Redirected from "functor with smash product".
Contents
Context
Stable Homotopy theory
Higher algebra
Contents
Idea
In suitable “coordinate-free ” presentations of spectra , the structure of a (commutative ) monoid with respect to the smash product of spectra (an A-infinity ring (E-infinity ring )) may be expressed directly as a lax monoidal functor on the indexing spaces, hence a functor that intertwines the smash product of indexing spaces with that of the component spaces, but without explicitly mentioning the smash product of spectra .
General
So a functor with smash products is a suitably well behaved functor
E : 𝒟 ⟶ Spaces * /
E \;\colon\; \mathcal{D} \longrightarrow Spaces^{\ast/}
from a monoidal category ( 𝒟 , ∧ ) (\mathcal{D},\wedge) to pointed topological spaces /pointed simplicial sets and equipped with natural transformations
𝕊 ( V ) ⟶ E ( V )
\mathbb{S}(V) \longrightarrow E(V)
and
E ( V ) ∧ E ( W ) ⟶ E ( V ∧ W )
E(V) \wedge E(W) \longrightarrow E(V \wedge W)
that are associative and unital in the evident sense.
For highly structured spectra
For the case of highly structured spectra such as orthogonal spectra , symmetric spectra and S-modules , the equivalence of FSPs with monoids with respect to the symmetric smash product of spectra is due to this proposition at Day convolution . (MMSS 00, prop. 22.1, prop. 22.6 ).
(For instance accounts such as (Kochmann 96, section 3.3 , Schwede 14 ) follow this perspective and define ring spectra first as FSPs, before or introducing the smash product on spectra)
For excisive functors
For the monoidal model structure for excisive functors , the fact that monoids with respect to the symmetric smash product of spectra are equivalently FSPs is discussed in (Lydakis 98, remark 5.12 ). See this proposition .
Ingredients
Topological ends and coends
For working with pointed topologically enriched functors , a certain shape of limits /colimits is particularly relevant: these are called (pointed topological enriched) ends and coends . We here introduce these and then derive some of their basic properties, such as notably the expression for topological left Kan extension in terms of coends (prop. below). Further below it is via left Kan extension along the ordinary smash product of pointed topological spaces (“Day convolution ”) that the symmetric monoidal smash product of spectra is induced.
Definition
Let 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 .
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 ) ⟶ ≃ τ 𝒞 ( Z , Y ) ∧ 𝒞 ( Y , X ) ⟶ ∘ Z , Y , X 𝒞 ( 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)
\,.
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 c ∈ 𝒞 c \in \mathcal{C} and d ∈ 𝒟 d\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.
F : 𝒞 × 𝒟 ⟶ Top 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. )
F : 𝒞 op × 𝒞 ⟶ Top 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 . Let
F : 𝒞 op × 𝒞 ⟶ Top 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. .
The coend of F F , denoted ∫ c ∈ 𝒞 F ( 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 F F via example :
∐ c , d ∈ 𝒞 𝒞 ( c , d ) ∧ F ( d , c ) AAAAAAAA ⟶ ⊔ c , d ρ ( d , c ) ( c ) ⟶ ⊔ c , d ρ ( c , d ) ( d ) ∐ c ∈ 𝒞 F ( c , c ) ⟶ coeq ∫ c ∈ 𝒞 F ( 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)
\,.
The end of F F , denoted ∫ c ∈ 𝒞 F ( 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 F F via example :
∫ c ∈ 𝒞 F ( c , c ) ⟶ equ ∏ c ∈ 𝒞 F ( c , c ) AAAAAAAA ⟶ ⊔ c , d ρ ˜ ( c , d ) ( c ) ⟶ ⊔ c , d ρ ˜ d , c ( d ) ∏ c ∈ 𝒞 Maps ( 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
\,.
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 ( ∫ c ∈ 𝒞 Maps ( F ( c ) , G ( c ) ) * ) ⟶ equ ∏ c ∈ 𝒞 Hom Top cg * / ( F ( c ) , G ( c ) ) AAAAAAAA ⟶ ⊔ c , d U ( ρ ˜ ( c , d ) ( d ) ) ⟶ ⊔ c , d U ( ρ ˜ d , c ( c ) ) ∏ c , d ∈ 𝒞 Hom 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 c G ( c ) } c ∈ 𝒞 \left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}} . The two parallel maps in the equalizer diagram take such a collection to the functions which send any c → f d c \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 } c ∈ 𝒞 \{\eta_c\}_{c\in \mathcal{C}} being in the equalizer, means precisley that for all c , d c,d and all f : c → d f\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 } c ∈ 𝒞 \{\eta_c\}_{c\in \mathcal{C}} be a natural transformation .
Conversely, example 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. ) of example :
[ 𝒞 , Top cg * / ] ( F , G ) ≔ ∫ c ∈ 𝒞 Maps ( 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
( ∫ c ∈ 𝒞 Maps ( F ( c ) , G ( c ) ) * ) ∧ ( ∫ c ∈ 𝒞 Maps ( G ( c ) , H ( c ) ) * ) ⟶ ∏ c ∈ 𝒞 Maps ( F ( c ) , G ( c ) ) * ∧ Maps ( G ( c ) , H ( c ) ) * ⟶ ( ∘ F ( c ) , G ( c ) , H ( c ) ) c ∈ 𝒞 ∏ c ∈ 𝒞 Maps ( 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 : 𝒞 → Top cg * / F \colon \mathcal{C}\to Top^{\ast/}_{cg} a pointed topologically enriched functor (def. ) and for c ∈ 𝒞 c\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. , from the functor represented by c c to F F , and the value of F F on c c .
In terms of the ends (def. ) defining these hom-spaces , this means that
∫ d ∈ 𝒞 Maps ( 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. is essentially dual to the proof of the next prop. .
Now that natural transformations are phrased in terms of ends (example ), as is the Yoneda lemma (prop. ), it is natural to consider the dual statement involving coends :
Proposition
(co-Yoneda lemma )
Let 𝒞 \mathcal{C} be a small pointed topologically enriched categories (def. ). For F : 𝒞 → Top cg * / F \colon \mathcal{C}\to Top^{\ast/}_{cg} a pointed topologically enriched functor (def. ) and for c ∈ 𝒞 c\in \mathcal{C} an object, there is a natural isomorphism
F ( − ) ≃ ∫ c ∈ 𝒞 𝒞 ( 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. 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 F F .
(e.g. MMSS 00, lemma 1.6 )
Proof
The coequalizer of pointed topological spaces that we need to consider has underlying it a coequalizer of underlying pointed sets (prop. , prop. , prop. ). That in turn is the colimit over the diagram of underlying sets with the basepointe adjoined to the diagram (prop. ). For a coequalizer diagram adding that extra point to the diagram clearly does not change the colimit, and so we need to consider the plain coequalizer of sets.
