The Stacks project

Definition 7.17.1. Let $\mathcal{C}$ be a site. An object $U$ of $\mathcal{C}$ is quasi-compact if given a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ there exists another covering $\mathcal{V} = \{ V_ j \to U\} _{j \in J}$ and a morphism $\mathcal{V} \to \mathcal{U}$ of families of maps with fixed target given by $\text{id} : U \to U$, $\alpha : J \to I$, and $V_ j \to U_{\alpha (j)}$ (see Definition 7.8.1) such that the image of $\alpha $ is a finite subset of $I$.