on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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.
When defining a commutative ring as an abelian group $A$ equipped with an associative, commutative and untial bilinear pairing
one evidently makes crucial use of the tensor product of abelian groups $\otimes_{\mathbb{Z}}$. 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 $A$ is a commutative monoid in the category Ab of abelian groups, and that this concept makes sense since $Ab$ 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 $Ho(Spectra)$ (def. \ref{TheStableHomotopyCategory}), 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.
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 1 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.
A (pointed) topologically enriched monoidal category is a (pointed) topologically enriched category $\mathcal{C}$ (def.) equipped with
a (pointed) topologically enriched functor (def.)
out of the (pointed) topologival product category of $\mathcal{C}$ with itself (def. 9), called the tensor product,
an object
called the unit object or tensor unit,
called the associator,
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:
(Kelly 64)
Let $(\mathcal{C}, \otimes, 1)$ be a monoidal category, def. 1. Then the left and right unitors $\ell$ and $r$ satisfy the following conditions:
$\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;
for all objects $x,y \in \mathcal{C}$ the following diagrams commutes:
and
For proof see at monoidal category this lemma and this lemma.
Just as for an associative algebra it is sufficient to demand $1 a = a$ and $a 1 = a$ and $(a b) c = a (b c)$ in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a coherence theorem for monoidal categories which states that all ways of composing the unitors and associators in a monoidal category (def. 1) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an associative algebra altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.
A (pointed) topological braided monoidal category, is a (pointed) topological monoidal category $\mathcal{C}$ (def. 1) 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 $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.
A (pointed) topological symmetric monoidal category is a (pointed) topological braided monoidal category (def. 2) for which the braiding
satisfies the condition:
for all objects $x, y$
Given a (pointed) topological symmetric monoidal category $\mathcal{C}$ with tensor product $\otimes$ (def. 3) it is called a closed monoidal category if for each $Y \in \mathcal{C}$ the functor $Y \otimes(-)\simeq (-)\otimes Y$ has a right adjoint, denoted $hom(Y,-)$
hence if there are natural bijections
for all objects $X,Z \in \mathcal{C}$.
Since for the case that $X = 1$ is the tensor unit of $\mathcal{C}$ this means that
the object $hom(Y,Z) \in \mathcal{C}$ is an enhancement of the ordinary hom-set $Hom_{\mathcal{C}}(Y,Z)$ to an object in $\mathcal{C}$. Accordingly, it is also called the internal hom between $Y$ and $Z$.
In a closed monoidal category, the adjunction isomorphism between tensor product and internal hom even holds internally:
In a symmetric closed monoidal category (def. 4) there are natural isomorphisms
whose image under $Hom_{\mathcal{C}}(1,-)$ are the defining natural bijections of def. 4.
Let $A \in \mathcal{C}$ be any object. By applying the defining natural bijections twice, there are composite natural bijections
Since this holds for all $A$, the Yoneda lemma (the fully faithfulness of the Yoneda embedding) says that there is an isomorphism $hom(X\otimes Y, Z) \simeq hom(X,hom(Y,Z))$. Moreover, by taking $A = 1$ in the above and using the left unitor isomorphisms $A \otimes (X \otimes Y) \simeq X \otimes Y$ and $A\otimes X \simeq X$ we get a commuting diagram
The category Set of sets and functions between them, regarded as enriched in discrete topological spaces, becomes a symmetric monoidal category according to def. 3 with tensor product the Cartesian product $\times$ of sets. The associator, unitor and braiding isomorphism are the evident (almost unnoticable but nevertheless nontrivial) canonical identifications.
Similarly the category $Top_{cg}$ of compactly generated topological spaces (def.) becomes a symmetric monoidal category with tensor product the corresponding Cartesian products, hence the operation of forming k-ified (cor.) product topological spaces (exmpl.). The underlying functions of the associator, unitor and braiding isomorphisms are just those of the underlying sets, as above.
Symmetric monoidal categories, such as these, for which the tensor product is the Cartesian product are called Cartesian monoidal categories.
Both examples are closed monoidal categories (def. 4), with internal hom the mapping spaces (prop.).
The category $Top_{cg}^{\ast/}$ of pointed compactly generated topological spaces with tensor product the smash product $\wedge$ (def.)
is a symmetric monoidal category (def. 3) with unit object the pointed 0-sphere $S^0$.
The components of the associator, the unitors and the braiding are those of Top as in example 1, descended to the quotient topological spaces which appear in the definition of the smash product. This works for pointed compactly generated spaces (but not for general pointed topological spaces) by this prop..
The category $Top^{\ast/}_{cg}$ is also a closed monoidal category (def. 4), with internal hom the pointed mapping space $Maps(-,-)_\ast$ (exmpl.)
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 $\otimes_{\mathbb{Z}}$ and with tensor unit the additive group $\mathbb{Z}$ of integers. Again the associator, unitor and braiding isomorphism are the evident ones coming from the underlying sets, as in example 1.
This is a closd monoidal cagory? with internal hom $hom(A,B)$ being the set of homomorphisms $Hom_{Ab}(A,B)$ equipped with the pointwise group structure for $\phi_1, \phi_2 \in Hom_{Ab}(A,B)$ then $(\phi_1 + \phi_2)(a) \coloneqq \phi_1(a) + \phi_2(b) \; \in B$.
This is the archetypical case that motivates the notation “$\otimes$” for the pairing operation in a monoidal category:
The category category of chain complexes $Ch_\bullet$, equipped with the tensor product of chain complexes is a symmetric monoidal category (def. 3).
In this case the braiding has a genuinely non-trivial aspect to it, beyond just the swapping of coordinates as in examples 1, 2 and def. 3, namely for $X, Y \in Ch_\bullet$ 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)
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is
an object $A \in \mathcal{C}$;
a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)
a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);
such that
(associativity) the following diagram commutes
where $a$ is the associator isomorphism of $\mathcal{C}$;
(unitality) the following diagram commutes:
where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.
Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a symmetric monoidal category (def. 3) $(\mathcal{C}, \otimes, 1, B)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, B)$ if in addition
(commutativity) the following diagram commutes
A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism
in $\mathcal{C}$, such that the following two diagrams commute
and
Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, then the tensor unit $1$ is a monoid in $\mathcal{C}$ (def. 5) with product given by either the left or right unitor
By lemma 1, these two morphisms coincide and define an associative product with unit the identity $id \colon 1 \to 1$.
If $(\mathcal{C}, \otimes , 1)$ is a symmetric monoidal category (def. 3), then this monoid is a commutative monoid.
Given a symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. 3), and given two commutative monoids $(E_i, \mu_i, e_i)$ $i \in \{1,2\}$ (def. {MonoidsInMonoidalCategory}), then the tensor product $E_1 \otimes E_2$ becomes itself a commutative monoid with unit morphism
(where the first isomorphism is, $\ell_1^{-1} = r_1^{-1}$ (lemma 1)) and with product morphism given by
(where we are notationally suppressing the associators and where $\tau$ denotes the braiding of $\mathcal{C} ).
That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of $(E_i,\mu_i, e_i)$, and from the hexagon identities for the braiding (def. 2) and from symmetry of the braiding.
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$ (def. 1), and given $(A,\mu,e)$ a monoid in $(\mathcal{C}, \otimes, 1)$ (def. 5), then a left module object in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is
an object $N \in \mathcal{C}$;
a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);
such that
(unitality) the following diagram commutes:
where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.
(action property) the following diagram commutes
A homomorphism of left $A$-module objects
is a morphism
in $\mathcal{C}$, such that the following diagram commutes:
For the resulting category of modules of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write
This is naturally a (pointed) topologically enriched category itself.
Given a monoidal category $(\mathcal{C},\otimes, 1)$ (def. 1) with the tensor unit $1$ regarded as a monoid in a monoidal category via example 5, then the left unitor
makes every object $C \in \mathcal{C}$ into a left module, according to def. 6, over $C$. The action property holds due to lemma 1. This gives an equivalence of categories
of $\mathcal{C}$ with the category of modules over its tensor unit.
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 3.
A monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. 5) is equivalently a ring.
A commutative monoid in in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. 5) is equivalently a commutative ring $R$.
An $R$-module object in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. 6) is equivalently an $R$-module;
The tensor product of $R$-module objects (def. 7) is the standard tensor product of modules.
The category of module objects $R Mod(Ab)$ (def. 7) is the standard category of modules $R Mod$.
Closely related to the example 8, but closer to the structure we will see below for spectra, are monoids in the category of chain complexes $(Ch_\bullet, \otimes, \mathbb{Z})$ from example 4. These monoids are equivalently differential graded algebras.
In the situation of def. 6, the monoid $(A,\mu, e)$ canonically becomes a left module over itself by setting $\rho \coloneqq \mu$. More generally, for $C \in \mathcal{C}$ any object, then $A \otimes C$ naturally becomes a left $A$-module by setting:
The $A$-modules of this form are called free modules.
The free functor $F$ constructing free $A$-modules is left adjoint to the forgetful functor $U$ which sends a module $(N,\rho)$ to the underlying object $U(N,\rho) \coloneqq N$.