That is just the set of equivalence classes of pairs
( c → c 0 , x ∈ F ( c ) ) ,
( c \overset{}{\to} c_0,\; x \in F(c) )
\,,
where two such pairs
( c → f c 0 , x ∈ F ( c ) ) , ( d → g c 0 , y ∈ F ( 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
c → ϕ d
c \overset{\phi}{\to} d
such that
f = g ∘ ϕ , and y = 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 0 d = c_0 and g = id c 0 g = id_{c_0} , so that f = ϕ f = \phi shows that any pair
( c → ϕ c 0 , x ∈ F ( 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 ) ⟶ ∫ c 𝒞 ( 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) .
It is this analogy that gives the name to the following statement:
Proposition
(Fubini theorem for (co)-ends)
For F F a pointed topologically enriched bifunctor on a small pointed topological product category 𝒞 1 × 𝒞 2 \mathcal{C}_1 \times \mathcal{C}_2 (def. ), 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. ) is equivalently formed consecutively over each variable, in either order:
∫ ( c 1 , c 2 ) F ( ( c 1 , c 2 ) , ( c 1 , c 2 ) ) ≃ ∫ c 1 ∫ c 2 F ( ( c 1 , c 2 ) , ( c 1 , c 2 ) ) ≃ ∫ c 2 ∫ c 1 F ( ( 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 1 ∫ c 2 F ( ( c 1 , c 2 ) , ( c 1 , c 2 ) ) ≃ ∫ c 2 ∫ c 1 F ( ( 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.
Proposition
(left Kan extension via coends)
Let 𝒞 , 𝒟 \mathcal{C}, \mathcal{D} be small pointed topologically enriched categories (def. ) and let
p : 𝒞 ⟶ 𝒟
p \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
be a pointed topologically enriched functor (def. ). Then precomposition with p p constitutes a functor
p * : [ 𝒟 , Top cg * / ] ⟶ [ 𝒞 , Top cg * / ]
p^\ast
\;\colon\;
[\mathcal{D}, Top^{\ast/}_{cg}]
\longrightarrow
[\mathcal{C}, Top^{\ast/}_{cg}]
G ↦ G ∘ p G\mapsto G\circ p . This functor has a left adjoint Lan p Lan_p , called left Kan extension along p p
[ 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. ):
( Lan p F ) : d ↦ ∫ c ∈ 𝒞 𝒟 ( 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 and def. ), then use the respect of Maps ( − , − ) * Maps(-,-)_\ast for ends/coends (remark ), use the smash/mapping space adjunction (cor. ), use the Fubini theorem (prop. ) and finally use Yoneda reduction (prop. ) to obtain a sequence of natural isomorphisms as follows:
[ 𝒟 , Top cg * / ] ( Lan p F , G ) = ∫ d ∈ 𝒟 Maps ( ( Lan p F ) ( d ) , G ( d ) ) * = ∫ d ∈ 𝒟 Maps ( ∫ c ∈ 𝒞 𝒟 ( p ( c ) , d ) ∧ F ( c ) , G ( d ) ) * ≃ ∫ d ∈ 𝒟 ∫ c ∈ 𝒞 Maps ( 𝒟 ( p ( c ) , d ) ∧ F ( c ) , G ( d ) ) * ≃ ∫ c ∈ 𝒞 ∫ d ∈ 𝒟 Maps ( F ( c ) , Maps ( 𝒟 ( p ( c ) , d ) , G ( d ) ) * ) * ≃ ∫ c ∈ 𝒞 Maps ( F ( c ) , ∫ d ∈ 𝒟 Maps ( 𝒟 ( p ( c ) , d ) , G ( d ) ) * ) * ≃ ∫ c ∈ 𝒞 Maps ( 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 below.
Definition
A (pointed) topologically enriched monoidal category is a (pointed) topologically enriched category 𝒞 \mathcal{C} (def. ) equipped with
a (pointed) topologically enriched functor (def. )
⊗ : 𝒞 × 𝒞 ⟶ 𝒞
\otimes
\;\colon\;
\mathcal{C} \times \mathcal{C}
\longrightarrow
\mathcal{C}
out of the (pointed) topologival product category of 𝒞 \mathcal{C} with itself (def. ), called the tensor product ,
an object
called the unit object or tensor unit ,
a natural isomorphism (def. )
a : ( ( − ) ⊗ ( − ) ) ⊗ ( − ) ⟶ ≃ ( − ) ⊗ ( ( − ) ⊗ ( − ) )
a
\;\colon\;
((-)\otimes (-)) \otimes (-)
\overset{\simeq}{\longrightarrow}
(-) \otimes ((-)\otimes(-))
called the associator ,
a natural isomorphism
ℓ : ( 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:
triangle identity :
( x ⊗ 1 ) ⊗ y ⟶ a x , 1 , y x ⊗ ( 1 ⊗ y ) ρ x ⊗ 1 y ↘ ↙ 1 x ⊗ λ y x ⊗ y
\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
&&
}
the pentagon identity :
Layer 1
(
w
⊗
x
)
⊗
(
y
⊗
z
)
(w\otimes x)\otimes(y\otimes z)
(
(
w
⊗
x
)
⊗
y
)
⊗
z
((w\otimes x)\otimes y)\otimes z
w
⊗
(
x
⊗
(
y
⊗
z
)
)
w\otimes (x\otimes(y\otimes z))
(
w
⊗
(
x
⊗
y
)
)
⊗
z
(w\otimes (x\otimes y))\otimes z
w
⊗
(
(
x
⊗
y
)
⊗
z
)
w\otimes ((x\otimes y)\otimes z)
a
w
⊗
x
,
y
,
z
a_{w\otimes x,y,z}
a
w
,
x
,
y
⊗
z
a_{w,x,y\otimes z}
a
w
,
x
,
y
⊗
1
z
a_{w,x,y}\otimes 1_{z}
1
w
⊗
a
x
,
y
,
z
1_w\otimes a_{x,y,z}
a
w
,
x
⊗
y
,
z
a_{w,x\otimes y,z}
Lemma
(Kelly 64 )
Let ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes, 1) be a monoidal category , def. . Then the left and right unitors ℓ \ell and r r satisfy the following conditions:
ℓ 1 = r 1 : 1 ⊗ 1 ⟶ ≃ 1 \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 ;
for all objects x , y ∈ 𝒞 x,y \in \mathcal{C} the following diagram commutes :
( 1 ⊗ x ) ⊗ y α 1 , x , y ↓ ↘ ℓ x y 1 ⊗ ( x ⊗ y ) ⟶ ℓ x ⊗ y x ⊗ y .
