# nLab symmetric smash product of spectra

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A symmetric smash product of spectra is a realization of the smash product of spectra such as to make a symmetric monoidal model category presentation of the symmetric monoidal (infinity,1)-category of spectra.

In higher algebra and stable homotopy theory one is interested in monoid objects in the stable (∞,1)-category of spectra – called $A_\infty$-rings – and commutative monoid objects – called $E_\infty$-rings. These monoid objects satisfy associativity, uniticity and, in the $E_\infty$-case, commutativity up to coherent higher homotopies.

For concretely working with these objects, it is often useful to have concrete 1-categorical algebraic models for these intricate higher categorical/homotopical entities. The symmetric monoidal smash product of spectra is a structure that allows to model A-infinity rings as ordinary monoids and E-infinity rings as ordinary commutative monoids in a suitable ordinary category – one speaks of highly structured ring spectra.

Historically, this had been desired but out of reach for a long time, due to the initial focus on the model by plain sequential spectra. By this remark at smash product of spectra, plain sequential spectra do not reflect the graded-commutativity implicit in the braiding of the smash product of n-spheres and thus do not admit a symmetric smash product of spectra.

When the relevant highly structured ring spectra were finally found that do admit symmetric smash products, the relief was substantial and led to terminology such as “brave new algebra”. More recently maybe the term higher algebra is becoming more popular.

Then, model structures were found which also admit symmetric monoidal smash products, but which are not of the form “highly structured spectra”: model structure for excisive functors.

As a first step one wants a model category of spectra $\mathcal{S}$ that presents the full (infinity,1)-category of spectra. This allows to model the notion of equivalence of spectra and of homotopies between maps of spectra. Several such model categories have been known for a long time; all are Quillen equivalent and their common homotopy category is called “the” stable homotopy category $Ho \mathcal{S}$.

Now, for some of the model categories $\mathcal{S}$ of spectra, the smash product on $Ho \mathcal{S}$ can be lifted to a functor

$\wedge\colon \mathcal{S} \times \mathcal{S} \to \mathcal{S} \,,$

but for the most part these functors were neither associative nor unital nor commutative at the level of the 1-category $\mathcal{S}$. In fact (Lewis 91) proved a theorem that there could be no symmetric monoidal category $\mathcal{S}$ modeling the stable homotopy category and satisfying a couple of other natural requirements.

However, in the 1990s it was realized that by dropping one or another of Lewis’ other requirements, symmetric monoidal categories of spectra could be produced. The first such category was the category of S-modules described by Elmendorf-Kriz-Mandell-May 97, but others soon followed, including symmetric spectra and orthogonal spectra. All of these form symmetric monoidal model categories which are symmetric-monoidally Quillen equivalent.

Moreover, in all of these cases, the monoidal structure on the model category $\mathcal{S}$ absorbs all the higher coherent homotopies that used to be supplied by the action of an $A_\infty$ or $E_\infty$ operad. Thus, honest (commutative) monoids in $\mathcal{S}$ model the same “(commutative) ring objects up to all coherent higher homotopies” that are modeled by the classical $A_\infty$ and $E_\infty$ ring spectra, and for this reason they are often still referred to as $A_\infty$ or $E_\infty$ ring spectra, respectively.

## Details

### For $S$-modules

The construction of S-modules by EKMM begins with the notion of coordinate free Lewis-May spectra. Using the linear isometries operad, one can construct a monad $\mathbb{L}$ on the category $\mathcal{S}$ of such spectra, and the category of $\mathbb{L}$-algebras is a well-behaved model for the stable homotopy category, and moreover admits a smash product which is associative up to isomorphism, but unital only up to weak equivalence. However, the subcategory of the $\mathbb{L}$-algebras for which the unit transformations are isomorphisms is again a well-behaved model for $Ho \mathbb{S}$, which is moreover symmetric monoidal.

Since the unit transformation is of the form $S\wedge E \to E$, where $S$ is the sphere spectrum, and this map looks like the action of a ring on a module, the objects of this subcategory are called $S$-modules and the category is called $Mod_S$. The intuition is that just as an abelian group is a module over the archetypical ring $\mathbb{Z}$ of integers, a spectrum should be regarded as a module over the archetypal ring spectrum, namely the sphere spectrum.

Similarly, just as an ordinary ring is a monoid in the category $Mod_\mathbb{Z}$ of $\mathbb{Z}$-modules, i.e. a $\mathbb{Z}$-algebra, an $A_\infty$ or $E_\infty$ ring spectrum is a (possibly commutative) monoid in the category of $S$-modules, and thus referred to as an $S$-algebra. More generally, for any $A_\infty$-ring spectrum $R$, there is a notion of $R$-module spectra forming a category $Mod_R$, which in turn carries an associative and commutative smash product $\wedge_R$ and a model category structure on $Mod_R$ such that $\wedge_R$ becomes unital in the homotopy category. All this is such that an $A_\infty$-algebra over $R$ is a monoid object in $(Mod_R, \wedge_R)$. Similarly $E_\infty$-algebras are commutative monoid objects in $(Mod_R, \wedge_R)$.

### For excisive functors

#### Topological ends and coends

For working with pointed topologically enriched functors, a certain shape of limits/colimits is particularly relevant: these are called (pointed topological enriched) ends and coends. We here introduce these and then derive some of their basic properties, such as notably the expression for topological left Kan extension in terms of coends (prop. below). Further below it is via left Kan extension along the ordinary smash product of pointed topological spaces (“Day convolution”) that the symmetric monoidal smash product of spectra is induced.

###### Definition

Let $\mathcal{C}, \mathcal{D}$ be pointed topologically enriched categories (def.), i.e. enriched categories over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example .

1. The pointed topologically enriched opposite category $\mathcal{C}^{op}$ is the topologically enriched category with the same objects as $\mathcal{C}$, with hom-spaces

$\mathcal{C}^{op}(X,Y) \coloneqq \mathcal{C}(Y,X)$

and with composition given by braiding followed by the composition in $\mathcal{C}$:

$\mathcal{C}^{op}(X,Y) \wedge \mathcal{C}^{op}(Y,Z) = \mathcal{C}(Y,X)\wedge \mathcal{C}(Z,Y) \underoverset{\simeq}{\tau}{\longrightarrow} \mathcal{C}(Z,Y) \wedge \mathcal{C}(Y,X) \overset{\circ_{Z,Y,X}}{\longrightarrow} \mathcal{C}(Z,X) = \mathcal{C}^{op}(X,Z) \,.$
2. 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

$(\mathcal{C}\times \mathcal{D})((c_1,d_1),\;(c_2,d_2) ) \coloneqq \mathcal{C}(c_1,c_2)\wedge \mathcal{D}(d_1,d_2)$

and whose composition operation is the braiding followed by the smash product of the separate composition operations:

$\array{ (\mathcal{C}\times \mathcal{D})((c_1,d_1), \; (c_2,d_2)) \wedge (\mathcal{C}\times \mathcal{D})((c_2,d_2), \; (c_3,d_3)) \\ {}^{\mathllap{=}}\downarrow \\ \left(\mathcal{C}(c_1,c_2) \wedge \mathcal{D}(d_1,d_2)\right) \wedge \left(\mathcal{C}(c_2,c_3) \wedge \mathcal{D}(d_2,d_3)\right) \\ \downarrow^{\mathrlap{\tau}}_{\mathrlap{\simeq}} \\ \left(\mathcal{C}(c_1,c_2)\wedge \mathcal{C}(c_2,c_3)\right) \wedge \left( \mathcal{D}(d_1,d_2)\wedge \mathcal{D}(d_2,d_3)\right) &\overset{ (\circ_{c_1,c_2,c_3})\wedge (\circ_{d_1,d_2,d_3}) }{\longrightarrow} & \mathcal{C}(c_1,c_3)\wedge \mathcal{D}(d_1,d_3) \\ && \downarrow^{\mathrlap{=}} \\ && (\mathcal{C}\times \mathcal{D})((c_1,d_1),\; (c_3,d_3)) } \,.$
###### Example

A pointed topologically enriched functor (def.) into $Top^{\ast/}_{cg}$ (exmpl.) out of a pointed topological product category as in def.

