American Mathematical Society

Symmetric spectra

By Mark Hovey, Brooke Shipley, 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 upper S -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

left-bracket upper X comma upper Y right-bracket right-arrow Overscript Endscripts left-bracket normal upper Sigma upper X comma normal upper Sigma upper Y right-bracket right-arrow Overscript Endscripts period period period right-arrow Overscript Endscripts left-bracket normal upper Sigma Superscript n Baseline upper X comma normal upper Sigma Superscript n Baseline upper Y right-bracket right-arrow Overscript Endscripts period period period

is eventually constant for finite-dimensional pointed CW-complexes upper X and upper Y , where normal upper Sigma upper X equals upper S Superscript 1 Baseline logical-and upper X is the reduced suspension of upper 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 spaces upper X Subscript n together with structure maps upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow Overscript Endscripts upper X Subscript n plus 1 . This was first done by Lima ReferenceLim59 and later generalized by Whitehead ReferenceWhi62. 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 ReferenceVog70 and Adams ReferenceAda74 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 ReferenceLew91, 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 upper 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 upper S Superscript 1 Baseline logical-and upper S Superscript 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 upper X Subscript n together with a pointed action of the permutation group normal upper Sigma Subscript n on upper X Subscript n and equivariant structure maps upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow Overscript Endscripts upper X Subscript n plus 1 . We must also require that the iterated structure maps upper S Superscript p Baseline logical-and upper X Subscript n Baseline right-arrow Overscript Endscripts upper X Subscript n plus p be normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript 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 ReferenceEKMM97 also constructed a symmetric monoidal category of spectra, called upper S -modules. Some generalizations of symmetric spectra appear in ReferenceMMSS98a. These many new symmetric monoidal categories of spectra, including upper S -modules and symmetric spectra, are shown to be equivalent in an appropriate sense in ReferenceMMSS98b and ReferenceSch98. Another symmetric monoidal category of spectra sitting between the approaches of ReferenceEKMM97 and of this paper is developed in ReferenceDS. We also point out that symmetric spectra are part of a more general theory of localization of model categories ReferenceHir99; we have not adopted this approach, but both ReferenceHir99 and ReferenceDHK have influenced us considerably.

Symmetric spectra have already proved useful. In ReferenceGH97, symmetric spectra are used to extend the definition of topological cyclic homology from rings to schemes. Similarly, in ReferenceShi, Bökstedt’s approach to topological Hochschild homology ReferenceBök85 is extended to symmetric ring spectra, without connectivity conditions. And in ReferenceSS, 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 ReferenceHov98b. In particular, symmetric spectra may be the logical way to construct a model structure for Voevodsky’s stable homotopy of schemes ReferenceVoe97.

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 upper Q upper X does not work in general. A good approximation to such a functor is constructed in ReferenceShi.

At present the theory of upper S -modules of ReferenceEKMM97 is considerably more developed than the theory of symmetric spectra. Their construction appears to be significantly different from symmetric spectra; however, ReferenceSch98 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 upper S -modules of ReferenceEKMM97; 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 upper S -modules appear to be tied to the category of topological spaces. There are also technical differences reflecting the result of ReferenceLew91 that there are limitations on any symmetric monoidal category of spectra. For example, the sphere spectrum upper S is cofibrant in the category of symmetric spectra, but is not in the category of upper S -modules. On the other hand, every upper S -module is fibrant, a considerable technical advantage. Also, the upper S -modules of ReferenceEKMM97 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 ReferenceSS.

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 normal upper Omega -spectra; i.e., symmetric spectra upper X such that each upper X Subscript n is a Kan complex and the adjoint upper X Subscript n Baseline right-arrow Overscript Endscripts upper X Subscript n plus 1 Superscript upper S Super Superscript 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 ReferenceBF78.

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 ReferenceSS97. 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 script upper C , the set of maps from upper X to upper Y is denoted script upper C left-parenthesis upper X comma upper Y right-parenthesis ; in a simplicial category script upper C , the simplicial set of maps from upper X to upper Y is denoted upper M a p Subscript script upper C Baseline left-parenthesis upper X comma upper Y right-parenthesis or upper M a p left-parenthesis upper X comma upper Y right-parenthesis ; in a category script upper C with an internal Hom, the object in script upper C of maps from upper X to upper Y is denoted upper H o m Subscript script upper C Baseline left-parenthesis upper X comma upper Y right-parenthesis or upper H o m left-parenthesis upper X comma upper Y right-parenthesis . In case script upper C is the category of modules over a commutative monoid upper S , we also use upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper Y right-parenthesis 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 normal upper Omega -spectra is described in Section 1.4.

1.1. Simplicial sets

We recall the basics. Consult ReferenceMay67 or ReferenceCur71 for more details.

The category normal upper Delta has the ordered sets left-bracket n right-bracket equals StartSet 0 comma 1 comma ellipsis comma n EndSet for n greater-than-or-equal-to 0 as its objects and the order preserving functions left-bracket n right-bracket right-arrow left-bracket m right-bracket as its maps. The category of simplicial sets, denoted script upper S , is the category of functors from normal upper Delta Superscript op to the category of sets. The set of n -simplices of the simplicial set upper X , denoted upper X Subscript n , is the value of the functor upper X at left-bracket n right-bracket . The standard n -simplex normal upper Delta left-bracket n right-bracket is the contravariant functor normal upper Delta left-parenthesis minus comma left-bracket n right-bracket right-parenthesis . Varying n gives a covariant functor normal upper Delta left-bracket minus right-bracket colon normal upper Delta right-arrow script upper S . By the Yoneda lemma, script upper S left-parenthesis normal upper Delta left-bracket n right-bracket comma upper X right-parenthesis equals upper X Subscript n and the contravariant functor script upper S left-parenthesis normal upper Delta left-bracket minus right-bracket comma upper X right-parenthesis is naturally isomorphic to upper X .

Let upper G be a discrete group. The category of upper G -simplicial sets is the category script upper S Superscript upper G of functors from upper G to script upper S , where upper G is regarded as a category with one object. A upper G -simplicial set is therefore a simplicial set upper X with a left simplicial upper G -action, i.e., a homomorphism upper G right-arrow script upper S left-parenthesis upper X comma upper X right-parenthesis .

A basepoint of a simplicial set upper X is a distinguished 0 -simplex asterisk element-of upper X 0 . The category of pointed simplicial sets and basepoint preserving maps is denoted script upper S Subscript asterisk . The simplicial set normal upper Delta left-bracket 0 right-bracket equals normal upper Delta left-parenthesis minus comma left-bracket 0 right-bracket right-parenthesis has a single simplex in each degree and is the terminal object in script upper S . A basepoint of upper X is the same as a map normal upper Delta left-bracket 0 right-bracket right-arrow upper X . The disjoint union upper X Subscript plus Baseline equals upper X amalgamation-or-coproduct normal upper Delta left-bracket 0 right-bracket adds a disjoint basepoint to the simplicial set upper X . For example, the 0 -sphere is upper S Superscript 0 Baseline equals normal upper Delta left-bracket 0 right-bracket Subscript plus . A basepoint of a upper G -simplicial set upper X is a upper G -invariant 0 -simplex of upper X . The category of pointed upper G -simplicial sets is denoted script upper S Subscript asterisk Superscript upper G .

The smash product upper X logical-and upper Y of the pointed simplicial sets upper X and upper Y is the quotient left-parenthesis upper X times upper Y right-parenthesis slash left-parenthesis upper X logical-or upper Y right-parenthesis that collapses the simplicial subset upper X logical-or upper Y equals upper X times asterisk union asterisk times upper Y to a point. For pointed upper G -simplicial sets upper X and upper Y , let upper X logical-and Subscript upper G Baseline upper Y be the quotient of upper X logical-and upper Y by the diagonal action of upper G . For pointed simplicial sets upper X , upper Y , and upper Z , there are natural isomorphisms left-parenthesis upper X logical-and upper Y right-parenthesis logical-and upper Z approximately-equals upper X logical-and left-parenthesis upper Y logical-and upper Z right-parenthesis , upper X logical-and upper Y approximately-equals upper Y logical-and upper X and upper X logical-and upper S Superscript 0 Baseline approximately-equals upper 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 ReferenceML71, VII or ReferenceBor94, 6.1.

Definition 1.1.1

A symmetric monoidal product on a category script upper C is: a bifunctor circled-times colon script upper C times script upper C right-arrow script upper C ; a unit upper U element-of script upper C ; and coherent natural isomorphisms left-parenthesis upper X circled-times upper Y right-parenthesis circled-times upper Z approximately-equals upper X circled-times left-parenthesis upper Y circled-times upper Z right-parenthesis (the associativity isomorphism), upper X circled-times upper Y approximately-equals upper Y circled-times upper X (the twist isomorphism), and upper U circled-times upper X approximately-equals upper X (the unit isomorphism). The product is closed if the functor upper X circled-times left-parenthesis minus right-parenthesis has a right adjoint upper H o m left-parenthesis upper X comma minus right-parenthesis for every upper X element-of script upper C . A (closed) symmetric monoidal category is a category script upper C with a (closed) symmetric monoidal product.

