If $X$ is an algebraic stack fibered in sets (and therefore essentially a sheaf), is it an algebraic space? It seems conceivable that at least when $X$ is Deligne-Mumford, it is actually an algebraic space. However, when $X$ is an Artin stack, it is only required to have an atlas of smooth maps of affines, so it seems less likely that this would be the case.
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asked Oct 11, 2010 at 20:58
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10$\begingroup$ A more practical version of the question is to impose the weaker-looking requirement that geometric points have trivial automorphism functors rather than the more "global" hypothesis that the groupoids are sets. This also has an affirmative answer, but it doesn't appear to be stated in the book by L. & M-B. (It is an exercise to show that the hypothesis on geometric pts implies the sheaf/set property for the fibered category, given that one is working with an Artin stack, understood to satisfy the diagonal requirements as in the L-MB book.) $\endgroup$– BCnrdCommented Oct 11, 2010 at 21:45
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Yes. The criterion for an Artin stack to be Deligne-Mumford is that it should have unramified diagonal (this is somewhere in Laumon Moret Bailly, I don't have it here). If the stack is fibered in sets, the diagonal is a monomorphism, and a monomorphism is certainly unramified.
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$\begingroup$ This shows that the stack is DM, but not that it is actually an algebraic space. $\endgroup$ Commented Oct 11, 2010 at 21:04
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3$\begingroup$ This shows that it has an étale atlas. Since it is a sheaf, it is an algbraic space. $\endgroup$– AngeloCommented Oct 11, 2010 at 21:07
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4$\begingroup$ Gotcha. Btw, this is exactly the content of Chapter 8 of Laumon and Moret-Bailly. It looks like the argument for showing that a DM stack which is a sheaf is an algebraic space is surprisingly involved though ... you need to prove that the diagonal of an algebraic stack is finite type. $\endgroup$ Commented Oct 11, 2010 at 21:16