Contents
Context
Model category theory
model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
for ∞-groupoids
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
Stable homotopy theory
Contents
Idea
A model structure on spectra for orthogonal spectra.
The category of orthogonal spectra is a presentation of the symmetric monoidal (∞,1)-category of spectra, with the special property that it implements the smash product of spectra such as to yield itself a symmetric monoidal model category of spectra: the model structure on orthogonal spectra. This implies in particular that with respect to this symmetric smash product of spectra an E-∞ ring is presented simply as a plain commutative monoid in orthogonal spectra.
Homotopy theory of orthogonal spectra
Categorical algebra
When defining a commutative ring as an abelian group equipped with an associative, commutative and untial bilinear pairing
one evidently makes crucial use of the tensor product of abelian groups . That tensor product itself gives the category Ab of all abelian groups a structure similar to that of a ring, namely it equips it with a pairing
that is a functor out of the product category of Ab with itself, satisfying category-theoretic analogs of the properties of associativity, commutativity and unitality.
One says that a ring is a commutative monoid in the category Ab of abelian groups, and that this concept makes sense since itself is a symmetric monoidal category.
Now in stable homotopy theory, as we have seen above, the category Ab is improved to the stable homotopy category (def. ), or rather to any stable model structure on spectra presenting it. Hence in order to correspondingly refine commutative monoids in Ab (namely commutative rings) to commutative monoids in Ho(Spectra) (namely commutative ring spectra), there needs to be a suitable symmetric monoidal category structure on the category of spectra. Its analog of the tensor product of abelian groups is to be called the symmetric monoidal smash product of spectra. The problem is how to construct it.
The theory for handling such a problem is categorical algebra. Here we discuss the minimum of categorical algebra that will allow us to elegantly construct the symmetric monoidal smash product of spectra.
Monoidal topological categories
We want to lift the concepts of ring and module from abelian groups to spectra. This requires a general idea of what it means to generalize these concepts at all. The abstract theory of such generalizations is that of monoid in a monoidal category.
We recall the basic definitions of monoidal categories and of monoids and modules internal to monoidal categories. We list archetypical examples at the end of this section, starting with example below. These examples are all fairly immediate. The point of the present discussion is to construct the non-trivial example of Day convolution monoidal stuctures below.
Definition
A (pointed) topologically enriched monoidal category is a (pointed) topologically enriched category (def.) equipped with
-
a (pointed) topologically enriched functor (def.)
out of the (pointed) topologival product category of with itself (def. ), called the tensor product,
-
an object
called the unit object or tensor unit,
-
a natural isomorphism (def.)
called the associator,
-
a natural isomorphism
called the left unitor, and a natural isomorphism
called the right unitor,
such that the following two kinds of diagrams commute, for all objects involved:
-
triangle identity:
-
the pentagon identity:
Lemma
(Kelly 64)
Let be a monoidal category, def. . Then the left and right unitors and satisfy the following conditions:
-
;
-
for all objects the following diagrams commutes:
and
For proof see at monoidal category this lemma and this lemma.
Definition
A (pointed) topological braided monoidal category, is a (pointed) topological monoidal category (def. ) equipped with a natural isomorphism
called the braiding, such that the following two kinds of diagrams commute for all objects involved (“hexagon identities”):
and
where denotes the components of the associator of .
Definition
Given a (pointed) topological symmetric monoidal category with tensor product (def. ) it is called a closed monoidal category if for each the functor has a right adjoint, denoted
hence if there are natural bijections
for all objects .
Since for the case that is the tensor unit of this means that
the object is an enhancement of the ordinary hom-set to an object in . Accordingly, it is also called the internal hom between and .
In a closed monoidal category, the adjunction isomorphism between tensor product and internal hom even holds internally:
Proof
Let be any object. By applying the defining natural bijections twice, there are composite natural bijections
Since this holds for all , the Yoneda lemma (the fully faithfulness of the Yoneda embedding) says that there is an isomorphism . Moreover, by taking in the above and using the left unitor isomorphisms and we get a commuting diagram
Example
The category Ab of abelian groups, regarded as enriched in discrete topological spaces, becomes a symmetric monoidal category with tensor product the actual tensor product of abelian groups and with tensor unit the additive group of integers. Again the associator, unitor and braiding isomorphism are the evident ones coming from the underlying sets, as in example .
This is a closd monoidal cagory? with internal hom being the set of homomorphisms equipped with the pointwise group structure for then .
This is the archetypical case that motivates the notation “” for the pairing operation in a monoidal category:
Example
The category category of chain complexes , equipped with the tensor product of chain complexes is a symmetric monoidal category (def. ).
In this case the braiding has a genuinely non-trivial aspect to it, beyond just the swapping of coordinates as in examples , and def. , namely for then
and in these components the braiding isomorphism is that of Ab, but with a minus sign thrown in whener two odd-graded components are commuted.
This is a first shadow of the graded-commutativity that also exhibited by spectra.
(e.g. Hovey 99, prop. 4.2.13)
Algebras and modules
Definition
Given a (pointed) topological monoidal category , then a monoid internal to is
-
an object ;
-
a morphism (called the unit)
-
a morphism (called the product);
such that
-
(associativity) the following diagram commutes
where is the associator isomorphism of ;
-
(unitality) the following diagram commutes:
where and are the left and right unitor isomorphisms of .
Moreover, if has the structure of a symmetric monoidal category (def. ) with symmetric braiding , then a monoid as above is called a commutative monoid in if in addition
A homomorphism of monoids is a morphism
in , such that the following two diagrams commute
and
Write for the category of monoids in , and for its subcategory of commutative monoids.
Example
Given a (pointed) topological monoidal category , then the tensor unit is a monoid in (def. ) with product given by either the left or right unitor
By lemma , these two morphisms coincide and define an associative product with unit the identity .
If is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.
Example
Given a symmetric monoidal category (def. ), and given two commutative monoids (def. {MonoidsInMonoidalCategory}), then the tensor product becomes itself a commutative monoid with unit morphism
(where the first isomorphism is, (lemma )) and with product morphism given by
(where we are notationally suppressing the associators and where denotes the braiding of ).
That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of , and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.
Definition
Given a (pointed) topological monoidal category (def. ), and given a monoid in (def. ), then a left module object in over is
-
an object ;
-
a morphism (called the action);
such that
-
(unitality) the following diagram commutes:
where is the left unitor isomorphism of .
-
(action property) the following diagram commutes
A homomorphism of left -module objects
is a morphism
in , such that the following diagram commutes:
For the resulting category of modules of left -modules in with -module homomorphisms between them, we write
This is naturally a (pointed) topologically enriched category itself.
Example
The archetypical case in which all these abstract concepts reduce to the basic familiar ones is the symmetric monoidal category Ab of abelian groups from example .
-
A monoid in (def. ) is equivalently a ring.
-
A commutative monoid in in (def. ) is equivalently a commutative ring .
-
An -module object in (def. ) is equivalently an -module;
-
The tensor product of -module objects (def. ) is the standard tensor product of modules.
-
The category of module objects (def. ) is the standard category of modules .
Proposition
In the situation of def. , the monoid canonically becomes a left module over itself by setting . More generally, for any object, then naturally becomes a left -module by setting:
The -modules of this form are called free modules.
The free functor constructing free -modules is left adjoint to the forgetful functor which sends a module to the underlying object .
Proof
A homomorphism out of a free -module is a morphism in of the form
fitting into the diagram (where we are notationally suppressing the associator)
Consider the composite
i.e. the restriction of to the unit “in” . By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor)
Pasting this square onto the top of the previous one yields
where now the left vertical composite is the identity, by the unit law in . This shows that is uniquely determined by via the relation
This natural bijection between and establishes the adjunction.
Definition
Given a (pointed) topological closed symmetric monoidal category (def. , def. ), given a commutative monoid in (def. ), and given and two left -module objects (def.), then
-
the tensor product of modules is, if it exists, the coequalizer
and if preserves these coequalizers, then this is equipped with the left -action induced from the left -action on
-
the function module is, if it exists, the equalizer
equipped with the left -action that is induced by the left -action on via
(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8)
Proposition
Given a (pointed) topological closed symmetric monoidal category (def. , def. ), and given a commutative monoid in (def. ). If all coequalizers exist in , then the tensor product of modules from def. makes the category of modules into a symmetric monoidal category, with tensor unit the object itself, regarded as an -module via prop. .