Coherence of the natural isomorphisms means that all reasonable diagrams built from the natural isomorphisms also commute ReferenceML71. When the product is closed, the pairing upper H o m left-parenthesis upper X comma upper Y right-parenthesis colon script upper C Superscript op Baseline times script upper C right-arrow script upper C is an internal Hom. For example, the smash product on the category script upper S Subscript asterisk of pointed simplicial sets is closed. For upper X comma upper Y element-of script upper S Subscript asterisk Baseline , the pointed simplicial set of maps from upper X to upper Y is upper M a p Subscript script upper S Sub Subscript asterisk Baseline left-parenthesis upper X comma upper Y right-parenthesis equals script upper S Subscript asterisk Baseline left-parenthesis upper X logical-and normal upper Delta left-bracket minus right-bracket Subscript plus Baseline comma upper Y right-parenthesis . For pointed upper G -simplicial sets upper X and upper Y , the simplicial subset of upper G -equivariant pointed maps is upper M a p Subscript upper G Baseline left-parenthesis upper X comma upper Y right-parenthesis equals script upper S Subscript asterisk Superscript upper G Baseline left-parenthesis upper X logical-and normal upper Delta left-bracket minus right-bracket Subscript plus Baseline comma upper Y right-parenthesis .

1.2. Symmetric spectra

Let upper S Superscript 1 be the simplicial circle normal upper Delta left-bracket 1 right-bracket slash partial-differential normal upper Delta left-bracket 1 right-bracket , obtained by identifying the two vertices of normal upper Delta left-bracket 1 right-bracket .

Definition 1.2.1

A spectrum is

(1)

a sequence upper X 0 comma upper X 1 comma ellipsis comma upper X Subscript n Baseline comma period period period of pointed simplicial sets; and

(2)

a pointed map sigma colon upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus 1 Baseline for each n greater-than-or-equal-to 0 .

The maps sigma are the structure maps of the spectrum. A map of spectra f colon upper X right-arrow upper Y is a sequence of pointed maps f Subscript n Baseline colon upper X Subscript n Baseline right-arrow upper Y Subscript n Baseline such that the diagram

StartLayout 1st Row 1st Column upper S Superscript 1 Baseline logical-and upper X Subscript n 2nd Column right-arrow Overscript sigma Endscripts 3rd Column upper X Subscript n plus 1 2nd Row 1st Column upper S Superscript 1 Baseline logical-and f Subscript n Baseline down-arrow 2nd Column Blank 3rd Column f Subscript n plus 1 Baseline down-arrow 4th Column Blank 3rd Row 1st Column upper S Superscript 1 Baseline logical-and upper Y Subscript n 2nd Column right-arrow Overscript sigma Endscripts 3rd Column upper Y Subscript n plus 1 EndLayout

is commutative for each n greater-than-or-equal-to 0 . Let upper S p Superscript double-struck upper 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 ReferenceBF78.

A symmetric spectrum is a spectrum to which symmetric group actions have been added. Let normal upper Sigma Subscript p be the group of permutations of the set p overbar equals StartSet 1 comma 2 comma ellipsis comma p EndSet , with ModifyingAbove 0 With bar equals normal empty-set . As usual, embed normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript q as the subgroup of normal upper Sigma Subscript p plus q with normal upper Sigma Subscript p acting on the first p elements of ModifyingAbove p plus q With bar and normal upper Sigma Subscript q acting on the last q elements of ModifyingAbove p plus q With bar . Let upper S Superscript p Baseline equals left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript logical-and p be the p -fold smash power of the simplicial circle with the left permutation action of normal upper Sigma Subscript p .

Definition 1.2.2

A symmetric spectrum is

(1)

a sequence upper X 0 comma upper X 1 comma ellipsis comma upper X Subscript n Baseline comma period period period of pointed simplicial sets;

(2)

a pointed map sigma colon upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus 1 Baseline for each n greater-than-or-equal-to 0 ; and

(3)

a basepoint preserving left action of normal upper Sigma Subscript n on upper X Subscript n such that the composition sigma Superscript p Baseline equals sigma ring left-parenthesis upper S Superscript 1 Baseline logical-and sigma right-parenthesis ring period period period ring left-parenthesis upper S Superscript p minus 1 Baseline logical-and sigma right-parenthesis colon upper S Superscript p Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus p Baseline

of the maps upper S Superscript i Baseline logical-and upper S Superscript 1 Baseline logical-and upper X Subscript n plus p minus i minus 1 Baseline right-arrow Overscript upper S Superscript i Baseline logical-and sigma Endscripts upper S Superscript i Baseline logical-and upper X Subscript n plus p minus i is normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript n -equivariant for p greater-than-or-equal-to 1 and n greater-than-or-equal-to 0 .

A map of symmetric spectra f colon upper X right-arrow upper Y is a sequence of pointed maps f Subscript n Baseline colon upper X Subscript n Baseline right-arrow upper Y Subscript n Baseline such that f Subscript n is normal upper Sigma Subscript n -equivariant and the diagram

StartLayout 1st Row 1st Column upper S Superscript 1 Baseline logical-and upper X Subscript n 2nd Column right-arrow Overscript sigma Endscripts 3rd Column upper X Subscript n plus 1 2nd Row 1st Column upper S Superscript 1 Baseline logical-and f Subscript n Baseline down-arrow 2nd Column Blank 3rd Column f Subscript n plus 1 Baseline down-arrow 4th Column Blank 3rd Row 1st Column upper S Superscript 1 Baseline logical-and upper Y Subscript n 2nd Column right-arrow Overscript sigma Endscripts 3rd Column upper Y Subscript n plus 1 EndLayout

is commutative for each n greater-than-or-equal-to 0 . Let upper S p Superscript normal upper Sigma 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 sigma colon upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus 1 Baseline and sigma squared colon upper S squared logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus 2 Baseline are equivariant; since the symmetric groups normal upper Sigma Subscript p are generated by transpositions left-parenthesis i comma i plus 1 right-parenthesis , if sigma and sigma squared are equivariant then all the maps sigma Superscript p are equivariant.

Example 1.2.4

The symmetric suspension spectrum normal upper Sigma Superscript normal infinity Baseline upper K of the pointed simplicial set upper K is the sequence of pointed simplicial sets upper S Superscript n Baseline logical-and upper K with the natural isomorphisms sigma colon upper S Superscript 1 Baseline logical-and upper S Superscript n Baseline logical-and upper K right-arrow upper S Superscript n plus 1 Baseline logical-and upper K as the structure maps and the diagonal action of normal upper Sigma Subscript n on upper S Superscript n Baseline logical-and upper K coming from the left permutation action on upper S Superscript n and the trivial action on upper K . The composition sigma Superscript p is the natural isomorphism which is normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript n -equivariant. The symmetric sphere spectrum upper S is the symmetric suspension spectrum of the 0 -sphere; upper S is the sequence of spheres upper S Superscript 0 Baseline comma upper S Superscript 1 Baseline comma upper S squared comma period period period with the natural isomorphisms upper S Superscript 1 Baseline logical-and upper S Superscript n Baseline right-arrow upper S Superscript n plus 1 as the structure maps and the left permutation action of normal upper Sigma Subscript n on upper S Superscript n .

Example 1.2.5

The Eilenberg-Mac Lane spectrum upper H double-struck upper Z is the sequence of simplicial abelian groups double-struck upper Z circled-times upper S Superscript n , where left-parenthesis double-struck upper Z circled-times upper S Superscript n Baseline right-parenthesis Subscript k is the free abelian group on the non-basepoint k -simplices of upper S Superscript n . We identify the basepoint with 0 . The symmetric group normal upper Sigma Subscript n acts by permuting the generators, and one can easily verify that the evident structure maps are equivariant. One could replace double-struck upper Z by any ring.

Remark 1.2.6

As explained in ReferenceGH97, Section 6, many other examples of symmetric spectra arise as the upper K -theory of a category with cofibrations and weak equivalences as defined by Waldhausen ReferenceWal85, p.330.

A symmetric spectrum with values in a simplicial category script upper C is obtained by replacing the sequence of pointed simplicial sets by a sequence of pointed objects in script upper C . 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 upper X be a symmetric spectrum. The underlying spectrum upper U upper X is the sequence of pointed simplicial sets left-parenthesis upper U upper X right-parenthesis Subscript n Baseline equals upper X Subscript n with the same structure maps sigma colon upper S Superscript 1 Baseline logical-and left-parenthesis upper U upper X right-parenthesis Subscript n Baseline right-arrow left-parenthesis upper U upper X right-parenthesis Subscript n plus 1 Baseline as upper X but ignoring the symmetric group actions. This gives a faithful functor upper U colon upper S p Superscript normal upper Sigma Baseline right-arrow upper S p Superscript double-struck upper N .

