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Let $Y$ be a complex algebraic variety, and let $n\in \mathbb Z_{\geq 1}\cup \{\infty\}$. How do I think about a proper $n$-etale morphism $X\to Y$?

If $n=1$, I think this should be a finite etale morphism (in the category of varieties).

If $n=2$, examples are $G$-gerbes over $Y$.

Do elements in $H^n(Y,G)$ give rise to proper $n$-etale morphisms $X\to Y$?

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    $\begingroup$ Could you define n-etale morphism? $\endgroup$ Feb 11, 2017 at 15:33
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    $\begingroup$ Perhaps Monsie means a locally trivial sheaf/stack/higher stack of n-1 truncated and n-1 connective sets/groupoids/higher groupoids? $\endgroup$
    – Joe Berner
    Feb 11, 2017 at 15:52
  • $\begingroup$ @JoeBerner Yes, that's what I mean. $\endgroup$
    – Monsie
    Feb 11, 2017 at 16:16
  • $\begingroup$ I don't think I can really answer your question, but you should read the preface of Higher Topos Theory. He talks about exactly this question there, and has some references math.harvard.edu/~lurie/papers/highertopoi.pdf $\endgroup$
    – Joe Berner
    Feb 11, 2017 at 18:41

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