Monoidal internalization

Convolution monoid

In a monoidal category , suppose is a comonoid and is a monoid. Then is a monoid in under the product

c

with the unit . #m/thm/cat This is called the convolution monoid on .

Proof

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

Note that when is a bimonoid, we have two in general distinct monoid structures on : Composition and convolution .

Submonoids

If is a bimonoid and is commutative, then is a submonoid of the convolution monoid . #m/thm/cat

Proof

By the definition of a bimonoid, , and clearly , so . On the other hand, using commutativity of we have

c

whence the convolution of monoid morphisms is a monoid morphism.

Moreover, if is a Hopf monoid with antipous , then is a group, where the inverse of is given by . #m/thm/cat

Proof

That is shown by

c

and the proof that is analogous.

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