If moreover all equalizers exist, then this is a closed monoidal category (def. ) with internal hom given by the function modules of def. .
(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8)
Proof sketch
The associators and braiding for are induced directly from those of and the universal property of coequalizers. That is the tensor unit for follows with the same kind of argument that we give in the proof of example below.
Example
For a monoid (def. ) in a symmetric monoidal category (def. ), the tensor product of modules (def. ) of two free modules (def. ) and always exists and is the free module over the tensor product in of the two generators:
Hence if has all coequalizers, so that the category of modules is a monoidal category (prop. ) then the free module functor (def. ) is a strong monoidal functor (def. )
Proof
It is sufficient to show that the diagram
is a coequalizer diagram (we are notationally suppressing the associators), hence that , hence that the claim holds for and .
To that end, we check the universal property of the coequalizer:
First observe that indeed coequalizes with , since this is just the associativity clause in def. . So for any other morphism with this property, we need to show that there is a unique morphism which makes this diagram commute:
We claim that
where the first morphism is the inverse of the right unitor of .
First to see that this does make the required triangle commute, consider the following pasting composite of commuting diagrams
Here the the top square is the naturality of the right unitor, the middle square commutes by the functoriality of the tensor product and the definition of the product category (def. ), while the commutativity of the bottom square is the assumption that coequalizes with .
Here the right vertical composite is , while, by unitality of , the left vertical composite is the identity on , Hence the diagram says that , which we needed to show.
It remains to see that is the unique morphism with this property for given . For that let be any other morphism with . Then consider the commuting diagram
where the top left triangle is the unitality condition and the two isomorphisms are the right unitor and its inverse. The commutativity of this diagram says that .
Definition
Given a monoidal category of modules as in prop. , then a monoid in (def. ) is called an -algebra.
Propposition
Given a monoidal category of modules in a monoidal category as in prop. , and an -algebra (def. ), then there is an equivalence of categories
between the category of commutative monoids in and the coslice category of commutative monoids in under , hence between commutative -algebras in and commutative monoids in that are equipped with a homomorphism of monoids .
(e.g. EKMM 97, VII lemma 1.3)
Proof
In one direction, consider a -algebra with unit and product . There is the underlying product
By considering a diagram of such coequalizer diagrams with middle vertical morphism , one find that this is a unit for and that is a commutative monoid in .
Then consider the two conditions on the unit . First of all this is an -module homomorphism, which means that
commutes. Moreover it satisfies the unit property
By forgetting the tensor product over , the latter gives
where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be pasted to the square above, to yield a commuting square
This shows that the unit is a homomorphism of monoids .
Now for the converse direction, assume that and are two commutative monoids in with a monoid homomorphism. Then inherits a left -module structure by
By commutativity and associativity it follows that coequalizes the two induced morphisms . Hence the universal property of the coequalizer gives a factorization through some . This shows that is a commutative -algebra.
Finally one checks that these two constructions are inverses to each other, up to isomorphism.
Topological ends and coends
For working with pointed topologically enriched functors, a certain shape of limits/colimits is particularly relevant: these are called (pointed topological enriched) ends and coends. We here introduce these and then derive some of their basic properties, such as notably the expression for topological left Kan extension in terms of coends (prop. below). Further below it is via left Kan extension along the ordinary smash product of pointed topological spaces (“Day convolution”) that the symmetric monoidal smash product of spectra is induced.
Definition
Let be pointed topologically enriched categories (def.), i.e. enriched categories over from example .
-
The pointed topologically enriched opposite category is the topologically enriched category with the same objects as , with hom-spaces
and with composition given by braiding followed by the composition in :
-
the pointed topological product category is the topologically enriched category whose objects are pairs of objects with and , whose hom-spaces are the smash product of the separate hom-spaces
and whose composition operation is the braiding followed by the smash product of the separate composition operations:
Example
A pointed topologically enriched functor (def.) into (exmpl.) out of a pointed topological product category as in def.
(a “pointed topological bifunctor”) has component maps of the form
By functoriality and under passing to adjuncts (cor.) this is equivalent to two commuting actions
and
In the special case of a functor out of the product category of some with its opposite category (def. )
then this takes the form of a “pullback action” in the first variable
and a “pushforward action” in the second variable
Definition
Let be a small pointed topologically enriched category (def.), i.e. an enriched category over from example . Let
be a pointed topologically enriched functor (def.) out of the pointed topological product category of with its opposite category, according to def. .
-
The coend of , denoted , is the coequalizer in (prop., exmpl., prop., cor.) of the two actions encoded in via example :
-
The end of , denoted , is the equalizer in (prop., exmpl., prop., cor.) of the adjuncts of the two actions encoded in via example :
Example
Let be a topological group. Write for the pointed topologically enriched category that has a single object , whose single hom-space is ( with a basepoint freely adjoined (def.))
and whose composition operation is the product operation in under adjoining basepoints (exmpl.)
Then a topologically enriched functor
is a pointed topological space equipped with a continuous function
satisfying the action property. Hence this is equivalently a continuous and basepoint-preserving left action (non-linear representation) of on .
The opposite category (def. ) comes from the opposite group
(The canonical continuous isomorphism induces a canonical euqivalence of topologically enriched categories .)
So a topologically enriched functor
is equivalently a basepoint preserving continuous right action of .
Therefore the coend of two such functors (def. ) coequalizes the relation
(where juxtaposition denotes left/right action) and hence is equivalently the canonical smash product of a right -action with a left -action, hence the quotient of the plain smash product by the diagonal action of the group :
Proof
The underlying pointed set functor preserves all limits (prop., prop., prop.). Therefore there is an equalizer diagram in of the form
Here the object in the middle is just the set of collections of component morphisms . The two parallel maps in the equalizer diagram take such a collection to the functions which send any to the result of precomposing
and of postcomposing
each component in such a collection, respectively. These two functions being equal, hence the collection being in the equalizer, means precisley that for all and all the square
is a commuting square. This is precisley the condition that the collection be a natural transformation.
Conversely, example says that ends over bifunctors of the form constitute hom-spaces between pointed topologically enriched functors:
Definition
Let be a small pointed topologically enriched category (def.). Define the structure of a pointed topologically enriched category on the category of pointed topologically enriched functors to (exmpl.) by taking the hom-spaces to be given by the ends (def. ) of example :
The composition operation on these is defined to be the one induced by the composite maps
where the first, morphism is degreewise given by projection out of the limits that defined the ends. This composite evidently equalizes the two relevant adjunct actions (as in the proof of example ) and hence defines a map into the end
The resulting pointed topologically enriched category is also called the -enriched functor category over with coefficients in .
This yields an equivalent formulation in terms of ends of the pointed topologically enriched Yoneda lemma (prop.):
Proposition
(topologically enriched Yoneda lemma)
Let be a small pointed topologically enriched categories (def.). For a pointed topologically enriched functor (def.) and for an object, there is a natural isomorphism
between the hom-space of the pointed topological functor category, according to def. , from the functor represented by to , and the value of on .
In terms of the ends (def. ) defining these hom-spaces, this means that
In this form the statement is also known as Yoneda reduction.
The proof of prop. is formally dual to the proof of the next prop. .
Now that natural transformations are expressed in terms of ends (example ), as is the Yoneda lemma (prop. ), it is natural to consider the dual statement involving coends:
Proposition
(co-Yoneda lemma)
Let be a small pointed topologically enriched category (def.). For a pointed topologically enriched functor (def.) and for an object, there is a natural isomorphism
Moreover, the morphism that hence exhibits as the coequalizer of the two morphisms in def. is componentwise the canonical action
which is adjunct to the component map of the topologically enriched functor .