A homomorphism out of a free $A$-module is a morphism in $\mathcal{C}$ of the form
fitting into the diagram (where we are notationally suppressing the associator)
Consider the composite
i.e. the restriction of $f$ to the unit “in” $A$. 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 $A$. This shows that $f$ is uniquely determined by $\tilde f$ via the relation
This natural bijection between $f$ and $\tilde f$ establishes the adjunction.
Given a (pointed) topological closed symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. 3, def. 4), given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. 5), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-module objects (def.5), then
the tensor product of modules $N_1 \otimes_A N_2$ is, if it exists, the coequalizer
and if $A \otimes (-)$ preserves these coequalizers, then this is equipped with the left $A$-action induced from the left $A$-action on $N_1$
the function module $hom_A(N_1,N_2)$ is, if it exists, the equalizer
equipped with the left $A$-action that is induced by the left $A$-action on $N_2$ via
(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8)
Given a (pointed) topological closed symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. 3, def. 4), and given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. 5). If all coequalizers exist in $\mathcal{C}$, then the tensor product of modules $\otimes_A$ from def. 7 makes the category of modules $A Mod(\mathcal{C})$ into a symmetric monoidal category, $(A Mod, \otimes_A, A)$ with tensor unit the object $A$ itself, regarded as an $A$-module via prop. 2.
If moreover all equalizers exist, then this is a closed monoidal category (def. 4) with internal hom given by the function modules $hom_A$ of def. 7.
(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8)
The associators and braiding for $\otimes_{A}$ are induced directly from those of $\otimes$ and the universal property of coequalizers. That $A$ is the tensor unit for $\otimes_{A}$ follows with the same kind of argument that we give in the proof of example 10 below.
For $(A,\mu,e)$ a monoid (def. 5) in a symmetric monoidal category $(\mathcal{C},\otimes, 1)$ (def. 1), the tensor product of modules (def. 7) of two free modules (def. 2) $A\otimes C_1$ and $A \otimes C_2$ always exists and is the free module over the tensor product in $\mathcal{C}$ of the two generators:
Hence if $\mathcal{C}$ has all coequalizers, so that the category of modules is a monoidal category $(A Mod, \otimes_A, A)$ (prop. 3) then the free module functor (def. 2) is a strong monoidal functor (def. 14)
It is sufficient to show that the diagram
is a coequalizer diagram (we are notationally suppressing the associators), hence that $A \otimes_A A \simeq A$, hence that the claim holds for $C_1 = 1$ and $C_2 = 1$.
To that end, we check the universal property of the coequalizer:
First observe that $\mu$ indeed coequalizes $id \otimes \mu$ with $\mu \otimes id$, since this is just the associativity clause in def. 5. So for $f \colon A \otimes A \longrightarrow Q$ any other morphism with this property, we need to show that there is a unique morphism $\phi \colon A \longrightarrow Q$ which makes this diagram commute:
We claim that
where the first morphism is the inverse of the right unitor of $\mathcal{C}$.
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 $\otimes \;\colon\; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C}$ and the definition of the product category (def. 9), while the commutativity of the bottom square is the assumption that $f$ coequalizes $id \otimes \mu$ with $\mu \otimes id$.
Here the right vertical composite is $\phi$, while, by unitality of $(A,\mu ,e)$, the left vertical composite is the identity on $A$, Hence the diagram says that $\phi \circ \mu = f$, which we needed to show.
It remains to see that $\phi$ is the unique morphism with this property for given $f$. For that let $q \colon A \to Q$ be any other morphism with $q\circ \mu = f$. 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 $q = \phi$.
Given a monoidal category of modules $(A Mod , \otimes_A , A)$ as in prop. 3, then a monoid $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. 5) is called an $A$-algebra.
Given a monoidal category of modules $(A Mod , \otimes_A , A)$ in a monoidal category $(\mathcal{C},\otimes, 1)$ as in prop. 3, and an $A$-algebra $(E,\mu,e)$ (def. 8), then there is an equivalence of categories
between the category of commutative monoids in $A Mod$ and the coslice category of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.
(e.g. EKMM 97, VII lemma 1.3)
In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$
By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.
Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that
commutes. Moreover it satisfies the unit property
By forgetting the tensor product over $A$, 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 $(\star)$ above, to yield a commuting square
This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.
Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-module structure by
By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the universal property of the coequalizer gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.
Finally one checks that these two constructions are inverses to each other, up to isomorphism.
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. 9 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.
Let $\mathcal{C}, \mathcal{D}$ be pointed topologically enriched categories (def.), i.e. enriched categories over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example 2.
The pointed topologically enriched opposite category $\mathcal{C}^{op}$ is the topologically enriched category with the same objects as $\mathcal{C}$, with hom-spaces
and with composition given by braiding followed by the composition in $\mathcal{C}$:
the pointed topological product category $\mathcal{C} \times \mathcal{D}$ is the topologically enriched category whose objects are pairs of objects $(c,d)$ with $c \in \mathcal{C}$ and $d\in \mathcal{D}$, 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:
A pointed topologically enriched functor (def.) into $Top^{\ast/}_{cg}$ (exmpl.) out of a pointed topological product category as in def. 9
(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 $\mathcal{C}$ with its opposite category (def. 9)
then this takes the form of a “pullback action” in the first variable
and a “pushforward action” in the second variable
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.), i.e. an enriched category over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example 2. Let
be a pointed topologically enriched functor (def.) out of the pointed topological product category of $\mathcal{C}$ with its opposite category, according to def. 9.
The coend of $F$, denoted $\overset{c \in \mathcal{C}}{\int} F(c,c)$, is the coequalizer in $Top_{cg}^{\ast}$ (prop., exmpl., prop., cor.) of the two actions encoded in $F$ via example 11:
The end of $F$, denoted $\underset{c\in \mathcal{C}}{\int} F(c,c)$, is the equalizer in $Top_{cg}^{\ast/}$ (prop., exmpl., prop., cor.) of the adjuncts of the two actions encoded in $F$ via example 11:
Let $G$ be a topological group. Write $\mathbf{B}(G_+)$ for the pointed topologically enriched category that has a single object $\ast$, whose single hom-space is $G_+$ ($G$ with a basepoint freely adjoined (def.))
and whose composition operation is the product operation $(-)\cdot(-)$ in $G$ under adjoining basepoints (exmpl.)
Then a topologically enriched functor
is a pointed topological space $X \coloneqq F(\ast)$ equipped with a continuous function
satisfying the action property. Hence this is equivalently a continuous and basepoint-preserving left action (non-linear representation) of $G$ on $X$.
The opposite category (def. 9) $(\mathbf{B}(G_+))^{op}$ comes from the opposite group
(The canonical continuous isomorphism $G \simeq G^{op}$ induces a canonical euqivalence of topologically enriched categories $(\mathbf{B}(G_+))^{op} \simeq \mathbf{B}(G_+)$.)
So a topologically enriched functor
is equivalently a basepoint preserving continuous right action of $G$.
Therefore the coend of two such functors (def. 10) coequalizes the relation
(where juxtaposition denotes left/right action) and hence is equivalently the canonical smash product of a right $G$-action with a left $G$-action, hence the quotient of the plain smash product by the diagonal action of the group $G$:
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). For $F,G \;\colon\; \mathcal{C} \longrightarrow Top^{\ast/}_{cg}$ two pointed topologically enriched functors, then the end (def. 10) of $Maps(F(-),G(-))_\ast$ is a topological space whose underlying pointed set is the pointed set of natural transformations $F\to G$ (def.):
The underlying pointed set functor $U\colon Top^{\ast/}_{cg}\to Set^{\ast/}$ preserves all limits (prop., prop., prop.). Therefore there is an equalizer diagram in $Set^{\ast/}$ of the form
Here the object in the middle is just the set of collections of component morphisms $\left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}}$. The two parallel maps in the equalizer diagram take such a collection to the functions which send any $c \overset{f}{\to} d$ to the result of precomposing
and of postcomposing
each component in such a collection, respectively. These two functions being equal, hence the collection $\{\eta_c\}_{c\in \mathcal{C}}$ being in the equalizer, means precisley that for all $c,d$ and all $f\colon c \to d$ the square
is a commuting square. This is precisley the condition that the collection $\{\eta_c\}_{c\in \mathcal{C}}$ be a natural transformation.
Conversely, example 13 says that ends over bifunctors of the form $Maps(F(-),G(-)))_\ast$ constitute hom-spaces between pointed topologically enriched functors:
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). Define the structure of a pointed topologically enriched category on the category $[\mathcal{C}, Top_{cg}^{\ast/}]$ of pointed topologically enriched functors to $Top^{\ast/}_{cg}$ (exmpl.) by taking the hom-spaces to be given by the ends (def. 10) of example 13:
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 13) and hence defines a map into the end
The resulting pointed topologically enriched category $[\mathcal{C},Top^{\ast/}_{cg}]$ is also called the $Top^{\ast/}_{cg}$-enriched functor category over $\mathcal{C}$ with coefficients in $Top^{\ast/}_{cg}$.
