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 of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra. -modules
Stable homotopy theory studies spectra as the linear approximation to spaces. Here, “stable” refers to the consideration of spaces after inverting the suspension functor. This approach is a general one: one can often create a simpler category by inverting an operation such as suspension. In this paper we study a particularly simple model for inverting such operations which preserves product structures. The combinatorial nature of this model means that it is easily transported, and hence may be useful in extending the methods of stable homotopy theory to other settings.
The idea of a spectrum is a relatively simple one: Freudenthal’s suspension theorem implies that the sequence of homotopy classes of maps
is eventually constant for finite-dimensional pointed CW-complexes and where , is the reduced suspension of This suggests forming a stable category where the suspension functor is an isomorphism. The standard way to do this is to define a spectrum to be a sequence of pointed spaces . together with structure maps This was first done by Lima .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 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. -theory
In this paper, we describe a symmetric monoidal category of spectra, called the category of symmetric spectra. The ordinary category of spectra as described above is the category of modules over the sphere spectrum. The sphere spectrum is a monoid in the category of sequences of spaces, but it is not a commutative monoid, because the twist map on is not the identity. This explains why the ordinary category of spectra is not symmetric monoidal, just as in algebra where the usual internal tensor product of modules is defined only over a commutative ring. To make the sphere spectrum a commutative monoid, we need to keep track of the twist map, and, more generally, of permutations of coordinates. We therefore define a symmetric spectrum to be a sequence of pointed simplicial sets together with a pointed action of the permutation group on and equivariant structure maps We must also require that the iterated structure maps . be This idea is due to the third author; the first and second authors joined the project later. -equivariant.
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 Some generalizations of symmetric spectra appear in -modules.ReferenceMMSS98a. These many new symmetric monoidal categories of spectra, including and symmetric spectra, are shown to be equivalent in an appropriate sense in -modulesReferenceMMSS98b 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 does not work in general. A good approximation to such a functor is constructed in ReferenceShi.
At present the theory of of -modulesReferenceEKMM97 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 of -modulesReferenceEKMM97; 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 appear to be tied to the category of topological spaces. There are also technical differences reflecting the result of -modulesReferenceLew91 that there are limitations on any symmetric monoidal category of spectra. For example, the sphere spectrum is cofibrant in the category of symmetric spectra, but is not in the category of On the other hand, every -modules. is fibrant, a considerable technical advantage. Also, the -module of -modulesReferenceEKMM97 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.
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 i.e., symmetric spectra -spectra; such that each is a Kan complex and the adjoint of the structure map is a weak equivalence. In Section 4, we prove that the stable homotopy theories of symmetric spectra and ordinary spectra are equivalent. More precisely, we construct a Quillen equivalence of model categories between symmetric spectra and the model category of ordinary spectra described in 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.
The authors would like to thank Dan Christensen, Bill Dwyer, Phil Hirschhorn, Dan Kan, Haynes Miller, John Palmieri, Charles Rezk, and Stefan Schwede for many helpful conversations about symmetric spectra.
We now establish some notation we will use throughout the paper. Many of the categories in this paper have an enriched Hom as well as a set-valued Hom. To distinguish them: in a category the set of maps from , to is denoted in a simplicial category ; the simplicial set of maps from , to is denoted or in a category ; with an internal Hom, the object in of maps from to is denoted or In case . is the category of modules over a commutative monoid we also use , for the internal Hom.
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 is described in Section -spectra1.4.
The category has the ordered sets for as its objects and the order preserving functions as its maps. The category of simplicial sets, denoted is the category of functors from , to the category of sets. The set of of the simplicial set -simplices denoted , is the value of the functor , at The standard . -simplex is the contravariant functor Varying . gives a covariant functor By the Yoneda lemma, . and the contravariant functor is naturally isomorphic to .
Let be a discrete group. The category of sets is the category -simplicial of functors from to where , is regarded as a category with one object. A set is therefore a simplicial set -simplicial with a left simplicial i.e., a homomorphism -action,.
A basepoint of a simplicial set is a distinguished -simplex The category of pointed simplicial sets and basepoint preserving maps is denoted . The simplicial set . has a single simplex in each degree and is the terminal object in A basepoint of . is the same as a map The disjoint union . adds a disjoint basepoint to the simplicial set For example, the . is -sphere A basepoint of a . set -simplicial is a -invariant of -simplex The category of pointed . sets is denoted -simplicial.