$F \;\colon\; \mathcal{C} \times \mathcal{D} \longrightarrow Top^{\ast/}_{cg}$

(a “pointed topological bifunctor”) has component maps of the form

$F_{(c_1,d_1),(c_2,d_2)} \;\colon\; \mathcal{C}(c_1,c_2) \wedge \mathcal{D}(d_1,d_2) \longrightarrow Maps(F_0((c_1,d_1)),F_0((c_2,d_2)))_\ast \,.$

By functoriallity and under passing to adjuncts (cor.) this is equivalent to two commuting actions

$\rho_{c_1,c_2}(d) \;\colon\; \mathcal{C}(c_1,c_2) \wedge F_0((c_1,d)) \longrightarrow F_0((c_2,d))$

and

$\rho_{d_1,d_2}(c) \;\colon\; \mathcal{D}(d_1,d_2) \wedge F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.$

In the special case of a functor out of the product category of some $\mathcal{C}$ with its opposite category (def. )

$F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Top^{\ast/}_{cg}$

then this takes the form

$\rho_{c_2,c_1}(d) \;\colon\; \mathcal{C}(c_1,c_2) \wedge F_0((c_2,d)) \longrightarrow F_0((c_1,d))$

and

$\rho_{d_1,d_2}(c) \;\colon\; \mathcal{C}(d_1,d_2) \wedge F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.$
###### Definition

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 . Let

$F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Top^{\ast/}_{cg}$

be a pointed topologically enriched functor (def.) out of the pointed topological product category of $\mathcal{C}$ with its opposite category, according to def. .

1. 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 :

$\underset{c,d \in \mathcal{C}}{\coprod} \mathcal{C}(c,d) \wedge F(d,c) \underoverset {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c \in \mathcal{C}}{\coprod} F(c,c) \overset{coeq}{\longrightarrow} \overset{c\in \mathcal{C}}{\int} F(c,c) \,.$
2. 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 :

$\underset{c\in \mathcal{C}}{\int} F(c,c) \overset{\;\;equ\;\;}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} F(c,c) \underoverset {\underset{\underset{c,d}{\sqcup} \tilde \rho_{(c,d)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \tilde\rho_{d,c}(d)}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c\in \mathcal{C}}{\prod} Maps\left( \mathcal{C}\left(c,d\right), \; F\left(c,d\right) \right)_\ast \,.$
###### Example

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. ) of $Maps(F(-),G(-))_\ast$ is a topological space whose underlying pointed set is the pointed set of natural transformations $F\to G$ (def.)

$U \left( \underset{c \in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast \right) \;\simeq\; Hom_{[\mathcal{C},Top^{\ast/}_{cg}]}(F,G) \,.$
###### Proof

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

$U \left( \underset{c\in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast \right) \overset{equ}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} Hom_{Top^{\ast/}_{cg}}(F(c),G(c)) \underoverset {\underset{\underset{c,d}{\sqcup} U(\tilde \rho_{(c,d)}(d)) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} U(\tilde\rho_{d,c}(c))}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c,d\in \mathcal{C}}{\prod} Hom_{Top^{\ast/}_{cg}}( \mathcal{C}(c,d), Maps(F(c),G(d))_\ast ) \,.$

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

$\array{ F(c) \\ {}^{\mathllap{f(f)}}\downarrow \\ F(d) &\underset{\eta_d}{\longrightarrow}& G(d) }$

and of postcomposing

$\array{ F(c) &\overset{\eta_c}{\longrightarrow}& G(c) \\ && \downarrow^{\mathrlap{G(f)}} \\ && G(d) }$

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

$\array{ F(c) &\overset{\eta_c}{\longrightarrow}& G(c) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(d) &\underset{\eta_d}{\longrightarrow}& G(g) }$

is a commuting square. This is precisley the condition that the collection $\{\eta_c\}_{c\in \mathcal{C}}$ be a natural transformation.

Conversely, example says that ends over bifunctors of the form $Maps(F(-),G(-)))_\ast$ constitute hom-spaces between pointed topologically enriched functors:

###### Definition

Let $\mathcal{C}$ be a small pointed topologically enriched categories (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. ) of example :

$[\mathcal{C},Top^{\ast/}_{cg}](F,G) \;\coloneqq\; \int_{c\in \mathcal{C}} Maps(F(c),G(c))_\ast$

and by taking the composition maps to be the morphisms induced by the maps

$\left( \underset{c\in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast \right) \wedge \left( \underset{c \in \mathcal{C}}{\int} Maps(G(c),H(c))_\ast \right) \overset{}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} Maps(F(c),G(c))_\ast \wedge Maps(G(c),H(c))_\ast \overset{(\circ_{F(c),G(c),H(c)})_{c\in \mathcal{C}}}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} Maps(F(c),H(c))_\ast$

by observing that these equalize the two actions in the definition of 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 $Top^{\ast/}_{cg}$.

First of all this yields a concise statement of the pointed topologically enriched Yoneda lemma (prop.)

###### Proposition

(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

$[\mathcal{C}, Top^{\ast/}_{cg}](\mathcal{C}(c,-),\; F) \;\simeq\; F(c)$

between the hom-space of the pointed topological functor category, according to def. , from the functor represented by $c$ to $F$, and the value of $F$ on $c$.

In terms of the ends (def. ) defining these hom-spaces, this means that

$\underset{d\in \mathcal{C}}{\int} Maps(\mathcal{C}(c,d), F(d))_\ast \;\simeq\; F(c) \,.$

In this form the statement is also known as Yoneda reduction.

The proof of prop. is essentially dual to the proof of the next prop. .

Now that natural transformations are phrased in terms of ends (example ), as is the Yoneda lemma (prop. ), it is natural to consider the dual statement involving coends:

###### Proposition

(co-Yoneda lemma)

Let $\mathcal{C}$ 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

$F(-) \simeq \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \wedge F(c) \,.$

Moreover, the morphism that hence exhibits $F(c)$ as the coequalizer of the two morphisms in def. is componentwise the canonical action

$\mathcal{C}(d,c) \wedge F(c) \longrightarrow F(d)$

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)

###### Proof

The coequalizer of pointed topological spaces that we need to consider has underlying it a coequalizer of underlying pointed sets (prop., prop., prop.). That in turn is the colimit over the diagram of underlying sets with the basepointe adjoined to the diagram (prop.). For a coequalizer diagram adding that extra point to the diagram clearly does not change the colimit, and so we need to consider the plain coequalizer of sets.

That is just the set of equivalence classes of pairs

$( c \overset{}{\to} c_0,\; x \in F(c) ) \,,$

where two such pairs

$( c \overset{f}{\to} c_0,\; x \in F(c) ) \,,\;\;\;\; ( d \overset{g}{\to} c_0,\; y \in F(d) )$

are regarded as equivalent if there exists

$c \overset{\phi}{\to} d$

such that

$f = g \circ \phi \,, \;\;\;\;\;and\;\;\;\;\; y = \phi(x) \,.$

(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

$( c \overset{\phi}{\to} c_0, \; x \in F(c))$

is identified, in the coequalizer, with the pair

$(id_{c_0},\; \phi(x) \in F(c_0)) \,,$

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

$\mathcal{C}(d,c) \wedge F(c) \longrightarrow \overset{c}{\int} \mathcal{C}(c,c_0) \wedge F(c)$

which we just found. But that system includes

$\mathcal{C}(c,c) \wedge F(c) \longrightarrow F(c)$

which is a retraction

$id \;\colon\; F(c) \longrightarrow \mathcal{C}(c,c) \wedge F(c) \longrightarrow F(c)$

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 $F(c)$.

###### Remark

The statement of the co-Yoneda lemma in prop. 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

$\int_{x \in X} \delta(x,x_0) f(x) = f(x_0) \,.$

It is this analogy that gives the name to the following statement:

###### Proposition

(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. ), i.e.

$F \;\colon\; \left( \mathcal{C}_1\times\mathcal{C}_2 \right)^{op} \times (\mathcal{C}_1 \times\mathcal{C}_2) \longrightarrow Top^{\ast/}_{cg}$

then its end and coend (def. ) is equivalently formed consecutively over each variable, in either order:

$\overset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_1}{\int} \overset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_2}{\int} \overset{c_1}{\int} F((c_1,c_2), (c_1,c_2))$

and

$\underset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_1}{\int} \underset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_2}{\int} \underset{c_1}{\int} F((c_1,c_2), (c_1,c_2)) \,.$
###### Proof

Because limits commute with limits, and colimits commute with colimits.

###### Remark

Because the pointed compactly generated mapping space functor (exmpl.)

