# The Stacks Project

## Tag: 02ZY

This tag has label algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids and it points to

The corresponding content:

Lemma 63.9.2. Let $S$ be a scheme contained in $\textit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. The following are necessary and sufficient conditions for $f$ to be representable by algebraic spaces:
1. for each scheme $U/S$ the functor $f_U : \mathcal{X}_U \longrightarrow \mathcal{Y}_U$ between fibre categories is faithful, and
2. for each $U$ and each $y \in \mathop{\rm Ob}\nolimits(\mathcal{Y}_U)$ the presheaf $$(h : V \to U) \longmapsto \{(x, \phi) \mid x \in \mathop{\rm Ob}\nolimits(\mathcal{X}_V), \phi : h^*y \to f(x)\}/\cong$$ is an algebraic space over $U$.
Here we have made a choice of pullbacks for $\mathcal{Y}$.

Proof. This follows from the description of fibre categories of the $2$-fibre products $(\textit{Sch}/U)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ in Categories, Lemma 4.38.3 combined with Lemma 63.8.2. $\square$

\begin{lemma}
\label{lemma-criterion-map-representable-spaces-fibred-in-groupoids}
Let $S$ be a scheme contained in $\Sch_{fppf}$.
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
The following are necessary and sufficient conditions for
$f$ to be representable by algebraic spaces:
\begin{enumerate}
\item for each scheme $U/S$ the
functor $f_U : \mathcal{X}_U \longrightarrow \mathcal{Y}_U$
between fibre categories is faithful, and
\item for each $U$ and each $y \in \Ob(\mathcal{Y}_U)$ the presheaf
$$(h : V \to U) \longmapsto \{(x, \phi) \mid x \in \Ob(\mathcal{X}_V), \phi : h^*y \to f(x)\}/\cong$$
is an algebraic space over $U$.
\end{enumerate}
Here we have made a choice of pullbacks for $\mathcal{Y}$.
\end{lemma}

\begin{proof}
This follows from the description of fibre categories of the $2$-fibre products
$(\Sch/U)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ in
Categories, Lemma \ref{categories-lemma-identify-fibre-product}
combined with
Lemma \ref{lemma-characterize-representable-by-space}.
\end{proof}


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## Comments (2)

Comment #69 by Quoc Ho on September 28, 2012 at 4:46 pm UTC

Probably you might want to point back to tag 02ZY (categories-lemma-criterion-representable-map-stack-in-groupoids)since the two are basically the same result.

Comment #70 by Johan (site) on September 30, 2012 at 6:13 pm UTC

@#69: I think you mean 02Y9. And the sentence just preceding 02ZY does point back to that lemma. So no change needed I think. Right?

Just to make sure: in the stacks project the base category is always the category of schemes, so "being representable" always refers to being representable by schemes''. Whereas being representable by algebraic spaces is what 02ZY is about.

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