Definition 66.18.2. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$, and let $\mathit{Sch}_{\acute{e}tale}$ be the corresponding big étale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The site $X_{spaces, {\acute{e}tale}}$ of $X$ is defined as follows:

1. An object of $X_{spaces, {\acute{e}tale}}$ is a morphism $\varphi : U \to X$ where $U$ is an algebraic space over $S$ and $\varphi$ is an étale morphism of algebraic spaces over $S$,

2. a morphism $(\varphi : U \to X) \to (\varphi ' : U' \to X)$ of $X_{spaces, {\acute{e}tale}}$ is given by a morphism of algebraic spaces $\chi : U \to U'$ such that $\varphi = \varphi ' \circ \chi$, and

3. a family of morphisms $\{ \varphi _ i : (U_ i \to X) \to (U \to X)\} _{i \in I}$ of $X_{spaces, {\acute{e}tale}}$ is a covering if and only if $|U| = \bigcup \varphi _ i(|U_ i|)$.

As usual we choose a set of coverings of this type, including at least the coverings in $X_{\acute{e}tale}$, as in Sets, Lemma 3.11.1 to turn $X_{spaces, {\acute{e}tale}}$ into a site.

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