This yields an equivalent formulation in terms of ends of the pointed topologically enriched Yoneda lemma (prop.):
(topologically enriched Yoneda lemma)
Let $\mathcal{C}$ be a small pointed topologically enriched categories (def.). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed topologically enriched functor (def.) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism
between the hom-space of the pointed topological functor category, according to def. 11, from the functor represented by $c$ to $F$, and the value of $F$ on $c$.
In terms of the ends (def. 10) defining these hom-spaces, this means that
In this form the statement is also known as Yoneda reduction.
The proof of prop. 5 is formally dual to the proof of the next prop. 6.
Now that natural transformations are expressed in terms of ends (example 13), as is the Yoneda lemma (prop. 5), it is natural to consider the dual statement involving coends:
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed topologically enriched functor (def.) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism
Moreover, the morphism that hence exhibits $F(c)$ as the coequalizer of the two morphisms in def. 10 is componentwise the canonical action
which is adjunct to the component map $\mathcal{C}(d,c) \to Maps(F(c),F(d))_{\ast}$ of the topologically enriched functor $F$.
(e.g. MMSS 00, lemma 1.6)
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 $(g,x)$ under the two morphisms being coequalized.)
But now considering the case that $d = c_0$ and $g = id_{c_0}$, so that $f = \phi$ shows that any pair
is identified, in the coequalizer, with the pair
hence with $\phi(x)\in F(c_0)$.
This shows the claim at the level of the underlying sets. To conclude it is now sufficient (prop.) to show that the topology on $F(c_0) \in Top^{\ast/}_{cg}$ 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 $F(c)$.
The statement of the co-Yoneda lemma in prop. 6 is a kind of categorification of the following statement in analysis (whence the notation with the integral signs):
For $X$ a topological space, $f \colon X \to\mathbb{R}$ a continuous function and $\delta(-,x_0)$ denoting the Dirac distribution, then
It is this analogy that gives the name to the following statement:
(Fubini theorem for (co)-ends)
For $F$ a pointed topologically enriched bifunctor on a small pointed topological product category $\mathcal{C}_1 \times \mathcal{C}_2$ (def. 9), i.e.
then its end and coend (def. 10) is equivalently formed consecutively over each variable, in either order:
and
Since the pointed compactly generated mapping space functor (exmpl.)
takes colimits in the first argument and limits in the second argument to limits (cor.), it in particular takes coends in the first argument and ends in the second argument, to ends (def. 10):
and
With this coend calculus in hand, there is an elegant proof of the defining universal property of the smash tensoring of topologically enriched functors $[\mathcal{C},Top^{\ast}_{cg}]$ (def.)
For $\mathcal{C}$ a pointed topologically enriched category, there are natural isomorphisms
and
for all $X,Y \in [\mathcal{C},Top^{\ast/}_{cg}]$ and all $K \in Top^{\ast/}_{cg}$.
In particular there is the combined natural isomorphism
exhibiting a pair of adjoint functors
Via the end-expression for $[\mathcal{C},Top^{\ast/}_{cg}](-,-)$ from def. 11 and the fact (remark 3) that the pointed mapping space construction $Maps(-,-)_\ast$ preserves ends in the second variable, this reduces to the fact that $Maps(-,-)_\ast$ is the internal hom in the closed monoidal category $Top^{\ast/}_{cg}$ (example 2) and hence satisfies the internal tensor/hom-adjunction isomorphism (prop. 1):
and
(left Kan extension via coends)
Let $\mathcal{C}, \mathcal{D}$ be small pointed topologically enriched categories (def.) and let
be a pointed topologically enriched functor (def.). Then precomposition with $p$ constitutes a functor
$G\mapsto G\circ p$. This functor has a left adjoint $Lan_p$, called left Kan extension along $p$
which is given objectwise by a coend (def. 10):
Use the expression of natural transformations in terms of ends (example 13 and def. 11), then use the respect of $Maps(-,-)_\ast$ for ends/coends (remark 3), use the smash/mapping space adjunction (cor.), use the Fubini theorem (prop. 7) and finally use Yoneda reduction (prop. 5) to obtain a sequence of natural isomorphisms as follows:
Given two functions $f_1, f_2 \colon G \longrightarrow \mathbb{C}$ on a group (or just a monoid) $G$, 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.
Let $(\mathcal{C}, \otimes, 1)$ be a small pointed topological monoidal category (def. 1).
Then the Day convolution tensor product on the pointed topological enriched functor category $[\mathcal{C},Top^{\ast/}_{cg}]$ (def. 11) is the functor
out of the pointed topological product category (def. 9) given by the following coend (def. 10)
Let $Seq$ 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. 17):
Regard this as a pointed topologically enriched category in the unique way. The operation of addition of natural numbers $\otimes = +$ makes this a monoidal category.
An object $X_\bullet \in [Seq, Top_{cg}^{\ast/}]$ is an $\mathbb{N}$-sequence of pointed topological spaces. Given two such, then their Day convolution according to def. 12 is
We observe now that Day convolution is equivalently a left Kan extension (def. 9). This will be key for understanding monoids and modules with respect to Day convolution.
Let $\mathcal{C}$ 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 $\mmathcal{C}$ to topologically enriched functors over the product category $\mathcal{C} \times \mathcal{C}$ (def. 9) given by
i.e.
For $(\mathcal{C}, \otimes 1)$ a pointed topologically enriched monoidal category (def. 1) the Day convolution product (def. 12) of two functors is equivalently the left Kan extension (def. 9) of their external tensor product (def. 13) along the tensor product $\otimes \colon \mathcal{C} \times \mathcal{C}$: 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).
By prop. 9 we may compute the left Kan extension as the following coend:
Proposition 10 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.
The operation of Day convolution $\otimes_{Day}$ (def. 12) is universally characterized by the property that there are natural isomorphisms
where $\overline{\wedge}$ is the external product of def. 13, hence that natural transformations of functors on $\mathcal{C}$ of the form
are in natural bijection with natural transformations of functors on the product category $\mmathcal{C}\times \mathcal{C}$ (def. 9) of the form
Write
for the $Top^{\ast/}_{cg}$-Yoneda embedding, so that for $c\in \mathcal{C}$ any object, $y(c)$ is the corepresented functor $y(c)\colon d \mapsto \mathcal{C}(c,d)$.
For $(\mathcal{C},\otimes, 1)$ a small pointed topological monoidal category (def. 1), the Day convolution tensor product $\otimes_{Day}$ of def. 12 makes the pointed topologically enriched functor category
into a pointed topological monoidal category (def. 1) with tensor unit $y(1)$ co-represented by the tensor unit $1$ of $\mathcal{C}$.
Moreover, if $(\mathcal{C}, \otimes, 1)$ is equipped with a (symmetric) braiding $\tau^{\mathcal{C}}$ (def. 2), then so is $([\mathcal{C}, Top^{\ast/}_{cg}],\otimes_{Day}, y(1))$.
Regarding associativity, observe that
where we used the Fubini theorem for coends (prop. 7) and then twice the co-Yoneda lemma (prop. 6). Similarly
So we obtain an associator by combining, in the integrand, the associator $\alpha^{\mathcal{C}}$ of $(\mathcal{C}, \otimes, 1)$ and $\tau^{Top_{cg}^{\ast/}}$ of $(Top^{\ast/}_{cg}, \wedge, S^0)$ (example 2):
It is clear that this satisfies the pentagon identity, since $\tau^{\mathcal{C}}$ and $\tau^{Top^{\ast/}_{cg}}$ do.
To see that $y(1)$ is the tensor unit for $\otimes_{Day}$, use the Fubini theorem for coends (prop. 7) and then twice the co-Yoneda lemma (prop. 6) to get for any $X \in [\mathcal{C},Top^{\ast/}_{cg}]$ that
Hence the right unitor of Day convolution comes from the unitor of $\mathcal{C}$ under the integral sign:
Analogously for the left unitor. Hence the triangle identity for $\otimes_{Day}$ follows from the triangle identity in $\mathcal{C}$ under the integral sign.
Similarly, if $\mathcal{C}$ has a braiding $\tau^{\mathcal{C}}$, it induces a braiding $\tau^{Day}$ under the integral sign:
and the hexagon identity for $\tau^{Day}$ follows from that for $\tau^{\mathcal{C}}$ and $\tau^{Top^{\ast/}_{cg}}$
Moreover:
For $(\mathcal{C}, \otimes ,1 )$ a small pointed topological symmetric monoidal category (def. 3), the monoidal category with Day convolution $([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$ from def. 11 is a closed monoidal category (def. 4). Its internal hom $[-,-]_{Day}$ is given by the end (def. 10)
Using the Fubini theorem (def. 7) and the co-Yoneda lemma (def. 6) and in view of definition 11 of the enriched functor category, there is the following sequence of natural isomorphisms:
In the situation of def. 11, the Yoneda embedding $c\mapsto \mathcal{C}(c,-)$ constitutes a strong monoidal functor (def. 14)
That the tensor unit is respected is part of prop. 11. To see that the tensor product is respected, apply the co-Yoneda lemma (prop. 6) twice to get the following natural isomorphism
Since the symmetric monoidal smash product of spectra discussed below is an instance of Day convolution (def. 12), and since ring spectra are going to be the monoids (def. 5) 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 $Top^{\ast/}_{cg}$. 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 9), 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.