Since the action of normal upper Sigma Subscript n on upper S Superscript n is non-trivial for n greater-than-or-equal-to 2 , it is usually impossible to obtain a symmetric spectrum from a spectrum by letting normal upper Sigma Subscript n act trivially on upper X Subscript 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 upper K is the underlying spectrum of the symmetric suspension spectrum of upper 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 script upper S Subscript asterisk -functor is a pointed functor upper R colon script upper S Subscript asterisk Baseline right-arrow script upper S Subscript asterisk Baseline and a natural transformation h colon upper R upper X logical-and upper K right-arrow upper R left-parenthesis upper X logical-and upper K right-parenthesis of bifunctors such that the composition upper R upper X logical-and upper S Superscript 0 Baseline right-arrow upper R left-parenthesis upper X logical-and upper S Superscript 0 Baseline right-parenthesis right-arrow upper R left-parenthesis upper X right-parenthesis is the unit isomorphism and the diagram of natural transformations

is commutative. A pointed simplicial natural transformation, or script upper S Subscript asterisk -natural transformation, from the script upper S Subscript asterisk -functor upper R to the script upper S Subscript asterisk -functor upper R prime is a natural transformation tau colon upper R right-arrow upper R prime such that tau h equals h prime left-parenthesis tau logical-and upper K right-parenthesis .

Definition 1.2.9

The prolongation of a script upper S Subscript asterisk -functor upper R colon script upper S Subscript asterisk Baseline right-arrow script upper S Subscript asterisk Baseline is the functor upper R colon upper S p Superscript normal upper Sigma Baseline right-arrow upper S p Superscript normal upper Sigma defined as follows. For upper X a symmetric spectrum, upper R upper X is the sequence of pointed simplicial sets upper R upper X Subscript n with the composition sigma colon upper S Superscript 1 Baseline logical-and upper R left-parenthesis upper X Subscript n Baseline right-parenthesis right-arrow upper R left-parenthesis upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-parenthesis right-arrow Overscript upper R sigma Endscripts upper R left-parenthesis upper X Subscript n plus 1 Baseline right-parenthesis as the structure map and the action of normal upper Sigma Subscript n on upper R left-parenthesis upper X Subscript n Baseline right-parenthesis obtained by applying the functor upper R to the action of normal upper Sigma Subscript n on upper X Subscript n . Since upper R is a script upper S Subscript asterisk -functor, each map sigma Superscript p is equivariant and so upper R upper X is a symmetric spectrum. For f a map of symmetric spectra, upper R f is the sequence of pointed maps upper R f Subscript n . Since upper R is an script upper S Subscript asterisk -functor, upper R f is a map of spectra. Similarly, we can prolong an script upper S Subscript asterisk -natural transformation to a natural transformation of functors on upper S p Superscript normal upper Sigma .

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 upper I , the limit and colimit functors script upper S Subscript asterisk Superscript upper I Baseline right-arrow script upper S Subscript asterisk are pointed simplicial functors; for upper K element-of script upper S Subscript asterisk and upper D element-of Set Superscript upper I there is a natural isomorphism

upper K logical-and c o l i m upper D approximately-equals c o l i m left-parenthesis upper K logical-and upper D right-parenthesis

and a natural map

upper K logical-and limit upper D right-arrow limit left-parenthesis upper K logical-and upper D right-parenthesis period

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 left-parenthesis limit upper D right-parenthesis Subscript n Baseline equals limit upper D Subscript n and the underlying sequence of the colimit is left-parenthesis c o l i m upper D right-parenthesis Subscript n Baseline equals c o l i m upper D Subscript n .

1.3. Simplicial structure on upper S p Superscript normal upper Sigma

For a pointed simplicial set upper K and a symmetric spectrum upper X , prolongation of the script upper S Subscript asterisk -functor left-parenthesis minus right-parenthesis logical-and upper K colon script upper S Subscript asterisk Baseline right-arrow script upper S Subscript asterisk Baseline defines the smash product upper X logical-and upper K and prolongation of the script upper S Subscript asterisk -functor left-parenthesis minus right-parenthesis Superscript upper K Baseline colon script upper S Subscript asterisk Baseline right-arrow script upper S Subscript asterisk Baseline defines the power spectrum upper X Superscript upper K . For symmetric spectra upper X and upper Y , the pointed simplicial set of maps from upper X to upper Y is upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis equals upper S p Superscript normal upper Sigma Baseline left-parenthesis upper X logical-and normal upper Delta left-bracket minus right-bracket Subscript plus Baseline comma upper Y right-parenthesis .

In the language of enriched category theory, the following proposition says that the smash product upper X logical-and upper K is a closed action of script upper S Subscript asterisk on upper S p Superscript normal upper Sigma . We leave the straightforward proof to the reader.

Proposition 1.3.1

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

(1)

There are coherent natural isomorphisms upper X logical-and left-parenthesis upper K logical-and upper L right-parenthesis approximately-equals left-parenthesis upper X logical-and upper K right-parenthesis logical-and upper L and upper X logical-and upper S Superscript 0 Baseline approximately-equals upper X .

(2)

left-parenthesis minus right-parenthesis logical-and upper K colon upper S p Superscript normal upper Sigma Baseline right-arrow upper S p Superscript normal upper Sigma is the left adjoint of the functor left-parenthesis minus right-parenthesis Superscript upper K Baseline colon upper S p Superscript normal upper Sigma Baseline right-arrow upper S p Superscript normal upper Sigma .

(3)

upper X logical-and left-parenthesis minus right-parenthesis colon script upper S Subscript asterisk Baseline right-arrow upper S p Superscript normal upper Sigma is the left adjoint of the functor upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis upper X comma minus right-parenthesis colon upper S p Superscript normal upper Sigma Baseline right-arrow script upper S Subscript asterisk Baseline .

The evaluation map upper X logical-and upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis right-arrow upper Y is the adjoint of the identity map on upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis . The composition pairing

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis logical-and upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-arrow upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Z right-parenthesis

is the adjoint of the composition

upper X logical-and upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis logical-and upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-arrow upper Y logical-and upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-arrow upper Z

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

Proposition 1.3.2

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

(1)

The composition pairing upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis logical-and upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-arrow upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Z right-parenthesis is associative.

(2)

The adjoint upper S Superscript 0 Baseline right-arrow upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper X right-parenthesis of the isomorphism upper X logical-and upper S Superscript 0 Baseline right-arrow upper X is a left and a right unit of the composition pairing.

(3)

There are natural isomorphisms upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis upper X logical-and upper K comma upper Y right-parenthesis approximately-equals upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis upper X comma upper Y Superscript upper K Baseline right-parenthesis approximately-equals upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis upper X comma upper Y right-parenthesis Superscript upper K Baseline period

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 colon upper U right-arrow upper V and g colon upper X right-arrow upper Y be maps of pointed simplicial sets. The pushout smash product f white medium square g is the natural map on the pushout

f white medium square g colon upper V logical-and upper X amalgamation-or-coproduct Subscript upper U logical-and upper X Baseline upper U logical-and upper Y right-arrow upper V logical-and upper Y

induced by the commutative square

StartLayout 1st Row 1st Column upper U logical-and upper X 2nd Column right-arrow Overscript f logical-and upper X Endscripts 3rd Column upper V logical-and upper X 2nd Row 1st Column upper U logical-and g down-arrow 2nd Column Blank 3rd Column upper V logical-and g 4th Column Blank 3rd Row 1st Column upper U logical-and upper Y 2nd Column right-arrow Underscript f logical-and upper Y Overscript Endscripts 3rd Column upper V logical-and upper Y period EndLayout

Let f be a map of symmetric spectra and let g be a map of pointed simplicial sets. The pushout smash product f white medium square g is defined by prolongation, left-parenthesis f white medium square g right-parenthesis Subscript n Baseline equals f Subscript n Baseline white medium square 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 ReferenceQui67, II.3, is:

Proposition 1.3.4

Let f and g be monomorphisms of pointed simplicial sets. Then f white medium square 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 Subscript n is a monomorphism of simplicial sets for each n greater-than-or-equal-to 0 .

Definition 1.3.5

A map f of symmetric spectra is a level equivalence if f Subscript n is a weak equivalence of simplicial sets for each n greater-than-or-equal-to 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 white medium square g is a monomorphism, which is a level equivalence if either f is a level equivalence or g is a weak equivalence.

By definition, a 0 -simplex of upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis is a map upper X logical-and normal upper Delta left-bracket 0 right-bracket Subscript plus Baseline right-arrow upper Y , but upper X logical-and normal upper Delta left-bracket 0 right-bracket Subscript plus Baseline approximately-equals upper X and so a 0 -simplex of upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis is a map upper X right-arrow upper Y . A 1 -simplex of upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis is a simplicial homotopy upper H colon upper X logical-and normal upper Delta left-bracket 1 right-bracket Subscript plus Baseline right-arrow upper Y from upper H ring left-parenthesis upper X logical-and i 0 right-parenthesis to upper H ring left-parenthesis upper X logical-and i 1 right-parenthesis where i 0 and i 1 are the two inclusions normal upper Delta left-bracket 0 right-bracket right-arrow normal upper Delta left-bracket 1 right-bracket . Simplicial homotopy generates an equivalence relation on upper S p Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis and the quotient is pi 0 upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis . A map f colon upper X right-arrow upper Y is a simplicial homotopy equivalence if it has a simplicial homotopy inverse, i.e., a map g colon upper Y right-arrow upper X such that g f is simplicially homotopic to the identity map on upper X and f g is simplicially homotopic to the identity map on upper Y . If f is a simplicial homotopy equivalence of symmetric spectra, then each of the maps f Subscript n is a simplicial homotopy equivalence, and so each of the maps f Subscript 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 normal upper Omega -spectra

The stable homotopy category can be defined using normal upper Omega -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 normal upper Omega -spectrum is a spectrum upper X such that for each n greater-than-or-equal-to 0 the simplicial set upper X Subscript n is a Kan complex and the adjoint upper X Subscript n Baseline right-arrow upper M a p Subscript script upper S Sub Subscript asterisk Baseline left-parenthesis upper S Superscript 1 Baseline comma upper X Subscript n plus 1 Baseline right-parenthesis of the structure map upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus 1 is a weak equivalence of simplicial sets.

