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Symmetric spectra

The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield and Friedlander's category of spectra. We show that the monoidal axiom holds, so that we get model categories of ring spectra and modules over a given ring spectrum.


Introduction
Stable homotopy theory studies spectra as the linear approximation to spaces. Here, "stable" refers to the consideration of spaces after inverting the suspension functor. This approach is a general one: one can often create a simpler category by inverting an operation such as suspension. In this paper we study a particularly simple model for inverting such operations which preserves product structures. The combinatorial nature of this model means that it is easily transported, and hence may be useful in extending the methods of stable homotopy theory to other settings.
The idea of a spectrum is a relatively simple one: Freudenthal's suspension theorem implies that the sequence of homotopy classes of maps 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 Lima [Lim59] and later generalized by Whitehead [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. Notice that this definition of a spectrum can be applied to any situation where one has an operation on a category that one would like to invert; however, this simplest construction does not preserve the smash product structure coming from spaces.
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 defined above 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 might 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, just as in algebra where the usual internal tensor product of modules is defined only over a commutative ring. 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, called S-modules. Some generalizations of symmetric spectra appear in [MMSS1]. These many new symmetric monoidal categories of spectra, including S-modules and symmetric spectra, are shown to be equivalent in an appropriate sense in [MMSS1] and [MMSS2]. 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 a model category of modules over a symmetric ring spectrum.
As mentioned above, since the construction of symmetric spectra is combinatorial in nature it may be applied in many different situations. Given any well-behaved symmetric monoidal model category, such as chain complexes, simplicial sets, or topological spaces, and an endofunctor on it that respects the monoidal structure, one can define symmetric spectra. This more general approach is explored in [Hov98b]. In particular, symmetric spectra may be the logical way to construct a model structure for Voevodsky's stable homotopy of schemes [Voe97].
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].
At present the theory of S-modules of [EKMM97] is considerably more developed than the theory of symmetric 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, as would be expected. 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 selfcontained, depending only on some standard results about simplicial sets and topological spaces. As discussed above, symmetric spectra can be built in many different circumstances, whereas S-modules appear to be tied to the category of topological spaces. There are also technical differences reflecting the result of [Lew91] that there are limitations on any symmetric monoidal category of spectra. For example, the sphere spectrum S is cofibrant in the category of symmetric spectra, but is not in the category of S-modules. On the other hand, every S-module is fibrant, a considerable technical advantage. Also, 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. For a first step in this direction see [SS].
Organization. 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 S 1 n+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.
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.

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 ∆ 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 ∆ 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 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 ∆[0] = ∆(−, [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 ∆[0] → X. The disjoint union X + = X ∐ ∆[0] adds a disjoint basepoint to the simplicial set X. For example, the 0-sphere is S 0 = ∆[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∧∆[−] + , 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 ∧ ∆[−] + , Y ). 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 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 p = {1, 2, . . . , p}, with 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 p + q and Σ q acting on the last q elements of 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 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 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 6.2.3. 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 U X is the sequence of pointed simplicial sets (U X) n = X n with the same structure maps σ : S 1 ∧ (U X) n → (U X) 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 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 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 : ) 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 and a natural map 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 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 is the adjoint of the composition 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 and a right unit of the composition pairing. 3. There are natural isomorphisms 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 induced by the commutative square 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.
. 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 f g 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 U X 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.

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 n = {1, 2, . . . , n} for n ≥ 0 (0 = ∅) as its objects and the automorphisms of the sets 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 Definition 2.1.3. The tensor product X⊗Y of the symmetric sequences X, Y ∈ S Σ * is the symmetric sequence The tensor product f ⊗ g : The tensor product of symmetric sequences has the universal property for "bilinear maps": 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 For an isomorphism f : 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(0, −) + = (S 0 , * , * , . . . ). The unit isomorphism is obvious. The associativity isomorphism is induced by the associativity isomorphism in S * and the natural isomorphism 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 . For each n ≥ 0, the free symmetric sequence is Σ[n] = Σ(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.
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 : By Proposition 2.1.8, the functor sending p to q Map Σq (Y q , Z p+q ) is the functor sending 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 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 / / 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 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 | | y y y y y y y y y 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 commutes (i.e., m is associative) and such that the compositions e ⊗ R 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-M od. If C is complete, the module category R-M od is complete and the forgetful functor R-M od → 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-M od is cocomplete and the forgetful functor R-M od → 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. 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 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(−,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: Proposition 2.2.7. Let f be a map of pointed simplicial sets.
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. 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 where m * is pullback along the multiplication m : R ⊗ M → M and m * is the composition 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 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 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 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, 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 We must also describe the structure maps of X from this point of view. Re- 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 Using this natural isomorphism, we find that the structure maps of Hom S (X, Y ) are the adjoints of the maps 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 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 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].

