → Ho(ΩSp^{Σ}) which is a natural inverse of Ho(U).

The category of symmetric Ω-spectra has major defects. It is not closed under limits and colimits, or even under pushouts and pullbacks. The smash product, defined in Section 2, of symmetric Ω-spectra is a symmetric spectrum but not an Ω-spectrum, except in trivial cases. For these reasons it is better to work with the category of all symmetric spectra. But then the notion of level equivalence is no longer adequate; the stable homotopy category is a retract of the homotopy category obtained from Sp^{Σ} by formally inverting the level equivalences but many symmetric spectra are not level equivalent to an Ω-spectrum. One must enlarge the class of equivalences. The stable equivalences of symmetric spectra are defined in Section 3.1. By Theorem 4.3.2, the homotopy category obtained from Sp^{Σ} by inverting the stable equivalences is naturally equivalent to the stable homotopy category.

2. The smash product of symmetric spectra

In this section we construct the closed symmetric monoidal product on the category of symmetric spectra. A symmetric spectrum can be viewed as a module over the symmetric sphere spectrum S, and the symmetric sphere spectrum (unlike the ordinary sphere spectrum) is a commutative monoid in an appropriate category. The smash product of symmetric spectra is the tensor product over S.

The closed symmetric monoidal category of symmetric sequences is constructed in Section 2.1. A reformulation of the definition of a symmetric spectrum is given in Section 2.2 where we recall the definition of monoids and modules in a symmetric monoidal category. In Section 2.3 we see that there is no closed symmetric monoidal smash product on the category of (non-symmetric) spectra.

2.1. Symmetric sequences. Every symmetric spectrum has an underlying sequence X_{0}, X_{1}, …, X_{n}, … of pointed simplicial sets with a basepoint preserving left action of Σ_{n} on X_{n}; these are called symmetric sequences. In this section we define the closed symmetric monoidal category of symmetric sequences of pointed simplicial sets.

Definition 2.1.1. The category Σ = ∐_{n≥0} Σ_{n} has the finite sets \overline{n} = \{1, 2, …, n\} for n ≥ 0 (\overline{0} = ∅) as its objects and the automorphisms of the sets \overline{n} as its maps. Let 𝒞 be a category. A symmetric sequence of objects in 𝒞 is a functor Σ → 𝒞, and the category of symmetric sequences of objects in 𝒞 is the functor category 𝒞^{Σ}.

A symmetric sequence X ∈ 𝒮_{∗}^{Σ} is a sequence X_{0}, X_{1}, …, X_{n}, … of pointed simplicial sets with a basepoint preserving left action of Σ_{n} on X_{n}. The category 𝒞^{Σ} is a product category. In particular, 𝒮_{∗}^{Σ}(X, Y) = ∏_{p} 𝒮_{∗}^{Σ_{p}}(X_{p}, Y_{p}).

Proposition 2.1.2. The category 𝒮_{∗}^{Σ} of symmetric sequences in 𝒮_{∗} is bicomplete.

Proof. The category 𝒮_{∗} is bicomplete, so the functor category 𝒮_{∗}^{Σ} is bicomplete.

Definition 2.1.3. The tensor product X⊗Y of the symmetric sequences X, Y ∈ 𝒮_{∗}^{Σ} is the symmetric sequence

(X ⊗ Y)_{n} = ⋁_{p+q=n}(Σ_{n})_{+} ∧_{Σ_{p}×Σ_{q}} (X_{p} ∧ Y_{q}).

The tensor product f ⊗ g: X ⊗ Y ⟶ X′ ⊗ Y′ of the maps f: X ⟶ X′ and g: Y ⟶ Y′ in 𝒮_{∗}^{Σ} is given by (f ⊗ g)(α, x, y) = (α, f_{p}x, g_{q}y) for α ∈ Σ_{p+q}, x ∈ X_{p} and y ∈ Y_{q}.

The tensor product of symmetric sequences has the universal property for “bilinear maps”:

Proposition 2.1.4. Let X, Y, Z ∈ 𝒮_{∗}^{Σ} be symmetric sequences. There is a natural isomorphism

𝒮_{∗}^{Σ}(X ⊗ Y, Z) ≅ ∏_{p,q} 𝒮_{∗}^{Σ_{p}×Σ_{q}}(X_{p} ∧ Y_{q}, Z_{p+q})

The twist isomorphism τ: X⊗Y → Y ⊗X for X, Y ∈ 𝒮_{∗}^{Σ} is the natural map given by τ(α, x, y) = (αρ_{q,p}, y, x) for α ∈ Σ_{p+q}, x ∈ X_{p}, and y ∈ Y_{q}, where ρ_{q,p} ∈ Σ_{p+q} is the (q, p)-shuffle given by ρ_{q,p}(i) = i + p for 1 ≤ i ≤ q and ρ_{q,p}(i) = i − q for q < i ≤ p + q. The map defined without the shuffle permutation is not a map of symmetric sequences.

Remark 2.1.5. There is another way of describing the tensor product and the twist isomorphism. The category Σ is a skeleton of the category of finite sets and isomorphisms. Hence every symmetric sequence has an extension, which is unique up to isomorphism, to a functor on the category of all finite sets and isomorphisms. The tensor product of two such functors X and Y is the functor defined on a finite set C as

(X ⊗ Y)(C) = ⋁_{A∪B=C,A∩B=∅} X(A) ∧ Y (B).

For an isomorphism f: C → D the map (X ⊗ Y)(f) is the coproduct of the isomorphisms X(A) ∧ Y (B) → X(fA) ∧ Y (fB). The twist isomorphism is the map that sends the summand X(A)∧Y (B) of (X ⊗Y)(C) to the summand Y (B)∧X(A) of (Y ⊗ X)(C) by switching the factors.

Lemma 2.1.6. The tensor product ⊗ is a symmetric monoidal product on the category of symmetric sequences 𝒮_{∗}^{Σ}.

Proof. The unit of the tensor product is the symmetric sequence Σ(\overline{0}, −)_{+} = (S^{0}, ∗, ∗, …). The unit isomorphism is obvious. The associativity isomorphism is induced by the associativity isomorphism in 𝒮_{∗} and the natural isomorphism

((X ⊗ Y) ⊗ Z)_{n} ≅ ⋁_{p+q+r=n}(Σ_{n})_{+} ∧_{Σ_{p}×Σ_{q}×Σ_{r}} (X_{p} ∧ Y_{q} ∧ Z_{r}).

The twist isomorphism is described in Remark 2.1.5. The coherence of the natural isomorphisms follows from coherence of the natural isomorphisms for the smash product in 𝒮_{∗}.

We now introduce several functors on the category of symmetric sequences.

Definition 2.1.7. The evaluation functor Ev_{n}: 𝒮_{∗}^{Σ} → 𝒮_{∗} is given by Ev_{n} X = X_{n} and Ev_{n} f = f_{n}. The free functor G_{n}: 𝒮_{∗} → 𝒮_{∗}^{Σ} is the left adjoint of the evaluation functor Ev_{n}. The smash product X ∧ K of X ∈ 𝒮_{∗}^{Σ} and K ∈ 𝒮_{∗} is the symmetric sequence (X ∧ K)_{n} = X_{n} ∧ K with the diagonal action of Σ_{n} that is trivial on K. The pointed simplicial set Map_{𝒮_{∗}^{Σ}} (X, Y) of maps from X to Y is the pointed simplicial set 𝒮_{∗}^{Σ}(X ∧ Δ[−]_{+}, Y).

For each n ≥ 0, the free symmetric sequence is Σ[n] = Σ(\overline{n}, −) and the free functor is G_{n} = Σ[n]_{+} ∧−: 𝒮_{∗} → 𝒮_{∗}^{Σ}. So, for a pointed simplicial set K, (G_{n}K)_{n} = (Σ_{n})_{+} ∧ K and (G_{n}K)_{k} = ∗ for k ≠ n. In particular, G_{n}S^{0} = Σ[n]_{+}, G_{0}K = (K, ∗, ∗, …) and G_{0}S^{0} is the unit of the tensor product ⊗.

We leave the proof of the following basic proposition to the reader.

Proposition 2.1.8. There are natural isomorphisms:

1. G_{p}K ⊗ G_{q}L ≅ G_{p+q}(K ∧ L) for K, L ∈ 𝒮_{∗}.

2. X ⊗ G_{0}K ≅ X ∧ K for K ∈ 𝒮_{∗} and X ∈ 𝒮_{∗}^{Σ}.

3. Map_{𝒮_{∗}^{Σ}} (G_{n}K, X) ≅ Map_{𝒮_{∗}} (K, X_{n}) for K ∈ 𝒮_{∗} and X ∈ 𝒮_{∗}^{Σ}.

4. Map_{𝒮_{∗}^{Σ}} (X ⊗ Y, Z) ≅ ∏_{p,q} Map_{Σ_{p}×Σ_{q}}(X_{p} ∧ Y_{q}, Z_{p+q}) for X, Y, Z ∈ Sp^{Σ}.

A map f of symmetric sequences is a level equivalence if each of the maps f_{n} is a weak equivalence. Since 𝒮_{∗}^{Σ} is a product category, a map f of symmetric sequences is a monomorphism if and only if each of the maps f_{n} is a monomorphism.

Proposition 2.1.9. Let X be a symmetric sequence, f be a map of symmetric sequences and g be a map of pointed simplicial sets.

1. X ⊗ (−) preserves colimits.

2. If f is a monomorphism then X ⊗ f is a monomorphism.

3. If f is a level equivalence then X ⊗ f is a level equivalence.

4. If g is a monomorphism then G_{n}g is a monomorphism for n ≥ 0.

5. If g is a weak equivalence then G_{n}g is a level equivalence for n ≥ 0.

Proof. Parts (1), (2) and (3) follow from the definition of ⊗ and the corresponding properties for the smash product of pointed simplicial sets. For Parts (4) and (5) use the isomorphism G_{n}K = Σ[n]_{+} ∧ K.

By part three of Proposition 2.1.8, Map(Σ[n]_{+}, X) ≅ X_{n}. As n varies, Σ[−]_{+} is a functor Σ^{op} → 𝒮_{∗}^{Σ}, and for X ∈ 𝒮_{∗}^{Σ}, the symmetric sequence Map_{𝒮_{∗}^{Σ}} (Σ[−]_{+}, X) is naturally isomorphic to X.

Definition 2.1.10. Let X and Y be symmetric sequences. The symmetric sequence of maps from X to Y is

Hom_{Σ}(X, Y) = Map_{𝒮_{∗}^{Σ}} (X ⊗ Σ[−]_{+}, Y).

Theorem 2.1.11. The tensor product is a closed symmetric monoidal product on the category of symmetric sequences.

Proof. The tensor product is a symmetric monoidal product by Lemma 2.1.6. The product is closed if there is a natural isomorphism

𝒮_{∗}^{Σ}(X ⊗ Y, Z) ≅ 𝒮_{∗}^{Σ} (X, Hom_{Σ}(Y, Z)) .

for symmetric sequences X, Y and Z.

By Proposition 2.1.4, a map of symmetric sequences f: X ⊗ Y → Z is a collection of Σ_{p} × Σ_{q}-equivariant maps f_{p,q}: X_{p} ∧ Y_{q} → Z_{p+q}. This is adjoint to a collection of Σ_{p}-equivariant maps g_{p,q}: X_{p} → Map_{Σ_{q}}(Y_{q}, Z_{p+q}). So there is a natural isomorphism

𝒮_{∗}^{Σ}(X ⊗ Y, Z) ≅ ∏_{p} 𝒮_{∗}^{Σ_{p}}(X_{p}, ∏_{q} Map_{Σ_{q}}(Y_{q}, Z_{p+q}))

By Proposition 2.1.8, the functor sending \overline{p} to ∏_{q} Map_{Σ_{q}}(Y_{q}, Z_{p+q}) is the functor sending \overline{p} to Map(Y ⊗Σ[p]_{+}, Z) which by definition is Hom_{Σ}(Y, Z). Combining the isomorphisms gives the natural isomorphism that finishes the proof.

2.2. Symmetric spectra. In this section we recall the language of “monoids” and “modules” in a symmetric monoidal category and apply it to the category of symmetric sequences. In this language, the symmetric sequence of spheres S = (S^{0}, S^{1}, …, S^{n}, …) is a commutative monoid in the category of symmetric sequences and a symmetric spectrum is a (left) S-module.

Consider the symmetric sphere spectrum S. By Proposition 2.1.4, the natural Σ_{p} × Σ_{q}-equivariant maps m_{p,q}: S^{p} ∧ S^{q} → S^{p+q} give a pairing m: S ⊗ S → S. The adjoint G_{0}S^{0} → S of the identity map S^{0} → Ev_{0} S = S^{0} is a two sided unit of the pairing. The diagram of natural isomorphisms

S^{p} ∧ S^{q} ∧ S^{r} ⟶ S^{p} ∧ S^{q+r}
↓  ↓
S^{p+q} ∧ S^{r} ⟶ S^{p+q+r}

commutes, showing that m is an associative pairing of symmetric sequences.

A symmetric spectrum X has an underlying symmetric sequence of pointed simplicial sets {X_{n}} and a collection of pointed Σ_{p} ×Σ_{q}-equivariant maps σ^{p}: S^{p} ∧ X_{q} → X_{p+q} for p ≥ 1 and q ≥ 0. Let σ^{0} : S^{0} ∧ X_{n} → X_{n} be the unit isomorphism. Then the diagram

[figure: S^{p} ∧ S^{q} ∧ X_{r}; S^{p}∧σ^{q}; ≅; S^{p} ∧ X_{q+r}; σ^{p}; S^{p+q} ∧ X_{r}; σ^{p+q}; X_{p+q+r}]

commutes for p, q, r ≥ 0. By Proposition 2.1.4, the equivariant maps σ^{p} give a pairing σ: S ⊗ X → X such that the composition G_{0}(S^{0}) ⊗ X → S ⊗ X → X is the unit isomorphism and the diagram

[figure: S ⊗ S ⊗ X; S⊗σ; m⊗X; S ⊗ X; σ; S ⊗ X; σ; X]

commutes.

In the language of monoidal categories, S is a monoid in the category of symmetric sequences and a symmetric spectrum is a left S-module. Moreover, S is a commutative monoid, i.e., the diagram

[figure: S ⊗ S; τ; m; S ⊗ S; m; S] ,

commutes, where τ is the twist isomorphism. To see this, one can use either the definition of the twist isomorphism or the description given in Remark 2.1.5. Then, as is the case for commutative monoids in the category of sets and for commutative monoids in the category of abelian groups (i.e., commutative rings), there is a tensor product ⊗_{S}, having S as the unit. This gives a symmetric monoidal product on the category of S-modules. The smash product X ∧ Y of X, Y ∈ Sp^{Σ} is the symmetric spectrum X ⊗_{S} Y.

We review the necessary background on monoidal categories. Monoids and modules can be defined in any symmetric monoidal category, see [ML71]. Let ⊗ be a symmetric monoidal product on a category 𝒞 with unit e ∈ 𝒞. A monoid in 𝒞 is an object R ∈ 𝒞, a multiplication μ: R ⊗ R → R, and a unit map η: e → R such that the diagram

R ⊗ R ⊗ R ⟶^{m⊗R} R ⊗ R
R⊗m↓  ↓^{m}
R ⊗ R ⟶_{m} R

commutes (i.e., m is associative) and such that the compositions e ⊗ R ⟶^{η⊗1} R ⊗ R ⟶^{μ} R and R ⊗ e ⟶^{η⊗1} R ⊗ R ⟶^{μ} R are the unit isomorphisms of the product ⊗. The monoid R is commutative if μ = μ ∘ τ where τ is the twist isomorphism of ⊗. A left R-module is an object M with an associative multiplication α: R ⊗ M → M that respects the unit; right R-modules are left R^{op}-modules, where R^{op} is R with the multiplication μ ∘ τ. A map of left R-modules M and N is a map f: M → N in 𝒞 that commutes with the left R-actions. The category of left R-modules is denoted R-Mod. If 𝒞 is complete, the module category R-Mod is complete and the forgetful functor R-Mod → 𝒞 preserves limits. If 𝒞 is cocomplete and the functor R ⊗ (−) preserves coequalizers (in fact it suffices that R ⊗ (−) preserve reflexive coequalizers) then the module category R-Mod is cocomplete and the forgetful functor R-Mod → 𝒞 preserves colimits.

The symmetric sequence of spheres S = (S^{0}, S^{1}, S^{2}, …) is a commutative monoid in the category of symmetric sequences.

Proposition 2.2.1. The category of symmetric spectra is naturally equivalent to the category of left S-modules.

Proof. A pairing m: S ⊗ X → X is the same as a collection of Σ_{p} × Σ_{q}-equivariant maps m_{p,q}: S^{p} ∧ X_{q} → X_{p+q}. If X is a left S-module, there is a spectrum for which X is the underlying symmetric sequence and the structure maps are the maps σ = m_{1,n}: S^{1}∧X_{n} → X_{n+1}. The compositions σ^{p} are the Σ_{p}×Σ_{q}-equivariant maps m_{p,q}. Conversely, for X a symmetric spectrum, the map of symmetric sequences m: S ⊗X → X corresponding to the collection of Σ_{p} ×Σ_{q}-equivariant maps m_{p,q} = σ^{p}: S^{p} ∧ X_{q} → X_{p+q}, where σ^{0} is the natural isomorphism S^{0} ∧ X_{n} → X_{n}, makes X a left S-module. These are inverse constructions and give a natural equivalence of categories.

The smash product on the category of symmetric spectra is a special case of the following lemma.

Lemma 2.2.2. Let 𝒞 be a symmetric monoidal category that is cocomplete and let R be a commutative monoid in 𝒞 such that the functor R ⊗ (−): 𝒞 → 𝒞 preserves coequalizers. Then there is a symmetric monoidal product ⊗_{R} on the category of R-modules with R as the unit.

Proof. Let Q be a monoid in 𝒞, M be a right Q-module, and N be a left Q-module. The tensor product M ⊗_{Q} N is the colimit in 𝒞 of the diagram

[figure: M ⊗ Q ⊗ N; m⊗1; 1⊗m; M ⊗ N] .

If M is an (R, Q)-bimodule (i.e., the right action of Q commutes with the left action of R) then this is a diagram of left R-modules. Since the functor R ⊗ (−) preserves coequalizers, the colimit M ⊗_{Q} N is a left R-module. If M is a left module over the commutative monoid R, the composition M ⊗R ⟶^{τ} R⊗M ⟶^{α} M is a right action of R; since R is commutative, the two actions commute and M is an (R, R)-bimodule. Hence, the tensor product M ⊗_{R} N is a left R-module. The unit of the product ⊗_{R} is R as a left R-module. The associativity, unit, and twist isomorphisms of the product on 𝒞 induce corresponding isomorphisms for the product ⊗_{R} on the category of left R-modules. Thus ⊗_{R} is a symmetric monoidal product.

Definition 2.2.3. The smash product X ∧ Y of symmetric spectra X and Y is the symmetric spectrum X ⊗_{S} Y.

Apply Lemma 2.2.2 to the commutative monoid S in the bicomplete category of symmetric sequences 𝒮_{∗}^{Σ} to obtain the following corollary.

Corollary 2.2.4. The smash product X ∧ Y is a symmetric monoidal product on the category of symmetric spectra.

Next, some important functors on the category of symmetric spectra.

Definition 2.2.5. The functor S ⊗ (−): 𝒮_{∗}^{Σ} → Sp^{Σ} gives the free S-module S ⊗ X generated by the symmetric sequence X. For each n ≥ 0, the evaluation functor Ev_{n}: Sp^{Σ} → 𝒮_{∗} is given by Ev_{n} X = X_{n} and Ev_{n} f = f_{n}. The free functor F_{n}: 𝒮_{∗} → Sp^{Σ} is the left adjoint of the evaluation functor Ev_{n}. The functor R_{n}: 𝒮_{∗} → Sp^{Σ} is the right adjoint of the evaluation functor Ev_{n}: Sp^{Σ} → 𝒮_{∗}.

The functor S ⊗ (−) is left adjoint to the forgetful functor Sp^{Σ} → 𝒮_{∗}^{Σ}. The free functor F_{n} is the composition S ⊗ G_{n} of the left adjoints G_{n}: 𝒮_{∗} → 𝒮_{∗}^{Σ} (Definition 2.1.7) and S ⊗ (−): 𝒮_{∗}^{Σ} → Sp^{Σ}. Thus, for X ∈ Sp^{Σ} and K ∈ 𝒮_{∗}, the left S-module X ∧ F_{n}K is naturally isomorphic to the left S-module X ⊗ G_{n}K. In particular, X ∧ F_{0}K is naturally isomorphic to the symmetric spectrum X ∧ K defined by prolongation in Section 1.3. Furthermore F_{0}K = S ∧ K is the symmetric suspension spectrum Σ^{∞}K of K, and F_{0}S^{0} is the symmetric sphere spectrum S. For a pointed simplicial set K, R_{n}K is the symmetric sequence Hom_{𝒮_{∗}^{Σ}} (S, K^{Σ(−,\overline{n})_{+}}), which is a left S-module since S is a right S-module.

We leave the proof of the following proposition to the reader.

Proposition 2.2.6. There are natural isomorphisms:

1. F_{m}(K) ∧ F_{n}(L) ≅ F_{m+n}(K ∧ L) for K, L ∈ 𝒮_{∗}.

2. Map_{Sp^{Σ}}(S ⊗ X, Y) ≅ Map_{𝒮_{∗}^{Σ}} (X, Y) for X ∈ 𝒮_{∗}^{Σ} and Y ∈ Sp^{Σ}.

3. Map_{Sp^{Σ}}(F_{n}K, X) ≅ Map_{𝒮_{∗}} (K, Ev_{n} X) for K ∈ 𝒮_{∗} and X ∈ Sp^{Σ}.

Proposition 2.2.7. Let f be a map of pointed simplicial sets.

1. F_{n}: 𝒮_{∗} → Sp^{Σ} preserves colimits.

2. If f is a monomorphism then F_{n}f is a monomorphism.

3. If f is a weak equivalence then F_{n}f is a level equivalence.

Proof. Use the isomorphism F_{n}f = S ⊗ G_{n}f and Proposition 2.1.9.

The internal Hom on the category of symmetric spectra is a special case of the following lemma.

Lemma 2.2.8. Let 𝒞 be a closed symmetric monoidal category that is bicomplete and let R be a commutative monoid in 𝒞. Then there is a function R-module Hom_{R}(M, N), natural for M, N ∈ 𝒞, such that the functor (−) ⊗_{R} M is left adjoint to the functor Hom_{R}(M, −)

Proof. Let R be a monoid in 𝒞 and let M and N be left R-modules. Then Hom_{R}(M, N) is the limit in 𝒞 of the diagram

[figure: Hom_{𝒞} (M, N); m^{∗}; m_{∗}; Hom_{𝒞} (R ⊗ M, N)]

where m^{∗} is pullback along the multiplication m: R ⊗ M → M and m_{∗} is the composition

Hom_{𝒞} (M, N) ⟶^{R⊗−} Hom_{𝒞} (R ⊗ M, R ⊗ N) ⟶^{m_{∗}} Hom_{𝒞} (R ⊗ M, N)

If Q is another monoid and N is an (R, Q)-bimodule then this is a diagram of right Q-modules and the limit Hom_{R}(M, N) is a right Q-module. If R is a commutative monoid and N is a left R-module then the left action is also a right action and N is an (R, R)-bimodule. So Hom_{R}(M, N) is a right R-module and hence a left R-module. It follows from the properties of the internal Hom in 𝒞 and the definition of ⊗_{R} that (−) ⊗_{R} M is left adjoint to Hom_{R}(M, −).

Definition 2.2.9. Let X and Y be symmetric spectra. The function spectrum Hom_{S}(X, Y) is the limit of the diagram in Sp^{Σ}

[figure: Hom_{Σ}(X, Y); m^{∗}; m_{∗}; Hom_{Σ}(S ⊗ X, Y)] .