Let normal upper Omega upper S p Superscript double-struck upper N subset-of-or-equal-to upper S p Superscript double-struck upper N be the full subcategory of normal upper Omega -spectra. The homotopy category upper H o left-parenthesis normal upper Omega upper S p Superscript double-struck upper N Baseline right-parenthesis is obtained from normal upper Omega upper S p Superscript double-struck upper N by formally inverting the level equivalences. By the results in ReferenceBF78, the category upper H o left-parenthesis normal upper Omega upper S p Superscript double-struck upper N Baseline right-parenthesis is naturally equivalent to Boardman’s stable homotopy category (or any other). Likewise, let normal upper Omega upper S p Superscript normal upper Sigma subset-of-or-equal-to upper S p Superscript normal upper Sigma be the full subcategory of symmetric normal upper Omega -spectra (i.e., symmetric spectra upper X for which upper U upper X is an normal upper Omega -spectrum). The homotopy category upper H o left-parenthesis normal upper Omega upper S p Superscript normal upper Sigma Baseline right-parenthesis is obtained from normal upper Omega upper S p Superscript normal upper Sigma by formally inverting the level equivalences. Since the forgetful functor upper U colon upper S p Superscript normal upper Sigma Baseline right-arrow upper S p Superscript double-struck upper N preserves normal upper Omega -spectra and level equivalences, it induces a functor upper H o left-parenthesis upper U right-parenthesis colon upper H o left-parenthesis normal upper Omega upper S p Superscript normal upper Sigma Baseline right-parenthesis right-arrow upper H o left-parenthesis normal upper Omega upper S p Superscript double-struck upper N Baseline right-parenthesis . As a corollary of Theorem 4.2.5, the functor upper H o left-parenthesis upper U right-parenthesis is a natural equivalence of categories. Thus the category upper H o left-parenthesis normal upper Omega upper S p Superscript normal upper Sigma Baseline right-parenthesis is naturally equivalent to Boardman’s stable homotopy category. To describe an inverse of upper H o left-parenthesis upper U right-parenthesis , let normal upper Omega Superscript normal infinity Baseline colon upper S p Superscript double-struck upper N Baseline right-arrow script upper S Subscript asterisk Baseline be the functor that takes a spectrum to the 0 -space of its associated normal upper Omega -spectrum. For any spectrum upper E element-of upper S p Superscript double-struck upper N , the symmetric spectrum upper V upper E equals normal upper Omega Superscript normal infinity Baseline left-parenthesis upper E logical-and upper S right-parenthesis is the value of the prolongation of the script upper S Subscript asterisk -functor normal upper Omega Superscript normal infinity Baseline left-parenthesis upper E logical-and minus right-parenthesis at the symmetric sphere spectrum upper S ; the underlying sequence is upper V upper E Subscript n Baseline equals normal upper Omega Superscript normal infinity Baseline left-parenthesis upper E logical-and upper S Superscript n Baseline right-parenthesis . The functor upper V preserves normal upper Omega -spectra, preserves level equivalences, and induces a functor upper H o left-parenthesis upper V right-parenthesis colon upper H o left-parenthesis normal upper Omega upper S p Superscript double-struck upper N Baseline right-parenthesis right-arrow upper H o left-parenthesis normal upper Omega upper S p Superscript normal upper Sigma Baseline right-parenthesis which is a natural inverse of upper H o left-parenthesis upper U right-parenthesis .

The category of symmetric normal upper Omega -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 normal upper Omega -spectra is a symmetric spectrum but not an normal upper Omega -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 upper S p Superscript normal upper Sigma by formally inverting the level equivalences but many symmetric spectra are not level equivalent to an normal upper Omega -spectrum. One must enlarge the class of equivalences. The stable equivalences of symmetric spectra are defined in Section 3.1. By Theorem 4.2.5, the homotopy category obtained from upper S p Superscript normal upper Sigma by inverting the stable equivalences is naturally equivalent to the stable homotopy category.

2. The smash product of symmetric spectra

In this section we construct the closed symmetric monoidal product on the category of symmetric spectra. A symmetric spectrum can be viewed as a module over the symmetric sphere spectrum upper 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 upper 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 upper X 0 comma upper X 1 comma ellipsis comma upper X Subscript n Baseline comma period period period of pointed simplicial sets with a basepoint preserving left action of normal upper Sigma Subscript n on upper X Subscript 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 normal upper Sigma equals coproduct Underscript n greater-than-or-equal-to 0 Endscripts normal upper Sigma Subscript n has the finite sets n overbar equals StartSet 1 comma 2 comma ellipsis comma n EndSet for n greater-than-or-equal-to 0 ( ModifyingAbove 0 With bar equals normal empty-set ) as its objects and the automorphisms of the sets n overbar as its maps. Let script upper C be a category. A symmetric sequence of objects in script upper C is a functor normal upper Sigma right-arrow script upper C , and the category of symmetric sequences of objects in script upper C is the functor category script upper C Superscript normal upper Sigma .

A symmetric sequence upper X element-of script upper S Subscript asterisk Superscript normal upper Sigma is a sequence upper X 0 comma upper X 1 comma ellipsis comma upper X Subscript n Baseline comma period period period of pointed simplicial sets with a basepoint preserving left action of normal upper Sigma Subscript n on upper X Subscript n . The category script upper C Superscript normal upper Sigma is a product category. In particular, script upper S Subscript asterisk Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis equals product Underscript p Endscripts script upper S Subscript asterisk Superscript normal upper Sigma Super Subscript p Baseline left-parenthesis upper X Subscript p Baseline comma upper Y Subscript p Baseline right-parenthesis .

Proposition 2.1.2

The category script upper S Subscript asterisk Superscript normal upper Sigma of symmetric sequences in script upper S Subscript asterisk is bicomplete.

Proof.

The category script upper S Subscript asterisk is bicomplete, so the functor category script upper S Subscript asterisk Superscript normal upper Sigma is bicomplete.

Definition 2.1.3

The tensor product upper X circled-times upper Y of the symmetric sequences upper X comma upper Y element-of script upper S Subscript asterisk Superscript normal upper Sigma is the symmetric sequence

left-parenthesis upper X circled-times upper Y right-parenthesis Subscript n Baseline equals logical-or Underscript p plus q equals n Endscripts left-parenthesis normal upper Sigma Subscript n Baseline right-parenthesis Subscript plus Baseline logical-and Subscript normal upper Sigma Sub Subscript p Subscript times normal upper Sigma Sub Subscript q Subscript Baseline left-parenthesis upper X Subscript p Baseline logical-and upper Y Subscript q Baseline right-parenthesis period

The tensor product f circled-times g colon upper X circled-times upper Y right-arrow Overscript Endscripts upper X prime circled-times upper Y prime of the maps f colon upper X right-arrow Overscript Endscripts upper X prime and g colon upper Y right-arrow Overscript Endscripts upper Y prime in script upper S Subscript asterisk Superscript normal upper Sigma is given by left-parenthesis f circled-times g right-parenthesis left-parenthesis alpha comma x comma y right-parenthesis equals left-parenthesis alpha comma f Subscript p Baseline x comma g Subscript q Baseline y right-parenthesis for alpha element-of normal upper Sigma Subscript p plus q , x element-of upper X Subscript p and y element-of upper Y Subscript q .

