Day convolution

Short description: 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 $\mathrm{H}\mathrm{o}\mathrm{m}(\mathbf{𝐂},\mathbf{𝐃})$ 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 $F,G:\mathbf{𝐂}\to \mathbf{𝐃}$ 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 $F,G:\mathbf{𝐂}\to V$, 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:

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

External links

• Day convolution in nLab