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

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.

arXiv comment17 pages

References in corpus (9)

• A torsion theory in the category of cocommutative Hopf algebras
• Pretorsion theories in general categories
• Factorization, Fibration and Torsion
• Auslander's formula and correspondence for exact categories
• The Yoneda isomorphism commutes with homology
• Torsion theories and coverings of preordered groups
• The stable category of preorders in a pretopos I: general theory
• A new Galois structure in the category of internal preorders
• Pretorsion theories in lextensive categories

Cited by in corpus (1)

• A 2-dimensional torsion theory on symmetric monoidal categories