On the semi-abelianness of cocommutative Hopf monoids

TL;DR

This paper proves that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, unifying previous results and showing that abelian objects are precisely the commutative cocommutative Hopf monoids.

Contribution

It generalizes the semi-abelianness of cocommutative Hopf algebras to monoids in any abelian symmetric monoidal category, using a new generalized bijective correspondence.

Findings

Category of cocommutative Hopf monoids is semi-abelian under certain conditions.

The category is action representable.

Abelian objects are exactly the commutative cocommutative Hopf monoids.

Abstract

By providing a suitable generalization of Newman's bijective correspondence known for cocommutative Hopf algebras, we prove that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, once faithful (co)flatness conditions are satisfied. This result unifies and generalizes the semi-abelianness of cocommutative Hopf algebras and of cocommutative color Hopf algebras known up to now. As a consequence of the semi-abelianness, the category of cocommutative Hopf monoids is also action representable. Finally, we prove that abelian objects in the category of cocommutative Hopf monoids coincide exactly with commutative and cocommutative Hopf monoids, which form so an abelian category.