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    • CommentRowNumber101.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 4th 2010
    #96 Urs


    G itself comes from an infty-Lie algebra cocycle.

    What does this mean? is G even a group here?
    • CommentRowNumber102.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 4th 2010
    #96 Urs
    BUT, for this construction, since it no longer works over Lie integraion,

    why not?

    and does Lie integration mean, as usual, thre exist a group such that..

    I no longer know currently how to prove that this still models Chern-Weil, i.e. that the evident composite



    BG←≃BG diff→B nU(1) diff→♭ dRB n+1U(1)

    still is a model for the curvature characteristic class we set out to model.

    Pardon my naivite' - what's missing for this to go through?

    From the looks of it it likely is. But I don't have control over this construction currently.
  1. @Jim #100

    Is there any way to print out e.g. 94 - 99 without 1-93?

    don’t think so, but actually don’t know.

    In the next version, would it be possible to recall the definitions of all these variants on BG\mathbf{B}G? though with the clear description of the morphisms between them, I guess it can be filled in

    sure. nForum is mosly a discussion place: clean version will appear (and are appeaing) on nLab

    Is there some (psychological?) implication to using H

    HH is a reminder of “cohomology”.

  2. and here is the (homotopy, of course) commutative diagram inducing the natural morphism B diffA dRB 2A\mathbf{B}_{diff}A\to\flat_{dR}\mathbf{B}^2 A:

    B diffA BEA B 2A BA BEA * B 2A \array{ \mathbf{B}_{diff}A &\to& \flat\mathbf{B}\mathbf{E}A&\to&\flat\mathbf{B}^2 A \\ \downarrow && \downarrow&&\downarrow \\ \mathbf{B}A &\to& \mathbf{B}\mathbf{E}A&& \\ \downarrow&&&\searrow&\downarrow \\ *&\to&\to&\to&\mathbf{B}^2 A }
    • CommentRowNumber105.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 5th 2010

    Jim,

    Is there some (psychological?) implication to using H

    HH is a reminder of “cohomology”.

    generally Urs seems to use H(a,b)H(a,b) for the generic hom-space between aa and bb in the (,1)(\infty,1)-cat context, with the understanding that all cohomology is just connected components for some hom-space. Zoran disagrees, having a large number of nonabelian models in mind, but that remains to be worked out.

    But Urs will have to confirm this.

    • CommentRowNumber106.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 5th 2010

    Jim,

    I’ve sent you via email a pdf file of just the comments you were interested in.

    • CommentRowNumber107.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 5th 2010
    • (edited Sep 6th 2010)

    I’ve been thinking of pseudoconnections a bit more, trying to look at them from a classical perspective. Maybe things can be seen as follows: in Chern-Weil theory we do not only want to map a connection to a closed differential form (this, in a sense, would be easy), but want to do that in a way such that two different connections give two closed forms which differ by an exact one (and this is difficult). But then the idea is that, since connections are an affine space for 1-forms, we could look instead of the set of connections to the groupoid of connections under the action of 1-forms. Actually this is only a rough approximation: connections are already a groupoid under gauge transformations, so what we are really willing to do is to “add” the action of 1-form to the action of gauge transformations. Doing this we move from the groupoid of connections to the groupoid of pseudoconnections.

    Now that I write it, I see that a more natural thing to do (or at least more in the spirit of Chern-Simons) would be to consider paths of connections as morphisms between connections. Then in particular any 1-form ω\omega defines a path A+tωA+t\omega which is the one appearing in the classical Chern-Simons formula.

