With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
Just as we can convolve functions where is a group, or more generally a monoid, we can convolve functors where is a monoidal category. So, for any monoidal category , the functor category becomes a monoidal category in its own right. The tensor product in is called Day convolution, named after Brian Day.
We can generalize this idea by replacing Set with a more general cocomplete symmetric monoidal category . The technical condition is that the tensor product must preserve colimits in the separate arguments and ; that is, that the functors and must preserve colimits. This occurs when for instance is symmetric monoidal closed (so that these functors are left adjoints).
For a monoidal category and two presheaves on , their Day convolution product is the presheaf given by the coend
Let be the Yoneda embedding.
With the tensor unit of , the presheaf is a unit for the Day convolution product.
Using the co-Yoneda lemma on the two coends we have
For a small monoidal category, regard the category of presheaves as a monoidal category with tensor product the Day convolution product and unit the unit of under the Yoneda embedding .
is a closed monoidal category;
the Yoneda embedding constitutes a strong monoidal functor .
In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in exists at all, (with right adjoint to ) then by the Yoneda lemma it has to be given by
In Day’s original paper, a stronger form of the Day convolution is discussed, in which is assumed only to be a promonoidal category.
Let be a Benabou cosmos, and a small -enriched category.
There is an equivalence of categories between the category of pro-monoidal structures on with strong pro-monoidal functors between them and the category of biclosed monoidal structures on with strong monoidal functors between them.
Then the above convolution product is
Notice that if we regard the presheaves and here, assuming they take values in finite sets, as categorifications of -valued functions , where is the cardinality operation on finite sets, then this reproduces precisely the ordinary convolution product of these -valued functions
This uses in particular that for every object the functor
is in this sense the Kronecker delta-function on the set supported at . Precisely because by assumption has only identity morphisms.