nLab
symmetric smash product of spectra

Context

Stable Homotopy theory

Higher 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 A_\infty-rings – and commutative monoid objects – called E E_\infty-rings. These monoid objects satisfy associativity, uniticity and, in the E 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 𝒮\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 Ho𝒮Ho \mathcal{S}.

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

:𝒮×𝒮𝒮, \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 𝒮\mathcal{S}. In fact (Lewis 91) proved a theorem that there could be no symmetric monoidal category 𝒮\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 𝒮\mathcal{S} absorbs all the higher coherent homotopies that used to be supplied by the action of an A A_\infty or E E_\infty operad. Thus, honest (commutative) monoids in 𝒮\mathcal{S} model the same “(commutative) ring objects up to all coherent higher homotopies” that are modeled by the classical A A_\infty and E E_\infty ring spectra, and for this reason they are often still referred to as A A_\infty or E E_\infty ring spectra, respectively.

Details

For SS-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 𝕃\mathbb{L} on the category 𝒮\mathcal{S} of such spectra, and the category of 𝕃\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 𝕃\mathbb{L}-algebras for which the unit transformations are isomorphisms is again a well-behaved model for Ho𝕊Ho \mathbb{S}, which is moreover symmetric monoidal.

Since the unit transformation is of the form SEES\wedge E \to E, where SS is the sphere spectrum, and this map looks like the action of a ring on a module, the objects of this subcategory are called SS-modules and the category is called Mod SMod_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 Mod_\mathbb{Z} of \mathbb{Z}-modules, i.e. a \mathbb{Z}-algebra, an A A_\infty or E E_\infty ring spectrum is a (possibly commutative) monoid in the category of SS-modules, and thus referred to as an SS-algebra. More generally, for any A A_\infty-ring spectrum RR, there is a notion of RR-module spectra forming a category Mod RMod_R, which in turn carries an associative and commutative smash product R\wedge_R and a model category structure on Mod RMod_R such that R\wedge_R becomes unital in the homotopy category. All this is such that an A A_\infty-algebra over RR is a monoid object in (Mod R, R)(Mod_R, \wedge_R). Similarly E E_\infty-algebras are commutative monoid objects in (Mod R, R)(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. 4 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 𝒞,𝒟\mathcal{C}, \mathcal{D} be pointed topologically enriched categories (def.), i.e. enriched categories over (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge, S^0) from example 4.

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

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

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

    𝒞 op(X,Y)𝒞 op(Y,Z)=𝒞(Y,X)𝒞(Z,Y)τ𝒞(Z,Y)𝒞(Y,X) Z,Y,X𝒞(Z,X)=𝒞 op(X,Z). \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 𝒞×𝒟\mathcal{C} \times \mathcal{D} is the topologically enriched category whose objects are pairs of objects (c,d)(c,d) with c𝒞c \in \mathcal{C} and d𝒟d\in \mathcal{D}, whose hom-spaces are the smash product of the separate hom-spaces

    (𝒞×𝒟)((c 1,d 1),(c 2,d 2))𝒞(c 1,c 2)𝒟(d 1,d 2) (\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:

    (𝒞×𝒟)((c 1,d 1),(c 2,d 2))(𝒞×𝒟)((c 2,d 2),(c 3,d 3)) = (𝒞(c 1,c 2)𝒟(d 1,d 2))(𝒞(c 2,c 3)𝒟(d 2,d 3)) τ (𝒞(c 1,c 2)𝒞(c 2,c 3))(𝒟(d 1,d 2)𝒟(d 2,d 3)) ( c 1,c 2,c 3)( d 1,d 2,d 3) 𝒞(c 1,c 3)𝒟(d 1,d 3) = (𝒞×𝒟)((c 1,d 1),(c 3,d 3)). \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 cg */Top^{\ast/}_{cg} (exmpl.) out of a pointed topological product category as in def. 1

F:𝒞×𝒟Top cg */ 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):𝒞(c 1,c 2)𝒟(d 1,d 2)Maps(F 0((c 1,d 1)),F 0((c 2,d 2))) *. 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

ρ c 1,c 2(d):𝒞(c 1,c 2)F 0((c 1,d))F 0((c 2,d)) \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

ρ d 1,d 2(c):𝒟(d 1,d 2)F 0((c,d 1))F 0((c,d 2)). \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 𝒞\mathcal{C} with its opposite category (def. 1)

F:𝒞 op×𝒞Top cg */ F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Top^{\ast/}_{cg}

then this takes the form

ρ c 2,c 1(d):𝒞(c 1,c 2)F 0((c 2,d))F 0((c 1,d)) \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

ρ d 1,d 2(c):𝒞(d 1,d 2)F 0((c,d 1))F 0((c,d 2)). \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 𝒞\mathcal{C} be a small pointed topologically enriched category (def.), i.e. an enriched category over (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge, S^0) from example 4. Let

F:𝒞 op×𝒞Top cg */ 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 𝒞\mathcal{C} with its opposite category, according to def. 1.

  1. The coend of FF, denoted c𝒞F(c,c)\overset{c \in \mathcal{C}}{\int} F(c,c), is the coequalizer in Top cg *Top_{cg}^{\ast} (prop., exmpl., prop., cor.) of the two actions encoded in FF via example 1:

    c,d𝒞𝒞(c,d)F(d,c)AAAAAAAAc,dρ (d,c)(c)c,dρ (c,d)(d)c𝒞F(c,c)coeqc𝒞F(c,c). \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 FF, denoted c𝒞F(c,c)\underset{c\in \mathcal{C}}{\int} F(c,c), is the equalizer in Top cg */Top_{cg}^{\ast/} (prop., exmpl., prop., cor.) of the adjuncts of the two actions encoded in FF via example 1:

    c𝒞F(c,c)equc𝒞F(c,c)AAAAAAAAc,dρ˜ (c,d)(c)c,dρ˜ d,c(d)c𝒞Maps(𝒞(c,d),F(c,d)) *. \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 𝒞\mathcal{C} be a small pointed topologically enriched category (def.). For F,G:𝒞Top cg */ F,G \;\colon\; \mathcal{C} \longrightarrow Top^{\ast/}_{cg} two pointed topologically enriched functors, then the end (def. 2) of Maps(F(),G()) *Maps(F(-),G(-))_\ast is a topological space whose underlying pointed set is the pointed set of natural transformations FGF\to G (def.)

U(c𝒞Maps(F(c),G(c)) *)Hom [𝒞,Top cg */](F,G). 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:Top cg */Set */U\colon Top^{\ast/}_{cg}\to Set^{\ast/} preserves all limits (prop., prop., prop.). Therefore there is an equalizer diagram in Set */Set^{\ast/} of the form

U(c𝒞Maps(F(c),G(c)) *)equc𝒞Hom Top cg */(F(c),G(c))AAAAAAAAc,dU(ρ˜ (c,d)(d))c,dU(ρ˜ d,c(c))c,d𝒞Hom Top cg */(𝒞(c,d),Maps(F(c),G(d)) *). 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 {F(c)η cG(c)} c𝒞\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 cfdc \overset{f}{\to} d to the result of precomposing

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

and of postcomposing

F(c) η c G(c) G(f) G(d) \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 {η c} c𝒞\{\eta_c\}_{c\in \mathcal{C}} being in the equalizer, means precisley that for all c,dc,d and all f:cdf\colon c \to d the square

F(c) η c G(c) F(f) G(f) F(d) η d G(g) \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 {η c} c𝒞\{\eta_c\}_{c\in \mathcal{C}} be a natural transformation.

