Double groupoids and $2$-groupoids in regular Mal'tsev categories

TL;DR

This paper investigates the structure of internal 2-groupoids within regular Mal'tsev categories, establishing their categorical properties, algebraic descriptions, and conditions for semi-abelian and action-representable structures.

Contribution

It proves that the category of internal 2-groupoids forms a Birkhoff subcategory of double groupoids in regular Mal'tsev categories and describes its algebraic theory in Mal'tsev varieties.

Findings

2-Groupoids form a Birkhoff subcategory of double groupoids.

When in a Mal'tsev variety, 2-groupoids constitute a Mal'tsev variety.

The category of 2-groupoids is semi-abelian when the base category is semi-abelian.

Abstract

We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ of internal $2$-groupoids is a Birkhoff subcategory of the category $ \mathrm{Grpd}^2(\mathscr{C}) $ of double groupoids in a regular Mal'tsev category $\mathscr{C}$ with finite colimits. In particular, when $\mathscr{C}$ is a Mal'tsev variety of universal algebras, the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is also a Mal'tsev variety, of which we describe the corresponding algebraic theory. When $\mathscr{C}$ is a naturally Mal'tsev category, the reflector from $ \mathrm{Grpd}^2(\mathscr{C}) $ to 2-$ \mathrm{Grpd}(\mathscr{C}) $ has an additional property related to the commutator of equivalence relations. We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ is semi-abelian when $\mathscr{C}$ is semi-abelian, and then provide sufficient conditions for 2-$ \mathrm{Grpd}(\mathscr{C}) $ to be action representable.