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two (pointed) topologically enriched monoidal categories (def. 1). A topologically enriched lax monoidal functor between them is
a topologically enriched functor
a morphism
for all $x,y \in \mathcal{C}$
satisfying the following conditions:
(associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes
where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;
(unitality) For all $x \in \mathcal{C}$ the following diagrams commutes
and
where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right unitors of the two monoidal categories, respectively.
If $\epsilon$ and alll $\mu_{x,y}$ are isomorphisms, then $F$ is called a strong monoidal functor.
If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of braided monoidal categories (def. 2) with braidings $\tau^{\mathcal{C}}$ and $\tau^{\mathcal{D}}$, respectively, then the lax monoidal functor $F$ is called a braided monoidal functor if in addition the following diagram commutes for all objects $x,y \in \mathcal{C}$
A homomorphism $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation $f_x \;\colon\; F_1(x) \longrightarrow F_2(x)$ of the underlying functors
compatible with the product and the unit in that the following diagrams commute for all objects $x,y \in \mathcal{C}$:
and
We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting category of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between braided monoidal categories, and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between symmetric monoidal categories.
In the literature the term “monoidal functor” often refers by default to what in def. 14 is called a strong monoidal functor. But for the purpose of the discussion of functors with smash product below, it is crucial to admit the generality of lax monoidal functors.
If $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are symmetric monoidal categories (def. 3) then a braided monoidal functor (def. 14) between them is often called a symmetric monoidal functor.
For $\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E}$ two composable lax monoidal functors (def. 14) between monoidal categories, then their composite $F \circ G$ becomes a lax monoidal functor with structure morphisms
and
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}})$ be two monoidal categories (def. 1) and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a lax monoidal functor (def. 14) between them.
Then for $(A,\mu_A,e_A)$ a monoid in $\mathcal{C}$ (def. 5), its image $F(A) \in \mathcal{D}$ becomes a monoid $(F(A), \mu_{F(A)}, e_{F(A)})$ by setting
(where the first morphism is the structure morphism of $F$) and setting
(where again the first morphism is the corresponding structure morphism of $F$).
This construction extends to a functor
from the category of monoids of $\mathcal{C}$ (def. 5) to that of $\mathcal{D}$.
Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal categories (def. 3) and $F$ is a braided monoidal functor (def. 14) and $A$ is a commutative monoid (def. 5) then so is $F(A)$, and this construction extends to a functor
This follows immediately from combining the associativity and unitality (and symmetry) constraints of $F$ with those of $A$.
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two (pointed) topologically enriched monoidal categories (def. 1), and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a topologically enriched lax monoidal functor between them, with product operation $\mu$.
Then a left module over the lax monoidal functor is
a topologically enriched functor
such that
(action property) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes
A homomorphism $f\;\colon\; (G_1, \rho_1) \longrightarrow (G_2,\rho_2)$ between two modules over a monoidal functor $(F,\mu,\epsilon)$ is
compatible with the action in that the following diagram commutes for all objects $x,y \in \mathcal{C}$:
We write $F Mod$ for the resulting category of modules over the monoidal functor $F$.
Now we may finally state the main proposition on functors with smash product:
Let $(\mathcal{C},\otimes, 1)$ be a pointed topologically enriched (symmetric) monoidal category (def. 1). Regard $(Top_{cg}^{\ast/}, \wedge , S^0)$ as a topological symmetric monoidal category as in example 2.
Then (commutative) monoids in (def. 5) the Day convolution monoidal category $([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))$ of prop. 11 are equivalent to (braided) lax monoidal functors (def. 14) of the form
called functors with smash products on $\mathcal{C}$, 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. 15).
This is stated in some form in (Day 70, example 3.2.2). It is highlighted again in (MMSS 00, prop. 22.1).
By definition 14, a lax monoidal functor $F \colon \mathcal{C} \to Top^{\ast/}_{cg}$ 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 $c_1,c_2 \in \mathcal{C}$.
Under the Yoneda lemma (prop. 5) the first of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form
Moreover, under the natural isomorphism of corollary 1 the second of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form
Translating the conditions of def. 14 satisfied by a lax monoidal functor through these identifications gives precisely the conditions of def. 5 on a (commutative) monoid in object $F$ under $\otimes_{Day}$.
Similarly for module objects and modules over monoidal functors.
Let $f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a lax monoidal functor (def. 14) between pointed topologically enriched monoidal categories (def. 1). Then the induced functor
given by $(f^\ast X)(c)\coloneqq X(f(c))$ preserves monoids under Day convolution
Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal categories (def. 3) and $f$ is a braided monoidal functor (def. 14), then $f^\ast$ also preserves commutative monoids
Similarly, for
any fixed monoid, then $f^\ast$ sends $A$-modules to $f^\ast(A)$-modules
This is an immediate corollary of prop. 16, since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor by prop. 14.
We give a unified discussion of the categories of
(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 $Top^{\ast/}_{fin}$.
This approach is due to (Mandell-May-Schwede-Shipley 00) following (Hovey-Shipley-Smith 00).
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)
$\,$
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. 11)
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 $\iota_{fin}$ itself:
We call this the standard incarnation of the sphere spectrum as a pre-excisive functor.
By prop. 11 the smash product of pointed compactly generated topological spaces induces the structure of a closed (def. 4) symmetric monoidal category (def. 3)
with
tensor unit the sphere spectrum $\mathbb{S}_{exc}$;
tensor product the Day convolution product $\otimes_{Day}$ from def. 12,
called the symmetric monoidal smash product of spectra for the model of pre-excisive functors;
internal hom the dual operation $[-,-]_{Day}$ from prop. 12,
called the mapping spectrum construction for pre-excisive functors.
By example 5 the sphere spectrum incarnated as a pre-excisive functor $\mathbb{S}_{exc}$ (according to def. 16) is canonically a commutative monoid in the category of pre-excisive functors (def. 5).
Moreover, by example 7, every object of $Exc(Top_{cg})$ (def. 16) is canonically a module object over $\mathbb{S}_{exc}$. We may therefore tautologically identify the category of pre-excisive functors with the module category over the sphere spectrum:
Identified as a functor with smash product under prop. 16, the pre-excisive sphere spectrum $\mathbb{S}_{exc}$ from def. 16 is given by the identity natural transformation
We claim that this is in fact the unique structure of a monoidal functor that may be imposed on the canonical inclusion $\iota \;\colon\; Top^{\ast/}_{cg,fin} \hookrightarrow Top^{\ast/}_{cg}$, hence it must be the one in question. To see the uniqueness, observe that naturality of the matural transformation $\mu$ in particular says that there are commuting squares of the form
where the vertical morphisms pick any two points in $K_1$ and $K_2$, respectively, and where the top morphism is necessarily the canonical identification since there is only one single isomorphism $S^0 \to S^0$, 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. 16.
Define the following pointed topologically enriched (def.) symmetric monoidal categories (def. 3):
$Seq$ 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, $\otimes = +$, and the tensor unit is 0. The braiding is, necessarily, the identity.
$Sym$ is the standard skeleton of the core of FinSet with zero morphisms adjoined: its objects are the finite sets $\overline{n} \coloneqq \{1, \cdots,n\}$ for $n \in \mathbb{N}$ (hence $\overline{0}$ is the empty set), all non-zero morphisms are automorphisms and the automorphism group of $\{1,\cdots,n\}$ is the symmetric group $\Sigma(n)$ on $n$ 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 $\Sigma(n_1+n_2)$ that shuffles the first $n_1$ elements past the remaining $n_2$ elements
$Orth$ has as objects the finite dimenional real linear inner product spaces $(\mathbb{R}^n, \langle -,-\rangle)$ and as non-zero morphisms the linear isometric isomorphisms between these; hence the automorphism group of the object $(\mathbb{R}^n, \langle -,-\rangle)$ is the orthogonal group $O(n)$; the monoidal product is direct sum of linear spaces, the tensor unit is the 0-vector space; again we turn this into a $Top^{\ast/}_{cg}$-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.
Notice that in the notation of example 12
the full subcategory of $Orth$ on $V$ is $\mathbf{B}(O(V)_+)$;
the full subcategory of $Sym$ on $\{1,\cdots,n\}$ is $\mathbf{B}(\Sigma(n)_+)$;
the full subcategory of $Seq$ on $n$ is $\mathbf{B}(1_+)$.
Moreover, after discarding the zero morphisms, then these categories are the disjoint union of categories of the form $\mathbf{B}O(n)$, $\mathbf{B}\Sigma(n)$ and $\mathbf{B}1 = \ast$, 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. 16).
Here $S^V$ denotes the one-point compactification of $V$. On morphisms $sym \colon (\Sigma_n)_+ \hookrightarrow (O(n))_+$ is the canonical inclusion of permutation matrices into orthogonal matrices and $orth \colon O(V)_+ \hookrightarrow Aut(S^V)$ is on $O(V)$ the topological subspace inclusions of the pointed homeomorphisms $S^V \to S^V$ that are induced under forming one-point compactification from linear isometries of $V$ (“representation spheres”).