The smash product of the pointed simplicial sets and is the quotient that collapses the simplicial subset to a point. For pointed sets -simplicial and let , be the quotient of by the diagonal action of For pointed simplicial sets . , and , there are natural isomorphisms , , and In the language of monoidal categories, the smash product is a symmetric monoidal product on the category of pointed simplicial sets. We recall the definition of symmetric monoidal product, but for more details see .ReferenceML71, VII or ReferenceBor94, 6.1.
A symmetric monoidal product on a category is: a bifunctor a unit ; and coherent natural isomorphisms ; (the associativity isomorphism), (the twist isomorphism), and (the unit isomorphism). The product is closed if the functor has a right adjoint for every A .(closed) symmetric monoidal category is a category with a (closed) symmetric monoidal product.
Coherence of the natural isomorphisms means that all reasonable diagrams built from the natural isomorphisms also commute ReferenceML71. When the product is closed, the pairing is an internal Hom. For example, the smash product on the category of pointed simplicial sets is closed. For the pointed simplicial set of maps from , to is For pointed . sets -simplicial and the simplicial subset of , pointed maps is -equivariant.
Let be the simplicial circle obtained by identifying the two vertices of ,.
A spectrum is
a sequence of pointed simplicial sets; and
a pointed map for each .
The maps are the structure maps of the spectrum. A map of spectra is a sequence of pointed maps such that the diagram
is commutative for each Let . denote the category of spectra.
Replacing the sequence of pointed simplicial sets by a sequence of pointed topological spaces in 1.2.1 gives the original definition of a spectrum (due to Whitehead and Lima). The categories of simplicial spectra and of topological spectra are discussed in the work of Bousfield and Friedlander ReferenceBF78.
A symmetric spectrum is a spectrum to which symmetric group actions have been added. Let be the group of permutations of the set with , As usual, embed . as the subgroup of with acting on the first elements of and acting on the last elements of Let . be the smash power of the simplicial circle with the left permutation action of -fold.
A symmetric spectrum is
a sequence of pointed simplicial sets;
a pointed map for each and ;
a basepoint preserving left action of on such that the composition
of the maps is for -equivariant and .
A map of symmetric spectra is a sequence of pointed maps such that is and the diagram -equivariant
is commutative for each Let . denote the category of symmetric spectra.
In part three of Definition 1.2.2, one need only assume that the maps and are equivariant; since the symmetric groups are generated by transpositions if , and are equivariant then all the maps are equivariant.
The symmetric suspension spectrum of the pointed simplicial set is the sequence of pointed simplicial sets with the natural isomorphisms as the structure maps and the diagonal action of on coming from the left permutation action on and the trivial action on The composition . is the natural isomorphism which is The symmetric sphere spectrum -equivariant. is the symmetric suspension spectrum of the -sphere; is the sequence of spheres with the natural isomorphisms as the structure maps and the left permutation action of on .
The Eilenberg-Mac Lane spectrum is the sequence of simplicial abelian groups where , is the free abelian group on the non-basepoint of -simplices We identify the basepoint with . The symmetric group . acts by permuting the generators, and one can easily verify that the evident structure maps are equivariant. One could replace by any ring.
As explained in ReferenceGH97, Section 6, many other examples of symmetric spectra arise as the of a category with cofibrations and weak equivalences as defined by Waldhausen -theoryReferenceWal85, p.330.
A symmetric spectrum with values in a simplicial category is obtained by replacing the sequence of pointed simplicial sets by a sequence of pointed objects in In particular, a topological symmetric spectrum is a symmetric spectrum with values in the simplicial category of topological spaces. .
By ignoring group actions, a symmetric spectrum is a spectrum and a map of symmetric spectra is a map of spectra. When no confusion can arise, the adjective “symmetric” may be dropped.
Let be a symmetric spectrum. The underlying spectrum is the sequence of pointed simplicial sets with the same structure maps as but ignoring the symmetric group actions. This gives a faithful functor .
