Lemma 96.4.4 (06TS)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 96: Sheaves on Algebraic Stacks
Section 96.4: Sheaves
Lemma 96.4.4 (cite)

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Lemma 96.4.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$ -morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ . Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$ . The functors ${}_ pf$ and $f^ p$ of (96.3.1.1) transform $\tau$ sheaves into $\tau$ sheaves and define a morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau )$ .

Proof. This follows immediately from Stacks, Lemma 8.10.3. $\square$

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