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 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 $(\mathbf{𝐂},{\otimes }_c)$ 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