$Maps(-,-)_\ast \;\colon\; \left(Top^{\ast/}_{cg}\right)^{op} \times Top^{\ast/}_{cg} \longrightarrow Top^{\ast/}_{cg}$

takes colimits in the first argument and limits in the second argument to limits (cor.), it also takes coends in the first argument and ends in the second argument, to ends (def. ):

$Maps( X, \; \int_{c} F(c,c))_\ast \simeq \int_c Maps(X, F(c,c)_\ast)$

and

$Maps( \int^{c} F(c,c),\; Y )_\ast \simeq \underset{c}{\int} Maps( F(c,c),\; Y )_\ast \,.$
###### Proposition

(left Kan extension via coends)

Let $\mathcal{C}, \mathcal{D}$ be small pointed topologically enriched categories (def.) and let

$p \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$

be a pointed topologically enriched functor (def.). Then precomposition with $p$ constitutes a functor

$p^\ast \;\colon\; [\mathcal{D}, Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}]$

$G\mapsto G\circ p$. This functor has a left adjoint $Lan_p$, called left Kan extension along $p$

$[\mathcal{D}, Top^{\ast/}_{cg}] \underoverset {\underset{p^\ast}{\longrightarrow}} {\overset{Lan_p }{\longleftarrow}} {\bot} [\mathcal{C}, Top^{\ast/}_{cg}]$

which is given objectwise by a coend (def. ):

$(Lan_p F) \;\colon\; d \;\mapsto \; \overset{c\in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \wedge F(c) \,.$
###### Proof

Use the expression of natural transformations in terms of ends (example and def. ), then use the respect of $Maps(-,-)_\ast$ for ends/coends (remark ), use the smash/mapping space adjunction (cor.), use the Fubini theorem (prop. ) and finally use Yoneda reduction (prop. ) to obtain a sequence of natural isomorphisms as follows:

\begin{aligned} [\mathcal{D},Top^{\ast/}_{cg}]( Lan_p F, \, G ) & = \underset{d \in \mathcal{D}}{\int} Maps( (Lan_p F)(d), \, G(d) )_\ast \\ & = \underset{d\in \mathcal{D}}{\int} Maps\left( \overset{c \in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \wedge F(c) ,\; G(d) \right)_\ast \\ &\simeq \underset{d \in \mathcal{D}}{\int} \underset{c \in \mathcal{C}}{\int} Maps( \mathcal{D}(p(c),d)\wedge F(c) \,,\; G(d) )_\ast \\ & \simeq \underset{c\in \mathcal{C}}{\int} \underset{d\in \mathcal{D}}{\int} Maps(F(c), Maps( \mathcal{D}(p(c),d) , \, G(d) )_\ast )_\ast \\ & \simeq \underset{c\in \mathcal{C}}{\int} Maps(F(c), \underset{d\in \mathcal{D}}{\int} Maps( \mathcal{D}(p(c),d) , \, G(d) )_\ast )_\ast \\ & \simeq \underset{c\in \mathcal{C}}{\int} Maps(F(c), G(p(c)) )_\ast \\ & = [\mathcal{C}, Top^{\ast/}_{cg}](F,p^\ast G) \end{aligned} \,.
##### Monoidal topological categories

We recall the basic definitions of monoidal categories and of monoids and modules internal to monoidal categories. All examples are at the end of this section, starting with example below.

###### Definition

A (pointed) topologically enriched monoidal category is a (pointed) topologically enriched category $\mathcal{C}$ (def.) equipped with

1. a (pointed) topologically enriched functor (def.)

$\otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$

out of the (pointed) topologival product category of $\mathcal{C}$ with itself (def. ), called the tensor product,

2. an object

$1 \in \mathcal{C}$

called the unit object or tensor unit,

3. $a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))$

called the associator,

4. $\ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-)$

called the left unitor, and a natural isomorphism

$r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)$

called the right unitor,

such that the following two kinds of diagrams commute, for all objects involved:

1. triangle identity:

$\array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }$
###### Lemma

(Kelly 64)

Let $(\mathcal{C}, \otimes, 1)$ be a monoidal category, def. . Then the left and right unitors $\ell$ and $r$ satisfy the following conditions:

1. $\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;

2. for all objects $x,y \in \mathcal{C}$ the following diagram commutes:

$\array{ (1 \otimes x) \otimes y & & \\ {}^\mathllap{\alpha_{1, x, y}} \downarrow & \searrow^\mathrlap{\ell_x y} & \\ 1 \otimes (x \otimes y) & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y } \,.$

Analogously for the right unitor.

###### Definition

A (pointed) topological braided monoidal category, is a (pointed) topological monoidal category $\mathcal{C}$ (def. ) equipped with a natural isomorphism

$\tau_{x,y} \colon x \otimes y \to y \otimes x$

called the braiding, such that the following two kinds of diagrams commute for all objects involved:

$\array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) }$

and

$\array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,$

where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.

###### Definition

A (pointed) topological symmetric monoidal category is a (pointed) topological braided monoidal category (def. ) for which the braiding

$\tau_{x,y} \colon x \otimes y \to y \otimes x$

satisfies the condition:

$\tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}$

for all objects $x, y$

###### Definition

Given a (pointed) topological symmetric monoidal category $\mathcal{C}$ with tensor product $\otimes$ (def. ) it is called a closed monoidal category if for each $Y \in \mathcal{C}$ the functor $Y \otimes(-)\simeq (-)\otimes X$ has a right adjoint, denoted $[Y,-]$

$\mathcal{C} \underoverset {\underset{[Y,-]}{\longrightarrow}} {\overset{(-) \otimes Y}{\longleftarrow}} {\bot} \mathcal{C} \,,$

hence if there are natural isomorphisms

$Hom_{\mathcal{C}}(X \otimes Y, Z) \;\simeq\; Hom_{\mathcal{C}}{C}(X, [Y,Z])$

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

$Hom_{\mathcal{C}}(1,[Y,Z]) \simeq Hom_{\mathcal{C}}(Y,Z) \,,$

the object $[Y,Z] \in \mathcal{C}$ 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 $Z$.

###### Example

The category Set of sets and functions between them, regarded as enriched in discrete topological spaces, becomes a symmetric monoidal category according to def. 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 $Top_{cg}$ 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.

###### Example

The category $Top_{cg}^{\ast/}$ of pointed compactly generated topological spaces with tensor product the smash product $\wedge$ (def.)

$X \wedge Y \coloneqq \frac{X\times Y}{X\vee Y}$

is a symmetric monoidal category (def. ) 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 , 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..

###### Example

The category Ab of abelian groups, regarded as enriched in discrete topological spaces, becomes a symmetric monoidal category with tensor product the actual tensor product of abelian groups $\otimes_{\mathbb{Z}}$ 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 .

This is the archetypical case that motivates the notation “$\otimes$” for the pairing operation in a monoidal category:

1. A monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a ring.

2. A commutative monoid in in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a commutative ring $R$.

3. An $R$-module object in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently an $R$-module;

4. The tensor product of $R$-module objects (def. ) is the standard tensor product of modules.

5. The category of module objects $R Mod(Ab)$ (def. ) is the standard category of modules $R Mod$.

##### Algebras and modules
###### Definition

Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is

1. an object $A \in \mathcal{C}$;

2. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)

3. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);

such that

1. (associativity) the following diagram commutes

$\array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,$

where $a$ is the associator isomorphism of $\mathcal{C}$;

2. (unitality) the following diagram commutes:

$\array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,$

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. ) $(\mathcal{C}, \otimes, 1, B)$ 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

$\array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.$

A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$f \;\colon\; A_1 \longrightarrow A_2$

in $\mathcal{C}$, such that the following two diagrams commute

$\array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }$

and

$\array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.$

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.

###### Example

Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, then the tensor unit $1$ is a monoid in $\mathcal{C}$ (def. ) with product given by either the left or right unitor

$\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.$

By lemma , 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. ), then this monoid is a commutative monoid.

###### Definition

Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(A,\mu,e)$ a monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), then a left module object in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

1. an object $N \in \mathcal{C}$;

2. a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);

such that

1. (unitality) the following diagram commutes:

$\array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && A } \,,$

where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.

2. (action property) the following diagram commutes

$\array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,$

A homomorphism of left $A$-module objects

$(N_1, \rho_1) \longrightarrow (N_2, \rho_2)$

is a morphism

$f\;\colon\; N_1 \longrightarrow N_2$

in $\mathcal{C}$, such that the following diagram commutes:

$\array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.$

For the resulting category of modules of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write

$A Mod(\mathcal{C}) \,.$

This is naturally a (pointed) topologically enriched category itself.

###### Example

Given a monoidal category $(\mathcal{C},\otimes, 1)$ (def. ) with the tensor unit $1$ regarded as a monoid in a monoidal category via example , then the left unitor

$\ell_C \;\colon\; 1\otimes C \longrightarrow C$

makes every object $C \in \mathcal{C}$ into a left module, according to def. , over $C$. The action property holds due to lemma . This gives an equivalence of categories

$\mathbb{C} \simeq 1 Mod(\mathcal{C})$

of $\mathcal{C}$ with the category of modules over its tensor unit.

###### Proposition

In the situation of def. , 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:

$\rho \;\colon\; A \otimes (A \otimes C) \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow} (A \otimes A) \otimes C \overset{\mu \otimes id}{\longrightarrow} A \otimes C \,.$

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 $U(N,\rho) \coloneqq N$.

