Non-Abelian Extensions of Groupoids and Their Groupoid Rings

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 \({\mathcal {N}}\rightarrow {\mathcal {E}}\rightarrow {\mathcal {G}}\) $\mathcal{N}\to \mathcal{E}\to \mathcal{G}$ gives rise to a groupoid crossed product of \({\mathcal {G}}\) $\mathcal{G}$ by the groupoid ring of \({\mathcal {N}}\) $\mathcal{N}$ which recovers the groupoid ring of  \({\mathcal {E}}\) $\mathcal{E}$ 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.