    But once I said “paths”, I’m naturally led to oo-paths, so I would now think of a pseudoconnection as the local datum of connections, paths between connections on the double intersections, 2-simplices of connections on the triple intersections,… and by the “every connection is flat” principle, this is the smplicial object of singular simplices in Maurer-Cartan elements of 𝔦𝔫𝔫(𝔤)\mathfrak{inn}(\mathfrak{g})-valued differential forms, or something closely related to that. And this is in turn the Hinitch nerve construction.

    edit: I see the above is discussed at Chern-Simons form

    • CommentRowNumber108.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2010
    • (edited Sep 6th 2010)

    Urs seems to use H(a,b)H(a,b) for the generic hom-space between aa and bb in the (,1)(\infty,1)-cat context,

    More precisely, I like to call my (,1)(\infty,1)-topos H\mathbf{H} since then the hom-\infty-groupoid H(X,A)\mathbf{H}(X,A) is indeed the “fat” version of the cohomology set H(X,A)H(X,A).

    with the understanding that all cohomology is just connected components for some hom-space.

    Yes.

    Zoran disagrees, having a large number of nonabelian models in mind, but that remains to be worked out.

    First of all, in the contexts that we are talking about here it is simply true that cohomology is just hom-spaces of oo-stacks.

    What Zoran did disagree with is something that does not affect the discussion in this thread here, namely the assertion that in fact essentially every definition of cohomology in every other context is of this form, too. If we continue discussing that, I suggest we do it for instance in this thread in order not to highjack tis one here.

    I have to say that I don’t think I have seen yet real counterexamples to the bigger claim. What Zoran did list in that thread were examples of categories with weak equivalences of chain complexes where the hom-spaces are hard to compute in that injective objcts are missing.

    • CommentRowNumber109.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2010
    • (edited Sep 6th 2010)

    And this is in turn the Hinitch nerve construction.

    Yes. Or maybe conversely, we get what I think of the central statement in \infty-Chern-Weil theory:

    We may integrate a Lie algebra or \infty-Lie algebra by looking at the simplicial object

    (U,[n]){C (U)Ω (Δ n)CE(𝔤)}. (U,[n]) \mapsto \{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) \leftarrow CE(\mathfrak{g}) \} \,.

    This describes paths in the space of flat “vertical” (not along UU) 𝔤\mathfrak{g}-valued forms.

    Now we may thicken this to the simplicial object

    (U,[n]){C (U)Ω (Δ n) CE(𝔤) Ω (U)Ω (Δ n) W(𝔤)}. (U,[n]) \mapsto \left\{ \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(\mathfrak{g}) } \right\} \,.

    This now describes paths of 𝔤\mathfrak{g}-valued forms, i.e. paths of local 𝔤\mathfrak{g}-connections. The \infty-Chern-Weil homomorphism is, up to the truncation issue, just the map that

    1. starts with an \infty-Lie algebra cocycle CE(𝔤)μCE(b k1)CE(\mathfrak{g}) \stackrel{\mu}{\leftarrow} CE(b^{k-1}\mathbb{R});

    2. picks an invariant polynomial and Chern-Simons element for it to extent it to a pasting diagram

      CE(𝔤) μ CE(b k1) 0 W(𝔤) (cs,P) W(b k1) P CE(b k) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) &\stackrel{}{\leftarrow}& 0 \\ \uparrow && \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs,P)}{\leftarrow}& W(b^{k-1}\mathbb{R}) &\stackrel{P}{\leftarrow}& CE(b^k \mathbb{R}) }
    3. and then considers the morphism on smplicial presheaves as above induced by composition with this one.

    edit: I see the above is discussed at Chern-Simons form

    Yes, there is a discussion, and a good example to keep in mind is also at Chern-Simons circle 3-bundle (which I now set out to further edit and polish)

    • CommentRowNumber110.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 7th 2010
    • (edited Sep 7th 2010)

    I’m trying to understand the cosk 3(exp𝔤)BGcosk_3(exp\mathfrak{g})\to\mathbf{B}G. does in a first very rough approximation looks like this description below?

