Day convolution


Monoidal categories

Category theory



Just as we can convolve functions f:Mf : M \to \mathbb{C} where MM is a group, or more generally a monoid, we can convolve functors f:MSetf: M \to Set where MM is a monoidal category. So, for any monoidal category MM, the functor category Set MSet^M becomes a monoidal category in its own right. The tensor product in Set MSet^M is called Day convolution, named after Brian Day.

We can generalize this idea by replacing Set with a more general cocomplete symmetric monoidal category VV. The technical condition is that the tensor product uvu \otimes v must preserve colimits in the separate arguments uu and vv; that is, that the functors uu \otimes - and v- \otimes v must preserve colimits. This occurs when for instance VV is symmetric monoidal closed (so that these functors are left adjoints).

Restricted Definition

For (C,)(C, \otimes) a monoidal category and F,G:C opSetF, G : C^{op} \to Set two presheaves on CC, their Day convolution product FGF \star G is the presheaf given by the coend

FG:= c,dCF(c)×G(d)×Hom C(,cd). F \star G := \int^{c,d \in C} F(c) \times G(d) \times Hom_C(-, c \otimes d) \,.


Let j:CPSh(C)j : C \to PSh(C) be the Yoneda embedding.


With ICI \in C the tensor unit of CC, the presheaf j(I)j(I) is a unit for the Day convolution product.


Using the co-Yoneda lemma on the two coends we have

Fj(I) c,dCF(c)×Hom C(d,I)×Hom C(,cd) cCF(c)×Hom C(,cI) cCF(c)×Hom C(,c) F(). \begin{aligned} F \star j(I) & \simeq \int^{c,d \in C} F(c) \times Hom_C(d,I) \times Hom_C(-, c\otimes d) \\ & \simeq \int^{c \in C} F(c) \times Hom_C(-, c \otimes I) \\ & \simeq \int^{c \in C} F(c) \times Hom_C(-, c) \\ & \simeq F(-) \end{aligned} \,.

For CC a small monoidal category, regard the category of presheaves (PSh(C),,j(I))(PSh(C), \star, j(I)) as a monoidal category with tensor product the Day convolution product and unit the unit of CC under the Yoneda embedding j:CPSh(C)j : C \hookrightarrow PSh(C).


  1. (PSh(C),,j(I))(PSh(C), \star, j(I)) is a closed monoidal category;

  2. the Yoneda embedding constitutes a strong monoidal functor (C,,I)(PSh(C),,j(I))(C,\otimes, I) \hookrightarrow (PSh(C), \star, j(I)).


In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in PSh(C)PSh(C) exists at all, (with [F,][F,-] right adjoint to ()F(-) \star F) then by the Yoneda lemma it has to be given by

[F,G](c) Hom C(j(c),[F,G]) Hom C(j(c)F,G). \begin{aligned} [F,G](c) & \simeq Hom_C(j(c), [F,G]) \\ &\simeq Hom_C(j(c)\star F, G) \end{aligned} \,.

Promonoidal Day Convolution

In Day’s original paper, a stronger form of the Day convolution is discussed, in which AA is assumed only to be a promonoidal category.

Let VV be a Benabou cosmos, and AA a small VV-enriched category.


There is an equivalence of categories between the category of pro-monoidal structures on AA with strong pro-monoidal functors between them and the category of biclosed monoidal structures on V A opV^{A^{op}} with strong monoidal functors between them.



Then the above convolution product is

FG:e cd=eF(c)×G(d). F \star G : e \mapsto \oplus_{c \cdot d = e} F(c) \times G(d) \,.

Notice that if we regard the presheaves FF and GG here, assuming they take values in finite sets, as categorifications of \mathbb{N}-valued functions |F|,|G|:C|F|, |G| : C \to \mathbb{N}, where ||:Set|\cdot| : Set \to \mathbb{N} is the cardinality operation on finite sets, then this reproduces precisely the ordinary convolution product of these \mathbb{N}-valued functions

|FG|:e c,dC|F(c)|×|G(d)|×δ(e,cd) = cd=e|F(c)||F(d)| \begin{aligned} |F \star G| : e &\mapsto \sum_{c,d \in C} |F(c)| \times |G(d)| \times \delta(e, c \otimes d) & = \sum_{c \cdot d = e} |F(c)| \cdot |F(d)| \end{aligned}

This uses in particular that for every object cCc \in C the functor

Hom C(c,)=δ c Hom_C(c,-) = \delta_c

is in this sense the Kronecker delta-function on the set CC supported at cCc \in C. Precisely because by assumption CC has only identity morphisms.

Hom C(c,d)={* ifc=d ifcd Hom_C(c,d) = \left\{ \array{ * & if c = d \\ \emptyset & if c \neq d } \right.


For (∞,1)-categories:

Revised on June 10, 2014 05:22:35 by Adeel Khan (