Combining Lemmas 2.2.2 and 2.2.8:

Theorem 2.2.10. The smash product is a closed symmetric monoidal product on the category of symmetric spectra. In particular, there is a natural adjunction isomorphism

Sp^{Σ}(X ∧ Y, Z) ≅ Sp^{Σ}(X, Hom_{S}(Y, Z)).

Proof. The smash product ∧ is a symmetric monoidal product by Corollary 2.2.4. The adjunction isomorphism follows from Lemma 2.2.8

The adjunction is also a simplicial adjunction and an internal adjunction.

Corollary 2.2.11. There are natural isomorphisms

Map_{Sp^{Σ}}(X ∧ Y, Z) ≅ Map_{Sp^{Σ}}(X, Hom_{S}(Y, Z)).

and

Hom_{S}(X ∧ Y, Z) ≅ Hom_{S}(X, Hom_{S}(Y, Z))

Remark 2.2.12. We use Proposition 2.2.6 to give another description of the function spectrum Hom_{S}(X, Y). For a symmetric spectrum X, the pointed simplicial set of maps Map_{Sp^{Σ}}(F_{n}S^{0}, X) is naturally isomorphic to Map_{𝒮_{∗}} (S^{0}, Ev_{n} X) = X_{n}. The symmetric spectrum F_{n}S^{0} is the S-module S⊗Σ[n]_{+} and as n varies, S⊗Σ[−]_{+} is a functor Σ^{op} → Sp^{Σ}. The symmetric sequence Map_{Sp^{Σ}}(S ⊗ Σ[−]_{+}, X) is the underlying symmetric sequence of X. In particular, the natural isomorphism X_{n} = Map_{Sp^{Σ}}(F_{n}S^{0}, X) is Σ_{n}-equivariant. Applying this to Hom_{S}(X, Y) and using Corollary 2.2.11, we find that the underlying symmetric sequence of Hom_{S}(X, Y) is the symmetric sequence Map_{Sp^{Σ}}(X ∧ (S ⊗ Σ[−]_{+}), Y).

We must also describe the structure maps of X from this point of view. Recall that Map_{Sp^{Σ}}(F_{n}S^{0}, X) = X_{n}, Map_{Sp^{Σ}}(F_{n}S^{1}, X) = Map_{𝒮_{∗}} (S^{1}, X_{n}). Let λ: F_{1}S^{1} → F_{0}S^{0} be the adjoint of the identity map S^{1} → Ev_{1} F_{0}S^{0} = S^{1}. The induced map Map_{Sp^{Σ}}(λ, X): X_{0} → Map_{𝒮_{∗}} (S^{1}, X_{1}) is adjoint to the structure map S^{1} ∧ X_{0} → X_{1}. The map

λ ∧ F_{n}S^{0} : F_{1}S^{1} ∧ F_{n}S^{0} = F_{n+1}S^{1} → F_{0}S^{0} ∧ F_{n}S^{0} = F_{n}S^{0}

is Σ_{1} × Σ_{n}-equivariant; the induced map

Map_{Sp^{Σ}}(λ ∧ F_{n}S^{0}, X): X_{n} → Map_{𝒮_{∗}} (S^{1}, X_{n+1})

is Σ_{1} × Σ_{n}-equivariant and is adjoint to the structure map σ: S^{1} ∧ X_{n} → X_{n+1}. In order to apply this to Hom_{S}(X, Y), use Proposition 2.2.6 and Corollary 2.2.11 to find a natural isomorphism

Map_{Sp^{Σ}}(X ∧ F_{n+1}S^{1}, Y) ≅ Map_{𝒮_{∗}} (S^{1}, Map_{Sp^{Σ}}(X ∧ F_{n+1}S^{0}, Y).

Using this natural isomorphism, we find that the structure maps of Hom_{S}(X, Y) are the adjoints of the maps

Map_{Sp^{Σ}}(X ∧ F_{n}S^{0}, Y) → Map_{Sp^{Σ}}(X ∧ F_{n+1}S^{1}, Y)

induced by λ ∧ F_{n}S^{0}.

For example, Hom_{S}(F_{k}S^{0}, X) is the k-shifted spectrum; its underlying symmetric sequence is the sequence of pointed simplicial sets

X_{k}, X_{1+k}, …, X_{n+k} …

with Σ_{n} acting on X_{n+k} by restricting the action of Σ_{n+k} to the copy of Σ_{n} that permutes the first n elements of \overline{n + k}. The structure maps of the k-shifted spectrum are the structure maps σ: S^{1} ∧ X_{n+k} → X_{n+k+1} of X. More generally, Hom_{S}(F_{k}K, X) is the k-shifted spectrum of X^{K}.

2.3. The ordinary category of spectra. An approach similar to the last two sections can be used to describe (non-symmetric) spectra as modules over the sphere spectrum in a symmetric monoidal category. But in this case the sphere spectrum is not a commutative monoid, which is why there is no closed symmetric monoidal smash product of spectra.

We begin as in Section 2.1 by considering a category of sequences.

Definition 2.3.1. The category ℕ is the category with the non-negative integers as its objects and with the identity maps of the objects as its only maps. The category of sequences 𝒮_{∗}^{ℕ} is the category of functors from ℕ to 𝒮_{∗}. An object of 𝒮_{∗}^{ℕ} is a sequence X_{0}, X_{1}, …, X_{n}, … of pointed simplicial sets and a map f: X ⟶ Y is a sequence of pointed simplicial maps f_{n}: X_{n} → Y_{n}.

Definition 2.3.2. The graded smash product of sequences X and Y is the sequence X ⊗ Y given in degree n by

(X ⊗ Y)_{n} = ⋁_{p+q=n} X_{p} ∧ Y_{q}

Lemma 2.3.3. The category of sequences is a bicomplete category and the graded smash product is a symmetric monoidal product on 𝒮_{∗}^{ℕ}.

Proof. Limits and colimits are defined at each object, so 𝒮_{∗}^{ℕ} has arbitrary limits and colimits since 𝒮_{∗} does. The graded tensor product is a symmetric monoidal product on 𝒮_{∗}^{ℕ}. The unit is the sequence S^{0}, ∗, ∗, · · · . The coherence isomorphisms for the graded smash product follow from the corresponding coherence isomorphisms in 𝒮_{∗}. The twist isomorphism takes (x, y) to (y, x).

Proposition 2.3.4. The sequence S whose nth level is S^{n} is a monoid in the category of sequences. The category of left S-modules is isomorphic to the ordinary category of spectra, Sp^{ℕ}.

Proof. By definition, a pairing of sequences X ⊗ Y → Z is the same as a collection of maps X_{p} ∧ Y_{q} → Z_{p+q} for p, q ≥ 0. The associative pairing S^{p} ∧ S^{q} → S^{p+q} gives an associative pairing of sequences S ⊗S → S with an obvious unit map. Thus S is a monoid. Since S^{p} = S^{1} ∧ … ∧ S^{1}, a left action of the monoid S on the sequence X is the same as a collection of maps S^{1} ∧ X_{q} → X_{q+1} for q ≥ 0.

Suppose there were a closed symmetric monoidal structure on Sp^{ℕ} with S as the unit; since Sp^{ℕ} is the category of left S-modules it would follow that S is a commutative monoid in the category of sequences. But the twist map on S^{1} ∧ S^{1} is not the identity map and so S is not a commutative monoid in 𝒮_{∗}^{ℕ}. Therefore there is no closed symmetric monoidal smash product of spectra. Note that the category of right S-modules is isomorphic to the category of left S-modules, even though S is not commutative. On the other hand, S is a graded commutative monoid up to homotopy, in fact up to E_{∞}-homotopy. This observation underlies Boardman’s construction of handicrafted smash products [Ada74].

3. Stable homotopy theory of symmetric spectra

To use symmetric spectra for the study of stable homotopy theory, one should have a stable model category of symmetric spectra such that the category obtained by inverting the stable equivalences is naturally equivalent to Boardman’s stable homotopy category of spectra (or to any other known to be equivalent to Boardman’s). In this section we define the stable model category of symmetric spectra. In Section 4 we show that it is Quillen equivalent to the stable model category of spectra discussed in [BF78].

In Section 3.1 we define the class of stable equivalences of symmetric spectra and discuss its non-trivial relationship to the class of stable equivalences of (nonsymmetric) spectra. In Section 3.2 we recall the axioms and basic theory of model categories. In Section 3.3 we discuss the level structure in Sp^{Σ}, and in Section 3.4 we define the stable model structure on the category of symmetric spectra which has the stable equivalences as the class of weak equivalences. The rest of the section is devoted to checking that the stable model structure satisfies the axioms of a model category.

3.1. Stable equivalence. One’s first inclination is to define stable equivalence using the forgetful functor U: Sp^{Σ} → Sp^{ℕ}; one would like a map f of symmetric spectra to be a stable equivalence if the underlying map Uf of spectra is a stable equivalence, i.e., if Uf induces an isomorphism of stable homotopy groups. The reader is warned: THIS WILL NOT WORK. Instead, stable equivalence is defined using cohomology; a map f of symmetric spectra is a stable equivalence if the induced map E^{∗}f of cohomology groups is an isomorphism for every generalized cohomology theory E. The two alternatives, using stable homotopy groups or using cohomology groups, give equivalent definitions on the category of (non-symmetric) spectra but not on the category of symmetric spectra.

It would be nice if the 0th cohomology group of the symmetric spectrum X with coefficients in the symmetric Ω-spectrum E could be defined as π_{0} Map_{Sp^{Σ}}(X, E), the set of simplicial homotopy classes of maps from X to E. But, even though the contravariant functor E^{0} = π_{0} Map_{Sp^{Σ}}(−, E) takes simplicial homotopy equivalences to isomorphisms, E^{0} may not take level equivalences to isomorphisms. This is a common occurrence in simplicial categories, but is a problem as every level equivalence should induce an isomorphism of cohomology groups; a level equivalence certainly induces an isomorphism of stable homotopy groups. We introduce injective spectra as a class of spectra E for which the functor E^{0} behaves correctly.

Definition 3.1.1. An injective spectrum is a symmetric spectrum E that has the extension property with respect to every monomorphism f of symmetric spectra that is a level equivalence. That is, for every diagram in Sp^{Σ}

\begin{matrix}X & ⟶^{g} & E \\ f & ↓ \\ Y \end{matrix}

where f is a monomorphism and a level equivalence there is a map h: Y → E such that g = hf.

Some examples of injective spectra follow. Recall that R_{n}: 𝒮_{∗} → Sp^{Σ} is the right adjoint of the evaluation functor Ev_{n}: Sp^{Σ} → 𝒮_{∗}. Also recall that a Kan complex has the extension property with respect to every map of pointed simplicial sets that is a monomorphism and a weak equivalence.

Lemma 3.1.2. If the pointed simplicial set K is a Kan complex then R_{n}K is an injective spectrum. If X is a symmetric sequence and E is an injective spectrum then Hom_{S}(S ⊗ X, E) is an injective spectrum.

Proof. Since Ev_{n} is left adjoint to R_{n}, the spectrum R_{n}K has the extension property with respect to the monomorphism and level equivalence f if and only if K has the extension property with respect to the monomorphism and weak equivalence Ev_{n} f. Since K is a Kan complex, it does have the extension property with respect to Ev_{n} f. Hence R_{n}K is injective.

Since (S⊗X)∧(−) is the left adjoint of Hom_{S}(S⊗X, −), the spectrum Hom_{S}(S⊗ X, E) has the extension property with respect to the monomorphism and level equivalence f if and only E has the extension property with respect to the map (S ⊗ X) ∧ f. There is a natural isomorphism of maps of symmetric sequences (S ⊗X)∧f ≅ X ⊗f. Since f is a monomorphism and a level equivalence, X ⊗f is a monomorphism and a level equivalence by Proposition 2.1.9. Thus (S⊗X)∧f is also a monomorphism and level equivalence of symmetric spectra. So Hom_{S}(S ⊗ X, E) is injective.

In fact, injective spectra are the fibrant objects of a model structure on Sp^{Σ} for which every object is cofibrant (Section 5.1). In particular, we will see in Corollary 5.1.3, there are enough injectives; every symmetric spectrum embeds in an injective spectrum by a map that is a level equivalence.

Definition 3.1.3. A map f: X → Y of symmetric spectra is a stable equivalence if E^{0}f: E^{0}Y → E^{0}X is an isomorphism for every injective Ω-spectrum E.

There are two other ways to define stable equivalence.

Proposition 3.1.4. Let f: X → Y be a map of symmetric spectra. The following conditions are equivalent:

• E^{0}f is an isomorphism for every injective Ω-spectrum E;

• Map_{Sp^{Σ}}(f, E) is a weak equivalence for every injective Ω-spectrum E;

• Hom_{S}(f, E) is a level equivalence for every injective Ω-spectrum E.

Proof. Let K be a pointed simplicial set and E be a symmetric Ω-spectrum. The adjoints of the structure maps of E are weak equivalences of Kan complexes. From Remark 2.2.12, for k, n ≥ 0 Ev_{k} Hom_{S}(F_{n}K, E) = E_{n+k}^{K}. The adjoints of the structure maps of Hom_{S}(F_{n}K, E) are the weak equivalences of Kan complexes E_{n+k}^{K} → E_{n+k+1}^{S^{1}∧K} induced by the adjoints of the structure maps of E. Therefore, Hom_{S}(F_{n}K, E) is an Ω-spectrum.

Now let E be an injective Ω-spectrum. Using the natural isomorphism F_{n}K ≅ S⊗G_{n}K and Lemma 3.1.2, Hom_{S}(F_{n}K, E) is an injective spectrum. By the preceding paragraph, Hom_{S}(F_{n}K, E) is an Ω-spectrum. Hence E^{S^{n}} = Hom_{S}(F_{0}(S^{n}), E) and the k-shifted spectrum Hom_{S}(F_{k}S^{0}, E) are injective Ω-spectra. Given a stable equivalence f: X ⟶ Y, we want to show that Map_{Sp^{Σ}}(f, E) is a weak equivalence. Since

π_{n} Map_{Sp^{Σ}}(f, E) = π_{0} Map_{Sp^{Σ}}(f, E^{S^{n}})

and the simplicial sets Map_{Sp^{Σ}}(Y, E) and Map_{Sp^{Σ}}(X, E) are Kan complexes by Lemma 3.1.5, Map_{Sp^{Σ}}(f, E) is a weak equivalence on the basepoint components.

We must extend this to all components. To do so, note that Map_{Sp^{Σ}}(f, E)^{S^{1}} is a weak equivalence for any injective Ω-spectrum E, since the loop space only depends on the basepoint component. Consider the commutative diagram

Map_{Sp^{Σ}}(Y, E) ⟶ Map_{Sp^{Σ}}(Y, Hom_{S}(F_{1}S^{0}, E))^{S^{1}}
Map_{Sp^{Σ}} (f,E)↓  ↓
Map_{Sp^{Σ}}(X, E) ⟶ Map_{Sp^{Σ}}(X, Hom_{S}(F_{1}S^{0}, E))^{S^{1}}

where the horizontal maps are induced by the map E ⟶ Hom_{S}(F_{1}S^{0}, E)^{S^{1}} adjoint to the structure map of E. Since E is an injective Ω-spectrum, this map is a level equivalence of injective spectra. By Lemma 3.1.6, it is a simplicial homotopy equivalence. Hence the horizontal maps in the diagram above are weak equivalences. Since the right-hand vertical map is a weak equivalence, so is the left-hand vertical map Map_{Sp^{Σ}}(f, E). Thus the first two conditions in the proposition are equivalent. Since Ev_{k} Hom_{S}(f, E) = Map_{Sp^{Σ}}(f, Hom_{S}(F_{k}S^{0}, E)), the second two conditions are equivalent.

Lemma 3.1.5. Suppose X is a symmetric spectrum and E is an injective spectrum. Then the pointed simplicial set Map_{Sp^{Σ}}(X, E) is a Kan complex. In particular, each pointed simplicial set E_{n} is a Kan complex.

Proof. Suppose that f: K ⟶ L is a monomorphism and weak equivalence of simplicial sets. We must show that Map_{Sp^{Σ}}(X, E) has the extension property with respect to f. By adjointness, this is equivalent to showing that E has the extension property with respect to X ∧f. But X ∧f is a monomorphism and level equivalence by Corollary 1.3.6 applied to the monomorphism ∗ ⟶ X and f, so E does have the required extension property.

The basic properties of injective spectra which are needed in the rest of this section are stated in the following lemma.

Lemma 3.1.6. Let f: X → Y be a map of symmetric spectra.

1. If E ∈ Sp^{Σ} is an injective spectrum and f is a level equivalence then E^{0}f is an isomorphism of sets.

2. If f: X → Y is a map of injective spectra, f is a level equivalence if and only if f is a simplicial homotopy equivalence.

The proof uses the following construction.

Construction 3.1.7 (Mapping cylinder construction). The mapping cylinder construction for maps of symmetric spectra is the prolongation of the reduced mapping cylinder construction for maps of pointed simplicial sets. Let i_{0} and i_{1} be the two inclusions Δ[0] → Δ[1]. The cylinder spectrum Cf of a map f: X → Y ∈ Sp^{Σ} is the corner of the pushout square

X ∧ Δ[0]_{+} = X ⟶^{f} Y
X∧i_{0}↓  ↓^{s}
X ∧ Δ[1]_{+}  ⟶_{g} Cf

Let i = g ∘(X ∧i_{1}): X → Cf. Let r: Cf → Y be the map on the pushout induced by the identity map on Y and the composition X ∧ Δ[1]_{+} → X → Y. Then f = ri, i is a monomorphism, rs = id_{Y}, and there is a simplicial homotopy from sr to the identity map of Cf.

Proof of Lemma 3.1.6. For part one of the lemma, let f: X → Y be a level equivalence and let Cf be the mapping cylinder of f. As above f = ri, i: X → Cf is a monomorphism, and r: Cf → Y is a simplicial homotopy equivalence. Then E^{0}r is an isomorphism and, if E^{0}i is an isomorphism, the composition E^{0}f is an isomorphism. The map i is a monomorphism which, by the 2-out-of-3 property, is a level equivalence. By the extension property of E with respect to i, the map E^{0}i is surjective . The inclusion of the boundary j: ∂Δ[1] → Δ[1] is a monomorphism. By Corollary 1.3.6, the map

i □ j: X ∧ Δ[1]_{+} ⨿_{X∧∂Δ[1]_{+}} Cf ∧ ∂Δ[1]_{+} → Cf ∧ Δ[1]_{+}

is a monomorphism and a level equivalence. The extension property of E with respect to i □ j implies that if g, h: Cf → E are maps such that gi and hi are simplicially homotopic, then g and h are simplicially homotopic. So E^{0}i is a monomorphism and hence E^{0}i is an isomorphism.

For the second part of the lemma, if f is a simplicial homotopy equivalence, each f_{n} is a simplicial homotopy equivalence of simplicial sets and so each f_{n} is a weak equivalence. Conversely, suppose f: X → Y is a level equivalence of injective spectra. By part one, X^{0}f: X^{0}Y → X^{0}X is an isomorphism. The inverse image of the equivalence class of the identity map X → X is an equivalence class of maps Y → X. Since Y is injective, Y^{0}f: Y^{0}Y → Y^{0}X is an isomorphism. Hence each map in the equivalence class is a simplicial homotopy inverse of f.

Restricting part one of Lemma 3.1.6 to injective Ω-spectra gives:

Corollary 3.1.8. Every level equivalence of symmetric spectra is a stable equivalence.

Next recall the definition of stable homotopy equivalence in the category of (nonsymmetric) spectra Sp^{ℕ}.

Definition 3.1.9. For each integer k the kth homotopy group of the spectrum (or symmetric spectrum) X is the colimit

π_{k}X = colim_{n} π_{k+n}X_{n}.

of the directed system given by the compositions

π_{k+n}X_{n} ⟶^{E} π_{k+n+1}(S^{1} ∧ X_{n}) ⟶^{π_{k+n+1}σ} π_{k+n+1}X_{n+1}

of the suspension homomorphism and the map induced by σ.

A map of spectra f ∈ Sp^{ℕ} is a stable homotopy equivalence if π_{∗}f is an isomorphism. For example, every level equivalence in Sp^{ℕ} is a stable homotopy equivalence as it induces an isomorphism of homotopy groups. We do not define stable equivalence of symmetric spectra in this way; as the following example shows, a stable equivalence of symmetric spectra need not induce an isomorphism of homotopy groups.

Example 3.1.10. The map λ: F_{1}S^{1} → F_{0}S^{0} (see 2.2.12) is the adjoint of the identity map S^{1} → Ev_{1} S = S^{1}. The nth space of F_{1}S^{1} is (Σ_{n})_{+} ∧_{Σ_{n−1}} S^{n}, which is a wedge of n copies of S^{n}. One can calculate that π_{0}F_{1}S^{1} is an infinite direct sum of copies of the integers ℤ, whereas π_{0}F_{0}S^{0} is ℤ. So π_{∗}λ is not an isomorphism of homotopy groups and thus Uλ is not a stable homotopy equivalence of (non-symmetric) spectra. But λ is a stable equivalence of symmetric spectra. For a symmetric Ω-spectrum E, Map_{Sp^{Σ}}(F_{1}S^{1}, E) = Map_{𝒮_{∗}} (S^{1}, E_{1}), Map_{Sp^{Σ}}(F_{0}S^{0}, E) = E_{0}, and the induced map Map_{Sp^{Σ}}(λ, E) = E_{0} → E_{1}^{S^{1}} is adjoint to the structure map S^{1}∧E_{0} → E_{1}. So Map_{Sp^{Σ}}(λ, E) is a weak equivalence for every Ω-spectrum E, including the injective ones, and so λ is a stable equivalence. By the same argument, the maps λ ∧ F_{n}S^{0} are stable equivalences.

The forgetful functor U: Sp^{Σ} → Sp^{ℕ} does not preserve stable equivalences. On the other hand, the functor U does reflect stable equivalences.

Theorem 3.1.11. Let f be a map of symmetric spectra such that π_{∗}f is an isomorphism of homotopy groups. Then f is a stable equivalence.

Proof. To ease notation, let RX = Hom_{S}(F_{1}S^{1}, X), so that R^{n}X = Hom_{S}(F_{n}S^{n}, X) for n ≥ 0 and Ev_{k} R^{n}X = Map_{𝒮_{∗}} (S^{n}, X_{n+k}). In particular R^{0}X = X and there is a natural map λ^{∗}: X → RX induced by the map λ: F_{1}S^{1} ⟶ F_{0}S^{0} discussed in Example 3.1.10 and Remark 2.2.12. The maps R^{n}(λ^{∗}): R^{n}X → R^{n+1}X give a directed system. Let

R^{∞}X = colim_{n≥0} R^{n}X = colim_{n≥0} Hom_{S}(F_{n}S^{n}, X)

and let r_{X}: X → R^{∞}X be the natural map from X to the colimit.

Let E be an injective Ω-spectrum. Since E is an Ω-spectrum, the map

R^{n}(λ^{∗}): R^{n}E → R^{n+1}E

is a level equivalence for each n and the map r_{E}: E → R^{∞}E is a level equivalence. Since E is injective, the proof of part one of Lemma 3.1.6 applied to r_{E} shows that there is a map g: R^{∞}E → E such that the composition gr_{E} is simplicially homotopic to the identity map on E, though the other composite r_{E}g may not be simplicially homotopic to the identity map on R^{∞}E. There is a natural transformation E^{0}X → E^{0}(R^{∞}X) sending the map f: X → E to the map gR^{∞}f; there is a natural transformation E^{0}(R^{∞}X) → E^{0}X induced by composition with the map r_{X}: X → R^{∞}X. Since gr_{E} is simplicially homotopic to the identity map on E, the composition of the natural transformations is the identity natural transformation of the functor E^{0}. In the diagram

[figure: E^{0}Y; E^{0}f; E^{0}R^{∞}Y; E^{0}R^{∞}f; E^{0}Y; E^{0}f; E^{0}X; E^{0}R^{∞}X; E^{0}X]

the composition of the horizontal maps is the identity, showing that E^{0}f is a retract of E^{0}R^{∞}f.

