The Stacks project

Lemma 97.11.3. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Then the following are equivalent

  1. $\mathcal{X}$ is a stack in setoids and $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects,

  2. $\mathcal{X}$ is a stack in setoids and limit preserving,

  3. $\mathcal{X}$ is representable by an algebraic space locally of finite presentation.

Proof. Under each of the three assumptions $\mathcal{X}$ is representable by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition 93.13.3. It is clear that (1) and (2) are equivalent as a functor between setoids is an equivalence if and only if it is surjective on isomorphism classes. Finally, (1) and (3) are equivalent by Limits of Spaces, Proposition 69.3.10. $\square$

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