nLab Day convolution

Contents

Context

Monoidal categories

monoidal categories

category theory

Contents

Idea

The category of functors on a monoidal category canonically inherits itself a monoidal category structure via a categorified convolution product. This holds generally in the context of enriched category theory. This was first observed by (Day 70) and accordingly these monoidal structures are called Day convolution products.

In more detail, just as there is convolution of functions $f : G \to \mathbb{C}$ whenever $G$ carries the structure of a group, or more generally just the structure of a monoid, so there is convolution of functors $f \colon \mathcal{G} \to Set$ whenever the category $\mathcal{G}$ carries the structure of a monoidal category.

This may be generalized by replacing Set with a more general cocomplete symmetric monoidal category $V$. The technical condition is that the tensor product $u \otimes v$ preserves colimits in its two arguments separately; hence that the functors $u \otimes -$ and $- \otimes v$ preserve colimits. This occurs notably when $V$ is symmetric closed monoidal (so that these functors are left adjoints).

Definition

For monoidal categories

Let $V$ be a closed symmetric monoidal category with all small limits and colimits.

For $\mathcal{C}$ a $V$-enriched category, write $[\mathcal{C},V]$ for the $V$-enriched functor category to $V$, etc.

We discuss two equivalent ways of defining Day convolution

In terms of coends

Definition

Let $(\mathcal{C}, \otimes, 1)$ be a small $V$-enriched monoidal category.

Then the Day convolution tensor product on $[\mathcal{C},V]$

$\otimes_{Day} \;\colon\; [\mathcal{C},V] \times [\mathcal{C},V] \longrightarrow [\mathcal{C},V]$

is given by the following coend

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

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

Definition

Let $\mathcal{C}$ be a small $V$-monoidal category. Its external tensor product is

$\overline{\otimes} \;\colon\; [\mathcal{C}, V] \times [\mathcal{C}, V] \longrightarrow [\mathcal{C}\times \mathcal{C}, V]$

given by

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

i.e.

$(X \overline\otimes Y)(c_1,c_2) = X(c_1)\otimes_V Y(c_2) \,.$
Proposition

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

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

Hence the adjunction unit is a natural transformation of the form

$\array{ \mathcal{C} \times \mathcal{C} && \overset{X \overline{\otimes} Y}{\longrightarrow} && V \\ & {}^{\mathllap{\otimes}}\searrow &\Downarrow& \nearrow_{\mathrlap{X \otimes_{Day} Y}} \\ && \mathcal{C} } \,.$

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

Proof

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

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

Proposition implies the following fact, which is the key for the identification of “functors with smash product”.

Corollary

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

$[\mathcal{C}, V](X \otimes_{Day} Y, Z) \simeq [\mathcal{C}\times \mathcal{C}, V]( X \overline{\otimes} Y,\; Z \circ \otimes_V ) \,,$

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

In terms of profunctors

The Day convolution can also be expressed in terms of profunctors. The tensor product $\otimes :\mathcal{C}\otimes \mathcal{C}\to \mathcal{C}$ induces a representable profunctor $\mathcal{C}(\otimes,1): \mathcal{C} ⇸ \mathcal{C}\otimes \mathcal{C}$. The above definition can be interpreted to say that if $X,Y\in [\mathcal{C},V]$ are regarded as profunctors $\mathcal{C} ⇸ I$, where $I$ is the unit $V$-category, then $X\otimes_{Day} Y$ is the composite of profunctors

$\mathcal{C} \xrightarrow{\mathcal{C}(\otimes,1)} \mathcal{C}\otimes \mathcal{C} \xrightarrow{X\otimes Y} I\otimes I \cong I.$

A more “global” way to say the same thing is to consider the “evaluation” functor $[\mathcal{C},V] \otimes \mathcal{C} \to V$ to be a profunctor $E:\mathcal{C}⇸[\mathcal{C},V]^{op}$. Then the profunctor composite

$\mathcal{C} \xrightarrow{\mathcal{C}(\otimes,1)} \mathcal{C}\otimes \mathcal{C} \xrightarrow{E\otimes E} [\mathcal{C},V]^{op} \otimes [\mathcal{C},V]^{op}$

is a functor $\mathcal{C}\otimes [\mathcal{C},V] \otimes [\mathcal{C},V] \to V$, which by exponential transpose gives a functor $[\mathcal{C},V] \otimes [\mathcal{C},V] \to [\mathcal{C},V]$; this is the Day convolution product.