Below we will often use these identifications to write just “$n$” for any of these objects, leaving implicit the identifications $n \mapsto \{1, \cdots, n\} \mapsto S^n$.
Consider the pointed topological diagram categries (def. 11, exmpl.) over these categories:
$[Seq,Top^{\ast/}_{cg}]$ is called the category of sequences of pointed topological spaces (e.g. HSS 00, def. 2.3.1);
$[Sym,Top^{\ast/}_{cg}]$ is called the category of symmetric sequences (e.g. HSS 00, def. 2.1.1);
$[Orth, Top^{\ast/}_{cg}]$ is called the category of orthogonal sequences.
Consider the sequence of restrictions of topological diagram categories, according to prop. 17 along the above inclusions:
Write
for the restriction of the excisive functor incarnation of the sphere spectrum (from def. 16) along these inclusions.
The functors $seq$, $sym$ and $orth$ in def. 17 become strong monoidal functors (def. 14) when equipped with the canonical isomorphisms
and
and
Moreover, $orth$ and $sym$ are braided monoidal functors (def. 14) (hence symmetric monoidal functors, remark 4). But $seq$ is not braided monoidal.
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 $Top^{\ast/}_{cg}$ is given by the maps induced from exchanging coordinates in the realization of $n$-spheres as one-point compactifications of Cartesian spaces $S^n \simeq (\mathbb{R}^n)^\ast$. This corresponds precisely to the action of the symmetric group inside the orthogonal group acting via the canonical action of the orthogonal group on $\mathbb{R}^n$. This shows that $sym$ and $orth$ are braided, for they include precisely these objects (the $n$-spheres) with these braidings on them. Finally it is clear that $seq$ is not braided, because the braiding on $Seq$ is trivial, while that on $Sym$ is not, so $seq$ necessrily fails to preserve precisely these non-trivial isomorphisms.
Since the standard excisive incarnation $\mathbb{S}_{exc}$ of the sphere spectrum (def. 16) is the tensor unit with repect to the Day convolution product on pre-excisive functors, and since it is therefore canonically a commutative monoid, by example 5, prop. 17 says that the restricted sphere spectra $\mathbb{S}_{orth}$, $\mathbb{S}_{sym}$ and $\mathbb{S}_{seq}$ are still monoids, and that under restriction every pre-excisive functor, regarded as a $\mathbb{S}_{exc}$-module via remark 5, canonically becomes a module under the restricted sphere spectrum:
Since all three functors $orth$, $sym$ and $seq$ are strong monoidal functors by prop. 18, all three restricted sphere spectra $\mathbb{S}_{orth}$, $\mathbb{S}_{sym}$ and $\mathbb{S}_{seq}$ canonically are monoids, by prop. 17. Moreover, according to prop. 18, $orth$ and $sym$ are braided monoidal functors, while functor $seq$ is not braided, therefore prop. 17 furthermore gives that $\mathbb{S}_{orth}$ and $\mathbb{S}_{sym}$ are commutative monoids, while $\mathbb{S}_{seq}$ is not commutative:
sphere spectrum | $\mathbb{S}_{exc}$ | $\mathbb{S}_{orth}$ | $\mathbb{S}_{sym}$ | $\mathbb{S}_{seq}$ |
---|---|---|---|---|
monoid | yes | yes | yes | yes |
commutative monoid | yes | yes | yes | no |
tensor unit | yes | no | no | no |
Explicitly:
The monoids $\mathbb{S}_{dia}$ from def. 17 are, when identified as functors with smash product via prop. 16 given by assigning
respectively, with product given by the canonical isomorphisms
By construction these functors with smash products are the composites, according to prop. 14, of the monoidal functors $seq$, $sym$, $orth$, respectively, with the lax monoidal functor corresponding to $\mathbb{S}_{exc}$. The former have as structure maps the canonical identifications by definition, and the latter has as structure map the canonical identifications by lemmma 2.
There is an equivalence of categories
which identifies the category of modules (def. 6) over the monoid $\mathbb{S}_{seq}$ (remark 6) in the Day convolution monoidal structure (prop. 11) over the topological functor category $[Seq,Top^{\ast/}_{cg}]$ from def. 17 with the category of sequential spectra (def.)
Under this equivalence, an $\mathbb{S}_{seq}$-module $X$ is taken to the sequential pre-spectrum $X^{seq}$ whose component spaces are the values of the pre-excisive functor $X$ on the standard n-sphere $S^n = (S^1)^{\wedge n}$
and whose structure maps are the images of the action morphisms
under the isomorphism of corollary 1
evaluated at $n_1 = 1$
(Hovey-Shipley-Smith 00, prop. 2.3.4)
After unwinding the definitions, the only point to observe is that due to the action property,
any $\mathbb{S}_{seq}$-action
is indeed uniquely fixed by the components of the form
This is because under corollary 1 the action property is identified with the componentwise property
where the left vertical morphism is an isomorphism by the nature of $\mathbb{S}_{seq}$. Hence this fixes the components $\rho_{n',n}$ to be the $n'$-fold composition of the structure maps $\sigma_n \coloneqq \rho(1,n)$.
However, since, by remark 19, $\mathbb{S}_{seq}$ is not commutative, there is no tensor product induced on $SeqSpec(Top_{cg})$ under the identification in prop. 19. But since $\mathbb{S}_{orth}$ and $\mathbb{S}_{sym}$ are commutative monoids by remark 19, it makes sense to consider the following definition.
In the terminology of remark 6 we say that
is the category of orthogonal spectra; and that
is the category of symmetric spectra.
By remark 6 and by prop. 3 these categories canonically carry a symmetric monoidal tensor product $\otimes_{\mathbb{S}_{orth}}$ and $\otimes_{\mathbb{S}_{seq}}$, 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.
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 18 from above.
Literature. ( Hovey-Shipley-Smith 00, section 1, section 2, Schwede 12, chapter I)
$\,$
A topological symmetric spectrum $X$ is
a sequence $\{X_n \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\}$ of pointed compactly generated topological spaces;
a basepoint preserving continuous right action of the symmetric group $\Sigma(n)$ on $X_n$;
a sequence of morphisms $\sigma_n \colon S^1 \wedge X_n \longrightarrow X_{n+1}$
such that
for all $n, k \in \mathbb{N}$ the composite
intertwines the $\Sigma(n) \times \Sigma(k)$-action.
A homomorphism of symmetric spectra $f\colon X \longrightarrow Y$ is
such that
each $f_n$ intetwines the $\Sigma(n)$-action;
the following diagrams commute
We write $SymSpec(Top_{cg})$ 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.
A topological orthogonal spectrum $X$ is
a sequence $\{X_n \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\}$ of pointed compactly generated topological spaces;
a basepoint preserving continuous right action of the orthogonal group $O(n)$ on $X_n$;
a sequence of morphisms $\sigma_n \colon S^1 \wedge X_n \longrightarrow X_{n+1}$
such that
for all $n, k \in \mathbb{N}$ the composite
intertwines the $O(n) \times Ok()$-action.
A homomorphism of orthogonal spectra $f\colon X \longrightarrow Y$ is
such that
each $f_n$ intetwines the $O(n)$-action;
the following diagrams commute
We write $OrthSpec(Top_{cg})$ for the resulting category of orthogonal spectra.
(e.g. Schwede 12, I, def. 7.2)
Definitions 19 and 20 are indeed equivalent to def. 18:
orthogonal spectra are euqivalently the module objects over the incarnation $\mathbb{S}_{orth}$ of the sphere spectrum
and symmetric spectra sre equivalently the module objects over the incarnation $\mathbb{S}_{sym}$ of the sphere spectrum
(Hovey-Shipley-Smith 00, prop. 2.2.1)
We discuss this for symmetric spectra. The proof for orthogonal spectra is of the same form.
First of all, by example 12 an object in $[Sym, Top^{\ast/}_{cg}]$ is equivalently a “symmetric sequence”, namely a sequence of pointed topological spaces $X_k$, for $k \in \mathbb{N}$, equipped with an action of $\Sigma(k)$ (def. 17).
By corollary 1 and lemma 3, the structure morphism of an $\mathbb{S}_{sym}$-module object on $X$
is equivalently (as a functor with smash products) a natural transformation
over $Sym \times Sym$. This means equivalently that there is such a morphism for all $n_1, n_2 \in \mathbb{N}$ and that it is $\Sigma(n_1) \times \Sigma(n_2)$-equivariant.
Hence it only remains to see that these natural transformations are uniquely fixed once the one for $n_1 = 1$ is given. To that end, observe that lemma 3 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 $S^{n_1 + n_2}$ has to be the composite of the actions of $S^{n_2}$ followed by that of $S^{n_1}$. 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.
Given $X,Y \in SymSpec(Top_{cg})$ two symmetric spectra, def. 19, then their smash product of spectra is the symmetric spectrum
with component spaces the coequalizer
where $\ell$ has components given by the structure maps
while $r$ has components given by the structure maps conjugated by the braiding in $Top^{\ast/}_{cg}$ and the permutation action $\chi_{p,1}$ (that shuffles the element on the right to the left)
Finally The structure maps of $X \wedge Y$ are those induced under the coequalizer by
Analogously for orthogonal spectra.
