Non-Abelian extensions of groupoids and their groupoid rings

TL;DR

This paper develops a geometric classification theory for non-Abelian extensions of groupoids, linking them to groupoid crossed products of rings, and extends existing theories for Abelian extensions and groups.

Contribution

It introduces a new classification framework for non-Abelian groupoid extensions and connects these to groupoid crossed product rings, broadening the understanding of their structure.

Findings

Each groupoid extension yields a corresponding crossed product ring.

The classification methods extend naturally to groupoid crossed products.

Provides a new perspective on natural examples of groupoid crossed products.

Abstract

We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids $\mathcal{N} \to \mathcal{E} \to \mathcal{G}$ gives rise to a groupoid crossed product of $\mathcal{G}$ by the groupoid ring of $\mathcal{N}$ which recovers the groupoid ring of $\mathcal{E}$ up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products.