Lemma 8.5.2 (02ZJ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 8: Stacks
Section 8.5: Stacks in groupoids
Lemma 8.5.2 (cite)

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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

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