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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    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

    Principal n-bundles in low dimension

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeSep 8th 2010

    There are some typos. For instance ’crossed complex’ leads to two terms of form G 2G_2. ’Useful’ is correct ’usefull’ is not. I will fix them if you are very busy, but if not … .

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    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!

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2010

    I have added a bit more to the section on “principal n-bundles in low dimension”.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2010

    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.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 13th 2010

    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) 1C(U)_1 in the diagram should be C(U) 0C(U)_0?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2010
    • (edited Sep 13th 2010)

    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) 1C(U)_1 in the diagram should be C(U) 0C(U)_0?

    Yes, that second C(U) 1C(U)_1 was a typo. Thanks, I have fixed that now.

    Concerning PP: 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) \array{ \tilde P &\to& \mathbf{E} \mathbf{B}U(1) \\ \downarrow && \downarrow \\ C(U) &\to& \mathbf{B}\mathbf{B} U(1) }

    then this P˜\tilde P will have nontrivial 2-morphisms (labeled by the value of the 2-cocycle g ijk(x)g_{i j k}(x)). But this 2-groupoid is nevertheless equivalent to just the 1-groupoid PP, 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) → BG \array{ \tilde P &\to& \mathbf{E} G \\ \downarrow && \downarrow \\ C(U) &\to& \mathbf{B} G }

    is a 1-groupoid with nontrivial 1-morphisms, but is nevertheless equivalent to a 0-groupoid, namely to the manifold PP (the total space of the GG-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.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 13th 2010

    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 Π 2(X)\mathbf{\Pi}_2(X) from P 2(X)\mathbf{P}_2(X)?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2010
    • (edited Sep 13th 2010)

    I still need to go over the part that starts talking about connections.

    In P 2(X)\mathbf{P}_2(X) the 2-morphisms are thin-homotopy-classes of surfaces, while in Π 2(X)\mathbf{\Pi}_2(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.)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2010
    • (edited Sep 13th 2010)

    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.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 14th 2010

    In ’Connections on principal 2-bundles’ I was expecting an imitation of the 1-story: (Π 1(X),BG)≃♭BG(\Pi_1(X),B G) \simeq \flat B G and (P 1(X),BG)≃BG conn(X)(P_1(X), B G) \simeq B G _{conn(X)}. Why then

    [CartSp op,Grpd](P 2(X),BG)≃♭BG[CartSp^op,Grpd](P_2(X), B G) \simeq \flat B G?

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2010
    • (edited Sep 14th 2010)

    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…

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2010

    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. )

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2010

    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.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2010
    • (edited Sep 14th 2010)

    I have now gone through the section

    Curvature characteristics of 1-bundles

    and its first subsection

    Of U(1)-principal bundles.

    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….

    • CommentRowNumber16.
    • CommentAuthorEric
    • CommentTimeSep 15th 2010

    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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2010
    • (edited Sep 15th 2010)

    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.

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 15th 2010

    @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, ℝ n→∼U↪M\mathbb{R}^n \stackrel{\sim}{\to} U \hookrightarrow 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.

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 15th 2010

    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\flat B B A?

    If it is necessarily flat when lifting in the nonabelian case to ♭BG\flat B G, isn’t G=BAG = B A just a special case?

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2010
    • (edited Sep 15th 2010)

    Why

    That’s a tyop! fixed it now. If should be [P 2(−),B 2A][\mathbf{P}_2(-), \mathbf{B}^2 A] instead of ♭B 2A=[Π 2(−),B 2A]\mathbf{\flat}\mathbf{B}^2 A= [\mathbf{\Pi}_2(-), \mathbf{B}^2 A] there.

    • CommentRowNumber21.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 15th 2010
    • (edited Sep 15th 2010)

    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.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2010
    • (edited Sep 15th 2010)

    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.

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 15th 2010

    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?

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2010
    • (edited Sep 15th 2010)

    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 nn-connection is an anafunctor out of P n(X)\mathbf{P}_n(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 nn-connections.

