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

We say an object $U$ of $\mathcal{C}$ is

*weakly contractible*if the equivalent conditions of Lemma 7.40.1 hold.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 $i$.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 $i$.

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