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Day convolution
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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 gives a symmetric monoidal structure on for two symmetric monoidal categories .
Another related version is that Day convolution acts as a tensor product for a monoidal category structure on the category of functors over some monoidal category .
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Definition
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First version
Given for two symmetric monoidal , we define their Day convolution as follows.
It is the left kan extension along of the composition
Thus evaluated on an object , intuitively we get a colimit in of along approximations of as a pure tensor
Left kan extensions are computed via coends, which leads to the version below.
Enriched version
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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