The Stacks project

Lemma 8.5.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. The following are equivalent

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

  2. $\mathcal{S}$ is a stack over $\mathcal{C}$ and all fibre categories are groupoids, and

  3. $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$ and is a stack over $\mathcal{C}$.

Proof. Omitted, but see Categories, Lemma 4.35.2. $\square$

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