Let X be a symmetric spectrum that is a level Kan complex, i.e., each X_{n} is a Kan complex. Since the functor π_{k} commutes with filtered colimits, the homotopy group π_{k} Ev_{n} R^{∞}X of the pointed simplicial set Ev_{n} R^{∞}X is naturally isomorphic to the homotopy group π_{k−n}X of the symmetric spectrum X. A warning: even though the groups π_{k} Ev_{n} R^{∞}X and π_{k+1} Ev_{n+1} R^{∞}X are abstractly isomorphic, the structure map of the symmetric spectrum R^{∞}X need not induce an isomorphism between them. In particular, despite its similarity to the standard construction of the Ω-spectrum associated to a (non-symmetric) spectrum, R^{∞}X need not be an Ω-spectrum and r_{X}: X → R^{∞}X need not induce an isomorphism of homotopy groups.

Now, let f be a map of symmetric spectra such that π_{∗}f is an isomorphism. Assume as well that X and Y are level Kan complexes. Then R^{∞}f is a level equivalence. By Proposition 3.1.8, E^{0}R^{∞}f is an isomorphism for every injective Ω-spectrum E. Thus E^{0}f, which is a retract of E^{0}R^{∞}f, is an isomorphism for every injective Ω-spectrum E, and so f is a stable equivalence.

To finish the proof, let f: X → Y be a map of arbitrary symmetric spectra such that π_{∗}f is an isomorphism. For every simplicial set X there is a natural weak equivalence X → KX where KX is a Kan complex. There are several such functors: K can be Kan’s Ex^{∞} functor; K can be the total singular complex of the geometric realization; or K can be constructed using a simplicial small object argument. In each case, K is a 𝒮_{∗}-functor and X → KX is a 𝒮_{∗}-natural transformation. By prolongation, for every symmetric spectrum X there is a natural level equivalence X → KX where KX is a level Kan complex. In the commutative diagram

X ⟶ KX
f↓  ↓^{Kf}
Y ⟶ KY

the horizontal maps are stable equivalences by 3.1.8; the map Kf is a stable equivalence by the preceding paragraph; hence f is a stable equivalence.

As a corollary some of the standard results about spectra translate into results about symmetric spectra.

Definition 3.1.12. Let (X, A) be a pair of symmetric spectra where A is a subspectrum of X. The kth relative homotopy group of the pair (X, A) is the colimit

π_{k}(X, A) = colim_{n} π_{k+n}(X_{n}, A_{n}; ∗).

of the relative homotopy groups of the pointed pairs of simplicial sets (X_{n}, A_{n}; ∗).

Lemma 3.1.13 (Stable excision). Let (X, A) be a pair of symmetric spectra with A a subspectrum of X. The map of homotopy groups π_{k}(X, A) → π_{k}(X/A) is an isomorphism.

Proof. Consider the diagram:

\xymatrix{π_{k+q}(X_{q}, A_{q}) \ar[r] \ar[d] & π_{k+q+p}(S^{p} ∧ X_{q}, S^{p} ∧ A_{q}) \ar[r] \ar[d] & π_{k+q+p}(X_{p+q}, A_{p+q}) \ar[d] \\ π_{k+q}(X_{q}/A_{q}) \ar[r] & π_{k+q+p}(S^{p} ∧ X_{q}/A_{q}) \ar[r] & π_{k+q+p}(X_{p+q}/A_{p+q})} .

Let (K, L) be a pair of pointed simplicial sets. By the homotopy excision theorem the map π_{n}(S^{p} ∧ K, S^{p} ∧ L) → π_{n}(S^{p} ∧ K/L) is an isomorphism when n < 2p. So the middle vertical arrow in the diagram is an isomorphism when p > k + q and hence the map of colimits π_{k}(X, A) → π_{k}(X/A) is an isomorphism.

Theorem 3.1.14. 1. Let f: X → B be a map of symmetric spectra such that f_{n}: X_{n} → B_{n} is a Kan fibration for each n ≥ 0 and let F be the fiber over the basepoint. Then the map X/F → B is a stable equivalence.

2. A map f ∈ Sp^{Σ} of symmetric spectra is a stable equivalence if and only if its suspension f ∧ S^{1} is a stable equivalence.

3. For symmetric spectra X and Y such that Y is a level Kan complex, a map X → Y^{S^{1}} is a stable equivalence if and only its adjoint X ∧ S^{1} → Y is a stable equivalence.

Proof. By stable excision, the map X/F → B induces an isomorphism of homotopy groups and hence is a stable equivalence by Theorem 3.1.11.

For part two, let E be an injective Ω-spectrum. By Lemma 3.1.2, the spectra E^{S^{1}}, Hom_{S}(F_{1}S^{0}, E) and Hom_{S}(F_{1}S^{1}, E) are injective Ω-spectra. If f is a stable equivalence of symmetric spectra then the map Map_{Sp^{Σ}}(f, E^{S^{1}}) = Map_{Sp^{Σ}}(f ∧ S^{1}, E) is a weak equivalence of simplicial sets and so f ∧ S^{1} is a stable equivalence. Conversely if f ∧ S^{1} is a stable equivalence then the map

Map_{Sp^{Σ}}(f ∧ S^{1}, Hom_{S}(F_{1}S^{0}, E)) = Map_{Sp^{Σ}}(f, Hom_{S}(F_{1}S^{1}, E))

is a weak equivalence. The map E → Hom_{S}(F_{1}S^{1}, E) is a level equivalence of injective spectra and thus a simplicial homotopy equivalence, by Lemma 3.1.6. So, for every symmetric spectrum X, the induced map

Map_{Sp^{Σ}}(X, E) → Map_{Sp^{Σ}}(X, Hom_{S}(F_{1}S^{1}, E))

is a simplicial homotopy equivalence. Therefore Map_{Sp^{Σ}}(f, E) is a weak equivalence, and so f is a stable equivalence.

For part three, let f: X → Y^{S^{1}} be a map and f^{a}: X ∧ S^{1} → Y be the adjoint of f. The diagram

[figure: X ∧ S^{1}; f∧S^{1}; f^{a}; Y^{S^{1}} ∧ S^{1}; ev; Y]

is commutative where the map ev is the evaluation map. By part one applied to the prolongation of the path fibration, the map ev is a stable equivalence; by part two, f is a stable equivalence if and only if f ∧S^{1} is a stable equivalence. Therefore f is a stable equivalence if and only if f^{a} is a a stable equivalence.

Once we have the stable model category of symmetric spectra, part three of this theorem tells us that it really is stable; i.e., that the suspension functor − ∧ S^{1} is an equivalence of model categories.

3.2. Model categories. In this section we recall the definition and the basic properties of model categories; see [DS95], [Hov97], or [DHK] for a more detailed introduction.

Definition 3.2.1. An ordered pair (f, g) of maps in the category 𝒞 has the lifting property if every commutative square

[figure: A; f; X; g; B; Y]

in 𝒞 extends to a commutative diagram

[figure: A; f; B; g; X; Y].

We also say that f has the left lifting property with respect to g and that g has the right lifting property with respect to f. More generally, if I and J are classes of maps in 𝒞, the pair (I, J) has the lifting property if every pair (f, g) with f ∈ I and g ∈ J has the lifting property. We also say that I has the left lifting property with respect to J and that J has the right lifting property with respect to I.

It would be more accurate to say that the pair (f, g) has the lifting-extension property but we prefer the shorter term.

Definition 3.2.2. Let f and g be maps in a category 𝒞. The map f is a retract of g if it is a retract in the category of arrows, i.e., if there is a commutative diagram

A ⟶ B ⟶ A
f↓  ^{g}↓  ^{f}↓
X ⟶ Y ⟶ X.

such that the horizontal compositions are the identity maps. A class of maps is closed under retracts if whenever f is a retract of g and g is in the class then f is in the class.

Definition 3.2.3. A model category is a category ℳ with three distinguished classes of maps; the class of weak equivalences, the class of cofibrations, and the class of fibrations; that satisfy the model category axioms below. We call a map that is both a cofibration and a weak equivalence a trivial cofibration, and we call a map that is both a fibration and a weak equivalence a trivial fibration. M1 Limit axiom The category ℳ is bicomplete (closed under arbitrary small

limits and colimits). M2 Two out of three axiom. Let f and g be maps in ℳ such that gf is defined. If

two of f, g and gf are weak equivalences then the third is a weak equivalence. M3 Retract axiom. The class of weak equivalences, the class of cofibrations, and

the class of fibrations are closed under retracts. M4 Lifting axiom. A cofibration has the left lifting property with respect to

every trivial fibration. A fibration has the right lifting property with respect to every trivial cofibration.

M5 Factorization axiom. Every map f ∈ ℳ has a factorization f = pi where i is a cofibration and p is a trivial fibration and a factorization f = qj where j is a trivial cofibration and q is a fibration.

Three classes of maps that satisfy axioms M2, M3, M4 and M5 are a model structure on the category. One should keep in mind that a category can have more than one model structure; there can even be distinct model structures with the same class of weak equivalences.

A bicomplete category has an initial object ∅ and a terminal object ∗. In a model category, an object X is cofibrant if the unique map ∅ → X is a cofibration and an object X is fibrant if the unique map X → ∗ is a fibration. A model category is pointed if the unique map ∅ → ∗ is an isomorphism.

Proposition 3.2.4 (The Retract Argument). Let 𝒞 be a category and let f = pi be a factorization in 𝒞.

1. If p has the right lifting property with respect to f then f is a retract of i.

2. If i has the left lifting property with respect to f then f is a retract of p.

Proof. We only prove the first part, as the second is similar. Since p has the right lifting property with respect to f, we have a lift g: Y ⟶ Z in the diagram

\begin{matrix}X & ⟶^{i} & Z \\ ^{f}↓ & & ^{p}↓ \\ Y & & Y \end{matrix}

This gives a diagram

X  X  X
f↓  ^{i}↓  ^{f}↓
Y ⟶^{g} Z ⟶^{p} Y

where the horizontal compositions are identity maps, showing that f is a retract of i.

The following proposition is a converse to the lifting axiom.

Proposition 3.2.5 (Closure property). In a model category:

1. The cofibrations are the maps having the left lifting property with respect to every trivial fibration.

2. The trivial cofibrations are the the maps having the left lifting property with respect to every fibration.

3. The fibrations are the maps having the right lifting property with respect to every trivial cofibration.

4. The trivial fibrations are the maps having the right lifting property with respect to every cofibration.

Proof. Use the factorization axiom and the retract argument.

In particular, any two of the three classes of maps in a model category determine the third. For example, a weak equivalence is a map that factors as a trivial cofibration composed with a trivial fibration.

Example 3.2.6. We recall the standard model structure on the category of simplicial sets [Qui67, II.3]. A weak equivalence is a map whose geometric realization is a homotopy equivalence of CW-complexes. The cofibrations are the monomorphisms and every simplicial set is cofibrant. Recall, the standard n-simplex is Δ[n] = Δ(−, \overline{n}). The boundary of Δ[n] is the subfunctor ∂Δ[n] ⊆ Δ[n] of nonsurjective maps. For 0 ≤ i ≤ n, the ith horn of Δ[n] is the subfunctor Λ^{i}[n] ⊆ ∂Δ[n] of maps for which i is not in the image. Geometrically, Λ^{i}[n] is obtained from the boundary of Δ[n] by removing the ith face. The fibrations are the Kan fibrations, the maps that have the right lifting property with respect to the maps Λ^{i}[n] → Δ[n] for n > 0 and 0 ≤ i ≤ n; the fibrant simplicial sets are the Kan complexes, the simplicial sets that satisfy the Kan extension condition. A map is a trivial fibration (a fibration and a weak equivalence) if and only if it has the right lifting property with respect to the maps ∂Δ[n] → Δ[n]. It follows that the pointed weak equivalences, the pointed monomorphisms, and the pointed (Kan) fibrations are a model structure on the category of pointed simplicial sets.

When constructing a model category, the factorization axiom can be the hardest to verify. After some preliminary definitions, Lemma 3.2.11 constructs functorial factorizations in the category of symmetric spectra.

Definition 3.2.7. Let I be a class of maps in a category 𝒞.

1. A map is I-injective if it has the right lifting property with respect to every map in I. The class of I-injective maps is denoted I-inj.

2. A map is I-projective if it has the left lifting property with respect to every map in I. The class of I-projective maps is denoted I-proj.

3. A map is an I-cofibration if it has the left lifting property with respect to every I-injective map. The class of I-cofibrations is the class (I-inj)-proj and is denoted I-cof.

4. A map is an I-fibration if it has the right lifting property with respect to every I-projective map. The class of I-fibrations is the class (I-proj)-inj and is denoted I-fib.

Injective and projective are dual notions; an I-injective map in 𝒞 is an Iprojective map in 𝒞^{op}; an I-fibration in 𝒞 is an I-cofibration in 𝒞^{op}. The class I-inj and the class I-proj are analogous to the orthogonal complement of a set vectors. That analogy helps explain the following proposition, whose proof we leave to the reader.

Proposition 3.2.8. Let I and J be classes of maps in a category 𝒞.

1. If I ⊆ J then J-inj ⊆ I-inj and J-proj ⊆ I-proj.

2. Repeating the operations: I ⊆ I-cof, I ⊆ I-fib, I-proj = (I-proj)-cof = (I-fib)-proj, and I-inj = (I-inj)-fib = (I-cof)-inj.

3. The following conditions are equivalent:

• The pair (I, J) has the lifting property.

• J ⊆ I-inj.

• I ⊆ J-proj.

• The pair (I-cof, J) has the lifting property.

• The pair (I, J-fib) has the lifting property.

4. The classes I-inj and I-proj are subcategories of 𝒞 and are closed under retracts.

5. The class I-inj is closed under base change. That is, if

A ⟶ B
f↓  ^{g}↓
X ⟶ Y

is a pullback square and g is an I-injective map then f is an I-injective map.

6. The class I-proj is closed under cobase change. That is, if

A ⟶ B
f↓  ^{g}↓
X ⟶ Y

is a pushout square and f is an I-projective map then g is an I-projective map.

Corollary 3.2.9. Let I be a class of maps in a category 𝒞. The class I-cof is a subcategory of 𝒞 that is closed under retracts and cobase change. The class I-fib is a subcategory of 𝒞 that is closed under retracts and base change.

Another useful elementary lemma about the lifting property is the following.

Lemma 3.2.10. Let L: 𝒞 → 𝒟 be a functor that is left adjoint to the functor R: 𝒟 → 𝒞. If I is a class of maps in 𝒞 and J is a class of maps in 𝒟, the pair (I, RJ) has the lifting property if and only if the pair (LI, J) has the lifting property.

The next lemma is used repeatedly to construct factorizations.

Lemma 3.2.11 (Factorization Lemma). Let I be a set of maps in the category Sp^{Σ}. There is a functorial factorization of every map of symmetric spectra as an I-cofibration followed by an I-injective map.

The factorization lemma is proved using the transfinite small object argument. We begin by showing that every symmetric spectrum is suitably small.

Recall that an ordinal is, by recursive definition, the well-ordered set of all smaller ordinals. In particular, we can regard an ordinal as a category. A cardinal is an ordinal of larger cardinality than all smaller ordinals.

Definition 3.2.12. Let γ be an infinite cardinal. An ordinal α is γ-filtered if every set A consisting of ordinals less than α such that sup A = α has cardinality greater than γ.

Every γ-filtered ordinal is a limit ordinal. In fact, since γ is infinite, every γfiltered ordinal is a limit ordinal α for which there is no countable set A of ordinals less than α such that sup A = α. The smallest γ-filtered ordinal is the first ordinal of cardinality greater than γ. For example, ω_{1} is the smallest ℵ_{0}-filtered ordinal. If γ < \overline{γ} and α is \overline{γ}-filtered, then α is γ-filtered.

Define the cardinality of a spectrum X to be the cardinality of its underlying set ⨿_{n} ⨿_{k} (Ev_{n}X)_{k}. Then the cardinality of X is always infinite, which is convenient for the following lemma.

Proposition 3.2.13. Let X be a simplicial spectrum of cardinality γ. Let α be a γ-filtered ordinal and let D: α → Sp^{Σ} be an α-indexed diagram of symmetric spectra. Then the natural map

colim_{α} Sp^{Σ}(X, D) → Sp^{Σ}(X, colim_{α} D)

is an isomorphism.

Proof. Every symmetric spectrum has a presentation as a coequalizer

\xymatrix{S ⊗ S ⊗ X \ar[r] \ar[r] & S ⊗ X \ar[r] & X}

of free symmetric spectra in the category Sp^{Σ}. The symmetric spectra X and S ⊗ X have the same cardinality. So the proposition follows once it is proved for free symmetric spectra. There is a natural isomorphism Sp^{Σ}(S ⊗ X, Y) = ∏_{p} 𝒮_{∗}^{Σ_{p}}(X_{p}, Y_{p}). The functors 𝒮_{∗}^{Σ_{p}}(X_{n}, −) have the property claimed for Sp^{Σ}(X, −).

This fact is the heart of the proposition. To begin the proof of it, suppose we have a map f: X_{n} → colim_{α} D, where D is an α-indexed diagram of Σ_{p}-simplicial sets. For each simplex x of X_{n}, we can choose an ordinal β_{x} < α and a simplex y_{x} ∈ D_{β_{x}} such that f(x) is the image of y_{x}. Because α is γ-filtered, we can then find one ordinal β < α and a map g: X ⟶ D_{β} factoring f. The map g may not be simplicial or equivariant, but, again using the fact that α is γ-filtered, we can go out far enough in the colimit so that g will be both simplicial and equivariant. We leave the details to the reader.

Since γ is infinite, for every countable set A of ordinals that are strictly less than α, the ordinal sup A is strictly less than α. Therefore, the countable product of functors 𝒮_{∗}^{Σ}(X, −) has the property claimed for Sp^{Σ}(X, −) and the proposition is proved.

Proof of Lemma 3.2.11. We begin by constructing a functorial factorization

X ⟶^{Ig} Eg ⟶^{P g} Y

of g: X → Y such that Ig is an I-cofibration and g = P g ∘ Ig. For a map f: b_{f} → c_{f} in I, let Df be the set of commutative squares

b_{f} ⟶ X
f↓  ↓^{g}
c_{f} ⟶ Y.

Let

B = ⨿_{f∈I} ⨿_{Df} b_{f}, C = ⨿_{f∈I} ⨿_{Df} c_{f}, and F = ⨿_{f∈I} ⨿_{Df} f: B → C.

By the definition of Df, there is a commutative square

B ⟶ X
F ↓  ↓^{g}
C ⟶ Y.

Let Eg be the pushout X ⨿_{B} C, Ig be the map X → Eg = X ⨿_{B} C, and P g: Eg → Y be the natural map on the pushout. By construction, the map Ig: X → X ⨿_{B} C is an I-cofibration and g = P g∘Ig. However, the map P g need not be an I-injective map.

Use transfinite induction to define functorial factorizations of g

X ⟶^{I^{α}g} E^{α}g ⟶^{P^{α}g} Y

for every ordinal α. The induction starts at 0 with E^{0}g = X, I^{0}g = id_{X}, and P^{0}g = g. For a successor ordinal α + 1, E^{α+1}g = E(P^{α}g), I^{α+1}g = I(P^{α}g) ∘ I^{α}g, and P^{α+1}g = P (P^{α}g). For β a limit ordinal, E^{β}g = colim_{α<β} E^{α}g, I^{β}g = colim_{α<β} I^{α}g and P^{β}g = colim_{α<β} P^{α}g.

The map I^{α}g: X → E^{α}g is an I-cofibration for each α; the required lift is constructed by transfinite induction. The proof of the lemma is completed by finding an ordinal β for which P_{β}g is an I-injective map. Let

b_{f} ⟶ E^{β}g
f↓  ↓_{P^{β}g}
c_{f} ⟶ Y

be a commutative square with f ∈ I. If the map b_{f} → E^{β}g factors as b_{f} → E^{α}g → E^{β}g for α < β then by construction there is a lift c_{f} → E^{α+1}g and, since α+1 ≤ β, a lift c_{f} → E^{β}g. Let γ_{f} be the cardinality of b_{f} and let γ = sup_{I} γ_{f}. Let β be a γ-filtered ordinal. Then Sp^{Σ}(b_{f}, P^{β}g) = colim_{α<β} Sp^{Σ}(b_{f}, P^{α}g) for every f ∈ I and hence P^{β}g is an I-injective map.

3.3. Level structure. Prolongation of the model structure on 𝒮_{∗} (see 3.2.6) gives the level structure on the category of symmetric spectra. It is not a model structure but it is a basic tool in the construction of the stable model structure. Its use is already implicit in Sections 1.3 and 3.1.

Definition 3.3.1. Let f: X ⟶ Y be a map of symmetric spectra.

1. The map f is a level equivalence if each map f_{n}: X_{n} → Y_{n} is a weak equivalence of simplicial sets.

2. The map f is a level (trivial) cofibration if each map f_{n}: X_{n} → Y_{n} is a (trivial) cofibration of simplicial sets.

3. The map f is a level (trivial) fibration if each map f_{n}: X_{n} → Y_{n} is a (trivial) fibration of simplicial sets.

The level cofibrations are the monomorphisms of symmetric spectra. Next, we characterize the level fibrations and trivial fibrations.

Definition 3.3.2. 1. Let I_{Λ} denote the set of maps Λ^{k}[r]_{+} ⟶ Δ[r]_{+} for r > 0

and 0 ≤ k ≤ r. Let F I_{Λ} = ∪_{n≥0}F_{n}(I_{Λ}).

2. Let I_{∂} denote the set of maps ∂Δ[r]_{+} ⟶ Δ[r]_{+} for r ≥ 0. Let F I_{∂} = ∪_{n≥0}F_{n}(I_{∂}).

Proposition 3.3.3. The level fibrations are the F I_{Λ}-injective maps. The level trivial fibrations are the F I_{∂}-injective maps.

Proof. A map g is a level (trivial) fibration if and only if Ev_{n} g = g_{n} is a (trivial) Kan fibration for each n ≥ 0. But Ev_{n} g is a (trivial) Kan fibration if and only if it has the right lifting property with respect to the class (I_{∂}) I_{Λ}. Then by adjunction, g is a level (trivial) fibration if and only if g has the right lifting property with respect to the class (F I_{∂}) F I_{Λ}.

The level structure is not a model structure; it satisfies the two-out-of-three axiom, the retract axiom, and the factorization axiom but not the lifting axiom. A model structure is determined by any two of its three classes and so the level structure is over determined. In Section 5.1 we prove there are two “level” model structures with the level equivalences as the weak equivalences: one that is generated by the level equivalences and the level cofibrations and one that is generated by the level equivalences and the level fibrations. In any case, the level homotopy category obtained by inverting the level equivalences is not the stable homotopy category of spectra.

The pushout smash product (Definition 1.3.3) has an adjoint construction.

Definition 3.3.4. Let f: U → V and g: X → Y be maps of pointed simplicial sets. The map

Map_{□}(f, g): Map(V, X) → Map(U, X) ×_{Map(U,Y)} Map(V, Y).

is the map to the fiber product induced by the commutative square

Map(V, X) ⟶^{f^{∗}} Map(U, X)
g_{∗}↓  ↓^{g_{∗}}
Map(V, Y) ⟶_{f^{∗}} Map(U, Y).

Let f be a map of pointed simplicial sets and g be a map of symmetric spectra. Then Hom_{□}(f, g) is the map of symmetric spectra that is defined by prolongation, Ev_{n} Hom_{□}(f, g) = Map_{□}(f, g_{n}).

