# On two notions of a Gerbe over a stack

## TL;DR

This paper investigates two different notions of gerbes over differentiable stacks, analyzing their relationship with Morita equivalence classes of Lie groupoid extensions.

## Contribution

It clarifies the connection between gerbes defined as morphisms of stacks and Lie groupoid extensions up to Morita equivalence.

## Key findings

- Establishes a correspondence between gerbes and Morita equivalence classes.
- Provides a framework linking stack morphisms to Lie groupoid extensions.
- Enhances understanding of gerbes in the context of differentiable stacks.

## 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 $F\colon \mathcal{D}\rightarrow \mathcal{C}$, 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.