For promonoidal categories

The above description in terms of profunctors makes it clear that the construction only depends on the representable profunctor induced by $\otimes : \mathcal{C}\otimes \mathcal{C}\to \mathcal{C}$, i.e. on the underlying promonoidal category of $\mathcal{C}$. In the original article (Day 70), a stronger form of the convolution is discussed, in which $\mathcal{C}$ is assumed only to be a promonoidal category.

Let $V$ be a Benabou cosmos, and $\mathcal{C}$ a small $V$-enriched category.

Proposition

There is an equivalence of categories between the category of pro-monoidal structures on $\mathcal{C}$ with strong pro-monoidal functors between them and the category of biclosed monoidal structures on $[\mathcal{C}^{op},V]$ with strong monoidal functors between them.

This is claimed without proof in (Day 70).

Properties

Closed monoidal structure

Proposition

For $(\mathcal{C}, \otimes_{\mathcal{C}}, I)$ a small monoidal $V$-enriched category, the Day convolution product $\otimes_{Day}$ of def. makes

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

a monoidal category with tensor unit $y(I)$ co-represented by the tensor unit $I$ of $\mathcal{C}$.

This may be deduced fairly abstractly from the above description of Day convolution in terms of profunctors, using the associativity of the promonoidal structure on $\mathcal{C}$.

Proposition

For $(\mathcal{C}, \otimes_{\mathcal{C}}, I)$ a small monoidal $V$-enriched category, the monoidal category with Day convolution $([\mathcal{C},V], \otimes_{Day}, y(I))$ from def. is a closed monoidal category. Its internal hom $[-,-]_{Day}$ is given by the end

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

In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in $[\mathcal{C},V]$ exists at all, (with $[-,X]_{Day}$ right adjoint to $(-) \otimes_{Day} X$) then by the enriched Yoneda lemma and by the end-expression for the hom-objects in the enriched functor category $[\mathcal{C},V]$ it has to be given by

\begin{aligned} [X,Y]_{Day}(c) & \simeq [\mathcal{C},V](y(c), [X,Y]) \\ & \simeq [\mathcal{C},V](y(c) \otimes_{Day} X, Y) \\ & \simeq \underset{c_1}{\int} V((y(c) \otimes_{Day} X)(c_1), Y(c_1)) \\ &\simeq \underset{c_1}{\int} V\left( \overset{d_2}{\int} \overset{d_1}{\int} \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_1) \otimes_V \mathcal{C}(c,d_1) \otimes_V X(d_2) , Y(c_1) \right) \\ & \simeq \underset{c_1}{\int} \underset{d_2}{\int} V\left( \underset{\simeq \mathcal{C}(c \otimes_{\mathcal{C}} d_2, c_1 )}{ \underbrace{ \overset{d_1}{\int} \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c_1) \otimes_V \mathcal{C}(c,d_1) } } \otimes_V X(d_2) , Y(c_1) \right) \\ & \simeq \underset{c_1,d_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} d_2,c_1), V(X(d_2), Y(c_1)) \right) \\ & = \underset{c_1,c_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2), V(X(c_1), Y(c_2)) \right) \end{aligned} \,.

