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{{cite journal |last1=Day |first1=Brian |title=On closed categories of functors |journal=Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics |date=1970 |volume=139 |pages=1–38}} in the general context of enriched functor categories.

Day convolution gives a symmetric monoidal structure on $\mathrm{Hom}(\mathbf{C},\mathbf{D})$ 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 $F,G \colon \mathbf{C} \to \mathbf{D}$ 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 $F,G \colon \mathbf{C} \to V$ , we define their Day convolution as the following coend.{{cite book |last1=Loregian |first1=Fosco |title=(Co)end Calculus |chapter= |arxiv=1501.02503 |page=51|year=2021 |doi=10.1017/9781108778657 |isbn=9781108778657 |s2cid=237839003 }}

: $F \otimes_d G = \int^{x,y \in \mathbf{C}} \mathbf{C}(x \otimes_c y , -) \otimes Fx \otimes Gy$

If $\otimes_c$ is symmetric, then $\otimes_d$ is also symmetric. We can show this defines an associative monoidal product:

: $\begin{aligned} & (F \otimes_d G) \otimes_d H \\[5pt]
\cong {} & \int^{c_1,c_2} (F \otimes_d G)c_1 \otimes Hc_2 \otimes \mathbf{C}(c_1 \otimes_c c_2, -) \\[5pt]
\cong {} & \int^{c_1,c_2} \left( \int^{c_3,c_4} Fc_3 \otimes Gc_4 \otimes \mathbf{C}(c_3 \otimes_c c_4 , c_1) \right) \otimes Hc_2 \otimes \mathbf{C}(c_1 \otimes_c c_2, -) \\[5pt]
\cong {} & \int^{c_1,c_2,c_3,c_4} Fc_3 \otimes Gc_4 \otimes Hc_2 \otimes \mathbf{C}(c_3 \otimes_c c_4 , c_1) \otimes \mathbf{C}(c_1 \otimes_c c_2, -) \\[5pt]
\cong {} & \int^{c_1,c_2,c_3,c_4} Fc_3 \otimes Gc_4 \otimes Hc_2 \otimes \mathbf{C}(c_3 \otimes_c c_4 \otimes_c c_2, -) \\[5pt]
\cong {} & \int^{c_1,c_2,c_3,c_4} Fc_3 \otimes Gc_4 \otimes Hc_2 \otimes \mathbf{C}(c_2 \otimes_c c_4 , c_1) \otimes \mathbf{C}(c_3 \otimes_c c_1, -) \\[5pt]
\cong {} & \int^{c_1,c_3} Fc_3 \otimes (G \otimes_d H)c_1 \otimes \mathbf{C}(c_3 \otimes_c c_1, -) \\[5pt]
\cong {} & F \otimes_d (G \otimes_d H)\end{aligned}$

References

External links

{{nlab|id=Day+convolution|title=Day convolution}}

Category:Category theory