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 U f of spectra is a stable equivalence, i.e., if U f 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 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.
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; 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 K n+k . The adjoints of the structure maps of Hom S (F n K, E) are the weak equivalences of Kan complexes E K n+k → E S 1 ∧K n+k+1 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 : 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 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.
, 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 Then f = ri, i is a monomorphism, rs = id Y , and there is a simplicial homotopy from sr to the identity map of M f .
Proof of Lemma 3.1.6. For part one of the lemma, let f : X → Y be a level equivalence and let M f be the mapping cylinder of f . As above f = ri, i : X → M f is a monomorphism, and r : M f → 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 : ∂∆[1] → ∆[1] is a monomorphism. By Corollary 1.3.6, the map is a monomorphism and a level equivalence. The extension property of E with respect to i j implies that if g, h : M f → 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 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 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 : 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 U λ 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, 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 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 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 transfor- 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 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 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: .
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.
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 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 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 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 : 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 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 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 : 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 of maps for which i is not in the image. Geometrically, Λ i [n] is obtained from the boundary of ∆[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] → ∆[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 ∂∆[n] → ∆[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. 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.
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. 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 < γ and α is 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 is an isomorphism.
Proof. Every symmetric spectrum has a presentation as a coequalizer 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 . 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 βx 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 By the definition of Df , there is a commutative square 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.
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 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.
fibration of simplicial sets.
The level cofibrations are the monomorphisms of symmetric spectra. Next, we characterize the level fibrations and trivial fibrations. 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. is the map to the fiber product induced by the commutative square 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 ).
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 coming from the commutative square A pair (f, g) has the lifting property if and only if C (f, g) is surjective. 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 v v m m m m m m m m m m m m m 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. Map (f, h)) has the lifting property. 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.
1. The class of stable cofibrations is the class F I ∂ -cof.

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 where k n is the inclusion on the second factor, i 0 and i 1 come from the two inclusions ∆[0] → ∆[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 ∂∆[1] + → ∆[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 where p and q are fibrations is a homotopy pullback square if the following equivalent conditions hold: 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 is a homotopy pullback square for each n ≥ 0, where the horizontal maps are the adjoints of the structure maps. 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 is naturally isomorphic to the map . 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 Let E be an injective Ω-spectrum. Since the map X ∐ F (F ∧ ∆[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 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 , is a homotopy pullback square for each n ≥ 0. The proof is completed by showing that p S 1 n+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 S 1 n : X S 1 n → Y S 1 n is a trivial Kan fibration for every n ≥ 0.
The next corollary finishes the proof of Lemma 3.4.8. 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 ∆[1] → E ×E, where j : ∂∆[1] + → ∆[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 ∆[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.

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.

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. 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. 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.
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 RiLX −−−→ 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. 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.
is a homotopy pullback square. 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. 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.  We prove this theorem by using Lemma 4.1.7. In particular, we need to understand stable equivalences between stably fibrant objects. 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 . 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 → U LV 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 F 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 F n : S * − → Sp N left adjoint to the evaluation functor Ev n .
Note that F 0 X = Σ ∞ X. Also, since U • Ev n = Ev n , the left adjoints satisfy 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 F 0 Y → U F 0 Y is a level equivalence and U F 0 Y → U LF 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.
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 : U LV X → U ΩLΣV X is a stable equivalence.
So g : X → U LV X is a stable equivalence if and only if f g : X → U ΩLΣV X is a stable equivalence. Since Ω and U commute, gf is a stable equivalence if and only if ΣX → U LΣV 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 U LΣV X → U LV ΣX. This completes the proof. Proof. Consider the following commuting square. 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 − → U LV X is a stable equivalence.
Proof. Suppose X is strictly bounded below at n. Then we have a map F 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 F n X n → U LF n X n is a stable equivalence. But there is an evident map F n X n ∧ S n ∼ = F 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 ∂∆[n] + and ∆[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 ∧ ∆[1] + are finite, homotopy commutes with filtered colimits of Kan complexes, and in particular with transfinite compositions of maps of Kan complexes.
In fact, homotopy commutes with transfinite compositions of arbitrary monomorphisms of simplicial sets. To see this, apply the geometric realization to get a sequence of cofibrations of CW complexes. Since homotopy commutes with such transfinite compositions, the result follows.
Lemma 4.3.11. Suppose X is a cofibrant object of Sp N . Then the map X − → U LV X is a stable equivalence.
Proof. Let X i denote the truncation of X at i. That is, we have X i n = X n for n ≤ i and X i n = 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 − → U LV X i is a stable equivalence.
We claim that the induced map X − → colim i U LV 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 j i Since homotopy commutes with transfinite compositions 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 U LV X i ∼ = colim i π n U LV X i . It follows that X − → colim i U LV X i is a stable homotopy equivalence, as required.
