Mike wrote:
I don't think it's correct, then, to say as Harry did that "hyper-descent is a generalization of a lower-dimensional process"; do you? It's only when you get to the oo-world that you can tell the difference between "satisfying descent" and "satisfying hyperdescent," even if you may want to look at n-step hypercovers when sheafifying n-presheaves.
Yes, I guess one should formulate this carefully enough.
Harry wrote:
I'm betting that with this scheme, we'll end up with two cases: toposes that act like higher grothendieck toposes, and toposes where whitehead's theorem holds.
I think I see what you are maning to get at, but maybe this requires a more detailed analysis. I don't think it is right to think of there just being two different types of descent for oo-stacks on a given Grothendieck topology. There should be many many more, one for every left exact localization of the oo-presheaf oo-category. In HTT for instance there is a whole section on n-toposes (meaning (n,1)-toposes) and all of them are examples of (oo,1)-toposes on the given site.
But I can't claim that I have a good systematic understanding of the different notions of descent. Certainly Cech descent is what one wants to look at in most cases. Notably for instance in Lurie's later developments such as in Structured Spaces, the main ambient oo-toposes are taken to come from Cech descent, as far as I am aware.
]]>Oh, okay, I see. I don't think it's correct, then, to say as Harry did that "hyper-descent is a generalization of a lower-dimensional process"; do you? It's only when you get to the oo-world that you can tell the difference between "satisfying descent" and "satisfying hyperdescent," even if you may want to look at n-step hypercovers when sheafifying n-presheaves.
]]>Ah, of course - as an n-groupoid is (n+1)-coskeletal (approximately - I may have the indexing out by one) it only sees a skeleton of a hypercover, and so a bounded hypercover. Cool!
]]>Right, thanks, I should have mentioned that, for completeness.
But I was just making the much more trivial comment in reply to something Harry said, that an n-groupoid valued presheaf will only care about n-step hypercovers being hommed into it, it can't tell if its an n+1-step hypercover, actually. That's why when talking about just sheaves, the question of hypercovers never arises.
But, yes, if you already know that the n-groupoid valued presheaf produces an equivalence after homming a projection out of any Cech-nerve into it, then the same will be true for homming all bounded hypercovers into it, as you point out.
What Beke's article shows is that if you start with an n-groupoid valued presheaf, then in order to n-stackify it (hence oo-stackify it) i.e. in order to refine it to something that does satisfy descent with respect to all Cech covbers, it is sufficient to do what for sheaves is called the "+-construction" at n-step hypercovers.
I should have been clearer about this. In fact, we should have all this cleanly said on some nLab page eventually. I suppose currently aspects of this must be hidden somewhere at descent and Cech cohomology.
]]>Cech covers are cofinal in bounded hypercovers, because the refinement process is guaranteed to stop after a finite number of steps, but not all hypercovers.
]]>Can you say exactly where in Beke's paper he talks about 1-stacks? Briefly glancing at it I don't see the connection. I am also confused because the DHI paper "Hypercovers and simplicial presheaves" claims that Cech descent implies descent for all "bounded hypercovers." Is that not contradictory to saying that for 1-stacks the hyperness already matters?
A stray thought: could 1-step hypercovers be identified with "sieves that are 2-subobjects" in the same way that 0-step, i.e. Cech, hypercovers are identified with "sieves that are 1-subobjects"?
]]>Ah, then the case is closed, at least from my perspective. Ordinary descent by sieves appears to generalize directly, and "hyper-descent " is a generalization of a lower-dimensional process as well. It's really nice when things generalize smoothly like that, because it gives some sort of direct confirmation that it's a canonical extension of the earlier theory.
One of the main reasons I was interested in the classical formulation of descent generalizing fully is that this is the form one would expect to generalize to "abstract (oo,1)-categories", which are something like our idealized notion of what an (oo,1)-category should look like. This formulation of descent doesn't depend on any of the underlying machinery of quasicategories, so it feels more "right". Formulating these kinds of things as abstract properties that use terms that are completely relative to the background seems like a very useful technique. Notions like equivalence, subobject, representable, etc., all have a well-defined meaning relative to your chosen model of (infinity,1) category. The distinction is something like the distinction between material and structural set theories, at least in my mind.
