Definition 64.6.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. An *algebraic space over $S$* is a presheaf

with the following properties

The presheaf $F$ is a sheaf.

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

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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