    given a Lie algebra 𝔤\mathfrak{g}, there is a (infinite dimensional) simplicial manifold 𝔤\int\mathfrak{g} integrating it, which turns out to be a Kan complex (and so we can think of as an oo-groupoid exp(𝔤)\int\mathfrak{g$exp(\mathfrak{g})). If GG is a Lie group with Lie algebra GG, then we have natural morphism exp𝔤BGexp\mathfrak{g}\to \mathbf{B}G, i.e., a morphism of simplicial manifolds 𝔤N(BG)\int\mathfrak{g}\to N(\mathbf{B}G). This is not an equivalence, and the topology of GG obstructs lifting maps to BG\mathbf{B}G to maps to exp(𝔤)exp(\mathfrak{g}). However, if GG is nn-connected, we can lift from BG\mathbf{B}G to cosk n+1(exp𝔤)cosk_{n+1}(exp\mathfrak{g}).

    • CommentRowNumber111.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 7th 2010

    What Zoran did disagree with is something that does not affect the discussion in this thread here,

    I agree, and did not mean to distract the conversation. It was just background for Jim’s interest.

    • CommentRowNumber112.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2010
    • (edited Sep 7th 2010)

    Domenico,

    yes, that’s the idea. A good way to start thinking about this is to think about how cosk 2exp(𝔤)\mathbf{cosk}_2 \exp(\mathfrak{g}) is equivalent to BG\mathbf{B}G, because that turns out to be a classical statement about intergration of Lie algebras.

    A morphism in exp(𝔤)exp(\mathfrak{g}) is a 𝔤\mathfrak{g}-valued 1-form on the interval. By its parallel transport this defines an element in the group. The thing is, by the “nonabelian Stokes theorem” two such 1-forms gives the same group element, precisely if there is a flat 𝔤\mathfrak{g}-valued 1-form on the disk that restricts to A_1 on the upper half circle and to A_2 on the lower half circle.

    This classical result is reviewed in lots of detail in the referenc by Crainic and Fernandes that is linked to at Lie integration.

    Next we observe that we can also describe the 2-group \Omega G \to P Gthisway,andthisisstillequivalenttothis way, and this is still equivalent to G.Thisisgivenby. This is given by \mathbf{cosk}_3 \exp(\mathfrak{g})$.

  3. I’m in course of adding a few details to Chern-Simons circle 3-bundle. As usual I’ll first discuss here the additions before editing the nLab entry.

    i) typo: in the first diagram after “thickening the simplicial presheaf for BG\mathbf{B}G to” there’s a spurious μ\mu in the upper horizontal arrow.

    ii) notation μCE(𝔤)\mu \in CE(\mathfrak{g}) may be confusing to the reader who has not worked a bit with this stuff. I would add a short digression to say that for 𝔤\mathfrak{g} a Lie algebra, Hom dgca(CE(b k1,CE(𝔤)Hom_{dgca}(CE(b^{k-1} \mathbb{R},CE(\mathfrak{g}) is naturally identified with the set of d CE(𝔤)d_{CE(\mathfrak{g})}-closed degree kk elements in CE(𝔤)CE(\mathfrak{g}), i.e., with d CE(𝔤)d_{CE(\mathfrak{g})}-closed elements of k𝔤 *\wedge^k\mathfrak{g}^*; moreover since the differential d CE(𝔤)d_{CE(\mathfrak{g})} is nothing but the Lie bracket of 𝔤\mathfrak{g}, being d CE(𝔤)d_{CE(\mathfrak{g})}-closed means to be 𝔤\mathfrak{g}-invariant (for the (co-)adjoint action of 𝔤\mathfrak{g} on k𝔤 *\wedge^k \mathfrak{g}^*. Also, I would recall that such a closed element in k𝔤 *\wedge^k \mathfrak{g}^*, seen as a map k𝔤\wedge^k \mathfrak{g}\to \mathbb{R}, is precisely an L L_\infty-morphism 𝔤b k1\mathfrak{g}\to b^{k-1}\mathbb{R}. This adds nothing to the mathematical content, but has a nice intuitive effect: one sees that one can use μ\mu to change a flat 𝔤\mathfrak{g}-connection into a b k1b^{k-1}\mathbb{R}-flat connection.

    iii) same kind of considerations for W(𝔤)\langle-\rangle\in W(\mathfrak{g}) and csW(𝔤)cs\in W(\mathfrak{g}): I would add a short digression to recall what is the datum of a morphism W(b k1)W(𝔤)W(b^{k-1} \mathbb{R})\to W(\mathfrak{g}).