Under the identification of prop. 20, the explicit smash product of spectra in def. 21 is equivalent to the abstractly defined tensor product in def. 18:
in the case of symmetric spectra:
in the case of orthogonal spectra:
By def. 7 the abstractly defined tensor product of two $\mathbb{S}_{sym}$-modules $X$ and $Y$ is the coequalizer
The Day convolution product appearing here is over the category $Sym$ from def. 17. By example 12 and unwinding the definitions, this is for any two symmetric spectra $A$ and $B$ 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. 21.
To see this, we apply the adjunction isomorphism between the Day convolution product and the external tensor product (cor. 1) twice, to find the following sequence of equivalent incarnations of morphisms:
This establishes the form of the morphism $\ell$. By the same reasoning as in the proof of prop. 20, we may restrict the coequalizer to $n_2 = 1$ without changing it.
The form of the morphism $r$ is obtained by the analogous sequence of identifications of morphisms, now with the parenthesis to the left. That it involves $\tau^{Top^{\ast/}_{cg}}$ and the permutation action $\tau^{sym}$ as shown above follows from the formula for the braiding of the Day convolution tensor product from the proof of prop. 11:
by translating it to the components of the precomposition
via the formula from the proof of prop. 9 for the left Kan extension $A \otimes_{Day} B \simeq Lan_{\otimes} A \overline{\wedge} B$ (prop. 10):
This last expression is the function on morphisms which precomposes components under the coend with the braiding $\tau_{X_{n_1}, S^{n_2} }^{Top^{\ast/}_{cg}}$ in topological spaces and postcomposes them with the image of the functor $X$ of the braiding in $Sym$. But the braiding in $Sym$ is, by def. 17, given by the respective shuffle permutations $\tau^{sym}_{n_1,n_2} = \chi_{n_1,n_2}$, and by prop. 20 the image of these under $X$ is via the given $\Sigma_{n_1+n_2}$-action on $X_{n_1+n_2}$.
Finally to see that the structure map is as claimed: By prop. 20 the structure morphisms are the degree-1 components of the $\mathbb{S}_{sym}$-action, and by prop. 7 the $\mathbb{S}_{sym}$-action on a tensor product of $\mathbb{S}_{sym}$-modules is induced via the action on the left tensor factor.
A commutative symmetric ring spectrum $E$ is
a sequence of component spaces $E_n \in Top^{\ast/}_{cg}$ for $n \in \mathbb{N}$;
a basepoint preserving continuous left action of the symmetric group $\Sigma(n)$ on $E_n$;
for all $n_1,n_2\in \mathbb{N}$ a multiplication map
(a morphism in $Top^{\ast/}_{cg}$)
two unit maps
such that
(equivariance) $\mu_{n_1,n_2}$ intertwines the $\Sigma(n_1) \times \Sigma(n_2)$-action;
(associativity) for all $n_1, n_2, n_3 \in \mathbb{N}$ the following diagram commutes (where we are notationally suppressing the associators of $(Top^{\ast/}_{cg}, \wedge, S^0)$)
(unitality) for all $n \in \mathbb{N}$ the following diagram commutes
and
where the diagonal morphisms $\ell$ and $r$ are the left and right unitors in $(Top^{\ast/}_{cg}, \wedge, S^0)$, respectively.
(commutativity) for all $n_1, n_2 \in \mathbb{N}$ the following diagram commutes
where the top morphism $\tau$ is the braiding in $(Top^{\ast/}_{cg}, \wedge, S^0)$ (def. 2) and where $\chi_{n_1,n_2} \in \Sigma(n_1 + n_2)$ denotes the permutation action which shuffles the first $n_1$ elements past the last $n_2$ elements.
A homomorphism of symmetric commutative ring spectra $f \colon E \longrightarrow E'$ is a sequence $f_n \;\colon\; E_n \longrightarrow E'_n$ of $\Sigma(n)$-equivariant pointed continuous functions such that the following diagrams commute for all $n_1, n_2 \in \mathbb{N}$
and $f_0 \circ \iota_0 = \iota_0$ and $f_1\circ \iota_1 = \iota_1$.
Write
for the resulting category of symmetric commutative ring spectra.
We regard a symmetric ring spectrum in particular as a symmetric spectrum (def. 19) 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.
The symmetric (orthogonal) commutative ring spectra in def. 22 are equivalently the commutative monoids in (def. 5) the symmetric monoidal category $\mathbb{S}_{sym}Mod$ ($\mathbb{S}_{orth}Mod$) of def. 18 with respect to the symmetric monoidal smash product of spectra $\wedge = \otimes_{\mathbb{S}_{sym}}$ ($\wedge = \otimes_{\mathbb{S}_{orth}}$). Hence there are equivalences of categories
and
Moreover, under these identifications the canonical forgetful functor
and
coincides with the forgetful functor defined in def. 22.
We discuss this for symmetric spectra. The proof for orthogonal spectra is directly analogous.
By prop. 4 and def. 18, the commutative monoids in $\mathbb{S}_{sym}Mod$ are equivalently commtutative monoids $E$ in $([Sym,Top^{\ast/}_{cg}], \otimes_{Day}, y(0))$ equipped with a homomorphism of monoids $\mathbb{S}_{sym} \longrightarrow E$. In turn, by prop. 16 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. 14)
The structure morphism of such a lax monoidal functor $E$ has as components precisely the morphisms $\mu_{n_1, n_2}\colon E_{n_1} \wedge E_{n_2} \to E_{n_1 + n_2}$. 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. 4 the $\mathbb{S}_{sym}$-module structure on an an $\mathbb{S}_{sym}$-algebra $E$ has action given by
which shows, via the identification in prop 20, 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 $\iota$ has components
for all $n \in \mathbb{N}$, and the unitality condition for $\iota_0$ and $\iota_1$ is manifestly as in the statement above.
We claim that the other components are uniquely fixed by these:
By lemma 3, the product structure in $\mathbb{S}_{sym}$ is by isomorphisms $S^{n_1} \wedge S^{n_2} \simeq S^{n_1 + n_2}$, so that the commuting square for the coherence condition of this monoidal natural transformation
says that $\iota_{n_1 + n_2} = \mu_{n_1,n_2} \circ (\iota_{n_1} \wedge \iota_{n_2})$. This means that $\iota_{n \geq 2}$ is uniquely fixed once $\iota_0$ and $\iota_1$ 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:
Given a symmetric (orthogonal) commutative ring spectrum $E$ (def. 22), then a left symmetric (orthogonal) module spectrum $N$ over $E$ is
a sequence of component spaces $N_n \in Top^{\ast/}_{cg}$ for $n \in \mathbb{N}$;
a basepoint preserving continuous left action of the symmetric group $\Sigma(n)$ on $N_n$;
for all $n_1,n_2\in \mathbb{N}$ an action map
(a morphism in $Top^{\ast/}_{cg}$)
such that
(equivariance) $\rho_{n_1,n_2}$ intertwines the $\Sigma(n_1) \times \Sigma(n_2)$-action;
(action property) for all $n_1, n_2, n_3 \in \mathbb{N}$ the following diagram commutes (where we are notationally suppressing the associators of $(Top^{\ast/}_{cg}, \wedge, S^0)$)
(unitality) for all $n \in \mathbb{N}$ the following diagram commutes
A homomorphism of left $E$-module spectra $f\;\colon\; N \longrightarrow N'$ is a sequence of pointed continuous functions $f_n \;\colon\; N_n \longrightarrow N'_n$ such that for all $n_1,n_2 \in \mathbb{N}$ the following diagrams commute:
We write
for the resulting category of symmetric (orthogonal) $E$-module spectra.
(e.g. Schwede 12, I, def. 1.5)
Under the identification, from prop. 22, of commutative ring spectra with commutative monoids with respect to the symmetric monoidal smash product of spectra, the $E$-module spectra of def. 23 are equivalently the left module objects (def. 6) over the respective monoids, i.e. there are equivalences of categories
and
where on the right we have the categories of modules from def. 6.
The proof is directly analogous to that of prop. 22. Now prop. 4 and prop. 16 give that the module objects in question are equivalently modules over a monoidal functor (def. 15) and writing these out in components yields precisely the above structures and properties.
In Introduction to Stable homotopy theory -- 1-1 we obtained the strict/level model structure on topological sequential spectra by identifying the category $SeqSpec(Top_{cg})$ of sequential spectra with a category of topologically enriched functors with values in $Top^{\ast/}_{cg}$ (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.
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ be a topologically enriched monoidal category (def. 1), and let $A \in Mon([\mathcal{C},Top^{\ast/}_{cg}],\otimes_{Day}, y(1_{\mathcal{C}}))$ be a monoid in (def. 5) the pointed topological Day convolution monoidal category over $\mathcal{C}$ from prop. 11.