(e.g. MMSS 00, lemma 1.6)
Proof
The coequalizer of pointed topological spaces that we need to consider has underlying it a coequalizer of underlying pointed sets (prop., prop., prop.). That in turn is the colimit over the diagram of underlying sets with the basepointe adjoined to the diagram (prop.). For a coequalizer diagram adding that extra point to the diagram clearly does not change the colimit, and so we need to consider the plain coequalizer of sets.
That is just the set of equivalence classes of pairs
where two such pairs
are regarded as equivalent if there exists
such that
(Because then the two pairs are the two images of the pair under the two morphisms being coequalized.)
But now considering the case that and , so that shows that any pair
is identified, in the coequalizer, with the pair
hence with .
This shows the claim at the level of the underlying sets. To conclude it is now sufficient (prop.) to show that the topology on is the final topology (def.) of the system of component morphisms
which we just found. But that system includes
which is a retraction
and so if all the preimages of a given subset of the coequalizer under these component maps is open, it must have already been open in .
It is this analogy that gives the name to the following statement:
Proposition
(Fubini theorem for (co)-ends)
For a pointed topologically enriched bifunctor on a small pointed topological product category (def. ), i.e.
then its end and coend (def. ) is equivalently formed consecutively over each variable, in either order:
and
Proof
Because limits commute with limits, and colimits commute with colimits.
With this coend calculus in hand, there is an elegant proof of the defining universal property of the smash tensoring of topologically enriched functors (def.)
Proposition
For a pointed topologically enriched category, there are natural isomorphisms
and
for all and all .
In particular there is the combined natural isomorphism
exhibiting a pair of adjoint functors
Proof
Via the end-expression for from def. and the fact (remark ) that the pointed mapping space construction preserves ends in the second variable, this reduces to the fact that is the internal hom in the closed monoidal category (example ) and hence satisfies the internal tensor/hom-adjunction isomorphism (prop. ):
and
Proposition
(left Kan extension via coends)
Let be small pointed topologically enriched categories (def.) and let
be a pointed topologically enriched functor (def.). Then precomposition with constitutes a functor
. This functor has a left adjoint , called left Kan extension along
which is given objectwise by a coend (def. ):
Proof
Use the expression of natural transformations in terms of ends (example and def. ), then use the respect of for ends/coends (remark ), use the smash/mapping space adjunction (cor.), use the Fubini theorem (prop. ) and finally use Yoneda reduction (prop. ) to obtain a sequence of natural isomorphisms as follows:
Topological Day convolution
Given two functions on a group (or just a monoid) , then their convolution product is, whenever well defined, given by the sum
The operation of Day convolution is the categorification of this situation where functions are replaced by functors and monoids by monoidal categories. Further below we find the symmetric monoidal smash product of spectra as the Day convolution of topologically enriched functors over the monoidal category of finite pointed CW-complexes, or over sufficiently rich subcategories thereof.
Definition
Let be a small pointed topological monoidal category (def. ).
Then the Day convolution tensor product on the pointed topological enriched functor category (def. ) is the functor
out of the pointed topological product category (def. ) given by the following coend (def. )
Example
Let denote the category with objects the natural numbers, and only the zero morphisms and identity morphisms on these objects (we consider this in a braoder context below in def. ):
Regard this as a pointed topologically enriched category in the unique way. The operation of addition of natural numbers makes this a monoidal category.
An object is an -sequence of pointed topological spaces. Given two such, then their Day convolution according to def. is
We observe now that Day convolution is equivalently a left Kan extension (def. ). This will be key for understanding monoids and modules with respect to Day convolution.
Definition
Let be a small pointed topologically enriched category (def.). Its external tensor product is the pointed topologically enriched functor
from pairs of topologically enriched functors over to topologically enriched functors over the product category (def. ) given by
i.e.
Proposition
For a pointed topologically enriched monoidal category (def. ) the Day convolution product (def. ) of two functors is equivalently the left Kan extension (def. ) of their external tensor product (def. ) along the tensor product : there is a natural isomorphism
Hence the adjunction unit is a natural transformation of the form
This perspective is highlighted in (MMSS 00, p. 60).
Proof
By prop. we may compute the left Kan extension as the following coend:
Proposition implies the following fact, which is the key for the identification of “functors with smash product” below and then for the description of ring spectra further below.
Corollary
The operation of Day convolution (def. ) is universally characterized by the property that there are natural isomorphisms
where is the external product of def. , hence that natural transformations of functors on of the form
are in natural bijection with natural transformations of functors on the product category (def. ) of the form
Write
for the -Yoneda embedding, so that for any object, is the corepresented functor .
Proposition
For a small pointed topological monoidal category (def. ), the Day convolution tensor product of def. makes the pointed topologically enriched functor category
into a pointed topological monoidal category (def. ) with tensor unit co-represented by the tensor unit of .
Moreover, if is equipped with a (symmetric) braiding (def. ), then so is .
Proof
Regarding associativity, observe that
where we used the Fubini theorem for coends (prop. ) and then twice the co-Yoneda lemma (prop. ). Similarly
So we obtain an associator by combining, in the integrand, the associator of and of (example ):
It is clear that this satisfies the pentagon identity, since and do.
To see that is the tensor unit for , use the Fubini theorem for coends (prop. ) and then twice the co-Yoneda lemma (prop. ) to get for any that
Hence the right unitor of Day convolution comes from the unitor of under the integral sign:
Analogously for the left unitor. Hence the triangle identity for follows from the triangle identity in under the integral sign.
Similarly, if has a braiding , it induces a braiding under the integral sign:
and the hexagon identity for follows from that for and
Moreover:
Proposition
For a small pointed topological symmetric monoidal category (def. ), the monoidal category with Day convolution from def. is a closed monoidal category (def. ). Its internal hom is given by the end (def. )
Proof
Using the Fubini theorem (def. ) and the co-Yoneda lemma (def. ) and in view of definition of the enriched functor category, there is the following sequence of natural isomorphisms:
Proposition
In the situation of def. , the Yoneda embedding constitutes a strong monoidal functor (def. )
Proof
That the tensor unit is respected is part of prop. . To see that the tensor product is respected, apply the co-Yoneda lemma (prop. ) twice to get the following natural isomorphism
Functors with smash product
Since the symmetric monoidal smash product of spectra discussed below is an instance of Day convolution (def. ), and since ring spectra are going to be the monoids (def. ) with respect to this tensor product, we are interested in characterizing the monoids with respect to Day convolution. These turn out to have a particularly transparent expression as what is called functors with smash product, namely lax monoidal functors from the base monoidal category to . Their components are pairing maps of the form
satisfying suitable conditions. This is the form in which the structure of ring spectra usually appears in examples. It is directly analogous to how a dg-algebra, which is equivalently a monoid with respect to the tensor product of chain complexes (example ), is given in components .
Here we introduce the concepts of monoidal functors and of functors with smash product and prove that they are equivalently the monoids with respect to Day convolution.
Definition
Let and be two (pointed) topologically enriched monoidal categories (def. ). A topologically enriched lax monoidal functor between them is
-
a topologically enriched functor
-
a morphism
-
a natural transformation
for all
satisfying the following conditions:
-
(associativity) For all objects the following diagram commutes
where and denote the associators of the monoidal categories;
-
(unitality) For all the following diagrams commutes
and
where , , , denote the left and right unitors of the two monoidal categories, respectively.
If and alll are isomorphisms, then is called a strong monoidal functor.
If moreover and are equipped with the structure of braided monoidal categories (def. ) with braidings and , respectively, then the lax monoidal functor is called a braided monoidal functor if in addition the following diagram commutes for all objects
A homomorphism between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation of the underlying functors
compatible with the product and the unit in that the following diagrams commute for all objects :
and
We write for the resulting category of lax monoidal functors between monoidal categories and , similarly for the category of braided monoidal functors between braided monoidal categories, and for the category of braided monoidal functors between symmetric monoidal categories.
Proposition
For two composable lax monoidal functors (def. ) between monoidal categories, then their composite becomes a lax monoidal functor with structure morphisms
and
Proposition
Let and be two monoidal categories (def. ) and let be a lax monoidal functor (def. ) between them.
