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I was wondering whether this sort of thing exists somewhere on the nLab (or can be derived as a special case of something), or if anyone knows of some references. Suppose is a space (I’m thinking smooth variety, but it could be rather general probably). By a cocycle argument, is in bijection with isomorphism classes of line bundles. One could think of this as a class in corresponds to some geometric thing.
Giraud’s theory shows that classes in correspond to certain gerbes. People actually use classes in there to construct very concrete geometric things like moduli spaces of twisted sheaves on surfaces by unwinding geometrically what a gerbe is (e.g. see Caldararu’s thesis).
Does anyone know if higher versions of this are written down anywhere? Maybe I don’t quite have the above right, but I think the general form of what I’m looking for should be clear. For example: gerbes are -categories, so is there an interpretation where classes in correspond to -categories that when unwound give some sort of geometric information?
People do talk about n-gerbes, but we don’t have much on them.
Duskin (Simplicial methods and the interpretation of “triple” cohomology) and his student Glenn (Realization of cohomology classes in arbitrary exact categories) showed there is a ’geometric’ interpretation if one is prepared to consider simplicial things. So I guess if one passes to simplicial presheaves one can do the same thing (see e.g. Beke’s Simplicial torsors)
Maybe it’s just worth mentioning Pawel Gajer’s “The Geometry of Deligne Classes” ?
http://arxiv.org/abs/alg-geom/9601025
He gives a geometric interpretation for not just , but for degree Deligne classes, which correspond to equivalence classes of geometric things classified by equipped with a “connection”. He covers both the smooth and holomorphic cases. But he doesn’t use “-gerbes”. He uses principal bundles, where is the iterated classifying space of . It’s been awhile since I’ve looked at the paper, so I can’t remember the specifics of the differentiable structure he uses for .
But maybe this isn’t helpful, if you are looking specifically for a higher categorical interpretation.
-Chris
One or two comments: (i) The idea of a bundle gerbe in work by Murray et al is worth following up to see if it helps. (ii) Breen handles 2-gerbes and Aldrovandi and Noohi have a neat approach to the general case, so far mostly illustrated by n=2. (I have been meaning to include more of this in the Menagerie but have been diverted to doing other things. :-()
Thanks. These look really good. If I start to read any of them in depth, I’ll find a place in the nLab to start writing what I find…how quickly I’ve developed a list of things I want to write up here :-0
First check out bundle gerbe in the Lab and ignore any Google ads for flowers, wreaths, hosiery and … I never knew there were so many sense of the word gerbe. :-)
This is discussed at principal infinity-bundle and at the references given there.
For the case with refinements also at circle n-bundle.
All other objects mentioned here (gerbes, 2-gerbes, etc) are special cases of this. I’d think the Lab entries on these should point this out, but probably I forgot to say it here and there.
Incidentally, by some coincidence, right this week I am (and last week I already was) visiting Danny Stevenson in Glasgow, together with Thomas Nikolaus, and we are busy writing up a detailed article on principal -bundle theory.
This was motivated originally by precisely the kind of questions that you are asking here: what’s the full -version of higher bundles and higher gerbes, and their classifcation by higher cohomology classes? (And for me eventually: their connections.)
After a bit of thinking it turns out that the answer is conceptually surprisingly simple: the principal -bundle corresponding to a cocycle in is simply the homotopy fiber of this cocycle, computed in the correct ambient -topos (for your applications there wil be some natural site in the game over which you are considering abelian sheaf cohomology and the -topos in question is that over this site.)
Moreover, it is precisely the Lurie-Giraud axioms characterizing -toposes that ensure that the -catgeory of -principal -bundles over is in fact equivalent to the cocycle -groupoid .
The fully general abstract formalization in (∞,1)-topos theory as discussed in the Lab entry principal infinity-bundle was first inducated in
A more comprehensive account (with more examples, etc.) is in
This general abstract theory is readily seen to recover all the special cases of bundles, gerbes and higher gerbes that one expects to see. I’d be happy to chat more about all this.
And with a little luck I can provide an early version of the detailed article that we are currently working on later this month. The main point of the article will be to supply the general abstract theory with a concrete model by a homotopical category of simplicial spaces that is useful for explicit computations.
If you let me know what speifically you are looking for, I can try to produce for you exactly that.
