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 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 $$F,G \colon \mathbf{C} \to V$$, 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{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}$$