Conversely, example 2 says that ends over bifunctors of the form Maps(F(),G())) *Maps(F(-),G(-)))_\ast constitute hom-spaces between pointed topologically enriched functors:

Definition

Let 𝒞\mathcal{C} be a small pointed topologically enriched categories (def.). Define the structure of a pointed topologically enriched category on the category [𝒞,Top cg */][\mathcal{C}, Top_{cg}^{\ast/}] of pointed topologically enriched functors to Top cg */Top^{\ast/}_{cg} (exmpl.) by taking the hom-spaces to be given by the ends (def. 2) of example 2:

[𝒞,Top cg */](F,G) c𝒞Maps(F(c),G(c)) * [\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

(c𝒞Maps(F(c),G(c)) *)(c𝒞Maps(G(c),H(c)) *)c𝒞Maps(F(c),G(c)) *Maps(G(c),H(c)) *( F(c),G(c),H(c)) c𝒞c𝒞Maps(F(c),H(c)) * \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 [𝒞,Top cg */][\mathcal{C},Top^{\ast/}_{cg}] is also called the Top cg */Top^{\ast/}_{cg}-enriched functor category over 𝒞\mathcal{C} with coefficients in Top cg */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 𝒞\mathcal{C} be a small pointed topologically enriched categories (def.). For F:𝒞Top cg */F \colon \mathcal{C}\to Top^{\ast/}_{cg} a pointed topologically enriched functor (def.) and for c𝒞c\in \mathcal{C} an object, there is a natural isomorphism

[𝒞,Top cg */](𝒞(c,),F)F(c) [\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. 3, from the functor represented by cc to FF, and the value of FF on cc.

In terms of the ends (def. 2) defining these hom-spaces, this means that

d𝒞Maps(𝒞(c,d),F(d)) *F(c). \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. 1 is essentially dual to the proof of the next prop. 2.

Now that natural transformations are phrased in terms of ends (example 2), as is the Yoneda lemma (prop. 1), it is natural to consider the dual statement involving coends:

Proposition

(co-Yoneda lemma)

Let 𝒞\mathcal{C} be a small pointed topologically enriched categories (def.). For F:𝒞Top cg */F \colon \mathcal{C}\to Top^{\ast/}_{cg} a pointed topologically enriched functor (def.) and for c𝒞c\in \mathcal{C} an object, there is a natural isomorphism

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

Moreover, the morphism that hence exhibits F(c)F(c) as the coequalizer of the two morphisms in def. 2 is componentwise the canonical action

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

which is adjunct to the component map 𝒞(d,c)Maps(F(c),F(d)) *\mathcal{C}(d,c) \to Maps(F(c),F(d))_{\ast} of the topologically enriched functor FF.

(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

(cc 0,xF(c)), ( c \overset{}{\to} c_0,\; x \in F(c) ) \,,

where two such pairs

(cfc 0,xF(c)),(dgc 0,yF(d)) ( 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

cϕd c \overset{\phi}{\to} d

such that

f=gϕ,andy=ϕ(x). f = g \circ \phi \,, \;\;\;\;\;and\;\;\;\;\; y = \phi(x) \,.

(Because then the two pairs are the two images of the pair (g,x)(g,x) under the two morphisms being coequalized.)

But now considering the case that d=c 0d = c_0 and g=id c 0g = id_{c_0}, so that f=ϕf = \phi shows that any pair

(cϕc 0,xF(c)) ( c \overset{\phi}{\to} c_0, \; x \in F(c))

is identified, in the coequalizer, with the pair

(id c 0,ϕ(x)F(c 0)), (id_{c_0},\; \phi(x) \in F(c_0)) \,,

hence with ϕ(x)F(c 0)\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)Top cg */F(c_0) \in Top^{\ast/}_{cg} is the final topology (def.) of the system of component morphisms

𝒞(d,c)F(c)c𝒞(c,c 0)F(c) \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

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

which is a retraction

id:F(c)𝒞(c,c)F(c)F(c) 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)F(c).

Remark

The statement of the co-Yoneda lemma in prop. 2 is a kind of categorification of the following statement in analysis (whence the notation with the integral signs):

For XX a topological space, f:Xf \colon X \to\mathbb{R} a continuous function and δ(,x 0)\delta(-,x_0) denoting the Dirac distribution, then

xXδ(x,x 0)f(x)=f(x 0). \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 FF a pointed topologically enriched bifunctor on a small pointed topological product category 𝒞 1×𝒞 2\mathcal{C}_1 \times \mathcal{C}_2 (def. 1), i.e.

F:(𝒞 1×𝒞 2) op×(𝒞 1×𝒞 2)Top cg */ 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. 2) is equivalently formed consecutively over each variable, in either order:

(c 1,c 2)F((c 1,c 2),(c 1,c 2))c 1c 2F((c 1,c 2),(c 1,c 2))c 2c 1F((c 1,c 2),(c 1,c 2)) \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

(c 1,c 2)F((c 1,c 2),(c 1,c 2))c 1c 2F((c 1,c 2),(c 1,c 2))c 2c 1F((c 1,c 2),(c 1,c 2)). \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(,) *:(Top cg */) op×Top cg */Top cg */ 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. 2):

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

and

Maps( cF(c,c),Y) *cMaps(F(c,c),Y) *. 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 𝒞,𝒟\mathcal{C}, \mathcal{D} be small pointed topologically enriched categories (def.) and let

p:𝒞𝒟 p \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

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

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

GGpG\mapsto G\circ p. This functor has a left adjoint Lan pLan_p, called left Kan extension along pp

[𝒟,Top cg */]p *Lan p[𝒞,Top cg */] [\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. 2):

(Lan pF):dc𝒞𝒟(p(c),d)F(c). (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 2 and def. 3), then use the respect of Maps(,) *Maps(-,-)_\ast for ends/coends (remark 2), use the smash/mapping space adjunction (cor.), use the Fubini theorem (prop. 3) and finally use Yoneda reduction (prop. 1) to obtain a sequence of natural isomorphisms as follows:

[𝒟,Top cg */](Lan pF,G) =d𝒟Maps((Lan pF)(d),G(d)) * =d𝒟Maps(c𝒞𝒟(p(c),d)F(c),G(d)) * d𝒟c𝒞Maps(𝒟(p(c),d)F(c),G(d)) * c𝒞d𝒟Maps(F(c),Maps(𝒟(p(c),d),G(d)) *) * c𝒞Maps(F(c),d𝒟Maps(𝒟(p(c),d),G(d)) *) * c𝒞Maps(F(c),G(p(c))) * =[𝒞,Top cg */](F,p *G). \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 3 below.