  4. iv) as I have already discussed with Urs, I don’t like very much the (A,F A)(A,F_A)-notation for morphisms stemming from a Weil algebra. I see the importance of stressing the presence of curvature, but I find this notation obscures the freeness property of the Weil algebra, and in the end makes it harder to read commutative diagarams. So, for instance, I would write only cscs in place of (cs,)(cs,\langle-\rangle).

    • CommentRowNumber115.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2010

    Hi Domenico,

    thanks for this! I removed the μ\mu. I am agnostic about the (A,F A)(A, F_A) versus AA. If you say the latter works better for the reader, then that’s good. For (cs,)(cs, \langle-\rangle) versus (cs)(cs) (same situation, different case) I would feel that it is important to remind the reader that the invariant polynomial is in the game, as the curvature of the cs-element, but if you say this is not so, I am fine with that.

    Concerning digressions: of course I am happy if you have the time to add these. Personally I feel that I don’t have the time to repeat everything that I have already typed in the dedicated entries. But of course I see that the reader may appreciate some information collected from different entries. If you have time and energy to add the digressions, I’d very much appreciate it!

    • CommentRowNumber116.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 7th 2010
    • (edited Sep 7th 2010)

    I would feel that it is important to remind the reader that the invariant polynomial is in the game

    absolutely. but that is not intrinsic to W(𝔤)W(\mathfrak{g}): it is an additional information involving inv(𝔤)inv(\mathfrak{g}). so I’d like to say this explicitely. anyway I have to think better to this, so I will leave the notation (cs,)(cs,\langle-\rangle) for the time being and will add only the digression on Lie algebra cocycles (which I’ll do in a minute).

    edit: in the end I decided for a minimal modification, just writing \langle-\rangle, μ\mu and cscs as morphisms instead of as elements. the other details on Lie algebra cocycles seemed out of their place there, instead I decide to add Lie algebra cocycles as an example in the entry on L_oo algebras.

    a major edit I would do is adding a bottom horizontal line on the commutative diagram in Chern-Simons circle 3-bundle I’m working around, with inv(𝔤)CE(b k)inv(\mathfrak{g})\stackrel{\langle-\rangle}{\leftarrow}CE(b^{k}\mathbb{R}), while leaving only cscs on the arrow W(𝔤)csW(b k1)W(\mathfrak{g})\stackrel{cs}{\leftarrow}W(b^{k-1}\mathbb{R}). but since the diagram would not fit neatly with “and then postcomposing with that”, I’ve left it unchangend for the time being.

    • CommentRowNumber117.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    at circle n-bundle with connection, the arrows C (;U(1))d dRΩ 1()C^\infty(-;U(1))\stackrel{d_{dR}}{\to}\Omega^1(-) should be C (;U(1))d dRlogΩ 1()C^\infty(-;U(1))\stackrel{d_{dR} log}{\to}\Omega^1(-). (or am I lost?)

    edit: also after “So by the above definition of differential cohomology…” the curvature morphism B diffU(1) dRBU(1)\mathbf{B}_{diff} U(1)\to \flat_{dR}\mathbf{B}U(1) seems to be written as B diffU(1)B dRU(1)\mathbf{B}_{diff} U(1)\to \mathbf{B}_{dR} U(1). Since this could be a misunderstanding of mine, I’m not correcting it, yet.

    • CommentRowNumber118.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    concerning the label of these morphisms:

    Yes, you are right. But I was being lazy. But also, If we think of U(1)U(1) as /\mathbb{R}/\mathbb{Z} then writing just d dRd_{dR} is actually more true to what happens than dlogd log, to my mind. But dlogd log is certainly the established convention.

