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Given a base site $S$ of schemes, consider a stack $\mathcal{C}$ on $S$ not assumed to be algebraic. Then in principle, given another stack $\mathcal{E}$ on $S$, together with a map of stacks $\mathcal{E} \to \mathcal{C}$ over $S$, we could ask for $\mathcal{E}$ to be algebraic over $\mathcal{C}$. This would entail, aside from having an appropriate surjective map from a scheme or algebraic space, demanding that $\mathcal{E} \to \mathcal{E} \times_{\mathcal{C}} \mathcal{E}$ was representable. At this point I'm being agnostic whether $\mathcal{E}$ is in fact a stack over $\mathcal{C}$, since the definition I give doesn't require this (and if $\mathcal{C}$ is not fibred in groupoids, which is conceivable, this is not always possible to organise).

Has this concept appeared in the literature? I cannot find anything about it in the Stacks Project.


EDIT In the comments Ariyan Javanpeykar identifies this concept as a map of stacks being representable by algebraic stacks, which appears a few times in the literature without being formally defined and I guess must be a kind of folklore definition (it's not too difficult to guess, but I cannot find a paper that uses the phrase that also gives a definition). What I would like to use such a thing for is to look at geometric properties of maps of stacks that are representable by algebraic stacks, much as one can talk about geometric properties of maps that are representable (by schemes, or by algebraic spaces, even). It may be one gains nothing new, but it might allow for a more flexible definition in some cases.

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  • $\begingroup$ If you like, this is kind of like asking whether $\mathcal{E}$ is 'relatively algebraic' over $\mathcal{C}$. $\endgroup$ – David Roberts Sep 21 '17 at 7:28
  • $\begingroup$ I don't know if it already appeared in the literature, but I think @Emerton might be someone to ask about this, as his group at the Stacks project workshop might have worked on something related to this. $\endgroup$ – pbelmans Sep 21 '17 at 11:18
  • $\begingroup$ Thanks, but I don't think he's active here anymore. If I get no response in a few days I'll ask directly (but I guess you know him better than me!) $\endgroup$ – David Roberts Sep 21 '17 at 11:48
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    $\begingroup$ @DavidRoberts Let $X\to Y$ be a morphism of categories fibred in groupoids over $(Sch/S)_{fppf}$ (with $S$ a scheme). The morphism $X\to Y$ is representable by algebraic stacks if, for any algebraic stack $Z$ over $S$ and any morphism $Z\to Y$, the fibre product $X\times_Y Z$ is an algebraic stack. Is this not the notion you're looking for? See TAG 04SX for the algebraic spaces version. $\endgroup$ – Ariyan Javanpeykar Sep 21 '17 at 20:30
  • $\begingroup$ @Ariyan I'm not assuming my categories are fibred in groupoids, as that case is rather easy. There are plenty of stacks like that. But thanks for the terminology, it didn't occur to me. $\endgroup$ – David Roberts Sep 21 '17 at 20:37

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