The Stacks project

Lemma 99.3.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be surjective, locally of finite presentation, and flat. The following are equivalent

  1. $f$ is representable by algebraic spaces, and

  2. $W \times _\mathcal {Y} \mathcal{X}$ is an algebraic space.

Proof. The implication (1) $\Rightarrow $ (2) is Algebraic Stacks, Lemma 93.9.8. Conversely, let $W \to \mathcal{Y}$ be as in (2). To prove (1) it suffices to show that $f$ is faithful on fibre categories, see Algebraic Stacks, Lemma 93.15.2. Assumption (2) implies in particular that $W \times _\mathcal {Y} \mathcal{X} \to W$ is faithful. Hence the faithfulness of $f$ follows from Stacks, Lemma 8.6.9. $\square$

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