Since the action of on is non-trivial for it is usually impossible to obtain a symmetric spectrum from a spectrum by letting , act trivially on However, many of the usual functors to the category of spectra lift to the category of symmetric spectra. For example, the suspension spectrum of a pointed simplicial set . is the underlying spectrum of the symmetric suspension spectrum of .
Many examples of symmetric spectra and of functors on the category of symmetric spectra are constructed by prolongation of simplicial functors.
A pointed simplicial functor or is a pointed functor -functor and a natural transformation of bifunctors such that the composition is the unit isomorphism and the diagram of natural transformations
is commutative. A pointed simplicial natural transformation, or transformation, from the -natural -functor to the -functor is a natural transformation such that .
The prolongation of a -functor is the functor defined as follows. For a symmetric spectrum, is the sequence of pointed simplicial sets with the composition as the structure map and the action of on obtained by applying the functor to the action of on Since . is a each map -functor, is equivariant and so is a symmetric spectrum. For a map of symmetric spectra, is the sequence of pointed maps Since . is an -functor, is a map of spectra. Similarly, we can prolong an transformation to a natural transformation of functors on -natural.
The category of symmetric spectra is bicomplete (every small diagram has a limit and a colimit).
For any small category the limit and colimit functors , are pointed simplicial functors; for and there is a natural isomorphism
and a natural map
A slight generalization of prolongation gives the limit and the colimit of a diagram of symmetric spectra.
In particular, the underlying sequence of the limit is and the underlying sequence of the colimit is .
For a pointed simplicial set and a symmetric spectrum prolongation of the , -functor defines the smash product and prolongation of the -functor defines the power spectrum For symmetric spectra . and the pointed simplicial set of maps from , to is .
In the language of enriched category theory, the following proposition says that the smash product is a closed action of on We leave the straightforward proof to the reader. .
Let be a symmetric spectrum. Let and be pointed simplicial sets.
There are coherent natural isomorphisms and .
is the left adjoint of the functor .
is the left adjoint of the functor .
The evaluation map is the adjoint of the identity map on The composition pairing .
is the adjoint of the composition
of two evaluation maps. In the language of enriched category theory, a category with a closed action of is the same as a tensored and cotensored The following proposition, whose proof we also leave to the reader, expresses this fact. -category.
Let , and , be symmetric spectra and let be a pointed simplicial set.
The composition pairing is associative.
The adjoint of the isomorphism is a left and a right unit of the composition pairing.
There are natural isomorphisms
The category of symmetric spectra satisfies Quillen’s axiom SM7 for simplicial model categories.
Let and be maps of pointed simplicial sets. The pushout smash product is the natural map on the pushout
induced by the commutative square
Let be a map of symmetric spectra and let be a map of pointed simplicial sets. The pushout smash product is defined by prolongation, .
Recall that a map of simplicial sets is a weak equivalence if its geometric realization is a homotopy equivalence of CW-complexes. One of the basic properties of simplicial sets, proved in ReferenceQui67, II.3, is:
Let and be monomorphisms of pointed simplicial sets. Then is a monomorphism, which is a weak equivalence if either or is a weak equivalence.
Prolongation gives a corollary for symmetric spectra. A map of symmetric spectra is a monomorphism if is a monomorphism of simplicial sets for each .
A map of symmetric spectra is a level equivalence if is a weak equivalence of simplicial sets for each .
Let be a monomorphism of symmetric spectra and let be a monomorphism of pointed simplicial sets. Then is a monomorphism, which is a level equivalence if either is a level equivalence or is a weak equivalence.
By definition, a of -simplex is a map but , and so a of -simplex is a map A . of -simplex is a simplicial homotopy from to where and are the two inclusions Simplicial homotopy generates an equivalence relation on . and the quotient is A map . is a simplicial homotopy equivalence if it has a simplicial homotopy inverse, i.e., a map such that is simplicially homotopic to the identity map on and is simplicially homotopic to the identity map on If . is a simplicial homotopy equivalence of symmetric spectra, then each of the maps is a simplicial homotopy equivalence, and so each of the maps is a weak equivalence. Every simplicial homotopy equivalence is therefore a level equivalence. The converse is false; a map can be a level equivalence and NOT a simplicial homotopy equivalence.
