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I’ve added a section called $\mathcal{A}-gerbes$ at gerbe (as a stack) in an attempt to add something about the differential geometry question that was raised. I’m just a lowly grad student so be gentle if I’ve accidentally written something crazy.
I propose we break off the section on differential geometry to a new page possibly called “gerbe (in differential geometry)” or something. There is a lot to be said here, and mostly it just feels a little awkward to impose some different fundamental assumptions than the rest of the article (namely that the base of the stack is a smooth manifold).
Please do!
There is an entry differentiable stack.
just for the record and to remind us to come back to it, I have created a stub for infinity-gerbe.
Maybe the definition of differentiable stack is more general than the one given (or I’m being overly pedantic), but I think moving the article to differentiable stack wouldn’t be good since the gerbes that show up in differential geometry aren’t necessarily differentiable stacks (since they are on the site defined by the manifold and not the entire site Diff, and maybe they aren’t geometric; I haven’t checked).
One can have a differentiable stack over a space - take it as being a stack having a map to Diff/X.
Oh, of course. Thanks. I’ve just moved stuff over and started a stub gerbe (in differential geometry).
In addition, a differentiable stack over X is presented by a groupoid over X. One example is if you have a bundle of groups with fibres isomorphic to a given (abelian) Lie group $A$ - then you get an $A$-gerbe which, even if trivial, is not $\mathbf{B}A$. This connects more to when people talk about a sheaf of groups as the ’structure group’ of the gerbe.
Yes, one needs to be aware of the two different contexts for stacks that are usually discussed, often withouth explicitly stating which one is which:
there is the site Diff or CartSp or similar for the big (2,1)-topos $Sh_{(2,1)}(Diff)$ of stacks on $Diff$. That’s where differentiable stacks are disucssed in. Then for every object $X \in Diff$ there is the site for the small topos of $X$, usually taken to be $(Diff/X)_{open embedding}$, that’s where “gerbes on $X$” are typically discussed in.
Of course both are related. A morphism $\mathcal{K} \to j(X)$ in the big stack topos $Sh_{(2,1)}(Diff)$ (for $j$ the yoneda embedding) determines an object in $Sh_{(2,1)}(Diff/X)$.
A little bit of technical discussion of this is at over-oo-topos: as oo-sheaves on the big oo-site of an object.
I guess it’s clear, but one has to keep in mind which (2,1)-topos one is speaking about. For instance when one says
if you have a bundle of groups with fibres isomorphic to a given (abelian) Lie group $A$ - then you get an $A$-gerbe which, even if trivial, is not $\mathbf{B}A$.
then what is implicitly meant is that this gerbe is not the deloopong of $A$ regarded in the big $(2,1)$-topos. It is however the delooping of $A$ in the small $(2,1)$-topos of $X$.
I have added to the entry gerbe the fully general definition, and then a little bit of unwinding it and a little bit of discussion of how the two different perspectives on them arise.
I hope this serves a purpose of leading over to the list of “sub-entries” that we are lsiting.
Now that I’ve hunted around for what already existed in the nLab, I’ve found that lots of the fundamental things I thought were missing actually aren’t. See my comment at gerbe (in differential geometry).
It's not bad to have distributed comments also assembled together. On the contrary, having access to a particular distillate can help focus study. There are many perspectives, even within the nPOV.
I’ve found that lots of the fundamental things I thought were missing actually aren’t.
Sorry, looking at your comment at the entry now, I see that I had not been aware that this is what you were looking for, or I would have said something.
For situatons like this we have the “floating context tables-of-contents”. I have added one to gerbe (in differential geometry) now.
On this general topic of higher analogs of bundles with connection we have quite a bit on the $n$Lab.
I wrote:
On this general topic of higher analogs of bundles with connection we have quite a bit on the $n$Lab.
I am in the process of polishing parts of these accounts. Given the latest exchanges here, It seems to me that the entry
would be the one most specifically addressing the kind of interest expressed here: “geometric models for higher classes in ordinary differential cohomology”.
I have just spent a few minutes with polishing and expanding the introduction to that entry. Maybe this is of interest here. I, for one, would be interested in whatever comments you might have.
(The remainder of the entry presents an ab initio derivation of higher bundle gerbes with connection from “first principles”, namely from the intrinsic differential cohomology inside any cohesive (∞,1)-topos applied to that of smooth ∞-groupoids. The exposition of that probably deserves further polishing, which I’ll try to get to, soon.)
Ah. Well I only took up the cause because I was looking at the gerbe (as a stack) article and found someone had asked specifically about those topics (looking at an old copy of the page it says “Daniel: Please, would someone mind explain this with detail? I am interested in understand quantization and somehow, this thing comes up all the time… I got it from the entry on wikipedia on gerbes, but it was not explained.” “Tim: I will get there but the route is not that short! It is however not that long either.”), so I just assumed they weren’t in other places already.
One thing led to another, … , but maybe they were curious not about the constructions but how they actually play a role in quantization? I have Jean-Luc Brylinski’s book, but that isn’t incredibly useful to answer this other question. He only talks about Dirac’s monopole construction. I’m not sure if there is some high level way to talk about quantization in these terms in general, since I don’t know anything about quantization. And in any case, there does seem to be several pages on exactly this including geometric quantization which seems to be what was asked about. Sorry about that.
(looking at an old copy of the page it says “Daniel: Please, would someone mind explain this with detail? I am interested in understand quantization and somehow, this thing comes up all the time… I got it from the entry on wikipedia on gerbes, but it was not explained.” “Tim: I will get there but the route is not that short! It is however not that long either.”)
Ah. I didn’t know that this is the question we are talking about. Nor did I know – or at least not remember – that this exchange was present on an earlier version of the page.
The bundle gerbes in quantum field theory as they are discussed by Stevenson, Mickelsson, Murray, Wang are all – as far as I am aware – realizations of the degree-3 part of the family index of a family of dirac operators, so they are one component of the quantum anomaly for certain QFTs.
This is a special case of the general refinement of the Chern character on differential K-theory to ordinary differential cohomology, hence to $n$-gerbes for all $n$. For instance section 6 of Bunke-Schick Differential K-theory, A survey.
This is stuff of which we have only somewhat vague indications on the $n$Lab so far. Would be nice to eventually have more details on this.
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