$A Mod(\mathcal{C}) \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} \mathcal{C} \,.$
###### Proof

A homomorphism out of a free $A$-module is a morphism in $\mathcal{C}$ of the form

$f \;\colon\; A\otimes C \longrightarrow N$

fitting into the diagram (where we are notationally suppressing the associator)

$\array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,.$

Consider the composite

$\tilde f \;\colon\; C \underoverset{\simeq}{\ell_C}{\longrightarrow} 1 \otimes C \overset{e\otimes id}{\longrightarrow} A \otimes C \overset{f}{\longrightarrow} N \,,$

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)

$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,.$

Pasting this square onto the top of the previous one yields

$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,,$

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

$f = \rho \circ (id_A \otimes \tilde f) \,.$

This natural bijection between $f$ and $\tilde f$ establishes the adjunction.

###### Definition

Given a (pointed) topological symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-module objects (def.), then the tensor product of modules $N_1 \otimes_A N_2$ is, if it exists, the coequalizer

$N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coequ}{\longrightarrow} N_1 \otimes_A N_2$
###### Proposition

Given a (pointed) topological symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ). If all coequalizers exist in $\mathcal{C}$, then the tensor product of modules $\otimes_A$ from def. 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. .

###### Definition

Given a monoidal category of modules $(A Mod , \otimes_A , A)$ as in prop. , then a monoid $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. ) is called an $A$-algebra.

###### Propposition

Given a monoidal category of modules $(A Mod , \otimes_A , A)$ in a monoidal category $(\mathcal{C},\otimes, 1)$ as in prop. , and an $A$-algebra $(E,\mu,e)$ (def. ), then there is an equivalence of categories

$A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/}$

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)

###### Proof

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 $\mu_E$

$\array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,.$

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

$(\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E }$

commutes. Moreover it satisfies the unit property

$\array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,.$

By forgetting the tensor product over $A$, the latter gives

$\array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,,$

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

$\array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,.$

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

$\rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,.$

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.

##### Day convolution
###### Definition

Let $\mathcal{C}$ be a small pointed topological monoidal category (def. ) with tensor product denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$.

Then the Day convolution tensor product on the pointed topological enriched functor category $[\mathcal{C},Top^{\ast/}_{cg}]$ (def. ) is the functor

$\otimes_{Day} \;\colon\; [\mathcal{C},Top^{\ast/}_{cg}] \times [\mathcal{C},Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C},Top^{\ast/}_{cg}]$

out of the pointed topological product category (def. ) given by the following coend (def. )

$X \otimes_{Day} Y \;\colon\; c \;\mapsto\; \overset{(c_1,c_2)\in \mathcal{C}\times \mathcal{C}}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c) \wedge X(c_1) \wedge Y(c_2) \,.$
###### Example

Let $Seq$ denote the category with objects the natural numbers, and only the zero morphisms and identity morphisms on these objects:

$Seq(n_1,n_2) \coloneqq \left\{ \array{ S^0 & if\; n_1 = n_2 \\ \ast & otherwise } \right. \,.$

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. is

\begin{aligned} (X \otimes_{Day} Y)_n & = \overset{(n_1,n_2)}{\int} Seq(n_1 + n_2 , n) \wedge X_{n_1} \wedge X_{n_2} \\ & = \underset{{n_1+n_2} \atop {= n}}{\coprod} \left(X_{n_1}\wedge X_{n_2}\right) \end{aligned} \,.

We observe now that Day convolution is equivalently a left Kan extension (def. ). This will be key for understanding monoids and modules with respect to Day convolution.

###### Definition

Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). Its external tensor product is the pointed topologically enriched functor

$\overline{\wedge} \;\colon\; [\mathcal{C},Top^{\ast/}_{cg}] \times [\mathcal{C},Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}\times \mathcal{C}, Top^{\ast/}_{cg}]$

given by

$X \overline{\wedge} Y \;\coloneqq\; \wedge \circ (X,Y) \,,$

i.e.

$(X \overline\wedge Y)(c_1,c_2) = X(c_1)\wedge X(c_2) \,.$
###### Proposition

The Day convolution product (def. ) of two functors is equivalently the left Kan extension (def. ) of their external tensor product (def. ) along the tensor product $\otimes_{\mathcal{C}}$: there is a natural isomorphism

$X \otimes_{Day} Y \simeq Lan_{\otimes_{\mathcal{C}}} (X \overline{\wedge} Y) \,.$

Hence the adjunction unit is a natural transformation of the form

$\array{ \mathcal{C} \times \mathcal{C} && \overset{X \overline{\wedge} Y}{\longrightarrow} && Top^{\ast/}_{cg} \\ & {}^{\mathllap{\otimes}}\searrow &\Downarrow& \nearrow_{\mathrlap{X \otimes_{Day} Y}} \\ && \mathcal{C} } \,.$

This perspective is highlighted in (MMSS 00, p. 60).

###### Proof

By prop. we may compute the left Kan extension as the following coend:

\begin{aligned} Lan_{\otimes_{\mathcal{C}}} (X\overline{\wedge} Y)(c) & \simeq \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c ) \wedge (X\overline{\wedge}Y)(c_1,c_2) \\ & = \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1\otimes c_2) \wedge X(c_1)\wedge X(c_2) \end{aligned} \,.
###### Corollary

The Day convolution $\otimes_{Day}$ (def. ) is universally characterized by the property that there are natural isomorphisms

$[\mathcal{C},Top^{\ast/}_{cg}](X \otimes_{Day} Y, Z) \simeq [\mathcal{C}\times \mathcal{C},Top^{\ast/}_{cg}]( X \overline{\wedge} Y,\; Z \circ \otimes ) \,,$

where $\overline{\wedge}$ is the external product of def. .

Write

$y \;\colon\; \mathcal{C}^{op} \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}]$

for the $Top^{\ast/}_{cg}$-Yoneda embedding, so that for $c\in \mathcal{C}$ any object, $y(c)$ is the corepresented functor $y(c)\colon d \mapsto \mathcal{C}(c,d)$.

###### Proposition

For $\mathcal{C}$ a small pointed topological monoidal category (def. ), the Day convolution tensor product $\otimes_{Day}$ of def. makes the pointed topologically enriched functor category

$( [\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$

into a pointed topological monoidal category (def. ) 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 braiding $\tau^{\mathcal{C}}$ (def. ), then $( [\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$ becomes itself a braided monoidal category with braiding given by

$\array{ (X \otimes_{Day} Y)(c) & = & \overset{c_1,c_2}{\int} \mathcal{C}(c_1 \otimes c_2) \wedge X(c_1) \wedge Y(c_2) \\ {}^{\mathllap{\tau}_{X,Y}(c)}\downarrow && \downarrow^{\mathrlap{\overset{c_1,c_2}{\int} \mathcal{C}(\tau^{\mathcal{C}}_{c_1,c_2}, c ) \wedge \tau^{Top^{\ast/}}_{X(c(1)), X(c_2)} }} \\ (Y \otimes_{Day} X)(c) & = & \overset{c_1,c_2}{\int} \mathcal{C}(c_2 \otimes c_1) \wedge Y(c_2) \wedge X(c_1) } \,.$
###### Proof

Regarding associativity, observe that

\begin{aligned} (X \otimes_{Day} ( Y \otimes_{Day} Z ))(c) & \simeq \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{D}} c_2, \,c) \wedge X(c_1) \wedge \overset{(d_1,d_2)}{\int} \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_2 ) (Y(d_2) \wedge Z(d_2)) \\ &\simeq \overset{c_1, d_1, d_2}{\int} \underset{\simeq \mathcal{C}(c_1 \otimes_{\mathcal{C}} d_1 \otimes_{\mathcal{C}} d_2, c )}{ \underbrace{ \overset{c_2}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{D}} c_2 , c) \wedge \mathcal{C}(d_1 \otimes_{\mathcal{C}}d_2, c_2 ) } } \wedge X(c_1) \wedge (Y(d_1) \wedge Z(d_2)) \\ &\simeq \overset{c_1, d_1, d_2}{\int} \mathcal{C}(c_1\otimes_{\mathcal{C}} d_1 \otimes_{\mathcal{C}} d_2, c ) \wedge X(c_1) \wedge (Y(d_1) \wedge Z(d_2)) \end{aligned} \,,

where we used the Fubini theorem for coends (prop. ) and then twice the co-Yoneda lemma (prop. ). An analogous formula follows for $X \otimes_{Day} (Y \otimes_{Day} Z)))(c)$, and so associativity follows via prop. from the associativity of the smash product and of the tensor product $\otimes_{\mathcal{C}}$.

Similarly, if $\mathcal{C}$ is braided then the hexagon identity for the braiding follows, under the coend, from the hexagon identities for the braidings in $\mathcal{C}$ and $Top^{\ast/}_{cg}$.