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

Proposition 2.1.4

Let upper X comma upper Y comma upper Z element-of script upper S Subscript asterisk Superscript normal upper Sigma be symmetric sequences. Then there is a natural isomorphism

script upper S Subscript asterisk Superscript normal upper Sigma Baseline left-parenthesis upper X circled-times upper Y comma upper Z right-parenthesis approximately-equals product Underscript p comma q Endscripts script upper S Subscript asterisk Superscript normal upper Sigma Super Subscript p Superscript times normal upper Sigma Super Subscript q Superscript Baseline left-parenthesis upper X Subscript p Baseline logical-and upper Y Subscript q Baseline comma upper Z Subscript p plus q Baseline right-parenthesis period

The twist isomorphism tau colon upper X circled-times upper Y right-arrow upper Y circled-times upper X for upper X comma upper Y element-of script upper S Subscript asterisk Superscript normal upper Sigma is the natural map given by tau left-parenthesis alpha comma x comma y right-parenthesis equals left-parenthesis alpha rho Subscript q comma p Baseline comma y comma x right-parenthesis for alpha element-of normal upper Sigma Subscript p plus q , x element-of upper X Subscript p , and y element-of upper Y Subscript q , where rho Subscript q comma p Baseline element-of normal upper Sigma Subscript p plus q is the left-parenthesis q comma p right-parenthesis -shuffle given by rho Subscript q comma p Baseline left-parenthesis i right-parenthesis equals i plus p for 1 less-than-or-equal-to i less-than-or-equal-to q and rho Subscript q comma p Baseline left-parenthesis i right-parenthesis equals i minus q for q less-than i less-than-or-equal-to p plus 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 normal upper Sigma 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 upper X and upper Y is the functor defined on a finite set upper C as

left-parenthesis upper X circled-times upper Y right-parenthesis left-parenthesis upper C right-parenthesis equals logical-or Underscript upper A union upper B equals upper C comma upper A intersection upper B equals normal empty-set Endscripts upper X left-parenthesis upper A right-parenthesis logical-and upper Y left-parenthesis upper B right-parenthesis period

For an isomorphism f colon upper C right-arrow upper D the map left-parenthesis upper X circled-times upper Y right-parenthesis left-parenthesis f right-parenthesis is the coproduct of the isomorphisms upper X left-parenthesis upper A right-parenthesis logical-and upper Y left-parenthesis upper B right-parenthesis right-arrow upper X left-parenthesis f upper A right-parenthesis logical-and upper Y left-parenthesis f upper B right-parenthesis . The twist isomorphism is the map that sends the summand upper X left-parenthesis upper A right-parenthesis logical-and upper Y left-parenthesis upper B right-parenthesis of left-parenthesis upper X circled-times upper Y right-parenthesis left-parenthesis upper C right-parenthesis to the summand upper Y left-parenthesis upper B right-parenthesis logical-and upper X left-parenthesis upper A right-parenthesis of left-parenthesis upper Y circled-times upper X right-parenthesis left-parenthesis upper C right-parenthesis by switching the factors.

Lemma 2.1.6

The tensor product circled-times is a symmetric monoidal product on the category of symmetric sequences script upper S Subscript asterisk Superscript normal upper Sigma .

Proof.

The unit of the tensor product is the symmetric sequence normal upper Sigma left-parenthesis ModifyingAbove 0 With bar comma minus right-parenthesis Subscript plus Baseline equals left-parenthesis upper S Superscript 0 Baseline comma asterisk comma asterisk comma period period period right-parenthesis . The unit isomorphism is obvious. The associativity isomorphism is induced by the associativity isomorphism in script upper S Subscript asterisk and the natural isomorphism

left-parenthesis left-parenthesis upper X circled-times upper Y right-parenthesis circled-times upper Z right-parenthesis Subscript n Baseline approximately-equals logical-or Underscript p plus q plus r equals n Endscripts left-parenthesis normal upper Sigma Subscript n Baseline right-parenthesis Subscript plus Baseline logical-and Subscript normal upper Sigma Sub Subscript p Subscript times normal upper Sigma Sub Subscript q Subscript times normal upper Sigma Sub Subscript r Subscript Baseline left-parenthesis upper X Subscript p Baseline logical-and upper Y Subscript q Baseline logical-and upper Z Subscript r Baseline right-parenthesis period

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 script upper S Subscript asterisk .

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

Definition 2.1.7

The evaluation functor upper E v Subscript n Baseline colon script upper S Subscript asterisk Superscript normal upper Sigma Baseline right-arrow script upper S Subscript asterisk Baseline is given by upper E v Subscript n Baseline upper X equals upper X Subscript n and upper E v Subscript n Baseline f equals f Subscript n . The free functor upper G Subscript n Baseline colon script upper S Subscript asterisk Baseline right-arrow script upper S Subscript asterisk Superscript normal upper Sigma is the left adjoint of the evaluation functor upper E v Subscript n . The smash product upper X logical-and upper K of upper X element-of script upper S Subscript asterisk Superscript normal upper Sigma and upper K element-of script upper S Subscript asterisk is the symmetric sequence left-parenthesis upper X logical-and upper K right-parenthesis Subscript n Baseline equals upper X Subscript n Baseline logical-and upper K with the diagonal action of normal upper Sigma Subscript n that is trivial on upper K . The pointed simplicial set upper M a p Subscript script upper S Sub Subscript asterisk Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis of maps from upper X to upper Y is the pointed simplicial set script upper S Subscript asterisk Superscript normal upper Sigma Baseline left-parenthesis upper X logical-and normal upper Delta left-bracket minus right-bracket Subscript plus Baseline comma upper Y right-parenthesis .

For each n greater-than-or-equal-to 0 , the free symmetric sequence is normal upper Sigma left-bracket n right-bracket equals normal upper Sigma left-parenthesis n overbar comma minus right-parenthesis and the free functor is upper G Subscript n Baseline equals normal upper Sigma left-bracket n right-bracket Subscript plus Baseline logical-and minus colon script upper S Subscript asterisk Baseline right-arrow script upper S Subscript asterisk Superscript normal upper Sigma . So, for a pointed simplicial set upper K , left-parenthesis upper G Subscript n Baseline upper K right-parenthesis Subscript n Baseline equals left-parenthesis normal upper Sigma Subscript n Baseline right-parenthesis Subscript plus Baseline logical-and upper K and left-parenthesis upper G Subscript n Baseline upper K right-parenthesis Subscript k Baseline equals asterisk for k not-equals n . In particular, upper G Subscript n Baseline upper S Superscript 0 Baseline equals normal upper Sigma left-bracket n right-bracket Subscript plus , upper G 0 upper K equals left-parenthesis upper K comma asterisk comma asterisk comma period period period right-parenthesis and upper G 0 upper S Superscript 0 is the unit of the tensor product circled-times .

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

Proposition 2.1.8

There are natural isomorphisms:

(1)

upper G Subscript p Baseline upper K circled-times upper G Subscript q Baseline upper L approximately-equals upper G Subscript p plus q Baseline left-parenthesis upper K logical-and upper L right-parenthesis for upper K comma upper L element-of script upper S Subscript asterisk Baseline .

(2)

upper X circled-times upper G 0 upper K approximately-equals upper X logical-and upper K for upper K element-of script upper S Subscript asterisk and upper X element-of script upper S Subscript asterisk Superscript normal upper Sigma .

(3)

upper M a p Subscript script upper S Sub Subscript asterisk Sub Superscript normal upper Sigma Baseline left-parenthesis upper G Subscript n Baseline upper K comma upper X right-parenthesis approximately-equals upper M a p Subscript script upper S Sub Subscript asterisk Baseline left-parenthesis upper K comma upper X Subscript n Baseline right-parenthesis for upper K element-of script upper S Subscript asterisk and upper X element-of script upper S Subscript asterisk Superscript normal upper Sigma .

(4)

upper M a p Subscript script upper S Sub Subscript asterisk Sub Superscript normal upper Sigma Baseline left-parenthesis upper X circled-times upper Y comma upper Z right-parenthesis approximately-equals product Underscript p comma q Endscripts upper M a p Subscript normal upper Sigma Sub Subscript p Subscript times normal upper Sigma Sub Subscript q Baseline left-parenthesis upper X Subscript p Baseline logical-and upper Y Subscript q Baseline comma upper Z Subscript p plus q Baseline right-parenthesis for upper X comma upper Y comma upper Z element-of upper S p Superscript normal upper Sigma .

A map f of symmetric sequences is a level equivalence if each of the maps f Subscript n is a weak equivalence. Since script upper S Subscript asterisk Superscript normal upper Sigma is a product category, a map f of symmetric sequences is a monomorphism if and only if each of the maps f Subscript n is a monomorphism.

Proposition 2.1.9

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

(1)

upper X circled-times left-parenthesis minus right-parenthesis preserves colimits.

(2)

If f is a monomorphism, then upper X circled-times f is a monomorphism.

(3)

If f is a level equivalence, then upper X circled-times f is a level equivalence.

(4)

If g is a monomorphism, then upper G Subscript n Baseline g is a monomorphism for n greater-than-or-equal-to 0 .

(5)

If g is a weak equivalence, then upper G Subscript n Baseline g is a level equivalence for n greater-than-or-equal-to 0 .

Proof.

Parts (1), (2) and (3) follow from the definition of circled-times and the corresponding properties for the smash product of pointed simplicial sets. For Parts (4) and (5) use the isomorphism upper G Subscript n Baseline upper K equals normal upper Sigma left-bracket n right-bracket Subscript plus Baseline logical-and upper K .

By part three of Proposition 2.1.8, upper M a p left-parenthesis normal upper Sigma left-bracket n right-bracket Subscript plus Baseline comma upper X right-parenthesis approximately-equals upper X Subscript n . As n varies, normal upper Sigma left-bracket minus right-bracket Subscript plus is a functor normal upper Sigma Superscript op Baseline right-arrow script upper S Subscript asterisk Superscript normal upper Sigma , and for upper X element-of script upper S Subscript asterisk Superscript normal upper Sigma , the symmetric sequence upper M a p Subscript script upper S Sub Subscript asterisk Sub Superscript normal upper Sigma Baseline left-parenthesis normal upper Sigma left-bracket minus right-bracket Subscript plus Baseline comma upper X right-parenthesis is naturally isomorphic to upper X .

