The Stacks project

Definition 94.9.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. A $1$-morphism $f : \mathcal{X} \to \mathcal{Y}$ of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ is called representable by algebraic spaces if for any $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and any $y : (\mathit{Sch}/U)_{fppf} \to \mathcal{Y}$ the category fibred in groupoids

\[ (\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X} \]

over $(\mathit{Sch}/U)_{fppf}$ is representable by an algebraic space over $U$.

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View Definition 94.9.1 as pdf