A Characterization of Groupoids in Terms of Their Category of C-Sets

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