The Stacks project

Lemma 94.8.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Then $\mathcal{X}$ is representable by an algebraic space over $S$ if and only if the following conditions are satisfied:

  1. $\mathcal{X}$ is fibred in setoids1, and

  2. the presheaf $U \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)/\! \! \cong $ is an algebraic space.

Proof. Omitted, but see Categories, Lemma 4.40.2. $\square$

[1] This means that it is fibred in groupoids and objects in the fibre categories have no nontrivial automorphisms, see Categories, Definition 4.39.2.

Comments (2)

Comment #8378 by Daniël Apol on

I think the footnote in Lemma 93.8.2 should refer to Definition 4.39.2 (Tag 04SA) and not to Definition 4.38.2, which defines a category fibred in sets, and not in setoids.

There are also:

  • 2 comment(s) on Section 94.8: Categories fibred in groupoids representable by algebraic spaces

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