Characterization of groupoid categories in terms of its category of $\mathcal{C}$-sets

TL;DR

This paper characterizes groupoids through their associated categories of $ ext{C}$-sets, offering insights into their structural properties via functor categories.

Contribution

It provides concise characterizations of groupoids based on the properties of their $ ext{C}$-set categories, linking algebraic and categorical structures.

Findings

Characterizations of groupoids via $ ext{C}$-set categories

Connections between groupoid properties and functor categories

Structural insights into groupoids from $ ext{C}$-set perspectives

Abstract

A $\mathcal{C}$-set is a functor from the category $\mathcal{C}$ to the category of finite sets and functions. The category of $\mathcal{C}$-sets, $\mathcal{C} - \operatorname*{set}$, is defined as the category whose objects are $\mathcal{C}$-sets, and whose morphisms are natural transformations between them. In this document we provide some concise characterizations of groupoids in terms of their category of $\mathcal{C}$-sets.