Sheaf surjections transmit quasi-compactness.

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

1. 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.

2. 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. $\square$

Comment #7339 by Alejandro González Nevado on

Slogan suggestion: sheaf surjections transmit quasi-compactness.

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