Day convolution

First version

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

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

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

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

Enriched version

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

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

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

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