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 [1] 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 [math]\displaystyle{ [\mathbf{C},V] }[/math] over some monoidal category [math]\displaystyle{ V }[/math].
Definition
Let [math]\displaystyle{ (\mathbf{C}, \otimes_c) }[/math] be a monoidal category enriched over a symmetric monoidal closed category [math]\displaystyle{ (V, \otimes) }[/math]. Given two functors [math]\displaystyle{ F,G \colon \mathbf{C} \to V }[/math], we define their Day convolution as the following coend.[2]
- [math]\displaystyle{ F \otimes_d G = \int^{x,y \in \mathbf{C}} \mathbf{C}(x \otimes_c y , -) \otimes Fx \otimes Gy }[/math]
If [math]\displaystyle{ \otimes_c }[/math] is symmetric, then [math]\displaystyle{ \otimes_d }[/math] is also symmetric. We can show this defines an associative monoidal product.
- [math]\displaystyle{ \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} }[/math]
References
- ↑ Day, Brian (1970). "On closed categories of functors". Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics 139: 1–38.
- ↑ Loregian, Fosco (2021). (Co)end Calculus. p. 51. doi:10.1017/9781108778657. ISBN 9781108778657.
External links
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