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 $F\colon \mathcal{ D}\rightarrow \mathcal{ C}$, such that $F$ and the diagonal map $\Delta_F\colon \mathcal{ D}\rightarrow \mathcal{ D}\times_{ \mathcal{ C}}\mathcal{ D}$ are epimorphisms. This paper explores the relationship between a gerbe defined above and a Morita equivalence class of a Lie groupoid extension.
Categories: math.DG math.CT