To see that $y(1)$ is the tensor unit for $\otimes_{Day}$, use the Fubini theorem for coends (prop. ) and then twice the co-Yoneda lemma (prop. ) to get for any $X \in [\mathcal{C},Top^{\ast/}_{cg}]$ that

\begin{aligned} X \otimes_{Day} y(1) & = \overset{c_1,c_2 \in \mathcal{C}}{\int} \mathcal{C}(c_1\otimes_{\mathcal{D}} c_2,-) \wedge X(c_1) \wedge \mathcal{C}(1,c_2) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} X(c_1) \wedge \overset{c_2 \in \mathcal{C}}{\int} \mathcal{C}(c_1\otimes_{\mathcal{C}} c_2,-) \wedge \mathcal{C}(1,c_2) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} X(c_1) \wedge \mathcal{C}(c_1 \otimes_{\mathcal{C}} 1, -) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} X(c_1) \wedge \mathcal{C}(c_1, -) \\ & \simeq X(-) \\ & \simeq X \end{aligned} \,.
###### Proposition

For $\mathcal{C}$ a small pointed topological monoidal category (def. ) with tensor product denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$, the monoidal category with Day convolution $([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$ from def. is a closed monoidal category (def. ). Its internal hom $[-,-]_{Day}$ is given by the end (def. )

$[X,Y]_{Day}(c) \simeq \underset{c_1,c_2}{\int} Maps\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2), \; Maps(X(c_1) , Y(c_2))_\ast \right)_\ast \,.$
###### Proof

Using the Fubini theorem (def. ) and the co-Yoneda lemma (def. ) and in view of definition of the enriched functor category, there is the following sequence of natural isomorphisms:

\begin{aligned} [\mathcal{C},V]( X, [Y,Z]_{Day} ) & \simeq \underset{c}{\int} Maps\left( X(c), \underset{c_1,c_2}{\int} Maps\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1 , c_2), Maps(Y(c_1), Z(c_2))_\ast \right)_\ast \right)_\ast \\ & \simeq \underset{c}{\int} \underset{c_1,c_2}{\int} Maps\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2) \wedge X(c) \wedge Y(c_1) ,\; Z(c_2) \right)_\ast \\ & \simeq \underset{c_2}{\int} Maps\left( \overset{c,c_1}{\int} \mathcal{C}(c \otimes_{\mathcal{C}} c_1, c_2) \wedge X(c) \wedge Y(c_1) ,\; Z(c_2) \right)_\ast \\ &\simeq \underset{c_2}{\int} Maps\left( (X \otimes_{Day} Y)(c_2), Z(c_2) \right)_\ast \\ &\simeq [\mathcal{C},V](X \otimes_{Day} Y, Z) \end{aligned} \,.
###### Proposition

In the situation of def. , the Yoneda embedding $c\mapsto \mathcal{C}(c,-)$ constitutes a strong monoidal functor

$(\mathcal{C},\otimes_{\mathcal{C}}, I) \hookrightarrow ([\mathcal{C},V], \otimes_{Day}, y(I)) \,.$
###### Proof

That the tensor unit is respected is part of prop. . To see that the tensor product is respected, apply the co-Yoneda lemma (prop ) twice to get the following natural isomorphism

\begin{aligned} (y(c_1) \otimes_{Day} y(c_2))(c) & \simeq \overset{d_1, d_2}{\int} \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c ) \wedge \mathcal{C}(c_1,d_1) \wedge \mathcal{C}(c_2,d_2) \\ & \simeq \mathcal{C}(c_1\otimes_{\mathcal{C}}c_2 , c ) \\ & = y(c_1 \otimes_{\mathcal{C}} c_2 )(c) \end{aligned} \,.
##### Functors with smash product
###### Definition

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. ). A topologically enriched lax monoidal functor between them is

1. $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,$
2. a morphism

$\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})$
3. $\mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)$

for all $x,y \in \mathcal{C}$

satisfying the following conditions:

1. (associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes

$\array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,$

where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;

2. (unitality) For all $x \in \mathcal{C}$ the following diagrams commutes

$\array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }$

and

$\array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,$

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. ), then the lax monoidal functor $F$ is called a braided monoidal functor if in addition the following diagram commutes for all objects $x,y \in \mathcal{C}$

$\array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.$

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 $x,y \in \mathcal{C}$:

$\array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }$

and

$\array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.$

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.

###### Remark

In the literature the term “monoidal functor” often refers by default to what in def. 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. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

###### Definition

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. ), 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

1. $G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,;$
2. $\rho_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} N(y) \longrightarrow N(x \otimes_{\mathcal{C}} y )$

such that

• (action property) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes

$\array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} G(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} G(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \rho_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} G(z) && F(x) \otimes_{\mathcal{D}} ( G(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\rho_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\rho_{ x, y \otimes_{\mathcal{C}} z }}} \\ G( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& G( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,$

A homomorphism $f\;\colon\; (G_1, \rho_1) \longrightarrow (G_2,\rho_2)$ between two modules over a monoidal functor $(F,\mu,\epsilon)$ is

• a natural transformation $f_x \;\colon\; N_1(x) \longrightarrow N_2(x)$ of the underlying functors

compatible with the action in that the following diagram commute for all objects $x,y \in \mathcal{C}$:

$\array{ F(x) \otimes_{\mathcal{D}} N_1(y) &\overset{id \otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F(x) \otimes_{\mathcal{D}} N_2(y) \\ {}^{\mathllap{(\rho_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\rhi_2)_{x,y}}} \\ N_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& N_2(x \otimes_{\mathcal{C}} y) }$

We write $F Mod$ for the resulting category of modules over the monoidal functor $F$.

###### Proposition

Let $(\mathcal{C},\otimes I)$ be a pointed topologically enriched category (symmetric monoidal category) monoidal category (def. ). Regard $(Top_{cg}^{\ast/}, \wedge , S^0)$ as a topological symmetric monoidal category as in example .

Then (commutative) monoids in (def. ) the Day convolution monoidal category $([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))$ of prop. are equivalent to (braided) lax monoidal functors (def. ) of the form

$(\mathcal{C},\otimes, I) \longrightarrow (Top^{\ast}_{cg}, \wedge, S^0) \,,$

called functors with smash products on $\mathcal{C}$, i.e. there are equivalences of categories of the form

\begin{aligned} Mon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) &\underoverset{\simeq}{\phi}{\longrightarrow} MonFunc(\mathcal{C},Top^{\ast/}_{cg}) \\ CMon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) &\simeq SymMonFunc(\mathcal{C},Top^{\ast/}_{cg}) \end{aligned} \,.

Furthermore, for $A \in Mon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))$ a given monoid object, then left $A$-module objects (def. ) are equivalent to left modules over monoidal functors (def. ):

$A Mod \simeq \phi(A) Mod \,.$

This is stated in some form in (Day 70, example 3.2.2). It is highlighted again in (MMSS 00, prop. 22.1).

###### Proof

By definition , a lax monoidal functor $F \colon \mathcal{C} \to Top^{\ast/}_{cg}$ is a topologically enriched functor equipped with a morphism of pointed topological spaces of the form

$S^0 \longrightarrow F(1_{\mathcal{C}})$

and equipped with a natural system of maps of pointed topological spaces of the form

$F(c_1) \wedge F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)$

for all $c_1,c_2 \in \mathcal{C}$.

Under the Yoneda lemma (prop. ) the first of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form

$y(S^0) \longrightarrow F \,.$

Moreover, under the natural isomorphism of corollary the second of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form

$F \otimes_{Day} F \longrightarrow F \,.$

Translating the conditions of def. satisfied by a lax monoidal functor through these identifications gives precisely the conditions of def. on a (commutative) monoid in object $F$ under $\otimes_{Day}$.

Similarly for module objects and modules over monoidal functors.

###### Proposition

Let $f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a lax monoidal functor (def. ) between pointed topologically enriched monoidal categories (def. ). Then the induced functor

$f^\ast \;\colon\; [\mathcal{D}, Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}, Top_{cg}^{\ast}]$

given by $(f^\ast X)(c)\coloneqq X(f(c))$ preserves monoids under Day convolution

$f^\ast \;\colon\; Mon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}})) \longrightarrow Mon([\mathcal{C}, Top_{cg}^{\ast}], \otimes_{Day}, y(1_{\mathcal{C}})$

Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal categories (def. ) and $f$ is a braided monoidal functor (def. ), then $f^\ast$ also preserves commutative monoids

$f^\ast \;\colon\; CMon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}})) \longrightarrow CMon([\mathcal{C}, Top_{cg}^{\ast}], \otimes_{Day}, y(1_{\mathcal{C}}) \,.$

Similarly, for

$A \in Mon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}}))$

any fixed monoid, then $f^\ast$ sends $A$-module object to $f^\ast(A)$-modules

$f^\ast \;\colon\; A Mod(\mathcal{D}) \longrightarrow (f^\ast A)Mod(\mathcal{C}) \,.$
###### Proof

This is an immediate corollary of prop. , since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor.

