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 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]

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

References

  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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