Lemma 91.15.5. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $j : \mathcal X \to \mathcal Y$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $j$ is representable by algebraic spaces. Then, if $\mathcal{Y}$ is a stack in groupoids (resp. an algebraic stack), so is $\mathcal{X}$.

Removing the hypothesis that $j$ is a monomorphism was observed in an email from Matthew Emerton dates June 15, 2016

**Proof.**
The statement on algebraic stacks will follow from the statement on stacks in groupoids by Lemma 91.15.4. If $j$ is representable by algebraic spaces, then $j$ is faithful on fibre categories and for each $U$ and each $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$ the presheaf

is an algebraic space over $U$. See Lemma 91.9.2. In particular this presheaf is a sheaf and the conclusion follows from Stacks, Lemma 8.6.11. $\square$

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## Comments (1)

Comment #4911 by Robot0079 on

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