Convolution and coconvolution
# Definition
In a monoidal category $(ℂ,⊗,I)$, let $(A,μ,η)$ be a monoid and let $(B, δ, ε)$ be a comonoid. The hom-set $\hom(B,A)$ has the structure of a monoid, given by $f \ast g \in \hom(B,A)$ and $u \in \hom(B,A)$ defined by $$(f \ast g) = \delta ⨾ (f ⊗ g) ⨾ μ; \qquad u = ε ⨾ η.$$ 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,
$$(P \ast Q)(A) = ∫^{X,Y} \hom(A, X ⊗ Y) × P(X) × Q(Y).$$
A more general version assumes two monoidal structures for two parallel profunctors,
$$(P \ast Q)(A,B) =
∫^{X,X’,Y,Y’} \hom(A, X ⊗ Y) × P(X,X’) × Q(Y,Y’) × \hom(X’⊗Y’,B).$$
Tags: monoids and comonoids