\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. ) equipped with a natural isomorphism
τ x , y : x ⊗ y → y ⊗ x
\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:
( x ⊗ y ) ⊗ z → a x , y , z x ⊗ ( y ⊗ z ) → τ x , y ⊗ z ( y ⊗ z ) ⊗ x ↓ τ x , y ⊗ Id ↓ a y , z , x ( y ⊗ x ) ⊗ z → a y , x , z y ⊗ ( x ⊗ z ) → Id ⊗ τ x , z y ⊗ ( z ⊗ x )
\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 ⊗ ( y ⊗ z ) → a x , y , z − 1 ( x ⊗ y ) ⊗ z → τ x ⊗ y , z z ⊗ ( x ⊗ y ) ↓ Id ⊗ τ y , z ↓ a z , x , y − 1 x ⊗ ( z ⊗ y ) → a x , z , y − 1 ( x ⊗ z ) ⊗ y → τ x , z ⊗ Id ( z ⊗ x ) ⊗ 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 : ( x ⊗ y ) ⊗ z → x ⊗ ( y ⊗ z ) 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
Given a (pointed) topological symmetric monoidal category 𝒞 \mathcal{C} with tensor product ⊗ \otimes (def. ) it is called a closed monoidal category if for each Y ∈ 𝒞 Y \in \mathcal{C} the functor Y ⊗ ( − ) ≃ ( − ) ⊗ X Y \otimes(-)\simeq (-)\otimes X has a right adjoint , denoted [ Y , − ] [Y,-]
𝒞 ⊥ ⟶ [ Y , − ] ⟵ ( − ) ⊗ Y 𝒞 ,
\mathcal{C}
\underoverset
{\underset{[Y,-]}{\longrightarrow}}
{\overset{(-) \otimes Y}{\longleftarrow}}
{\bot}
\mathcal{C}
\,,
hence if there are natural isomorphisms
Hom 𝒞 ( X ⊗ Y , Z ) ≃ Hom 𝒞 C ( X , [ Y , Z ] )
Hom_{\mathcal{C}}(X \otimes Y, Z)
\;\simeq\;
Hom_{\mathcal{C}}{C}(X, [Y,Z])
for all objects X , Z ∈ 𝒞 X,Z \in \mathcal{C} .
Since for the case that X = 1 X = 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 Y Y and Z Z .
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 ⊗ ℤ \otimes_{\mathbb{Z}} and with tensor unit the additive group ℤ \mathbb{Z} of integers . Again the associator , unitor and braiding isomorphism are the evident ones coming from the underlying sets, as in example .
This is the archetypical case that motivates the notation “⊗ \otimes ” for the pairing operation in a monoidal category :
A monoid in ( Ab , ⊗ ℤ , ℤ ) (Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a ring .
A commutative monoid in in ( Ab , ⊗ ℤ , ℤ ) (Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a commutative ring R R .
An R R -module object in ( Ab , ⊗ ℤ , ℤ ) (Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently an R R -module ;
The tensor product of R R -module objects (def. ) is the standard tensor product of modules .
The category of module objects R Mod ( Ab ) R Mod(Ab) (def. ) is the standard category of modules R Mod R 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
an object A ∈ 𝒞 A \in \mathcal{C} ;
a morphism e : 1 ⟶ A e \;\colon\; 1 \longrightarrow A (called the unit )
a morphism μ : A ⊗ A ⟶ A \mu \;\colon\; A \otimes A \longrightarrow A (called the product );
such that
(associativity ) the following diagram commutes
( A ⊗ A ) ⊗ A ⟶ ≃ a A , A , A A ⊗ ( A ⊗ A ) ⟶ A ⊗ μ A ⊗ A μ ⊗ A ↓ ↓ μ A ⊗ A ⟶ ⟶ 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 a a is the associator isomorphism of 𝒞 \mathcal{C} ;
(unitality ) the following diagram commutes :
1 ⊗ A ⟶ e ⊗ id A ⊗ A ⟵ id ⊗ e A ⊗ 1 ℓ ↘ ↓ 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 ℓ \ell and r r 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. ) ( 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
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 1 ⟶ A 2
f \;\colon\; A_1 \longrightarrow A_2
in 𝒞 \mathcal{C} , such that the following two diagrams commute
A 1 ⊗ A 1 ⟶ f ⊗ f A 2 ⊗ A 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 1 1 is a monoid in 𝒞 \mathcal{C} (def. ) with product given by either the left or right unitor
ℓ 1 = r 1 : 1 ⊗ 1 ⟶ ≃ 1 .
\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1
\,.
By lemma , these two morphisms coincide and define an associative product with unit the identity id : 1 → 1 id \colon 1 \to 1 .
If ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes , 1) is a symmetric monoidal category (def. ), then this monoid is a commutative monoid .
Definition
Given a (pointed) topological monoidal category ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes, 1) (def. ), and given ( A , μ , e ) (A,\mu,e) a monoid in ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes, 1) (def. ), then a left module object in ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes, 1) over ( A , μ , e ) (A,\mu,e) is
an object N ∈ 𝒞 N \in \mathcal{C} ;
a morphism ρ : A ⊗ N ⟶ N \rho \;\colon\; A \otimes N \longrightarrow N (called the action );
such that
(unitality ) the following diagram commutes :
1 ⊗ N ⟶ e ⊗ id A ⊗ N ℓ ↘ ↓ ρ A ,
\array{
1 \otimes N
&\overset{e \otimes id}{\longrightarrow}&
A \otimes N
\\
& {}_{\mathllap{\ell}}\searrow
& \downarrow^{\mathrlap{\rho}}
\\
&& A
}
\,,
where ℓ \ell is the left unitor isomorphism of 𝒞 \mathcal{C} .
(action property) the following diagram commutes
( A ⊗ A ) ⊗ N ⟶ ≃ a A , A , N A ⊗ ( A ⊗ N ) ⟶ A ⊗ ρ A ⊗ N μ ⊗ N ↓ ↓ ρ A ⊗ N ⟶ ⟶ 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 A A -module objects
( N 1 , ρ 1 ) ⟶ ( N 2 , ρ 2 )
(N_1, \rho_1) \longrightarrow (N_2, \rho_2)
is a morphism
f : N 1 ⟶ N 2
f\;\colon\; N_1 \longrightarrow N_2
in 𝒞 \mathcal{C} , such that the following diagram commutes :
A ⊗ N 1 ⟶ A ⊗ f A ⊗ N 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 A A -modules in 𝒞 \mathcal{C} with A A -module homomorphisms between them, we write
A Mod ( 𝒞 ) .
A Mod(\mathcal{C})
\,.
This is naturally a (pointed) topologically enriched category itself.
