Some remarks on groupoids and small categories

TL;DR

This note explores properties of groupoids and small categories, proving that any groupoid is a group bundle over an equivalence relation, and defining actions and semi-direct products of categories, with implications for related mathematical work.

Contribution

It provides detailed proofs and definitions for the structure of groupoids as group bundles and the construction of semi-direct product categories, clarifying their properties and interactions.

Findings

Any groupoid is a group bundle over an equivalence relation.

The action of a category on another category and the semi-direct product are defined.

If both categories are groupoids, their semi-direct product is also a groupoid.

Abstract

This unpublished note contains some materials taken from my old study note on groupoids and small categories. It contains a proof for the fact that any groupoid is a group bundle over an equivalence relation. Moreover, the action of a category $G$ on a category $H$ as well as the resulting semi-direct product category $H\times_\alpha G$ will be defined (when either $G$ is a groupoid or $H^{(0)} = G^{(0)}$). If both $G$ and $H$ are groupoids, then $H\times_\alpha G$ is also a groupoid. The reason of producing this note is for people who want to check some details in a recent work of Li.