Then for a monoid in (def. ), its image becomes a monoid by setting
(where the first morphism is the structure morphism of ) and setting
(where again the first morphism is the corresponding structure morphism of ).
This construction extends to a functor
from the category of monoids of (def. ) to that of .
Moreover, if and are symmetric monoidal categories (def. ) and is a braided monoidal functor (def. ) and is a commutative monoid (def. ) then so is , and this construction extends to a functor
Proof
This follows immediately from combining the associativity and unitality (and symmetry) constraints of with those of .
Definition
Let and be two (pointed) topologically enriched monoidal categories (def. ), and let be a topologically enriched lax monoidal functor between them, with product operation .
Then a left module over the lax monoidal functor is
-
a topologically enriched functor
-
a natural transformation
such that
-
(action property) For all objects the following diagram commutes
A homomorphism between two modules over a monoidal functor is
- a natural transformation of the underlying functors
compatible with the action in that the following diagram commutes for all objects :
We write for the resulting category of modules over the monoidal functor .
Now we may finally state the main proposition on functors with smash product:
Proposition
Let be a pointed topologically enriched (symmetric) monoidal category (def. ). Regard as a topological symmetric monoidal category as in example .
Then (commutative) monoids in (def. ) the Day convolution monoidal category of prop. are equivalent to (braided) lax monoidal functors (def. ) of the form
called functors with smash products on , i.e. there are equivalences of categories of the form
Moreover, module objects over these monoid objects are equivalent to the corresponding modules over monoidal functors (def. ).
This is stated in some form in (Day 70, example 3.2.2). It is highlighted again in (MMSS 00, prop. 22.1).
Proof
By definition , a lax monoidal functor is a topologically enriched functor equipped with a morphism of pointed topological spaces of the form
and equipped with a natural system of maps of pointed topological spaces of the form
for all .
Under the Yoneda lemma (prop. ) the first of these is equivalently a morphism in of the form
Moreover, under the natural isomorphism of corollary the second of these is equivalently a morphism in of the form
Translating the conditions of def. satisfied by a lax monoidal functor through these identifications gives precisely the conditions of def. on a (commutative) monoid in object under .
Similarly for module objects and modules over monoidal functors.
Proposition
Let be a lax monoidal functor (def. ) between pointed topologically enriched monoidal categories (def. ). Then the induced functor
given by preserves monoids under Day convolution
Moreover, if and are symmetric monoidal categories (def. ) and is a braided monoidal functor (def. ), then also preserves commutative monoids
Similarly, for
any fixed monoid, then sends -modules to -modules
Proof
This is an immediate corollary of prop. , since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor by prop. .
-Modules
We give a unified discussion of the categories of
-
sequential spectra
-
symmetric spectra
-
orthogonal spectra
-
pre-excisive functors
(all in topological spaces) as categories of modules with respect to Day convolution monoidal structures on Top-enriched functor categories over restrictions to faithful sub-sites of the canonical representative of the sphere spectrum as a pre-excisive functor on .
This approach is due to (Mandell-May-Schwede-Shipley 00) following (Hovey-Shipley-Smith 00).
Pre-Excisive functors
We consider an almost tautological construction of a pointed topologically enriched category equipped with a closed symmetric monoidal product: the category of pre-excisive functors. Then we show that this tautological category restricts, in a certain sense, to the category of sequential spectra. However, under this restriction the symmetric monoidal product breaks, witnessing the lack of a functorial smash product of spectra on sequential spectra. However from inspection of this failure we see that there are categories of structured spectra “in between” those of all pre-excisive functors and plain sequential spectra, notably the categories of orthogonal spectra and of symmetric spectra. These intermediate categories retain the concrete tractable nature of sequential spectra, but are rich enough to also retain the symmetric monoidal product inherited from pre-excisive functors: this is the symmetric monoidal smash product of spectra that we are after.
Literature (MMSS 00, Part I and Part III)
Definition
Write
for the full subcategory of pointed compactly generated topological spaces (def.) on those that admit the structure of a finite CW-complex (a CW-complex (def.) with a finite number of cells).
We say that the pointed topological enriched functor category (def. )
is the category of pre-excisive functors. (We had previewed this in Part P, this example).
Write
for the functor co-represented by 0-sphere. This is equivalently the inclusion itself:
We call this the standard incarnation of the sphere spectrum as a pre-excisive functor.
By prop. the smash product of pointed compactly generated topological spaces induces the structure of a closed (def. ) symmetric monoidal category (def. )
with
-
tensor unit the sphere spectrum ;
-
tensor product the Day convolution product from def. ,
called the symmetric monoidal smash product of spectra for the model of pre-excisive functors;
-
internal hom the dual operation from prop. ,
called the mapping spectrum construction for pre-excisive functors.
Lemma
Identified as a functor with smash product under prop. , the pre-excisive sphere spectrum from def. is given by the identity natural transformation
Proof
We claim that this is in fact the unique structure of a monoidal functor that may be imposed on the canonical inclusion , hence it must be the one in question. To see the uniqueness, observe that naturality of the matural transformation in particular says that there are commuting squares of the form
where the vertical morphisms pick any two points in and , respectively, and where the top morphism is necessarily the canonical identification since there is only one single isomorphism , namely the identity. This shows that the bottom horizontal morphism has to be the identity on all points, hence has to be the identity.
We now consider restricting the domain of the pre-excisive functors of def. .
Definition
Define the following pointed topologically enriched (def.) symmetric monoidal categories (def. ):
-
is the category whose objects are the natural numbers and which has only identity morphisms and zero morphisms on these objects, hence the hom-spaces are
The tensor product is the addition of natural numbers, , and the tensor unit is 0. The braiding is, necessarily, the identity.
-
is the standard skeleton of the core of FinSet with zero morphisms adjoined: its objects are the finite sets for (hence is the empty set), all non-zero morphisms are automorphisms and the automorphism group of is the symmetric group on elements, hence the hom-spaces are the following discrete topological spaces:
The tensor product is the disjoint union of sets, tensor unit is the empty set. The braiding
is given by the canonical permutation in that shuffles the first elements past the remaining elements
(MMSS 00, example 4.2)
-
has as objects the finite dimenional real linear inner product spaces and as non-zero morphisms the linear isometric isomorphisms between these; hence the automorphism group of the object is the orthogonal group ; the monoidal product is direct sum of linear spaces, the tensor unit is the 0-vector space; again we turn this into a -enriched category by adjoining a basepoint to the hom-spaces;
The tensor product is the direct sum of linear inner product spaces, tensor unit is the 0-vector space. The braiding
is the canonical orthogonal transformation that switches the summands.
(MMSS 00, example 4.4)
Notice that in the notation of example
-
the full subcategory of on is ;
-
the full subcategory of on is ;
-
the full subcategory of on is .
Moreover, after discarding the zero morphisms, then these categories are the disjoint union of categories of the form , and , respectively.
There is a sequence of canonical faithful pointed topological subcategory inclusions
into the pointed topological category of pointed compactly generated topological spaces of finite CW-type (def. ).
Here denotes the one-point compactification of . On morphisms is the canonical inclusion of permutation matrices into orthogonal matrices and is on the topological subspace inclusions of the pointed homeomorphisms that are induced under forming one-point compactification from linear isometries of (“representation spheres”).
Below we will often use these identifications to write just “” for any of these objects, leaving implicit the identifications .
Consider the pointed topological diagram categries (def. , exmpl.) over these categories:
-
is called the category of sequences of pointed topological spaces (e.g. HSS 00, def. 2.3.1);
-
is called the category of symmetric sequences (e.g. HSS 00, def. 2.1.1);
-
is called the category of orthogonal sequences.
Consider the sequence of restrictions of topological diagram categories, according to prop. along the above inclusions:
Write
for the restriction of the excisive functor incarnation of the sphere spectrum (from def. ) along these inclusions.
Proposition
The functors , and in def. become strong monoidal functors (def. ) when equipped with the canonical isomorphisms
and
and
Moreover, and are braided monoidal functors (def. ) (hence symmetric monoidal functors, remark ). But is not braided monoidal.
