Coalgebraic models of Omega-groups

TL;DR

This paper explores coalgebraic models of Omega-groups within cocommutative coalgebras, establishing their categorical properties and introducing Omega-Hopf algebra categories as semi-abelian structures.

Contribution

It introduces a new class of coalgebraic models called Omega-Hopf algebras and characterizes their categorical properties, extending the understanding of algebraic theories in coalgebraic contexts.

Findings

Categories of Omega-Hopf algebras are semi-abelian.

Coalgebraic models of Omega-groups can be characterized within cocommutative coalgebras.

Established categorical properties similar to algebraic varieties.

Abstract

We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of coalgebraic models of algebraic theories endowed with an underlying structure of cocommutative Hopf algebra, and show that these categories are semi-abelian. We call them ``categories of Omega-Hopf algebras'', since it is possible to characterize them as coalgebraic models of algebraic theories of Omega-groups.