We now examine the relationship between U LV X and colim i U LV 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 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 ) S 1 n+1 = colim i ((LV X i ) S 1 n+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 U LV X − → colim i U LV X i is also a level equivalence. We have seen above that the map X − → colim i U LV X i is a stable equivalence, so X − → U LV 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.

Description of V . This short section is devoted to the construction of the left adjoint
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 G n , where G n is X at level n and the basepoint everywhere else.
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 U S = 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 (G1S 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 :

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 Λcofibration is a projective cofibration which is a level equivalence and an F I Λinjective 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 * . 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 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 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 : Then f s is the identity map on B. To study the composite sf , let j be the monomor- is commutative since f sf = 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 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 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.
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 nsimplicial 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 transfinite compositions of monomorphisms. The same methods imply that relative homotopy commutes with transfinite compositions 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 : Let D = colim n F n C. The spectrum D is countable. Since relative homotopy commutes with transfinite compositions 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 S to be the symmetric spectrum such that S n = S n for n > 0 and 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 ∧ S).
There is a map of symmetric spectra i : 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 ∐ LnX 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 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 is the natural map on the pushout defined by the commutative square If C is a closed symmetric monoidal category, is the natural map to the fiber product defined by the commutative square Definition 5.3.2. 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. Since the unit, S, is cofibrant in symmetric spectra this condition also ensures that the symmetric monoidal structure induces a symmetric monoidal structure on the homotopy category. 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 5.3.3. Let f, g and h be maps in a closed symmetric monoidal category C. There is a natural isomorphism Hom (g, h)) Proof. Use the argument in the proof of Proposition 3.3.8. 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 5.3.6. 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 5.3.7. 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 5.3.5. Part 1: Let I = J = K = F I ∂ . Then K-cof is the class of stable cofibrations. We have a natural isomorphism 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 where the first is taken in Sp Σ and the second is taken in S Σ * . In degree n, 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 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 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 , 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 : ∂∆[1] + → ∆[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 Cf ∧ Cg of the cofiber Cf of f , which is S-cofibrant, and the cofiber Cg of g. Let E be an injective Ω-spectrum. We will show that Hom S (Cf ∧ Cg, E) is a level trivial spectrum, and thus that Cf ∧ Cg is stably trivial.
First suppose that f is a stable equivalence. Then Hom S (Cf, E) is a level trivial spectrum which is also injective, by part 4 and the fact that Cf is S-cofibrant. Therefore Hom S (Cf ∧ Cg, E) ∼ = Hom S (Cg, Hom S (Cf, E)) is a level trivial spectrum, so Cf ∧ Cg is stably trivial and thus f g is a stable equivalence. Now suppose that g is a stable equivalence, so that Cg is stably trivial. By adjunction Hom S (Cf ∧ Cg, E) = Hom S (Cg, Hom S (Cf, E)). We claim that D = Hom S (Cf, E) is an injective Ω-spectrum. Indeed, we have already seen that D is injective. From Remark 2.2.12, we have 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 (Cf, E) is an injective Ω-spectrum. Since Cg is stably trivial, Hom S (Cf ∧ Cg, E) ∼ = Hom S (Cg, Hom S (Cf, E)) is a level trivial spectrum, so Cf ∧ Cg is stably trivial and thus f g is a stable equivalence.
Corollary 5.3.8. 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 5.3.9. 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 5.3.10. If X is an S-cofibrant symmetric spectrum, the functor X ∧ − preserves level equivalences and it preserves stable equivalences.
Proof. Part four of Theorem 5.3.7 implies that X ∧ − preserves level trivial cofibrations. Lemma 4.1.3 then implies that it preserves level equivalences, since every symmetric spectrum is level cofibrant. An arbitrary stable equivalence can be factored as a stable trivial cofibration followed by a level equivalence. Part five of Theorem 5.3.7 implies that X ∧ − takes stable trivial cofibrations to stable equivalences.
with g a fibration and f a weak equivalence, h is a weak equivalence.

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.

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

Let
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 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) 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 results of [SS97] then immediately give a model structure on symmetric ring spectra. See also [Shi97] for more about symmetric ring spectra. After proving the monoid axiom, we discuss the homotopy invariance of the resulting model categories of modules and algebras.
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. Transfinite compositions 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 [SS97, Theorem 3.1].
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). Moreover, if R is a commutative monoid then this is a monoidal model category satisfying the monoid axiom.
Corollary 5.5.3. Suppose R is a commutative monoid in the category of symmetric spectra. Then there is a model structure on the category of R-algebras where a homomorphism f : X − → Y is a weak equivalence (fibration) if and only if the underlying map of symmetric spectra is a stable equivalence (stable fibration). Any cofibration of R-algebras whose source is cofibrant as an R-module is a cofibration of R-modules.
Taking R = S gives a model structure on the category of monoids of symmetric spectra, the symmetric ring spectra.
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 5.3.7 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 model 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.
