Day convolution

Last updated January 29, 2025

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.

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.^[2]

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}}

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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. arXiv: 1501.02503 . doi:10.1017/9781108778657. ISBN 9781108778657. S2CID 237839003.

External links

Day convolution at the nLab

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[1] Day, Brian (1970). "On closed categories of functors". Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics. 139: 1–38.

[2] Loregian, Fosco (2021). (Co)end Calculus. p. 51. arXiv: 1501.02503 . doi:10.1017/9781108778657. ISBN 9781108778657. S2CID 237839003.

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