Jump to content

Day convolution

From Wikipedia, the free encyclopedia

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[edit]

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]

If is symmetric, then is also symmetric. We can show this defines an associative monoidal product.

References[edit]

  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.

External links[edit]