Convolution [mde-9MBD]

In its classical form, (Day) convolution refers to the monoidal structure defined on the \((\mathcal {V})\)-category of presheaves \([\hat {C},\mathcal {V}]\) whenever \(C\) has promonoidal structure.

More generally, for a pseudomonoid \(M\) and a pseudocomonoid \(C\) in a closed monoidal bicategory \(\mathcal {K}\), the internal hom \([C,M]\) is a pseudomonoid. This furthermore induces a monoidal structure on the hom-category \(\mathcal {K}(C,M)\).