Lemma 94.9.3 (0457)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 94: Algebraic Stacks
Section 94.9: Morphisms representable by algebraic spaces
Lemma 94.9.3 (cite)

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Lemma 94.9.3. Let $S$ be an object of $\mathit{Sch}_{fppf}$ . Consider a $2$ -commutative diagram

$\xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Y}' \ar[r] & \mathcal{Y} }$

of $1$ -morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ . Assume the horizontal arrows are equivalences. Then $f$ is representable by algebraic spaces if and only if $f'$ is representable by algebraic spaces.

Proof. Omitted. $\square$

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