Mario Román


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Convolution and coconvolution

Last updated Nov 22, 2022

# 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 $B$.

Tags: Monoids and comonoids