Proposition
In the situation of def. , the monoid ( A , μ , e ) (A,\mu, e) canonically becomes a left module over itself by setting ρ ≔ μ \rho \coloneqq \mu . More generally, for C ∈ 𝒞 C \in \mathcal{C} any object, then A ⊗ C A \otimes C naturally becomes a left A A -module by setting:
ρ : A ⊗ ( A ⊗ C ) ⟶ ≃ a A , A , C − 1 ( A ⊗ A ) ⊗ C ⟶ μ ⊗ id A ⊗ C .
\rho
\;\colon\;
A \otimes (A \otimes C)
\underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow}
(A \otimes A) \otimes C
\overset{\mu \otimes id}{\longrightarrow}
A \otimes C
\,.
The A A -modules of this form are called free modules .
The free functor F F constructing free A A -modules is left adjoint to the forgetful functor U U which sends a module ( N , ρ ) (N,\rho) to the underlying object U ( N , ρ ) ≔ N U(N,\rho) \coloneqq N .
A Mod ( 𝒞 ) ⊥ ⟶ U ⟵ F 𝒞 .
A Mod(\mathcal{C})
\underoverset
{\underset{U}{\longrightarrow}}
{\overset{F}{\longleftarrow}}
{\bot}
\mathcal{C}
\,.
Proof
A homomorphism out of a free A A -module is a morphism in 𝒞 \mathcal{C} of the form
f : A ⊗ C ⟶ N
f \;\colon\; A\otimes C \longrightarrow N
fitting into the diagram (where we are notationally suppressing the associator )
A ⊗ A ⊗ C ⟶ A ⊗ f A ⊗ N μ ⊗ id ↓ ↓ ρ A ⊗ C ⟶ 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 ⟶ ≃ ℓ C 1 ⊗ C ⟶ e ⊗ id A ⊗ C ⟶ f N ,
\tilde f
\;\colon\;
C
\underoverset{\simeq}{\ell_C}{\longrightarrow}
1 \otimes C
\overset{e\otimes id}{\longrightarrow}
A \otimes C
\overset{f}{\longrightarrow}
N
\,,
i.e. the restriction of f f to the unit “in” A A . By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor )
A ⊗ C ⟶ id ⊗ f ˜ A ⊗ N id ⊗ e ⊗ id ↓ ↓ = A ⊗ A ⊗ C ⟶ id ⊗ f A ⊗ 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
&\underset{id \otimes f}{\longrightarrow}&
A \otimes N
}
\,.
Pasting this square onto the top of the previous one yields
A ⊗ C ⟶ id ⊗ f ˜ A ⊗ N id ⊗ e ⊗ id ↓ ↓ = A ⊗ A ⊗ C ⟶ A ⊗ f A ⊗ N μ ⊗ id ↓ ↓ ρ A ⊗ C ⟶ 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 A A . This shows that f f is uniquely determined by f ˜ \tilde f via the relation
f = ρ ∘ ( id A ⊗ f ˜ ) .
f = \rho \circ (id_A \otimes \tilde f)
\,.
This natural bijection between f f and f ˜ \tilde f establishes the adjunction.
Definition
Given a (pointed) topological symmetric monoidal category ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes, 1) (def. ), given ( A , μ , e ) (A,\mu,e) a commutative monoid in ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes, 1) (def. ), and given ( N 1 , ρ 1 ) (N_1, \rho_1) and ( N 2 , ρ 2 ) (N_2, \rho_2) two left A A -module objects (def. ), then the tensor product of modules N 1 ⊗ A N 2 N_1 \otimes_A N_2 is, if it exists, the coequalizer
N 1 ⊗ A ⊗ N 2 AAAA ⟶ ρ 1 ∘ ( τ N 1 , A ⊗ N 2 ) ⟶ N 1 ⊗ ρ 2 N 1 ⊗ N 1 ⟶ coequ N 1 ⊗ A N 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. ), and given ( A , μ , e ) (A,\mu,e) a commutative monoid in ( 𝒞 , ⊗ , 1 ) (\mathcal{C}, \otimes, 1) (def. ). If all coequalizers exist in 𝒞 \mathcal{C} , then the tensor product of modules ⊗ A \otimes_A from def. makes the category of modules A Mod ( 𝒞 ) A Mod(\mathcal{C}) into a symmetric monoidal category , ( A Mod , ⊗ A , A ) (A Mod, \otimes_A, A) with tensor unit the object A A itself, regarded as an A A -module via prop. .
Definition
Given a monoidal category of modules ( A Mod , ⊗ A , A ) (A Mod , \otimes_A , A) as in prop. , then a monoid ( E , μ , e ) (E, \mu, e) in ( A Mod , ⊗ A , A ) (A Mod , \otimes_A , A) (def. ) is called an A A -algebra .
Proposition
Given a monoidal category of modules ( A Mod , ⊗ A , A ) (A Mod , \otimes_A , A) in a monoidal category ( 𝒞 , ⊗ , 1 ) (\mathcal{C},\otimes, 1) as in prop. , and an A A -algebra ( E , μ , e ) (E,\mu,e) (def. ), then there is an equivalence of categories
A Alg comm ( 𝒞 ) ≔ CMon ( A Mod ) ≃ CMon ( 𝒞 ) A /
A Alg_{comm}(\mathcal{C})
\coloneqq
CMon(A Mod)
\simeq
CMon(\mathcal{C})^{A/}
between the category of commutative monoids in A Mod A Mod and the coslice category of commutative monoids in 𝒞 \mathcal{C} under A A , hence between commutative A A -algebras in 𝒞 \mathcal{C} and commutative monoids E E in 𝒞 \mathcal{C} that are equipped with a homomorphism of monoids A ⟶ E A \longrightarrow E .
(e.g. EKMM 97, VII lemma 1.3 )
Proof
In one direction, consider a A A -algebra E E with unit e E : A ⟶ E e_E \;\colon\; A \longrightarrow E and product μ E / A : E ⊗ A E ⟶ E \mu_{E/A} \colon E \otimes_A E \longrightarrow E . There is the underlying product μ E \mu_E
E ⊗ A ⊗ E AAA ⟶ ⟶ E ⊗ E ⟶ coeq E ⊗ A E 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 E ∘ e A e_E\circ e_A , one find that this is a unit for μ E \mu_E and that ( E , μ E , e E ∘ e 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 : A ⟶ E e_E \colon A \longrightarrow E . First of all this is an A A -module homomorphism, which means that
( ⋆ ) A ⊗ A ⟶ id ⊗ e E A ⊗ E 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 ⊗ A E ⟶ e A ⊗ id E ⊗ A E ≃ ↘ ↓ 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 A A , the latter gives
A ⊗ E ⟶ e ⊗ id E ⊗ E ↓ ↓ A ⊗ A E ⟶ e E ⊗ id E ⊗ A E ≃ ↓ ↓ μ E / A E = E ≃ A ⊗ E ⟶ e E ⊗ id E ⊗ E 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
A ⊗ A ⟶ id ⊗ e E A ⊗ E ⟶ e E ⊗ id E ⊗ E μ A ↓ ρ ↓ ↓ μ E A ⟶ e E E ⟶ id E = A ⊗ A ⟶ e E ⊗ e E E ⊗ E 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 A e_A is a homomorphism of monoids ( A , μ A , e A ) ⟶ ( E , μ E , e E ∘ e 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 : A → E e_E \;\colon\; A \to E a monoid homomorphism. Then E E inherits a left A A -module structure by
ρ : A ⊗ E ⟶ e A ⊗ id E ⊗ E ⟶ μ E E .
