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
over some monoidal category
.
Definition
Let
be a monoidal category enriched over a symmetric monoidal closed category
. Given two functors
, we define their Day convolution as the following coend.[2]
![{\displaystyle F\otimes _{d}G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes _{c}y,-)\otimes Fx\otimes Gy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88654f19e681d60ebb96b864e671698a5ce1e32a)
If
is symmetric, then
is also symmetric. We can show this defines an associative monoidal product.
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5539d6a3ddbe543a5a1691cc60953d17a5765282)
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