    Concerning that typo: from the source code of your comment I think you are right that this is a typo. but I cannot currently find this in the entry! Maybe I am looking at the wrong entry.

    • CommentRowNumber119.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    concerning d dRd_{dR} vs. dlogd log I agree the best solution is a one-line remark, which I’m going to add. Concerning the missing \flat, the entry is circle n-bundle with connection. I’ve now corrected it.

    • CommentRowNumber120.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    as a trivial exercise in the definitions, I’m working out what the prestacks exp(b 2)exp(b^2\mathbb{R}) and exp(b 2) diffexp(b^2\mathbb{R})_{diff} are. By dimensional reasons it seems that for n=0,1,2,3n=0,1,2,3 we just have

    exp(b 2):(U,[n])0 exp(b^2\mathbb{R}): (U,[n])\mapsto 0 for n=0,1,2n=0,1,2

    exp(b 2):(U,[3])C (U)Ω 3(Δ 3) exp(b^2\mathbb{R}): (U,[3])\mapsto C^\infty(U)\otimes\Omega^3(\Delta^3)

    and

    exp(b 2) diff:(U,[n])Ω 3(U×Δ n) exp(b^2\mathbb{R})_{diff}:(U,[n])\mapsto \Omega^3(U\times \Delta^n)

    Is this correct?

    • CommentRowNumber121.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    concerning d dRd_{dR} vs. dlogd log I agree the best solution is a one-line remark, which I’m going to add.

    Thanks! Looks good.

    Concerning the missing \flat, the entry is circle n-bundle with connection. I’ve now corrected it.

    Ah, now I see it from the diff-functionality of the page. Yes, thanks, now it’s correct.

    I’m working out what the prestacks exp(b 2)\exp(b^2 \mathbb{R}) and exp(b 2) diff\exp(b^2 \mathbb{R})_{diff} are

    Okay. By the way, these are actually stacks (satisfy descent) on the site CartSpCartSp.! That’s part of the point of using this “small” site instead of all of DiffDiff. This restriction makes more objects fibrant. (It also makes less objects cofibrant, but we have good control over the cofibrant objects, since we know these come from good covers and so this is perfect for applying methods of Cech cohomology).

    By dimensional reasons it seems that for n=0,1,2,3n=0,1,2,3 we just have

    exp(b 2):(U,[n])0exp(b^2\mathbb{R}): (U,[n])\mapsto 0 for n=0,1,2n=0,1,2

    exp(b 2):(U,[3])C (U)Ω 3(Δ 3)exp(b^2\mathbb{R}): (U,[3])\mapsto C^\infty(U)\otimes\Omega^3(\Delta^3)

    Yes. But it is important to go at least one degree higher and also look at

    exp(b 2):(U,[4])C (U)Ω closed 3(Δ 4)exp(b^2\mathbb{R}): (U,[4])\mapsto C^\infty(U)\otimes\Omega^3_{closed}(\Delta^4)

    where it is important that it is closed 3-forms on the 4-simplex. That is what makes the 3-morphisms compose as they should, because a 3-form on S 3S^3 may be extended to a closed 3-form on D 3D^3 precisely if its integral vanishes. The integral of the 3-form is the actual real number it represents under the equivalence

    Δ :exp(b 2)B 3. \int_{\Delta^\bullet} : \exp(b^2 \mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^3 \mathbb{R} \,.

    and exp(b 2) diff:(U,[n])Ω 3(U×Δ n)exp(b^2\mathbb{R})_{diff}:(U,[n])\mapsto \Omega^3(U\times \Delta^n)

    This is not quite correct: on the right we have the subset of Ω 3(U×Δ n)\Omega^3(U \times \Delta^n) of precisely those forms that have the property that the component of their 4-form curvature which sits in C (U)Ω (Δ n)C^\infty(U)\otimes \Omega^\bullet(\Delta^n) (all “legs” in the simplicial direction) vanishes.