###### Proposition

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ be a topologically enriched monoidal category (def. ), and let $A \in Mon([\mathcal{C},Top^{\ast/}_{cg}],\otimes_{Day}, y(1_{\mathcal{C}}))$ be a monoid in (def. ) the pointed topological Day convolution monoidal category over $\mathcal{C}$ from prop. .

Then the category of left A-modules (def. ) is a pointed topologically enriched functor category category (exmpl.)

$A Mod \;\simeq\; [ A Free_{\mathcal{C}}Mod^{op}, \; Top_{cg}^{\ast/} ] \,,$

over the category of free modules over $A$ (def. ) 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

$A Free_{\mathcal{C}}Mod( c_1, c_2) \;=\; A Mod( A \otimes_{Day} y(c_1),\; A \otimes_{Day} y(c_2) ) \,.$
###### Proof idea

Use the identification from prop. 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

$A(c_1)\otimes N(c_2) \overset{\rho_{c_1,c_2}}{\longrightarrow} N(c_1 \otimes c_2) \,.$

Notice that these transformations have just the same structure as those of the enriched functoriality of $N$ (def.) of the form

$\mathcal{C}(c_1,c_2) \otimes N(c_1) \overset{}{\longrightarrow} N(c_2) \,.$

Hence we may unify these two kinds of transformations into a single kind of the form

$\mathcal{C}(c_3 \otimes c_1 , c_2) \otimes A(c_3) \otimes N(c_1) \overset{id \otimes \rho_{c_3,c_1}}{\longrightarrow} \mathcal{C}(c_3 \otimes c_1, c_2) \otimes N(c_3 \otimes c_2) \longrightarrow N(c_2)$

and subject to certain identifications.

Now observe that the hom-objects of $A Free_{\mathcal{C}}Mod$ have just this structure:

\begin{aligned} A Free_{\mathcal{C}}Mod(c_2,c_1) & = A Mod( A \otimes_{Day} y(c_2) , A \otimes_{Day} y(c_1) ) \\ & \simeq [\mathcal{C},Top^{\ast/}_{cg}](y(c_2), A \otimes_{Day} y(c_1) ) \\ & \simeq (A \otimes_{Day} y(c_1) )(c_2) \\ & \simeq \overset{c_3,c_4}{\int} \mathcal{C}(c_3 \otimes c_4,c_2) \wedge A(c_3) \wedge \mathcal{C}(c_1, c_4) \\ & \simeq \overset{c_3}{\int} \mathcal{C}(c_3 \otimes c_1, c_2) \wedge A (c_3) \end{aligned} \,.

Here we used first the free-forgetful adjunction of prop. , then the enriched Yoneda lemma (prop. ), then the coend-expression for Day convolution (def. ) and finally the co-Yoneda lemma (prop. ).

We claim that under this identification, composition in $A Free_{\mathcal{C}}Mod$ is given by the following composite.

\begin{aligned} A Free_{\mathcal{C}}Mod(c_3, c_2) \wedge A Free_{\mathcal{C}}Mod(c_2, c_1) & = \left( \overset{c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3 ) \wedge A(c_5) \right) \wedge \left( \overset{c_4}{\int} \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1, c_2) \wedge A(c_4) \right) \\ & \simeq \overset{c_4, c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3) \wedge \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1, c_2 ) \wedge A(c_5) \wedge A(c_4) \\ & \longrightarrow \overset{c_4,c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3) \wedge \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_1, c_5 \otimes_{\mathcal{C}} c_2 ) \wedge A(c_5 \otimes_{\mathcal{C}} c_4 ) \\ & \longrightarrow \overset{c_4, c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_1, c_5 \otimes_{\mathcal{C}} c_2 ) \wedge A(c_5 \otimes_{\mathcal{C}} c_4 ) \\ & \longrightarrow \overset{c_4}{\int} \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1 , c_3) \otimes_V A(c_4 ) \end{aligned} \,,

where

1. the equivalence is braiding in the integrand (and the Fubini theorem, prop. );

2. the first morphism is, in the integrand, the smash product of

1. forming the tensor product of hom-objects of $\mathcal{C}$ with the identity morphism on $c_5$;

2. 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 $A$;

3. the second morphism is, in the integrand, given by composition in $\mathcal{C}$;

4. 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).

It is fairly straightforward to see that, under the above identifications, functoriality under this composition is equivalently functoriality in $\mathcal{C}$ together with the action property over $A$.

#### Pre-Excisive functors

###### Definition

Write

$\iota_{fin}\;\colon\; Top^{\ast/}_{cg,fin} \hookrightarrow Top^{\ast/}_{cg}$

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. )

$Exc(Top_{cg}) \coloneqq [Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}]$

is the category of pre-excisive functors.

Write

$\mathbb{S}_{exc} \coloneqq y(S^0) \coloneqq Top^{\ast/}_{cg,fin}(S^0,-)$

for the functor co-represented by 0-sphere. This is equivalently the inclusion $\iota_{fin}$ itself:

$\mathbb{S}_{exc} = \iota_{fin} \;\colon\; K \mapsto K \,.$

We call this the standard incarnation of the sphere spectrum as a pre-excisive functor.

By prop. the smash product of pointed compactly generated topological spaces induces the structure of a closed (def. ) symmetric monoidal category (def. )

$\left( Exc(Top_{cg}) ,\; \wedge_{Day} ,\; \mathbb{S}_{exc} \right)$

with

1. tensor unit the sphere spectrum $\mathbb{S}_{exc}$;

2. tensor product the Day convolution product $\otimes_{Day}$ from def. ,

called the symmetric monoidal smash product of spectra for the model of pre-excisive functors;

3. internal hom the dual operation $[-,-]_{Day}$ from prop. ,

called the mapping spectrum construction for pre-excisive functors.

###### Remark

By example the sphere spectrum incarnated as a pre-excisive functor $\mathbb{S}_{exc}$ (according to def. ) is canonically a commutative monoid in the category of pre-excisive functors (def. )

Moreover, by example , every object of $Exc(Top_{cg})$ (def. ) 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:

$Exc(Top_{cg}) \simeq \mathbb{S}_{exc}Mod \,.$

We now consider restricting the domain of the pre-excisive functors of def. .

###### Definition

Define the following pointed topologically enriched (def.) symmetric monoidal categories (def. ):

1. $Seq$ 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

$Seq(n_1,n_2) = \left\{ \array{ S^0 & for\; n_1 = n_2 \\ \ast & otherwise } \right.$

The tensor product is the addition of natural numbers, $\otimes = +$, and the tensor unit is 0. The braiding is, necessarily, the identity.

2. $Sym$ 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 $n \in \mathbb{N}$, all non-zero morphisms are automorphisms and the automorphism group of $\{1,\cdots,n\}$ is the symmetric group $\Sigma_n$, hence the hom-spaces are the following discrete topological spaces:

$Sym(n_1, n_2) = \left\{ \array{ (\Sigma_{n_1})_+ & for \; n_1 = n_2 \\ \ast & otherwise } \right.$

The tensor product is the disjoint union of sets, tensor unit is the empty set. The braiding

$\tau_{n_1 , n_2} \;\colon\; \overline{n_1} \cup \overline{n_2} \overset{}{\longrightarrow} \overline{n_2} \cup \overline{n_1}$

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.

3. $Orth$ has as objects finite dimenional real linear inner product spaces $(V, \langle -,-\rangle)$ and as non-zero morphisms the linear isometric isomorphisms between these; hence the automorphism group of the object $(V, \langle -,-\rangle)$ is the orthogonal group $O(V)$; the monoidal product is direct sum of linear spaces, the tensor unit is the 0-vector space; again we turn this into a $Top^{\ast/}$-enriched category by adjoining a basepoint to the hom-spaces;

$Orth(V_1,V_2) \simeq \left\{ \array{ O(V_1)_+ & for \; dim(V_1) = dim(V_2) \\ \ast & otherwise } \right.$

The tensor product is the direct sum of linear inner product spaces, tensor unit is the 0-vector space. The braiding is that of $Sym$, under the canonical embedding $\Sigma_{n_1+n_2} \hookrightarrow O(n_1+n_2)$ of the symmetric group into the orthogonal group.

There is a sequence of canonical faithful pointed topological subcategory inclusions

$\array{ Seq &\stackrel{seq}{\hookrightarrow}& Sym &\stackrel{sym}{\hookrightarrow}& Orth &\stackrel{orth}{\hookrightarrow}& Top_{cg,fin}^{\ast/} \\ n &\mapsto& \{1,\cdots, n\} &\mapsto& \mathbb{R}^n &\mapsto& S^n \\ && && V &\mapsto& S^V } \,,$

into the pointed topological category of pointed compactly generated topological spaces of finite CW-type (def. ).

