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.
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