Hi Urs,
what overlap is there between this work and the Roberts-Stevenson preprint? :)
firstname.lastname at adelaide.edu.au if too sensitive for public eyes.
what overlap is there between this work and the Roberts-Stevenson preprint? :)
There is not so much overlap than that we are building on it. And – I am thinking and hoping – maybe catalyze its appearance… ;-)
In rough outline, the plan of what we are doing is running like this:
give a general abstract definition of principal oo-bundles in any oo-topos and show that they are classified by the intrinsic cohomology of that oo-topos (a more detailed discussion of material in my cohesive notes);
give criteria for the coefficient objects such that in a model category presentation of the oo-topos they are presented by topological or Lie simplicial groups (or their ); and give criteria for when the intrinsic cohomology with these coefficients is inded given by the Cech-hypercohomology that Roberts-Steveson applies to – theryby giving sufficient conditions for when Roberts-Stevenson provides a classification of genuine oo-bundles (a kind of strictification result: when are simplicial bundles over simplicial topological groups a model for general weak oo-bundles)
refine the Roberts-Stevenson classifying spaces from the topological to the smooth case (by generalizing what Nikolaus-Waldorf do in their “Four models for non-abelian gerbes.”)
(maybe the main point) provide a model for principal oo-bundles by “strict G but weakly principal” simplicial bundles, that generally models the intrinsically defined oo-bundles.
Cool! Was just mentioning the R-S preprint today to Ray Vozzo, who is visiting Danny soon.
@Urs No 8. There is, I think, a conceptual difficulty for some people to go from the n=1 case of bog-standard gerbes (whether or otherwise) to general -gerbes. This is perhaps because even 2-gerbes seem difficult when viewed from ‘below’ but viewed from above (i.e. from the nPOV, Lurie etc.) they seem almost ’standard’. This leads to a fear that something is missing from the second viewpoint. (I do not feel this myself, just that as that viewpoint is fairly new, there is a lot to discuss and to push further, although the formulation is such that ALL of that depth is there from the start if you peel off the layers.)
I remember years ago trying to convince someone at a category theory conference that simplicial principal fibre bundles already did a lot of what was needed to do infinity stacks etc. , as they had Kan complexes (so infinity groupoids) as their fibres, but even though Jack Duskin was with me and was pushing exactly the same point, the others were not convinced. I did not see what the blockage is, but perhaps we need some effort made to unblock this on the lab. I do not know how, perhaps with some entries that take the general definitions and deconstruct them in low dimensions. (We already have quite a lot of stuff along these lines but perhaps need more .. or different???)
(I should say that at about the same time I was putting forward the idea (as was Jack) that Kan complexes were weak infinity groupoids, and was met with blank stares a lot of the time. I was hoping for an amalgamation of the approaches to infinity category theory and methods from homotopy theory at the time. It eventually came I suppose, although the homotopy theorists would probably deny that … I don’t know.)
I agree. As someone who is just getting to the point where throwing an “infinity” in front of a word isn’t that scary anymore, it is hard for me to look at Breen’s treatment (which I’m finding fantastic!) of 2-gerbes which is essentially an entire book unwinding the correspondence between H^3 and 2-gerbes using all the horrifying cocycles and then see that all of this is just some special case of something that can be stated more cleanly easily with infinty things.
Although I have sympathy for people that feel this way, I’m not skeptical that it becomes the equivalent thing. The thing that concerns me is when I go to principal infinity-bundle I don’t really see how to get the special case of 2-gerbes out of that. I really want to build something concrete using actual classes in H^3. I can try to explain briefly what this is, but I haven’t figured out exactly yet what everything should be. Basically, for a projective scheme you can take a Brauer class such that the universal family for the moduli space of stable sheaves is realized as a -twisted sheaf.
Now in the situation I care about, there is a trick using which basically works when is a surface. My situation has a 3-fold and the exact same trick should work if I can figure out what sort of geometric thing is going on with . Maybe it is some moduli space of gerbes, and there should be a notion of -twisted gerbes since the class will correspond to a 2-gerbe or something.
The thing that concerns me is when I go to principal infinity-bundle I don’t really see how to get the special case of 2-gerbes out of that. I really want to build something concrete using actual classes in
For 1-gerbes I have spelled this out in a bit of detail at bundle gerbe in the section As the total space of a principal 2-bundle.
I can give the analogous discussion in all detail you want for bundle 2-gerbes, too. I’d only need to find a free hour to type out the somewhat more lengthy formulas. But it is all straightforward.