\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 E ⊗ A ⊗ E AA ⟶ ⟶ E ⊗ E E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E . Hence the universal property of the coequalizer gives a factorization through some μ E / A : E ⊗ A E ⟶ E \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 A A -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. ) with tensor product denoted ⊗ 𝒞 : 𝒞 × 𝒞 → 𝒞 \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. ) 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. ) given by the following coend (def. )
X ⊗ Day Y : c ↦ ∫ ( c 1 , c 2 ) ∈ 𝒞 × 𝒞 𝒞 ( c 1 ⊗ 𝒞 c 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 Seq Seq denote the category with objects the natural numbers , and only the zero morphisms and identity morphisms on these objects:
Seq ( n 1 , n 2 ) ≔ { S 0 if n 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 ℕ \mathbb{N} -sequence of pointed topological spaces. Given two such, then their Day convolution according to def. is
( X ⊗ Day Y ) n = ∫ ( n 1 , n 2 ) Seq ( n 1 + n 2 , n ) ∧ X n 1 ∧ X n 2 = ∐ n 1 + n 2 = n ( X n 1 ∧ X 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. ). 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. ) of two functors is equivalently the left Kan extension (def. ) of their external tensor product (def. ) along the tensor product ⊗ 𝒞 \otimes_{\mathcal{C}} : there is a natural isomorphism
X ⊗ Day Y ≃ Lan ⊗ 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 ⊗ Day Y 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. we may compute the left Kan extension as the following coend :
Lan ⊗ 𝒞 ( X ∧ ¯ Y ) ( c ) ≃ ∫ ( c 1 , c 2 ) 𝒞 ( c 1 ⊗ 𝒞 c 2 , c ) ∧ ( X ∧ ¯ Y ) ( c 1 , c 2 ) = ∫ ( c 1 , c 2 ) 𝒞 ( c 1 ⊗ c 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. ) is universally characterized by the property that there are natural isomorphisms
[ 𝒞 , Top cg * / ] ( X ⊗ Day Y , 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. .
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 c ∈ 𝒞 c\in \mathcal{C} any object , y ( c ) y(c) is the corepresented functor y ( c ) : d ↦ 𝒞 ( c , d ) y(c)\colon d \mapsto \mathcal{C}(c,d) .
Proof
Regarding associativity , observe that
( X ⊗ Day ( Y ⊗ Day Z ) ) ( c ) ≃ ∫ ( c 1 , c 2 ) 𝒞 ( c 1 ⊗ 𝒟 c 2 , c ) ∧ X ( c 1 ) ∧ ∫ ( d 1 , d 2 ) 𝒞 ( d 1 ⊗ 𝒞 d 2 , c 2 ) ( Y ( d 2 ) ∧ Z ( d 2 ) ) ≃ ∫ c 1 , d 1 , d 2 ∫ c 2 𝒞 ( c 1 ⊗ 𝒟 c 2 , c ) ∧ 𝒞 ( d 1 ⊗ 𝒞 d 2 , c 2 ) ⏟ ≃ 𝒞 ( c 1 ⊗ 𝒞 d 1 ⊗ 𝒞 d 2 , c ) ∧ X ( c 1 ) ∧ ( Y ( d 1 ) ∧ Z ( d 2 ) ) ≃ ∫ c 1 , d 1 , d 2 𝒞 ( c 1 ⊗ 𝒞 d 1 ⊗ 𝒞 d 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. ) and then twice the co-Yoneda lemma (prop. ). An analogous formula follows for X ⊗ Day ( Y ⊗ Day Z ) ) ) ( c ) X \otimes_{Day} (Y \otimes_{Day} Z)))(c) , and so associativity follows via prop. from the associativity of the smash product and of the tensor product ⊗ 𝒞 \otimes_{\mathcal{C}} .
To see that y ( 1 ) y(1) is the tensor unit for ⊗ Day \otimes_{Day} , use the Fubini theorem for coends (prop. ) and then twice the co-Yoneda lemma (prop. ) to get for any X ∈ [ 𝒞 , Top cg * / ] X \in [\mathcal{C},Top^{\ast/}_{cg}] that
X ⊗ Day y ( 1 ) = ∫ c 1 , c 2 ∈ 𝒞 𝒞 ( c 1 ⊗ 𝒟 c 2 , − ) ∧ X ( c 1 ) ∧ 𝒞 ( 1 , c 2 ) ≃ ∫ c 1 ∈ 𝒞 X ( c 1 ) ∧ ∫ c 2 ∈ 𝒞 𝒞 ( c 1 ⊗ 𝒞 c 2 , − ) ∧ 𝒞 ( 1 , c 2 ) ≃ ∫ c 1 ∈ 𝒞 X ( c 1 ) ∧ 𝒞 ( c 1 ⊗ 𝒞 1 , − ) ≃ ∫ c 1 ∈ 𝒞 X ( 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. ) with tensor product denoted ⊗ 𝒞 : 𝒞 × 𝒞 → 𝒞 \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. is a closed monoidal category (def. ). Its internal hom [ − , − ] Day [-,-]_{Day} is given by the end (def. )
[ X , Y ] Day ( c ) ≃ ∫ c 1 , c 2 Maps ( 𝒞 ( c ⊗ 𝒞 c 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. ) and the co-Yoneda lemma (def. ) and in view of definition of the enriched functor category , there is the following sequence of natural isomorphisms :
[ 𝒞 , V ] ( X , [ Y , Z ] Day ) ≃ ∫ c Maps ( X ( c ) , ∫ c 1 , c 2 Maps ( 𝒞 ( c ⊗ 𝒞 c 1 , c 2 ) , Maps ( Y ( c 1 ) , Z ( c 2 ) ) * ) * ) * ≃ ∫ c ∫ c 1 , c 2 Maps ( 𝒞 ( c ⊗ 𝒞 c 1 , c 2 ) ∧ X ( c ) ∧ Y ( c 1 ) , Z ( c 2 ) ) * ≃ ∫ c 2 Maps ( ∫ c , c 1 𝒞 ( c ⊗ 𝒞 c 1 , c 2 ) ∧ X ( c ) ∧ Y ( c 1 ) , Z ( c 2 ) ) * ≃ ∫ c 2 Maps ( ( X ⊗ Day Y ) ( c 2 ) , Z ( c 2 ) ) * ≃ [ 𝒞 , V ] ( X ⊗ Day Y , 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. , the Yoneda embedding c ↦ 𝒞 ( 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. . To see that the tensor product is respected, apply the co-Yoneda lemma (prop ) twice to get the following natural isomorphism