Definition

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

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

    :𝒞×𝒞𝒞 \otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}

    out of the (pointed) topologival product category of 𝒞\mathcal{C} with itself (def. 1), called the tensor product,

  2. an object

    1𝒞 1 \in \mathcal{C}

    called the unit object or tensor unit,

  3. a natural isomorphism (def.)

    a:(()())()()(()()) a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))

    called the associator,

  4. a natural isomorphism

    :(1())() \ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-)

    called the left unitor, and a natural isomorphism

    r:()1() 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:

    (x1)y a x,1,y x(1y) ρ x1 y 1 xλ y xy \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 && }
  2. the pentagon identity:

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Lemma

(Kelly 64)

Let (𝒞,,1)(\mathcal{C}, \otimes, 1) be a monoidal category, def. 4. Then the left and right unitors \ell and rr satisfy the following conditions:

  1. 1=r 1:111\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1;

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

    (1x)y α 1,x,y xy 1(xy) xy xy. \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 𝒞\mathcal{C} (def. 4) equipped with a natural isomorphism

τ x,y:xyyx \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:

(xy)z a x,y,z x(yz) τ x,yz (yz)x τ x,yId a y,z,x (yx)z a y,x,z y(xz) Idτ x,z y(zx) \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

x(yz) a x,y,z 1 (xy)z τ xy,z z(xy) Idτ y,z a z,x,y 1 x(zy) a x,z,y 1 (xz)y τ x,zId (zx)y, \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:(xy)zx(yz)a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z) denotes the components of the associator of 𝒞 \mathcal{C}^\otimes.

Definition

A (pointed) topological symmetric monoidal category is a (pointed) topological braided monoidal category (def. 5) for which the braiding

τ x,y:xyyx \tau_{x,y} \colon x \otimes y \to y \otimes x

satisfies the condition:

τ y,xτ x,y=1 xy \tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}

for all objects x,yx, y

Definition

Given a (pointed) topological symmetric monoidal category 𝒞\mathcal{C} with tensor product \otimes (def. 6) it is called a closed monoidal category if for each Y𝒞Y \in \mathcal{C} the functor Y()()XY \otimes(-)\simeq (-)\otimes X has a right adjoint, denoted [Y,][Y,-]

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

hence if there are natural isomorphisms

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

for all objects X,Z𝒞X,Z \in \mathcal{C}.

Since for the case that X=1X = 1 is the tensor unit of 𝒞\mathcal{C} this means that

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

the object [Y,Z]𝒞[Y,Z] \in \mathcal{C} is an enhancement of the ordinary hom-set Hom 𝒞(Y,Z)Hom_{\mathcal{C}}(Y,Z) to an object in 𝒞\mathcal{C}. Accordingly, it is also called the internal hom between YY and ZZ.

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

XYX×YXY X \wedge Y \coloneqq \frac{X\times Y}{X\vee Y}

is a symmetric monoidal category (def. 6) with unit object the pointed 0-sphere S 0S^0.

The components of the associator, the unitors and the braiding are those of Top as in example 3, 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 3.

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

  1. A monoid in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. 8) is equivalently a ring.

  2. A commutative monoid in in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. 8) is equivalently a commutative ring RR.

  3. An RR-module object in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. 9) is equivalently an RR-module;

  4. The tensor product of RR-module objects (def. 10) is the standard tensor product of modules.

  5. The category of module objects RMod(Ab)R Mod(Ab) (def. 10) is the standard category of modules RModR Mod.

Algebras and modules
Definition

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

  1. an object A𝒞A \in \mathcal{C};

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

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

such that

  1. (associativity) the following diagram commutes

    (AA)A a A,A,A A(AA) Aμ AA μA μ AA μ A, \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 aa is the associator isomorphism of 𝒞\mathcal{C};

  2. (unitality) the following diagram commutes:

    1A eid AA ide A1 μ r A, \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 rr are the left and right unitor isomorphisms of 𝒞\mathcal{C}.

Moreover, if (𝒞,,1)(\mathcal{C}, \otimes , 1) has the structure of a symmetric monoidal category (def. 6) (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) with symmetric braiding τ\tau, then a monoid (A,μ,e)(A,\mu, e) as above is called a commutative monoid in (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) if in addition

  • (commutativity) the following diagram commutes

    AA τ A,A AA μ μ A. \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,μ 1,e 1)(A 2,μ 2,f 2)(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2) is a morphism

f:A 1A 2 f \;\colon\; A_1 \longrightarrow A_2

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

A 1A 1 ff A 2A 2 μ 1 μ 2 A 1 f A 2 \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

1 𝒸 e 1 A 1 e 2 f A 2. \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.

Write Mon(𝒞,,1)Mon(\mathcal{C}, \otimes,1) for the category of monoids in 𝒞\mathcal{C}, and CMon(𝒞,,1)CMon(\mathcal{C}, \otimes, 1) for its subcategory of commutative monoids.

Example

Given a (pointed) topological monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1), then the tensor unit 11 is a monoid in 𝒞\mathcal{C} (def. 8) with product given by either the left or right unitor

1=r 1:111. \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.

By lemma 1, these two morphisms coincide and define an associative product with unit the identity id:11id \colon 1 \to 1.

If (𝒞,,1)(\mathcal{C}, \otimes , 1) is a symmetric monoidal category (def. 6), then this monoid is a commutative monoid.

Definition

Given a (pointed) topological monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 4), and given (A,μ,e)(A,\mu,e) a monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 8), then a left module object in (𝒞,,1)(\mathcal{C}, \otimes, 1) over (A,μ,e)(A,\mu,e) is

  1. an object N𝒞N \in \mathcal{C};

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

such that

  1. (unitality) the following diagram commutes:

    1N eid AN ρ A, \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 𝒞\mathcal{C}.

  2. (action property) the following diagram commutes

    (AA)N a A,A,N A(AN) Aρ AN μN ρ AN ρ N, \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 AA-module objects

(N 1,ρ 1)(N 2,ρ 2) (N_1, \rho_1) \longrightarrow (N_2, \rho_2)

is a morphism

f:N 1N 2 f\;\colon\; N_1 \longrightarrow N_2

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

AN 1 Af AN 2 ρ 1 ρ 2 N 1 f N 2. \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 AA-modules in 𝒞\mathcal{C} with AA-module homomorphisms between them, we write

AMod(𝒞). A Mod(\mathcal{C}) \,.

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

Example

Given a monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) (def. 4) with the tensor unit 11 regarded as a monoid in a monoidal category via example 6, then the left unitor

C:1CC \ell_C \;\colon\; 1\otimes C \longrightarrow C

makes every object C𝒞C \in \mathcal{C} into a left module, according to def. 9, over CC. The action property holds due to lemma 1. This gives an equivalence of categories

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

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

Proposition

In the situation of def. 9, the monoid (A,μ,e)(A,\mu, e) canonically becomes a left module over itself by setting ρμ\rho \coloneqq \mu. More generally, for C𝒞C \in \mathcal{C} any object, then ACA \otimes C naturally becomes a left AA-module by setting:

ρ:A(AC)a A,A,C 1(AA)CμidAC. \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 AA-modules of this form are called free modules.

The free functor FF constructing free AA-modules is left adjoint to the forgetful functor UU which sends a module (N,ρ)(N,\rho) to the underlying object U(N,ρ)NU(N,\rho) \coloneqq N.

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

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

f:ACN f \;\colon\; A\otimes C \longrightarrow N

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

AAC Af AN μid ρ AC f N. \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

f˜:C C1CeidACfN, \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 ff to the unit “in” AA. By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor)

AC idf˜ AN ideid = AAC idf AN. \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

AC idf˜ AN ideid = AAC Af AN μid ρ AC f N, \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 AA. This shows that ff is uniquely determined by f˜\tilde f via the relation

f=ρ(id Af˜). f = \rho \circ (id_A \otimes \tilde f) \,.