Definition 2.1.10

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

upper H o m Subscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis equals upper M a p Subscript script upper S Sub Subscript asterisk Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis normal upper Sigma left-bracket minus right-bracket Subscript plus Baseline circled-times upper X comma upper Y right-parenthesis period

Theorem 2.1.11

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

Proof.

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

script upper S Subscript asterisk Superscript normal upper Sigma Baseline left-parenthesis upper X circled-times upper Y comma upper Z right-parenthesis approximately-equals script upper S Subscript asterisk Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper H o m Subscript normal upper Sigma Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-parenthesis

for symmetric sequences upper X comma upper Y and upper Z .

By Proposition 2.1.4, a map of symmetric sequences f colon upper X circled-times upper Y right-arrow upper Z is a collection of normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript q -equivariant maps f Subscript p comma q Baseline colon upper X Subscript p Baseline logical-and upper Y Subscript q Baseline right-arrow upper Z Subscript p plus q Baseline . This is adjoint to a collection of normal upper Sigma Subscript p -equivariant maps g Subscript p comma q Baseline colon upper X Subscript p Baseline right-arrow upper M a p Subscript normal upper Sigma Sub Subscript q Subscript Baseline left-parenthesis upper Y Subscript q Baseline comma upper Z Subscript p plus q Baseline right-parenthesis . So there is a natural isomorphism

script upper S Subscript asterisk Superscript normal upper Sigma Baseline left-parenthesis upper X circled-times upper Y comma upper Z right-parenthesis approximately-equals product Underscript p Endscripts script upper S Subscript asterisk Superscript normal upper Sigma Super Subscript p Superscript Baseline left-parenthesis upper X Subscript p Baseline comma product Underscript q Endscripts upper M a p Subscript normal upper Sigma Sub Subscript q Subscript Baseline left-parenthesis upper Y Subscript q Baseline comma upper Z Subscript p plus q Baseline right-parenthesis right-parenthesis period

By Proposition 2.1.8, the functor sending p overbar to product Underscript q Endscripts upper M a p Subscript normal upper Sigma Sub Subscript q Baseline left-parenthesis upper Y Subscript q Baseline comma upper Z Subscript p plus q Baseline right-parenthesis is the functor sending p overbar to upper M a p left-parenthesis normal upper Sigma left-bracket p right-bracket Subscript plus Baseline circled-times upper Y comma upper Z right-parenthesis which by definition is upper H o m Subscript normal upper Sigma Baseline left-parenthesis upper Y comma upper Z right-parenthesis . Combining the isomorphisms gives the natural isomorphism that finishes the proof.

2.2. Symmetric spectra

In this section we apply the language of “monoids” and “modules” in a symmetric monoidal category to the category of symmetric sequences. See ReferenceML71ReferenceBor94 for background on monoidal categories. In this language, the symmetric sequence of spheres upper S equals left-parenthesis upper S Superscript 0 Baseline comma upper S Superscript 1 Baseline comma ellipsis comma upper S Superscript n Baseline comma period period period right-parenthesis is a commutative monoid in the category of symmetric sequences and a symmetric spectrum is a (left) upper S -module.

Consider the symmetric sphere spectrum upper S . By Proposition 2.1.4, the natural normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript q -equivariant maps m Subscript p comma q Baseline colon upper S Superscript p Baseline logical-and upper S Superscript q Baseline right-arrow upper S Superscript p plus q give a pairing m colon upper S circled-times upper S right-arrow upper S . The adjoint upper G 0 upper S Superscript 0 Baseline right-arrow upper S of the identity map upper S Superscript 0 Baseline right-arrow upper E v 0 upper S equals upper S Superscript 0 is a two-sided unit of the pairing. The diagram of natural isomorphisms

StartLayout 1st Row 1st Column upper S Superscript p Baseline logical-and upper S Superscript q Baseline logical-and upper S Superscript r 2nd Column right-arrow Overscript Endscripts 3rd Column upper S Superscript p Baseline logical-and upper S Superscript q plus r 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column down-arrow 4th Column Blank 3rd Row 1st Column upper S Superscript p plus q Baseline logical-and upper S Superscript r 2nd Column right-arrow Overscript Endscripts 3rd Column upper S Superscript p plus q plus r EndLayout

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

In the language of monoidal categories, upper S is a monoid in the category of symmetric sequences and a symmetric spectrum is a left upper S -module.

Proposition 2.2.1

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

Proof.

A pairing m colon upper S circled-times upper X right-arrow upper X is the same as a collection of normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript q -equivariant maps m Subscript p comma q Baseline colon upper S Superscript p Baseline logical-and upper X Subscript q Baseline right-arrow upper X Subscript p plus q Baseline . If upper X is a left upper S -module, there is a spectrum for which upper X is the underlying symmetric sequence and the structure maps are the maps sigma equals m Subscript 1 comma n Baseline colon upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus 1 Baseline . The compositions sigma Superscript p are the normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript q -equivariant maps m Subscript p comma q . Conversely, for upper X a symmetric spectrum, the map of symmetric sequences m colon upper S circled-times upper X right-arrow upper X corresponding to the collection of normal upper Sigma Subscript p Baseline times normal upper Sigma Subscript q -equivariant maps m Subscript p comma q Baseline equals sigma Superscript p Baseline colon upper S Superscript p Baseline logical-and upper X Subscript q Baseline right-arrow upper X Subscript p plus q Baseline , where sigma Superscript 0 is the natural isomorphism upper S Superscript 0 Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n , makes upper X a left upper S -module. These are inverse constructions and give a natural equivalence of categories.

Moreover, upper S is a commutative monoid, i.e., m equals m ring tau , where tau 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 circled-times Subscript upper S Baseline , having upper S as the unit. This gives a symmetric monoidal product on the category of upper S -modules. The smash product upper X logical-and upper Y of upper X comma upper Y element-of upper S p Superscript normal upper Sigma is the symmetric spectrum upper X circled-times Subscript upper S Baseline upper Y .

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

Lemma 2.2.2

Let script upper C be a symmetric monoidal category that is cocomplete and let upper R be a commutative monoid in script upper C such that the functor upper R circled-times left-parenthesis minus right-parenthesis colon script upper C right-arrow script upper C preserves coequalizers. Then there is a symmetric monoidal product circled-times Subscript upper R Baseline on the category of upper R -modules with upper R as the unit.

We leave the proof of this lemma to the reader; the main point is the following definition.

Definition 2.2.3

The smash product upper X logical-and upper Y of symmetric spectra upper X and upper Y is the symmetric spectrum upper X circled-times Subscript upper S Baseline upper Y . The tensor product upper X circled-times Subscript upper S Baseline upper Y is the colimit in symmetric sequences of the diagram

period

For upper X a left upper S -module the composition upper X circled-times upper S right-arrow Overscript tau Endscripts upper S circled-times upper X right-arrow Overscript alpha Endscripts upper X is the right action of upper S ; since upper S is commutative, the two actions commute and upper X is an left-parenthesis upper S comma upper S right-parenthesis -bimodule. Hence, the tensor product upper X circled-times Subscript upper S Baseline upper Y is a left upper S -module.

Apply Lemma 2.2.2 to the commutative monoid upper S in the bicomplete category of symmetric sequences script upper S Subscript asterisk Superscript normal upper Sigma to obtain the following corollary.

Corollary 2.2.4

The smash product upper X logical-and upper 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 upper S circled-times left-parenthesis minus right-parenthesis colon script upper S Subscript asterisk Superscript normal upper Sigma Baseline right-arrow upper S p Superscript normal upper Sigma gives the free upper S -module upper S circled-times upper X generated by the symmetric sequence upper X . For each n greater-than-or-equal-to 0 , the evaluation functor upper E v Subscript n Baseline colon upper S p Superscript normal upper Sigma Baseline right-arrow script upper S Subscript asterisk Baseline is given by upper E v Subscript n Baseline upper X equals upper X Subscript n and upper E v Subscript n Baseline f equals f Subscript n . The free functor upper F Subscript n Baseline colon script upper S Subscript asterisk Baseline right-arrow upper S p Superscript normal upper Sigma is the left adjoint of the evaluation functor upper E v Subscript n . The functor upper R Subscript n Baseline colon script upper S Subscript asterisk Baseline right-arrow upper S p Superscript normal upper Sigma is the right adjoint of the evaluation functor upper E v Subscript n Baseline colon upper S p Superscript normal upper Sigma Baseline right-arrow script upper S Subscript asterisk Baseline .