( y ( c 1 ) ⊗ Day y ( c 2 ) ) ( c ) ≃ ∫ d 1 , d 2 𝒞 ( d 1 ⊗ 𝒞 d 2 , c ) ∧ 𝒞 ( c 1 , d 1 ) ∧ 𝒞 ( c 2 , d 2 ) ≃ 𝒞 ( c 1 ⊗ 𝒞 c 2 , c ) = y ( c 1 ⊗ 𝒞 c 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. ). A topologically enriched lax monoidal functor between them is
a topologically enriched functor
F : 𝒞 ⟶ 𝒟 ,
F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
\,,
a morphism
ϵ : 1 𝒟 ⟶ F ( 1 𝒞 )
\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
a natural transformation
μ x , y : F ( x ) ⊗ 𝒟 F ( y ) ⟶ F ( x ⊗ 𝒞 y )
\mu_{x,y}
\;\colon\;
F(x) \otimes_{\mathcal{D}} F(y)
\longrightarrow
F(x \otimes_{\mathcal{C}} y)
for all x , y ∈ 𝒞 x,y \in \mathcal{C}
satisfying the following conditions:
(associativity ) For all objects x , y , z ∈ 𝒞 x,y,z \in \mathcal{C} the following diagram commutes
( F ( x ) ⊗ 𝒟 F ( y ) ) ⊗ 𝒟 F ( z ) ⟶ ≃ a F ( x ) , F ( y ) , F ( z ) 𝒟 F ( x ) ⊗ 𝒟 ( F ( y ) ⊗ 𝒟 F ( z ) ) μ x , y ⊗ id ↓ ↓ id ⊗ μ y , z F ( x ⊗ 𝒞 y ) ⊗ 𝒟 F ( z ) F ( x ) ⊗ 𝒟 ( F ( x ⊗ 𝒞 y ) ) μ x ⊗ 𝒞 y , z ↓ ↓ μ x , y ⊗ 𝒞 z F ( ( x ⊗ 𝒞 y ) ⊗ 𝒞 z ) ⟶ F ( a x , y , z 𝒞 ) F ( x ⊗ 𝒞 ( y ⊗ 𝒞 z ) ) ,
\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 𝒞 a^{\mathcal{C}} and a 𝒟 a^{\mathcal{D}} denote the associators of the monoidal categories;
(unitality ) For all x ∈ 𝒞 x \in \mathcal{C} the following diagrams commutes
1 𝒟 ⊗ 𝒟 F ( x ) ⟶ ϵ ⊗ id F ( 1 𝒞 ) ⊗ 𝒟 F ( x ) ℓ F ( x ) 𝒟 ↓ ↓ μ 1 𝒞 , x F ( x ) ⟵ F ( ℓ x 𝒞 ) F ( 1 ⊗ 𝒞 x )
\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 ) ⊗ 𝒟 1 𝒟 ⟶ id ⊗ ϵ F ( x ) ⊗ 𝒟 F ( 1 𝒞 ) r F ( x ) 𝒟 ↓ ↓ μ x , 1 𝒞 F ( x ) ⟵ F ( r x 𝒞 ) F ( x ⊗ 𝒞 1 ) ,
\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 ℓ 𝒞 \ell^{\mathcal{C}} , ℓ 𝒟 \ell^{\mathcal{D}} , r 𝒞 r^{\mathcal{C}} , r 𝒟 r^{\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 F F 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. ), then the lax monoidal functor F F is called a braided monoidal functor if in addition the following diagram commutes for all objects x , y ∈ 𝒞 x,y \in \mathcal{C}
F ( x ) ⊗ 𝒞 F ( y ) ⟶ τ F ( x ) , F ( y ) 𝒟 F ( y ) ⊗ 𝒟 F ( x ) μ x , y ↓ ↓ μ y , x F ( x ⊗ 𝒞 y ) ⟶ F ( τ x , y 𝒞 ) F ( y ⊗ 𝒞 x ) .
\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 , y ∈ 𝒞 x,y \in \mathcal{C} :
F 1 ( x ) ⊗ 𝒟 F 1 ( y ) ⟶ f ( x ) ⊗ 𝒟 f ( y ) F 2 ( x ) ⊗ 𝒟 F 2 ( y ) ( μ 1 ) x , y ↓ ↓ ( μ 2 ) x , y F 1 ( x ⊗ 𝒞 y ) ⟶ f ( x ⊗ 𝒞 y ) F 2 ( x ⊗ 𝒞 y )
\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 .
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. ), and let F : 𝒞 ⟶ 𝒟 F \;\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
a topologically enriched functor
G : 𝒞 ⟶ 𝒟 ;
G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
\,;
a natural transformation
ρ x , y : F ( x ) ⊗ 𝒟 G ( y ) ⟶ G ( x ⊗ 𝒞 y )
\rho_{x,y}
\;\colon\;
F(x) \otimes_{\mathcal{D}} G(y)
\longrightarrow
G(x \otimes_{\mathcal{C}} y )
such that
(action property) For all objects x , y , z ∈ 𝒞 x,y,z \in \mathcal{C} the following diagram commutes
( F ( x ) ⊗ 𝒟 F ( y ) ) ⊗ 𝒟 G ( z ) ⟶ ≃ a F ( x ) , F ( y ) , F ( z ) 𝒟 F ( x ) ⊗ 𝒟 ( F ( y ) ⊗ 𝒟 G ( z ) ) μ x , y ⊗ id ↓ ↓ id ⊗ ρ y , z F ( x ⊗ 𝒞 y ) ⊗ 𝒟 G ( z ) F ( x ) ⊗ 𝒟 ( G ( x ⊗ 𝒞 y ) ) ρ x ⊗ 𝒞 y , z ↓ ↓ ρ x , y ⊗ 𝒞 z G ( ( x ⊗ 𝒞 y ) ⊗ 𝒞 z ) ⟶ F ( a x , y , z 𝒞 ) G ( x ⊗ 𝒞 ( y ⊗ 𝒞 z ) ) ,
\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 : G 1 ( x ) ⟶ G 2 ( x ) f_x \;\colon\; G_1(x) \longrightarrow G_2(x) of the underlying functors
compatible with the action in that the following diagram commutes for all objects x , y ∈ 𝒞 x,y \in \mathcal{C} :
F ( x ) ⊗ 𝒟 G 1 ( y ) ⟶ id ⊗ 𝒟 f ( y ) F ( x ) ⊗ 𝒟 G 2 ( y ) ( ρ 1 ) x , y ↓ ↓ ( rhi 2 ) x , y G 1 ( x ⊗ 𝒞 y ) ⟶ f ( x ⊗ 𝒞 y ) G 2 ( x ⊗ 𝒞 y )
\array{
F(x) \otimes_{\mathcal{D}} G_1(y)
&\overset{id \otimes_{\mathcal{D}} f(y)}{\longrightarrow}&
F(x) \otimes_{\mathcal{D}} G_2(y)
\\
{}^{\mathllap{(\rho_1)_{x,y}}}\downarrow
&&
\downarrow^{\mathrlap{(\rhi_2)_{x,y}}}
\\
G_1(x\otimes_{\mathcal{C}} y)
&\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}&
G_2(x \otimes_{\mathcal{C}} y)
}
We write F Mod F Mod for the resulting category of modules over the monoidal functor F F .