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: Then g = p ′′ i ′′ . We will show that q, q ′′ , and p ′ i ′ are stable equivalences. The two out of three property will then guarantee that g = p ′′ i ′′ is a stable equivalence. The map q is a level equivalence by hypothesis. The map p ′ i ′ is a pushout of a level cofibration, so is a level cofibration. The properness of the injective model structure then implies that q ′′ is a level equivalence. Since p ′ i ′ is a pushout of the stable equivalence f though the stable cofibration j, it is a stable equivalence by the properness of the stable model structure. Thus 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 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 transfinite compositions of monomorphisms.
We now show that the model categories of modules and algebras are homotopy invariant.
Lemma 5.5.7. Suppose R is a monoid in Sp Σ , f : M − → N is a level equivalence of right R-modules, and P is cofibrant in the model category of left R-modules.
Then the map f ∧ R P : M ∧ R P − → N ∧ R P is a level equivalence.
Proof. The factorizations in the model category of R-modules are obtained by applying the argument of Lemma 3.2.11 to the sets R ∧ F I ∂ and R ∧ F I Λ . In particular, P is a retract, as an R-module, of a transfinite composition of pushouts of coproducts of R ∧ F I ∂ . Thus we can assume that we have an ordinal α and an α-indexed diagram X of R-modules with colimit P , such that X 0 = 0 and each map i β : X β − → X β+1 is a pushout of a coproduct of maps of R ∧ F I ∂ . We will show that the map f ∧ R X β is a level equivalence for all β ≤ α by transfinite induction.
Getting the induction started is trivial. For the successor ordinal case of the induction, suppose f ∧ R X β is a level equivalence. We have a pushout square and an analogous pushout square with N replacing M . There is a map from the pushout square with M to the pushout square with N . By assumption, this map is a level equivalence on the lower right corner. Since M ∧ R (R ∧ F n K) = M ∧ F n K and similarly for N , this map is also a level equivalence on the upper right and upper left corners by Corollary 5.3.10 . Furthermore the map M ∧ R g is a level cofibration by part three of Theorem 5.3.7. Dan Kan's cubes lemma (see [Hov97,Section 5.2] or [DHK]), applied one level at a time, then implies that f ∧ R X β+1 is a level equivalence. Now suppose β is a limit ordinal and f ∧ R X γ is a level equivalence for all γ < β. Since each map M ∧ R i γ is a level cofibration, since it is a pushout of M ∧ R g, the homotopy of M ∧ R X β is the colimit of the homotopy of the M ∧ R X γ . A similar comment holds for N replacing M . By comparing homotopy, we find that f ∧ R X β is a level equivalence, as required.
Proposition 5.5.8. Suppose R is a monoid in Sp Σ , f : M − → N is a stable equivalence of right R-modules, and P is cofibrant in the model category of left R-modules.
Then the map f ∧ R P : M ∧ R P − → N ∧ R P is a stable equivalence.
Proof. Factor f in the model category of right R-modules into a stable trivial cofibration followed by a level trivial fibration. In view of Lemma 5.5.7, we can then assume f is a stable trivial cofibration. Any stable trivial cofibration is in (J ∧ R)-cof, so is a retract of a transfinite composition of pushouts of coproducts of J ∧ R. Since retracts of stable equivalences are stable equivalences, we may as well assume that f is the transfinite composition of an α-indexed sequence X β , where X 0 = M , colim β<α X β = N , and each map X β − → X β+1 is a pushout of a coproduct of maps of J ∧ R. Note that (J ∧ R) ∧ R P = J ∧ P consists of maps which are level cofibrations (by part 3 of Theorem 5.3.7 applied to the level cofibration 0 − → P and the stable cofibrations of J) and stable equivalences (by part 5 of Theorem 5.3.7). It follows from Lemma 5.5.5 that the map X β ∧ R P − → X β+1 ∧ R P is a level cofibration and a stable equivalence. Then Lemma 5.5.6 shows that f ∧ R P is a level cofibration and stable equivalence, completing the proof.
Since S, the sphere spectrum, is cofibrant in Sp Σ , Theorems 3.3 and 3.4 of [SS97], apply to give the following theorem.
Theorem 5.5.9. Suppose f : R − → R ′ is a stable equivalence of monoids of symmetric spectra. Then induction and restriction induce a Quillen equivalence from the model category of R-modules to the model category of R ′ -modules. If, in addition, R and R ′ are commutative, then induction and restriction induce a Quillen equivalence from the model category of R-algebras to the model category of R ′algebras.
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.
Before proving this proposition we need the following lemma.
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 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. For an example of this phenomenon, note that we could have π k X n ∼ = Z n−k , with the horizontal maps being the usual inclusions, but the vertical maps begin the inclusions that take (x 1 , . . . , x n−k ) to (0, x 1 , x 2 , . . . , x n−k ). Then the element (1, 0, . . . , 0, . . . ) of the colimit is not in the image of the colimit of the vertical maps.