Proposition 3.3.5. 1. If f ∈ 𝒮_{∗} is a monomorphism and g ∈ 𝒮_{∗} is a Kan fibration then Map_{□}(f, g) is a Kan fibration. If, in addition, either f or g is a weak equivalence, then Map_{□}(f, g) is a weak equivalence.

2. If f ∈ 𝒮_{∗} is a monomorphism and g ∈ Sp^{Σ} is a level fibration then Hom_{□}(f, g) is a level fibration. If, in addition, either f is a weak equivalence or g is a level equivalence, then Hom_{□}(f, g) is a level equivalence.

Proof. Part one is a standard property of simplicial sets, proved in [Qui67, II.3]. Part two follows from part one by prolongation.

Definition 3.3.6. Let f: U → V and g: X → Y be maps in a category 𝒞. Then 𝒞_{□}(f, g) is the natural map of sets to the fiber product

𝒞(V, X) → 𝒞(U, X) ×_{𝒞(U,Y)} 𝒞(V, Y)

coming from the commutative square

𝒞(V, X) ⟶^{f^{∗}} 𝒞(U, X)
g_{∗}↓  ↓^{g_{∗}}
𝒞(V, Y) ⟶_{f^{∗}} 𝒞(U, Y)

A pair (f, g) has the lifting property if and only if 𝒞_{□}(f, g) is surjective.

Definition 3.3.7. Let f: U → V and g: X → Y be maps of symmetric spectra. Then Map_{□}(f, g) is the natural map to the fiber product

Sp_{□}^{Σ}(f ∧ Δ[−]_{+}, g): Map(V, X) → Map(U, X) ×_{Map(U,Y)} Map(V, Y)

Proposition 3.3.8. Let f and h be maps of symmetric spectra and g be a map of pointed simplicial sets. There are natural isomorphisms

Sp_{□}^{Σ}(f □ g, h) ≅ (𝒮_{∗})_{□}(g, Map_{□}(f, h)) ≅ Sp_{□}^{Σ}(f, Hom_{□}(g, h))

In fact this proposition holds in any simplicial model category.

Proof. Let f: U → V and h: X → Y be maps in Sp^{Σ} and g: K → L be a map in 𝒮_{∗}. Using adjunction and the defining property of pushouts and of pullbacks, each of the three maps in the proposition is naturally isomorphic to the map from Sp^{Σ}(V ∧ L, X) to the limit of the diagram

\xymatrix{Sp^{Σ}(U ∧ L, X) \ar[d] \ar[dr] & Sp^{Σ}(V ∧ K, X) \ar[dl] \ar[dr] & Sp^{Σ}(V ∧ L, Y) \ar[d] \ar[dl] \\ Sp^{Σ}(U ∧ K, X) \ar[dr] & Sp^{Σ}(U ∧ L, Y) \ar[d] & Sp^{Σ}(V ∧ K, Y) \ar[dl] \\  & Sp^{Σ}(U ∧ K, Y)}

Corollary 3.3.9. Let f and h be maps of symmetric spectra and g be a map of pointed simplicial sets. The following are equivalent:

• (f □ g, h) has the lifting property.

• (g, Map_{□}(f, h)) has the lifting property.

• (f, Hom_{□}(g, h)) has the lifting property.

3.4. Stable model category. In this section we define the stable cofibrations and the stable fibrations of symmetric spectra. The main result is that the class of stable equivalences, the class of stable cofibrations, and the class of stable fibrations are a model structure on Sp^{Σ}.

Recall that f is a level trivial fibration if f_{n} is a trivial Kan fibration for each n ≥ 0.

Definition 3.4.1. A map of symmetric spectra is a stable cofibration if it has the left lifting property with respect to every level trivial fibration. A map of symmetric spectra is a stable trivial cofibration if it is a stable cofibration and a stable equivalence. A symmetric spectrum X is stably cofibrant if ∗ → X is a stable cofibration.

The basic properties of the class of stable cofibrations are next.

Proposition 3.4.2. 1. The class of stable cofibrations is the class F I_{∂}-cof.

2. The class of stable cofibrations is a subcategory that is closed under retracts and closed under cobase change.

3. If f is a cofibration of pointed simplicial sets and n ≥ 0 then F_{n}f is a stable cofibration. In particular, F_{n}K is stably cofibrant for K ∈ 𝒮_{∗}.

4. If f ∈ Sp^{Σ} is a stable cofibration and g ∈ 𝒮_{∗} is a cofibration then the pushout smash product f □ g is a stable cofibration.

5. If f ∈ Sp^{Σ} is a stable cofibration and h ∈ Sp^{Σ} is a level fibration then Map_{□}(f, h) is a Kan fibration.

Proof. The stable cofibrations are the maps having the left lifting property with respect to the level trivial fibrations, which by Proposition 3.3.3 are the F I_{∂}-injective maps. So the stable cofibrations are the F I_{∂}-cofibrations.

Every class I-cof has the properties stated in part two.

Suppose g ∈ Sp^{Σ} is a level trivial fibration, and f ∈ 𝒮_{∗} is a cofibration. Then f has the left lifting property with respect to the trivial Kan fibration Ev_{n} g. By adjunction, F_{n}f has the left lifting property with respect to g. Hence F_{n}f is a stable cofibration. In particular, for every pointed simplicial set K, the map ∗ → F_{n}K is a stable cofibration, and so F_{n}K is stably cofibrant.

Now suppose f ∈ Sp^{Σ} is a stable cofibration and g ∈ 𝒮_{∗} is a cofibration. Then, given a level trivial fibration h ∈ Sp^{Σ}, the map Hom_{□}(g, h) is a level trivial fibration by Proposition 3.3.5. Therefore the pair (f, Hom_{□}(g, h)) has the lifting property. Then by Corollary 3.3.9, the pair (f □ g, h) has the lifting property, and so f □ g is a stable cofibration.

Finally, suppose f ∈ Sp^{Σ} is a stable cofibration and h ∈ Sp^{Σ} is a level fibration. Given a trivial cofibration g ∈ 𝒮_{∗}, Hom_{□}(g, h) is a level trivial fibration by Proposition 3.3.5. Therefore, the pair (f, Hom_{□}(g, h)) has the lifting property. Then, by Corollary 3.3.9, the pair (g, Map_{□}(f, h)) has the lifting property. Therefore Map_{□}(f, h) is a Kan fibration.

The next definition is natural in view of the closure properties in a model category, see Proposition 3.2.5.

Definition 3.4.3. A map of symmetric spectra is a stable fibration if it has the right lifting property with respect to every map that is a stable trivial cofibration. A map of symmetric spectra is a stable trivial fibration if it is a stable fibration and a stable equivalence. A spectrum X is stably fibrant if the map X → ∗ is a stable fibration.

Theorem 3.4.4. The category of symmetric spectra with the class of stable equivalences, the class of stable cofibrations, and the class of stable fibrations is a model category.

Proof. The category Sp^{Σ} is bicomplete by Proposition 1.2.10. The two out of three axiom and the retract axiom are immediate consequences of the definitions. By definition, (i, p) has the lifting property when i is a stable trivial cofibration and p is a stable fibration. The lifting axiom for i a stable cofibration and p a stable trivial fibration is verified in Corollary 3.4.7. The two parts of the factorization axiom are verified in Corollary 3.4.6 and Lemma 3.4.8.

Lemma 3.4.5. A map is a stable trivial fibration if and only if it is a level trivial fibration.

Proof. Suppose g is a level trivial fibration. By definition, every stable cofibration has the left lifting property with respect to g and in particular every stable trivial cofibration has the left lifting property with respect to g. So g is a stable fibration which is a level equivalence and hence a stable equivalence. So g is a stable trivial fibration.

Conversely, suppose g is a stable trivial fibration. Recall that at this point we do not know that g has the right lifting property with respect to stable cofibrations. By Lemma 3.2.11, g can be factored as g = pi with i an F I_{∂}-cofibration and p an F I_{∂}-injective map. Since p is a level equivalence, it is a stable equivalence. By the two out of three property, i is a stable equivalence. Therefore, i is a stable trivial cofibration and has the left lifting property with respect to g. By the Retract Argument 3.2.4, g is a retract of p and so g is a level trivial fibration.

Corollary 3.4.6. Every map f of symmetric spectra has a factorization f = pi as a stable cofibration i followed by a stable trivial fibration p.

Proof. By the Factorization Lemma 3.2.11, every map f in Sp^{Σ} can be factored as f = pi with i an F I_{∂}-cofibration and p an F I_{∂}-injective map. Then i is a stable cofibration and p is level trivial fibration, which, by Lemma 3.4.5, means that p is a stable trivial fibration.

Corollary 3.4.7. A stable cofibration has the left lifting property with respect to every stable trivial fibration.

Proof. By Lemma 3.4.5 every stable trivial fibration is a level trivial fibration. By definition, stable cofibrations have the left lifting property with respect to every level trivial fibration.

The following lemma will finish the proof of Theorem 3.4.4.

Lemma 3.4.8. Every map f of symmetric spectra has a factorization f = pi as a stable trivial cofibration i followed by a stable fibration p.

To prove the lemma we need a set of maps J such that a J-cofibration is a stable trivial cofibration and a J-injective map is a stable fibration. Using the Factorization Lemma with the set J will prove Lemma 3.4.8. The set J is defined in 3.4.9 and Corollary 3.4.16 verifies its properties. This takes up the rest of the section.

The maps λ ∧ F_{n}S^{0} used in the definition below appeared in the description of the function spectrum in Remark 2.2.12. They are stable equivalences (see Example 3.1.10) but are not stable cofibrations or even level cofibrations. We modify them to get the set J.

Definition 3.4.9. Let λ: F_{1}S^{1} → F_{0}S^{0} be the adjoint of the identity map S^{1} → Ev_{1} F_{0}S^{0} = S^{1} and let λ_{n} be the map λ ∧ F_{n}S^{0} : F_{n+1}S^{1} → F_{n}S^{0}, so that λ_{0} = λ. The mapping cylinder construction 3.1.7 gives a factorization λ_{n} = r_{n}c_{n} where r_{n}: Cλ_{n} → F_{n}S^{0} is a simplicial homotopy equivalence and c_{n}: F_{n+1}S^{1} → Cλ_{n} is a level cofibration. For n ≥ 0, let K_{n} = c_{n} □ I_{∂}, i.e., K_{n} is the set of maps c_{n} □ j for j ∈ I_{∂}. Let K = ∪_{n}K_{n} and finally, let J = F I_{Λ} ∪ K.

Lemma 3.4.10. For each n ≥ 0, the map c_{n}: F_{n+1}S^{1} → Cλ_{n} is a stable trivial cofibration.

Proof. The map λ_{n} is a stable equivalence (Example 3.1.10) and the simplicial homotopy equivalence r_{n} is a stable equivalence. Using the factorization λ_{n} = r_{n}c_{n} and the two out of three property of stable equivalences, c_{n} is a stable equivalence.

Next we show that c_{n} is a stable cofibration. The mapping cylinder Cλ_{n} can also be defined as the corner in the pushout square

F_{n+1}S^{1} ∨ F_{n+1}S^{1} ⟶^{i_{0}∨i_{1}} F_{n+1}S^{1} ∧ Δ[1]_{+}
λ_{n}∨1↓  ↓
F_{n+1}S^{1} ⟶^{k_{n}} F_{n}S^{0} ∨ F_{n+1}S^{1} ⟶^{g_{n}}  Cλ_{n},

where k_{n} is the inclusion on the second factor, i_{0} and i_{1} come from the two inclusions Δ[0] → Δ[1], and g_{n} is the natural map to the pushout. Using the properties of stable cofibrations in Proposition 3.4.2, we find that the map ∗ → F_{n}S^{0} is a stable cofibration and, by cobase change, that k_{n} is a stable cofibration. Let j be the cofibration ∂Δ[1]_{+} → Δ[1]_{+}. Then (∗ → F_{n+1}S^{1})□j = i_{0}∨i_{1} is a stable cofibration and, by cobase change, g_{n} is a stable cofibration. Thus the composition c_{n} = g_{n}k_{n} is a stable cofibration.

Next we characterize the J-injective maps.

Definition 3.4.11. A commutative square of simplicial sets

X ⟶ Z
p↓  ^{q}↓
Y ⟶_{f} W

where p and q are fibrations is a homotopy pullback square if the following equivalent conditions hold:

• The induced map X ⟶ Y ×_{W} Z is a weak equivalence.

• For every 0-simplex v ∈ Y_{0}, the map of fibers p^{−1}v → q^{−1}fv is a weak equivalence.

Lemma 3.4.12. A map of symmetric spectra p: E → B is J-injective if and only if p is a level fibration and the diagram

\begin{matrix}E_{n} & ⟶^{σ^{a}} & E_{n+1}^{S^{1}} \\ ^{p_{n}}↓ & & ↓^{p_{n+1}} \\ B_{n} & ⟶^{σ^{a}} & B_{n+1}^{S^{1}}, \end{matrix} (∗)

is a homotopy pullback square for each n ≥ 0, where the horizontal maps are the adjoints of the structure maps.

Proof. Since J = F I_{Λ} ∪ K and K = ∪_{n}K_{n}, a map is J-injective if and only if it is F I_{Λ}-injective and K_{n}-injective for each n ≥ 0. By Proposition 3.3.3, the F I_{Λ}- injective maps are the level fibrations. By definition, p ∈ Sp^{Σ} is a K_{n}-injective map if and only if p has the right lifting property with respect to the class c_{n} □ I_{∂}. Then, by Corollary 3.3.9, p is K_{n}-injective if and only if Map_{□}(c_{n}, p) has the right lifting property with respect to the class I_{∂}. Hence, p is K_{n}-injective if and only if Map_{□}(c_{n}, p) is a trivial Kan fibration. If p is a level fibration, Map_{□}(c_{n}, p) is a Kan fibration by Proposition 3.4.2. So, a level fibration p is K_{n}-injective if and only if Map_{□}(c_{n}, p) is a weak equivalence. Taken together, p is J-injective if and only if p is a level fibration and Map_{□}(c_{n}, p) is a weak equivalence for each n ≥ 0.

For each n ≥ 0, the map r_{n}: Cλ_{n} → F_{n}S^{0} has a simplicial homotopy inverse s_{n}: F_{n}S^{0} → Cλ_{n} for which r_{n}s_{n} is the identity map on F_{n}S^{0} (see 3.1.7). Then Map_{□}(c_{n}, p) is simplicially homotopic to Map_{□}(s_{n}λ_{n}, p). Since F_{n}S^{0} is a simplicial deformation retract of Cλ_{n}, λ_{n} is a simplicial deformation retract of s_{n}λ_{n} and Map_{□}(λ_{n}, p) is a simplicial deformation retract of Map_{□}(s_{n}λ_{n}, p). Therefore, Map_{□}(c_{n}, p) is a weak equivalence if and only if Map_{□}(λ_{n}, p) is a weak equivalence.

The map

Map_{□}(λ_{n}, p): Map(F_{n}S^{0}, E) → Map(F_{n}S^{0}, B) ×_{Map(F_{n+1}S^{1},B)} Map(F_{n+1}S^{1}, E)

is naturally isomorphic to the map

E_{n} → B_{n} ×_{B_{n+1}^{S^{1}}} E_{n+1}^{S^{1}}.

induced by the diagram (∗). If p is a level fibration then by definition the diagram (∗) is a homotopy pullback square if and only if the map Map_{□}(λ_{n}, p) is a weak equivalence.

Combining the conclusions of the three paragraphs completes the proof.

Corollary 3.4.13. The map F → ∗ is J-injective if and only if F is an Ωspectrum.

We also get the following corollary, which is not needed in the sequel. Its proof uses properness (see Section 5.4).

Corollary 3.4.14. A level fibration between two Ω-spectra is J-injective.

Lemma 3.4.15. Let p: X → Y be a map of symmetric spectra. If p is J-injective and p is a stable equivalence then p is a level equivalence.

Proof. Suppose p: X → Y is a J-injective stable equivalence. In particular, p is a level fibration. Let F be the fiber over the basepoint. Since the class J-inj is closed under base change, the map F → ∗ is J-injective and F is an Ω-spectrum. The map p factors as X → X/F → Y. The map X/F → Y is a stable equivalence by Theorem 3.1.14. Since p: X → Y is a stable equivalence, q: X → X/F is a stable equivalence.

A Barratt-Puppe type sequence for symmetric spectra is constructed by prolongation to give the diagram

X → X ⨿_{F} (F ∧ Δ[1]_{+}) → F ∧ S^{1} → X ∧ S^{1} → (X ⨿_{F} (F ∧ Δ[1]_{+})) ∧ S^{1}.

Let E be an injective Ω-spectrum. Since the map X ⨿_{F} (F ∧ Δ[1]_{+}) → X/F is a level equivalence, after applying E^{0}(−) to this sequence we can rewrite the terms involving the homotopy cofiber as E^{0}(X/F). This gives an exact sequence

E^{0}X ⟵^{E^{0}q} E^{0}(X/F) ← E^{0}(F ∧ S^{1}) ← E^{0}(X ∧ S^{1}) ⟵^{E^{0}(q∧S^{1})} E^{0}(X/F ∧ S^{1}).

Since q: X → X/F is a stable equivalence, E^{0}q is an isomorphism by definition, and E^{0}(q∧S^{1}) is an isomorphism by part two of Theorem 3.1.14. Hence, E^{0}(F ∧S^{1}) = ∗ for every injective Ω-spectrum E, and so, by part two of Theorem 3.1.14, E^{0}F = ∗ for every injective Ω-spectrum E. By Corollary 5.1.3, there is a level equivalence F → E where E is an injective spectrum; since F is an Ω-spectrum, E is an injective Ω-spectrum. By Lemma 3.1.6, E^{0}E = E^{0}F = ∗. So E is simplicially homotopic to ∗ and F is level equivalent to ∗.

This does not finish the argument as the base of the fibration X_{n} → Y_{n} need not be connected. Since p is J-injective

X_{n} ⟶^{σ^{a}} X_{n+1}^{S^{1}}
p_{n}↓  ↓_{p_{n+1}^{S^{1}}}
Y_{n} ⟶^{σ^{a}} Y_{n+1}^{S^{1}},

is a homotopy pullback square for each n ≥ 0. The proof is completed by showing that p_{n+1}^{S^{1}} is a trivial Kan fibration for every n ≥ 0 which implies that p_{n} is a trivial Kan fibration for every n ≥ 0. For a pointed simplicial set K, let cK denote the connected component of the basepoint. If E → B is a pointed Kan fibration, then cE → cB is a Kan fibration; if the fiber over the basepoint ∗ ∈ B is contractible then cE → cB is a trivial Kan fibration. In particular, cX_{n} → cY_{n} is a trivial Kan fibration and therefore, (cX_{n})^{S^{1}} → (cY_{n})^{S^{1}} is a trivial fibration. Since K^{S^{1}} = (cK)^{S^{1}} for any pointed simplicial set K, p_{n}^{S^{1}} : X_{n}^{S^{1}} → Y_{n}^{S^{1}} is a trivial Kan fibration for every n ≥ 0.

The next corollary finishes the proof of Lemma 3.4.8.

Corollary 3.4.16. The J-cofibrations are the stable trivial cofibrations and the J-injective maps are the stable fibrations.

Proof. Every level trivial fibration is J-injective since it satisfies the condition in Lemma 3.4.12. Thus, a J-cofibration has the left lifting property with respect to every level trivial fibration, and hence a J-cofibration is a stable cofibration. Let E be an Ω-spectrum. The maps p: E → ∗ and q = Hom(j, E): E^{Δ[1]} → E×E, where j: ∂Δ[1]_{+} → Δ[1]_{+} is the inclusion, are J-injective by Lemma 3.4.12. Let E be an injective Ω-spectrum and f be a J-cofibration. Since f has the left lifting property with respect to p: E → ∗, E^{0}f is surjective. Since f has the left lifting property with respect to q: E^{Δ[1]} → E × E, E^{0}f is injective. So E^{0}f is an isomorphism and every J-cofibration is a stable trivial cofibration.

Conversely, let f be a stable trivial cofibration. By the Factorization Lemma 3.2.11, f factors as f = pi where i is a J-cofibration and p is a J-injective map. We have just seen that i is a stable equivalence. So, the J-injective map p is a stable equivalence and by Lemma 3.4.15, p is a level equivalence. Therefore the stable cofibration f has the left lifting property with respect to the map p. By the Retract Argument 3.2.4, f is a J-cofibration.

Let F be the class of stable fibrations. Since J-cof is the class of stable trivial cofibrations, one has by the definition of stable fibrations that F = (J-cof)-inj. But (J-cof)-inj = J-inj by Proposition 3.2.8 (2). In other words, the stable fibrations are the J-injective maps.

In particular, Lemma 3.4.12 characterizes the stable fibrations. The stably fibrant objects are the Ω-spectra by Corollary 3.4.13. Corollary 3.4.16 finishes the proof of Lemma 3.4.8 and the verification of the axioms for the stable model category of symmetric spectra.

4. Comparison with the Bousfield-Friedlander category

The goal of this section is to show that the stable homotopy theory of symmetric spectra and the stable homotopy theory of spectra are equivalent. We begin in Section 4.1 by recalling the general theory of Quillen equivalences of model categories. In Section 4.2 we provide a brief recap of the stable homotopy theory of (non-symmetric) spectra. In Section 4.3 we show that the forgetful functor U from symmetric spectra to spectra is part of a Quillen equivalence. The left adjoint V of U plays very little role in this proof, beyond its existence, so we postpone its construction to Section 4.4.

4.1. Quillen equivalences. In this section, we briefly recall Quillen functors and Quillen equivalences between model categories.

Definition 4.1.1. Let 𝒞 and 𝒟 be model categories. Let L: 𝒞 → 𝒟 and R: 𝒟 → 𝒞 be functors such that L is left adjoint to R. The adjoint pair of functors L and R is a Quillen adjoint pair if L preserves cofibrations and R preserves fibrations. We refer to the functors in such a pair as left and right Quillen functors. A Quillen adjoint pair is a Quillen equivalence if for every cofibrant object X ∈ 𝒞 and every fibrant object Y ∈ 𝒟, a map LX ⟶ Y is a weak equivalence if and only if its adjoint X ⟶ RY is a weak equivalence.

The definition of a Quillen adjoint pair can be reformulated.

Lemma 4.1.2. Let L and R be a pair of functors between model categories such that L is left adjoint to R.

1. L preserves cofibrations if and only if R preserves trivial fibrations.

2. L preserves trivial cofibrations if and only if R preserves fibrations.

This lemma is an immediate corollary of Lemma 3.2.10; see also [DS95, 9.8]. A useful lemma associated to these questions is Ken Brown’s lemma.

Lemma 4.1.3 (Ken Brown’s Lemma). Let F be a functor between model categories.

1. If F takes trivial cofibrations between cofibrant objects to weak equivalences, then F preserves all weak equivalences between cofibrant objects.

2. If F takes trivial fibrations between fibrant objects to weak equivalences, then F preserves all weak equivalences between fibrant objects.

For the proof of this lemma see [DS95, 9.9].

In particular, a left Quillen functor L preserves weak equivalences between cofibrant objects, and a right Quillen functor R preserves weak equivalences between fibrant objects.

The following proposition is the reason Quillen equivalences are important.

Proposition 4.1.4. A Quillen adjoint pair of functors between model categories induces an adjoint pair of functors on the homotopy categories which is an adjoint equivalence if and only if the adjoint pair of functors is a Quillen equivalence.

For the proof of this proposition, see [DS95, Theorem 9.7].

We now describe a useful sufficient condition for a Quillen adjoint pair to be a Quillen equivalence.

Definition 4.1.5. Suppose F: 𝒞 → 𝒟 is a functor between model categories. For any full subcategory 𝒞′ of 𝒞, we say that F detects and preserves weak equivalences of 𝒞′ if a map f in 𝒞′ is a weak equivalence if and only if F f is.

In practice, very few functors detect and preserve weak equivalences on the whole category. However, many functors detect and preserve weak equivalences between cofibrant objects or fibrant objects, so the next lemma is often useful. Before stating it, we need a definition.

