Definition 63.6.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. An algebraic space over $S$ is a presheaf

$F : (\mathit{Sch}/S)^{opp}_{fppf} \longrightarrow \textit{Sets}$

with the following properties

1. The presheaf $F$ is a sheaf.

2. The diagonal morphism $F \to F \times F$ is representable.

3. There exists a scheme $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a map $h_ U \to F$ which is surjective, and étale.

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