Day convolution

In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 in the general context of enriched functor categories.

Day convolution gives a symmetric monoidal structure on $\mathrm {Hom} (\mathbf {C} ,\mathbf {D} )$ for two symmetric monoidal categories $\mathbf {C} ,\mathbf {D}$.

Another related version is that Day convolution acts as a tensor product for a monoidal category structure on the category of functors $[\mathbf {C} ,V]$ over some monoidal category $V$.

Definition

First version

Given $F,G\colon \mathbf {C} \to \mathbf {D}$ for two symmetric monoidal $\mathbf {C} ,\mathbf {D}$, we define their Day convolution as follows.

It is the left kan extension along $\mathbf {C} \times \mathbf {C} \to ^{\otimes }\mathbf {C}$ of the composition $\mathbf {C} \times \mathbf {C} \to ^{F,G}\mathbf {D} \times \mathbf {D} \to ^{\otimes }\mathbf {D}$

Thus evaluated on an object $O\in \mathbf {C}$, intuitively we get a colimit in $\mathbf {D}$ of $F(x)\otimes G(y)$ along approximations of $O\in \mathbf {C}$ as a pure tensor $x\otimes y$

Left kan extensions are computed via coends, which leads to the version below.

Enriched version

Let $(\mathbf {C} ,\otimes _{c})$ be a monoidal category enriched over a symmetric monoidal closed category $(V,\otimes )$. Given two functors $F,G\colon \mathbf {C} \to V$, we define their Day convolution as the following coend.

$F\otimes _{d}G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes _{c}y,-)\otimes Fx\otimes Gy$

If $\otimes _{c}$ is symmetric, then $\otimes _{d}$ is also symmetric. We can show this defines an associative monoidal product:

${\begin{aligned}&(F\otimes _{d}G)\otimes _{d}H\\[5pt]\cong {}&\int ^{c_{1},c_{2}}(F\otimes _{d}G)c_{1}\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2}}\left(\int ^{c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\right)\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{2}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&\int ^{c_{1},c_{3}}Fc_{3}\otimes (G\otimes _{d}H)c_{1}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&F\otimes _{d}(G\otimes _{d}H)\end{aligned}}$

External links

• Day convolution at the nLab