# nLab Day convolution

### Context

#### Monoidal categories

monoidal categories

category theory

# Contents

## Idea

Just as we can convolve functions $f : M \to \mathbb{C}$ where $M$ is a group, or more generally a monoid, we can convolve functors $f: M \to Set$ where $M$ is a monoidal category. So, for any monoidal category $M$, the functor category $Set^M$ becomes a monoidal category in its own right. The tensor product in $Set^M$ is called Day convolution, named after Brian Day.

We can generalize this idea by replacing Set with a more general cocomplete symmetric monoidal category $V$. The technical condition is that the tensor product $u \otimes v$ must preserve colimits in the separate arguments $u$ and $v$; that is, that the functors $u \otimes -$ and $- \otimes v$ must preserve colimits. This occurs when for instance $V$ is symmetric monoidal closed (so that these functors are left adjoints).

## Restricted Definition

For $(C, \otimes)$ a monoidal category and $F, G : C^{op} \to Set$ two presheaves on $C$, their Day convolution product $F \star G$ is the presheaf given by the coend

$F \star G := \int^{c,d \in C} F(c) \times G(d) \times Hom_C(-, c \otimes d) \,.$

## Properties

Let $j : C \to PSh(C)$ be the Yoneda embedding.

###### Lemma

With $I \in C$ the tensor unit of $C$, the presheaf $j(I)$ is a unit for the Day convolution product.

###### Proof

Using the co-Yoneda lemma on the two coends we have

\begin{aligned} F \star j(I) & \simeq \int^{c,d \in C} F(c) \times Hom_C(d,I) \times Hom_C(-, c\otimes d) \\ & \simeq \int^{c \in C} F(c) \times Hom_C(-, c \otimes I) \\ & \simeq \int^{c \in C} F(c) \times Hom_C(-, c) \\ & \simeq F(-) \end{aligned} \,.
###### Proposition

For $C$ a small monoidal category, regard the category of presheaves $(PSh(C), \star, j(I))$ as a monoidal category with tensor product the Day convolution product and unit the unit of $C$ under the Yoneda embedding $j : C \hookrightarrow PSh(C)$.

Then

1. $(PSh(C), \star, j(I))$ is a closed monoidal category;

2. the Yoneda embedding constitutes a strong monoidal functor $(C,\otimes, I) \hookrightarrow (PSh(C), \star, j(I))$.

###### Proof

In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in $PSh(C)$ exists at all, (with $[F,-]$ right adjoint to $(-) \star F$) then by the Yoneda lemma it has to be given by

\begin{aligned} [F,G](c) & \simeq Hom_C(j(c), [F,G]) \\ &\simeq Hom_C(j(c)\star F, G) \end{aligned} \,.

## Promonoidal Day Convolution

In Day’s original paper, a stronger form of the Day convolution is discussed, in which $A$ is assumed only to be a promonoidal category.

Let $V$ be a Benabou cosmos, and $A$ a small $V$-enriched category.

###### Proposition

There is an equivalence of categories between the category of pro-monoidal structures on $A$ with strong pro-monoidal functors between them and the category of biclosed monoidal structures on $V^{A^{op}}$ with strong monoidal functors between them.

## Examples

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

Then the above 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.

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

## References

For (∞,1)-categories:

Revised on June 10, 2014 05:22:35 by Adeel Khan (132.252.63.24)