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It is known, thanks to Gabber, that algebraic spaces are sheaves in the fpqc topology:

Stacks project 03W8

Is the analogous statement for algebraic (Artin) stacks true? If not, is it true under some reasonable hypotheses?

AAK
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    "analogous statement" has a least two meanings 1 algebraic (Artin) stacks are sheaves in the fpqc topology and better 2 algebraic (Artin) stacks are stacks in the fpqc topology could you clarify ? – Niels Mar 25 '15 at 08:52

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It may be helpful to have a look at these notes by Anatoly Preygel (see also MO/15910/2503). In particular, Proposition 3.3.6 says that an algebraic stack is an fpqc sheaf if the diagonal is quasi-affine.

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    Both the question and your answer are ambiguous. Usually by sheaf one means sheaf of sets, whereas here you obviously mean fpqc "sheaf of groupoids", usually called fpqc stack. – Niels Mar 25 '15 at 08:51
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    @Niels, for me an algebraic stack is defined to be a sheaf of groupoids (satisfying some conditions), so I interpret the question as whether or not this sheaf satisfies fpqc descent. – AAK Mar 25 '15 at 09:33
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    This is not the classical definition, for instance not the definition in the stacks project. What you call sheaf is simply misleading with the current terminology. – Niels Mar 25 '15 at 13:15
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    @Niels This is, however, the standard terminology in homotopy theory (sometimes you see them referred as "homotopy sheaves" but the "homotopy" part is dropped more often than not). – Denis Nardin Mar 25 '15 at 13:33
  • @Denis the question was formulated with a clear reference to the stacks project and to algebraic stacks, so I don't understand the point of your remark. – Niels Mar 26 '15 at 08:26
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    I guess my problem is that I don't understand how Adeel's answer can be misleading. In which sense could you misunderstand the sentence "algebraic stacks with quasi-affine diagonal are fpqc sheaves"? Also the fact that it referenced the stacks project does not mean that the person in question is interested mainly in algebraic geometry (even if I admit that's probably the case): homotopy theorists use algebraic stacks all the time! – Denis Nardin Mar 26 '15 at 12:32
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    @Denis as I already mentioned in the first comment "algebraic stacks with quasi-affine diagonal are fpqc sheaves" is misleading since this can be understood as : a sheaf of sets. That is what most algebraic geometers will understand when you claim a stack is a sheaf. If the answer uses a different terminology than the question, there should be a warning. – Niels Mar 26 '15 at 13:30
  • I see your point, but it is not clear to me that the question is using a different terminology than the answer. Anyway I don't think this is worth discussing further – Denis Nardin Mar 26 '15 at 14:12
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    Thank you very much - this is exactly what I wanted (and I find it hard to see how an algebraic geometer could interpret the claim that "algebraic stacks with quasi-affine diagonal are fpqc sheaves" to mean that they are then forced to be sheaves in sets, but...). – beginner Mar 27 '15 at 15:14