Groupoids and skeletal categories form a pretorsion theory in $\mathsf{Cat}$

TL;DR

This paper establishes a pretorsion theory in the category of small categories, identifying groupoids as torsion objects and skeletal categories as torsion-free objects, based on properties of coequalizers.

Contribution

It introduces a novel pretorsion theory in Cat, linking groupoids and skeletal categories through new insights into coequalizer properties.

Findings

Coequalizers in Cat are faithful.

Coequalizers in Cat reflect isomorphisms.

Groupoids and skeletal categories form a pretorsion theory.

Abstract

We describe a pretorsion theory in the category $Cat$ of small categories: the torsion objects are the groupoids, while the torsion-free objects are the skeletal categories, i.e., those categories in which every isomorphism is an automorphism. We infer these results from two unexpected properties of coequalizers in $Cat$ that identify pairs of objects: they are faithful and reflect isomorphisms.