Hopf measuring comonoids and enrichment

TL;DR

This paper investigates the existence and properties of universal measuring comonoids in braided monoidal categories, revealing conditions under which these structures are bimonoids or Hopf monoids, with applications to vector and graded spaces.

Contribution

It establishes conditions for universal measuring comonoids to be bimonoids or Hopf monoids, extending the theory in braided monoidal categories and providing new descriptions of comodules.

Findings

$P(A,B)$ is a bimonoid when $A$ is a bimonoid and $B$ is commutative

If $A$ is a cocommutative Hopf monoid, then $P(A,B)$ is Hopf

Examples include universal measuring comonoids in vector and graded spaces

Abstract

We study the existence of universal measuring comonoids $P(A,B)$ for a pair of monoids $A$, $B$ in a braided monoidal closed category, and the associated enrichment of a category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if $A$ is a bimonoid and $B$ is a commutative monoid, then $P(A,B)$ is a bimonoid; in addition, if $A$ is a cocommutative Hopf monoid then $P(A,B)$ always is Hopf. If $A$ is a Hopf monoid, not necessarily cocommutative, then $P(A,B)$ is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative description of the dualizable $P(A,B)$-comodules and use the theory of Hopf (co)monads. We explore the examples of universal measuring comonoids in vector spaces and graded spaces.