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

Definition

Given a (pointed) topological symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 6), given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 8), and given (N 1,ρ 1)(N_1, \rho_1) and (N 2,ρ 2)(N_2, \rho_2) two left AA-module objects (def.8), then the tensor product of modules N 1 AN 2N_1 \otimes_A N_2 is, if it exists, the coequalizer

N 1AN 2AAAAρ 1(τ N 1,AN 2)N 1ρ 2N 1N 1coequN 1 AN 2 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 (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 6), and given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 8). If all coequalizers exist in 𝒞\mathcal{C}, then the tensor product of modules A\otimes_A from def. 10 makes the category of modules AMod(𝒞)A Mod(\mathcal{C}) into a symmetric monoidal category, (AMod, A,A)(A Mod, \otimes_A, A) with tensor unit the object AA itself, regarded as an AA-module via prop. 5.

Definition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) as in prop. 6, then a monoid (E,μ,e)(E, \mu, e) in (AMod, A,A)(A Mod , \otimes_A , A) (def. 8) is called an AA-algebra.

Propposition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) in a monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) as in prop. 6, and an AA-algebra (E,μ,e)(E,\mu,e) (def. 11), then there is an equivalence of categories

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

between the category of commutative monoids in AModA Mod and the coslice category of commutative monoids in 𝒞\mathcal{C} under AA, hence between commutative AA-algebras in 𝒞\mathcal{C} and commutative monoids EE in 𝒞\mathcal{C} that are equipped with a homomorphism of monoids AEA \longrightarrow E.

(e.g. EKMM 97, VII lemma 1.3)

Proof

In one direction, consider a AA-algebra EE with unit e E:AEe_E \;\colon\; A \longrightarrow E and product μ E/A:E AEE\mu_{E/A} \colon E \otimes_A E \longrightarrow E. There is the underlying product μ E\mu_E

EAE AAA EE coeq E AE μ E μ E/A 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 Ee Ae_E\circ e_A, one find that this is a unit for μ E\mu_E and that (E,μ E,e Ee A)(E, \mu_E, e_E \circ e_A) is a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1).

Then consider the two conditions on the unit e E:AEe_E \colon A \longrightarrow E. First of all this is an AA-module homomorphism, which means that

()AA ide E AE μ A ρ A e E E (\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

A AE e Aid E AE μ E/A E. \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 AA, the latter gives

AE eid EE A AE e Eid E AE μ E/A E = EAE e Eid EE ρ μ E E id E, \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

AA ide E AE e Eid EE μ A ρ μ E A e E E id E=AA e Ee E EE μ A μ E A e E E. \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 Ae_A is a homomorphism of monoids (A,μ A,e A)(E,μ E,e Ee A)(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A).

Now for the converse direction, assume that (A,μ A,e A)(A,\mu_A, e_A) and (E,μ E,e E)(E, \mu_E, e'_E) are two commutative monoids in (𝒞,,1)(\mathcal{C}, \otimes, 1) with e E:AEe_E \;\colon\; A \to E a monoid homomorphism. Then EE inherits a left AA-module structure by

ρ:AEe AidEEμ EE. \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 μ E\mu_E coequalizes the two induced morphisms EAEAAEEE \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E. Hence the universal property of the coequalizer gives a factorization through some μ E/A:E AEE\mu_{E/A}\colon E \otimes_A E \longrightarrow E. This shows that (E,μ E/A,e E)(E, \mu_{E/A}, e_E) is a commutative AA-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

Day convolution
Definition

Let 𝒞\mathcal{C} be a small pointed topological monoidal category (def. 4) with tensor product denoted 𝒞:𝒞×𝒞𝒞\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 [𝒞,Top cg */][\mathcal{C},Top^{\ast/}_{cg}] (def. 3) is the functor

Day:[𝒞,Top cg */]×[𝒞,Top cg */][𝒞,Top cg */] \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. 1) given by the following coend (def. 2)

X DayY:c(c 1,c 2)𝒞×𝒞𝒞(c 1 𝒞c 2,c)X(c 1)Y(c 2). 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 SeqSeq denote the category with objects the natural numbers, and only the zero morphisms and identity morphisms on these objects:

Seq(n 1,n 2){S 0 ifn 1=n 2 * otherwise. 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 [Seq,Top cg */]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

(X DayY) n =(n 1,n 2)Seq(n 1+n 2,n)X n 1X n 2 =n 1+n 2=n(X n 1X n 2). \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. 4). This will be key for understanding monoids and modules with respect to Day convolution.

Definition

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

¯:[𝒞,Top cg */]×[𝒞,Top cg */][𝒞×𝒞,Top cg */] \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¯Y(X,Y), X \overline{\wedge} Y \;\coloneqq\; \wedge \circ (X,Y) \,,

i.e.

(X¯Y)(c 1,c 2)=X(c 1)X(c 2). (X \overline\wedge Y)(c_1,c_2) = X(c_1)\wedge X(c_2) \,.
Proposition

The Day convolution product (def. 12) of two functors is equivalently the left Kan extension (def. 4) of their external tensor product (def. 13) along the tensor product 𝒞\otimes_{\mathcal{C}}: there is a natural isomorphism

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

Hence the adjunction unit is a natural transformation of the form

𝒞×𝒞 X¯Y Top cg */ X DayY 𝒞. \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. 4 we may compute the left Kan extension as the following coend:

Lan 𝒞(X¯Y)(c) (c 1,c 2)𝒞(c 1 𝒞c 2,c)(X¯Y)(c 1,c 2) =(c 1,c 2)𝒞(c 1c 2)X(c 1)X(c 2). \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 Day\otimes_{Day} (def. 12) is universally characterized by the property that there are natural isomorphisms

[𝒞,Top cg */](X DayY,Z)[𝒞×𝒞,Top cg */](X¯Y,Z), [\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. 13.

Write

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

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

Proposition

For 𝒞\mathcal{C} a small pointed topological monoidal category (def. 4), the Day convolution tensor product Day\otimes_{Day} of def. 12 makes the pointed topologically enriched functor category

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

into a pointed topological monoidal category (def. 4) with tensor unit y(1)y(1) co-represented by the tensor unit 11 of 𝒞\mathcal{C}.

Moreover, if (𝒞,,1)(\mathcal{C}, \otimes, 1) is equipped with a braiding τ 𝒞\tau^{\mathcal{C}} (def. 5), then ([𝒞,Top cg */], Day,y(1))( [\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1)) becomes itself a braided monoidal category with braiding given by

(X DayY)(c) = c 1,c 2𝒞(c 1c 2)X(c 1)Y(c 2) τ X,Y(c) c 1,c 2𝒞(τ c 1,c 2 𝒞,c)τ X(c(1)),X(c 2) Top */ (Y DayX)(c) = c 1,c 2𝒞(c 2c 1)Y(c 2)X(c 1). \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

(X Day(Y DayZ))(c) (c 1,c 2)𝒞(c 1 𝒟c 2,c)X(c 1)(d 1,d 2)𝒞(d 1 𝒞d 2,c 2)(Y(d 2)Z(d 2)) c 1,d 1,d 2c 2𝒞(c 1 𝒟c 2,c)𝒞(d 1 𝒞d 2,c 2)𝒞(c 1 𝒞d 1 𝒞d 2,c)X(c 1)(Y(d 1)Z(d 2)) c 1,d 1,d 2𝒞(c 1 𝒞d 1 𝒞d 2,c)X(c 1)(Y(d 1)Z(d 2)), \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. 3) and then twice the co-Yoneda lemma (prop. 2). An analogous formula follows for X Day(Y DayZ)))(c)X \otimes_{Day} (Y \otimes_{Day} Z)))(c), and so associativity follows via prop. 8 from the associativity of the smash product and of the tensor product 𝒞\otimes_{\mathcal{C}}.

