A Characterization of Groupoids in Terms of Their Category of \(\mathcal {C}\)-Sets

Abstract

A \(\mathcal {C}-\text {set}\) is a functor from a category \(\mathcal {C}\) to the category of finite sets, and \(\mathcal {C}-\text {set}\) denotes the category of such functors with natural transformations as morphisms. In this work, we prove that \(\mathcal {C}\) is a groupoid if and only if \(\mathcal {C}-\text {set}\)  has finitely many indecomposable objects, which is the categorical analog of transitive G-sets in classical group actions.