Proof of Proposition 5.6.2. First we show that 1 implies 2 by using the following diagram.
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.
By 3 and 4 the left arrow is a stable homotopy equivalence to an Ω 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.
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.

Topological spectra
In this final section, we describe symmetric spectra based on topological spaces. The advantage of symmetric spectra over ordinary spectra is that symmetric spectra form a symmetric monoidal category, whereas spectra do not. Thus, we are interested in a setting where symmetric spectra of topological spaces also form a symmetric monoidal category. This requires that we begin with a symmetric monoidal category of pointed topological spaces. The category Top * of all pointed topological spaces is not symmetric monoidal under the smash product; the associativity isomorphism is not always continuous. So we begin in Section 6.1 by describing compactly generated spaces, a convenient category of topological spaces which is closed symmetric monoidal. Then in Section 6.2 we define topological spectra, and in Section 6.3 we discuss the stable model structure on topological symmetric spectra. In Section 6.4 we show that the stable model structure is monoidal and proper. We also consider monoids and modules over them, though here our results are much less strong than in the simplicial case. Indeed, in this regard, we do not know more than we do in a general monoidal model category, where we can appeal to the results of [Hov98a].
6.1. Compactly generated spaces. In this section, we describe a closed symmetric monoidal category of topological spaces. There are many different choices for such a category [Wyl73], and any one of them suffices to construct topological symmetric spectra. However, in order to get model categories of modules over a monoid, one needs especially good properties of the underlying category of topological spaces. For this reason, we work in the category of compactly generated spaces, for which the basic reference is the appendix of [Lew78]. This is the same category used in [LMS86] and [EKMM97]. We also discuss the model structure on the category of compactly generated spaces.
is open. We denote by kX the set X with the topology consisting of the compactly open sets, and we say that X is a k-space if kX = X. We denote the full subcategory of Top * consisting of pointed k-spaces by K * . The space X is weak Hausdorff if for every continuous map f : K − → X where K is compact Hausdorff, the image f (K) is closed in X. We denote the full subcategory of K * consisting of weak Hausdorff spaces by T * , and refer to its objects as compactly generated pointed spaces.
Note that every locally compact Hausdorff space is compactly generated. The functor k : Top * − → K * is a right adjoint to the inclusion functor. It follows that K * is bicomplete. Colimits are taken in Top * , so in particular a colimit of k-spaces is again a k-space. One must apply k to construct limits. Thus, the product in K * is the topological space k(X × Y ). If X is locally compact Hausdorff, then One can easily see that limits of weak Hausdorff spaces are again weak Hausdorff. Thus, T * is complete and the inclusion functor T * − → K * preserves all limits. Freyd's adjoint functor theorem then guarantees the existence of a left adjoint w : K * − → T * to the inclusion functor. The space wX is the maximal weak Hausdorff quotient of X, but it is very hard to describe wX in general. The functor w creates colimits in T * , so that T * is cocomplete. However, very often the colimit of a diagram in T * is the same as the colimit of the diagram in K * , so that we never need to apply w in practice. This is the case for directed systems of injective maps [Lew78, Proposition A.9.3], and for pushouts of closed inclusions [Lew78, Proposition A.7.5]. In particular, cofibrations are both injective and closed inclusions [Lew78, Proposition A.8.2], so these remarks will apply to transfinite compositions and pushouts of cofibrations.
Both the category K of unpointed k-spaces and the category T of unpointed compactly generated spaces are closed symmetric monoidal under the k-space version of the product [Lew78, Theorem A.5.5]. The right adjoint of X × − is the functor kC(X, −). Here C(X, Y ) is the set of continuous maps from X to Y . A subbasis for the topology on C(X, Y ) is given by 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, then this topology on C(X, Y ) is the same as the compact-open topology. However, there is no guarantee that C(X, Y ) is a k-space, even if X is locally compact Hausdorff, and so we must apply k. If X and Y are both second countable Hausdorff, and X is locally compact, then kC(X, Y ) = C(X, Y ) [Lew78, p. 161]. Fortunately, C(X, Y ) is weak Hausdorff as long as Y is [Lew78, Lemma A.5.2], so kC(X, Y ) is already weak Hausdorff and there is no need to apply w to get the closed structure in T.
The product induces a smash product X ∧ Y = k(X × Y )/(X ∨ Y ) on K * . Then K * is a closed symmetric monoidal category; the right adjoint of X ∧ − is kC * (X, −), k applied to the subspace of kC(X, Y ) consisting of pointed continuous maps. The subcategory T * of K * is closed under the smash product [Lew78, Lemma A.6.2], so T * is also a closed symmetric monoidal category. In fact, the functor w : K * − → T * is symmetric monoidal. This fact does not appear in [Lew78]; its proof uses adjointness and the fact that kC * (X, Y ) is already weak Hausdorff if Y is so.
