Convolution monoid

In a monoidal category $𝖢$ , suppose $𝑅$ is a comonoid and $𝐵$ is a monoid. Then $𝖢 (𝑅, 𝐵)$ is a monoid in $𝖲 𝖾 𝗍$ under the product

with the unit $𝜂 𝜖$ . #m/thm/cat This is called the convolution monoid on $𝖢 (𝑅, 𝐵)$ .

Proof

Associativity follows directly from the coässociative law law of $𝑅$ and associative law of $𝐵$ . Unitality follows directly from the coünit law of $𝑅$ and unit law of $𝐵$ .

Note that when $𝑅 = 𝐵$ is a bimonoid, we have two in general distinct monoid structures on $E n d 𝖢 (𝑅)$ : Composition $(\circ)$ and convolution $(*)$ .

Submonoids

If $𝑅$ is a bimonoid and $𝐵$ is commutative, then $𝖬 𝗈 𝗇 𝖢 (𝑅, 𝐵)$ is a submonoid of the convolution monoid $𝖢 (𝑅, 𝐵)$ . #m/thm/cat

Proof

By the definition of a bimonoid, $𝜖 \in 𝖬 𝗈 𝗇 𝖢 (𝑅, 𝟙)$ , and clearly $𝜂 \in 𝖬 𝗈 𝗇 𝖢 (𝟙, 𝐵)$ , so $𝜂 𝜖 \in 𝖬 𝗈 𝗇 𝖢 (𝑅, 𝐵)$ . On the other hand, using commutativity of $𝐵$ we have

whence the convolution of monoid morphisms is a monoid morphism.

Moreover, if $𝑅$ is a Hopf monoid with antipous $𝜎$ , then $𝖬 𝗈 𝗇 𝖢 (𝑅, 𝐵)$ is a group, where the inverse of $𝑓 \in 𝖬 𝗈 𝗇 𝖢 (𝑅, 𝐵)$ is given by $𝑓 𝜎$ . #m/thm/cat

Proof

That $𝑓 𝜎 * 𝑓 = 𝜂 𝜖$ is shown by

and the proof that $𝑓 * 𝑓 𝜎 = 𝜂 𝜖$ is analogous.

#state/tidy| #lang/en | #SemBr