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.
Groupoids and skeletal categories form a pretorsion theory in Cat
Campanini, Federico;
2023
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.
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