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

To see that y(1)y(1) is the tensor unit for Day\otimes_{Day}, use the Fubini theorem for coends (prop. 3) and then twice the co-Yoneda lemma (prop. 2) to get for any X[𝒞,Top cg */]X \in [\mathcal{C},Top^{\ast/}_{cg}] that

X Dayy(1) =c 1,c 2𝒞𝒞(c 1 𝒟c 2,)X(c 1)𝒞(1,c 2) c 1𝒞X(c 1)c 2𝒞𝒞(c 1 𝒞c 2,)𝒞(1,c 2) c 1𝒞X(c 1)𝒞(c 1 𝒞1,) c 1𝒞X(c 1)𝒞(c 1,) X() X. \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 𝒞\mathcal{C} a small pointed topological monoidal category (def. 4) with tensor product denoted 𝒞:𝒞×𝒞𝒞\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}, the monoidal category with Day convolution ([𝒞,Top cg */], Day,y(1))([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1)) from def. 9 is a closed monoidal category (def. 7). Its internal hom [,] Day[-,-]_{Day} is given by the end (def. 2)

[X,Y] Day(c)c 1,c 2Maps(𝒞(c 𝒞c 1,c 2),Maps(X(c 1),Y(c 2)) *) *. [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. 3) and the co-Yoneda lemma (def. 2) and in view of definition 3 of the enriched functor category, there is the following sequence of natural isomorphisms:

[𝒞,V](X,[Y,Z] Day) cMaps(X(c),c 1,c 2Maps(𝒞(c 𝒞c 1,c 2),Maps(Y(c 1),Z(c 2)) *) *) * cc 1,c 2Maps(𝒞(c 𝒞c 1,c 2)X(c)Y(c 1),Z(c 2)) * c 2Maps(c,c 1𝒞(c 𝒞c 1,c 2)X(c)Y(c 1),Z(c 2)) * c 2Maps((X DayY)(c 2),Z(c 2)) * [𝒞,V](X DayY,Z). \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. 9, the Yoneda embedding c𝒞(c,)c\mapsto \mathcal{C}(c,-) constitutes a strong monoidal functor

(𝒞, 𝒞,I)([𝒞,V], Day,y(I)). (\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. 9. To see that the tensor product is respected, apply the co-Yoneda lemma (prop 2) twice to get the following natural isomorphism

(y(c 1) Dayy(c 2))(c) d 1,d 2𝒞(d 1 𝒞d 2,c)𝒞(c 1,d 1)𝒞(c 2,d 2) 𝒞(c 1 𝒞c 2,c) =y(c 1 𝒞c 2)(c). \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 (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two (pointed) topologically enriched monoidal categories (def. 4). A topologically enriched lax monoidal functor between them is

  1. a topologically enriched functor

    F:𝒞𝒟, F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,
  2. a morphism

    ϵ:1 𝒟F(1 𝒞) \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
  3. a natural transformation

    μ x,y:F(x) 𝒟F(y)F(x 𝒞y) \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)

    for all x,y𝒞x,y \in \mathcal{C}

satisfying the following conditions:

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

    (F(x) 𝒟F(y)) 𝒟F(z) a F(x),F(y),F(z) 𝒟 F(x) 𝒟(F(y) 𝒟F(z)) μ x,yid idμ y,z F(x 𝒞y) 𝒟F(z) F(x) 𝒟(F(x 𝒞y)) μ x 𝒞y,z μ x,y 𝒞z F((x 𝒞y) 𝒞z) F(a x,y,z 𝒞) F(x 𝒞(y 𝒞z)), \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 𝒞a^{\mathcal{C}} and a 𝒟a^{\mathcal{D}} denote the associators of the monoidal categories;

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

    1 𝒟 𝒟F(x) ϵid F(1 𝒞) 𝒟F(x) F(x) 𝒟 μ 1 𝒞,x F(x) F( x 𝒞) F(1 𝒞x) \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

    F(x) 𝒟1 𝒟 idϵ F(x) 𝒟F(1 𝒞) r F(x) 𝒟 μ x,1 𝒞 F(x) F(r x 𝒞) F(x 𝒞1), \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 𝒞\ell^{\mathcal{C}}, 𝒟\ell^{\mathcal{D}}, r 𝒞r^{\mathcal{C}}, r 𝒟r^{\mathcal{D}} denote the left and right unitors of the two monoidal categories, respectively.

If ϵ\epsilon and alll μ x,y\mu_{x,y} are isomorphisms, then FF is called a strong monoidal functor.

If moreover (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are equipped with the structure of braided monoidal categories (def. 5), then the lax monoidal functor FF is called a braided monoidal functor if in addition the following diagram commutes for all objects x,y𝒞x,y \in \mathcal{C}

F(x) 𝒞F(y) τ F(x),F(y) 𝒟 F(y) 𝒟F(x) μ x,y μ y,x F(x 𝒞y) F(τ x,y 𝒞) F(y 𝒞x). \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:(F 1,μ 1,ϵ 1)(F 2,μ 2,ϵ 2)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:F 1(x)F 2(x)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,y𝒞x,y \in \mathcal{C}:

F 1(x) 𝒟F 1(y) f(x) 𝒟f(y) F 2(x) 𝒟F 2(y) (μ 1) x,y (μ 2) x,y F 1(x 𝒞y) f(x 𝒞y) F 2(x 𝒞y) \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

1 𝒟 ϵ 1 ϵ 2 F 1(1 𝒞) f(1 𝒞) F 2(1 𝒞). \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(𝒞,𝒟)MonFun(\mathcal{C},\mathcal{D}) for the resulting category of lax monoidal functors between monoidal categories 𝒞\mathcal{C} and 𝒟\mathcal{D}, similarly BraidMonFun(𝒞,𝒟)BraidMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between braided monoidal categories, and SymMonFun(𝒞,𝒟)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. 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 (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are symmetric monoidal categories (def. 6) then a braided monoidal functor (def. 14) between them is often called a symmetric monoidal functor.

Definition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two (pointed) topologically enriched monoidal categories (def. 4), and let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a topologically enriched lax monoidal functor between them, with product operation μ\mu.

Then a left module over the lax monoidal functor is

  1. a topologically enriched functor

    G:𝒞𝒟; G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,;
  2. a natural transformation

    ρ x,y:F(x) 𝒟N(y)N(x 𝒞y) \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,z𝒞x,y,z \in \mathcal{C} the following diagram commutes

    (F(x) 𝒟F(y)) 𝒟G(z) a F(x),F(y),F(z) 𝒟 F(x) 𝒟(F(y) 𝒟G(z)) μ x,yid idρ y,z F(x 𝒞y) 𝒟G(z) F(x) 𝒟(G(x 𝒞y)) ρ x 𝒞y,z ρ x,y 𝒞z G((x 𝒞y) 𝒞z) F(a x,y,z 𝒞) G(x 𝒞(y 𝒞z)), \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:(G 1,ρ 1)(G 2,ρ 2)f\;\colon\; (G_1, \rho_1) \longrightarrow (G_2,\rho_2) between two modules over a monoidal functor (F,μ,ϵ)(F,\mu,\epsilon) is

  • a natural transformation f x:N 1(x)N 2(x)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,y𝒞x,y \in \mathcal{C}:

F(x) 𝒟N 1(y) id 𝒟f(y) F(x) 𝒟N 2(y) (ρ 1) x,y (rhi 2) x,y N 1(x 𝒞y) f(x 𝒞y) N 2(x 𝒞y) \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 FModF Mod for the resulting category of modules over the monoidal functor FF.

