Covering morphisms of internal groupoids in the models of a semi-abelian theory

TL;DR

This paper explores the relationship between topological models of semi-abelian theories and their internal groupoid structures, establishing criteria for lifting these structures to covering groupoids.

Contribution

It introduces a functor from topological T-algebras to internal groupoids and provides a criterion for lifting internal groupoid structures to coverings.

Findings

Fundamental groups of topological T-algebras are characterized.

A functor from topological T-algebras to internal groupoids is constructed.

A criterion for lifting internal groupoid structures to coverings is proved.

Abstract

In this paper, for given an algebraic theory $T$ whose category $C$ of models is semi-abelian, we consider the topological models of $T$ called topological $T$-algebras and obtain some results related to the fundamental groups of topological $T$-algebras. We also deal with the internal groupoid structure in the category of models providing that the fundamental groupoid deduces a functor from topological $T$-algebras to the internal groupoids in $C$ and prove a criterion for the lifting of such an internal groupoid structure to the covering groupoids.