The stable homotopy category can be defined using and level equivalences. -spectra
A Kan complex (see Example 3.2.6) is a simplicial set that satisfies the Kan extension condition. An is a spectrum -spectrum such that for each the simplicial set is a Kan complex and the adjoint of the structure map is a weak equivalence of simplicial sets.
Let be the full subcategory of The homotopy category -spectra. is obtained from by formally inverting the level equivalences. By the results in ReferenceBF78, the category is naturally equivalent to Boardman’s stable homotopy category (or any other). Likewise, let be the full subcategory of symmetric (i.e., symmetric spectra -spectra for which is an The homotopy category -spectrum). is obtained from by formally inverting the level equivalences. Since the forgetful functor preserves and level equivalences, it induces a functor -spectra As a corollary of Theorem .4.2.5, the functor is a natural equivalence of categories. Thus the category is naturally equivalent to Boardman’s stable homotopy category. To describe an inverse of let , be the functor that takes a spectrum to the of its associated -space For any spectrum -spectrum. the symmetric spectrum , is the value of the prolongation of the -functor at the symmetric sphere spectrum the underlying sequence is ; The functor . preserves preserves level equivalences, and induces a functor -spectra, which is a natural inverse of .
The category of symmetric has major defects. It is not closed under limits and colimits, or even under pushouts and pullbacks. The smash product, defined in Section -spectra2, of symmetric is a symmetric spectrum but not an -spectra 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 -spectrum, by formally inverting the level equivalences but many symmetric spectra are not level equivalent to an One must enlarge the class of equivalences. The stable equivalences of symmetric spectra are defined in Section -spectrum.3.1. By Theorem 4.2.5, the homotopy category obtained from by inverting the stable equivalences is naturally equivalent to the stable homotopy category.
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 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 ,.
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.
Every symmetric spectrum has an underlying sequence of pointed simplicial sets with a basepoint preserving left action of on these are called symmetric sequences. In this section we define the closed symmetric monoidal category of symmetric sequences of pointed simplicial sets. ;
The category has the finite sets for ( as its objects and the automorphisms of the sets ) as its maps. Let be a category. A symmetric sequence of objects in is a functor and the category of symmetric sequences of objects in , is the functor category .
A symmetric sequence is a sequence of pointed simplicial sets with a basepoint preserving left action of on The category . is a product category. In particular, .
The category of symmetric sequences in is bicomplete.
The category is bicomplete, so the functor category is bicomplete.
The tensor product of the symmetric sequences is the symmetric sequence
The tensor product of the maps and in is given by for , and .
The tensor product of symmetric sequences has the universal property for “bilinear maps”:
Let be symmetric sequences. Then there is a natural isomorphism
The twist isomorphism for is the natural map given by for , and , where , is the given by -shuffle for and for The map defined without the shuffle permutation is not a map of symmetric sequences. .
There is another way of describing the tensor product and the twist isomorphism. The category 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 and is the functor defined on a finite set as
For an isomorphism the map is the coproduct of the isomorphisms The twist isomorphism is the map that sends the summand . of to the summand of by switching the factors.
The tensor product is a symmetric monoidal product on the category of symmetric sequences .
The unit of the tensor product is the symmetric sequence The unit isomorphism is obvious. The associativity isomorphism is induced by the associativity isomorphism in . and the natural isomorphism
The twist isomorphism is described in Remark 2.1.5. The coherence of the natural isomorphisms follows from coherence of the natural isomorphisms for the smash product in .
We now introduce several functors on the category of symmetric sequences.
The evaluation functor is given by and The free functor . is the left adjoint of the evaluation functor The smash product . of and is the symmetric sequence with the diagonal action of that is trivial on The pointed simplicial set . of maps from to is the pointed simplicial set .
For each the free symmetric sequence is , and the free functor is So, for a pointed simplicial set . , and for In particular, . , and is the unit of the tensor product .
We leave the proof of the following basic proposition to the reader.
There are natural isomorphisms:
for and .
for and .
A map of symmetric sequences is a level equivalence if each of the maps is a weak equivalence. Since is a product category, a map of symmetric sequences is a monomorphism if and only if each of the maps is a monomorphism.