This exists, by the assumption that $\mathcal{C}$ is small and that $V$ has all small limits. Now to check that this really gives a right adjoint:

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

The Yoneda embedding constitutes a strong monoidal functor $(\mathcal{C},\otimes_{\mathcal{C}}, I) \hookrightarrow ([\mathcal{C},V], \otimes_{Day}, y(I))$.

Proof

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

\begin{aligned} (y(c_1) \otimes_{Day} y(c_2))(c) & \simeq \overset{d_1, d_2}{\int} \mathcal{C}(d_1 \otimes_{\mathcal{C}} d_2, c ) \otimes_V \mathcal{C}(c_1,d_1) \otimes_V \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} \,.

Monoids with respect to Day convolution

Given any monoidal category then one may consider monoid objects and module objects inside it.

Proposition

For $(\mathcal{C}, \otimes)$ a small (symmetric) monoidal $V$-enriched category, then (commutative) monoid objects in the Day convolution monoidal category $([\mathcal{C},V], \otimes_{Day}, y(I))$ of prop. are equivalent to (symmetric) lax monoidal functors $\mathcal{C} \to V$:

$Mon([\mathcal{C},V], \otimes_{Day}, y(I)) \simeq MonFunc(\mathcal{C},V)$
$CMon([\mathcal{C},V], \otimes_{Day}, y(I)) \simeq SymMonFunc(\mathcal{C},V) \,.$

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 was highlighted again in (MMSS 00, prop. 22.1). See also MO discussion here.

Proof

A lax monoidal functor $F \colon \mathcal{C} \to V$ is given by natural transformations

$I_V \longrightarrow F(I_{\mathcal{C}})$
$F(c_1) \otimes_V F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)$

satisfying compatibility conditions. Under the natural isomorphism of corollary these are identified with natural transformations

$y(I) \to F$
$F \otimes_{Day} F \longrightarrow F$

satisfying analogous conditions. This is just the structure of a monoid object on $F$ under $\otimes_{Day}$.

Similarly for module objects and modules over monoidal functors.

Example

In the case that $V$ is pointed topological spaces or pointed simplicial sets equipped with the smash product of pointed objects and that $\mathcal{C}$ is a diagram category for spectra, then monoids in prop. are known as ring spectra and the lax monoidal functors in prop. are known as the incarnation of ring spectra as “functors with smash product”.

Modules with respect to Day convolution

Definition

For $(\mathcal{C},\otimes, I)$ a small monoidal $V$-enriched category, and for $R \in Mon([\mathcal{C}, V],\otimes_{Day})$ a monoid object with respect to Day convolution over $\mathcal{C}$, write

$R Free_{\mathcal{C}}Mod \hookrightarrow R Mod$

for the full subcategory of the category of modules over $R$ on those that are free modules and moreover free on objects in $\mathcal{C}$ (under the Yoneda embedding). Hence the objects of $R Free_{\mathcal{C}}Mod$ are those of $\mathcal{C}$ and the hom-objects are

$R Free_{\mathcal{C}}Mod(c_1,c_2) \;\coloneqq\; R Mod( y(c_1) \otimes_{Day} R , y(c_2) \otimes_{Day} R) \,.$
Proposition

For $(\mathcal{C},\otimes, I)$ a small $V$-enriched category, and for $R \in Mon([\mathcal{C}, V],\otimes_{Day})$ a monoid object with respect to Day convolution over $\mathcal{C}$, then there is an equivalence of categories

$Mod_R \simeq [R Free_{\mathcal{C}}Mod^{op}, V]$

between the category of right modules over $R$ and the enriched functor category out of the opposite category of that of free $R$-modules from def. .

Proof idea

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

$N(c_1) \otimes R(c_2) \longrightarrow N(c_1 \otimes c_2) \,.$

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

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

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

$\mathcal{C}(c_1 \otimes c_4, c_2) \otimes R(c_4) \otimes N(c_1) \longrightarrow \mathcal{C}(c_1 \otimes c_4, c_2) \otimes N(c_1 \otimes c_4) \longrightarrow N(c_2)$

and subject to certain identifications.