Definition 4.1.6. Suppose 𝒞 is a model category. A fibrant replacement functor on 𝒞 is a functor K: 𝒞 → 𝒞 whose image lies in the full subcategory of fibrant objects, together with a natural weak equivalence i: X → KX.

There is a dual notion of a cofibrant replacement functor, but we do not use it. Fibrant replacement functors are usually obtained by using a version of the Factorization Lemma 3.2.11 appropriate for 𝒞 to functorially factor the map X ⟶ 1 into a trivial cofibration followed by a fibration. We have already used fibrant replacement functors in 𝒮_{∗} in the proof of Theorem 3.1.11.

Lemma 4.1.7. Suppose L: 𝒞 → 𝒟 is a left Quillen functor with right adjoint R, and suppose K is a fibrant replacement functor on 𝒟. Suppose R detects and preserves weak equivalences between fibrant objects and the composition X → RLX ⟶^{Ri} RKLX is a weak equivalence for all cofibrant objects X of 𝒞. Then the pair (L, R) is a Quillen equivalence.

There is also a dual statement, but this is the criterion we use.

Proof. Suppose f: LX ⟶ Y is a map, where X is cofibrant and Y is fibrant. Consider the commutative diagram below.

X ⟶ RLX ⟶^{Rf} RY
Ri_{LX} ↓  ↓^{Ri_{Y}}
RKLX ⟶_{RKf} > RKY

The top composite is the adjoint g: X ⟶ RY of f. The map i_{Y} is a weak equivalence between fibrant objects, so Ri_{Y} is a weak equivalence. The composite X ⟶ RLX ⟶^{Ri_{LX}} RKLX is a weak equivalence by hypothesis. Thus g is a weak equivalence if and only if RKf is a weak equivalence. But R detects and preserves weak equivalences between fibrant objects, so RKf is a weak equivalence if and only if Kf is a weak equivalence. Since i is a natural weak equivalence, Kf is a weak equivalence if and only if f is a weak equivalence.

4.2. The stable Bousfield-Friedlander category. In this section we describe the stable homotopy theory of (non-symmetric) spectra. The basic results are proved in [BF78], but we use the approach of Section 3. In particular, we can define injective spectra and Ω-spectra just as we did for symmetric spectra.

Definition 4.2.1. Suppose f: X ⟶ Y is a map of spectra.

1. The map f is a stable equivalence if E^{0}f is an isomorphism for every injective Ω-spectrum E.

2. The map f is a stable cofibration if f has the left lifting property with respect to every level trivial fibration.

3. The map f is a stable fibration if f has the right lifting property with respect to every map which is both a stable cofibration and a stable equivalence.

We can then attempt to carry out the program of Section 3 with these definitions. The category Sp^{ℕ} of non-symmetric spectra is neither symmetric monoidal nor closed, so we must be careful in some places. However, if X is a sequence and Y is a spectrum, we can form the sequences X ⊗ Y and Hom_{ℕ}(X, Y), and these are S-modules in a canonical way. With this minor change, all of the results of Section 3 go through without difficulty. There is one important change: in the proof of Theorem 3.1.11, we considered the spectrum R^{∞}X, which looks like an Ωspectrum but is not in general. The analogous spectrum in Sp^{ℕ} is an Ω-spectrum, so we obtain the following result.

Theorem 4.2.2. A map in Sp^{ℕ} is a stable equivalence if and only if it is a stable homotopy equivalence.

We also obtain the usual characterizations of the stable trivial fibrations and the stable fibrations.

Proposition 4.2.3. A map in Sp^{ℕ} is a stable trivial fibration if and only if it is a level trivial fibration.

Proposition 4.2.4. A map f: E ⟶ B in Sp^{ℕ} is a stable fibration if and only if it is a level fibration and, for all n ≥ 0, the diagram

E_{n} ⟶ E_{n+1}^{S^{1}}
↓  ↓
B_{n} ⟶ B_{n+1}^{S^{1}}

is a homotopy pullback square.

Corollary 4.2.5. A spectrum X in Sp^{ℕ} is fibrant in the stable model category if and only if X is an Ω-spectrum.

Finally, we recover the theorem of Bousfield and Friedlander.

Theorem 4.2.6 ( [BF78]). The stable equivalences, stable cofibrations, and stable fibrations define a model structure on Sp^{ℕ}.

Note that the model structure we have just described is the same as the model structure in [BF78], since the stable equivalences and stable cofibrations are the same.

In particular, we have the following characterization of stable cofibrations.

Proposition 4.2.7. A map f: X ⟶ Y in Sp^{ℕ} is a stable cofibration if and only if f_{0} : X_{0} ⟶ Y_{0} is a monomorphism and the induced map X_{n}⨿_{X_{n−1}∧S^{1}} (Y_{n−1}∧S^{1}) ⟶ Y_{n} is a monomorphism for all n > 0. In particular, X is cofibrant if and only if the structure maps are monomorphisms.

This proposition is proved in [BF78]; the proof is very similar to the proof of the analogous fact for symmetric spectra, proved in Proposition 5.2.2.

4.3. The Quillen equivalence. The goal of this section is to show that the forgetful functor U from symmetric spectra to spectra is part of a Quillen equivalence. Obviously this requires that U have a left adjoint V : Sp^{ℕ} ⟶ Sp^{Σ}. We will assume the existence of V in this section, and construct V in Section 4.4.

Proposition 4.3.1. The functors U: Sp^{Σ} ⟶ Sp^{ℕ} and its left adjoint V are a Quillen adjoint pair.

Proof. Proposition 4.2.4 implies that U preserves stable fibrations. The stable trivial fibrations in Sp^{Σ} and in Sp^{ℕ} are the level trivial fibrations, so U preserves stable trivial fibrations as well.

Theorem 4.3.2. The functor U: Sp^{Σ} → Sp^{ℕ} and its left adjoint V form a Quillen equivalence of the stable model categories.

We prove this theorem by using Lemma 4.1.7. In particular, we need to understand stable equivalences between stably fibrant objects.

Lemma 4.3.3. Suppose f: X ⟶ Y is a stable equivalence between stably fibrant objects in either Sp^{Σ} or Sp^{ℕ}. Then f is a level equivalence.

Proof. Factor f as a stably trivial cofibration, i, followed by a stable fibration, p. Since f is a stable equivalence, p is also. Hence, p is a level trivial fibration by Lemma 3.4.5. Also, i is a stably trivial cofibration between stably fibrant objects, hence it is a strong deformation retract, see [Qui67, II p. 2.5]. To see this, note that i has the left lifting property with respect to X → ∗, so the lift constructs a homotopy inverse to i. Because the simplicial structure is given on levels, a strong deformation retract here is a level equivalence. So both i and p are level equivalences, hence so is f.

Corollary 4.3.4. U: Sp^{Σ} ⟶ Sp^{ℕ} detects and preserves stable equivalences between stably fibrant objects.

Let L denote a fibrant replacement functor in Sp^{Σ}, obtained by factoring X ⟶ ∗ into a stable trivial cofibration followed by a stable fibration. By Lemma 4.1.7 and Corollary 4.3.4, to prove Theorem 4.3.2 it suffices to show that X → ULV X is a stable equivalence for all cofibrant (non-symmetric) spectra X. We prove this in several steps.

Definition 4.3.5. Given a simplicial set X, define F̃_{n}(X) to be the (non-symmetric) spectrum whose mth level is S^{m−n} ∧ X for m ≥ n and the basepoint otherwise, with the obvious structure maps. This defines a functor F̃: 𝒮_{∗} ⟶ Sp^{ℕ} left adjoint to the evaluation functor Ev_{n}.

Note that F̃_{0}X = Σ^{∞}X. Also, since U ∘ Ev_{n} = Ev_{n}, the left adjoints satisfy V ∘ F̃_{n} = F_{n}.

Lemma 4.3.6. The map X ⟶ ULV X is a stable equivalence when X = Σ^{∞}Y = F̃_{0}Y for any Y ∈ 𝒮_{∗}.

Proof. Consider the functor on simplicial sets QZ = colim Ω^{n}KΣ^{n}Z, where K is a simplicial fibrant replacement functor. Because Q is simplicial we can prolong it to a functor on Sp^{Σ}. The map F_{0}Y ⟶ QF_{0}Y induces an isomorphism on stable homotopy. Also QF_{0}Y is an Ω-spectrum since QZ → ΩQΣZ is a weak equivalence for any Z ∈ 𝒮_{∗}. Hence QF_{0}Y is level equivalent to LF_{0}Y, so F_{0}Y → LF_{0}Y induces an isomorphism in stable homotopy. Since F̃_{0}Y → UF_{0}Y is a level equivalence and UF_{0}Y → ULF_{0}Y is a stable homotopy equivalence, the lemma follows.

Because both Sp^{Σ} and Sp^{ℕ} are stable model categories, the following lemma is expected.

Lemma 4.3.7. Suppose X is a cofibrant spectrum in Sp^{ℕ}. Then the map X ⟶ ULV X is a stable equivalence if and only if X ∧ S^{1} ⟶ ULV (X ∧ S^{1}) is a stable equivalence.

Proof. For notational convenience, we write ΣX for X ∧S^{1} and ΩX for X^{S^{1}} in this proof, for X a (possibly symmetric) spectrum. Consider the stable trivial cofibration ΣV X ⟶ LΣV X in Sp^{Σ}. By Theorem 3.1.14 part three, V X → ΩLΣV X is also a stable equivalence. By the lifting property of the stable trivial cofibration V X → LV X and the 2-out-of-3 property, there is a stable equivalence LV X → ΩLΣV X. This map is a stable equivalence between stably fibrant objects, so by Corollary 4.3.4, f: ULV X → UΩLΣV X is a stable equivalence.

So g: X → ULV X is a stable equivalence if and only if fg: X → UΩLΣV X is a stable equivalence. Since Ω and U commute, gf is a stable equivalence if and only if ΣX → ULΣV X is a stable equivalence by part three of Theorem 3.1.14 for (non-symmetric) spectra. But, since U commutes with Ω, the left adjoint V commutes with Σ, so we have a natural isomorphism ULΣV X → ULV ΣX. This completes the proof.

Lemma 4.3.8. Let f: X → Y be a stable equivalence between cofibrant spectra in Sp^{ℕ}. Then X → ULV X is a stable equivalence if and only if Y → ULV Y is a stable equivalence.

Proof. Consider the following commuting square.

X ⟶ Y
↓  ↓
ULV X ⟶ ULV Y

Since V is a left Quillen functor by Proposition 4.3.1, it preserves trivial cofibrations. Hence, by Ken Brown’s Lemma 4.1.3, V preserves stable equivalences between cofibrant objects. Hence V X → V Y is a stable equivalence. L takes stable equivalences to level equivalences, by Lemma 4.3.3. So ULV X → ULV Y is a level equivalence since U preserves level equivalences. Hence the top and bottom maps are stable equivalences, so the right map is a stable equivalence if and only if the left map is.

Using the preceding three lemmas we can extend Lemma 4.3.6 to any cofibrant strictly bounded below spectrum.

Definition 4.3.9. Define a spectrum X ∈ Sp^{ℕ} to be strictly bounded below if there is an n such that for all m ≥ n the structure map S^{1} ∧ X_{m} ⟶ X_{m+1} is an isomorphism.

Lemma 4.3.10. Suppose X ∈ Sp^{ℕ} is cofibrant and strictly bounded below. Then the map X ⟶ ULV X is a stable equivalence.

Proof. Suppose X is strictly bounded below at n. Then we have a map F̃_{n}X_{n} ⟶^{g} X which is the identity on all levels ≥ n. In particular, g is a stable homotopy equivalence. Applying Lemma 4.3.8, this shows that to prove the lemma it is enough to show that F̃_{n}X_{n} → ULF_{n}X_{n} is a stable equivalence. But there is an evident map F̃_{n}X_{n} ∧ S^{n} ≅ F̃_{n}(X_{n} ∧ S^{n}) → Σ^{∞}X_{n} which is the identity map above level n − 1, and so is a stable equivalence. Lemmas 4.3.6, 4.3.7, and 4.3.8 complete the proof.

We now extend this lemma to all cofibrant objects, completing the proof of Theorem 4.3.2. First, we need to recall a basic fact about simplicial sets. Recall that the homotopy group π_{n}X of a pointed simplicial set X is defined to be π_{0} Map_{𝒮_{∗}} (S^{n}, KX), where K is a fibrant replacement functor. This ensures that weak equivalences are homotopy isomorphisms. If X is already a Kan complex, X is simplicially homotopy equivalent to KX, and so π_{n}X ≅ π_{0} Map_{𝒮_{∗}} (S^{n}, X). Since the simplicial sets ∂Δ[n]_{+} and Δ[n]_{+} are finite, the colimit of a sequence of Kan complexes is again a Kan complex. Since the simplicial sets S^{n} and S^{n} ∧ Δ[1]_{+} are finite, homotopy commutes with sequential (or filtered) colimits of Kan complexes.

In fact, homotopy commutes with sequential colimits of arbitrary monomorphisms of simplicial sets. To see this, apply the geometric realization to get a sequence of closed inclusions of CW complexes. Since homotopy commutes with such sequential colimits, the result follows.

Lemma 4.3.11. Suppose X is a cofibrant object of Sp^{ℕ}. Then the map X ⟶ ULV X is a stable equivalence.

Proof. Let X^{i} denote the truncation of X at i. That is, we have X_{n}^{i} = X_{n} for n ≤ i and X_{n}^{i} = X_{i} ∧ S^{n−i} for n ≥ i. Then the X^{i} are strictly bounded below and cofibrant, and there are monomorphisms X^{i} ⟶ X^{i+1} with colim_{i} X^{i} = X. Thus each map X^{i} ⟶ ULV X^{i} is a stable equivalence.

We claim that the induced map X ⟶ colim_{i} ULV X^{i} is a stable equivalence. To see this, note that

π_{n}X = colim_{i} π_{n+i}X_{i} = colim_{i} π_{n+i} colim_{j} X_{i}^{j}

Since homotopy commutes with sequential colimits of monomorphisms, we find that π_{n}X ≅ colim_{j} π_{n}X^{j}. Similarly, since homotopy of Kan complexes commutes with arbitrary filtered colimits, we find π_{n} colim_{i} ULV X^{i} ≅ colim_{i} π_{n}ULV X^{i}. It follows that X ⟶ colim_{i} ULV X^{i} is a stable homotopy equivalence, as required.

We now examine the relationship between ULV X and colim_{i} ULV X^{i}. Since V is a left adjoint, V X ≅ colim_{i} V X^{i}. Each map V X^{i} ⟶ LV X^{i} is a stable trivial cofibration; we claim that the induced map colim_{i} V X^{i} ⟶ colim_{i} LV X^{i} is a stable equivalence. To see this, we define a new sequence Y^{i} and maps of sequences V X^{i} ⟶ Y^{i} and Y^{i} ⟶ LV X^{i} inductively. Define Y^{0} = LV X^{0}. Having defined Y^{i}, define Y^{i+1} by factoring the map Y^{i} ⨿_{V X^{i}} V X^{i+1} ⟶ LV X^{i+1} into a stable trivial cofibration Y^{i} ⨿_{V X^{i}} V X^{i+1} ⟶ Y^{i+1} followed by a stable fibration Y^{i+1} ⟶ LV X^{i+1}. Then the induced map colim_{i} V X^{i} ⟶ colim_{i} Y^{i} is a stable trivial cofibration, by a lifting argument. On the other hand, each map Y^{i} ⟶ LV X^{i} is a stable equivalence, by the two out of three axiom. Since LV X^{i} and hence Y^{i} are stably fibrant, the maps Y^{i} ⟶ LV X^{i} are level equivalences, by Lemma 4.3.3. Since homotopy of level Kan complexes commutes with filtered colimits, we find that colim Y^{i} ⟶ colim_{i} LV X^{i} is a level equivalence, and therefore that V X ≅ colim_{i} V X^{i} ⟶ colim_{i} LV X^{i} is a stable equivalence.

We now claim that colim_{i} LV X^{i} is an Ω-spectrum, and thus is stably fibrant. Indeed, colim_{i} LV X^{i} is a level Kan complex by the comments preceding this lemma. Similarly, (colim LV X^{i})_{n+1}^{S^{1}} = colim_{i}((LV X^{i})_{n+1}^{S^{1}}). Since homotopy of Kan complexes commutes with filtered colimits, it follows that colim_{i} LV X^{i} is an Ω-spectrum.

Hence the stable equivalence V X ⟶ colim_{i} LV X^{i} extends to a stable equivalence LV X ⟶ colim_{i} LV X^{i}. By Lemma 4.3.3, this map is actually a level equivalence. Since U preserves level equivalences and colimits, the map ULV X ⟶ colim_{i} ULV X^{i} is also a level equivalence. We have seen above that the map X ⟶ colim_{i} ULV X^{i} is a stable equivalence, so X ⟶ ULV X must also be a stable equivalence.

Remark 4.3.12. It follows from the results of Section 5.3 that the smash product on Sp^{Σ} induces a smash product on Ho Sp^{Σ}. The handicrafted smash products of [Ada74] induce a smash product on Ho Sp^{ℕ}. We now consider to what extent the equivalence RU: Ho Sp^{Σ} ⟶ Ho Sp^{ℕ} induced by U preserves these smash products. Since U is a simplicial functor, there is a natural isomorphism RU(X ∧ Y) ≅ (RU)(X) ∧ (RU)(Y) for all (arbitrary desuspensions of) suspension spectra X. On the other hand, in either Ho Sp^{Σ} or Ho Sp^{ℕ}, X ∧Y is determined by the collection of F ∧Y for all finite spectra F mapping to X. To be precise, X∧Y is the minimal weak colimit [HPS97] of the F ∧Y. As an equivalence of categories, RU preserves minimal weak colimits, so there is an isomorphism RU(X ∧ Y) ≅ (RU)(X) ∧ (RU)(Y), However, we do not know if this is natural, as the minimal weak colimit is only a weak colimit. This isomorphism is natural up to phantom maps, however.

4.4. Description of V. This short section is devoted to the construction of the left adjoint V : Sp^{ℕ} ⟶ Sp^{Σ} to the forgetful functor U: Sp^{Σ} ⟶ Sp^{ℕ}.

Recall that, in any cocomplete symmetric monoidal category 𝒞, the free monoid or tensor algebra generated by an object X is T (X) = e∨X ∨X^{⊗2} ∨⋯∨X^{⊗n} ∨⋯, where e is the unit and ∨ is the coproduct. The multiplication on T (X) is the concatenation X^{⊗n} ⊗ X^{⊗m} → X^{⊗(n+m)}. Similarly, the free commutative monoid on an object X is Sym(X) = e ∨ X ∨ (X^{⊗2}/Σ_{2}) ∨ ⋯ ∨ (X^{⊗n}/Σ_{n}) ∨ · · ·.

Recall that the evaluation functor Ev_{n}: 𝒮_{∗}^{Σ} ⟶ 𝒮_{∗} has a left adjoint G_{n}, where G_{n}X is (Σ_{n})_{+} ∧ X at level n and the basepoint everywhere else. Similarly, the evaluation functor Ev_{n}: 𝒮_{∗}^{ℕ} ⟶ 𝒮_{∗} has a left adjoint G̃_{n}, where G̃_{n} is X at level n and the basepoint everywhere else.

Lemma 4.4.1. In the category 𝒮_{∗}^{ℕ} of sequences, the sphere spectrum S is the tensor algebra on the sequence G̃_{1}S^{1} = (∗, S^{1}, ∗, …, ∗, …). In the category 𝒮_{∗}^{Σ} of symmetric sequences, the sphere symmetric spectrum S is the free commutative monoid on the symmetric sequence G_{1}S^{1} = (∗, S^{1}, ∗, …, ∗, …).

Proof. The first statement follows directly from the definitions. In the category of symmetric sequences, (G_{1}S^{1})^{⊗n} = G_{n}S^{n}, so T (G_{1}S^{1}) is (Σ_{n})_{+} ∧ S^{n} in degree n. Therefore Sym(G_{1}S^{1}) is S^{n} in degree n. Since we already know S is a commutative monoid, the map G_{1}S^{1} ⟶ S induces a map Sym(G_{1}S^{1}) ⟶ S which is an isomorphism.

This lemma explains why left S-modules and right S-modules are equivalent in the category of sequences, since this is true for any tensor algebra. This lemma also explains why Remark 1.2.3 holds, since an analogous statement holds for any free commutative monoid.

Now, the forgetful functor U: 𝒮_{∗}^{Σ} ⟶ 𝒮_{∗}^{ℕ} has a left adjoint G, defined by GX = ⋁ G_{n}X_{n}, so that the nth level of GX is just (Σ_{n})_{+}∧X_{n}. The functor G is monoidal; that is, there is a natural isomorphism G(X)⊗G(Y) ⟶ G(X ⊗Y) compatible with the associativity and unit isomorphisms. However, G is definitely not a symmetric monoidal functor; this natural isomorphism is not compatible with the commutativity isomorphisms. This explains how S can be commutative in 𝒮_{∗}^{Σ} yet US = S is not commutative in 𝒮_{∗}^{ℕ}.

Since G is a monoidal functor, G preserves monoids and modules, and so defines a functor G: Sp^{ℕ} ⟶ T (G_{1}S^{1})-mod, left adjoint to the forgetful functor T (G_{1}S^{1})-mod ⟶ Sp^{ℕ}. On the other hand, the map of monoids T (G_{1}S^{1}) ⟶^{p} Sym(G_{1}S^{1}) = S defines the usual adjoint pair of induction and restriction. Induction takes a (left) T (G_{1}S^{1})-module X to S ⊗_{T (G_{1}S^{1})} X, where the tensor product uses the right action of T (G_{1}S^{1}) on S determined by p. It follows that the left adjoint V : Sp^{ℕ} ⟶ Sp^{Σ} of the forgetful functor U: Sp^{Σ} ⟶ Sp^{ℕ} is V (X) = S ⊗_{T (G_{1}S^{1})} GX.

5. Additional properties of symmetric spectra

In this section we discuss some properties of the category of symmetric spectra. In Section 5.1, we consider the level model structures on Sp^{Σ}. In particular, we show that every symmetric spectrum embeds in an injective spectrum by a level equivalence, completing the proof that the stable structures define a model structure on Sp^{Σ}. In Section 5.2 we characterize the stable cofibrations. In Sections 5.3 and 5.5, we study the relationship between the stable model structure on Sp^{Σ} and the smash product. This is necessary for constructing model categories of monoids, algebras, and modules, as is done in [SS97]. In Section 5.4, we show that the stable model structure on Sp^{Σ} is proper. Finally, in Section 5.6 we define semistable spectra and investigate their relationship to stable homotopy equivalences.

5.1. Level model structure. In this section we construct the two level model structures on the category of symmetric spectra.

Definition 5.1.1. A projective cofibration of symmetric spectra is a map that has the left lifting property with respect to every level trivial fibration. The projective cofibrations are the stable cofibrations from Section 3.4. The projective level structure on Sp^{Σ} is the class of level equivalences, the class of projective cofibrations, and the class of level fibrations. An injective fibration of symmetric spectra is a map that has the right lifting property with respect to every level trivial cofibration (the adjective “injective” refers to the lifting properties of the map and not to its being a monomorphism). The injective level structure is the class of level equivalences, the class of level cofibrations, and the class of injective fibrations.

Theorem 5.1.2. The projective level structure and the injective level structure are model structures on the category of symmetric spectra.

Proof. The category of symmetric spectra is bicomplete. The class of level equivalences has the two-out-of-three property. The retract axiom holds by construction in both the projective and injective level structures.

