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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?

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    $\begingroup$ "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 ? $\endgroup$
    – Niels
    Commented Mar 25, 2015 at 8:52

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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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    $\begingroup$ 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. $\endgroup$
    – Niels
    Commented Mar 25, 2015 at 8:51
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    $\begingroup$ @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. $\endgroup$
    – AAK
    Commented Mar 25, 2015 at 9:33
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    $\begingroup$ 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. $\endgroup$
    – Niels
    Commented Mar 25, 2015 at 13:15
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    $\begingroup$ @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). $\endgroup$
    – Denis Nardin
    Commented Mar 25, 2015 at 13:33
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    $\begingroup$ 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! $\endgroup$
    – Denis Nardin
    Commented Mar 26, 2015 at 12:32

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