The Stacks project

Definition 7.40.2. Let $\mathcal{C}$ be a site.

1. We say an object $U$ of $\mathcal{C}$ is weakly contractible if the equivalent conditions of Lemma 7.40.1 hold.

2. We say a site has enough weakly contractible objects if every object $U$ of $\mathcal{C}$ has a covering $\{ U_ i \to U\}$ with $U_ i$ weakly contractible for all $i$.

3. More generally, if $P$ is a property of objects of $\mathcal{C}$ we say that $\mathcal{C}$ has enough $P$ objects if every object $U$ of $\mathcal{C}$ has a covering $\{ U_ i \to U\}$ such that $U_ i$ has $P$ for all $i$.