Or I could be completely wrong. ;) I'm sure someone will correct any egregious errors I've made in my assessment.
]]>Yes, for 1-stacks it already matters. 1-stacks see "1-step" hypercovers. In general, n-stacks will see "n-step hypercovers". The good reference for this is the article by Tibor Beke that is referenced at Cech cohomology.
]]>I meant for 1-stacks, which is really the only place we might be able to get a glimpse of it. Either that or in one of Street's papers on 2-stacks (if such papers even exist. I assume they do, but I may have misunderstood the other people in this thread).
]]>You mean for 0-stacks = sheaves? There one can't see the difference: homming a Cech nerve into a set-valued presheaf yields the same result as homming any of its hypercover refinements into it.
]]>Is there an analogous condition for 1-sites to have descent by hypercovers?
]]>My contention that sieves as 1-subobjects gives you Cech descent is supported by appendix A of Hypercovers and simplicial presheaves, although they only consider 1-sites rather than (oo,1)-sites.
]]>Hypercovers come a lot and essentially in Hodge theory for algebraic varieties (as in 1970s papers of Deligne), though I am not competent to defend the particulars of this.
]]>Notice that one main point that one might feel is wrong about Cech localizations is that the Whitehead theorem generally fails in them. It is precisely their further hypercompletion to hypercover localizations that makes the Whitehead theorem come out. If in your applicaiton Whitehead's theorem is crucial, then hypercover descent will induce the "right" notion of localization for you.
]]>Lurie's conclusion that the Cech covering definition is the correct one,
I wouldn't quite put it that way. A priori every localization is as "right" as any other. What he does, though, it spend six pages on pointing out that the Cech localizations are particularly well behaved with respect to several other constructions. So maybe these are the "most natural" ones.
]]>@Mike: Well, whatever characterization of 2-toposes has brought you to that (admittedly informal) conclusion seems like it is a promising one. I hope I've appropriately disclaimed you from the next paragraph.
Deferring to Mike, who has not written up a formal proof of this statement (I only qualify this because I don't want to claim something that might oblige him to actually write up a proof [or force him to repudiate earlier statements that he did not make]), this agrees with Lurie's conclusion that the Cech covering definition is the correct one, at least for (infinity,1)-categories. Perhaps for (infinity,n)-categories, we need the hypercover definition more, but from what I've seen, the case of (infinty,1)-categories is a canonical extension of 1-categorical intuition. It may turn out that for (infinity,n)-categories, we will need the more precise information given by hypercovers.
]]>I think that with oo-sieve = 1-subobject of a representable in the sense I advocated, you will end up with Cech descent.
]]>Oh, quick question. Does the duality between fibered categories and pseudopresheaves generalize to right fibrations of simplicial sets and simplicial presheaves? I use the term "duality" loosely here.
@Mike: If we restrict our sieves to be fully faithful (in the appropriate infinity,1-category sense), can we recover either of the covering methods Urs was talking about?
]]>The reason I say that 1-subobjects = fully faithful maps are the "right" notion of subobject for 2-topoi is that using them you get the 2-Giraud theorem. Cf. Street "characterization of bicategories of stacks". I certainly agree that 2-subobjects = faithful maps are useful for other purposes.
]]>You've convinced me :) And you're right about the 'up to equivalence' bit - we certainly would want homotopy pullbacks.
]]>Well, yeah, but we're looking at (infinity,1)-categories, which means that we can't just keep constructing examples like that. Our notion of equivalence requires us to have an SSet map in that induces an isomorphism in the homotopy category, so I think our notion of equivalence is more robust than that. Also, the notion of a sieve being a subfunctor really only needs to be "up to equivalence" because we require our covering sieve system to be stable under (homotopy?)-pullbacks anyway.
]]>That's true, but there are 1-sub-oo-categories, 2-sub-oo-categories and so on. There is an m-poset of k-sub-oo-categories for some appropriate values of m and k that I can't think of right now. I suppose the choice has to be application driven, and Mike's suggestion is as good as any unless it is necessary to change (in my own work, I take a map 'including' a sub-2-group to be a faithful functor (or locally faithful, if dealing with the deloopings) - this would be a 2-subobject perhaps)
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