Proposition

Let (𝒞,I)(\mathcal{C},\otimes I) be a pointed topologically enriched category (symmetric monoidal category) monoidal category (def. 4). Regard (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge , S^0) as a topological symmetric monoidal category as in example 4.

Then (commutative) monoids in (def. 8) the Day convolution monoidal category ([𝒞,Top cg */], Day,y(1 𝒞))([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) of prop. 9 are equivalent to (braided) lax monoidal functors (def. 14) of the form

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

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

Mon([𝒞,Top cg */], Day,y(1 𝒞)) ϕMonFunc(𝒞,Top cg */) CMon([𝒞,Top cg */], Day,y(1 𝒞)) SymMonFunc(𝒞,Top cg */). \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 AMon([𝒞,Top cg */], Day,y(1 𝒞))A \in Mon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) a given monoid object, then left AA-module objects (def. 9) are equivalent to left modules over monoidal functors (def. 15):

AModϕ(A)Mod. 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 14, a lax monoidal functor F:𝒞Top cg */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 0F(1 𝒞) 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)F(c 2)F(c 1 𝒞c 2) F(c_1) \wedge F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)

for all c 1,c 2𝒞c_1,c_2 \in \mathcal{C}.

Under the Yoneda lemma (prop. 1) the first of these is equivalently a morphism in [𝒞,Top cg */][\mathcal{C}, Top^{\ast/}_{cg}] of the form

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

Moreover, under the natural isomorphism of corollary 1 the second of these is equivalently a morphism in [𝒞,Top cg */][\mathcal{C}, Top^{\ast/}_{cg}] of the form

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

Translating the conditions of def. 14 satisfied by a lax monoidal functor through these identifications gives precisely the conditions of def. 8 on a (commutative) monoid in object FF under Day\otimes_{Day}.

Similarly for module objects and modules over monoidal functors.

Proposition

Let f:𝒞𝒟f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a lax monoidal functor (def. 14) between pointed topologically enriched monoidal categories (def. 4). Then the induced functor

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

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

f *:Mon([𝒟,Top cg */], Day,y(1 𝒟))Mon([𝒞,Top cg *], Day,y(1 𝒞) 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 𝒞\mathcal{C} and 𝒟\mathcal{D} are symmetric monoidal categories (def. 6) and ff is a braided monoidal functor (def. 14), then f *f^\ast also preserves commutative monoids

f *:CMon([𝒟,Top cg */], Day,y(1 𝒟))CMon([𝒞,Top cg *], Day,y(1 𝒞). 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

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

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

f *:AMod(𝒟)(f *A)Mod(𝒞). f^\ast \;\colon\; A Mod(\mathcal{D}) \longrightarrow (f^\ast A)Mod(\mathcal{C}) \,.
Proof

This is an immediate corollary of prop. 12, since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor.

Proposition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) be a topologically enriched monoidal category (def. 4), and let AMon([𝒞,Top cg */], Day,y(1 𝒞))A \in Mon([\mathcal{C},Top^{\ast/}_{cg}],\otimes_{Day}, y(1_{\mathcal{C}})) be a monoid in (def. 8) the pointed topological Day convolution monoidal category over 𝒞\mathcal{C} from prop. 9.

Then the category of left A-modules (def. 9) is a pointed topologically enriched functor category category (exmpl.)

AMod[AFree 𝒞Mod op,Top cg */], A Mod \;\simeq\; [ A Free_{\mathcal{C}}Mod^{op}, \; Top_{cg}^{\ast/} ] \,,

over the category of free modules over AA (def. 5) on objects in 𝒞\mathcal{C} (under the Yoneda embedding y:𝒞 op[𝒞,Top cg */]y \colon \mathcal{C}^{op} \to [\mathcal{C}, Top^{\ast/}_{cg}]). Hence the objects of AFree 𝒞ModA Free_{\mathcal{C}}Mod are identified with those of 𝒞\mathcal{C}, and its hom-spaces are

AFree 𝒞Mod(c 1,c 2)=AMod(A Dayy(c 1),A Dayy(c 2)). A Free_{\mathcal{C}}Mod( c_1, c_2) \;=\; A Mod( A \otimes_{Day} y(c_1),\; A \otimes_{Day} y(c_2) ) \,.

(MMSS 00, theorem 2.2)

Proof idea

Use the identification from prop. 12 of AA with a lax monoidal functor and of any AA-module object NN as a functor with the structure of a module over a monoidal functor, given by natural transformations

A(c 1)N(c 2)ρ c 1,c 2N(c 1c 2). 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 NN (def.) of the form

𝒞(c 1,c 2)N(c 1)N(c 2). \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

𝒞(c 3c 1,c 2)A(c 3)N(c 1)idρ c 3,c 1𝒞(c 3c 1,c 2)N(c 3c 2)N(c 2) \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 AFree 𝒞ModA Free_{\mathcal{C}}Mod have just this structure:

AFree 𝒞Mod(c 2,c 1) =AMod(A Dayy(c 2),A Dayy(c 1)) [𝒞,Top cg */](y(c 2),A Dayy(c 1)) (A Dayy(c 1))(c 2) c 3,c 4𝒞(c 3c 4,c 2)A(c 3)𝒞(c 1,c 4) c 3𝒞(c 3c 1,c 2)A(c 3). \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. 5, then the enriched Yoneda lemma (prop. 1), then the coend-expression for Day convolution (def. 12) and finally the co-Yoneda lemma (prop. 2).

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

AFree 𝒞Mod(c 3,c 2)AFree 𝒞Mod(c 2,c 1) =(c 5𝒞(c 5 𝒞c 2,c 3)A(c 5))(c 4𝒞(c 4 𝒞c 1,c 2)A(c 4)) c 4,c 5𝒞(c 5 𝒞c 2,c 3)𝒞(c 4 𝒞c 1,c 2)A(c 5)A(c 4) c 4,c 5𝒞(c 5 𝒞c 2,c 3)𝒞(c 5 𝒞c 4 𝒞c 1,c 5 𝒞c 2)A(c 5 𝒞c 4) c 4,c 5𝒞(c 5 𝒞c 4 𝒞c 1,c 5 𝒞c 2)A(c 5 𝒞c 4) c 4𝒞(c 4 𝒞c 1,c 3) VA(c 4), \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. 3);

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

    1. forming the tensor product of hom-objects of 𝒞\mathcal{C} with the identity morphism on c 5c_5;

    2. the monoidal functor incarnation A(c 5)A(c 4)A(c 5 𝒞c 4)A(c_5) \wedge A(c_4)\longrightarrow A(c_5 \otimes_{\mathcal{C}} c_4 ) of the monoid structure on AA;

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

  4. the last morphism is the morphism induced on coends by regarding extranaturality in c 4c_4 and c 5c_5 separately as a special case of extranaturality in c 6c 4c 5c_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 𝒞\mathcal{C} together with the action property over AA.

Pre-Excisive functors

Definition

Write

ι fin:Top cg,fin */Top cg */ \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. 3)

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

is the category of pre-excisive functors.

