Day convolution

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 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 .

Definition

Summarize
Perspective

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:

References

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.