Proof
The first statement is clear from inspection.
For the second statement it is sufficient to observe that all the nontrivial braiding of n-spheres in is given by the maps induced from exchanging coordinates in the realization of -spheres as one-point compactifications of Cartesian spaces . This corresponds precisely to the action of the symmetric group inside the orthogonal group acting via the canonical action of the orthogonal group on . This shows that and are braided, for they include precisely these objects (the -spheres) with these braidings on them. Finally it is clear that is not braided, because the braiding on is trivial, while that on is not, so necessrily fails to preserve precisely these non-trivial isomorphisms.
Explicitly:
Lemma
The monoids from def. are, when identified as functors with smash product via prop. given by assigning
respectively, with product given by the canonical isomorphisms
Proof
By construction these functors with smash products are the composites, according to prop. , of the monoidal functors , , , respectively, with the lax monoidal functor corresponding to . The former have as structure maps the canonical identifications by definition, and the latter has as structure map the canonical identifications by lemmma .
Proposition
There is an equivalence of categories
which identifies the category of modules (def. ) over the monoid (remark ) in the Day convolution monoidal structure (prop. ) over the topological functor category from def. with the category of sequential spectra (def.)
Under this equivalence, an -module is taken to the sequential pre-spectrum whose component spaces are the values of the pre-excisive functor on the standard n-sphere
and whose structure maps are the images of the action morphisms
under the isomorphism of corollary
evaluated at
(Hovey-Shipley-Smith 00, prop. 2.3.4)
Proof
After unwinding the definitions, the only point to observe is that due to the action property,
any -action
is indeed uniquely fixed by the components of the form
This is because under corollary the action property is identified with the componentwise property
where the left vertical morphism is an isomorphism by the nature of . Hence this fixes the components to be the -fold composition of the structure maps .
However, since, by remark , is not commutative, there is no tensor product induced on under the identification in prop. . But since and are commutative monoids by remark , it makes sense to consider the following definition.
Definition
In the terminology of remark we say that
is the category of orthogonal spectra; and that
is the category of symmetric spectra.
By remark and by prop. these categories canonically carry a symmetric monoidal tensor product and , respectively. This we call the symmetric monoidal smash product of spectra. We usually just write for short
and
In the next section we work out what these symmetric monoidal categories of orthogonal and of symmetric spectra look like more explicitly.
Symmetric and orthogonal spectra
We now define symmetric spectra and orthogonal spectra and their symmetric monoidal smash product. We proceed by giving the explicit definitions and then checking that these are equivalent to the abstract definition from above.
Literature. ( Hovey-Shipley-Smith 00, section 1, section 2, Schwede 12, chapter I)
Definition
A topological symmetric spectrum is
-
a sequence of pointed compactly generated topological spaces;
-
a basepoint preserving continuous right action of the symmetric group on ;
-
a sequence of morphisms
such that
-
for all the composite
intertwines the -action.
A homomorphism of symmetric spectra is
- a sequence of maps
such that
-
each intetwines the -action;
-
the following diagrams commute
We write for the resulting category of symmetric spectra.
(Hovey-Shipley-Smith 00, def. 1.2.2, Schwede 12, I, def. 1.1)
The definition of orthogonal spectra has the same structure, just with the symmetric groups replaced by the orthogonal groups.
Definition
A topological orthogonal spectrum is
-
a sequence of pointed compactly generated topological spaces;
-
a basepoint preserving continuous right action of the orthogonal group on ;
-
a sequence of morphisms
such that
-
for all the composite
intertwines the -action.
A homomorphism of orthogonal spectra is
- a sequence of maps
such that
-
each intetwines the -action;
-
the following diagrams commute
We write for the resulting category of orthogonal spectra.
(e.g. Schwede 12, I, def. 7.2)
Proposition
Definitions and are indeed equivalent to def. :
orthogonal spectra are euqivalently the module objects over the incarnation of the sphere spectrum
and symmetric spectra sre equivalently the module objects over the incarnation of the sphere spectrum
(Hovey-Shipley-Smith 00, prop. 2.2.1)
Proof
We discuss this for symmetric spectra. The proof for orthogonal spectra is of the same form.
First of all, by example an object in is equivalently a “symmetric sequence”, namely a sequence of pointed topological spaces , for , equipped with an action of (def. ).
By corollary and lemma , the structure morphism of an -module object on
is equivalently (as a functor with smash products) a natural transformation
over . This means equivalently that there is such a morphism for all and that it is -equivariant.
Hence it only remains to see that these natural transformations are uniquely fixed once the one for is given. To that end, observe that lemma says that in the following commuting squares (exhibiting the action property on the level of functors with smash product, where we are notationally suppressing the associators) the left vertical morphisms are isomorphisms:
This says exactly that the action of has to be the composite of the actions of followed by that of . Hence the statement follows by induction.
Finally, the definition of homomorphisms on both sides of the equivalence are just so as to preserve precisely this structure, hence they conincide under this identification.
Definition
Given two symmetric spectra, def. , then their smash product of spectra is the symmetric spectrum
with component spaces the coequalizer
where has components given by the structure maps
while has components given by the structure maps conjugated by the braiding in and the permutation action (that shuffles the element on the right to the left)
Finally The structure maps of are those induced under the coequalizer by
Analogously for orthogonal spectra.
(Schwede 12, p. 82)
(Schwede 12, E.1.16)
Proof
By def. the abstractly defined tensor product of two -modules and is the coequalizer
The Day convolution product appearing here is over the category from def. . By example and unwinding the definitions, this is for any two symmetric spectra and given degreewise by the wedge sum of component spaces summing to that total degree, smashed with the symmetric group with basepoint adjoined and then quotiented by the diagonal action of the symmetric group acting on the degrees separately:
This establishes the form of the coequalizer diagram. It remains to see that under this identification the two abstractly defined morphisms are the ones given in def. .
To see this, we apply the adjunction isomorphism between the Day convolution product and the external tensor product (cor. ) twice, to find the following sequence of equivalent incarnations of morphisms:
This establishes the form of the morphism . By the same reasoning as in the proof of prop. , we may restrict the coequalizer to without changing it.
The form of the morphism is obtained by the analogous sequence of identifications of morphisms, now with the parenthesis to the left. That it involves and the permutation action as shown above follows from the formula for the braiding of the Day convolution tensor product from the proof of prop. :
by translating it to the components of the precomposition
via the formula from the proof of prop. for the left Kan extension (prop. ):
This last expression is the function on morphisms which precomposes components under the coend with the braiding in topological spaces and postcomposes them with the image of the functor of the braiding in . But the braiding in is, by def. , given by the respective shuffle permutations , and by prop. the image of these under is via the given -action on .
Finally to see that the structure map is as claimed: By prop. the structure morphisms are the degree-1 components of the -action, and by prop. the -action on a tensor product of -modules is induced via the action on the left tensor factor.
Definition
A commutative symmetric ring spectrum is
-
a sequence of component spaces for ;
-
a basepoint preserving continuous left action of the symmetric group on ;
-
for all a multiplication map
(a morphism in )
-
two unit maps
such that
-
(equivariance) intertwines the -action;
-
(associativity) for all the following diagram commutes (where we are notationally suppressing the associators of )
-
(unitality) for all the following diagram commutes
and
where the diagonal morphisms and are the left and right unitors in , respectively.
-
(commutativity) for all the following diagram commutes
where the top morphism is the braiding in (def. ) and where denotes the permutation action which shuffles the first elements past the last elements.
A homomorphism of symmetric commutative ring spectra is a sequence of -equivariant pointed continuous functions such that the following diagrams commute for all
and and .
Write
for the resulting category of symmetric commutative ring spectra.
We regard a symmetric ring spectrum in particular as a symmetric spectrum (def. ) by taking the structure maps to be
This defines a forgetful functor
There is an analogous definition of orthogonal ring spectrum and we write
for the category that these form.
(e.g. Schwede 12, def. 1.3)
We discuss examples below in a dedicated section Examples.