We now prove the lifting and factorization axioms, beginning with the projective level structure. We use the sets of maps F I_{∂} and F I_{Λ} defined in Definition 3.3.2. The lifting axiom for a projective cofibration and a level trivial fibration holds by definition. The other lifting and factorization axioms follow by identifying the respective classes in terms of F I_{∂} and F I_{Λ}. By part 4 of Lemma 5.1.4, an F I_{Λ}- cofibration is a projective cofibration which is a level equivalence and an F I_{Λ}- injective map is a level fibration. Since (J-cof, J-inj) has the lifting property for any class J, the lifting axiom for a map that is both a level equivalence and a projective cofibration and a map that is a level fibration follows by setting J = F I_{Λ}. Moreover, every map can be factored as the composition of an F I_{Λ}-cofibration and an F I_{Λ}-injective map. Similarly, every map can be factored as the composition of an F I_{∂}-cofibration and an F I_{∂}-injective map, by Lemma 3.2.11. By part 3 of Lemma 5.1.4, an F I_{∂}-cofibration is a projective cofibration and an F I_{∂}-injective map is a level trivial fibration.

Now consider lifting and factorization for the injective level model structure. Here we use a set C containing a map from each isomorphism class of monomorphisms i: X → Y with Y a countable symmetric spectrum, and a set tC containing a map from each isomorphism class of level trivial cofibrations i: X → Y with Y a countable symmetric spectrum. The lifting axiom for a level trivial cofibration i and an injective fibration p holds by definition. By part 5 of Lemma 5.1.4, a Ccofibration is a level cofibration and a C-injective map is an injective fibration that is a level equivalence. Since (J-cof, J-inj) has the lifting property for any class J, the lifting axiom for a level cofibration and a map that is both an injective fibration and a level equivalence follows with J = C. Also, every map can be factored as the composition of a C-cofibration followed by a C-injective map, by Lemma 3.2.11. Similarly, every map can be factored as the composition of a tC-cofibration and a tC-injective map. By part 6 of Lemma 5.1.4, a tC-cofibration is a level trivial cofibration and a tC-injective map is an injective fibration.

Corollary 5.1.3. Every symmetric spectrum embeds in an injective spectrum by a map that is a level equivalence.

Proof. For a symmetric spectrum X, the map X → ∗ is the composition of a level trivial cofibration X → E and an injective fibration E → ∗. The fibrant object E is an injective spectrum.

Some parts of the next lemma have already been proved. They are repeated for easy reference. Recall that R_{n}: 𝒮_{∗} → Sp^{Σ} is the right adjoint of the evaluation functor Ev_{n}: Sp^{Σ} → 𝒮_{∗}.

Lemma 5.1.4.

1. Let K ⊆ 𝒮_{∗} be the class of Kan fibrations and let RK = ∪_{n}R_{n}K. Then a map is RK-projective if and only if it is a level trivial cofibration.

2. Let tK ⊆ 𝒮_{∗} be the class of trivial Kan fibrations and let R(tK) = ∪_{n}R_{n}(tK). Then a map is R(tK)-projective if and only if it is a level cofibration.

3. Let F I_{∂} be the set defined in 3.3.2. Then a map is F I_{∂}-injective if and only if it is a level trivial fibration. A map is an F I_{∂}-cofibration if and only if it is a projective cofibration.

4. Let F I_{Λ} be the set defined in 3.3.2. Then a map is F I_{Λ}-injective if and only if it is a level fibration. A map is an F I_{Λ}-cofibration if and only if it is a projective cofibration and a level equivalence.

5. Let C be a set containing a map from each isomorphism class of monomorphisms i: X → Y with Y a countable symmetric spectrum. Then a map is C-injective if and only if it is an injective fibration and a level equivalence. A map is a C-cofibration if and only if it is a level cofibration.

6. Let tC be a set containing a map from each isomorphism class of level trivial cofibrations i: X → Y with Y a countable symmetric spectrum. Then a map is tC-injective if and only if it is a injective fibration. A map is a tCcofibration if and only if it is a level trivial cofibration.

Proof. Parts 1 and 2: By adjunction, a map g has the left lifting property with respect to the class RK (R(tK)) if and only if for each n ≥ 0 the map Ev_{n} g has the left lifting property with respect to K (tK). But Ev_{n} g has the left lifting property with respect to K (tK) if and only if Ev_{n} g is a trivial cofibration (arbitrary cofibration), i.e., if and only if g is a level trivial cofibration (arbitrary level cofibration).

Part 3 is proved in Propositions 3.3.3 and 3.4.2.

Part 4: The first claim is proved in Proposition 3.3.3. Every F I_{Λ}-cofibration has the left lifting property with respect to level fibrations, so is in particular a projective cofibration by Part 3. Every map in F I_{Λ} is a level trivial cofibration by Proposition 2.2.7, so is RK-projective by Part 1. So every F I_{Λ}-cofibration is also RK-projective and hence is a level trivial cofibration by Part 1 again. So in particular it is a level equivalence.

Conversely, suppose f is a projective cofibration and a level equivalence. We can factor f as the composition of an F I_{Λ}-cofibration i and an F I_{Λ}-injective map p, by Lemma 3.2.11. By the two-out-of-three property, p is a level equivalence. Therefore the projective cofibration f has the left lifting property with respect to the level trivial fibration p. By the Retract Argument 3.2.4, f is a retract of i, and so is an F I_{Λ}-cofibration.

For part 5, first note that, by part 2, every C-cofibration is a level cofibration. Conversely, suppose f: X ⟶ Y is a level cofibration. Then f is a C-cofibration if, for every C-injective map g and commutative square

X ⟶ E
f↓  ↓^{g}
Y ⟶ Z

there is a lift h: Y → E making the diagram commute. Let 𝒫 be the partially ordered set of partial lifts: an object of 𝒫 is a pair (U, h_{U}) such that X ⊆ U ⊆ Y and the diagram

[figure: X; i_{U}; E; g; U; h_{U}; Z,]

is commutative. We define (U, h_{U}) ≤ (V, h_{V}) if U ⊆ V and h_{V} extends h_{U}. Every chain in 𝒫 has an upper bound and so Zorn’s lemma gives a maximum (M, h_{M}). Suppose M is strictly contained in Y. Then, by taking the subspectrum generated by a simplex not in M, we find a countable subspectrum D (by Lemma 5.1.6 below) such that the level cofibration D∩M → D is not an isomorphism. By construction, the map D ∩ M → D is isomorphic to a map in C. By cobase change, M → D ∪ M is a C-cofibration. Thus h_{M} extends to a partial lift on D ∪ M, contradicting the maximality of (M, h_{M}). Therefore M = Y, and so f is a C-cofibration.

We now identify C-inj. Since (C-cof)-inj = C-inj, every C-injective map has the right lifting property with respect to every monomorphism. In particular, every C-injective map is an injective fibration. Let f: E → B be a map having the right lifting property with respect to every monomorphism. Let s: B → E be a lift in the diagram

\begin{matrix}∗ & ⟶ & E \\ ↓ & & ↓^{f} \\ B & & B \end{matrix}

Then fs is the identity map on B. To study the composite sf, let j be the monomorphism ∂Δ[1] → Δ[1]. The diagram

E ∨ E ⟶^{sf∨1} E
E∧j_{+}↓  ↓^{f}
E ∧ Δ[1]_{+} ⟶ B

is commutative since fsf = f and has a lift since E ∧ j_{+} is a monomorphism. The lift is a simplicial homotopy from sf to the identity on E. Therefore f is a simplicial homotopy equivalence and in particular f is a level equivalence. Conversely suppose f is both an injective fibration and a level equivalence. We can factor f as the composition of a C-cofibration i and a C-injective map p. By the two-out-of-three property, i is a level equivalence. The level trivial cofibration i has the left lifting property with respect to the injective fibration f. By the Retract Argument 3.2.4, f is a retract of p and so is a C-injective map.

The proof of part 6 is similar, though slightly more complex. By part 1, every tC-cofibration is a level trivial cofibration. Conversely, suppose f: X → Y is a level trivial cofibration. Then f is a tC-cofibration if, for every tC-injective map g and commutative square

X ⟶ E
f↓  ↓^{g}
Y ⟶ Z

there is a lift h: Y → E making the diagram commute. We again let 𝒫 be the partially ordered set of partial lifts: an object of 𝒫 is a pair (U, h_{U}) such that X ⊆ U ⊆ Y, and the diagram

[figure: X; i_{U}; E; g; U; h_{U}; Z,]

is commutative, but we also require that the inclusion i_{U}: X → U is a weak equivalence. We define (U, h_{U}) ≤ (V, h_{V}) as before. Every chain in 𝒫 has an upper bound (using the fact that a transfinite composition of level trivial cofibrations is a level trivial cofibration) and so Zorn’s lemma gives a maximum (M, h_{M}). The inclusion X → M is a level trivial cofibration, so, by the two-out-of-three property, the inclusion M → Y is a weak equivalence. If M is strictly contained in Y, Lemma 5.1.7, proved below, applied to the countable subspectrum of Y generated by a simplex not in M, gives a countable subspectrum D of Y such that the monomorphism D ∩ M → D is a weak equivalence but is not an isomorphism. By construction, D ∩ M → D is isomorphic to a map in tC. By cobase change, M → D∪M is a tC-cofibration. So h_{M} extends to a partial lift on D∪M. This is a contradiction since (M, h_{M}) is maximal. Thus M = Y, and so f is a tC-cofibration.

Since (tC-cof)-inj = tC-inj, the tC-injective maps are the injective fibrations.

Corollary 5.1.5. Every injective fibration is a level fibration and every projective cofibration is a level cofibration.

Proof. By Proposition 2.2.7, every map in F I_{∂} is a level cofibration. Therefore, by part 2 of Lemma 5.1.4, every projective cofibration is a level cofibration. By Proposition 2.2.7, every map in F I_{Λ} is a level trivial cofibration. Therefore every injective fibration is a level fibration.

The following lemmas are used in the proof of Lemma 5.1.4.

Lemma 5.1.6. Let X be a spectrum, and suppose x is a simplex of X_{n} for some n ≥ 0. Then the smallest subspectrum of X containing x is countable.

Proof. First note that if L is a countable collection of simplices in a simplicial set K, then the smallest subsimplicial set of K containing L is also countable. Indeed, we need only include all degeneracies of all faces of simplices in L, of which there are only countably many. Similarly, if L is a countable collection of simplices in a Σ_{n}- simplicial set K, then the smallest sub-Σ_{n}-simplicial set containing L is countable.

Indeed, we only need to include the orbits of all degeneracies of all faces of simplices in L.

Now, let Y_{n} denote the sub-Σ_{n}-simplicial set of X_{n} generated by x. We have just seen that Y_{n} is countable. We then inductively define Y_{n+k} to be the smallest sub-Σ_{n+k}-simplicial set of X_{n+k} containing the image of S^{1} ∧ Y_{n+k−1}. Then each Y_{n+k} is countable, and the Y_{n+k} define a subspectrum of X containing x.

It follows in similar fashion that the smallest subspectrum of a spectrum X containing any countable collection of simplices of X is countable.

We need a similar lemma for inclusions which are level equivalences. To prove such a lemma, we need to recall from the comments before Lemma 4.3.11 that homotopy of simplicial sets commutes with sequential colimits of monomorphisms. The same methods imply that relative homotopy commutes with sequential colimits of monomorphisms.

Lemma 5.1.7. Let f: X → Y be a level trivial cofibration of symmetric spectra. For every countable subspectrum C of Y there is a countable subspectrum D of Y such that C ⊆ D and D ∩ X → D is a level trivial cofibration.

Proof. Let K ⊆ L be a pair of pointed simplicial sets and v be a 0-simplex of K. For n ≥ 1, let π_{n}(L, K; v) denote the relative homotopy set of the pair with the null element as the basepoint (ignore the group structure when n ≥ 2). To ease notation let π_{0}(L, K; v) be the pointed set π_{0}L/π_{0}K. The inclusion K → L is a weak equivalence if and only if π_{n}(L, K; v) = ∗ for every v ∈ K_{0} and n ≥ 0.

Now, construct a countable spectrum F C such that the map π_{∗}(C_{n}, C_{n}∩X_{n}; v) → π_{∗}(F C_{n}, F C_{n} ∩ X_{n}; v) factors through the basepoint ∗ for every 0-simplex v of C_{n} ∩ X_{n} and integer n ≥ 0. Since π_{∗}(Y, X; v) = ∗ and π_{∗} commutes with filtered colimits, for each homotopy class α ∈ π_{∗}(C_{n}, C_{n} ∩ X_{n}; v) there is a finite simplicial subset K_{α} ⊆ Y_{n} such that π_{∗}(C_{n}, C_{n} ∩ X_{n}; v) → π_{∗}(K_{α} ∪ C_{n}, (K_{α} ∪ C_{n}) ∩ X_{n}; v) sends α to the basepoint. Since C_{n} is countable, the set π_{∗}(C_{n}, C_{n} ∩ X_{n}; v) is countable. Let B_{n} be the union of C_{n} with all the finite simplicial sets K_{α}. The B_{n} are countable simplicial sets and generate a countable subspectrum F C of Y having the desired property.

Repeat the construction to get a sequence of countable subspectra of Y:

C → F C → F^{2}C → ⋯ → F^{n}C → …

Let D = colim_{n} F^{n}C. The spectrum D is countable. Since relative homotopy commutes with sequential colimits of monomorphisms, the set π_{∗}(D_{n}, D_{n} ∩ X_{n}; v) has only one element. Therefore the inclusion D_{n} ∩X_{n} → D_{n} is a weak equivalence, and so D ∩ X → D is a level equivalence.

5.2. Stable cofibrations. The object of this section is to give a characterization of stable cofibrations in Sp^{Σ}. To this end, we introduce the latching space.

Definition 5.2.1. Define \overline{S} to be the symmetric spectrum such that \overline{S}_{n} = S^{n} for n > 0 and \overline{S}_{0} = ∗. The structure maps are the evident ones. Given a symmetric spectrum X, define the nth latching space, L_{n}X, to be Ev_{n}(X ∧ \overline{S}).

There is a map of symmetric spectra i: \overline{S} → S which is the identity on positive levels. This induces a natural transformation L_{n}X → X_{n} of pointed Σ_{n} simplicial sets.

The following proposition uses a model structure on the category of pointed Σ_{n} simplicial sets. A map f: X → Y of pointed Σ_{n} simplicial sets is a Σ_{n}-fibration if it is a Kan fibration of the underlying simplicial sets. Similarly, f is a weak equivalence if it is a weak equivalence of the underlying simplicial sets. The map f is a Σ_{n}-cofibration if it is a monomorphism such that Σ_{n} acts freely on the simplices of Y not in the image of f. It is well-known, and easy to check, that the Σ_{n}-cofibrations, the Σ_{n}-fibrations, and the weak equivalences define a model structure on the category of pointed Σ_{n}-simplicial sets.

Proposition 5.2.2. A map f: X → Y in Sp^{Σ} is a stable cofibration if and only if for all n ≥ 0 the induced map Ev_{n}(f □ i): X_{n} ⨿_{L_{n}X} L_{n}Y → Y_{n} is a Σ_{n}-cofibration.

Proof. Suppose first that Ev_{n}(f□i) is a Σ_{n}-cofibration for all n. Suppose g: E → B is a level trivial fibration. We show that f has the left lifting property with respect to g by constructing a lift using induction on n. A partial lift defines a commutative square

X_{n} ⨿_{L_{n}X} L_{n}Y ⟶ E_{n}
↓  ↓
Y_{n}  ⟶ B_{n}

Since the left vertical map is a Σ_{n}-cofibration and the right vertical map is a trivial Σ_{n}-fibration, there is a lift in this diagram and so we can extend our partial lift. Hence f has the left lifting property with respect to g, and so f is a stable cofibration.

To prove the converse, note that Ev_{n} is a left adjoint as a functor to pointed Σ_{n} simplicial sets. Since the class of stable cofibrations is the class F I_{∂}-cof, it suffices to check that Ev_{n}(f □ i) is a Σ_{n}-cofibration for f ∈ F I_{∂}. More generally, suppose g: A → B is a monomorphism of pointed simplicial sets. Since F_{m}g = S ⊗ G_{m}g, we have F_{m}g □ i = G_{m}g □ i, where the second □ is taken in 𝒮_{∗}^{Σ}. One can easily check that Ev_{n}(G_{m}g □ i) is an isomorphism when n ≠ m and is the map (Σ_{n})_{+} ∧ g when n = m. In both cases, Ev_{n}(G_{m}g □ i) is a Σ_{n}-cofibration, as required.

5.3. Pushout smash product. In this section we consider the pushout smash product in an arbitrary symmetric monoidal category and apply our results to Sp^{Σ}. We show that the projective level structure and the stable model structure on Sp^{Σ} are both compatible with the symmetric monoidal structure. A monoid E in Sp^{Σ} is called a symmetric ring spectrum, and is similar to an A_{∞}-ring spectrum. Thus, there should be a stable model structure on the category of E-modules. Similarly, there should be a model structure on the category of symmetric ring spectra and the category of commutative symmetric ring spectra. These issues are dealt with more fully in [SS97] and in work in progress of the third author. Their work depends heavily on the results in this section and in Section 5.5. The results of this section alone suffice to construct a stable model structure on the category of modules over a symmetric ring spectrum which is stably cofibrant. This section also contains brief descriptions of two other stable model structures on Sp^{Σ}.

Definition 5.3.1. Let f: U → V and g: X → Y be maps in a symmetric monoidal category 𝒞. The pushout smash product

f □ g: V ∧ X ⨿_{U∧X} U ∧ Y → V ∧ Y.

is the natural map on the pushout defined by the commutative square

U ∧ X ⟶^{f∧X} V ∧ X
U∧g↓  ↓^{V ∧g}
U ∧ Y ⟶_{f∧Y} V ∧ Y.

If 𝒞 is a closed symmetric monoidal category,

Hom_{□}(f, g): Hom(V, X) → Hom(U, X) ×_{Hom(U,Y)} Hom(V, Y).

is the natural map to the fiber product defined by the commutative square

Hom(V, X) ⟶^{f^{∗}} Hom(U, X)
g_{∗}↓  ↓^{g_{∗}}
Hom(V, Y) ⟶_{f^{∗}} Hom(U, Y).

A model structure on a symmetric monoidal category is called monoidal if the pushout smash product f □ g of two cofibrations f and g is a cofibration which is trivial if either f or g is. In our situation, this is the correct condition to require so that the model structure is compatible with the symmetric monoidal structure. For a more general discussion of monoidal model structures, see [Hov97].

Recall, from Definition 3.3.6, the map of sets 𝒞_{□}(f, g).

Proposition 5.3.2. Let f, g and h be maps in a closed symmetric monoidal category 𝒞. There is a natural isomorphism

𝒞_{□}(f □ g, h) ≅ 𝒞_{□}(f, Hom_{□}(g, h))

Proof. Use the argument in the proof of Proposition 3.3.8.

Proposition 5.3.3. Let I and J be classes of maps in a closed symmetric monoidal category 𝒞. Then

I-cof □ J-cof ⊆ (I □ J)-cof

Proof. Let K = (I □ J)-inj. By hypothesis, (I □ J, K) has the lifting property. By adjunction, (I, Hom_{□}(J, K)) has the lifting property. By Proposition 3.2.8, (I-cof, Hom_{□}(J, K)) has the lifting property. Then (J, Hom_{□}(I-cof, K)) has the lifting property, by using adjunction twice. Thus (J-cof, Hom_{□}(I-cof, K)) has the lifting property, by Proposition 3.2.8. By adjunction, (I-cof □ J-cof, K) has the lifting property. So I-cof □ J-cof ⊆ (I □ J)-cof and the proposition is proved.

Corollary 5.3.4. For classes I, J and K in Sp^{Σ}, if I □ J ⊆ K-cof then I-cof □ J-cof ⊆ K-cof.

We now examine to what extent the pushout smash product preserves stable cofibrations and stable equivalences. To do so, we introduce a new class of maps in Sp^{Σ}.

Definition 5.3.5. Let M be the class of monomorphisms in the category of symmetric sequences 𝒮_{∗}^{Σ}. A map of symmetric spectra is an S-cofibration if it is an S ⊗ M-cofibration. A symmetric spectrum X is S-cofibrant if the map ∗ → X is an S-cofibration. A map is an S-fibration if it has the right lifting property with respect to every map which is both an S-cofibration and a stable equivalence.

Note that every stable cofibration is an S-cofibration, since F I_{∂} = S ⊗ ⋃_{n} G_{n}I_{∂}. On the other hand, by Proposition 2.1.9, every element of S ⊗ M is a monomorphism, and so every S-cofibration is a level cofibration. There is a model structure on Sp^{Σ}, called the S model structure, where the cofibrations are the Scofibrations and the weak equivalences are the stable equivalences. The fibrations, called S-fibrations are those maps with the right lifting property with respect to S-cofibrations which are also stable equivalences. Every S-fibration is a stable fibration. This model structure will be used in a forthcoming paper by the third author to put a model structure on certain commutative S-algebras.

We mention as well that there is a third model structure on Sp^{Σ} where the weak equivalences are the stable equivalences, called the injective (stable) model structure. The injective cofibrations are the level cofibrations and the injective stable fibrations are all maps which are both injective fibrations and stable fibrations. In particular, the fibrant objects are the injective Ω-spectra. The interested reader can prove this is a model structure using the methods of Section 3.4, replacing the set I with the union of I and the countable level cofibrations.

Theorem 5.3.6. Let f and g be maps of symmetric spectra.

1. If f and g are stable cofibrations then f □ g is a stable cofibration.

2. If f and g are S-cofibrations then f □ g is an S-cofibration.

3. If f is an S-cofibration and g is a level cofibration, then f □ g is a level cofibration.

4. If f is an S-cofibration, g is a level cofibration, and either f or g is a level equivalence, then f □ g is a level equivalence.

5. If f is an S-cofibration, g is a level cofibration, and either f or g is a stable equivalence, then f □ g is a stable equivalence.

Proof. Parts 1 through 4 of the Proposition are proved using Corollary 5.3.4.

Part 1: Let I = J = K = F I_{∂}. Then K-cof is the class of stable cofibrations. We have a natural isomorphism

F_{p}f □ F_{q}g = F_{p+q}(f □ g)

for f, g ∈ 𝒮_{∗}. By Proposition 1.3.4, f □ g is a monomorphism when f and g are. Part 3 of Proposition 3.4.2 the shows that I □ J ⊆ K-cof. Now use the corollary.

Part 2: Let I = J = K = S ⊗ M (recall that M is the class of monomorphisms in 𝒮_{∗}^{Σ}). By definition, K-cof is the class of S-cofibrations. For f and g in 𝒮_{∗}^{Σ}, we have a natural isomorphism

S ⊗ f □ S ⊗ g = S ⊗ (f □ g)

where the first □ is taken in Sp^{Σ} and the second □ is taken in 𝒮_{∗}^{Σ}. In degree n,

(f □ g)_{n} = ⋀_{p+q=n}(Σ_{p+q})_{+} ∧_{Σ_{p}×Σ_{q}} (f_{p} □ g_{q})

For f, g ∈ M, each map f_{p} □ g_{q} is a monomorphism, so it follows that f □ g is a monomorphism. Thus I □ J ⊆ K-cof. Now use the corollary.

Part 3: Let I = S ⊗ M and let J = K be the class of level cofibrations. By Part 2 of Lemma 5.1.4 K-cof = K. For f ∈ 𝒮_{∗}^{Σ} and g ∈ Sp^{Σ}, we have a natural isomorphism of maps of symmetric sequences

(S ⊗ f) □ g = f □ g

where the first □ is taken in Sp^{Σ} and the second □ is taken in 𝒮_{∗}^{Σ}. We have seen in the proof of part 2 that f □ g is a monomorphism of symmetric sequences if f and g are monomorphisms. Hence I □ J ⊆ K-cof. Now use the corollary.

