We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor-Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.

No abstract

No abstract

We prove that the category of cocommutative bialgebras in any locally presentable symmetric monoidal category is an [Formula: see text]-protomodular category with respect to a particular class of split extensions of cocommutative bialgebras. We also obtain the “partial” well-known Smith is Huq condition, meaning that two effective [Formula: see text]-equivalence relations centralize each other as soon as the normal subobjects associated with them commute in the sense of Huq.