The Stacks project

Lemma 8.4.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category over $\mathcal{C}$. The following are equivalent

  1. $\mathcal{S}$ is a stack over $\mathcal{C}$, and

  2. for any covering $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ of the site $\mathcal{C}$ the functor

    \[ \mathcal{S}_ U \longrightarrow DD(\mathcal{U}) \]

    which associates to an object its canonical descent datum is an equivalence.

Proof. Omitted. $\square$

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