    That’s precisely what the cmmutativity of this diagram

    C (U)Ω (Δ n) CE(b 2) Ω (U)Ω (Δ n) W(b 2) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& CE(b^2 \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(b^2 \mathbb{R}) }

    asserts. What you wrote down is the set of morphisms in the bottom of the diagram. What we want is just the subset that fits into such a diagram.

    You see, this condition ensures that underlying this “pseudo-connection” is indeed the cocycle for a B 3\mathbf{B}^3 \mathbb{R}-bundle (or B 3/\mathbf{B}^3 \mathbb{R}/\mathbb{Z}-bundle) which, as just mentioned above, needs to have closed 3-forms on the simplices.

    We should add this kind of dicussion as examples to Lie integration.

    • CommentRowNumber122.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    What you wrote down is the set of morphisms in the bottom of the diagram

    yes, but it seems to me that for n=0,1,2n=0,1,2 the upper horizontal arrow and the leftmost vertical arrow are zero, while for n=3n=3 one only needs that the 3-form component in C (U)Ω 3(Δ 3)C^\infty(U)\otimes\Omega^3(\Delta^3) is closed, but this is trivially true for dimensional reasons. This is no more true for higher nn’s and there one actually finds a subset of what I write. and as you remarked, in what I was considering I was missing the n=4n=4 step.

    • CommentRowNumber123.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    Yes, right.

    I am working on Lie integration now, planning to write out more examples there. But some of them are at Lie infinity-groupoid already, as you know.

  5. At Chern-Simons circle 3-bundle which is the role of the invariant polynomial \langle-\rangle? I find it is a beautiful fact that the Killing form of 𝔤\mathfrak{g} fits the "big diagram" involving also inv(𝔤)inv(\mathfrak{g}), and calling it in shows how Killing form, Chern-Simons term and the Lie algebra 3-cocycle are interrelated, which is an important point. Indeed, in a sense, everything starts with the Killing form: as it is written somewhere in the nLab, the other two terms follow from the acyclicity of the Weil algebra. So I find it is important to stress the role played by the Killing form here. But after \langle-\rangle has played this role, we are only concerned with the upper part of the diagram, i.e., with μ\mu and cscs. And so (I know I’m repeating myself, but now I’m more convinced of what I’m saying) I would omit \langle-\rangle from the morphism stemming out of the Weil algebra. Note also that \langle-\rangle plays no role in the Cech-Deligne cocycle one obtains in the end, where only μ\mu and cscs appear; so omitting it from thenotations would make this manifest (clearly, it plays no explicit role: implicitly, as I said, everything is based on \langle-\rangle).

    So I would write something like:

    thickening the Lie algebra cocycle by its Chern-Simons element

    CE(𝔤) μ CE(b k1) W(𝔤) cs W(b k1) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) }

    and then postcomposing with that. Note that the above diagram is part of a larger diagram involving the invariant polynomial \langle-\rangle for μ\mu and exhibiting the Chern-Simons element as a transgression element between these two:

    CE(𝔤) μ CE(b k1) W(𝔤) cs W(b k1) inv(𝔤) CE(b k) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& CE(b^{k} \mathbb{R}) }
    • CommentRowNumber125.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    Yes, I agree with what you say. I am not sure, but in case you are waiting for me to approve of changes to the entry that you want to make to prettify it, you need wait no longer! :-)

    One way to think of the invariant polynomial is this: the invariant polynomial \langle -\rangle may be thought of (under Lie integration) as the curvature of the canonical nn-gerbe with connection on BGB G.

    One sees it more explicitly if we consider the “unrefined Chern-Weil” morphism

    BGB nU(1) dRB n+1U(1) \mathbf{B}G \to \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)

    that forgets the circle n-bundle and just remembers its curvature. This is where \langle -\rangle makes it’s appearance.

