Groupoids: unifying internal and external symmetry

TL;DR

This paper introduces groupoids as a unifying framework for understanding both internal and external symmetries across various mathematical contexts, illustrated through elementary and advanced examples.

Contribution

It provides an accessible explanation of groupoids and demonstrates their role in describing symmetry, bridging discrete and continuous cases with illustrative examples.

Findings

Groupoids unify internal and external symmetry concepts.

Lie groupoids extend symmetry notions to differentiable settings.

Lie algebroids serve as infinitesimal counterparts to Lie groupoids.

Abstract

The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which involve Lie groupoids and their corresponding infinitesimal objects, Lie algebroids.