A weakly contractible object of a site is one that satisfies the equivalent conditions of the following lemma.
Lemma 7.40.1. Let \mathcal{C} be a site. Let U be an object of \mathcal{C}. The following conditions are equivalent
For every covering \{ U_ i \to U\} there exists a map of sheaves h_ U^\# \to \coprod h_{U_ i}^\# right inverse to the sheafification of \coprod h_{U_ i} \to h_ U.
For every surjection of sheaves of sets \mathcal{F} \to \mathcal{G} the map \mathcal{F}(U) \to \mathcal{G}(U) is surjective.
Proof. Assume (1) and let \mathcal{F} \to \mathcal{G} be a surjective map of sheaves of sets. For s \in \mathcal{G}(U) there exists a covering \{ U_ i \to U\} and t_ i \in \mathcal{F}(U_ i) mapping to s|_{U_ i}, see Definition 7.11.1. Think of t_ i as a map t_ i : h_{U_ i}^\# \to \mathcal{F} via (7.12.3.1). Then precomposing \coprod t_ i : \coprod h_{U_ i}^\# \to \mathcal{F} with the map h_ U^\# \to \coprod h_{U_ i}^\# we get from (1) we obtain a section t \in \mathcal{F}(U) mapping to s. Thus (2) holds.
Assume (2) holds. Let \{ U_ i \to U\} be a covering. Then \coprod h_{U_ i}^\# \to h_ U^\# is surjective (Lemma 7.12.4). Hence by (2) there exists a section s of \coprod h_{U_ i}^\# mapping to the section \text{id}_ U of h_ U^\# . This section corresponds to a map h_ U^\# \to \coprod h_{U_ i}^\# which is right inverse to the sheafification of \coprod h_{U_ i} \to h_ U which proves (1). \square
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.
The small étale site of \mathbf{A}^1_\mathbf {C} does not have any weakly contractible objects. On the other hand, the small pro-étale site of any scheme has enough contractible objects.
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