In a previous paper, we showed that the category of cocommutative color Hopf algebras is semi-abelian in case the group G is abelian and finitely generated and the characteristic of the base field is different from 2 (not needed if G is finite of odd cardinality). Here we describe the commutator of cocommutative color Hopf algebras and we explain the Hall's criterion for nilpotence and the Zassenhaus Lemma. Furthermore, we introduce the category of color Hopf crossed modules and we explicitly show that this is equivalent to the category of internal crossed modules in the category of cocommutative color Hopf algebras and to the category of simplicial cocommutative color Hopf algebras with Moore complex of length 1.
Keywords: semi-abelian categories, color Hopf algebras, commutators, semi-direct products, crossed modules, simplicial Hopf algebras, Zassenhaus Lemma, Hall's criterion
2020 MSC: Primary 18E13, 16T05; Secondary 18D40, 18G45, 18N50, 16S40
Theory and Applications of Categories, Vol. 45, 2026, No. 32, pp 1321-1356.
Published 2026-05-25.
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