We now discuss the geometric realization. One source for the geometric realization is [GZ67,III]. Note that the geometric realization of a finite simplicial set is compact Hausdorff, and so is a k-space. Since the geometric realization is a left adjoint, it commutes with colimits, and so the geometric realization of any simplicial set is a k-space. Also, the geometric realization of any simplicial set is a cell complex, so is Hausdorff. We can therefore consider the geometric realization as a functor to T * . We denote the geometric realization by | − | : S * − → T * , and its right adjoint, the singular functor, by Sing : The geometric realization has a number of properties not expected in a general left adjoint. For example, it preserves finite limits [GZ67, III.3]. Gabriel and Zisman prove this using a different category of topological spaces, but their proof relies on first working with certain finite simplicial sets and their geometric realizations, and then using the closed symmetric monoidal structure to extend to general simplicial sets. Thus their argument works in any closed symmetric monoidal category of topological spaces that contains finite CW complexes. It follows easily from this that the geometric realization preserves the smash product, and so defines a symmetric monoidal functor | − | : S * − → T * .
The categories T * , K * and Top * are all model categories, and both w : T * − → K * and the inclusion K * − → Top * are Quillen equivalences. The easiest way to describe the model structure is to define f to be a weak equivalence (fibration) if and only Sing f is a weak equivalence (fibration) of simplicial sets. Then f is a cofibration if and only if it has the left lifting property with respect to all trivial fibrations. Of course, the weak equivalences are the maps which induce isomorphisms on homotopy groups at all possible basepoints. All inclusions of relative CW-complexes are cofibrations, and any cofibration is a retract of a transfinite composition of inclusions of relative CW-complexes. In fact, the cofibrations are the class |I ∂ |-cof and the trivial cofibrations are the class |I Λ |-cof. Every topological space is fibrant, and the geometric realization is a Quillen equivalence. Both the geometric realization and the singular functor preserve and reflect weak equivalences, and the maps | Sing X| − → X and K − → Sing |K| are weak equivalences.
In proving the claims of the last paragraph, one needs the small object argument in each of the categories involved. The most obvious smallness statement is the following which states that all spaces are small relative to inclusions.
Lemma 6.1.2. Suppose K is a topological space with cardinality γ, and suppose X : α − → Top * is a colimit-preserving functor, where α is a γ-filtered ordinal, and suppose that each map X β − → X β+1 is an inclusion. Then the map is an isomorphism.
Proof. One can verify using transfinite induction that each map X β − → colim α X is an inclusion. Given a map f : K − → colim α X, there is a β < α such that f factors as a map of sets through X β , since α is γ-filtered. But then f is automatically continuous as a map to X β since the map X β − → colim α X is an inclusion.
For Lemma 6.1.3 we need the class of inclusions to be preserved under various constructions. Inclusions are not well behaved in general. However, in T * , closed inclusions are well behaved: they are closed under coproducts [Lew78, Proposition A.7.1(c)], pushouts [Lew78, Proposition A.7.5], and transfinite compositions [Lew78, Proposition A.9.3]. Furthermore, the smash product preserves closed inclusions [Lew78, Proposition A.7.3]. Given this, the proof of the Factorization Lemma 3.2.11 applies to prove the following analogue.
Lemma 6.1.3. Let I be a set of closed inclusions in the category T * . There is a functorial factorization of every map in T * as an I-cofibration followed by an I-injective map.
The model structure on T * is monoidal and proper. The fact that it is monoidal follows immediately from the corresponding fact for simplicial sets, since the geometric realization is monoidal and the generating cofibrations and trivial cofibrations are in the image of the geometric realization. Properness for T * is a little harder: right properness is formal, since all objects are fibrant, but left properness requires some care with local coefficients [Hir97, Proposition 11.1.11]. The monoid axiom holds in both T * and K * . We get a model category of monoids in T * , but not in K * . This is the reason we choose to use T * here and is explained in [Hov98a].
6.2. Topological spectra. In this section we define the category of topological symmetric spectra. The basic definitions are completely analogous to the definitions for simplicial symmetric spectra.
The basic definitions for topological spectra and topological symmetric spectra are straightforward. Let S 1 ∈ T * be the unit circle in R 2 with basepoint (1, 0). Definition 6.2.1. A topological spectrum is 1. a sequence X 0 , X 1 , . . . , X n , . . . in T * ; 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 is commutative for each n ≥ 0. Let Sp N T * denote the category of topological spectra. Let S p = (S 1 ) ∧p with the (continuous) left permutation action of Σ p .
Definition 6.2.2. A topological symmetric spectrum is 1. a sequence X 0 , X 1 , . . . , X n , . . . in T * ; 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 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 is commutative for each n ≥ 0. Let Sp Σ T * denote the category of 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 closed symmetric monoidal category.
