Definition 8.11.4. Let $\mathcal{C}$ be a site. Let $\mathcal{X}$ and $\mathcal{Y}$ be stacks in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories over $\mathcal{C}$. We say $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ if the equivalent conditions of Lemma 8.11.3 are satisfied.

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