    Because, if you take a general 2-group GG and look at 2-anafunctors P 2(X)→BG\mathbf{P}_2(X) \to \mathbf{B}G 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.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2010

    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 GG. The reason is that for general GG 2-anafunctors out of P 2(X)\mathbf{P}_2(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 GG of the form BA\mathbf{B}A does the 2-form curvature necessarily vanish anyway, so that in this case the definition by morphisms out of P 2(X)\mathbf{P}_2(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 ∞\infty-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 ∞\infty-connections.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2010

    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.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2010
    • (edited Sep 19th 2010)

    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..)

    • CommentRowNumber28.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 20th 2010

    P 2(X)P_2(X) the fundamental path 2-groupoid (2-morphisms are thin homotopy-classes of disks). Π 2(X)\Pi_2(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?

    • CommentRowNumber29.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 20th 2010

    This needs fixing:

    Moreover, we have a natural equivalence of bicategories

    [CartSp op,2Grpd](P 2(C(U),[P 2(−),BG])≃U(1)Gerb ∇(X), [CartSp^{op}, 2Grpd](\mathbf{P}_2(C(U), [\mathbf{P}_2(-), \mathbf{B}G]) \simeq U(1) Gerb_\nabla(X) \,,

    So, BG\mathbf{B}G should be B 2U(1)\mathbf{B}^2 U(1), and that first P 2\mathbf{P}_2 shouldn’t be there?

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2010
    • (edited Sep 20th 2010)

    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\mathbf{B}G should be B 2U(1)\mathbf{B}^2 U(1), and that first P 2\mathbf{P}_2 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!!

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2010

    I reordered and polished a bit the material in Summary – oo-Lie algebra-valued connections

  32. Hi. In case anyone (but Urs) is wondering, I’m here, working in the background :)

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2010
    • (edited Sep 27th 2010)

    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 “∞\infty-connections from Lie integration”.

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2011

    have been going through infinity-Chern-Weil theory the last days and polished plenty of minor things. Turned it all into LaTeX, too. (phew)

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeMar 14th 2011

    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.

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2011
    • (edited Mar 23rd 2011)

    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.

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2011
    • (edited Mar 23rd 2011)

    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.

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2011
    • (edited Sep 16th 2011)

    In reaction to a question by Anton Kapustin, I have considerably expanded the infinity-Chern-Weil theory introduction … but currently not on the nnLab but in my pdf-version at differential cohomology in a cohesive topos (schreiber):

    • section 1.3.1 Principal n-bundles for low nn , 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.

  39. Hi Urs,

    somehow related to this, which is a path object for H(X,B nU(1) conn)\mathbf{H}(X,\mathbf{B}^n U(1)_{conn}), where XX 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× Z hYX\times^h_Z Y to Z IZ^I, isn’t it?

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2011

    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.

  41. Hi Urs,

    at Cech cohomology there’s only the version with B nU(1)\mathbf{B}^n U(1)-coefficients, and not the refined one with B nU(1) conn\mathbf{B}^n U(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,BG 1,conn) ↓ H(X,BG 2,conn) → H(X,B nU(1) conn) \begin{aligned} &&\mathbf{H}(X,\mathbf{B}G_{1,conn})\\ &&\downarrow\\ \mathbf{H}(X,\mathbf{B}G_{2,conn})&\to&\mathbf{H}(X,\mathbf{B}^n U(1)_{conn}) \end{aligned}

    where BG 1,conn→B nU(1) conn\mathbf{B}G_{1,conn}\to \mathbf{B}^n U(1)_{conn} and BG 2,conn→B nU(1) conn\mathbf{B}G_{2,conn}\to \mathbf{B}^n U(1)_{conn} are two differential cocycles. behind the scenes we are doing this by taking the (ordinary) limit with a fibrant replacement of BG 2,conn→B nU(1) conn\mathbf{B}G_{2,conn}\to \mathbf{B}^n U(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,B nU(1) conn)\mathbf{H}(X,\mathbf{B}^n U(1)_{conn}).

    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2011

    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.

  43. 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.

1 to 43 of 43

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