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Recall that a map of schemes, algebraic spaces, stacks, etc is called submersive if the associated map on underlying topological spaces is a quotient map. Recall moreover that a map is called universally submersive if it is submersive and all of its base changes are submersive.

Recall that a map $f:b\to b^\prime$ is called an $E$-descent morphism relative to a Cartesian fibration $E\to B$ if the natural map $E_{b^\prime}\to E_{S_f}$ is fully faithful, where $E_{S_f}$ is the value of $E$ on the $b^\prime$-sieve $S_f$ generated by $f$. A map $b\to b^\prime$ is a universal E-descent morphism if all of its base-changes are E-descent morphisms.

In SGA4 VIII 9.3, there is a theorem demonstrating that universally submersive maps of schemes are universal descent morphisms for the Cartesian fibration on the category of schemes sending a scheme $S$ to the category of étale sheaves of sets on $S$ with transition morphisms given by the pullback functors.

Two questions:

1.) Are universally submersive maps of schemes also universal descent maps for the $(2,1)$-stack sending a scheme $S$ to the $(2,1)$-category of étale sheaves of groupoids on $S$? How about the $(n+1,1)$-stack of sheaves of n-groupoids?

2.) Can this be extended to the case where we replace the category of schemes with the $(2,1)$-category of Artin stacks (or Deligne-Mumford stacks)? Do we have to add more adjectives to 'universally submersive' to make it hold in this more general context (for example, representable universally submersive, locally separated universally submersive, etc)?

The original proof in SGA uses a very cute trick, but it doesn't seem to apply once you throw stacks in the mix (it uses the fact that any map admitting a section is a mono, and this fails very badly for stacks).

Edit: I added a bounty. What's left to understand is the case of a universal submersion of ordinary schemes (whether these are universal descent morphisms for the fibration of étale n-sheaves)

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  • $\begingroup$ Does the map E_{b'} \to E_{S_f} not have to be an equivalence? It seems you're omitting the "descent data are effective" part of the stack condition. I believe 1). 2) should follow from the usual yoga of "cover the artin stack with a scheme, give data on the cover plus gluing data on overlaps" etc. The difficulty is defining "univ subm" for stacks, but if they're representable, this should be clear. Also you may mean the lis-et site in the artin stack case. $\endgroup$ – Leo Herr Nov 19 '20 at 1:15
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    $\begingroup$ @LeoHerr That's effective descent (or -2-descent (Lurie), or 0-descent (Giraud)). I'm talking about -1-descent (Lurie)/1-descent (Giraud) (i.e. ineffective descent) (there is an off-by-2 indexing change in modern sources). $\endgroup$ – Harry Gindi Nov 19 '20 at 1:56
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    $\begingroup$ Also this stack of étale sheaves can be defined without reference to a topology. The objects are precisely the n-representable étale maps over your stack. $\endgroup$ – Harry Gindi Nov 19 '20 at 2:05
  • $\begingroup$ Does SGA actually prove that \'etale sheaves of sets form a stack over schemes for the universal submersion topology? Unless I'm misunderstanding, the effectivity of the descent is only proved in certain cases (in 9.4 of SGA4 VIII 9.3). $\endgroup$ – Akhil Mathew Dec 29 '20 at 19:15
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    $\begingroup$ @HarryGindi: indeed, I missed the definition of universal descent (vs. universal effective descent). I'm curious how generally effectivity might hold. $\endgroup$ – Akhil Mathew Dec 29 '20 at 20:30
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A partial answer: The second claim follows from the first:

Let $X$ be an algebraic stack, and let $f:Y\to X$ be a universal submersion. Then to show that $Y\to X$ is universal descent for $E_n$ the (n+1,1)-stack of étale $n$-sheaves, find an fppf cover $U\to X$ by $U$ a separated scheme. Then pulling back $f$ to $U$, we have a universal submersion $f_U:Y_U \to U$. Now we can choose an fppf cover $V\to Y_U$ by a separated scheme $V$, which is a universal submersion, and which now makes the composite $V\to Y_U \to U$ a universal submersion as well.

Then by the first claim, if it is true, we see that $V\to U$ is universal $E_n$-descent. It follows then that $Y_U\to U$ is universal $E_n$-descent, by the usual stuff about factorization of universal descent maps. Moreover, the map $U\to X$ is universal effective $E_n$-descent by Toën's theorem on fppf descent for relative n-stacks together with a computation involving the cotangent complex. In particular, it is universal $E_n$-descent, and such maps are closed under composition.

So it follows that $Y_U\to X$ is universal $E_n$-descent, but $Y_U\to X$ factors as $Y_U\to Y\to X$, and using the lemma on factorization of universal descent morphisms, we see that $Y\to X$ is now universal $E_n$-descent. In particular, we are reduced to proving the claim of universal $E_n$-descent in the case of a universal submersion of separated schemes.

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  • $\begingroup$ This reduction step works for $X$ even an algebraic $m$-stack. $\endgroup$ – Harry Gindi Nov 17 '20 at 15:59

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