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I am working on further bringing the entry
infinity-Chern-Weil theory introduction
into shape. Now I have spent a bit of time on the (new) subsection that exposes just the standard notion of principal bundles, but in the kind of language (Lie groupoids, anafunctors, etc) that eventually leads over to the description of smooth principal oo-bundles.
I want to ask beta-testers to check this out, and let me know just how dreadful this still is ! ;-) The section I mean is at
There are some typos. For instance ’crossed complex’ leads to two terms of form G2. ’Useful’ is correct ’usefull’ is not. I will fix them if you are very busy, but if not … .
Thanks. I fixed the “useful[l]”-thing right now. (I know this, but sometimes my fingers don’t). Will have to go offline any second now. So if you feel like fixing typos now, I’d be very grateful, indeed!
I have added a bit more to the section on “principal n-bundles in low dimension”.
I set out this morning to work on the next main section Parallel transport in low dimensions but ended up working until now on the previous unfinished subsection:
A model for principal oo-bundles.
As before, I’d be grateful for beta-tester comments.
This is a very useful intro. I’ve worked through about a third, correcting typos, etc.
Just to check before I change, in Circle n-bundles and bundle (n−1)-gerbes,
Unwinding what this means, one see that P is a groupoid…
should be 2-groupoid, and the second C(U)1 in the diagram should be C(U)0?
This is a very useful intro.
Thanks, that’s a useful bit of information for me.
I’ve worked through about a third, correcting typos, etc.
Thanks!
Just to check before I change, in Circle n-bundles and bundle (n−1)-gerbes,
Unwinding what this means, one see that P is a groupoid…
should be 2-groupoid, and the second C(U)1 in the diagram should be C(U)0?
Yes, that second C(U)1 was a typo. Thanks, I have fixed that now.
Concerning P: it is correct as stated there, but there is a subtlety to be kept in mind here: when we compute the 2-groupoid that is the strict pullback
˜P→EBU(1)↓↓C(U)→BBU(1)then this ˜P will have nontrivial 2-morphisms (labeled by the value of the 2-cocycle gijk(x)). But this 2-groupoid is nevertheless equivalent to just the 1-groupoid P, that is presented by the bundle gerbe.
This is the higher analog of the analogous situation discussed further above in the entry, for a 1-bundle. The strict pullback
˜P→EG↓↓C(U)→BGis a 1-groupoid with nontrivial 1-morphisms, but is nevertheless equivalent to a 0-groupoid, namely to the manifold P (the total space of the G-bundle) itself.
I’ll think about ways to make this subtlety more explicit in the entry. Right now I will have to take care of something else.
Something like that would be good to include.
I just nipped to the end where there is in For connections on G-principal 1-bundles
Do the fundamental 2-groupoid and the path 2-groupoid coincide, or is there a need to distinguish Π2(X) from P2(X)?
I still need to go over the part that starts talking about connections.
In P2(X) the 2-morphisms are thin-homotopy-classes of surfaces, while in Π2(X) they are full homotopy classes.
(Am sitting in the last talk for today, will probably not have a chance to look into this before tomorrow morning. But then I wil try to continue polishing this. Thanks for all your input.)
Okay, I found a minute to implement the things we mentioned above and to edit a little bit more beyond that.
Now I have to rush off to pick up somebody from the station. Will get back to you later.
In ’Connections on principal 2-bundles’ I was expecting an imitation of the 1-story: (Π1(X),BG)≃♭BG and (P1(X),BG)≃BGconn(X). Why then
[CartSpop,Grpd](P2(X),BG)≃♭BG?
In ’Connections on principal 2-bundles’
Give me a minute, I am going to start working on that right now. Just finished with a considerable rewrite and expansion of the previous section Connections on a principle bundle
Why then
That looks like a typo. Let me check…
now I have gone editing through Connections on a 2-bundle. Not yet perfect, but should be better.
(And yes, the above was a typo, introduced last night when I decided to change the exposition strategy. )
started working on the next subsection Curvature characteristics of 1-bundles but only managed to make a few remarks. Now I have to interrupt to hear a talk.
I have now gone through the section
Curvature characteristics of 1-bundles
and its first subsection
I ended up entirely rewriting this. For whatever that’s worth. But I am hoping it serves some genuine introductory purpose now.
Again, I’d bre grateful for hearing just how dreadful this still is….
I was able to nod my way down to the mention of Cech groupoids without too much of a hitch. Nice so far.
It made me think that if I ever truly grok what a sheaf is, this stuff would come a lot easier. It will be a while before I can return to actually studying this stuff again unfortunately.
Nice so far.
Thanks for the information.
