Symmetric spectra

By Mark Hovey, Brooke Shipley, and Jeff Smith

Abstract

The stable homotopy category, much studied by algebraic topologists, is a closed symmetric monoidal category. For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category. In this paper, we present such a category of spectra; the category of symmetric spectra. Our method can be used more generally to invert a monoidal functor, up to homotopy, in a way that preserves monoidal structure. Symmetric spectra were discovered at about the same time as the category of -modules of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra.

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 and , where is the reduced suspension of . 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 spaces together with structure maps . This was first done by Lima Reference Lim59 and later generalized by Whitehead Reference 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 Reference Vog70 and Adams Reference 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 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 Reference Lew91, where it is shown that a symmetric monoidal category of spectra cannot 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 -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 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 together with a pointed action of the permutation group on and equivariant structure maps . We must also require that the iterated structure maps be -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 Reference EKMM97 also constructed a symmetric monoidal category of spectra, called -modules. Some generalizations of symmetric spectra appear in Reference MMSS98a. These many new symmetric monoidal categories of spectra, including -modules and symmetric spectra, are shown to be equivalent in an appropriate sense in Reference MMSS98b and Reference Sch98. Another symmetric monoidal category of spectra sitting between the approaches of Reference EKMM97 and of this paper is developed in Reference DS. We also point out that symmetric spectra are part of a more general theory of localization of model categories Reference Hir99; we have not adopted this approach, but both Reference Hir99 and Reference DHK have influenced us considerably.

Symmetric spectra have already proved useful. In Reference GH97, symmetric spectra are used to extend the definition of topological cyclic homology from rings to schemes. Similarly, in Reference Shi, Bökstedt’s approach to topological Hochschild homology Reference Bök85 is extended to symmetric ring spectra, without connectivity conditions. And in Reference 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 Reference Hov98b. In particular, symmetric spectra may be the logical way to construct a model structure for Voevodsky’s stable homotopy of schemes Reference 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 does not work in general. A good approximation to such a functor is constructed in Reference Shi.

At present the theory of -modules of Reference EKMM97 is considerably more developed than the theory of symmetric spectra. Their construction appears to be significantly different from symmetric spectra; however, Reference Sch98 shows 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 -modules of Reference EKMM97; this paper is almost entirely self-contained, depending only on some standard results about simplicial sets. As discussed above, symmetric spectra can be built in many different circumstances, whereas -modules appear to be tied to the category of topological spaces. There are also technical differences reflecting the result of Reference Lew91 that there are limitations on any symmetric monoidal category of spectra. For example, the sphere spectrum is cofibrant in the category of symmetric spectra, but is not in the category of -modules. On the other hand, every -module is fibrant, a considerable technical advantage. Also, the -modules of Reference 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 Reference SS.

Organization

The paper is organized as follows. We choose to work in the category of 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 cannot 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 -spectra; i.e., symmetric spectra such that each is a Kan complex and the adjoint 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 Reference 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 5.5, and use this to verify the monoid axiom in Section 5.4. The monoid axiom is required to construct model categories of monoids and of modules over a given monoid; see Reference SS97. In Section 5.6, we define semistable spectra, which are helpful for understanding the difference between stable equivalences and stable homotopy equivalences.

Acknowledgments

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

Notation.

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

1. Symmetric spectra

In this section we construct the category of symmetric spectra over simplicial sets. 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 -spectra is described in Section 1.4.

1.1. Simplicial sets

We recall the basics. Consult Reference May67 or Reference Cur71 for more details.

The category has the ordered sets for as its objects and the order preserving functions as its maps. The category of simplicial sets, denoted , is the category of functors from to the category of sets. The set of -simplices of the simplicial set , denoted , is the value of the functor at . The standard -simplex is the contravariant functor . Varying gives a covariant functor . By the Yoneda lemma, and the contravariant functor is naturally isomorphic to .

Let be a discrete group. The category of -simplicial sets is the category of functors from to , where is regarded as a category with one object. A -simplicial set is therefore a simplicial set with a left simplicial -action, i.e., a homomorphism .

A basepoint of a simplicial set is a distinguished -simplex . The category of pointed simplicial sets and basepoint preserving maps is denoted . The simplicial set has a single simplex in each degree and is the terminal object in . A basepoint of is the same as a map . The disjoint union adds a disjoint basepoint to the simplicial set . For example, the -sphere is . A basepoint of a -simplicial set is a -invariant -simplex of . The category of pointed -simplicial sets is denoted .

The smash product of the pointed simplicial sets and is the quotient that collapses the simplicial subset to a point. For pointed -simplicial sets and , let be the quotient of by the diagonal action of . For pointed simplicial sets , , and , there are natural isomorphisms , and . 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 Reference ML71, VII or Reference Bor94, 6.1.

Definition 1.1.1.

A symmetric monoidal product on a category is: a bifunctor ; a unit ; and coherent natural isomorphisms (the associativity isomorphism), (the twist isomorphism), and (the unit isomorphism). The product is closed if the functor has a right adjoint for every . A (closed) symmetric monoidal category is a category with a (closed) symmetric monoidal product.

Coherence of the natural isomorphisms means that all reasonable diagrams built from the natural isomorphisms also commute Reference ML71. When the product is closed, the pairing is an internal Hom. For example, the smash product on the category of pointed simplicial sets is closed. For , the pointed simplicial set of maps from to is . For pointed -simplicial sets and , the simplicial subset of -equivariant pointed maps is .

1.2. Symmetric spectra

Let be the simplicial circle , obtained by identifying the two vertices of .