Proposition
Let ( 𝒞 , ⊗ I ) (\mathcal{C},\otimes I) be a pointed topologically enriched category (symmetric monoidal category ) monoidal category (def. ). Regard ( Top cg * / , ∧ , S 0 ) (Top_{cg}^{\ast/}, \wedge , S^0) as a topological symmetric monoidal category as in example .
Then (commutative ) monoids in (def. ) the Day convolution monoidal category ( [ 𝒞 , Top cg * / ] , ⊗ Day , y ( 1 𝒞 ) ) ([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) of prop. are equivalent to (braided ) lax monoidal functors (def. ) 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 ) ) ≃ MonFunc ( 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}}))
&\simeq
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}
\,.
Moreover, module objects over these monoid objects are equivalent to the corresponding modules over monoidal functors .
This is stated in some form in (Day 70, example 3.2.2 ). It is highlighted again in (MMSS 00, prop. 22.1 ).
Proof
By definition , a lax monoidal functor F : 𝒞 → Top 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 0 ⟶ F ( 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 ⊗ 𝒞 c 2 )
F(c_1) \wedge F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)
for all c 1 , c 2 ∈ 𝒞 c_1,c_2 \in \mathcal{C} .
Under the Yoneda lemma (prop. ) 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 the second of these is equivalently a morphism in [ 𝒞 , Top cg * / ] [\mathcal{C}, Top^{\ast/}_{cg}] of the form
F ⊗ Day F ⟶ F .
F \otimes_{Day} F \longrightarrow F
\,.
Translating the conditions of def. satisfied by a lax monoidal functor through these identifications gives precisely the conditions of def. on a (commutative ) monoid in object F F under ⊗ Day \otimes_{Day} .
Similarly for module objects and modules over monoidal functors .
Proposition
Let f : 𝒞 ⟶ 𝒟 f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a lax monoidal functor (def. ) between pointed topologically enriched monoidal categories (def. ). Then the induced functor
f * : [ 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. ) and f f is a braided monoidal functor (def. ), 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}})
\,.
Proof
This is an immediate corollary of prop. , since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor.
Examples
For 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. )
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 exc ≔ y ( 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 : K ↦ K .
\mathbb{S}_{exc} = \iota_{fin}
\;\colon\;
K \mapsto K
\,.
We call this the standard incarnation of the sphere spectrum as a pre-excisive functor.
By prop. the smash product of pointed compactly generated topological spaces induces the structure of a closed (def. ) symmetric monoidal category (def. )
( Exc ( Top cg ) , ∧ Day , 𝕊 exc )
\left(
Exc(Top_{cg})
,\;
\wedge_{Day}
,\;
\mathbb{S}_{exc}
\right)
with
tensor unit the sphere spectrum 𝕊 exc \mathbb{S}_{exc} ;
tensor product the Day convolution product ⊗ Day \otimes_{Day} from def. ,
called the symmetric monoidal smash product of spectra for the model of pre-excisive functors;
internal hom the dual operation [ − , − ] Day [-,-]_{Day} from prop. ,
called the mapping spectrum construction for pre-excisive functors.
We now consider restricting the domain of the pre-excisive functors of def. .
Definition
Define the following pointed topologically enriched (def. ) symmetric monoidal categories (def. ):
Seq Seq is the category whose objects are the natural numbers and which has only identity morphisms and zero morphisms on these objects, hence the hom-spaces are
Seq ( n 1 , n 2 ) = { S 0 for n 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.
Sym Sym is the standard skeleton of the core of FinSet with zero morphisms adjoined: its objects are the finite sets { 1 , ⋯ , n } \{1, \cdots,n\} for n ∈ ℕ n \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 ) + for n 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 .
Orth Orth 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 ) + for dim ( 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.
There is a sequence of canonical faithful pointed topological subcategory inclusions
Seq ↪ seq Sym ↪ sym Orth ↪ orth Top cg , fin * / n ↦ { 1 , ⋯ , n } ↦ ℝ 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. ).
Here S V S^V denotes the one-point compactification of V V . 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 V → S V S^V \to S^V that are induced under forming one-point compactification from linear isometries of V V (“representation spheres ”).
Consider the sequence of restrictions of topological diagram categories, according to prop. 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 Orth ≔ orth * 𝕊 exc , 𝕊 Sym ≔ sym * 𝕊 orth , 𝕊 Seq ≔ seq * 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. ) along these inclusions.
Therefore we may consider module objects over the restrictions of the sphere spectrum from def. .
Proof
Write 𝕊 dia \mathbb{S}_{dia} for any of the three monoids. By prop. , left modules with respect to Day convolution are equivalently modules over monoidal functors over the monoidal functor corresponding to 𝕊 dia \mathbb{S}_{dia} . This means that for 𝕊 Sym \mathbb{S}_{Sym} and 𝕊 Seq \mathbb{S}_{Seq} they are functors X : Sym ⟶ sSet * / X \colon Sym \longrightarrow sSet^{\ast/} or X : Seq ⟶ sSet * / X \colon Seq \longrightarrow sSet^{\ast/} , respectively equipped with natural transformations
S 1 ∧ X p ⟶ X p + q
S^1 \wedge X_p \longrightarrow X_{p+q}
satisfying the evident categorified action property. In the present case this action property says that these morphisms are determined by
S 1 ∧ X p ⟶ X p + 1
S^1 \wedge X_p \longrightarrow X_{p+1}
under the isomorphisms S p ≃ S 1 ∧ S p − 1 S^p \simeq S^1 \wedge S^{p-1} . Naturality of all these morphisms as functors on Sym Sym is the equivariance under the symmetric group actions in the definition of symmetric spectra .