For topological symmetric spectra X and Y , we can form the mapping space C Sp Σ T * (X, Y ) whose points are maps of symmetric spectra from X to Y , topologized as k applied to the subspace of k kC * (X n , Y n ). We get the usual adjunction properties. Similarly, we can form function objects Hom Σ (X, Y ) and Hom S (X, Y ). Indeed, given topological symmetric sequences X and Y , the nth space of the symmetric sequence Hom Σ (X, Y ) is C T Σ * (X ⊗ G n S 0 , Y ). A point is a map of symmetric sequences that raises degree by n; the leftover Σ n action defines a symmetric sequence. Similarly, given topological symmetric spectra X and Y , the nth space of the symmetric spectrum Hom S (X, Y ) is C Sp Σ T * (X ∧ F n S 0 , Y ). A point is a map of symmetric spectra that raises degree by n, and the topology is k applied to the subspace topology inherited from C T Σ * (X ⊗ G n S 0 , Y ). The structure map where t ∈ S 1 and f is a map of degree n, and σ t (f )(x) = f (σ X (t, x)) = σ Y (t, f (x)).
6.3. 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 that both the geometric realization and the singular functor reflect and preserve weak equivalences, and that maps A − → Sing |A| and the counit | Sing X| − → X of the adjunction are both natural weak equivalences.
The following proposition is then immediate.
adjoint to the evaluation functor Ev n . Thus, it suffices to show that each map in |F I ∂ | is a level cofibration, but this is clear.
To characterize the stable cofibrations, we can introduce the latching space L n X and the natural transformation i : L n X − → X n just as we did in Section 5.2. In the category of Σ n -spaces, a Σ n -cofibration is an equivariant map f : X − → Y which has the left lifting property with respect to all equivariant maps that are both weak equivalences and fibrations. In particular, if g : X − → Y is an equivariant relative CW complex on which Σ n acts cellularly and also acts freely on the cells not in X, then f is a Σ n -cofibration. Lemma 6.3.4. A map f : X − → Y of topological symmetric spectra is a stable cofibration if and only if for all n ≥ 0 the induced map Ev n (f i) : X n ∐ LnX L n Y − → Y n is a Σ n -cofibration.
The proof of this lemma is exactly the same as the proof of Proposition 5.2.2. In order to prove that these classes define a model structure on Sp Σ T * , the first thing we need is a version of the Factorization Lemma 3.2.11. The following proposition is the analogue of Proposition 3.2.13, and its proof is similar, given Lemma 6.1.2. Define the cardinality of a topological symmetric spectrum X to be the cardinality of the disjoint union of its spaces X n . Proposition 6.3.5. Let X be a topological symmetric spectrum of cardinality γ. Let α be a γ-filtered ordinal and let D : α − → Sp Σ T * be colimit-preserving α-indexed diagram of level inclusions. Then the natural map colim α Sp Σ T * (X, D) − → Sp Σ T * (X, colim α D) is an isomorphism.
This proposition immediately leads to a factorization lemma for topological symmetric spectra, analogous to the Factorization Lemmas 3.2.11 and 6.1.3 for simplicial symmetric spectra and for compactly generated spaces. Lemma 6.3.6. Let I be a set of level closed inclusions in Sp Σ T * . There is a functorial factorization of every map of topological symmetric spectra as an I-cofibration followed by an I-injective.
We can now establish the projective and stable model structures on Sp Σ T * . We begin with the projective model structure.
Theorem 6.3.7. The stable cofibrations, level fibrations, and level equivalences define a model structure on Sp Σ T * , called the projective level structure. The geometric realization | − | : Sp Σ − → Sp Σ T * defines a Quillen equivalence between the projective level structure on Sp Σ and the projective level structure on Sp Σ T * . Proof. Certainly Sp Σ T * is bicomplete, and the retract and two out of three axioms are immediate consequences of the definitions. The lifting axiom for stable cofibrations and level trivial fibrations follows from the definitions as well. Adjointness implies that the level trivial fibrations are the class |F I ∂ |-inj. Part two of Lemma 6.3.3 then implies that the stable cofibrations are the class |F I ∂ |-cof. Lemma 6.3.6 produces the required factorization into a stable cofibration followed by a level trivial fibration.
Adjointness also implies that the level fibrations are the class |F I Λ |-inj. Each map of |F I Λ | is a stable cofibration and a level equivalence, and in particular a level trivial cofibration. Just as in part 1 of Lemma 5.1.4, the level trivial cofibrations are the RK-projective maps. It follows that every map of |F I Λ |-cof is a level trivial cofibration. We knew already that each map of |F I Λ |-cof is a stable cofibration. Thus, Lemma 6.3.6 gives us a factorization of any map into a stable cofibration and level equivalence followed by a level fibration.
