S-protomodularity of the category of cocommutative bialgebras

TL;DR

This paper establishes that the category of cocommutative bialgebras in symmetric monoidal categories with equalizers is S-protomodular, revealing structural properties and a partial Smith is Huq condition relevant to algebraic theory.

Contribution

It proves S-protomodularity of cocommutative bialgebras and introduces a partial Smith is Huq condition within this context.

Findings

Category of cocommutative bialgebras is S-protomodular

Partial Smith is Huq condition holds for these categories

Normal subobjects associated with S-equivalence relations commute

Abstract

We prove that the category of cocommutative bialgebras in any symmetric monoidal category (that has equalizers) is an S-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 S-equivalence relations centralize each other as soon as the normal subobjects associated with them commute in the sense of Huq.