Fundamental groupoids for simplicial objects in Mal'tsev categories

TL;DR

This paper explores the structure of internal groupoids within Mal'tsev categories, establishing their properties, factorization systems, and connections to Kan complexes and weighted commutators.

Contribution

It characterizes the category of internal groupoids as a reflective subcategory and links simplicial objects with weighted commutators in Mal'tsev categories.

Findings

Internal groupoids form a reflective Birkhoff subcategory.

Regular epimorphisms admit a monotone-light factorization system.

Connections are drawn between simplicial objects, Kan complexes, and weighted commutators.

Abstract

We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and in fact a Birkhoff subcategory of the category of simplicial objects. We then characterize the central extensions of the corresponding Galois structure, and show that regular epimorphisms admit a relative monotone-light factorization system in the sense of Chikhladze. We also draw some comparison with Kan complexes. By comparing the reflections of simplicial objects and reflexive graphs into groupoids, we exhibit a connection with weighted commutators (as defined by Gran, Janelidze and Ursini).