Something meant as an expository survey of these issues is at infinity-Chern-Weil theory introduction in the section Principal n-bundles in low dimension. (This is also at the beginning of my pdf that is linked to there.)
Let me know if there are further details that you need. There is much more to be said than has been typed up on the Lab, of course.
Thanks! There is no rush to type anything out. I have plenty of sources to keep me busy for awhile. I’ll keep reading these and if I come up with anything more specific I’ll ask.
This is an issue more generally with -category theory for large (including and ) – we have standard and more explicit descriptions of -categories for small (like and sometimes ), and we’d like to make sure the large- case reduces to the small- one when is small. Sometimes it takes a substantial amount of work to make the connection.
But for small there is more in available works: the infinite bundles are about -case, while in small dimensions there are some facts about -principal bundles etc.
This is an issue more generally with n-category theory for large n (including ∞ and (∞,1)) – we have standard and more explicit descriptions of n-categories for small n (like n=2 and sometimes n=3), and we’d like to make sure the large-n case reduces to the smalln one when n is small. Sometimes it takes a substantial amount of work to make the connection.
There is also a converse aspect to this situation: often explicit low--constructions are done in the hope to achieve some goal, and often it is hard to even figure out if the goal has indeed been achieved, since everything is written out in components.
The case of higher gerbes and higher principal bundles is an example of that: whatever higher -principal bundle are, they are supposed to be classified by cohomology with coefficients in the higher group . This is usually far from obvious in the component-based low- constructions. On the other hand, this goes through very smoothly in the full -case. Therefore – I have to say – if a notion of -principal -bundles is such that the abstract notion of -principal -bundles does not reduce to it, up to equivalence, then it is simply wrong.
My personal painful experience may maybe serve as an illustration of what I mean: as some of you may remember, I started out doing 2-bundles and then trying 3-bundles and later strict -bundles for low in the components version. But it can be a bit painful. Then at some point there was this observation that later materialized in the article with David Roberts on what I’d now call universal principal infinity-bundles: the component-constructions of principal -bundles that we had been using were nothing but models for the homotopy fiber of their classifying cocycle, computed in the correct ambient -context.
From that point on it was clear what’s really going on.
In a more pronounced way I could say:
doing abstract -category theoretic constructions is like writing consistency-checked computer code specifications: you get a structure outline that is guaranteed to do what you want. Then finding the low--component realizations is like writing the code that realizes this code specification.
No argument here. But whatever order you do it in, you then have to prove that your small- calculations actually do implement the -categorical specifications, which can sometimes be tricky.
I would say more. The problem is explaining how the link occurs. This is sort of the ‘homotopy coherent’ yields ’infinity homotopic’ (in a more model category sense) route.
Mike,
we agree. But not everyone does. As Tim mentions:
This is perhaps because even 2-gerbes seem difficult when viewed from ‘below’ but viewed from above (i.e. from the nPOV, Lurie etc.) they seem almost ’standard’. This leads to a fear that something is missing from the second viewpoint.
I was meaning to point out that – in contrast to this – one should fear that there is something missing from the first viewpoint!
Good point. Especially for 2-gerbes! :-) although some version of the Aldrovandi-Noohi stuff would seem so near the general case that that fear seems unfounded.
I think my thought on 2-gerbes is more that even with gerbes, there is so much written ’out there’ which claims to be general but is like proving things about Abelian groups and then saying you have proved them about groupoids… it is true but not much help. The infinity nPOV although difficult avoids unnecessary barriers and produces ’the reason why’.
My thoughts, for what it is worth, on the link between the infinity and the low dimensional case is that the latter uses something akin to Algebraic Homotopy whilst the other is Homotopical Algebra, i.e. the Whitehead view of the subject against the Quillen one. This does not fit exactly but gives a feeling for the relationship.
By the way, there are “historical” examples for this:
for instance the first definition of morphisms of bundle gerbes that appeared in the literature was wrong. It was too restrictive: it failed to capture the correct notion of morphisms of smooth groupoids/smooth stacks. The mistake happened because the wrong notion of morphisms seemed to be the correct one given the component-presentation of bundle gerbes. Today, the correct notion of morphisms of bundle gerbes goes, somewhate unfortunately, by the name “stable morphism”.
Today we might think: this was an easy mistake to avoid. But try writing out the morphisms of bundle 3-gerbes directly. I think there is a good chance that you will make a similar mistake.
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