Write

𝕊 excy(S 0)Top cg,fin */(S 0,) \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 ι fin\iota_{fin} itself:

𝕊 exc=ι fin:KK. \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. 9 the smash product of pointed compactly generated topological spaces induces the structure of a closed (def. 7) symmetric monoidal category (def. 6)

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

with

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

  2. tensor product the Day convolution product Day\otimes_{Day} from def. 12,

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

  3. internal hom the dual operation [,] Day[-,-]_{Day} from prop. 10,

    called the mapping spectrum construction for pre-excisive functors.

Remark

By example 6 the sphere spectrum incarnated as a pre-excisive functor 𝕊 exc\mathbb{S}_{exc} (according to def. 16) is canonically a commutative monoid in the category of pre-excisive functors (def. 8)

Moreover, by example 7, every object of Exc(Top cg)Exc(Top_{cg}) (def. 16) is canonically a module object over 𝕊 exc\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)𝕊 excMod. Exc(Top_{cg}) \simeq \mathbb{S}_{exc}Mod \,.

We now consider restricting the domain of the pre-excisive functors of def. 16.

Definition

Define the following pointed topologically enriched (def.) symmetric monoidal categories (def. 6):

  1. SeqSeq 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)={S 0 forn 1=n 2 * otherwise 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. SymSym is the standard skeleton of the core of FinSet with zero morphisms adjoined: its objects are the finite sets n¯{1,,n}\overline{n} \coloneqq \{1, \cdots,n\} for nn \in \mathbb{N}, all non-zero morphisms are automorphisms and the automorphism group of {1,,n}\{1,\cdots,n\} is the symmetric group Σ n\Sigma_n, hence the hom-spaces are the following discrete topological spaces:

    Sym(n 1,n 2)={(Σ n 1) + forn 1=n 2 * otherwise 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

    τ n 1,n 2:n 1¯n 2¯n 2¯n 1¯ \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 Σ n 1+n 2\Sigma_{n_1+n_2} that shuffles the first n 1n_1 elements past the remaining n 2n_2 elements.

  3. OrthOrth has as objects finite dimenional real linear inner product spaces (V,,)(V, \langle -,-\rangle) and as non-zero morphisms the linear isometric isomorphisms between these; hence the automorphism group of the object (V,,)(V, \langle -,-\rangle) is the orthogonal group O(V)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 */Top^{\ast/}-enriched category by adjoining a basepoint to the hom-spaces;

    Orth(V 1,V 2){O(V 1) + fordim(V 1)=dim(V 2) * otherwise 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 SymSym, under the canonical embedding Σ n 1+n 2O(n 1+n 2)\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

Seq seq Sym sym Orth orth Top cg,fin */ n {1,,n} n S n V S V, \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. 16).

Here S VS^V denotes the one-point compactification of VV. On morphisms sym:(Σ n) +(O(n)) +sym \colon (\Sigma_n)_+ \hookrightarrow (O(n))_+ is the canonical inclusion of permutation matrices into orthogonal matrices and orth:O(V) +Aut(S V)orth \colon O(V)_+ \hookrightarrow Aut(S^V) is on O(V)O(V) the topological subspace inclusions of the pointed homeomorphisms S VS VS^V \to S^V that are induced under forming one-point compactification from linear isometries of VV (“representation spheres”).

Consider the sequence of restrictions of topological diagram categories, according to prop. 13 along the above inclusions:

Exc(Top cg)orth *[Orth,Top cg */]sym *[Sym,Top cg */]seq *[Seq,Top cg */]. 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

𝕊 orthorth *𝕊 exc,𝕊 symsym *𝕊 orth,𝕊 seqseq *𝕊 sym \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. 16) along these inclusions.

Remark

Since 𝕊 exc\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 6, prop. 13 says that all these restricted sphere spectra are still monoids, and that under restriction every pre-excisive functor, regarded as a 𝕊 exc\mathbb{S}_{exc}-module via remark 4, canonically becomes a module under the restricted sphere spectrum:

orth * :Exc(Top cg)𝕊 excMod𝕊 orthMod sym * :Exc(Top cg)𝕊 excMod𝕊 symMod seq * :Exc(Top cg)𝕊 excMod𝕊 seqMod. \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 orthorth and symsym are braided monoidal functors, the functor seqseq is not braided, hence 𝕊 orth\mathbb{S}_{orth} and 𝕊 sym\mathbb{S}_{sym} are commutative monoids, but 𝕊 Seq\mathbb{S}_{Seq} is not commutative. Hence prop. 13 gives the following situation

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

There is an equivalence of categories

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

which identifies the category of modules (def. 9) over the monoid 𝕊 seq\mathbb{S}_{seq} (remark 5) in the Day convolution monoidal structure (prop. 9) over the topological functor category [Seq,Top cg */][Seq,Top^{\ast/}_{cg}] from def. 17 with the category of sequential spectra (def.)

Under this equivalence, an 𝕊 seq\mathbb{S}_{seq}-module XX is taken to the sequential pre-spectrum X seqX^{seq} whose component spaces are the values of the pre-excisive functor XX on the standard n-sphere S n=(S 1) nS^n = (S^1)^{\wedge n}

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

and whose structure maps are the images of the action morphisms

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

under the isomorphism of corollary 1

𝕊 seq(n 1)X(n 1)X n 1+n 2 \mathbb{S}_{seq}(n_1) \wedge X(n_1) \longrightarrow X_{n_1 + n_2}

evaluated at n 1=1n_1 = 1

𝕊 seq(1)X(n) X n+1 S 1X n X n+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,

𝕊 seq Day𝕊 seq DayX id Dayρ 𝕊 seq DayX μ Dayid ρ 𝕊 seq DayX ρ X \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 𝕊 seq\mathbb{S}_{seq}-action

ρ:𝕊 seq DayXX \rho \;\colon\; \mathbb{S}_{seq} \otimes_{Day} X \longrightarrow X

is indeed uniquely fixed by the components of the form

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

This is because under corollary 1 the action property is identified with the componentwise property

S n 1S n 2X n 3 idρ n 2,n 3 S n 1X n 2+n 3 ρ n 1,n 2+n 3 S n 1+n 2X n 3 ρ n 1+n 2,n 3 X n 1+n 2+n 3, \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 𝕊 seq\mathbb{S}_{seq}. Hence this fixes the components ρ n,n\rho_{n',n} to be the nn'-fold composition of the structure maps σ nρ(1,n)\sigma_n \coloneqq \rho(1,n).

However, since, by remark 15, 𝕊 seq\mathbb{S}_{seq} is not commutative, there is no tensor product induced on SeqSpec(Top cg)SeqSpec(Top_{cg}) under the identification in prop. 15. But since 𝕊 orth\mathbb{S}_{orth} and 𝕊 sym\mathbb{S}_{sym} are commutative monoids by remark 15, it makes sense to consider the following definition.

Definition

In the terminology of remark 5 we say that

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

is the category of orthogonal spectra; and that

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

is the category of symmetric spectra.

By remark 5 and by prop. 6 these categories canonically carry a symmetric monoidal tensor product 𝕊 orth\otimes_{\mathbb{S}_{orth}} and 𝕊 seq\otimes_{\mathbb{S}_{seq}}, respectively. This we call the symmetric monoidal smash product of spectra. We usually just write for short

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

and

𝕊 sym:SymSpec(Top cg)×SymSpec(Top cg)SymSpec(Top cg) \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 18 from above.