Then the category of left A-modules (def. 6) is a pointed topologically enriched functor category category (exmpl.)
over the category of free modules over $A$ (prop. 2) on objects in $\mathcal{C}$ (under the Yoneda embedding $y \colon \mathcal{C}^{op} \to [\mathcal{C}, Top^{\ast/}_{cg}]$). Hence the objects of $A Free_{\mathcal{C}}Mod$ are identified with those of $\mathcal{C}$, and its hom-spaces are
Use the identification from prop. 16 of $A$ with a lax monoidal functor and of any $A$-module object $N$ 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 $N$ (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 $A Free_{\mathcal{C}}Mod$ have just this structure:
Here we used first the free-forgetful adjunction of prop. 2, then the enriched Yoneda lemma (prop. 5), then the coend-expression for Day convolution (def. 12) and finally the co-Yoneda lemma (prop. 6).
Then define a topologically enriched category $\mathcal{D}$ to have objects and hom-spaces those of $A Free_{\mathcal{C}}Mod^{op}$ as above, and whose composition operation is defined as follows:
where
the equivalence is braiding in the integrand (and the Fubini theorem, prop. 7);
the first morphism is, in the integrand, the smash product of
forming the tensor product of hom-objects of $\mathcal{C}$ with the identity morphism on $c_5$;
the monoidal functor incarnation $A(c_5) \wedge A(c_4)\longrightarrow A(c_5 \otimes_{\mathcal{C}} c_4 )$ of the monoid structure on $A$;
the second morphism is, in the integrand, given by composition in $\mathcal{C}$;
the last morphism is the morphism induced on coends by regarding extranaturality in $c_4$ and $c_5$ separately as a special case of extranaturality in $c_6 \coloneqq c_4 \otimes c_5$ (and then renaming).
With this it is fairly straightforward to see that
because, by the above definition of composition, functoriality over $\mathcal{D}$ manifestly encodes the $A$-action property together with the functoriality over $\mathcal{C}$.
This way we are reduced to showing that actually $\mathcal{D} \simeq A Free_{\mathcal{C}}Mod^{op}$.
But by construction, the image of the objects of $\mathcal{D}$ under the Yoneda embedding are precisely the free $A$-modules over objects of $\mathcal{C}$:
Since the Yoneda embedding is fully faithful, this shows that indeed
For the sequential case $Dia = Seq$ in def. 17, then the opposite category of free modules on objects in $Seq$ over $\mathbb{S}_{seq}$ (def.) is identified as the category $StdSpheres$ (def.):
Accordingly, in this case prop. 24 reduces to the identification (prop.) of sequential spectra as topological diagrams over $StdSpheres$:
There is one object $y(n)$ for each $n \in \mathbb{N}$. Moreover, from the expression in the proof of prop. 24 we compute the hom-spaces between these to be
These are the objects and hom-spaces of the category $StdSpheres$. It is straightforward to check that the definition of composition agrees, too.
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
For $Dia \in \{Top^{\ast/}_{cg,fin}, Orth, Sym, Seq\}$ one of the shapes of structured spectra from def. 17, let $\mathbb{S}_{dia}Mod$ be the corresponding category of structured spectra (def. 16, prop. 19, def. 18).
The stable homotopy groups of an object $X \in \mathbb{S}_{dia}Mod$ are those of the underlying sequential spectrum (def.):
An object $X \in \mathbb{S}_{dia}Mod$ is a structured Omega-spectrum if the underlying sequential spectrum $seq^\ast X$ (def. 17) is a sequential Omega spectrum (def.)
A morphism $f$ in $\mathbb{S}_{dia}Mod$ is a stable weak homotopy equivalence (or: $\pi_\bullet$-isomorphism) if the underlying morphism of sequential spectra $seq^\ast(f)$ is a stable weak homotopy equivalence of sequential spectra (def.);
a morphism $f$ is a stable cofibration if it is a cofibration in the strict model structure $OrthSpec(Top_{cg})_{strict}$ from prop. 1.
(MMSS 00, def. 8.3 with the notation from p. 21, Mandell-May 02, III, def. 3.1, def. 3.2)
Given a morphism $f\;\colon\; X \longrightarrow Y$ in $\mathbb{S}_{dia}Mod$, then there are long exact sequences of stable homotopy groups (def. 24) of the form
and
where $Cone(f)$ denotes the mapping cone and $Path_\ast(f)$ the mapping cocone of $f$ (def.) formed with respect to the standard cylinder spectrum $X \wedge (I_+)$ hence formed degreewise with respect to the standard reduced cylinder of pointed topological spaces.
Since limits and colimits in the diagram category $\mathbb{S}_{dia}Mod$ are computed objectwise, the functor $seq^\ast$ that restricts $\mathbb{S}_{dia}$-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 $n$ 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 $Path_\ast(f) \longrightarrow \Omega Cone(f)$ is a stable weak homotopy equivalence and is compatible with the map $f$ (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.
For $K \in Top^{\ast/}_{cg,fin}$ a CW-complex then the operation of smash tensoring $(-) \wedge K$ preserves stable weak homotopy equivalences in $\mathbb{S}_{dia}Mod$.
Since limits and colimits in the diagram category $\mathbb{S}_{dia}Mod$ are computed objectwise, the functor $seq^\ast$ that restricts $\mathbb{S}_{dia}$-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 $K$.
Write
for the stages of the cell complex $K$, so that for each $i$ there is a pushout diagram in $Top^{}_{cg}$ of the form
Equivalently these are pushoutdiagrams in $Top^{\ast/}_{cg}$ of the form
Notice that it is indeed $S^{n_i}$ that appears in the top right, not $S^{n_i}_+$.
Now forming the smash tensoring of any morphism $f\colon X \longrightarrow Y$ in $\mathbb{S}_{dia}Mod(Top_{cg})$ by the morphisms in the pushout on the right yields a commuting diagram in $\mathbb{S}_{dia}Mod$ of the form
Here the horizontal morphisms on the left are degreewise cofibrations in $Top^{\ast/}_{cg}$, hence the morphism on the right is degreewise their homotopy cofiber. This way lemma 4 implies that there are commuting diagrams
where the top and bottom are long exact sequences of stable homotopy groups.
Now proceed by induction. For $i = 0$ then clearly smash tensoring with $K_0 = \ast$ preserves stable weak homotopy equivalences. So assume that smash tensoring with $K_i$ does, too. Observe that $(-)\wedge S^n$ preserves stable weak homotopy equivalences, since $\Sigma X[1]\to X$ 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 $f \wedge K_{i+1}$ 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.).
The pushout in $\mathbb{S}_{dia}Mod$ of a stable weak homotopy equivalence along a morphism that is degreewise a cofibration in $(Top^{\ast/}_{cg})_{Quillen}$ is again a stable weak homotopy equivalence.
Given a pushout square
observe that the pasting law implies an isomorphism between the horizontal cofibers
Moreover, since cofibrations in $(Top^{\ast/}_{cg})_{Quillen}$ 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 $(Top^{\ast/}_{cg})_{Quillen}$. Hence lemma 4 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.
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. 27 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 11 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.
For $Dia \in \{Top^{\ast/}_{fin}, Orth, Sym, Seq\}$ any one of the four diagram shapes of def. 17, and for each $n \in \mathbb{N}$, the functor
that sends a structured spectrum to the $n$th component space of its underlying sequential spectrum has, by prop. 9, a left adjoint
This is called the free structured spectrum-functor.
For the special case $n = 0$ it is also called the structured suspension spectrum functor and denoted
(Hovey-Shipley-Smith 00, def. 2.2.5, MMSS 00, section 8)
Let $Dia \in \{Top^{\ast/}_{fin}, Orth, Sym, Seq\}$ be any one of the four diagram shapes of def. 17. Then
the free spectrum on $K \in Top^{\ast/}_{cg}$ (def. 25) is equivalently the smash tensoring with $K$ (def.) of the free module (def. 2) over $\mathbb{S}_{dia}$ (remark 6) on the representable $y(n) \in [Dia, Top^{\ast/}_{cg}]$
on $n' \in Dia^{op} \stackrel{y}{\hookrightarrow} [Dia, Top^{\ast/}_{cg}]$ its value is given by the following coend expression (def. 10)
In particular the structured sphere spectrum is the free spectrum in degree 0 on the 0-sphere:
and generally for $K \in Top^{\ast/}_{cg}$ then
is the smash tensoring of the strutured sphere spectrum with $K$.
(Hovey-Shipley-Smith 00, below def. 2.2.5, MMSS00, p. 7 with theorem 2.2)
Under the equivalence of categories
from prop. 24, the expression for $F^{dia}_n K$ is equivalently the smash tensoring with $K$ of the functor that $n$ represents over $\mathbb{S}_{dia}Free_{dia}Mod$:
(by fully faithfulness of the Yoneda embedding).
This way the first statement is a special case of the following general fact: For $\mathcal{C}$ a pointed topologically enriched category, and for $c \in \mathcal{C}$ any object, then there is an adjunction
(saying that evaluation at $c$ is right adjoint to smash tensoring the functor represented by $c$) witnessed by the following composite natural isomorphism:
The first is the characteristic isomorphism of tensoring from prop. 8, while the second is the enriched Yoneda lemma of prop. 5.
From this, the second statement follows by the proof of prop. 24.