Proposition
The symmetric (orthogonal) commutative ring spectra in def. are equivalently the commutative monoids in (def. ) the symmetric monoidal category () of def. with respect to the symmetric monoidal smash product of spectra (). Hence there are equivalences of categories
and
Moreover, under these identifications the canonical forgetful functor
and
coincides with the forgetful functor defined in def. .
Proof
We discuss this for symmetric spectra. The proof for orthogonal spectra is directly analogous.
By prop. and def. , the commutative monoids in are equivalently commtutative monoids in equipped with a homomorphism of monoids . In turn, by prop. this are equivalently braided lax monoidal functors (which we denote by the same symbols, for convenience) of the form
equipped with a monoidal natural transformation (def. )
The structure morphism of such a lax monoidal functor has as components precisely the morphisms . In terms of these, the associativity and braiding condition on the lax monoidal functor are manifestly the above associativity and commutativity conditions.
Moreover, by the proof of prop. the -module structure on an an -algebra has action given by
which shows, via the identification in prop , that the forgetful functors to underlying symmetric spectra coincide as claimed.
Hence it only remains to match the nature of the units. The above unit morphism has components
for all , and the unitality condition for and is manifestly as in the statement above.
We claim that the other components are uniquely fixed by these:
By lemma , the product structure in is by isomorphisms , so that the commuting square for the coherence condition of this monoidal natural transformation
says that . This means that is uniquely fixed once and are given.
Finally it is clear that homomorphisms on both sides of the equivalence precisely respect all this structure under both sides of the equivalence.
Similarly:
Definition
Given a symmetric (orthogonal) commutative ring spectrum (def. ), then a left symmetric (orthogonal) module spectrum over is
-
a sequence of component spaces for ;
-
a basepoint preserving continuous left action of the symmetric group on ;
-
for all an action map
(a morphism in )
such that
-
(equivariance) intertwines the -action;
-
(action property) for all the following diagram commutes (where we are notationally suppressing the associators of )
-
(unitality) for all the following diagram commutes
A homomorphism of left -module spectra is a sequence of pointed continuous functions such that for all the following diagrams commute:
We write
for the resulting category of symmetric (orthogonal) -module spectra.
(e.g. Schwede 12, I, def. 1.5)
Proposition
Under the identification, from prop. , of commutative ring spectra with commutative monoids with respect to the symmetric monoidal smash product of spectra, the -module spectra of def. are equivalently the left module objects (def. ) over the respective monoids, i.e. there are equivalences of categories
and
where on the right we have the categories of modules from def. .
Proof
The proof is directly analogous to that of prop. . Now prop. and prop. give that the module objects in question are equivalently modules over a monoidal functor (def. ) and writing these out in components yields precisely the above structures and properties.
As diagram spectra
In Introduction to Stable homotopy theory – 1-1 we obtained the strict/level model structure on topological sequential spectra by identifying the category of sequential spectra with a category of topologically enriched functors with values in (prop.) and then invoking the general existence of the projective model structure on functors (thm.).
Here we discuss the analogous construction for the more general structured spectra from above.
Proposition
Let be a topologically enriched monoidal category (def. ), and let be a monoid in (def. ) the pointed topological Day convolution monoidal category over from prop. .
Then the category of left A-modules (def. ) is a pointed topologically enriched functor category category (exmpl.)
over the category of free modules over (prop. ) on objects in (under the Yoneda embedding ). Hence the objects of are identified with those of , and its hom-spaces are
(MMSS 00, theorem 2.2)
Proof
Use the identification from prop. of with a lax monoidal functor and of any -module object as a functor with the structure of a module over a monoidal functor, given by natural transformations
Notice that these transformations have just the same structure as those of the enriched functoriality of (def.) of the form
Hence we may unify these two kinds of transformations into a single kind of the form
and subject to certain identifications.
Now observe that the hom-objects of have just this structure:
Here we used first the free-forgetful adjunction of prop. , then the enriched Yoneda lemma (prop. ), then the coend-expression for Day convolution (def. ) and finally the co-Yoneda lemma (prop. ).
Then define a topologically enriched category to have objects and hom-spaces those of as above, and whose composition operation is defined as follows:
where
-
the equivalence is braiding in the integrand (and the Fubini theorem, prop. );
-
the first morphism is, in the integrand, the smash product of
-
forming the tensor product of hom-objects of with the identity morphism on ;
-
the monoidal functor incarnation of the monoid structure on ;
-
the second morphism is, in the integrand, given by composition in ;
-
the last morphism is the morphism induced on coends by regarding extranaturality in and separately as a special case of extranaturality in (and then renaming).
With this it is fairly straightforward to see that
because, by the above definition of composition, functoriality over manifestly encodes the -action property together with the functoriality over .
This way we are reduced to showing that actually .
But by construction, the image of the objects of under the Yoneda embedding are precisely the free -modules over objects of :
Since the Yoneda embedding is fully faithful, this shows that indeed
Example
For the sequential case in def. , then the opposite category of free modules on objects in over (def.) is identified as the category (def.):
Accordingly, in this case prop. reduces to the identification (prop.) of sequential spectra as topological diagrams over :
Proof
There is one object for each . Moreover, from the expression in the proof of prop. we compute the hom-spaces between these to be
These are the objects and hom-spaces of the category . It is straightforward to check that the definition of composition agrees, too.
Stable weak homotopy equivalences
We consider the evident version of stable weak homotopy equivalences for structured spectra and prove a few technical lemmas about them that are needed in the proof of the stable model structure below
Definition
For one of the shapes of structured spectra from def. , let be the corresponding category of structured spectra (def. , prop. , def. ).
-
The stable homotopy groups of an object are those of the underlying sequential spectrum (def.):
-
An object is a structured Omega-spectrum if the underlying sequential spectrum (def. ) is a sequential Omega spectrum (def.)
-
A morphism in is a stable weak homotopy equivalence (or: -isomorphism) if the underlying morphism of sequential spectra is a stable weak homotopy equivalence of sequential spectra (def.);
-
a morphism is a stable cofibration if it is a cofibration in the strict model structure from prop. .
(MMSS 00, def. 8.3 with the notation from p. 21, Mandell-May 02, III, def. 3.1, def. 3.2)
Lemma
Given a morphism in , then there are long exact sequences of stable homotopy groups (def. ) of the form
and
where denotes the mapping cone and the mapping cocone of (def.) formed with respect to the standard cylinder spectrum hence formed degreewise with respect to the standard reduced cylinder of pointed topological spaces.
(MMSS 00, theorem 7.4 (vi))
Proof
Since limits and colimits in the diagram category are computed objectwise, the functor that restricts -modules to their underlying sequential spectra preserves both limits and colimits, hence it is sufficient to consider the statement for sequential spectra.
For the first case, there is degreewise the long exact sequence of homotopy groups to the left of pointed topological spaces (exmpl.)
Observe that the sequential colimit that defines the stable homotopy groups (def.) preserves exact sequences of abelian groups, because generally filtered colimits in Ab are exact functors (prop.). This implies that by taking the colimit over in the above sequences, we obtain a long exact sequence of stable homotopy groups as shown.
Now use that in sequential spectra the canonical morphism morphism is a stable weak homotopy equivalence and is compatible with the map (prop.) so that there is a commuting diagram of the form
Since the top sequence is exact, and since all vertical morphisms are isomorphisms, it follows that also the bottom sequence is exact.
Proof
Since limits and colimits in the diagram category are computed objectwise, the functor that restricts -modules to their underlying sequential spectra preserves both limits and colimits, and it also preserves smash tensoring. Hence it is sufficient to consider the statement for sequential spectra.
Fist consider the case of a finite cell complex .
Write
for the stages of the cell complex , so that for each there is a pushout diagram in of the form
Equivalently these are pushoutdiagrams in of the form
Notice that it is indeed that appears in the top right, not .
Now forming the smash tensoring of any morphism in by the morphisms in the pushout on the right yields a commuting diagram in of the form
Here the horizontal morphisms on the left are degreewise cofibrations in , hence the morphism on the right is degreewise their homotopy cofiber. This way lemma implies that there are commuting diagrams
where the top and bottom are long exact sequences of stable homotopy groups.
