Английская Википедия:Day convolution

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Шаблон:Short description 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 <math>[\mathbf{C},V]</math> over some monoidal category <math>V</math>.

Definition

Let <math>(\mathbf{C}, \otimes_c)</math> be a monoidal category enriched over a symmetric monoidal closed category <math>(V, \otimes)</math>. Given two functors <math>F,G \colon \mathbf{C} \to V</math>, we define their Day convolution as the following coend.

<math>F \otimes_d G = \int^{x,y \in \mathbf{C}} \mathbf{C}(x \otimes_c y , -) \otimes Fx \otimes Gy</math>

If <math>\otimes_c</math> is symmetric, then <math>\otimes_d</math> is also symmetric. We can show this defines an associative monoidal product.

<math>\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>

External links

• Шаблон:Nlab