For the last statement it is sufficient to observe that $y(0)$ is the tensor unit under Day convolution by prop. 11 (since $0$ is the tensor unit in $Dia$), so that
Explicitly, the free spectra according to def. 25, look as follows:
For sequential spectra:
for symmetric spectra:
for orthogonal spectra:
where “$\wedge_G$” is as in example 12.
(e.g. Schwede 12, example 3.20)
With the formula in item 2 of lemma 7 we have for the case of orthogonal spectra
where in the second line we used that the coend collapses to $n_1 = q-n$ ranging in the full subcategory
on the object $\mathbb{R}^{q-n}$ and then we applied example 12. 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.
For $Dia \in \{ Orth, Sym, Seq\}$ the diagram shape for orthogonal spectra, symmetric spectra or sequential spectra, then the free structured spectra
from def. 25 have component spaces that admit the structure of CW-complexes.
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. 25 we have to show that for all $q \geq n \in \mathbb{N}$ 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 $O(q-n)$ induces a smooth action on $D^{q-n}$ (which happens to be constant on $\partial D^{q-n}$).
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 $O(q)\times D^{q-n}$ has the structure of a smooth manifold with boundary and equipped with a smooth action of the compact Lie group $O(q-n)$. Under these conditions (Illman 83, corollary 7.2) states that $O(q) \times D^{q-n}$ admits the structure of a G-CW complex for $G = O(q-n)$, and moreover (Illman 83, line above theorem 7.1) states that this may be chosen such that the boundary is a $G$-CW subcomplex.
Now the quotient of a $G$-CW complex by $G$ 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.
Let $Dia \in \{Top^{\ast/}_{cg,fin}, Orth, Sym\}$ be the diagram shape of either pre-excisive functors, orthogonal spectra or symmetric spectra. Then under the symmetric monoidal smash product of spectra (def. 16, def. 16, def.18) the free structured spectra of def. 25 behave as follows
In particular for structured suspension spectra $\Sigma^\infty_{dia}\coloneqq F_0^{dia}$ (def. 25) this gives isomorphisms
Hence the structured suspension spectrum functor $\Sigma^\infty_{dia}$ is a strong monoidal functor (def. 14) and in fact a braided monoidal functor (def. \ref{braided monoidal functor}) 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 $X \in \mathbb{S}_{dia}Mod$ then
where on the right we have the smash tensoring of $X$ with $K \in Top^{\ast/}_{cg}$.
(MMSS 00, lemma 1.8 with theorem 2.2, Mandell-May 02, prop. 2.2.6)
By lemma 7 the free spectra are free modules over the structured sphere spectrum $\mathbb{S}_{dia}$ of the form $F^{dia}_n(K) \simeq \mathbb{S}_{dia} \otimes_{Day} ( y(n) \wedge K )$. By example 10 the tensor product of such free modules is given by
Using the co-Yoneda lemma (prop. 6) the expression on the right is
For the last statement we may use that $\Sigma^\infty_{dia} K \simeq \mathbb{S}_{dia} \wedge K$, by lemma 7), and that $\mathbb{S}_{dia}$ is the tensor unit for $\otimes_{\mathbb{S}_{dia}}$ by prop. 3.
To see that $\Sigma^\infty_{dia}$ is braided, write $\Sigma^\infty_{dia}K\simeq \mathbb{S} \wedge K$. We need to see that
commutes. Chasing the smash factors through this diagram and using symmetry (def. 3) and the hexagon identities (def. 2) 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.
For each $n \in \mathbb{N}$ there exists a morphism
between free spectra (def. 25) such that for every structured spectrum $X\in \mathbb{S}_{dia} Mod$ precomposition $\lambda_n^\ast$ 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 $(\Sigma \dashv \Omega)$-adjunct of the structure map $\sigma_n$ of the sequential spectrum $seq^\ast X$ underlying $X$ (def. 17).
Since all prescribed morphisms in the diagram are natural transformations, this is in fact a diagram of copresheaves on $\mathbb{S}_{dia} Mod$
With this the statement follows by the Yoneda lemma.
Now we say explicitly what these maps are:
For $n \in \mathbb{N}$, write
for the adjunct under the (free structured spectrum $\dashv$ $n$-component)-adjunction in def. 25 of the composite morphism
where the first morphism is via prop. 25 and the second comes from the adjunction units according to def. 25.
(MMSS 00, def. 8.4, Schwede 12, example 4.26)
(MMSS 00, lemma 8.5, following Hovey-Shipley-Smith 00, remark 2.2.12)
Consider the case $Dia = Seq$ and $n = 0$. All other cases work analogously.
By lemma 25, in this case the morphism $\lambda_0$ has components like so:
Now for $X$ any sequential spectrum, then a morphism $f \colon F_0 S^0 \to X$ is uniquely determined by its 0th components $f_0 \colon S^0 \to X_0$ (that’s of course the very free property of $F_0 S^0$); as the compatibility with the structure maps forces the first component, in particular, to be $\sigma_0^X\circ \Sigma f$:
But that first component is just the component that similarly determines the precompositon of $f$ with $\lambda_0$, hence $\lambda_0^\ast f$ is fully fixed as being the map $\sigma_0^X \circ \Sigma f$. Therefore $\lambda_0^\ast$ is the function
It remains to see that this is the $(\Sigma \dashv \Omega)$-adjunct of $\sigma_0^X$. By the general formula for adjuncts, this is
To compare to the above, we check what this does on points: $S^0 \stackrel{f_0}{\longrightarrow} X_0$ is sent to the composite
To identify this as a map $S^1 \to X_1$ we use the adjunction isomorphism once more to throw all the $\Omega$-s on the right back to $\Sigma$-s the left, to finally find that this is indeed
The maps $\lambda_n \;\colon\; F_{n+1} S^1 \longrightarrow F_n S^0$ in def. 26 are
stable weak homotopy equivalences for sequential spectra, orthogonal spectra and pre-excisive functors, i.e. for ${Dia} \in \{Top^{\ast/}, Orth, Seq\}$;
not stable weak homotopy equivalences for the case of symmetric spectra ${Dia} = {Sym}$.
(Hovey-Shipley-Smith 00, example 3.1.10, MMSS 00, lemma 8.6, Schwede 12, example 4.26)
This follows by inspection of the explicit form of the maps, via prop. 25. We discuss each case separately:
sequential case
Here the components of the morphism eventually stabilize to isomorphisms
and this immediately gives that $\lambda_n$ is an isomorphism on stable homotopy groups.
orthogonal case
Here for $q \geq n+1$ the $q$-component of $\lambda_n$ is the quotient map
By the suspension isomorphism for stable homotopy groups, $\lambda_n$ is a stable weak homotopy equivalence precisely if any of its suspensions is. Hence consider instead $\Sigma^n \lambda_n \coloneqq S^n \wedge \lambda_n$, whose $q$-component is
Now due to the fact that $O(q-k)$-action on $S^q$ lifts to an $O(q)$-action, the quotients of the diagonal action of $O(q-k)$ 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 $(g,s) \in O(q)_+ \wedge S^q$ in the quotient $Q(q)_+ \wedge_{Q(q-k)} S^q$ is labeled by $([g],s)$.
It follows that $(\Sigma^n\lambda_n)_q$ 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 $(q - n -1 )$-connected (see here). Hence its smash product with $S^q$ is $(2q - n -1 )$-connected.
The key here is the fast growth of the connectivity with $q$. This implies that for each $s$ there exists $q$ such that $\pi_{s+q}((\Sigma^n \lambda_n)_q)$ becomes an isomorphism. Hence $\Sigma^n \lambda_n$ is a stable weak homotopy equivalence and therefore so is $\lambda_n$.
symmetric case
Here the morphism $\lambda_n$ 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 $q$ increases, due to the discrete topological space underlying the symmetric group. Accordingly the conclusion now is the opposite: $\lambda_n$ 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.
The symmetric monoidal smash product of spectra of the free spectrum constructions (def. 25) on the generating cofibrations $\{S^{n-1}\overset{i_n}{\hookrightarrow} D^n\}_{n \in \mathbb{B}}$ of the classical model structure on topological spaces is given by addition of indices
By lemma 26 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 four categories of
pre-excisive functors $Exc(Top_{cg})$;
orthogonal spectra $OrthSpec(Top_{cg}) = \mathbb{S}_{orth} Mod$;
symmetric spectra $SymSpec(Top_{cg}) = \mathbb{S}_{sym}Mod$;
sequential spectra $SeqSpec(Top_{cg}) = \mathbb{S}_{seq}Mod$
(from def. 16, prop. 19, def. 18) 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 $(Top^{\ast/}_{cg})_{Quillen}$. (thm., prop.). These are called the strict model structures (or level model structures) on structured spectra.
Moreover, under the equivalences of categories of prop. 19 and prop. 20, the restriction functors in def. 17 constitute right adjoints of Quillen adjunctions (def.) between these model structures:
By prop. 24 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 $dia^\ast$ each have a left adjoint $dia_!$ by topological left Kan extension (prop. 9).
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.
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. 25, on the morphisms in $I_{Top^{\ast/}}$ from above. Similarly, write
for the set of images under forming free spectra of the morphisms in ${J}_{{\mathrm{Top}}_{\mathrm{cg}}^{*/}}$