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.

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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 (2015). "This is the (co)end, my only (co)friend". p. 51. arXiv:1501.02503 [math.CT].
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