Let be a symmetric sequence, let be a map of symmetric sequences and let be a map of pointed simplicial sets.
If is a monomorphism, then is a monomorphism.
If is a level equivalence, then is a level equivalence.
If is a monomorphism, then is a monomorphism for .
If is a weak equivalence, then is a level equivalence for .
Parts (1), (2) and (3) follow from the definition of and the corresponding properties for the smash product of pointed simplicial sets. For Parts (4) and (5) use the isomorphism .
By part three of Proposition 2.1.8, As . varies, is a functor and for , the symmetric sequence , is naturally isomorphic to .
Let and be symmetric sequences. The symmetric sequence of maps from to is
The tensor product is a closed symmetric monoidal product on the category of symmetric sequences.
The tensor product is a symmetric monoidal product by Lemma 2.1.6. The product is closed if there is a natural isomorphism
for symmetric sequences and .
By Proposition 2.1.4, a map of symmetric sequences is a collection of maps -equivariant This is adjoint to a collection of . maps -equivariant So there is a natural isomorphism .
By Proposition 2.1.8, the functor sending to is the functor sending to which by definition is Combining the isomorphisms gives the natural isomorphism that finishes the proof. .
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 is a commutative monoid in the category of symmetric sequences and a symmetric spectrum is a (left) -module.
Consider the symmetric sphere spectrum By Proposition .2.1.4, the natural maps -equivariant give a pairing The adjoint . of the identity map is a two-sided unit of the pairing. The diagram of natural isomorphisms
commutes, showing that is an associative pairing of symmetric sequences.
In the language of monoidal categories, is a monoid in the category of symmetric sequences and a symmetric spectrum is a left -module.
The category of symmetric spectra is naturally equivalent to the category of left -modules.
A pairing is the same as a collection of maps -equivariant If . is a left there is a spectrum for which -module, is the underlying symmetric sequence and the structure maps are the maps The compositions . are the maps -equivariant Conversely, for . a symmetric spectrum, the map of symmetric sequences corresponding to the collection of maps -equivariant where , is the natural isomorphism makes , a left These are inverse constructions and give a natural equivalence of categories. -module.
Moreover, is a commutative monoid, i.e., where , 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 having , as the unit. This gives a symmetric monoidal product on the category of The smash product -modules. of is the symmetric spectrum .
The smash product on the category of symmetric spectra is a special case of the following lemma.
Let be a symmetric monoidal category that is cocomplete and let be a commutative monoid in such that the functor preserves coequalizers. Then there is a symmetric monoidal product on the category of with -modules as the unit.
We leave the proof of this lemma to the reader; the main point is the following definition.
The smash product of symmetric spectra and is the symmetric spectrum The tensor product . is the colimit in symmetric sequences of the diagram
For a left the composition -module is the right action of since ; is commutative, the two actions commute and is an Hence, the tensor product -bimodule. is a left -module.
Apply Lemma 2.2.2 to the commutative monoid in the bicomplete category of symmetric sequences to obtain the following corollary.
The smash product is a symmetric monoidal product on the category of symmetric spectra.
Next, some important functors on the category of symmetric spectra.
The functor gives the free -module generated by the symmetric sequence For each . the evaluation functor , is given by and The free functor . is the left adjoint of the evaluation functor The functor . is the right adjoint of the evaluation functor .
The functor is left adjoint to the forgetful functor The free functor . is the composition of the left adjoints (Definition 2.1.7) and Thus, for . and the left , -module is naturally isomorphic to the left -module In particular, . is naturally isomorphic to the symmetric spectrum defined by prolongation in Section 1.3. Furthermore is the symmetric suspension spectrum of and , is the symmetric sphere spectrum For a pointed simplicial set . , is the symmetric sequence which is a left , since -module is a right -module.
We leave the proof of the following proposition to the reader.
There are natural isomorphisms:
for and .
for and .
Let be a map of pointed simplicial sets.
If is a monomorphism, then is a monomorphism.
If is a weak equivalence, then is a level equivalence.
Use the isomorphism and Proposition 2.1.9.
The internal Hom on the category of symmetric spectra is a special case of the following lemma.
Let be a closed symmetric monoidal category that is bicomplete and let be a commutative monoid in Then there is a function . -module natural for , such that the functor , is left adjoint to the functor .
