Day convolution

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 acts as a tensor product for a monoidal category structure on the category of functors $[\mathbf {C} ,V]$ over some monoidal category $V$.

Definition

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

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

If the category $V$ is a symmetric monoidal closed category, we can show this defines an associative monoidal product.

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