Definition 1.2.1.

A spectrum is

(1)

a sequence of pointed simplicial sets; and

(2)

a pointed map for each .

The maps are the structure maps of the spectrum. A map of spectra is a sequence of pointed maps such that the diagram

is commutative for each . Let 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 Reference BF78.

A symmetric spectrum is a spectrum to which symmetric group actions have been added. Let be the group of permutations of the set , with . As usual, embed as the subgroup of with acting on the first elements of and acting on the last elements of . Let be the -fold smash power of the simplicial circle with the left permutation action of .

Definition 1.2.2.

A symmetric spectrum is

(1)

a sequence of pointed simplicial sets;

(2)

a pointed map for each ; and

(3)

a basepoint preserving left action of on such that the composition

of the maps is -equivariant for and .

A map of symmetric spectra is a sequence of pointed maps such that is -equivariant and the diagram

is commutative for each . Let 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 and are equivariant; since the symmetric groups are generated by transpositions , if and are equivariant then all the maps are equivariant.

Example 1.2.4.

The symmetric suspension spectrum of the pointed simplicial set is the sequence of pointed simplicial sets with the natural isomorphisms as the structure maps and the diagonal action of on coming from the left permutation action on and the trivial action on . The composition is the natural isomorphism which is -equivariant. The symmetric sphere spectrum is the symmetric suspension spectrum of the -sphere; is the sequence of spheres with the natural isomorphisms as the structure maps and the left permutation action of on .

Example 1.2.5.

The Eilenberg-Mac Lane spectrum is the sequence of simplicial abelian groups , where is the free abelian group on the non-basepoint -simplices of . We identify the basepoint with . The symmetric group acts by permuting the generators, and one can easily verify that the evident structure maps are equivariant. One could replace by any ring.

Remark 1.2.6.

As explained in Reference GH97, Section 6, many other examples of symmetric spectra arise as the -theory of a category with cofibrations and weak equivalences as defined by Waldhausen Reference Wal85, p.330.

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

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 be a symmetric spectrum. The underlying spectrum is the sequence of pointed simplicial sets with the same structure maps as but ignoring the symmetric group actions. This gives a faithful functor .

Since the action of on is non-trivial for , it is usually impossible to obtain a symmetric spectrum from a spectrum by letting act trivially on . 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 is the underlying spectrum of the symmetric suspension spectrum of .

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 -functor is a pointed functor and a natural transformation of bifunctors such that the composition is the unit isomorphism and the diagram of natural transformations

is commutative. A pointed simplicial natural transformation, or -natural transformation, from the -functor to the -functor is a natural transformation such that .

Definition 1.2.9.

The prolongation of a -functor is the functor defined as follows. For a symmetric spectrum, is the sequence of pointed simplicial sets with the composition as the structure map and the action of on obtained by applying the functor to the action of on . Since is a -functor, each map is equivariant and so is a symmetric spectrum. For a map of symmetric spectra, is the sequence of pointed maps . Since is an -functor, is a map of spectra. Similarly, we can prolong an -natural transformation to a natural transformation of functors on .

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 , the limit and colimit functors are pointed simplicial functors; for and 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 and the underlying sequence of the colimit is .

1.3. Simplicial structure on

For a pointed simplicial set and a symmetric spectrum , prolongation of the -functor defines the smash product and prolongation of the -functor defines the power spectrum . For symmetric spectra and , the pointed simplicial set of maps from to is .

In the language of enriched category theory, the following proposition says that the smash product is a closed action of on . We leave the straightforward proof to the reader.

Proposition 1.3.1.

Let be a symmetric spectrum. Let and be pointed simplicial sets.

(1)

There are coherent natural isomorphisms and .

(2)

is the left adjoint of the functor .

(3)

is the left adjoint of the functor .

The evaluation map is the adjoint of the identity map on . The composition pairing

is the adjoint of the composition

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

Proposition 1.3.2.

Let , , and be symmetric spectra and let be a pointed simplicial set.

(1)

The composition pairing is associative.

(2)

The adjoint of the isomorphism is a left 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 and be maps of pointed simplicial sets. The pushout smash product is the natural map on the pushout

induced by the commutative square

Let be a map of symmetric spectra and let be a map of pointed simplicial sets. The pushout smash product is defined by prolongation, .

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 Reference Qui67, II.3, is:

Proposition 1.3.4.

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

Prolongation gives a corollary for symmetric spectra. A map of symmetric spectra is a monomorphism if is a monomorphism of simplicial sets for each .

Definition 1.3.5.

A map of symmetric spectra is a level equivalence if is a weak equivalence of simplicial sets for each .

Corollary 1.3.6.

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

By definition, a -simplex of is a map , but and so a -simplex of is a map . A -simplex of is a simplicial homotopy from to where and are the two inclusions . Simplicial homotopy generates an equivalence relation on and the quotient is . A map is a simplicial homotopy equivalence if it has a simplicial homotopy inverse, i.e., a map such that is simplicially homotopic to the identity map on and is simplicially homotopic to the identity map on . If is a simplicial homotopy equivalence of symmetric spectra, then each of the maps is a simplicial homotopy equivalence, and so each of the maps 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 -spectra

The stable homotopy category can be defined using -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 -spectrum is a spectrum such that for each the simplicial set is a Kan complex and the adjoint of the structure map is a weak equivalence of simplicial sets.

Let be the full subcategory of -spectra. The homotopy category is obtained from by formally inverting the level equivalences. By the results in Reference BF78, the category is naturally equivalent to Boardman’s stable homotopy category (or any other). Likewise, let