Literature. ( Hovey-Shipley-Smith 00, section 1, section 2, Schwede 12, chapter I)

\,

Definition

A topological symmetric spectrum XX is

  1. a sequence {X nTop cg */|n}\{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 Σ(n)\Sigma(n) on X nX_n;

  3. a sequence of morphisms σ n:S 1X nX n+1\sigma_n \colon S^1 \wedge X_n \longrightarrow X_{n+1}

such that

  • for all n,kn, k \in \mathbb{N} the composite

    S kX nS k1S 1X nidσ nS k1X n+1S k2S 1X n+2idσ n+1σ n+k1X n+k 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 Σ(n)×Σ(k)\Sigma(n) \times \Sigma(k)-action.

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

  • a sequence of maps f n:X nY nf_n \colon X_n \longrightarrow Y_n

such that

  1. each f nf_n intetwines the Σ(n)\Sigma(n)-action;

  2. the following diagrams commute

    S 1X n f nid S 1Y n σ n X σ n Y X n+1 f n+1 Y n+1. \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)SymSpec(Top_{cg}) for the resulting category of symmetric spectra.

(Hovey-Shipley-Smith 00, def. 1.2.2, Schwede 12, def. 1.1)

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

Definition

A topological orthogonal spectrum XX is

  1. a sequence {X nTop cg */|n}\{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)O(n) on X nX_n;

  3. a sequence of morphisms σ n:S 1X nX n+1\sigma_n \colon S^1 \wedge X_n \longrightarrow X_{n+1}

such that

  • for all n,kn, k \in \mathbb{N} the composite

    S kX nS k1S 1X nidσ nS k1X n+1S k2S 1X n+2idσ n+1σ n+k1X n+k 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)×Ok()O(n) \times Ok()-action.

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

  • a sequence of maps f n:X nY nf_n \colon X_n \longrightarrow Y_n

such that

  1. each f nf_n intetwines the O(n)O(n)-action;

  2. the following diagrams commute

    S 1X n f nid S 1Y n σ n X σ n Y X n+1 f n+1 Y n+1. \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)OrthSpec(Top_{cg}) for the resulting category of orthogonal spectra.

Proposition

Definitions 19 and 20 are indeed equivalent to def. 18:

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

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

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

SymSpec(Top cg)𝕊 symMod. SymSpec(Top_{cg}) \simeq \mathbb{S}_{sym} Mod \,.

(Hovey-Shipley-Smith 00, prop. 2.2.1)

Proof

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

First of all, by example \ref{CoendGivesQuotientByDiagonalGroupAction} an object in [Sym,Top cg */][Sym, Top^{\ast/}_{cg}] is equivalently a “symmetric sequence”, namely a sequence of pointed topological spaces X kX_k, for kk \in \mathbb{N}, equipped with an action of Σ(k)\Sigma(k) (def. 17).

By corollary 1 and lemma \ref{FSPStructuredSphereSpectra}, the structure morphism of an 𝕊 sym\mathbb{S}_{sym}-module object on XX

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

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

S n 1X n 2X n 1+n 2 S^{n_1} \wedge X_{n_2} \longrightarrow X_{n_1 + n_2}

over Sym×SymSym \times Sym. This means equivalently that there is such a morphism for all n 1,n 2n_1, n_2 \in \mathbb{N} and that it is Σ(n 1)×Σ(n 2)\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=1n_1 = 1 is given. To that end, observe that lemma \ref{FSPStructuredSphereSpectra} 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:

S n 1S n 2X n 3 S n 1X n 2+n 3 S n 1+n 2X n 3 X n 1+n 2+n 3. \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 2S^{n_1 + n_2} has to be the composite of the actions of S n 2S^{n_2} followed by that of S n 1S^{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,YSymSpec(Top cg)X,Y \in SymSpec(Top_{cg}) two symmetric spectra, def. 19, then their smash product of spectra is the symmetric spectrum

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

with component spaces the coequalizer

p+1+q=nΣ(p+1+q) +Σ p×Σ 1×Σ qX pS 1Y qAAAArp+q=nΣ(p+q) +Σ p×Σ qX pY qcoeq(XY)(n) \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 pS 1Y qidσ qX pY q X_p \wedge S^1 \wedge Y_q \overset{id \wedge \sigma_{q}}{\longrightarrow} X_p \wedge Y_q

while rr has components given by the structure maps conjugated by the braiding in Top cg */Top^{\ast/}_{cg} and the permutation action χ p,1\chi_{p,1} (that shuffles the element on the right to the left)

X pS 1X qτ X p,S 1 Top cg */idS 1X pX qσ pidX p+1X qχ p,1idX 1+pX q. 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 XYX \wedge Y are those induced under the coequalizer by

X pY qS 1idσ pX pY q+1. X_p \wedge Y_q \wedge S^1 \overset{id \wedge \sigma_{p}}{\longrightarrow} X_{p} \wedge Y_{q+1} \,.

Analogously for orthogonal spectra.

(Schwede 12, p. 82)

Proposition

Under the identification of prop. 16, 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:

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

in the case of orthogonal spectra:

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

(Schwede 12, E.1.16)

Proof

By def. 10 the abstractly defined tensor product of two 𝕊 sym\mathbb{S}_{sym}-modules XX and YY is the coequalizer

X Day𝕊 sym DayYAAAAρ 1(τ X,𝕊 sym Dayid)Xρ 2XYcoeqX 𝕊 symY. 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 SymSym from def. 17. By example \ref{CoendGivesQuotientByDiagonalGroupAction} and unwinding the definitions, this is for any two symmetric spectra AA and BB 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:

(A DayB)(n) =n 1,n 2Σ(n 1+n 2,n)={Σ(n 1+n 2,n) + ifn 1+n 2=n * otherwise +A n 1B n 1 n 1+n 2=nΣ(n 1+n 2) +O(n 1)×O(n 2)(A n 1B n 2). \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. 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:

(X Day(𝕊 orth DayY))(n) (X DayY)(n) Z n X n 1(𝕊 sym DayY)(n 2) X n 1Y(n 2) Z n 1+n 2 (𝕊 sym DayY)(n 2) Y(n 2) Maps(X n 1,Z n 1+n 2) S n 2Y n 3 Y n 2+n 3 Maps(X n 1,Z n 1+n 2+n 3) X n 1S n 2Y n 3 X n 1Y n 2+n 3 Z n 1+n 2+n 3. \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. 16, we may restrict the coequalizer to n 2=1n_2 = 1 without changing it.

The form of the morphism rr is obtained by the analogous sequence of identifications of morphisms, now with the parenthesis to the left. That it involves τ Top cg */\tau^{Top^{\ast/}_{cg}} and the permutation action τ sym\tau^{sym} as shown above follows from the formula for the braiding of the Day convolution tensor product from the proof of prop. 9:

τ A,B Day(n)=n 1,n 2Sym(τ n 1,n 2 Sym,n)τ A n 1,B n 2 Top cg */ \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 Day𝕊 symτ X,𝕊 sym Day𝕊 sym DayXX 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. 4 for the left Kan extension A DayBLan A¯BA \otimes_{Day} B \simeq Lan_{\otimes} A \overline{\wedge} B (prop. 8):

[Sym,Top cg */](τ X,𝕊 sym Day,X) nMaps(n 1,n 2Sym(τ n 1,n 2 sym,n)τ X n 1,S n 2 Top cg */,X(n)) * n 1,n 2Maps(τ X n 1,S n 2 Top cg */,X(τ n 1,n 2 sym)) *. \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

See also

Revised on July 13, 2016 13:06:40 by Urs Schreiber (131.220.184.222)