Again, we leave the proof of this lemma to the reader, but the main definition follows.
Let and be symmetric spectra. The function spectrum is the limit of the diagram in
The smash product is a closed symmetric monoidal product on the category of symmetric spectra. In particular, there is a natural adjunction isomorphism
The adjunction is also a simplicial adjunction and an internal adjunction.
There are natural isomorphisms
We use Proposition 2.2.6 to give another description of the function spectrum For a symmetric spectrum . the pointed simplicial set of maps , is naturally isomorphic to The symmetric spectrum . is the -module and as varies, is a functor The symmetric sequence . is the underlying symmetric sequence of In particular, the natural isomorphism . is Applying this to -equivariant. and using Corollary 2.2.11, we find that the underlying symmetric sequence of is the symmetric sequence .
We must also describe the structure maps of from this point of view. Recall that , Let . be the adjoint of the identity map The induced map . is adjoint to the structure map The map .
is the induced map -equivariant;
Using this natural isomorphism, we find that the structure maps of are the adjoints of the maps
induced by .
For example, is the spectrum; its underlying symmetric sequence is the sequence of pointed simplicial sets -shifted
with acting on by restricting the action of to the copy of that permutes the first elements of The structure maps of the . spectrum are the structure maps -shifted of More generally, . is the spectrum of -shifted.
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.
The category 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 is the category of functors from to An object of . is a sequence of pointed simplicial sets and a map is a sequence of pointed simplicial maps .
The graded smash product of sequences and is the sequence given in degree by
The category of sequences is a bicomplete category and the graded smash product is a symmetric monoidal product on .
The sequence whose level is th is a monoid in the category of sequences. The category of left is isomorphic to the ordinary category of spectra, -modules.
The twist map on is not the identity map and thus is not a commutative monoid in In fact, . 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.
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 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.
One’s first inclination is to define stable equivalence using the forgetful functor one would like a map ; of symmetric spectra to be a stable equivalence if the underlying map of spectra is a stable equivalence, i.e., if induces an isomorphism of stable homotopy groups. The reader is warned: THIS WILL NOT WORK. Instead, stable equivalence is defined using cohomology; a map of symmetric spectra is a stable equivalence if the induced map of cohomology groups is an isomorphism for every generalized cohomology theory 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 cohomology group of the symmetric spectrum th with coefficients in the symmetric -spectrum could be defined as the set of simplicial homotopy classes of maps from , to But, even though the contravariant functor . takes simplicial homotopy equivalences to isomorphisms, 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 for which the functor behaves correctly.
An injective spectrum is a symmetric spectrum that has the extension property with respect to every monomorphism of symmetric spectra that is a level equivalence. That is, for every diagram in
where is a monomorphism and a level equivalence there is a map such that .
Some examples of injective spectra follow. Recall that is the right adjoint of the evaluation functor 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. .
If the pointed simplicial set is a Kan complex, then is an injective spectrum. If is a symmetric sequence and is an injective spectrum, then is an injective spectrum.
Since is left adjoint to the spectrum , has the extension property with respect to the monomorphism and level equivalence if and only if has the extension property with respect to the monomorphism and weak equivalence Since . is a Kan complex, it does have the extension property with respect to Hence . is injective.
Since the functor is the left adjoint of the spectrum , has the extension property with respect to the monomorphism and level equivalence if and only if has the extension property with respect to the map There is a natural isomorphism of maps of symmetric sequences . Since . is a monomorphism and a level equivalence, is a monomorphism and a level equivalence by Proposition 2.1.9. Thus is also a monomorphism and level equivalence of symmetric spectra. So is injective.
In fact, injective spectra are the fibrant objects of a model structure on 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.
A map of symmetric spectra is a stable equivalence if is an isomorphism for every injective -spectrum.
There are two other ways to define stable equivalence.
Let be a map of symmetric spectra. The following conditions are equivalent:
is an isomorphism for every injective -spectrum;
is a weak equivalence for every injective -spectrum;
is a level equivalence for every injective -spectrum.
Let be a pointed simplicial set and let be a symmetric The adjoints of the structure maps of -spectrum. are weak equivalences of Kan complexes. From Remark 2.2.12, for