The functor upper S circled-times left-parenthesis minus right-parenthesis is left adjoint to the forgetful functor upper S p Superscript normal upper Sigma Baseline right-arrow script upper S Subscript asterisk Superscript normal upper Sigma . The free functor upper F Subscript n is the composition upper S circled-times upper G Subscript n of the left adjoints upper G Subscript n Baseline colon script upper S Subscript asterisk Baseline right-arrow script upper S Subscript asterisk Superscript normal upper Sigma (Definition 2.1.7) and upper S circled-times left-parenthesis minus right-parenthesis colon script upper S Subscript asterisk Superscript normal upper Sigma Baseline right-arrow upper S p Superscript normal upper Sigma . Thus, for upper X element-of upper S p Superscript normal upper Sigma and upper K element-of script upper S Subscript asterisk , the left upper S -module upper X logical-and upper F Subscript n Baseline upper K is naturally isomorphic to the left upper S -module upper X circled-times upper G Subscript n Baseline upper K . In particular, upper X logical-and upper F 0 upper K is naturally isomorphic to the symmetric spectrum upper X logical-and upper K defined by prolongation in Section 1.3. Furthermore upper F 0 upper K equals upper S logical-and upper K is the symmetric suspension spectrum normal upper Sigma Superscript normal infinity Baseline upper K of upper K , and upper F 0 upper S Superscript 0 is the symmetric sphere spectrum upper S . For a pointed simplicial set upper K , upper R Subscript n Baseline upper K is the symmetric sequence upper H o m Subscript script upper S Sub Subscript asterisk Sub Superscript normal upper Sigma Baseline left-parenthesis upper S comma upper K Superscript normal upper Sigma left-parenthesis minus comma n overbar right-parenthesis Super Subscript plus Superscript Baseline right-parenthesis , which is a left upper S -module since upper S is a right upper S -module.

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

Proposition 2.2.6

There are natural isomorphisms:

(1)

upper F Subscript m Baseline left-parenthesis upper K right-parenthesis logical-and upper F Subscript n Baseline left-parenthesis upper L right-parenthesis approximately-equals upper F Subscript m plus n Baseline left-parenthesis upper K logical-and upper L right-parenthesis for upper K comma upper L element-of script upper S Subscript asterisk Baseline .

(2)

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper S circled-times upper X comma upper Y right-parenthesis approximately-equals upper M a p Subscript script upper S Sub Subscript asterisk Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper Y right-parenthesis for upper X element-of script upper S Subscript asterisk Superscript normal upper Sigma and upper Y element-of upper S p Superscript normal upper Sigma .

(3)

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper F Subscript n Baseline upper K comma upper X right-parenthesis approximately-equals upper M a p Subscript script upper S Sub Subscript asterisk Baseline left-parenthesis upper K comma upper E v Subscript n Baseline upper X right-parenthesis for upper K element-of script upper S Subscript asterisk and upper X element-of upper S p Superscript normal upper Sigma .

Proposition 2.2.7

Let f be a map of pointed simplicial sets.

(1)

upper F Subscript n Baseline colon script upper S Subscript asterisk Baseline right-arrow upper S p Superscript normal upper Sigma preserves colimits.

(2)

If f is a monomorphism, then upper F Subscript n Baseline f is a monomorphism.

(3)

If f is a weak equivalence, then upper F Subscript n Baseline f is a level equivalence.

Proof.

Use the isomorphism upper F Subscript n Baseline f equals upper S circled-times upper G Subscript n Baseline f and Proposition 2.1.9.

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

Lemma 2.2.8

Let script upper C be a closed symmetric monoidal category that is bicomplete and let upper R be a commutative monoid in script upper C . Then there is a function upper R -module upper H o m Subscript upper R Baseline left-parenthesis upper M comma upper N right-parenthesis , natural for upper M comma upper N element-of script upper C , such that the functor left-parenthesis minus right-parenthesis circled-times Subscript upper R Baseline upper M is left adjoint to the functor upper H o m Subscript upper R Baseline left-parenthesis upper M comma minus right-parenthesis .

Again, we leave the proof of this lemma to the reader, but the main definition follows.

Definition 2.2.9

Let upper X and upper Y be symmetric spectra. The function spectrum upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper Y right-parenthesis is the limit of the diagram in upper S p Superscript normal upper Sigma

period

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

upper S p Superscript normal upper Sigma Baseline left-parenthesis upper X logical-and upper Y comma upper Z right-parenthesis approximately-equals upper S p Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper H o m Subscript upper S Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-parenthesis period

Proof.

The smash product logical-and 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

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X logical-and upper Y comma upper Z right-parenthesis approximately-equals upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper H o m Subscript upper S Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-parenthesis

and

upper H o m Subscript upper S Baseline left-parenthesis upper X logical-and upper Y comma upper Z right-parenthesis approximately-equals upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper H o m Subscript upper S Baseline left-parenthesis upper Y comma upper Z right-parenthesis right-parenthesis period

Remark 2.2.12

We use Proposition 2.2.6 to give another description of the function spectrum upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper Y right-parenthesis . For a symmetric spectrum upper X , the pointed simplicial set of maps upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper F Subscript n Baseline upper S Superscript 0 Baseline comma upper X right-parenthesis is naturally isomorphic to upper M a p Subscript script upper S Sub Subscript asterisk Baseline left-parenthesis upper S Superscript 0 Baseline comma upper E v Subscript n Baseline upper X right-parenthesis equals upper X Subscript n . The symmetric spectrum upper F Subscript n Baseline upper S Superscript 0 is the upper S -module upper S circled-times normal upper Sigma left-bracket n right-bracket Subscript plus and as n varies, upper S circled-times normal upper Sigma left-bracket minus right-bracket Subscript plus is a functor normal upper Sigma Superscript op Baseline right-arrow upper S p Superscript normal upper Sigma . The symmetric sequence upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper S circled-times normal upper Sigma left-bracket minus right-bracket Subscript plus Baseline comma upper X right-parenthesis is the underlying symmetric sequence of upper X . In particular, the natural isomorphism upper X Subscript n Baseline equals upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper F Subscript n Baseline upper S Superscript 0 Baseline comma upper X right-parenthesis is normal upper Sigma Subscript n -equivariant. Applying this to upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper Y right-parenthesis and using Corollary 2.2.11, we find that the underlying symmetric sequence of upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper Y right-parenthesis is the symmetric sequence upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X logical-and left-parenthesis upper S circled-times normal upper Sigma left-bracket minus right-bracket Subscript plus Baseline right-parenthesis comma upper Y right-parenthesis .

We must also describe the structure maps of upper X from this point of view. Recall that upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper F Subscript n Baseline upper S Superscript 0 Baseline comma upper X right-parenthesis equals upper X Subscript n , upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper F Subscript n Baseline upper S Superscript 1 Baseline comma upper X right-parenthesis equals upper M a p Subscript script upper S Sub Subscript asterisk Baseline left-parenthesis upper S Superscript 1 Baseline comma upper X Subscript n Baseline right-parenthesis . Let lamda colon upper F 1 upper S Superscript 1 Baseline right-arrow upper F 0 upper S Superscript 0 be the adjoint of the identity map upper S Superscript 1 Baseline right-arrow upper E v 1 upper F 0 upper S Superscript 0 Baseline equals upper S Superscript 1 . The induced map upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis lamda comma upper X right-parenthesis colon upper X 0 right-arrow upper M a p Subscript script upper S Sub Subscript asterisk Subscript Baseline left-parenthesis upper S Superscript 1 Baseline comma upper X 1 right-parenthesis is adjoint to the structure map upper S Superscript 1 Baseline logical-and upper X 0 right-arrow upper X 1 . The map

lamda logical-and upper F Subscript n Baseline upper S Superscript 0 Baseline colon upper F 1 upper S Superscript 1 Baseline logical-and upper F Subscript n Baseline upper S Superscript 0 Baseline equals upper F Subscript n plus 1 Baseline upper S Superscript 1 Baseline right-arrow upper F 0 upper S Superscript 0 Baseline logical-and upper F Subscript n Baseline upper S Superscript 0 Baseline equals upper F Subscript n Baseline upper S Superscript 0

is normal upper Sigma 1 times normal upper Sigma Subscript n -equivariant; the induced map

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis lamda logical-and upper F Subscript n Baseline upper S Superscript 0 Baseline comma upper X right-parenthesis colon upper X Subscript n Baseline right-arrow upper M a p Subscript script upper S Sub Subscript asterisk Subscript Baseline left-parenthesis upper S Superscript 1 Baseline comma upper X Subscript n plus 1 Baseline right-parenthesis

is normal upper Sigma 1 times normal upper Sigma Subscript n -equivariant and is adjoint to the structure map sigma colon upper S Superscript 1 Baseline logical-and upper X Subscript n Baseline right-arrow upper X Subscript n plus 1 Baseline . In order to apply this to upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper Y right-parenthesis , use Proposition 2.2.6 and Corollary 2.2.11 to find a natural isomorphism

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis upper X logical-and upper F Subscript n plus 1 Baseline upper S Superscript 1 Baseline comma upper Y right-parenthesis approximately-equals upper M a p Subscript script upper S Sub Subscript asterisk Subscript Baseline left-parenthesis upper S Superscript 1 Baseline comma upper M a p Subscript upper S p Sub Superscript normal upper Sigma Subscript Baseline left-parenthesis upper X logical-and upper F Subscript n plus 1 Baseline upper S Superscript 0 Baseline comma upper Y right-parenthesis right-parenthesis period

Using this natural isomorphism, we find that the structure maps of upper H o m Subscript upper S Baseline left-parenthesis upper X comma upper Y right-parenthesis are the adjoints of the maps

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X logical-and upper F Subscript n Baseline upper S Superscript 0 Baseline comma upper Y right-parenthesis right-arrow upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X logical-and upper F Subscript n plus 1 Baseline upper S Superscript 1 Baseline comma upper Y right-parenthesis

induced by lamda logical-and upper F Subscript n Baseline upper S Superscript 0 .

