S-protomodularity of the category of cocommutative bialgebras

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.