---
tag: 457T9T7Y75
kind: concept
name: Day convolution
msc: [18M05, 18D20]
sources:
  - url: https://en.wikipedia.org/wiki/Day_convolution
same_as:
  wikidata: Q85755931
  wikipedia: Day convolution
---

## Summary

Day convolution is a tensor product on the category of set-valued functors on a small [[monoidal category|M3MSQ3NKAE]].
It is built from the tensor product of the source category.
It makes the [[functor category|1X0WKFQX8K]] into a closed monoidal category.
The Yoneda embedding becomes a strong monoidal functor.
So Day convolution is the universal way to extend a monoidal structure from a category to its category of set-valued functors.
It generalizes the convolution of functions on a group and the Cauchy product of power series.

## Explanations

### L1 human

Think of a functor $F$ from a monoidal category to sets as a family of sets $F(a)$, one for each object $a$.
To combine two such families at an object $c$, one collects all pairs of elements $x \in F(a)$, $y \in G(b)$ together with a way to map $a \otimes b$ to $c$.
For example, if the objects are the natural numbers with $+$ and only identity arrows, a functor is a graded set.
The Day convolution of two graded sets has in degree $n$ the pairs of elements whose degrees add up to $n$, exactly like multiplying power series.