Similarly, modules over 𝕊 Orth \mathbb{S}_{Orth} are equivalently functors
S W ∧ X V ⟶ X V ⊕ W
S^W \wedge X_V \longrightarrow X_{V \oplus W}
etc. and their functoriality embodies the orthogonal group -equivariance in the definition of orthogonal spectra .
For orthogonal spectra
Consider the non-full inclusion of topologically enriched categories
Orth ↪ Top cg , fin * /
Orth \hookrightarrow Top^{\ast/}_{cg,fin}
on the standard n-spheres S n ≔ ( S 1 ) ∧ n S^n \coloneqq (S^1)^{\wedge^n} , with hom-spaces given by the orthogonal groups with basepoint adjoint, acting on these spheres as their canonical representation spheres
Orth ( S n 1 , S n 2 ) ≔ { O ( n 1 ) + if n 1 = n 2 * otherwise .
Orth(S^{n_1}, S^{n_2})
\coloneqq
\left\{
\array{
O(n_1)_+ & if \; n_1= n_2
\\
\ast & otherwise
}
\right.
\,.
Regard Orth as a monoidal category with monoidal structure induced form ( Top cg * / , ∧ , S 0 ) (Top^{\ast/}_{cg}, \wedge, S^0) (via example ) under the restriction. This makes the inclusion a braided monoidal functor .
Restricting the standard pre-excisive model y ( S 0 ) y(S^0) of the sphere spectrum yields 𝕊 orth \mathbb{S}_{orth} . Since restriction is a monoidal functor, and since y ( S 0 ) y(S^0) is the tensor unit and hence canonically a monoid, prop. says that 𝕊 orth \mathbb{S}_{orth} is still a commutative monoid with respect to Day convolution:
CMon ( [ Top cg , fin * / , Top cg * / ] , ⊗ Day , y ( S 0 ) ) ⟶ CMon ( [ Orth , Top cg * / ] , ⊗ Day , y ( S 0 ) )
CMon([Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}], \otimes_{Day}, y(S^0))
\longrightarrow
CMon([Orth, Top^{\ast/}_{cg}], \otimes_{Day}, y(S^0))
( 𝕊 exc , μ = id , e = id ) ↦ ( 𝕊 orth , μ , e ) .
(\mathbb{S}_{exc},\mu = id,e = id) \mapsto (\mathbb{S}_{orth}, \mu, e)
\,.
The category of orthogonal spectra is the category of 𝕊 orth \mathbb{S}_{orth} -modules (def. ):
OrthSpec ( Top cg ) = 𝕊 orth Mod ( [ Top cg , fin * / , Top cg * / ] ) ,
\begin{aligned}
OrthSpec(Top_{cg})
&=
\mathbb{S}_{orth}Mod( [Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}] )
\end{aligned}
\,,
Since 𝕊 orth \mathbb{S}_{orth} is a commutative monoid, prop. says that there is a symmetric monoidal category structure ⊗ 𝕊 orth \otimes_{\mathbb{S}_{orth}} on OrthSpec ( Top cg ) OrthSpec(Top_{cg}) . This is the symmetric monoidal smash product of spectra for orthogonal spectra.
An orthogonal ring spectrum E E is a monoid with respect to ⊗ 𝕊 orth \otimes_{\mathbb{S}_{orth}} , hence an 𝕊 orth \mathbb{S}_{orth} -algebra (def. ). By prop. , such E E is equivalently a monoid with respect to ⊗ Day \otimes_{Day} and equipped with a monoid homomorphism 𝕊 orth ⟶ E \mathbb{S}_{orth} \longrightarrow E . Finally, by prop. this is equivalently a functor with smash products
E : Orth ⟶ Top cg * /
E \;\colon\; Orth \longrightarrow Top^{\ast/}_{cg}
equipped with a natural transformation of functors with smash product
𝕊 orth ⟶ E .
\mathbb{S}_{orth} \longrightarrow E
\,.
In the terminology of MMSS 00, def. 22.5 this is an “Orth Orth -FSP over 𝕊 Orth \mathbb{S}_{Orth} ”.
Symmetric spectra
Restrict further along the non-full inclusion
Sym ↪ Orth ↪ Top cg , fin * / ,
Sym \hookrightarrow Orth \hookrightarrow Top^{\ast/}_{cg,fin}
\,,
where Sym Sym has the same objects, but the hom-spaces are now just the symmetric groups (with basepoint adjoint)
Σ n ↪ O ( n ) .
\Sigma_n \hookrightarrow O(n)
\,.
Then proceed as for orthogonal spectra.
For sequential spectra (non-example)
Restrict further along
Seq ↪ Sym ↪ Orth ↪ Top cg * / ,
Seq \hookrightarrow Sym \hookrightarrow Orth \hookrightarrow Top^{\ast/}_{cg}
\,,
where Seq Seq still has the same objects, the n n -spheres, but no non-trivial morphisms (just the identity morphisms and the zero morphisms).
Now the inclusion Seq ⟶ Top cg * / Seq \longrightarrow Top^{\ast/}_{cg} is no longer a braided monoidal functor , for the braiding on Seq Seq is trivial, while on Top cg * / Top^{\ast/}_{cg} it is not. Accordingly the assumption of the second clause in prop. is vialoted.
Indeed, restricting 𝕊 \mathbb{S} along this inclusion yields the stndard sequential sphere spectrum 𝕊 seq \mathbb{S}_{seq} which is still a monoid with respect to Day convolution, but not a commutative monoid anymore (see at smash product of spectra – graded commutativity ) and hence the assumption of prop. is violated.
The 𝕊 seq \mathbb{S}_{seq} -module objects (def. ) are equivalently the sequential spectra .
But since 𝕊 seq \mathbb{S}_{seq} is not a commutative monoid, the assumption of prop. there is no induced tensor product on 𝕊 seq Mod \mathbb{S}_{seq}Mod and hence the story ends here.
Examples
Reference
The concept was introduced (before symmetric smash products of spectra had been found) in
Marcel Bökstedt , Topological Hochschild homology . Preprint, Bielefeld, 1986
Restricted to spheres as “FSPs defined on spheres” they were considered in
and identified there as the monoids in symmetric spectra as previously introduced by Jeff Smith .
In the model structure for excisive functors the concept was recovered in
Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf )
and in the model of connective spectra by Gamma-spaces in
Lydakis, Smash products and Γ \Gamma -spaces , Math. Proc. Cam. Phil. Soc. 126 (1999), 311-328 (pdf )
A systematic account is in
Based on discussion in