# The Stacks Project

## Tag 07XN

Lemma 88.11.3. 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 $\mathcal{X} \to (\textit{Sch}/S)_{fppf}$ 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 84.13.3. 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 61.3.9. $\square$

The code snippet corresponding to this tag is a part of the file artin.tex and is located in lines 1374–1385 (see updates for more information).

\begin{lemma}
\label{lemma-limit-preserving-algebraic-space}
Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$.
Then the following are equivalent
\begin{enumerate}
\item $\mathcal{X}$ is a stack in setoids and
$\mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects,
\item $\mathcal{X}$ is a stack in setoids and limit preserving,
\item $\mathcal{X}$ is representable by an algebraic space
locally of finite presentation.
\end{enumerate}
\end{lemma}

\begin{proof}
Under each of the three assumptions $\mathcal{X}$ is representable
by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition
\ref{algebraic-proposition-algebraic-stack-no-automorphisms}.
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
\ref{spaces-limits-proposition-characterize-locally-finite-presentation}.
\end{proof}

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