Now proceed by induction. For then clearly smash tensoring with preserves stable weak homotopy equivalences. So assume that smash tensoring with does, too. Observe that preserves stable weak homotopy equivalences, since is a stable weak homotopy equivalence (lemma). Hence in the above the two vertical morphisms on the left and the two on the right are isomorphism. Now the five lemma implies that also is an isomorphism.
Finally, the statement for a non-finite cell complex follows with these arguments and then using that spheres are compact and hence maps out of them into a transfinite composition factor through some finite stage (prop.).
Lemma
The pushout in of a stable weak homotopy equivalence along a morphism that is degreewise a cofibration in is again a stable weak homotopy equivalence.
Proof
Given a pushout square
observe that the pasting law implies an isomorphism between the horizontal cofibers
Moreover, since cofibrations in are preserves by pushout, and since pushout of spectra are computed degreewise, both the top and the bottom horizontal sequences here are degreewise homotopy cofiber sequence in . Hence lemma applies and gives a commuting diagram
where the top and the bottom row are both long exact sequences of stable homotopy groups. Hence the claim now follows by the five lemma.
Free spectra and Suspension spectra
The concept of free spectrum is a generalization of that of suspension spectrum. In fact the stable homotopy types of free spectra are precisely those of iterated loop space objects of suspension spectra. But for the development of the theory what matters is free spectra before passing to stable homotopy types, for as such they play the role of the basic cells for the stable model structures on spectra analogous to the role of the n-spheres in the classical model structure on topological spaces (def. below).
Moreover, while free sequential spectra are just re-indexed suspension spectra, free symmetric spectra and free orthogonal spectra in addition come with suitably freely generated actions of the symmetric group and the orthogonal group. It turns out that this is not entirely trivial; it leads to a subtle issue (lemma below) where the adjuncts of certain canonical inclusions of free spectra are stable weak homotopy equivalences for sequential and orthogonal spectra, but not for symmetric spectra.
Definition
For any one of the four diagram shapes of def. , and for each , the functor
that sends a structured spectrum to the th component space of its underlying sequential spectrum has, by prop. , a left adjoint
This is called the free structured spectrum-functor.
For the special case it is also called the structured suspension spectrum functor and denoted
(Hovey-Shipley-Smith 00, def. 2.2.5, MMSS 00, section 8)
Lemma
Let be any one of the four diagram shapes of def. . Then
-
the free spectrum on (def. ) is equivalently the smash tensoring with (def.) of the free module (def. ) over (remark ) on the representable
-
on its value is given by the following coend expression (def. )
In particular the structured sphere spectrum is the free spectrum in degree 0 on the 0-sphere:
and generally for then
is the smash tensoring of the strutured sphere spectrum with .
(Hovey-Shipley-Smith 00, below def. 2.2.5, MMSS00, p. 7 with theorem 2.2)
Proof
Under the equivalence of categories
from prop. , the expression for is equivalently the smash tensoring with of the functor that represents over :
(by fully faithfulness of the Yoneda embedding).
This way the first statement is a special case of the following general fact: For a pointed topologically enriched category, and for any object, then there is an adjunction
(saying that evaluation at is right adjoint to smash tensoring the functor represented by ) witnessed by the following composite natural isomorphism:
The first is the characteristic isomorphism of tensoring from prop. , while the second is the enriched Yoneda lemma of prop. .
From this, the second statement follows by the proof of prop. .
For the last statement it is sufficient to observe that is the tensor unit under Day convolution by prop. (since is the tensor unit in ), so that
(e.g. Schwede 12, example 3.20)
Proof
With the formula in item 2 of lemma we have for the case of orthogonal spectra
where in the second line we used that the coend collapses to ranging in the full subcategory
on the object and then we applied example . The case of symmetric spectra is verbatim the same, with the symmetric group replacing the orthogonal group, and the case of sequential spectra is again verbatim the same, with the orthogonal group replaced by the trivial group.
Proof
We consider the case of orthogonal spectra. The case of symmetric spectra works verbatim the same, and the case of sequential spectra is tivial.
By prop. we have to show that for all the topological spaces of the form
admit the structure of CW-complexes.
To that end, use the homeomorphism
which is a diffeomorphism away from the basepoint and hence such that the action of the orthogonal group induces a smooth action on (which happens to be constant on ).
Also observe that we may think of the above quotient by the group action
as being the quotient by the diagonal action
given by
Using this we may rewrite the space in question as
Here has the structure of a smooth manifold with boundary and equipped with a smooth action of the compact Lie group . Under these conditions (Illman 83, corollary 7.2) states that admits the structure of a G-CW complex for , and moreover (Illman 83, line above theorem 7.1) states that this may be chosen such that the boundary is a -CW subcomplex.
Now the quotient of a -CW complex by is a CW complex, and so the last expression above exhibits the quotient of a CW-complex by a subcomplex, hence exhibits CW-complex structure.
Proposition
Let be the diagram shape of either pre-excisive functors, orthogonal spectra or symmetric spectra. Then under the symmetric monoidal smash product of spectra (def. , def. , def.) the free structured spectra of def. behave as follows
In particular for structured suspension spectra (def. ) this gives isomorphisms
Hence the structured suspension spectrum functor is a strong monoidal functor (def. ) and in fact a braided monoidal functor (def. ) from pointed topological spaces equipped with the smash product of pointed objects, to structured spectra equipped with the symmetric monoidal smash product of spectra
More generally, for then
where on the right we have the smash tensoring of with .
(MMSS 00, lemma 1.8 with theorem 2.2, Mandell-May 02, prop. 2.2.6)
Proof
By lemma the free spectra are free modules over the structured sphere spectrum of the form . By example the tensor product of such free modules is given by
Using the co-Yoneda lemma (prop. ) the expression on the right is
For the last statement we may use that , by lemma ), and that is the tensor unit for by prop. .
To see that is braided, write . We need to see that
commutes. Chasing the smash factors through this diagram and using symmetry (def. ) and the hexagon identities (def. ) shows that indeed it does.
One use of free spectra is that they serve to co-represent adjuncts of structure morphisms of spectra. To this end, first consider the following general existence statement.
Lemma
For each there exists a morphism
between free spectra (def. ) such that for every structured spectrum precomposition forms a commuting diagram of the form
where the horizontal equivalences are the adjunction isomorphisms and the canonical identification, and where the right morphism is the -adjunct of the structure map of the sequential spectrum underlying (def. ).
Proof
Since all prescribed morphisms in the diagram are natural transformations, this is in fact a diagram of copresheaves on
With this the statement follows by the Yoneda lemma.
Now we say explicitly what these maps are:
Definition
For , write
for the adjunct under the (free structured spectrum -component)-adjunction in def. of the composite morphism
where the first morphism is via prop. and the second comes from the adjunction units according to def. .
(MMSS 00, def. 8.4, Schwede 12, example 4.26)
Lemma
The morphisms of def. are those whose existence is asserted by prop. .
(MMSS 00, lemma 8.5, following Hovey-Shipley-Smith 00, remark 2.2.12)
Proof
Consider the case and . All other cases work analogously.
By lemma , in this case the morphism has components like so:
Now for any sequential spectrum, then a morphism is uniquely determined by its 0th components (that’s of course the very free property of ); as the compatibility with the structure maps forces the first component, in particular, to be :
But that first component is just the component that similarly determines the precompositon of with , hence is fully fixed as being the map . Therefore is the function
It remains to see that this is the -adjunct of . By the general formula for adjuncts, this is
To compare to the above, we check what this does on points: is sent to the composite
To identify this as a map we use the adjunction isomorphism once more to throw all the -s on the right back to -s the left, to finally find that this is indeed
(Hovey-Shipley-Smith 00, example 3.1.10, MMSS 00, lemma 8.6, Schwede 12, example 4.26)
Proof
This follows by inspection of the explicit form of the maps, via prop. . We discuss each case separately:
sequential case
Here the components of the morphism eventually stabilize to isomorphisms
and this immediately gives that is an isomorphism on stable homotopy groups.
orthogonal case
Here for the -component of is the quotient map
By the suspension isomorphism for stable homotopy groups, is a stable weak homotopy equivalence precisely if any of its suspensions is. Hence consider instead , whose -component is
Now due to the fact that -action on lifts to an -action, the quotients of the diagonal action of equivalently become quotients of just the left action. Formally this is due to the existence of the commuting diagram
which says that the image of any in the quotient is labeled by .