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”).

Consider the sequence of restrictions of topological diagram categories, according to prop. along the above inclusions:

$Exc(Top_{cg}) \overset{orth^\ast}{\longrightarrow} [Orth,Top^{\ast/}_{cg}] \overset{sym^\ast}{\longrightarrow} [Sym,Top^{\ast/}_{cg}] \overset{seq^\ast}{\longrightarrow} [Seq,Top^{\ast/}_{cg}] \,.$

Write

$\mathbb{S}_{orth} \coloneqq orth^\ast \mathbb{S}_{exc} \,, \;\;\;\;\;\;\;\; \mathbb{S}_{sym} \coloneqq sym^\ast \mathbb{S}_{orth} \,, \;\;\;\;\;\;\;\; \mathbb{S}_{seq} \coloneqq seq^\ast \mathbb{S}_{sym}$

for the restriction of the excisive functor incarnation of the sphere spectrum (from def. ) along these inclusions.

###### Remark

Since $\mathbb{S}_{exc}$ 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 , prop. says that all these restricted sphere spectra are still monoids, and that under restriction every pre-excisive functor, regarded as a $\mathbb{S}_{exc}$-module via remark , canonically becomes a module under the restricted sphere spectrum:

\begin{aligned} orth^\ast & \colon\; Exc(Top_{cg}) \simeq \mathbb{S}_{exc} Mod \longrightarrow \mathbb{S}_{orth} Mod \\ sym^\ast &\colon\; Exc(Top_{cg}) \simeq \mathbb{S}_{exc} Mod \longrightarrow \mathbb{S}_{sym} Mod \\ seq^\ast &\colon\; Exc(Top_{cg}) \simeq \mathbb{S}_{exc} Mod \longrightarrow \mathbb{S}_{seq} Mod \end{aligned} \,.

However, while $orth$ and $sym$ are braided monoidal functors, the functor $seq$ is not braided, hence $\mathbb{S}_{orth}$ and $\mathbb{S}_{sym}$ are commutative monoids, but $\mathbb{S}_{Seq}$ is not commutative. Hence prop. gives the following situation

sphere spectrum$\mathbb{S}_{exc}$$\mathbb{S}_{orth}$$\mathbb{S}_{sym}$$\mathbb{S}_{seq}$
monoidyesyesyesyes
commutative monoidyesyesyesno
tensor unityesnonono
###### Proposition

There is an equivalence of categories

$(-)^{seq} \;\colon\; \mathbb{S}_{seq} Mod \overset{}{\longrightarrow} SeqSpec(Top_{cg})$

which identifies the category of modules (def. ) over the monoid $\mathbb{S}_{seq}$ (remark ) in the Day convolution monoidal structure (prop. ) over the topological functor category $[Seq,Top^{\ast/}_{cg}]$ from def. 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 $S^n = (S^1)^{\wedge n}$

$(X^{seq})_n \coloneqq X(seq(n)) = X(S^n)$

and whose structure maps are the images of the action morphisms

$\mathbb{S}_{seq} \otimes_{Day} X \longrightarrow X$

under the isomorphism of corollary

$\mathbb{S}_{seq}(n_1) \wedge X(n_1) \longrightarrow X_{n_1 + n_2}$

evaluated at $n_1 = 1$

$\array{ \mathbb{S}_{seq}(1) \wedge X(n) &\longrightarrow& X_{n+1} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ S^1 \wedge X_n &\longrightarrow& X_{n+1} } \,.$
###### Proof

After unwinding the definitions, the only point to observe is that due to the action property,

$\array{ \mathbb{S}_{seq} \otimes_{Day} \mathbb{S}_{seq} \otimes_{Day} X &\overset{id \otimes_{Day} \rho}{\longrightarrow}& \mathbb{S}_{seq} \otimes_{Day} X \\ {}^{\mathllap{\mu \otimes_{Day} id } }\downarrow && \downarrow^{\mathrlap{\rho}} \\ \mathbb{S}_{seq} \otimes_{Day} X &\underset{\rho}{\longrightarrow}& X }$

any $\mathbb{S}_{seq}$-action

$\rho \;\colon\; \mathbb{S}_{seq} \otimes_{Day} X \longrightarrow X$

is indeed uniquely fixed by the components of the form

$\mathbb{S}_{seq}(1) \wedge X(n) \longrightarrow X(n) \,.$

This is because under corollary the action property is identified with the componentwise property

$\array{ S^{n_1} \wedge S^{n_2} \wedge X_{n_3} &\overset{id \wedge \rho_{n_2,n_3}}{\longrightarrow}& S^{n_1} \wedge X_{n_2 + n_3} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\rho_{n_1,n_2+n_3}}} \\ S^{n_1 + n_2} \wedge X_{n_3} &\underset{\rho_{n_1+n_2,n_3}}{\longrightarrow}& X_{n_1 + n_2 + n_3} } \,,$

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 , $\mathbb{S}_{seq}$ is not commutative, there is no tensor product induced on $SeqSpec(Top_{cg})$ under the identification in prop. . But since $\mathbb{S}_{orth}$ and $\mathbb{S}_{sym}$ are commutative monoids by remark , it makes sense to consider the following definition.

###### Definition

In the terminology of remark we say that

$OrthSpec(Top_{cg}) \coloneqq \mathbb{S}_{orth} Mod$

is the category of orthogonal spectra; and that

$SymSpec(Top_{cg}) \coloneqq \mathbb{S}_{sym} Mod$

is the category of symmetric spectra.

By remark and by prop. 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

$\wedge \coloneqq \otimes_{\mathbb{S}_{orth}} \;\colon\; OrthSpec(Top_{cg}) \times OrthSpec(Top_{cg}) \longrightarrow OrthSpec(Top_{cg})$

and

$\wedge \coloneqq \otimes_{\mathbb{S}_{sym}} \;\colon\; SymSpec(Top_{cg}) \times SymSpec(Top_{cg}) \longrightarrow SymSpec(Top_{cg})$

In the next section we work out what these symmetric monoidal categories of orthogonal and of symmetric spectra look like more explicitly.

### For symmetric and orthogonal spectra

We now define symmetric spectra and orthogonal spectra and their symmetric monoidal smash product. We proceed by giving the explicit definitions and then checking that these are equivalent to the abstract definition from above.

$\,$

###### Definition

A topological symmetric spectrum $X$ is

1. a sequence $\{X_n \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\}$ of pointed compactly generated topological spaces;

2. a basepoint preserving continuous right action of the symmetric group $\Sigma(n)$ on $X_n$;

3. 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

$S^{k} \wedge X_n \simeq S^{k-1} \wedge S^1 \wedge X_n \stackrel{id \wedge \sigma_n }{\longrightarrow} S^{k-1} \wedge X_{n+1} \simeq S^{k-2}\wedge S^1 \wedge X_{n+2} \stackrel{id \wedge \sigma_{n+1}}{\longrightarrow} \cdots \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}$

intertwines the $\Sigma(n) \times \Sigma(k)$-action.

A homomorphism of symmetric spectra $f\colon X \longrightarrow Y$ is

• a sequence of maps $f_n \colon X_n \longrightarrow Y_n$

such that

1. each $f_n$ intetwines the $\Sigma(n)$-action;

2. the following diagrams commute

$\array{ S^1 \wedge X_n &\stackrel{f_n \wedge id}{\longrightarrow}& S^1 \wedge Y_n \\ \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.$

We write $SymSpec(Top_{cg})$ for the resulting category of symmetric spectra.

The definition of orthogonal spectra has the same structure, just with the symmetric groups replaced by the orthogonal groups.

###### Definition

A topological orthogonal spectrum $X$ is

1. a sequence $\{X_n \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\}$ of pointed compactly generated topological spaces;

2. a basepoint preserving continuous right action of the orthogonal group $O(n)$ on $X_n$;

3. 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

$S^{k} \wedge X_n \simeq S^{k-1} \wedge S^1 \wedge X_n \stackrel{id \wedge \sigma_n }{\longrightarrow} S^{k-1} \wedge X_{n+1} \simeq S^{k-2}\wedge S^1 \wedge X_{n+2} \stackrel{id \wedge \sigma_{n+1}}{\longrightarrow} \cdots \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}$

intertwines the $O(n) \times Ok()$-action.

A homomorphism of orthogonal spectra $f\colon X \longrightarrow Y$ is

• a sequence of maps $f_n \colon X_n \longrightarrow Y_n$

such that

1. each $f_n$ intetwines the $O(n)$-action;

2. the following diagrams commute

$\array{ S^1 \wedge X_n &\stackrel{f_n \wedge id}{\longrightarrow}& S^1 \wedge Y_n \\ \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.$

We write $OrthSpec(Top_{cg})$ for the resulting category of orthogonal spectra.