Part 4: First assume g is a level trivial cofibration. Let I = S ⊗M and let J = K be the class of level trivial cofibrations. By Part 1 of Lemma 5.1.4, K = K-cof. Proposition 1.3.4 and the method used in the proof of part 2 imply that, if f and g are monomorphisms of symmetric sequences and g is a level equivalence, then f □ g is a level equivalence. Recall that, for h ∈ 𝒮_{∗}^{Σ} and g ∈ Sp^{Σ}, we have a natural isomorphism of maps of symmetric sequences

(S ⊗ h) □ g = h □ g

where the first □ is taken in Sp^{Σ} and the second □ is in 𝒮_{∗}^{Σ}. Hence I □ J ⊆ K-cof. Now use the corollary to prove part 4 when g is a level equivalence.

It follows that, for any injective spectrum E and an arbitrary S-cofibration h, the map Hom_{S}(h, E) is an injective fibration. Indeed, if g is a level cofibration and a level equivalence, Sp^{Σ}(g, Hom_{S}(h, E)) ≅ Sp^{Σ}(g □ h, E), and we have just seen that g □ h is a level cofibration and a level equivalence, so (g □ h, E) has the lifting property.

Now suppose f is an S-cofibration and a level equivalence. Then the map Hom_{Sp^{Σ}}(f, E) is an injective fibration and a level equivalence. Indeed, we have Ev_{k} Hom_{Sp^{Σ}}(f, E) = Map_{Sp^{Σ}}(f ∧ (S ⊗ Σ[k]_{+}), E), by Remark 2.2.12. Since S ⊗ Σ[k]_{+} is S-cofibrant, and f is both a level equivalence and a level cofibration, we have just proved that f ∧ (S ⊗ Σ[k]_{+}) = f □ (∗ → S ⊗ Σ[k]_{+}) is a level equivalence and a level cofibration. This shows that π_{0} Ev_{k} Hom_{Sp^{Σ}}(f, E) is an isomorphism; smashing with F_{0}S^{n} and using a similar argument shows that π_{n} Ev_{k} Hom_{Sp^{Σ}}(f, E) is an isomorphism.

Thus every level cofibration g has the left lifting property with respect to the map Hom_{Sp^{Σ}}(f, E). By adjunction, f □ g and f □ (g □ j) where j: ∂Δ[1]_{+} → Δ[1]_{+} is the inclusion, have the extension property with respect to every injective spectrum E. It follows that E^{0}(f □ g) is an isomorphism for every injective spectrum E and hence that f □ g is a level equivalence.

Part 5: Because we are working in a stable situation, a level cofibration i: X → Y is a stable equivalence if and only if its cofiber C_{i} = Y/X is stably trivial. The map f □ g is a level cofibration by part (3). By commuting colimits, the cofiber of f □ g is the smash product C_{f} ∧ C_{g} of the cofiber C_{f} of f, which is S-cofibrant, and the cofiber C_{g} of g. Let E be an injective Ω-spectrum. We will show that Hom_{S}(C_{f} ∧ C_{g}, E) is a level trivial spectrum, and thus that C_{f} ∧ C_{g} is stably trivial.

First suppose that f is a stable equivalence. Then Hom_{S}(C_{f}, E) is a level trivial spectrum which is also injective, by part 4 and the fact that C_{f} is S-cofibrant. Therefore Hom_{S}(C_{f} ∧C_{g}, E) ≅ Hom_{S}(C_{g}, Hom_{S}(C_{f}, E)) is a level trivial spectrum, so C_{f} ∧ C_{g} is stably trivial and thus f □ g is a stable equivalence.

Now suppose that g is a stable equivalence, so that C_{g} is stably trivial. By adjunction Hom_{S}(C_{f} ∧ C_{g}, E) = Hom_{S}(C_{g}, Hom_{S}(C_{f}, E)). We claim that D = Hom_{S}(C_{f}, E) is an injective Ω-spectrum. Indeed, we have already seen that D is injective. From Remark 2.2.12, we have

Ev_{n} D ≅ Map_{Sp^{Σ}}(C_{f} ∧ F_{n}S^{0}, E) ≅ Map_{Sp^{Σ}}(C_{f}, Hom_{S}(F_{n}S^{0}, E))

Similarly, we have

(Ev_{n+1} D)^{S^{1}} ≅ Map_{Sp^{Σ}}(C_{f}, Hom_{S}(F_{n+1}S^{1}, E)).

Since E is an Ω-spectrum, (F_{n}S^{0} ∧λ)^{∗}: Hom_{S}(F_{n}S^{0}, E) ⟶ Hom_{S}(F_{n+1}S^{1}, E) is a level equivalence. Since E is injective, both the source and target are injective, and so this map is a simplicial homotopy equivalence by Lemma 3.1.6. Hence Ev_{n} D ⟶ (Ev_{n+1} D)^{S^{1}} is still a level equivalence, so D = Hom_{S}(C_{f}, E) is an injective Ωspectrum. Since C_{g} is stably trivial, Hom_{S}(C_{f} ∧C_{g}, E) ≅ Hom_{S}(C_{g}, Hom_{S}(C_{f}, E)) is a level trivial spectrum, so C_{f} ∧ C_{g} is stably trivial and thus f □ g is a stable equivalence.

Corollary 5.3.7. The projective model structure and the stable model structure on Sp^{Σ} are monoidal.

It also follows that the S model structure on Sp^{Σ} is monoidal, once it is proven to be a model structure. Neither the injective level structure nor the injective stable structure is monoidal.

Adjunction then gives the following corollary.

Corollary 5.3.8. Let f and g be maps of symmetric spectra.

1. If f is a stable cofibration and g is a stable fibration, then Hom_{□}(f, g) is a stable fibration, which is a level equivalence if either f or g is a stable equivalence.

2. If f is a stable cofibration and g is a level fibration, then Hom_{□}(f, g) is a level fibration, which is a level equivalence if either f or g is a level equivalence.

3. If f is an S-cofibration and g is an S-fibration, then Hom_{□}(f, g) is an Sfibration, which is a level equivalence if either f or g is a stable equivalence.

4. If f is an S-cofibration and g is an injective fibration, then Hom_{□}(f, g) is an injective fibration, which is a level equivalence if either f or g is a level equivalence.

Corollary 5.3.9. If X is an S-cofibrant symmetric spectrum, the functor X ∧ − preserves level equivalences and it preserves stable equivalences between stably cofibrant objects.

Proof. Use Lemma 4.1.3 and parts four and five of Theorem 5.3.6.

5.4. Proper model categories. In this section we recall the definition of a proper model category and show that the stable model category of symmetric spectra is proper.

Definition 5.4.1. 1. A model category is left proper if for every pushout square

A ⟶^{g} B
f↓  ↓^{h}
X ⟶ Y

with g a cofibration and f a weak equivalence, h is a weak equivalence.

2. A model category is right proper if for every pullback square

A ⟶ B
h↓  ↓^{f}
X ⟶_{g} Y

with g a fibration and f a weak equivalence, h is a weak equivalence.

3. A model category is proper if it is both left proper and right proper.

The category of simplicial sets is a proper model category [BF78] (see [Hir97] for more details). Hence the category of pointed simplicial sets and both level model structures on Sp^{Σ} are proper.

Theorem 5.4.2. The stable model category of symmetric spectra is proper.

Proof. Since every stable cofibration is a level cofibration, the stable model category of symmetric spectra is left proper by part one of Lemma 5.4.3. Since every stable fibration is a level fibration, the stable model category of symmetric spectra is right proper by part two of Lemma 5.4.3.

Lemma 5.4.3.

1. Let

A ⟶^{g} B
f↓  ↓^{h}
X ⟶ Y

be a pushout square with g a level cofibration and f a stable equivalence. Then h is a stable equivalence.

2. Let

A ⟶^{k} B
f↓  ↓^{h}
X ⟶_{g} Y

be a pullback square with g a level fibration and h a stable equivalence. Then f is a stable equivalence.

Proof. Part one: Let E be an injective Ω-spectrum. Apply the functor Map_{Sp^{Σ}}(−, E) to the pushout square. The resulting commutative square

Map(A, E) ⟵^{Map(g,E)} Map(B, E)
Map(f,E)↑  ↑^{Map(h,E)}
Map(X, E) ⟵ Map(Y, E)

is a pullback square of pointed simplicial sets with Map(f, E) a weak equivalence, by Proposition 3.1.4. We claim that Map(g, E) is a Kan fibration. Indeed, let k: E → ∗ denote the obvious map, and let c denote a trivial cofibration of pointed simplicial sets. Then Map(g, E) = Map_{□}(g, k). We must show that (c, Map_{□}(g, k)) has the lifting property. By Corollary 3.3.9, this is equivalent to showing that (c □ g, k) has the lifting property. But, by Proposition 1.3.4, c □ g is a level equivalence and a level cofibration. Since E is injective, it follows that (c □ g, k) has the lifting property, and so Map(g, E) is a Kan fibration. By properness for simplicial sets, Map(h, E) is a weak equivalence. It follows that h is a stable equivalence.

Part two: Let F be the fiber over the basepoint of the map g: X → Y. Since k is a pullback of g, F is isomorphic to the fiber over the basepoint of the map k: A → B. The maps X/F → Y, A/F → B and B → Y are stable equivalences; so A/F → X/F is a stable equivalence. Consider the Barratt-Puppe sequence (considered in the proof of Lemma 3.4.15)

F → A → A/F → F ∧ S^{1} → A ∧ S^{1} → A/F ∧ S^{1} → F ∧ S^{2}

and the analogous sequence for the pair (F, X). Given an injective Ω-spectrum E, apply the functor E^{0}(−), and note that E^{0}(Z ∧ S^{1}) ≅ π_{1} Map(Z, E) is naturally a group. The five-lemma then implies that f ∧ S^{1} : A ∧ S^{1} ⟶ X ∧ S^{1} is a stable equivalence. Part two of Theorem 3.1.14 shows that f is a stable equivalence.

5.5. The monoid axiom. In [SS97], techniques are developed to form model category structures for categories of monoids, algebras, and modules over a monoidal model category. One more axiom is required which is referred to as the monoid axiom. In this section we verify the monoid axiom for symmetric spectra. The methods of [SS97] then immediately give a model structure on symmetric ring spectra. See also [Shi97] for more about symmetric ring spectra.

Let K denote the class in Sp^{Σ} consisting of all maps f ∧ X, where f is a stable trivial cofibration and X is some symmetric spectrum. The following theorem is the monoid axiom for symmetric spectra.

Theorem 5.5.1. Sequential colimits of pushouts of maps of K are stable equivalences. That is, suppose α is an ordinal and A: α → Sp^{Σ} is a functor which preserves colimits and such that each map A_{i} ⟶ A_{i+1} is a pushout of a map of K. Then the map A_{0} ⟶ colim_{i<α} A_{i} is a stable equivalence ( and a level cofibration ).

We then have the following two corollaries, which follow from the general result of [SS97].

Corollary 5.5.2. Suppose R is a monoid in the category of symmetric spectra. Then there is a model structure on the category of R-modules where a map f: X ⟶ Y is a weak equivalence (fibration) if and only if the underlying map of symmetric spectra is a stable equivalence (stable fibration).

Corollary 5.5.3. There is a model structure on the category of monoids of symmetric spectra. If f: X ⟶ Y is a homomorphism of monoids, then f is a weak equivalence (fibration) if and only if the underlying map of symmetric spectra is a stable equivalence (stable fibration).

We will prove Theorem 5.5.1 in a series of lemmas.

Lemma 5.5.4. Every map of K is a level cofibration and a stable equivalence.

Proof. This follows from parts three and five of Theorem 5.3.6 applied to a Scofibration and stable equivalence f and the level cofibration ∗ → X.

This lemma means that every map of K is a trivial cofibration in the injective stable model structure. Thus every colimit of pushouts of maps of K will also be a trivial cofibration in the injective stable mode structure. This is the real reason that the monoid axiom is true for Sp^{Σ}, but since we have not proved that the injective stable structure really is a model structure, we give a direct proof of Theorem 5.5.1.

Lemma 5.5.5. Let

\begin{matrix}A & ⟶^{f} & B \\ ^{h}↓ & & ↓ \\ C & ⟶_{g} & D \end{matrix}

be a pushout square in Sp^{Σ} where f is a level cofibration and a stable equivalence. Then g is a level cofibration and a stable equivalence.

Proof. The map g is certainly a level cofibration. There are factorizations f = pi and h = qj where i and j are stable cofibrations and p and q are level trivial fibrations. Consider the resulting diagram of pushout squares:

A ⟶^{i} A′ ⟶^{p} B
j↓  ^{j′}↓  ↓^{j′′}
C′ ⟶^{i′} Q′ ⟶^{p′} Q
q↓  ^{q′}↓  ↓^{q′′}
C ⟶_{i′′} D′ ⟶_{p′′} D

Then g = p′′i′′. The map p′i′ is a pushout of f = pi, so is a level cofibration. Since q is a level equivalence, properness of the level structures on Sp^{Σ} guarantees that q′′ is a level equivalence. On the other hand, i′ is a pushout of the stable trivial cofibration i, so is a stable equivalence. The map p′ is a pushout of the level equivalence p through the level cofibration j′, so properness of the level structures implies that p′ is a level equivalence. Therefore the composite p′i′ is a stable equivalence. But then the two out of three axiom ensures that g = p′′i′′ is a stable equivalence, as required.

The following lemma completes the proof of Theorem 5.5.1.

Lemma 5.5.6. Suppose α is an ordinal and A: α ⟶ Sp^{Σ} is a functor that preserves colimits such that the map A_{i} ⟶ A_{i+1} is a level cofibration and a stable equivalence for all i. Then the composite A_{0} ⟶ colim A is a level cofibration and a stable equivalence.

Proof. Inductively define a new sequence B_{i} by letting B_{0} = A_{0} and factoring the composite B_{i} ⟶ A_{i} ⟶ A_{i+1} into a stable cofibration B_{i} ⟶ B_{i+1} followed by a level equivalence B_{i+1} ⟶ A_{i+1}. For limit ordinals i, define B_{i} to be the colimit of the B_{j} for j < i. Then each map B_{i} ⟶ B_{i+1} is a stable trivial cofibration, so the composite A_{0} = B_{0} ⟶ colim B is a stable trivial cofibration. On the other hand, the map colim B ⟶ colim A is a level equivalence, since homotopy of simplicial sets commutes with sequential colimits.

5.6. Semistable spectra. In this section we consider a subcategory of symmetric spectra called the semistable spectra. This subcategory sheds light on the difference between stable equivalences and stable homotopy equivalences of symmetric spectra. As in Section 1.4, the stable homotopy category is equivalent to the homotopy category of semistable spectra obtained by inverting the stable homotopy equivalences. Semistable spectra also play a role in [Shi97].

Because stable equivalences are not always stable homotopy equivalences, the stable homotopy groups are not algebraic invariants of stable homotopy types. So the stable homotopy groups of a spectrum may not be “correct.” For any symmetric spectrum X, though, if there is a map from X to an Ω-spectrum which induces an isomorphism in stable homotopy, then the stable homotopy groups of X are the “correct” stable homotopy groups. In other words, these groups are isomorphic to the stable homotopy groups of the stable fibrant replacement of X. Spectra with this property are called semistable.

Let L denote a fibrant replacement functor in Sp^{Σ}, obtained by factoring X → ∗ into a stable trivial cofibration followed by a stable fibration, as in Section 4.3.

Definition 5.6.1. A semistable symmetric spectrum is one for which the stable fibrant replacement map, X ⟶ LX, is a stable homotopy equivalence.

Of course X ⟶ LX is always a stable equivalence, but not all spectra are semistable. For instance, F_{1}S^{1} is not semistable. Certainly any stably fibrant spectrum, i.e., an Ω-spectrum, is semistable. In Section 3.1 we defined the functor R^{∞} and noted that, although it is similar to the standard Ω-spectrum construction for (non-symmetric) spectra, it is not always an Ω-spectrum and X ⟶ R^{∞}X is not always a stable homotopy equivalence, even if X is level fibrant. Let K be a level fibrant replacement functor, the prolongation of a fibrant replacement functor for simplicial sets. The following proposition shows that on semistable spectra R^{∞}K does have these expected properties.

Proposition 5.6.2. The following are equivalent.

1. The symmetric spectrum X is semistable.

2. The map X ⟶ RKX = Hom(F_{1}S^{1}, KX) is a stable homotopy equivalence.

3. X ⟶ R^{∞}KX is a stable homotopy equivalence.

4. R^{∞}KX is an Ω-spectrum.

Before proving this proposition we need the following lemma.

Lemma 5.6.3. Let X ∈ Sp^{Σ}. Then π_{k}(R^{∞}KX)_{n} and π_{k+1}(R^{∞}KX)_{n+1} are isomorphic groups, and i_{R^{∞}KX}: (R^{∞}KX)_{n} → Hom(F_{1}S^{1}, R^{∞}KX)_{n+1} induces a monomorphism π_{k}(R^{∞}KX)_{n} ⟶ π_{k+1}(R^{∞}KX)_{n+1}.

Proof. As noted in the proof of Theorem 3.1.11, π_{k}(R^{∞}KX)_{n} and π_{k+1}(R^{∞}KX)_{n+1} are isomorphic to the (k −n)th classical stable homotopy group of X. However, the map π_{k}i_{R^{∞}KX} need not be an isomorphism. Indeed, π_{k}i_{R^{∞}KX} is the map induced on the colimit by the vertical maps in the diagram

π_{k}X_{n} ⟶ π_{k+1}X_{n+1} ⟶ π_{k+2}X_{n+2} ⟶ ⋯
↓  ↓  ↓
π_{k+1}X_{n+1} ⟶ π_{k+2}X_{n+2} ⟶ π_{k+3}X_{n+3} ⟶ ⋯

where the vertical maps are not the same as the horizontal maps, but differ from them by isomorphisms. The induced map on the colimit is injective in such a situation, though not necessarily surjective.

Proof of Proposition 5.6.2. First we show that 1 implies 2 by using the following diagram.

X ⟶ Hom(F_{1}S^{1}, KX)
↓  ↓
LX ⟶ Hom(F_{1}S^{1}, KLX)

Since Hom(F_{1}S^{1}, K(−)) preserves stable homotopy equivalences, both vertical arrows are stable homotopy equivalences. The bottom map is a level equivalence since LX is an Ω-spectrum. Hence the top map is also a stable homotopy equivalence.

Also, 2 easily implies 3. Since Hom(F_{1}S^{1}, K(−)) preserves stable homotopy equivalences, X → R^{∞}KX is a colimit of stable homotopy equivalences provided X → Hom(F_{1}S^{1}, KX) is a stable homotopy equivalence.

Next we show that 3 and 4 are equivalent. The map π_{∗}X → π_{∗}R^{∞}KX factors as π_{∗}X → π_{∗}(R^{∞}KX)_{0} → π_{∗}R^{∞}KX where the first map here is an isomorphism by definition. Then by Lemma 5.6.3 we see that π_{∗}(R^{∞}KX)_{0} → π_{∗}R^{∞}KX is an isomorphism if and only if π_{∗}(R^{∞}KX)_{n} → π_{∗+1}(R^{∞}KX)_{n+1} is an isomorphism for each n.

To see that 3 implies 1, consider the following diagram.

X ⟶ LX
↓  ↓
R^{∞}KX ⟶ R^{∞}KLX

By 3 and 4 the left arrow is a stable homotopy equivalence to an omega spectrum, R^{∞}KX. Since LX is an Ω-spectrum the right arrow is a level equivalence. Since X → LX is a stable equivalence, the bottom map must also be a stable equivalence. But a stable equivalence between Ω-spectra is a level equivalence, so the bottom map is a level equivalence. Hence the top map is a stable homotopy equivalence.

Two classes of semistable spectra are described in the following proposition. The second class includes the connective and convergent spectra.

Proposition 5.6.4. 1. If the classical stable homotopy groups of X are all finite

then X is semistable.

2. Suppose that X is a level fibrant symmetric spectrum and there exists some α > 1 such that X_{n} ⟶ ΩX_{n+1} induces an isomorphism π_{k}X_{n} ⟶ π_{k+1}X_{n+1} for all k ≤ αn for sufficiently large n. Then X is semistable.

Proof. By Lemma 5.6.3, π_{k}(R^{∞}KX)_{n} → π_{k+1}(R^{∞}KX)_{n+1} is a monomorphism between two groups which are isomorphic. In the first case these groups are finite, so this map must be an isomorphism. Hence R^{∞}KX is an Ω-spectrum, so X is semistable.

For the second part we also show that R^{∞}KX is an Ω-spectrum. Since for fixed k the maps π_{k+i}X_{n+i} → π_{k+1+i}X_{n+1+i} are isomorphisms for large i, π_{k}(R^{∞}KX)_{n} → π_{k+1}(R^{∞}KX)_{n+1} is an isomorphism for each k and n.

The next proposition shows that stable equivalences between semistable spectra are particularly easy to understand.

Proposition 5.6.5. Let f: X ⟶ Y be a map between two semistable symmetric spectra. Then f is a stable equivalence if and only if it is a stable homotopy equivalence.

Proof. Every stable homotopy equivalence is a stable equivalence, by Theorem 3.1.11. Conversely, if f is a stable equivalence, so is Lf. Since stable equivalences between stably fibrant objects are level equivalences by Lemma 4.3.3, Lf is in particular a stable homotopy equivalence. Since X and Y are semistable, both maps X → LX and Y → LY are stable homotopy equivalences. Hence f is a stable homotopy equivalence.

6. Topological spectra

In this final section, we describe symmetric spectra based on the category of pointed topological spaces 𝒯_{∗}.

6.1. Topological spectra. In this section we discuss the category of topological symmetric spectra. We point out the two main differences between the topological situation and the simplicial situation: function objects work less well in the topological situation, and monomorphisms must be replaced by level cofibrations.

The basic definitions for topological spectra and topological symmetric spectra are straightforward. Let 𝒯_{∗} denote the category of pointed topological spaces and pointed continuous maps. Then 𝒯_{∗} is a symmetric monoidal category under the smash product: X ∧ Y = (X × Y)/(X ∨ Y). Let S^{1} ∈ 𝒯_{∗} be the unit circle in R^{2} with basepoint (1, 0).

Definition 6.1.1. A topological spectrum is

1. a sequence X_{0}, X_{1}, …, X_{n}, … of pointed topological spaces; and

2. a pointed continuous map σ: S^{1} ∧ X_{n} → X_{n+1} for each n ≥ 0.

The maps σ are the structure maps of the spectrum. A map of topological spectra f: X → Y is a sequence of pointed continuous maps f_{n}: X_{n} → Y_{n} such that the diagram

S^{1} ∧ X_{n} ⟶^{σ} X_{n+1}
S^{1}∧f_{n}↓  ^{f_{n+1}}↓
S^{1} ∧ Y_{n} ⟶^{σ} Y_{n+1}

is commutative for each n ≥ 0. Let Sp_{𝒯_{∗}}^{ℕ} denote the category of topological spectra.

Let S^{p} = (S^{1})^{∧p} with the (continuous) left permutation action of Σ_{p}.

Definition 6.1.2. A topological symmetric spectrum is

1. a sequence X_{0}, X_{1}, …, X_{n}, … of pointed topological spaces;

2. a pointed continuous map σ: S^{1} ∧ X_{n} → X_{n+1} for each n ≥ 0; and

3. a basepoint preserving continuous left action of Σ_{n} on X_{n} such that the composition

σ^{p} = σ ∘ (S^{1} ∧ σ) ∘ ⋯ ∘ (S^{p−1} ∧ σ): S^{p} ∧ X_{n} → X_{n+p},

of the maps S^{i} ∧ S^{1} ∧ X_{n+p−i−1} ⟶^{S^{i}∧σ} S^{i} ∧ X_{n+p−i} is Σ_{p} × Σ_{n}-equivariant

for p ≥ 1 and n ≥ 0.

A map of topological symmetric spectra f: X → Y is a sequence of pointed continuous maps f_{n}: X_{n} → Y_{n} such that f_{n} is Σ_{n}-equivariant and the diagram

S^{1} ∧ X_{n} ⟶^{σ} X_{n+1}
S^{1}∧f_{n}↓  ^{f_{n+1}}↓
S^{1} ∧ Y_{n} ⟶^{σ} Y_{n+1}

is commutative for each n ≥ 0. Let Sp_{𝒯_{∗}}^{Σ} denote the category of symmetric spectra.