It made me think that if I ever truly grok what a sheaf is,
For understanding the deeper inner workings of this theory, you need to know this. But for the purpose of this introduction, you can get away with ignoring the notion of sheaves and just remembering that whenever it says – in this introduction – that some groupoid or 2-groupoid is regarded as a sheaf or presheaf, this just means that we remember a smooth structure on this object. The sheaf-tecnology is just a way to keep track of what this means.
@Eric
Indeed, a manifold is a certain sort of sheaf, but in thinking of it like this, we remember not just the charts (as maps in, ℝn∼→U↪M)) but all maps from all cartesian spaces. It’s just that since manifolds don’t form a nice category, we embed them in a slightly bigger category (a category of sheaves) which has all pullbacks, quotients etc.
Why in ’Connections on principal 2-bundles’ , do you have
For G=BA, a connection on a 2-bundle (not necessarily flat) is a lift… to ♭BBA?
If it is necessarily flat when lifting in the nonabelian case to ♭BG, isn’t G=BA just a special case?
Why
That’s a tyop! fixed it now. If should be [P2(−),B2A] instead of ♭B2A=[Π2(−),B2A] there.
I haven’t read through much of infinity-Chern-Weil theory introduction yet, but I like what I’m seeing.
A small comment on infinity-Chern-Weil theory: could we put the Motivation section ahead of the Idea section? It’s very short but very helpful, especially for putting the reader in an appropriate frame of mind to read on.
but I like what I’m seeing.
Thanks for letting me know.
could we put the Motivation section ahead of the Idea section? It’s very short but very helpful, especially for putting the reader in an appropriate frame of mind to read on.
I see, good point. I have done that switch now. Maybe the idea-section should be shortened to something less long-winded.
Re #20 I still don’t see why in that definition you give flat connections in the case G, and not necessarily flat connections in the case BA. What’s wrong with not necessarily flat connections for G, and flat connections for BA?
David,
right, good point, I need to add more on this.
The thing is that at this part in the discussion we are still looking at the idea that an n-connection is an anafunctor out of Pn(X). But this idea is too simple minded, as I try to mention. Currently the flow of the introduction is meant to do 1-connections in this way, then flat 2-connections and then discuss how these combined give what will actually be the correct way to think of n-connections.
Because, if you take a general 2-group G and look at 2-anafunctors P2(X)→BG you get a curious hyprid between flat and non-flat, namely 2-connections whose 3-form curvature is unconstrained but whose 2-form curvature vanishes.
There is a huge amount to be said about this case, but I felt that for the flow of the introduction it would be a distraction. It certainly distracts leisurely-listening people when I talk about this. Typically their attention span is up (and understandably so) once I am through with talking about this subtlety, and then I haven’t even mentioned what should be the main take-home message.
But since attentive readers like you are left wondering what’s going on with the present state of the entry, I should at least add some kind of remark, saying that the reason for the possibly odd-seeming omission of one obvious construction will be explained in a separate section later on.
When I have a bit more time, I’ll continue working on the entry and expand on this point. For the moment I just added this remark
The attentive reader will wonder why we do not state the last definition for general Lie 2-groups G. The reason is that for general G 2-anafunctors out of P2(X) do not produce the fully general notion of 2-connections that we are after, but yield a special case in between flatness and non-flatness: the case where precisely the 2-form curvature-components vanish, while the 3-form curvature part is unrestricted. This case is important in itself and discussed in detail below.
Only for G of the form BA does the 2-form curvature necessarily vanish anyway, so that in this case the definition by morphisms out of P2(X) happens to already coincide with the proper general one. This serves in the following theorem as an illustration for the toolset that we are exposing, but for the purposes of introducing the full notion of ∞-Chern-Weil theory we will rather focus on flat 2-conenctions, and then show in the next sections how using these one does arrive at a functorial definition of 1-connections that does generalize to the fully general definition of ∞-connections.
I worked today on bringing the next (and last) main section into better shape, the section oo-Lie algebra valued connections.
Agaín, I ended up effectively rewriting it. It should be better now than it was before. But I am not quite done yet. Have to interrupt now.
spent a good part of the day further polishing the section oo-Lie algebra valued connections.
It should be getting better now. This is the critical part, I suppose. As Domenico observed elsewhere, the central statement about that big diagram is very nice and has a nice flow to it once one sees it, but apparently my previous attempts to help see it had been somewhat ghastly.
There are still plenty improvements to be made that even I can see, but I’d already enjoy critical feedback from beta testers, in case anyone is brave enough and has nothing better to do (cough). Let me know where in the discussion you feel you get stuck.