For example, upper H o m Subscript upper S Baseline left-parenthesis upper F Subscript k Baseline upper S Superscript 0 Baseline comma upper X right-parenthesis is the k -shifted spectrum; its underlying symmetric sequence is the sequence of pointed simplicial sets

upper X Subscript k Baseline comma upper X Subscript 1 plus k Baseline comma ellipsis comma upper X Subscript n plus k Baseline comma period period period

with normal upper Sigma Subscript n acting on upper X Subscript n plus k by restricting the action of normal upper Sigma Subscript n plus k to the copy of normal upper Sigma Subscript n that permutes the first n elements of ModifyingAbove n plus k With bar . The structure maps of the k -shifted spectrum are the structure maps sigma colon upper S Superscript 1 Baseline logical-and upper X Subscript n plus k Baseline right-arrow upper X Subscript n plus k plus 1 Baseline of upper X . More generally, upper H o m Subscript upper S Baseline left-parenthesis upper F Subscript k Baseline upper K comma upper X right-parenthesis is the k -shifted spectrum of upper X Superscript upper 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.

Definition 2.3.1

The category double-struck upper 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 script upper S Subscript asterisk Superscript double-struck upper N is the category of functors from double-struck upper N to script upper S Subscript asterisk . An object of script upper S Subscript asterisk Superscript double-struck upper N is a sequence upper X 0 comma upper X 1 comma ellipsis comma upper X Subscript n Baseline comma period period period of pointed simplicial sets and a map f colon upper X right-arrow Overscript Endscripts upper Y is a sequence of pointed simplicial maps f Subscript n Baseline colon upper X Subscript n Baseline right-arrow upper Y Subscript n Baseline .

Definition 2.3.2

The graded smash product of sequences upper X and upper Y is the sequence upper X circled-times upper Y given in degree n by

left-parenthesis upper X circled-times upper Y right-parenthesis Subscript n Baseline equals logical-or Underscript p plus q equals n Endscripts upper X Subscript p Baseline logical-and upper Y Subscript q Baseline period

Lemma 2.3.3

The category of sequences is a bicomplete category and the graded smash product is a symmetric monoidal product on script upper S Subscript asterisk Superscript double-struck upper N .

Proposition 2.3.4

The sequence upper S whose n th level is upper S Superscript n is a monoid in the category of sequences. The category of left upper S -modules is isomorphic to the ordinary category of spectra, upper S p Superscript double-struck upper N .

The twist map on upper S Superscript 1 Baseline logical-and upper S Superscript 1 is not the identity map and thus upper S is not a commutative monoid in script upper S Subscript asterisk Superscript double-struck upper N . In fact, upper S is a free monoid (Section 4.3). Therefore the approach taken in Section 2.2 does not provide a closed symmetric monoidal smash product on the ordinary category of spectra.

3. Stable homotopy theory of symmetric spectra

To use symmetric spectra for the study of stable homotopy theory, one should have a stable model category of symmetric spectra such that the category obtained by inverting the stable equivalences is naturally equivalent to Boardman’s stable homotopy category of spectra (or to any other known to be equivalent to Boardman’s). In this section we define the stable model category of symmetric spectra. In Section 4 we show that it is Quillen equivalent to the stable model category of spectra discussed in ReferenceBF78.

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 (non-symmetric) 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 upper S p Superscript normal upper Sigma , 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 upper U colon upper S p Superscript normal upper Sigma Baseline right-arrow upper S p Superscript double-struck upper N ; one would like a map f of symmetric spectra to be a stable equivalence if the underlying map upper U f of spectra is a stable equivalence, i.e., if upper 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 upper E Superscript asterisk Baseline f of cohomology groups is an isomorphism for every generalized cohomology theory upper 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 0 th cohomology group of the symmetric spectrum upper X with coefficients in the symmetric normal upper Omega -spectrum upper E could be defined as pi 0 upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis upper X comma upper E right-parenthesis , the set of simplicial homotopy classes of maps from upper X to upper E . But, even though the contravariant functor upper E Superscript 0 Baseline equals pi 0 upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis minus comma upper E right-parenthesis takes simplicial homotopy equivalences to isomorphisms, upper E Superscript 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 upper E for which the functor upper E Superscript 0 behaves correctly.

Definition 3.1.1

An injective spectrum is a symmetric spectrum upper 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 upper S p Superscript normal upper Sigma

StartLayout 1st Row 1st Column upper X 2nd Column right-arrow Overscript g Endscripts 3rd Column upper E 2nd Row 1st Column f down-arrow 2nd Column Blank 3rd Column Blank 4th Column Blank 3rd Row 1st Column upper Y 2nd Column Blank 3rd Column Blank EndLayout

where f is a monomorphism and a level equivalence there is a map h colon upper Y right-arrow upper E such that g equals h f .

Some examples of injective spectra follow. Recall that upper R Subscript n Baseline colon script upper S Subscript asterisk Baseline right-arrow upper S p Superscript normal upper Sigma is the right adjoint of the evaluation functor upper E v Subscript n Baseline colon upper S p Superscript normal upper Sigma Baseline right-arrow script upper S Subscript asterisk Baseline . 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 upper K is a Kan complex, then upper R Subscript n Baseline upper K is an injective spectrum. If upper X is a symmetric sequence and upper E is an injective spectrum, then upper H o m Subscript upper S Baseline left-parenthesis upper S circled-times upper X comma upper E right-parenthesis is an injective spectrum.

Proof.

Since upper E v Subscript n is left adjoint to upper R Subscript n , the spectrum upper R Subscript n Baseline upper K has the extension property with respect to the monomorphism and level equivalence f if and only if upper K has the extension property with respect to the monomorphism and weak equivalence upper E v Subscript n Baseline f . Since upper K is a Kan complex, it does have the extension property with respect to upper E v Subscript n Baseline f . Hence upper R Subscript n Baseline upper K is injective.

Since the functor left-parenthesis upper S circled-times upper X right-parenthesis logical-and left-parenthesis minus right-parenthesis is the left adjoint of upper H o m Subscript upper S Baseline left-parenthesis upper S circled-times upper X comma minus right-parenthesis , the spectrum upper H o m Subscript upper S Baseline left-parenthesis upper S circled-times upper X comma upper E right-parenthesis has the extension property with respect to the monomorphism and level equivalence f if and only if upper E has the extension property with respect to the map left-parenthesis upper S circled-times upper X right-parenthesis logical-and f . There is a natural isomorphism of maps of symmetric sequences left-parenthesis upper S circled-times upper X right-parenthesis logical-and f approximately-equals upper X circled-times f . Since f is a monomorphism and a level equivalence, upper X circled-times f is a monomorphism and a level equivalence by Proposition 2.1.9. Thus left-parenthesis upper S circled-times upper X right-parenthesis logical-and f is also a monomorphism and level equivalence of symmetric spectra. So upper H o m Subscript upper S Baseline left-parenthesis upper S circled-times upper X comma upper E right-parenthesis is injective.

In fact, injective spectra are the fibrant objects of a model structure on upper S p Superscript normal upper Sigma for which every object is cofibrant (Section 5.1). In particular, as we will see in Corollary 5.1.3, there are enough injectives; every symmetric spectrum embeds in an injective spectrum by a map that is a level equivalence.

Definition 3.1.3

A map f colon upper X right-arrow upper Y of symmetric spectra is a stable equivalence if upper E Superscript 0 Baseline f colon upper E Superscript 0 Baseline upper Y right-arrow upper E Superscript 0 Baseline upper X is an isomorphism for every injective normal upper Omega -spectrum upper E .

There are two other ways to define stable equivalence.

Proposition 3.1.4

Let f colon upper X right-arrow upper Y be a map of symmetric spectra. The following conditions are equivalent:

upper E Superscript 0 Baseline f is an isomorphism for every injective normal upper Omega -spectrum upper E ;

upper M a p Subscript upper S p Sub Superscript normal upper Sigma Baseline left-parenthesis f comma upper E right-parenthesis is a weak equivalence for every injective normal upper Omega -spectrum upper E ;

upper H o m Subscript upper S Baseline left-parenthesis f comma upper E right-parenthesis is a level equivalence for every injective normal upper Omega -spectrum upper E .

Proof.

Let upper K be a pointed simplicial set and let upper E be a symmetric normal upper Omega -spectrum. The adjoints of the structure maps of upper E are weak equivalences of Kan complexes. From Remark 2.2.12, for k comma n greater-than-or-equal-to 0 upper E v Subscript k Baseline upper H o m Subscript upper S Baseline left-parenthesis upper F Subscript n Baseline upper K comma upper E right-parenthesis equals upper E Subscript n plus k Superscript upper K