###### Proposition

Definitions and are indeed equivalent to def. :

orthogonal spectra are euqivalently the module objects over the incarnation $\mathbb{S}_{orth}$ of the sphere spectrum

$OrthSpec(Top_{cg}) \simeq \mathbb{S}_{orth} Mod$

and symmetric spectra sre equivalently the module objects over the incarnation $\mathbb{S}_{sym}$ of the sphere spectrum

$SymSpec(Top_{cg}) \simeq \mathbb{S}_{sym} Mod \,.$
###### Proof

We discuss this for symmetric spectra. The proof for orthogonal spectra is of the same form.

First of all, (by this example) 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. ).

By corollary and this lemma, the structure morphism of an $\mathbb{S}_{sym}$-module object on $X$

$\mathbb{S}_{sym} \otimes_{Day} X \longrightarrow X$

is equivalently (as a functor with smash products) a natural transformation

$S^{n_1} \wedge X_{n_2} \longrightarrow X_{n_1 + n_2}$

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 this lemma says that in the following commuting squares (exhibiting the action property on the level of functors with smash product, where we are notationally suppressing the associators) the left vertical morphisms are isomorphisms: a

$\array{ S^{n_1}\wedge S^{n_2} \wedge X_{n_3} &\longrightarrow& S^{n_1} \wedge X_{n_2 + n_3} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow \\ S^{n_1+ n_2} \wedge X_{n_3} &\longrightarrow& X_{n_1 + n_2 + n_3} } \,.$

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.

###### Definition

Given $X,Y \in SymSpec(Top_{cg})$ two symmetric spectra, def. , then their smash product of spectra is the symmetric spectrum

$X \wedge Y \; \in SymSpec(Top_{cg})$

with component spaces the coequalizer

$\underset{p+1+q = n}{\bigvee} \Sigma(p+1+q)_+ \underset{\Sigma_p \times \Sigma_1 \times \Sigma_q}{\wedge} X_p \wedge S^1 \wedge Y_q \underoverset {\underset{r}{\longrightarrow}} {\overset{\ell}{\longrightarrow}} {\phantom{AAAA}} \underset{p+q=n}{\bigvee} \Sigma(p+q)_+ \underset{\Sigma_p \times \Sigma_q}{\wedge} X_p \wedge Y_q \overset{coeq}{\longrightarrow} (X \wedge Y)(n)$

where $\ell$ has components given by the structure maps

$X_p \wedge S^1 \wedge Y_q \overset{id \wedge \sigma_{q}}{\longrightarrow} X_p \wedge Y_q$

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)

$X_p \wedge S^1 \wedge X_q \overset{\tau^{Top^{\ast/}_{cg}}_{X_p,S^1} \wedge id}{\longrightarrow} S^1 \wedge X_p \wedge X_q \overset{\sigma_p\wedge id}{\longrightarrow} X_{p+1} \wedge X_q \overset{\chi_{p,1} \wedge id}{\longrightarrow} X_{1+p} \wedge X_q \,.$

The structure maps of $X \wedge Y$ are those induced under the coequalizer by

$X_p \wedge Y_q \wedge S^1 \overset{id \wedge \sigma_{p}}{\longrightarrow} X_{p} \wedge Y_{q+1} \,.$

Analogously for orthogonal spectra.

###### Proposition

Under the identification of prop. , the explicit smash product of spectra in def. is equivalent to the abstractly defined tensor product in def. :

in the case of symmetric spectra:

$\wedge \simeq \otimes_{\mathbb{S}_{sym}}$

in the case of orthogonal spectra:

$\wedge \simeq \otimes_{\mathbb{S}_{orth}} \,.$
###### Proof

By def. the abstractly defined tensor product of two $\mathbb{S}_{sym}$-modules $X$ and $Y$ is the coequalizer

$X \otimes_{Day} \mathbb{S}_{sym} \otimes_{Day} Y \underoverset {\underset{\rho_{1}\circ (\tau^{Day}_{X, \mathbb{S}_{sym}} \otimes id)}{\longrightarrow}} {\overset{X \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} X \otimes Y \overset{coeq}{\longrightarrow} X \otimes_{\mathbb{S}_{sym}} Y \,.$

The Day convolution product appearing here is over the category $Sym$ from def. . By this example 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:

\begin{aligned} (A \otimes_{Day} B)(n) & = \overset{n_1,n_2}{\int} \underset{ = \left\{ \array{ \Sigma(n_1 + n_2,n)_+ & if \; n_1+n_2 = n \\ \ast & otherwise } \right. }{ \underbrace{\Sigma(n_1 + n_2, n)} }_+ \wedge A_{n_1} \wedge B_{n_1} \\ & \simeq \underset{n_1 + n_2 = n}{\bigvee} \Sigma(n_1+n_2)_+ \underset{O(n_1) \times O(n_2) }{\wedge} \left( A_{n_1} \wedge B_{n_2} \right) \end{aligned} \,.

This establishes the form of the coequalizer diagram. It remains to see that under this identification the two abstractly defined morphisms are the ones given in def. .

To see this, we apply the adjunction isomorphism between the Day convolution product and the external tensor product (cor. ) twice, to find the following sequence of equivalent incarnations of morphisms:

$\array{ \arrayopts{\rowlines{solid}} (X \otimes_{Day} ( \mathbb{S}_{orth} \otimes_{Day} Y ))(n) &\longrightarrow& (X \otimes_{Day} Y)(n) &\longrightarrow& Z_n \\ X_{n_1} \wedge (\mathbb{S}_{sym} \otimes_{Day} Y)(n'_2) &\longrightarrow& X_{n_1}\wedge Y(n'_2) &\longrightarrow& Z_{n_1 + n'_2} \\ (\mathbb{S}_{sym} \otimes_{Day} Y)(n'_2) &\longrightarrow& Y(n'_2) &\longrightarrow& Maps(X_{n_1}, Z_{n_1 + n'_2}) \\ S^{n_2} \wedge Y_{n_3} &\longrightarrow& Y_{n_2 + n_3} &\longrightarrow& Maps(X_{n_1}, Z_{n_1 + n_2 + n_3}) \\ X_{n_1} \wedge S^{n_2} \wedge Y_{n_3} &\longrightarrow& X_{n_1} \wedge Y_{n_2 + n_3} &\longrightarrow& Z_{n_1 + n_2 + n_3} } \,.$

This establishes the form of the morphism $\ell$. By the same reasoning as in the proof of prop. , 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. :

$\tau^{Day}_{A,B}(n) = \overset{n_1,n_2}{\int} Sym( \tau^{Sym}_{n_1,n_2}, n ) \wedge \tau^{Top^{\ast/}_{cg}}_{A_{n_1}, B_{n_2}}$

by translating it to the components of the precomposition

$X \otimes_{Day} \mathbb{S}_{sym} \overset{\tau^{Day}_{X,\mathbb{S}_{sym}}}{\longrightarrow} \mathbb{S}_{sym} \otimes_{Day} X \overset{}{\longrightarrow} X$

via the formula from the proof of prop. for the left Kan extension $A \otimes_{Day} B \simeq Lan_{\otimes} A \overline{\wedge} B$ (prop. ):

\begin{aligned} [Sym, Top^{\ast/}_{cg}]( \tau^{Day}_{X,\mathbb{S}_{sym}}, X) & \simeq \underset{n}{\int} Maps( \overset{n_1, n_2}{\int} Sym( \tau^{sym}_{n_1,n_2}, n ) \wedge \tau^{Top^{\ast/}_{cg}}_{X_{n_1}, S^{n_2}} , X(n) )_\ast \\ & \simeq \underset{n_1,n_2}{\int} Maps( \tau_{X_{n_1}, S^{n_2} }^{Top^{\ast/}_{cg}} , X( \tau^{sym}_{n_1,n_2} ) )_\ast \end{aligned} \,.

model structure on spectra

Also the

carries a symmetric monoida smash product.

## References

### Original sources

The original no-go theorem for a well-behave smash product of spectra is

• Gaunce Lewis, Is there a conveinient category of spectra?, Journal of Pure and Applied Algebra Volume 73, Issue 3, 30 August 1991, Pages 233–246

In the mid-1990s, several categories of spectra with nice smash products were discovered, and simultaneously, model categories experienced a major renaissance.

The definition of S-modules and their theory originates in

and around 1993 Jeff Smith gave the first talks about symmetric spectra; the details of the model structure were later worked out and written up in

Discussion that makes the Day convolution structure on the symmetric smash product of spectra manifest is in

### Reviews and introductions

Surveys of the history are in

A textbook account of the theory of symmetric spectra is

Seminar notes on symmetric spectra are in