(Hm, looking back at the entry, maybe you should better wait until I went through it at least once more tonight. Well, whatever..)
P2(X) the fundamental path 2-groupoid (2-morphisms are thin homotopy-classes of disks). Π2(X) the fundamental path 2-groupoid (2-morphisms are homotopy-classes of disks).
Should that be ’path 2-groupoid’ and ’fundamental 2-groupoid’ respectively? And shouldn’t the second of those link to ’fundamental n-groupoid’ or whatever we have?
This needs fixing:
Moreover, we have a natural equivalence of bicategories
[CartSpop,2Grpd](P2(C(U),[P2(−),BG])≃U(1)Gerb∇(X),
So, BG should be B2U(1), and that first P2 shouldn’t be there?
David, thanks a lot!!
Should that be ’path 2-groupoid’ and ’fundamental 2-groupoid’ respectively?
Yes, right. Fixed it.
And shouldn’t the second of those link to ’fundamental n-groupoid’ or whatever we have?
I thought about this. Currently we don’t have an entry of this title. What I have is the entry of path infinity-groupoid (schreiber), but that uses theory that in this introduction I want to avoid. I’ll see what to do about it.
So, BG should be B2U(1), and that first P2 shouldn’t be there?
Oops, yes. I must have copy-and-pasted something and then forgot about editing this line. Yes, both right. Fixed now.
Thanks again!!
I reordered and polished a bit the material in Summary – oo-Lie algebra-valued connections
Hi. In case anyone (but Urs) is wondering, I’m here, working in the background :)
Due to the magnificent work of Andrew Stacey, I can now again edit the entry “oo-Chern-Weil theory introduction” while on the train across Europe.
First thing I did now is that I pasted in the more polished discussion of oo-connections from infinity-Lie algebroid-valued forms into the section “∞-connections from Lie integration”.
have been going through infinity-Chern-Weil theory the last days and polished plenty of minor things. Turned it all into LaTeX, too. (phew)
I did some proof-reading on infinity-Chern-Weil theory introduction (and my LaTeX copy thereof) and fixed a bunch of trivial typos and tried to fix a bunch of awkward formulations.
am trying to bring the section The oo-Chern-Weil homomorphism itself into shape.
So far I have written a lead-in sub-section Motivating examples.
Have also done a major re-arrangement of some subsections. Should be better now.
made various further additions in an attempt to bring out the storyline better. Added some remarks explicitly about Deligne cohomology, polished the discussion of the integraded differential characteristic classes. I guess more deserves to be done, but have to stop for the moment.
In reaction to a question by Anton Kapustin, I have considerably expanded the infinity-Chern-Weil theory introduction … but currently not on the nLab but in my pdf-version at differential cohomology in a cohesive topos (schreiber):
section 1.3.1 Principal n-bundles for low n , starting on p. 24 now goes on to discuss principal 3-bundles, and twisted principal 2-bundles;
and a new section 1.3.4 Characteristic classes in low degree , starting on p. 60 looks into construction of first Chern-classes, Dixmier-Douady classes and the first Pontryagin class in terms of higher twisted bundles.
Hi Urs,
somehow related to this, which is a path object for H(X,BnU(1)conn), where X is a smooth manifold? or better, how can I decribe this path object by differential cocycle data?
by the way, it seems to me that in the first diagram in “Concrete construction” in homotopy pullback there is a missing \searrow from X×hZY to ZI, isn’t it?
Hi Domenico,
I think at Cech cohomology I once typed out some details about how the 0-cells in this path object give Cech coboundaries. Is that what you might be looking for?
Concerning that diagram at homotopy pullback: yes, thanks, a searrow should be there. I have included it now.
Hi Urs,
at Cech cohomology there’s only the version with BnU(1)-coefficients, and not the refined one with BnU(1)conn-coefficients: it seems I’ll have to work the latter out myself :)
as you may have been imagining by my question above, what I’d like to write are local differential forms data for an object in the homotopy limit of
H(X,BG1,conn)↓H(X,BG2,conn)→H(X,BnU(1)conn)where BG1,conn→BnU(1)conn and BG2,conn→BnU(1)conn are two differential cocycles. behind the scenes we are doing this by taking the (ordinary) limit with a fibrant replacement of BG2,conn→BnU(1)conn, as at differential string structure, but I think it would be interesting to explicitly describe that also in terms of the path object of H(X,BnU(1)conn).
I think the argument at Cech cohomology works immediately for coefficients that are complexes of sheaves of abelian groups. I can try to look into it later. But I think if you just do it, you will see it goes through.
surely it works. it’s only me being lazy and hoping it was already worked out somewhere on the Lab. I’ll work it out as I have a minute.
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