Example 6.1.3. The real bordism spectrum MO is the sequence of spaces MO(n) = S^{n} ∧_{O_{n}} (EO_{n})_{+}. Here we require that EG be a functorial construction of a free contractible G-space with a left and right action of G, for example Milnor’s construction, [Mil56]. Then the symmetric group acts via the inclusion into O_{n} and the right action of O_{n} on EO_{n}. One can check that the usual structure maps S^{1} ∧MO(n) ⟶ MO(n+1) make MO into a topological symmetric spectrum. Similarly, the other bordism spectra such as MU are naturally topological symmetric spectra.

With these definitions, most of the basic results of Sections 1 and 2 go through without difficulty, replacing simplicial sets by topological spaces. That is, the category of topological symmetric spectra is the category of S-modules, where S is the topological symmetric sequence (S^{0}, S^{1}, S^{2}, …), and S^{n} = (S^{1})^{∧n} with the permutation action of Σ_{n}. The category of topological symmetric spectra is a symmetric monoidal category.

However, the category of topological symmetric spectra is not a closed symmetric monoidal category. Indeed, the category 𝒯_{∗} is not a closed symmetric monoidal category. We can form the function space C_{𝒯_{∗}} (X, Y) of pointed continuous maps, given the compact-open topology. There is an injective adjunction map 𝒯_{∗}(Z ∧ X, Y) ⟶ 𝒯_{∗}(Z, C_{𝒯_{∗}} (X, Y)), but it need not be surjective in general. It is surjective when X is locally compact Hausdorff. We can get around this problem by modifying 𝒯_{∗}; we discuss this approach in Section 6.3. For the moment, we stay with 𝒯_{∗}. Then, if we have a symmetric spectrum X and a topological space K, there is no difficulty in forming X ∧ K, or the power spectrum X^{K} (though one does have to check that the functor X ↦ X^{K} is a 𝒯_{∗}-functor even when K is not locally compact Hausdorff). Then there is an injective adjunction map Sp_{𝒯_{∗}}^{Σ} (X ∧ K, Y) ⟶ Sp_{𝒯_{∗}}^{Σ} (X, Y^{K}) which is an isomorphism when K is locally compact Hausdorff, but not in general. Similarly, for symmetric spectra X and Y, we can form the mapping space C_{Sp_{𝒯∗}^{Σ}} (X, Y) whose points are maps of symmetric spectra from X to Y, topologized as a subspace of ∏ C_{𝒯_{∗}} (X_{n}, Y_{n}). We have an injective adjunction map Sp_{𝒯_{∗}}^{Σ} (A ∧ X, Y) ⟶ 𝒯_{∗}(A, C_{Sp_{𝒯∗}^{Σ}} (X, Y)) which is an isomorphism when X is level locally compact Hausdorff; i.e., when each X_{n} is locally compact Hausdorff. Note that there is no difficulty in defining Ω-spectra, since S^{1} is locally compact Hausdorff.

Hence, to translate a statement about function spaces from simplicial sets or spectra to topological spaces or spectra, we need to add the assumption that the domain be (level) locally compact Hausdorff whenever adjointness is used.

Although neither the category of topological symmetric sequences nor the category of topological symmetric spectra is a closed symmetric monoidal category, we can still form function objects Hom_{Σ}(X, Y) and Hom_{S}(X, Y). Indeed, given topological symmetric sequences X and Y, the kth space of the symmetric sequence Hom_{Σ}(X, Y) is C_{𝒯_{∗}^{Σ}} (X ⊗ G_{k}S^{0}, Y). A point is a map of symmetric sequences that raises degree by k; the leftover Σ_{k} action defines a symmetric sequence. Similarly, given topological symmetric spectra X and Y, the kth space of the symmetric spectrum Hom_{S}(X, Y) is C_{Sp_{𝒯∗}^{Σ}} (X ∧ F_{k}S^{0}, Y). A point is a map of symmetric spectra that raises degree by k, and the topology is the subspace topology inherited from C_{𝒯_{∗}^{Σ}} (X ⊗ G_{k}S^{0}, Y). The structure map S^{1} ∧ C_{Sp_{𝒯∗}^{Σ}} (X ∧ F_{k}S^{0}, Y) ⟶ C_{Sp_{𝒯∗}^{Σ}} (X ∧ F_{k+1}S^{0}, Y) takes (t, f) to σ_{t}(f), where t ∈ S^{1} and f is a map of degree k, and σ_{t}(f)(x) = f(σ_{X}(t, x)) = σ_{Y} (t, f(x)). For both of these function objects, there is an injective adjunction map which is an isomorphism when X is level locally compact Hausdorff, but not in general.

6.2. Stable model structure. In this section we discuss the stable homotopy theory of topological symmetric spectra. Rather than proceeding analogously to Section 3, we use the geometric realization and singular functor to lift the stable model structure on simplicial symmetric spectra to topological symmetric spectra.

Recall the geometric realization |−|: 𝒮_{∗} ⟶ 𝒯_{∗}, which we have already used, and its right adjoint Sing: 𝒯_{∗} ⟶ 𝒮_{∗}. The best source for the geometric realization is probably [GZ67, III]. The geometric realization has a number of properties shared by general left adjoints; in particular, if X is locally finite, i.e., has only finitely many simplices in each dimension, then the natural map |X × Y | ⟶ |X| × |Y | is a homeomorphism. To see this, begin with the fact [GZ67, p. 51] that the natural map |Δ[p] × Δ[q]| ⟶ |Δ[p]| × |Δ[q]| is a homeomorphism. Since |Δ[q]| is locally compact Hausdorff, and every simplicial set is a colimit of simplices, it follows that |X ×Δ[p]| ⟶ |X|×|Δ[p]| is a homeomorphism. Now, if X is locally finite, then |X| is locally compact Hausdorff, so we get the desired result by again taking colimits. It follows that the natural map |X ∧ Y | ⟶ |X| ∧ |Y | is a homeomorphism when X is locally compact Hausdorff.

Another important property of the geometric realization and singular complex is that they both preserve and detect weak equivalences. That is, a map f of simplicial sets is a weak equivalence (by definition) if and only if |f| is a weak equivalence, and a map g of topological spaces is a weak equivalence if and only if Sing g is. The unit A ⟶ Sing |A| and the counit | Sing X| ⟶ X of the adjunction are both natural weak equivalences.

The following proposition is then immediate.

Proposition 6.2.1. The geometric realization induces a functor |−|: Sp^{Σ} ⟶ Sp_{𝒯_{∗}}^{Σ} such that |X|_{n} = |X_{n}|. Furthermore, there is a natural map |X ∧ Y | ⟶ |X| ∧ |Y | which is an isomorphism when X is level locally finite. The singular functor induces a functor Sing: Sp_{𝒯_{∗}}^{Σ} ⟶ Sp^{Σ}, right adjoint to |−|, such that (Sing X)_{n} = Sing X_{n}. Both the geometric realization and the singular complex detect and preserve level equivalences, and the unit and counit maps of the adjunction are level equivalences.

We then make the following definition.

Definition 6.2.2. Suppose f: X ⟶ Y is a map of topological symmetric spectra. 1. Define f to be a stable equivalence if Sing f is a stable equivalence in Sp^{Σ}.

2. Define f to be a stable fibration if Sing f is a stable fibration in Sp^{Σ}.

3. Define f to be a stable cofibration if f has the left lifting property with respect to all stable trivial fibrations (i.e., maps which are both stable equivalences and stable fibrations).

Note that every level equivalence is a stable equivalence. We will prove that these structures form a model structure on Sp_{𝒯_{∗}}^{Σ} , called the stable model structure.

The first thing we need is a version of the Factorization Lemma 3.2.11.

Definition 6.2.3. Suppose X is a cofibrant pointed topological space. Then X is a retract of at least one cell complex Y. Such a cell complex Y can be written as the colimit of an α-indexed colimit-preserving diagram D: α ⟶ 𝒯_{∗} for some ordinal α, where each map D_{β} ⟶ D_{β+1} is obtained by attaching a cell to D_{β}. We define the number of cells of Y to be the least ordinal α so that Y can be written as the colimit of such a diagram. We then define the cellular cardinality of X to be the cardinality of the least α such that X is a retract of a cell complex with α cells. For convenience, we define the cellular cardinality of a one-point space to be 1.

This definition respects the standard intuitions: the cellular cardinality of S^{n} is 1, and the cellular cardinality of CP^{2} is 2.

Lemma 6.2.4. Let X be a cofibrant pointed topological space of cellular cardinality γ. If γ is infinite, let α be a γ-filtered ordinal, see 3.2.12; otherwise let α be an ω-filtered ordinal. Let D: α ⟶ 𝒯_{∗} be an α-indexed diagram of cofibrations, so that each map D_{β} ⟶ D_{β+1} is a cofibration. Then the natural map

colim_{α} 𝒯_{∗}(X, D) ⟶ 𝒯_{∗}(X, colim_{α} D)

is an isomorphism.

We do not know if an arbitrary pointed topological space has this property for sufficiently large ordinals α.

Proof. We can assume that X is a cell complex with γ cells. It is somewhat more difficult to verify that we can assume that D is a sequence of cellular inclusions; every cofibration is a retract of a cellular inclusion, but one must choose these cellular inclusions carefully; see [Hir97]. By transfinite induction, using the fact that α is γ-filtered, we reduce to the case when X has only one cell. Here the result is well-known and is the basis for the model structure on 𝒯_{∗}.

We can then define the cellular cardinality of a level cofibrant topological symmetric spectrum X to be the sum of the cellular cardinalities of the X_{n}. Then the cellular cardinality is always infinite, since we took the cellular cardinality of the one-point space to be 1. The following proposition is the analogue of Proposition 3.2.13, and its proof is similar given Lemma 6.2.4.

Proposition 6.2.5. Let X be a level cofibrant topological symmetric spectrum of cellular cardinality γ. Let α be a γ-filtered ordinal and let D: α ⟶ Sp_{𝒯_{∗}}^{Σ} be an α-indexed diagram of level cofibrations. Then the natural map

colim_{α} Sp_{𝒯_{∗}}^{Σ} (X, D) ⟶ Sp_{𝒯_{∗}}^{Σ} (X, colim_{α} D)

is an isomorphism.

This proposition immediately leads to a factorization lemma for topological symmetric spectra, analogous to the Factorization Lemma 3.2.11.

Lemma 6.2.6. Let I be a set of level cofibrations between level cofibrant objects of Sp_{𝒯_{∗}}^{Σ} . There is a functorial factorization of every map of topological symmetric spectra as an I-cofibration followed by an I-injective.

The proof of this lemma uses the transfinite small object argument, and we leave it to the reader.

Theorem 6.2.7. The stable model structure on Sp_{𝒯_{∗}}^{Σ} is a model structure. We have the following properties:

1. A map f ∈ Sp^{Σ} is a stable equivalence if and only if |f| is a stable equivalence in Sp_{𝒯_{∗}}^{Σ} ;

2. A map g ∈ Sp_{𝒯_{∗}}^{Σ} is a stable equivalence if and only if Sing g is a stable equivalence in Sp^{Σ}; and

3. The geometric realization and singular functor define a Quillen equivalence between the stable model structure on simplicial symmetric spectra and the stable model structure on topological symmetric spectra.

Proof. The category Sp_{𝒯_{∗}}^{Σ} is cocomplete, and the two out of three and retract axioms are immediate consequences of the definitions. By definition, (i, p) has the lifting property when i is a stable cofibration and p is a stable trivial fibration. One can easily verify using adjointness that the stable trivial fibrations form the class |F I_{∂}|-inj. It follows that the stable cofibrations form the class |F I_{∂}|-cof. The Factorization Lemma 6.2.6, applied to |F I_{∂}|, then gives a factorization of an arbitrary map in Sp_{𝒯_{∗}}^{Σ} into a stable cofibration followed by a stable trivial fibration. It also follows from this that every stable cofibration is a level cofibration.

Similarly, the stable fibrations form the class |J|-inj, where J is the set of generating stable trivial cofibrations used in Section 3.4. In Proposition 6.2.10, we show that every map of |J|-cof is a stable cofibration and a stable equivalence. Hence the Factorization Lemma 6.2.6 gives us the required (functorial) factorization of an arbitrary map into a stable trivial cofibration followed by a stable fibration.

Only one lifting property remains. Suppose i is a stable trivial cofibration. We can factor i as i = qi′, where q is a stable fibration and i′ ∈ |J|-cof. By the preceding paragraph, i′ is a stable equivalence. By the two out of three axiom, q is a stable equivalence. Thus i has the left lifting property with respect to q, and so the Retract Argument 3.2.4 implies that i is a retract of i′. Thus i ∈ |J|-cof, and so i has the left lifting property with respect to every stable fibration p. This completes the proof that Sp_{𝒯_{∗}}^{Σ} is a model category.

Now, by definition, Sing: Sp_{𝒯_{∗}}^{Σ} ⟶ Sp^{Σ} is a right Quillen functor, and f ∈ Sp_{𝒯_{∗}}^{Σ} is a stable equivalence if and only if Sing f is a stable equivalence in Sp^{Σ}. Now let K be a fibrant replacement functor on Sp_{𝒯_{∗}}^{Σ} , obtained, for example, from the Factorization Lemma 6.2.6 applied to |J|. Then in order to show that the pair (| − |, Sing) is a Quillen equivalence, it suffices by Lemma 4.1.7 to show that the map X ⟶ Sing K|X| is a stable equivalence for all simplicial symmetric spectra X. However, the map X ⟶ Sing |X| is a level equivalence. Since the map |X| ⟶ K|X| is a stable equivalence, so is the map Sing |X| ⟶ Sing K|X|. By the two out of three axiom, the composite X ⟶ Sing K|X| is a stable equivalence, as required.

Finally, we show that | − | detects and preserves stable equivalences. Given a map f: X ⟶ Y, consider the commutative square below.

X  ⟶^{f} Y
↓  ↓
Sing |X| ⟶_{Sing |f|} Sing |Y |

The vertical maps are level equivalences. Now Sing |f| is a stable equivalence if and only if |f| is a stable equivalence. Hence f is a stable equivalence if and only if |f| is a stable equivalence.

We still owe the reader a proof that the maps of |J|-cof are stable cofibrations and stable equivalences. We begin with the following lemma.

Lemma 6.2.8. Suppose f: A ⟶ B is an element of J, and suppose the following square is a pushout square in Sp_{𝒯_{∗}}^{Σ} .

|A| ⟶^{|f|} |B|
r↓  ↓^{s}
C ⟶_{g} D

Then g is a stable equivalence and a stable cofibration.

Proof. It is clear that |f|, and hence g, is a stable cofibration. Note that |f| is itself a stable equivalence, since Sing |f| is level equivalent to the stable equivalence f, so is a stable equivalence.

We first suppose that r is a level cofibration. Consider the pushout square in Sp^{Σ} below.

Sing |A| ⟶^{Sing |f|} Sing |B|
Sing r↓  ↓_{s′}
Sing C ⟶_{g′} E

Since Sing preserves level cofibrations (since it preserves injections), each map in this square is a level cofibration. Furthermore, Sing |f| is a stable equivalence. By the properness result of Theorem 5.4.2, we find that g′ is a stable equivalence. There is a map h: E ⟶ Sing D such that hg′ = Sing g, so it will suffice to prove that h is a level equivalence.

To see this, we apply the geometric realization again to obtain a pushout square

| Sing |A|| ⟶^{| Sing |f||} | Sing |B||
| Sing r|↓  ↓^{|s′|}
| Sing C| ⟶_{|g′|}  |E|

Again, each of the maps in this diagram is a level cofibration. There is an obvious map from this pushout square to our original pushout square induced by the natural level equivalence | Sing X| ⟶ X. This map is a level equivalence everywhere except possibly the bottom right. It follows from a general model category argument that the induced map |E| ⟶ D is a level equivalence. To see this, we can work one level at a time. Let 𝒟 denote the category with three objects i, j, k and two non-identity maps i ⟶ j and i ⟶ k. Then the pushout defines a functor 𝒯_{∗}^{𝒟} ⟶ 𝒯_{∗} left adjoint to the diagonal functor. There is a model structure on 𝒯_{∗}^{𝒟} such that a map f from X_{j} ⟵ X_{i} ⟶ X_{k} to Y_{j} ⟵ Y_{i} ⟶ Y_{k} is a weak equivalence (fibration) if and only if each map f_{i}, f_{j}, and f_{k} is a weak equivalence (fibration). Cofibrations are defined by lifting, and one can check that f is a cofibration if and only if f_{i} and the induced maps X_{j} ⨿_{X_{i}} Y_{i} ⟶ Y_{j} and X_{k} ⨿_{X_{i}} Y_{i} ⟶ Y_{k} are cofibrations. The diagonal is easily seen to be a right Quillen functor, so the pushout is a left Quillen functor, and hence preserves weak equivalences between cofibrant objects. Hence the map |E|_{n} ⟶ D_{n} is a weak equivalence as desired.

Since the level equivalence |E| ⟶ D is the composite |E| ⟶^{Sing h} Sing |D| ⟶ D, it follows that Sing h is a level equivalence. Thus h is a level equivalence as required, and the proof is complete in the case where r is a level cofibration.

In the general case, we can factor r = r′′ ∘ r′, where r′ is a stable cofibration and r′′ is a stable trivial fibration, and hence a level equivalence. Then we get a commutative diagram

|A| ⟶^{|f|} |B|
r′↓  ↓_{s′}
C′ ⟶^{g′} D′
r′′↓  ↓_{s′′}
C ⟶_{g} D

where each square is a pushout square. Since r′ is a stable cofibration, and in particular a level cofibration, g′ is a stable equivalence (and a level cofibration). The map r′′ is a level equivalence. Since the model category 𝒯_{∗} is proper, it follows that s′′ is a level equivalence. The two out of three property then guarantees that g is a stable equivalence as required.

Now we need to understand stable equivalences and colimits.

Lemma 6.2.9. Suppose α is an ordinal, and X: α ⟶ Sp_{𝒯_{∗}}^{Σ} is a functor such that each map X_{i} ⟶ X_{i+1} is a stable cofibration and a stable equivalence. Then the induced map X_{0} ⟶ colim_{i} X_{i} is a stable cofibration and a stable equivalence.

Proof. It is easy to see that X_{0} ⟶ colim_{i} X_{i} is a stable cofibration. Consider the functor Sing X: α ⟶ Sp^{Σ}. Then each map Sing X_{i} ⟶ Sing X_{i+1} is a level cofibration and a stable equivalence. Let Y = colim_{i} Sing X_{i}. Then the map Sing X_{0} ⟶ Y is a stable equivalence, by Lemma 5.5.6. Thus it suffices to show that the map Y ⟶^{f} Sing colim_{i} X_{i} is a level equivalence.

Note that |Y | = colim_{i} | Sing X_{i}|. There is a natural map |Y | ⟶ colim_{i} X_{i} induced by the level equivalences | Sing X_{i}| ⟶ X_{i}. Since homotopy groups of topological spaces commute with colimits of cofibrations, it follows that |Y | ⟶ colim_{i} X_{i} is

a level equivalence. Since this map factors as |Y | ⟶^{|f|} | Sing colim_{i} X_{i}| ⟶ colim_{i} X_{i}, we see that |f| is a level equivalence. Hence f is a level equivalence as required.

Proposition 6.2.10. Every map in |J|-cof is a stable cofibration and a stable equivalence.

Proof. Lemma 6.2.8 and Lemma 6.2.9 show that every colimit of pushouts of maps of |J| is a stable cofibration and a stable equivalence. Now consider an arbitrary map f in |J|-cof. The proof of the Factorization Lemma 6.2.6 shows that we can factor f = qi, where i is a colimit of pushouts of maps of |J| and q ∈ |J|-inj. Since f has the left lifting property with respect to q, the Retract Argument 3.2.4 shows that f is a retract of i. Since i is a stable cofibration and a stable equivalence, so is f.

We note that it is also possible to develop the homotopy theory of topological symmetric spectra in a parallel way to our development for simplicial symmetric spectra. There are level model structures, and there is a similar characterization of stable fibrations. One must be careful with function spaces, of course, and in addition, one must take into account that not every topological symmetric spectrum is level cofibrant. Because of the lack of adjointness, it is a little more difficult to verify the analogues of Theorem 5.3.6 and Theorem 5.5.1, but they do hold.

6.3. Compactly generated spectra. In this section, we describe a closed symmetric monoidal category of topological symmetric spectra. The idea is to replace topological spaces by a large full subcategory which is closed symmetric monoidal. There are many such subcategories, which all go by the name of compactly generated spaces; see [Wyl73] for details. In the one we choose, colimits are simply colimits of the underlying topological spaces, which is not true in the compactly generated weak Hausdorff spaces of [EKMM97].

Definition 6.3.1. Suppose X is a (pointed) topological space. A subset A of X is called compactly open if, for every continuous map f: K ⟶ X where K is compact Hausdorff, f^{−1}(A) is open. We denote by kX the set X with the topology consisting of the compactly open sets, and we say that X is compactly generated if kX = X. We denote the full subcategory of 𝒯_{∗} consisting of pointed compactly generated spaces by 𝒯′_{∗}.

We now discuss some of the basic facts about compactly generated spaces. Every locally compact Hausdorff space is compactly generated. The functor k: 𝒯_{∗} ⟶ 𝒯′_{∗} is a right adjoint to the inclusion functor. It follows that 𝒯′_{∗} is bicomplete. Colimits are taken in 𝒯_{∗}, so in particular a colimit of compactly generated spaces is again compactly generated. One must apply k to construct limits. Thus, the product in 𝒯′_{∗} is the topological space k(X × Y). If X is locally compact Hausdorff, then k(X × Y) = X × Y. With the resulting compactly generated smash product k(X ∧ Y) = k(X × Y)/(X ∨ Y), the category 𝒯′_{∗} is a closed symmetric monoidal category. The adjoint of X × − is C_{∗}(X, −), where the topology on C_{∗}(X, Y) has a sub-basis consisting of the sets S(f, U), where f: K ⟶ X is a continuous map, K is compact Hausdorff, and U is open in Y. A map g: X ⟶ Y is in S(f, U) if and only if (g ∘ f)(K) ⊆ U. If X is locally compact Hausdorff, this is the same as the usual compact-open topology. Again, see [Wyl73] for details on compactly generated spaces.

There is then no difficulty in constructing a category of compactly generated symmetric spectra, with an obvious inclusion functor to topological symmetric spectra. This category is closed symmetric monoidal, unlike the category of topological symmetric spectra.

The stable homotopy theory of compactly generated symmetric spectra is the same as that of topological symmetric spectra. Indeed, the geometric realization of a simplicial set is always compactly generated, since it is the colimit of compact Hausdorff spaces. Furthermore, the geometric realization 𝒮_{∗} ⟶ 𝒯′_{∗} is a symmetric monoidal functor, by the same argument involving colimits that we used to show that |X ∧ Y | ⟶ |X| ∧ |Y | is a homeomorphism for locally compact Hausdorff X.

We can define stable equivalences, stable fibrations, and stable cofibrations for compactly generated symmetric spectra just as we did for topological symmetric spectra. We then have the following theorem, whose proof is the same as the proof of Theorem 6.2.7.

Theorem 6.3.2. The category of compactly generated symmetric spectra, together with the stable equivalences, stable fibrations, and stable cofibrations, is a model category. The geometric realization defines a monoidal Quillen equivalence from simplicial symmetric spectra to compactly generated symmetric spectra, and the inclusion functor defines a Quillen equivalence from compactly generated symmetric spectra to topological symmetric spectra.

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Department of Mathematics, Wesleyan University, Middletown, CT

   E-mail address: hovey@member.ams.org

Department of Mathematics, University of Chicago, Chicago, IL

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

Department of Mathematics, Purdue University, West Lafayette, IN

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