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

Lemma 93.15.5. 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 $\mathcal{X}$.

Proof. The statement on algebraic stacks will follow from the statement on stacks in groupoids by Lemma 93.15.4. 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

$(h : V \to U) \longmapsto \{ (x, \phi ) \mid x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V), \phi : h^*y \to f(x)\} /\cong$

is an algebraic space over $U$. See Lemma 93.9.2. In particular this presheaf is a sheaf and the conclusion follows from Stacks, Lemma 8.6.11. $\square$

Comment #4911 by Robot0079 on

If I didn't make any mistakes, the following general argument illustrates why this is true. We will show that two definitions of being representable by stack coincide.

Note that effective descent can be easily proved by the notion of sieve and properties of 2-pullback. So what we really need to show is the following fact.

Let $j: X \to Y$ be a morphism such that the pullback by any scheme is stack. If diagonal $\Delta_Y: Y \to Y\times Y$ is representable by sheaf, then $\Delta_X$ is also representable by sheaf.

Fix $T \to X\times X$, then $u: Isom_X \to Isom_Y$ is pull back of $\Delta_{X/Y}$. Given any scheme U, the fact that $X_U$ is a stack tells us u is representable by sheaf after fibered producting with U over Y. However, the target of this morphism is sheaf, hence $Isom_X \times_Y U$ is also a sheaf. In other words, $Isom_X \to Y$ is representable by sheaf. Note that this morphism factor through sheaf $Isom_Y$, we conclude that $Isom_X$ is sheaf.

Comment #5181 by on

@#4911: This type of comment isn't so useful. If you think you have an essentially different argument and especially if the argument uses less machinery, then by all means suggest replacing the proof (but it has to use machinery that is already in the Stacks project of course). This is useful for everybody. But in this comment I can't even understand what "the two definitions of being representable by stack" means. Is it related to the lemma?

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• 2 comment(s) on Section 93.15: Algebraic stacks, overhauled

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