Semi-abelian condition for color Hopf algebras

Abstract

Recently it was shown that the category of cocommutative Hopf algebras over an arbitrary field $\Bbbk$ is semi-abelian. We extend this result to the category of cocommutative color Hopf algebras, i.e. of cocommutative Hopf monoids in the symmetric monoidal category of $G$-graded vector spaces with $G$ an abelian group, given an arbitrary skew-symmetric bicharacter on $G$, when $G$ is finitely generated and the characteristic of $\Bbbk$ is different from 2 (not needed if $G$ is finite of odd cardinality). We also prove that this category is action representable and locally algebraically cartesian closed, then algebraically coherent. In particular, these results hold for the category of cocommutative super Hopf algebras by taking $G=\mathbb{Z}_{2}$. Furthermore, we prove that, under the same assumptions on $G$ and $\Bbbk$, the abelian category of abelian objects in the category of cocommutative color Hopf algebras is given by those cocommutative color Hopf algebras which are also commutative.