The Stacks project

Lemma 91.9.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The following are necessary and sufficient conditions for $f$ to be representable by algebraic spaces:

  1. for each scheme $U/S$ the functor $f_ U : \mathcal{X}_ U \longrightarrow \mathcal{Y}_ U$ between fibre categories is faithful, and

  2. for each $U$ and each $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$ the presheaf

    \[ (h : V \to U) \longmapsto \{ (x, \phi ) \mid x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V), \phi : h^*y \to f(x)\} /\cong \]

    is an algebraic space over $U$.

Here we have made a choice of pullbacks for $\mathcal{Y}$.

Proof. This follows from the description of fibre categories of the $2$-fibre products $(\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ in Categories, Lemma 4.40.3 combined with Lemma 91.8.2. $\square$


Comments (2)

Comment #69 by Quoc Ho on

Probably you might want to point back to tag 02ZY (categories-lemma-criterion-representable-map-stack-in-groupoids)since the two are basically the same result.

Comment #70 by on

@#69: I think you mean 4.40.7. And the sentence just preceding 91.9.2 does point back to that lemma. So no change needed I think. Right?

Just to make sure: in the stacks project the base category is always the category of schemes, so "being representable" always refers to being representable ``by schemes''. Whereas being representable by algebraic spaces is what 91.9.2 is about.

There are also:

  • 2 comment(s) on Section 91.9: Morphisms representable by algebraic spaces

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