Now observe that the hom-objects of $R Free_{\mathcal{C}}Mod$ (def. ) have just this structure:

\begin{aligned} R Free_{\mathcal{C}}Mod(c_2,c_1) & = R Mod( y(c_2) \otimes_{Day} R , y(c_1) \otimes_{Day} R) \\ & \simeq [\mathcal{C},V](y(c_2), y(c_1) \otimes_{Day} R) \\ & \simeq (y(c_1) \otimes_{Day} R)(c_2) \\ & \simeq \overset{c_3,c_4}{\int} \mathcal{C}(c_3 \otimes c_4,c_2) \otimes_V \mathcal{C}(c_1, c_3) \otimes_V R(c_4) \\ & \simeq \overset{c_4}{\int} \mathcal{C}(c_1 \otimes c_4,c_2) \otimes_V R(c_4) \end{aligned} \,.

We claim that under this identification, composition in $R Free_{\mathcal{C}}Mod$ is given by

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

where

1. the first morphism is, in the integrand, the tensor product of

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

$\mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \simeq \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes_V 1_V \overset{}{\longrightarrow} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes \mathcal{C}(c_5,c_5) \longrightarrow \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5, c_2 \otimes_{\mathcal{C}} c_5)$
2. the monoidal functor incarnation $R(c_4) \otimes_V R(c_5)\longrightarrow R(c_4 \otimes_{\mathcal{C}} c_5 )$ of the monoid structure on $R$;

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

3. the last morphism is the morphism induced on coends by regarding extranaturality in $c_4$ and $c_5$ separately as a special case of extranaturality in $c_6 \coloneqq c_4 \otimes c_5$ (and then renaming).

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

Examples

Example

Let $C$ be a discrete category over a set, which is hence a monoid (for instance a group) with product $\cdot$.

Then the Day convolution product is

$F \star G : e \mapsto \oplus_{c \cdot d = e} F(c) \times G(d) \,.$

Notice that if we regard the presheaves $F$ and $G$ here, assuming they take values in finite sets, as categorifications of $\mathbb{N}$-valued functions $|F|, |G| : C \to \mathbb{N}$, where $|\cdot| : Set \to \mathbb{N}$ is the cardinality operation on finite sets, then this reproduces precisely the ordinary convolution product of these $\mathbb{N}$-valued functions

\begin{aligned} |F \star G| : e &\mapsto \sum_{c,d \in C} |F(c)| \times |G(d)| \times \delta(e, c \otimes d) & = \sum_{c \cdot d = e} |F(c)| \cdot |F(d)| \end{aligned}

This uses in particular that for every object $c \in C$ the functor

$Hom_C(c,-) = \delta_c$

is in this sense the Kronecker delta-function on the set $C$ supported at $c \in C$. Precisely because by assumption $C$ has only identity morphisms.

$Hom_C(c,d) = \left\{ \array{ * & if c = d \\ \emptyset & if c \neq d } \right.$

Further examples:

Example

The symmetric smash product of spectra on, in particular, symmetric spectra and orthogonal spectra is the Day convolution product for Top-enriched functors on monoidal categories of symmetric groups of orthogonal groups, respectively (MMSS 00, theorem 1.7 and section 21.).

Similarly the symmetric smash product of spectra on the model structure for excisive functors is Day convolution for sSet-enriched functors on the plain smash product of finite pointed simplicial sets (Lydakis 98).

The concept originates in

• Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 (pdf)

The universal property of the Day convolution, in the sense of free monoidal cocompletion, is discussed in

• Geun Bin Im and G. M. Kelly, A universal property of the convolution monoidal structure, J. Pure Appl. Algebra 43 (1986), no. 1, 75-88.

General discussion includes

The application of Day convolution to the construction of symmetric smash products of spectra for highly structured spectra is due to

and for excisive functors due to

• Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)