The Stacks project

Definition 8.11.1. A gerbe over a site $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ such that

1. $p : \mathcal{S} \to \mathcal{C}$ is a stack in groupoids over $\mathcal{C}$ (see Definition 8.5.1),

2. for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ such that $\mathcal{S}_{U_ i}$ is nonempty, and

3. for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ such that $x|_{U_ i} \cong y|_{U_ i}$ in $\mathcal{S}_{U_ i}$.