Definition 7.17.4. An object $\mathcal{F}$ of a topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is quasi-compact if for any surjective map $\coprod _{i \in I} \mathcal{F}_ i \to \mathcal{F}$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ there exists a finite subset $J \subset I$ such that $\coprod _{i \in J} \mathcal{F}_ i \to \mathcal{F}$ is surjective. A topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is said to be quasi-compact if its final object $*$ is a quasi-compact object.
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