It follows that is the smash product of a projection map of coset spaces with the identity on the sphere:
Now finally observe that this projection function
is -connected (see here). Hence its smash product with is -connected.
The key here is the fast growth of the connectivity with . This implies that for each there exists such that becomes an isomorphism. Hence is a stable weak homotopy equivalence and therefore so is .
symmetric case
Here the morphism has the same form as in the orthogonal case above, except that all occurences of orthogonal groups are replaced by just their sub-symmetric groups.
Accordingly, the analysis then proceeds entirely analogously, with the key difference that the projection
does not become highly connected as increases, due to the discrete topological space underlying the symmetric group. Accordingly the conclusion now is the opposite: is not a stable weak homotopy equivalence in this case.
Another use of free spectra is that their pushout products may be explicitly analyzed, and checking the pushout-product axiom for general cofibrations may be reduced to checking it on morphisms between free spectra.
Proof
By lemma the commuting diagram defining the pushout product of free spectra
is equivalent to this diagram:
Since the free spectrum construction is a left adjoint, it preserves pushouts, and so
where in the second step we used this lemma.
The strict model structure on structured spectra
Theorem
The four categories of
-
pre-excisive functors;
-
orthogonal spectra;
-
symmetric spectra;
-
sequential spectra
(from def. , prop. , def. ) each admit a model category structure (def.) whose weak equivalences and fibrations are those morphisms which induce on all component spaces weak equivalences or fibrations, respectively, in the classical model structure on pointed topological spaces . (thm., prop.). These are called the strict model structures (or level model structures) on structured spectra.
Moreover, under the equivalences of categories of prop. and prop. , the restriction functors in def. constitute right adjoints of Quillen adjunctions (def.) between these model structures:
(MMSS 00, theorem 6.5)
Proof
By prop. all four categories are equivalently categories of pointed topologically enriched functors
and hence the existence of the model structures with componentwise weak equivalences and fibrations is a special case of the general existence of the projective model structure on enriched functors (thm.).
The three restriction functors each have a left adjoint by topological left Kan extension (prop. ).
Moreover, the three right adjoint restriction functors are along inclusions of objects, hence evidently preserve componentwise weak equivalences and fibrations. Hence these are Quillen adjunctions.
Definition
Recall the sets
of generating cofibrations and generating acyclic cofibrations, respectively, of the classical model structure on pointed topological spaces (def.)
Write
for the set of images under forming free spectra, def. , on the morphisms in from above. Similarly, write
for the set of images under forming free spectra of the morphisms in .
Proposition
The sets and from def. are, respectively sets of generating cofibrations and generating acyclic cofibrations that exhibit the strict model structure from theorem as a cofibrantly generated model category (def.).
(MMSS 00, theorem 6.5)
Proof
By theorem the strict model structure is equivalently the projective pointed model structure on topologically enriched functors
of the opposite of the category of free spectra on objects in .
By the general discussion in Part P – Classical homotopy theory (this theorem) the projective model structure on functors is cofibrantly generated by the smash tensoring of the representable functors with the elements in and . By the proof of lemma , these are precisely the morphisms of free spectra in and , respectively.
Topological enrichment
By the general properties of the projective model structure on topologically enriched functors, theorem implies that the strict model category of structured spectra inherits the structure of an enriched model category, enriched over the classical model structure on pointed topological spaces. This proceeds verbatim as for sequential spectra (in part 1.1 – Topological enrichement), but for ease of reference we here make it explicit again.
Definition
Let one of the shapes for structured spectra from def. .
Let be a morphism in (as in prop. ) and let a morphism in .
Their pushout product with respect to smash tensoring is the universal morphism
in
where
denotes the smash tensoring of pointed topologically enriched functors with pointed topological spaces (def.)
Dually, their pullback powering is the universal morphism
in
where
denotes the smash powering (def.).
Finally, for and both morphisms in , then their pullback powering is the universal morphism
in
where now is the hom-space functor of from def. .
Proposition
The operations of forming pushout products and pullback powering with respect to smash tensoring in def. is compatible with the strict model structure on structured spectra from theorem and with the classical model structure on pointed topological spaces (thm., prop.) in that pushout product takes two cofibrations to a cofibration, and to an acyclic cofibration if at least one of the inputs is acyclic, and pullback powering takes a fibration and a cofibration to a fibration, and to an acylic one if at least one of the inputs is acyclic:
Dually, the pullback powering (def. ) satisfies
Proof
The statement concering the pullback powering follows directly from the analogous statement for topological spaces (prop.) by the fact that, via theorem , the fibrations and weak equivalences in are degree-wise those in , and since smash tensoring and powering is defined degreewise. From this the statement about the pushout product follows dually by Joyal-Tierney calculus (prop.).
We record some immediate consequences of prop. that will be useful.
Proposition
Let be a retract of a cell complex (def.), then the smash-tensoring/powering adjunction from prop. is a Quillen adjunction (def.) for the strict model structure from theorem
Proof
By assumption, is a cofibrant object in the classical model structure on pointed topological spaces (thm., prop.), hence is a cofibration in . Observe then that the the pushout product of any morphism with is equivalently the smash tensoring of with :
This way prop. implies that preserves cofibrations and acyclic cofibrations, hence is a left Quillen functor.
Lemma
Let be a structured spectrum, regarded in the strict model structure of theorem .
-
The smash powering of with the standard topological interval (exmpl.) is a good path space object (def.)
-
If is cofibrant, then its smash tensoring with the standard topological interval (exmpl.) is a good cylinder object (def.)
Proof
It is clear that we have weak equivalences as shown ( is even a homotopy equivalence), what requires proof is that the path object is indeed good in that is a fibration, and the cylinder object is indeed good in that is indeed a cofibration.
For the first statement, notice that the pullback powering (def. ) of into the terminal morphism is the same as the powering :
But since every object in is fibrant, so that is a fibration, and since is a relative cell complex inclusion and hence a cofibration in , prop. says that is a fibration.
Dually, observe that
Hence if is assumed to be cofibrant, so that is a cofibration, then prop. implies that is a cofibration.
Proposition
For a structured spectrum, any morphism of structured spectra, and for a morphism of pointed topological spaces, then the hom-spaces of def. (via prop. ) interact with the pushout-product and pullback-powering from def. in that there is a natural isomorphism
Proof
Since the pointed compactly generated mapping space functor (exmpl.)
takes colimits in the first argument to limits (cor.) and ends in the second argument to ends (remark ), and since limits and colimits in are computed objectswise (this prop. via prop. ) this follows with the end-formula for the mapping space (def. ):
Proposition
For two structured spectra with cofibrant in the strict model structure of def. , then there is a natural bijection
between the connected components of the hom-space (def. via prop. ) and the hom-set in the homotopy category (def.) of the strict model structure from theorem .
Proof
By prop. the path components of the hom-space are the left homotopy classes of morphisms of structured spectra with respect to the standard cylinder spectrum :
Moreover, by lemma the degreewise standard reduced cylinder of structured spectra is a good cylinder object on in . Hence hom-sets in the strict homotopy category out of a cofibrant into a fibrant object are given by standard left homotopy classes of morphisms
(this lemma). Since is cofibrant by assumption and since every object is fibrant in , this is the case. Hence the notion of left homotopy here is that seen by the standard interval, and so the claim follows.
Monoidal model structure
We now combine the concepts of model category (def.) and monoidal category (def. ).
Given a category that is equipped both with the structure of a monoidal category and of a model category, then one may ask whether these two structures are compatible, in that the left derived functor (def.) of the tensor product exists to equip also the homotopy category with the structure of a monoidal category. If so, then one may furthermore ask if the localization functor is a monoidal