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 $\mathrm{H}\mathrm{o}\mathrm{m}(𝐂,𝐃)$ for two symmetric monoidal categories $𝐂,𝐃$.

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

Definition

First version

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

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

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

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

Enriched version

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

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

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

$\begin{array}{cc}& (F{\otimes }_{d}G){\otimes }_{d}H\\ \cong & {\int }^{c_1,c_2}(F{\otimes }_{d}G)c_1\otimes H{c}_2\otimes 𝐂(c_1{\otimes }_c{c}_2,-)\\ \cong & {\int }^{c_1,c_2}({\int }^{c_3,c_4}F{c}_3\otimes G{c}_4\otimes 𝐂(c_3{\otimes }_c{c}_4,c_1))\otimes H{c}_2\otimes 𝐂(c_1{\otimes }_c{c}_2,-)\\ \cong & {\int }^{c_1,c_2,c_3,c_4}F{c}_3\otimes G{c}_4\otimes H{c}_2\otimes 𝐂(c_3{\otimes }_c{c}_4,c_1)\otimes 𝐂(c_1{\otimes }_c{c}_2,-)\\ \cong & {\int }^{c_1,c_2,c_3,c_4}F{c}_3\otimes G{c}_4\otimes H{c}_2\otimes 𝐂(c_3{\otimes }_c{c}_4{\otimes }_c{c}_2,-)\\ \cong & {\int }^{c_1,c_2,c_3,c_4}F{c}_3\otimes G{c}_4\otimes H{c}_2\otimes 𝐂(c_2{\otimes }_c{c}_4,c_1)\otimes 𝐂(c_3{\otimes }_c{c}_1,-)\\ \cong & {\int }^{c_1,c_3}F{c}_3\otimes (G{\otimes }_{d}H)c_1\otimes 𝐂(c_3{\otimes }_c{c}_1,-)\\ \cong & F{\otimes }_d(G{\otimes }_{d}H)\end{array}$

External links

• Day convolution at the nLab