On two notions of a Gerbe over a stack

Abstract

Let $\mathcal{G}$ be a Lie groupoid. The category $B\mathcal{G}$ of principal $\mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $\mathcal{D}$, there exists a Lie groupoid $\mathcal{H}$ such that $B\mathcal{H}$ is isomorphic to $\mathcal{D}$. Define a gerbe over a stack as a morphism of stacks $F:\mathcal{D}\to \mathcal{C}$, such that $F$ and the diagonal map ${\Delta }_F:\mathcal{D}\to \mathcal{D}{\times }_{\mathcal{C}}\mathcal{D}$ are epimorphisms. This paper explores the relationship between a gerbe defined above and a Morita equivalence class of a Lie groupoid extension.

(A shorter version of this is) accepted for publication in Bulletin des Sciences Mathématiques

Cited by in corpus (1)

• Extension of topological groupoids and Serre, Hurewicz morphisms