Lemma 98.11.3. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Then the following are equivalent
$\mathcal{X}$ is a stack in setoids and $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects,
$\mathcal{X}$ is a stack in setoids and limit preserving,
$\mathcal{X}$ 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 94.13.3. 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 70.3.10. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: