Convolution and coconvolution

Definition. In a monoidal category $(C,\otimes ,I)$, let $(A,\mu ,\eta )$ be a monoid and let $(B,\delta ,\epsilon )$ be a comonoid. The hom-set $\mathrm{hom}(B,A)$ has the structure of a monoid, given by $f\ast g\in \mathrm{hom}(B,A)$ and $u\in \mathrm{hom}(B,A)$ defined by $(f\ast g)=\delta ⨾(f\otimes g)⨾\mu ;u=\epsilon ⨾\eta .$ This is the convolution monoidal structure of $A$ and $B$.

• When we particularize to profunctors (presheaves) with the monoidal structure of the base category, we recover the Day convolution of presheaves, $(P\ast Q)(A)={\int }^{X,Y}\mathrm{hom}(A,X\otimes Y)\times P(X)\times Q(Y).$ A more general version assumes two monoidal structures for two parallel profunctors, ∫^{X,X',Y,Y'} \hom(A, X ⊗ Y) × P(X,X') × Q(Y,Y') × \hom(X'⊗Y',B).$$

• See Raudsilla Seminar, November 2022, Lax twisted monoids and convolution.

Tags: monoid