Lemma 7.17.5 (0GMP)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.17: Quasi-compact objects and colimits
Lemma 7.17.5 (cite)

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Sheaf surjections transmit quasi-compactness.

Lemma 7.17.5. Let $\mathcal{C}$ be a site.

If $U \to V$ is a morphism of $\mathcal{C}$ such that $h_ U^\# \to h_ V^\#$ is surjective and $U$ is quasi-compact, then $V$ is quasi-compact.
If $\mathcal{F} \to \mathcal{G}$ is a surjection of sheaves of sets and $\mathcal{F}$ is quasi-compact, then $\mathcal{G}$ is quasi-compact.

Proof. Omitted. $\square$

Comments (1)

Comment #7339 by Alejandro González Nevado on May 06, 2022 at 17:57

Slogan suggestion: sheaf surjections transmit quasi-compactness.

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