We are left with the lifting axiom for stable cofibrations and level equivalences, and level fibrations. Let f be a stable cofibration and level equivalence. Write f = pg, where g ∈ |F I Λ |-cof and p is a level fibration. Then p is a level equivalence, so f has the left lifting property with respect to p. It follows from Proposition 3.2.4 that f is a retract of g, so f has the left lifting property with respect to level fibrations.
The singular functor is obviously a right Quillen functor, and reflects all level equivalences. Now let K be a fibrant replacement functor on Sp Σ T * , obtained, for example, from the Factorization Lemma 6.3.6 applied to |F I Λ |. 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 level 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 level 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.
We do not know if the injective level structure on Sp Σ T * is a model structure. Theorem 6.3.8. The stable model structure on Sp Σ T * 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 Σ T * ; 2. A map g ∈ Sp Σ T * 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 Σ T * is bicomplete, and the two out of three and retract axioms are immediate consequences of the definitions. The stable trivial fibrations and the level trivial fibrations coincide, by Lemma 6.3.3. Thus the lifting and factorization axioms for stable cofibrations follow from the corresponding axioms in the projective level structure.
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 6.3.11, we show that every map of |J|-cof is a stable cofibration and a stable equivalence. Hence the Factorization Lemma 6.3.6 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 Σ T * is a model category.
Now, by definition, Sing : Sp Σ T * − → Sp Σ is a right Quillen functor, and f ∈ Sp Σ T * 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 Σ T * , obtained, for example, from the Factorization Lemma 6.3.6 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.
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. 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 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 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 ∐ Xi Y i − → Y j and X k ∐ Xi 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 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 6.3.10. Suppose α is an ordinal, and X : α − → Sp Σ T * 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.
|f | |g| = |f g|, so the result follows from the corresponding result for simplicial symmetric spectra.
For part 2, we use Corollary 5.3.5 with I = |F I ∂ | and J = K the class of level cofibrations. Then J-cof = J, since J = R(tK)-inj, as we have already seen in the proof of Lemma 6.3.3. Any map in I is of the form S ⊗ f , where f is a level cofibration of topological symmetric sequences. Then, just as in the proof of part 2 of Theorem 5.3.7, we have (S ⊗ f ) g = f g, where the second is taken in the category of topological symmetric sequences. But in T Σ * , we have (f g) n = p+q=n (Σ p+q ) + ∧ Σp×Σq (f p g q ).
Each map f p g q is a cofibration, since T * is a monoidal model category, f g is a level cofibration as required.
For part 3, if g is a level equivalence, then we can use the same method as in part 2 to conclude that f g is a level equivalence for f ∈ |F I ∂ |. Since the class of level trivial cofibrations is RK-inj (by the analogue of part 1 of Lemma 5.1.4), this completes the proof in case g is a level equivalence.
To do the case when f is a level equivalence, we use Corollary 5.3.5 with I = |F I Λ |, J the class of level cofibrations, and K the class of level trivial cofibrations. Then I-cof is the class of stable cofibrations and level equivalences, as these are the cofibrations in the projective level structure. Then the proof that I J ⊆ K = K-cof proceeds just as in part 2.
Finally, for part 4 we use Corollary 5.3.5 with I = |F I ∂ |, and J = K = |J|, where the second J is the set of generating stable trivial cofibrations in Sp Σ . Then, as in part 1, we use the fact that the geometric realization commutes with the box product and the corresponding fact for simplicial symmetric spectra.
be a pullback square of topological symmetric spectra with g a level fibration and h a stable equivalence. Then f is a stable equivalence.
Proof. Part 2 follows immediately from the fact that Sing preserves pullbacks, level fibrations, and stable equivalences, the corresponding fact for simplicial symmetric spectra 5.4.3, and the fact that Sing detects stable equivalences. Part 1 would also follow in the same way if Sing preserved level cofibrations and pushouts. The functor Sing does preserve level cofibrations, since it preserves monomorphisms. But Sing only preserves pushouts up to level equivalence. More precisely, suppose C is the pushout of Sing g and Sing f in the diagram in part 1. Then there is a map C j − → Sing Y , and we claim this map is a level equivalence. To prove this, it suffices to prove that |j| is a level equivalence. Since the geometric realization does preserve pushouts, it suffices to verify that the map |C| − → Y induced by the adjunction level equivalences | Sing D| − → D for D = A, X and B is a level equivalence. We can work one level at a time. We are then reduced to showing that, if we have a diagram form a monoidal model category. Also, the proof of part 3 of Theorem 6.4.5 is quite similar to the proof of Lemma 5.5.7. We interpret this theorem as asserting that, in order to study modules over R, one must first replace R by a stably cofibrant monoid R ′ .
Since we do not know whether the monoid axiom is true in topological symmetric spectra, we cannot expect a model category of topological symmetric ring spectra. We do have the following result, also taken from [Hov98a].