    In the cocycle for the circle bundle with connection it does not appear explicitly, because there it is implicitly encoded in the connections forms. (You know this well, I am just saying it for the record or maybe for some lurkers or maybe just because I have a tendency to ramble ;-)

    • CommentRowNumber126.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010
    • (edited Sep 10th 2010)

    I find it is a beautiful fact that the Killing form of 𝔤\mathfrak{g} fits the “big diagram” involving also inv(𝔤)inv(\mathfrak{g})

    By the way, I quite share your enthusiasm for that double-square diagram

    C (U)Ω (Δ n) A vert CE(𝔤) μ CE(b k1) Ω (U)Ω (Δ n) A W(𝔤) cs W(b k1) Ω (U)Ω (Δ n) F A inv(𝔤) CE(b k) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& CE(b^{k} \mathbb{R}) }

    The observation of the relevance and beauty of this diagram is what underlies my first and third article with Jim and Hisham. I was pretty thrilled by this in the beginning. Hisham keeps in his office a big dinA2-size version of this diagram that I drew, which encodes already in it all the seeds of the stuff about Chern-Simons n-bundles and differential String structures that we are talking about here.

    If you are brave and have a good connection, you can get a jpeg-scan of this diagram here. But beware, this is around 6MB and in order to view it you need to zoom around a bit.

    I am quite thrilled now you tell me you appreciate this diagram. My promotion of it so far didn’t quite have the success that it deserves. Of course I know part of the reason is that after writing this down and saying “hey, now we just need to wave the wand of Lie integration over this” it took me so long (until now) to work out and write the full theory to make that real. But now it finally is.

    edit: in the case where we are looking at ordinary connections instead of pseudo-connections, the bottom left bit of the above diagram looks as in the following diagram

    C (U)Ω (Δ n) A vert CE(𝔤) μ CE(b k1) Ω (U)Ω (Δ n) A W(𝔤) cs W(b k1) Ω (U)C (Δ n) F A inv(𝔤) CE(b k) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes C^\infty(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& CE(b^{k} \mathbb{R}) }

    if we allow the left bottom morphism to be just the gca-inclusion, not a dgca morphism.

    • CommentRowNumber127.
    • CommentAuthorEric
    • CommentTimeSep 8th 2010

    This sounds awesome. I hope I can understand it someday.

    • CommentRowNumber128.
    • CommentAuthorzskoda
    • CommentTimeSep 8th 2010

    My promotion of it so far didn’t quite have the success that it deserves.

    If analysed in peace of one’s room it is indeed really beautiful and insightful, but when presented in a lecture to somebody who did not think through it before, it is dreadful, complicated and incomprehensible; I hope you are aware now of this “duality”. I heard a dreadful impression upon your big diagram even from one of the greatest geometers of present day.

  6. you can get a jpeg-scan of this diagram

    Wow! That’s a diagram!

    Before working on Chern-Simons circle 3-bundle let me add a consideration on the big diagram: since the L L_\infty-algebra b k1b^{k-1}\mathbb{R} has trivial differential and brackets, every polynomial in the shifted generators in W(b k1)W(b^{k-1}\mathbb{R}) is D WD_W-closed, and so is an invariant polynomial. This says that inv(b k1)=CE(b k)inv(b^{k-1}\mathbb{R})=CE(b^k\mathbb{R}) and the big diagram above can be rewritten in the following more coherent form:

    C (U)Ω (Δ n) A vert CE(𝔤) μ CE(b k1) Ω (U)Ω (Δ n) A W(𝔤) cs W(b k1) Ω (U)Ω (Δ n) F A inv(𝔤) inv(b k1) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }

    Also, I see from this I’m not completely satisfied by our current definition of invariant polynomials. Namely W(𝔤)W(\mathfrak{g}) is a shorthand notation for CE(inn(𝔤)CE(inn(\mathfrak{g}). as a graded vector space, inn(𝔤)=𝔤𝔤[1]inn(\mathfrak{g})=\mathfrak{g}\oplus\mathfrak{g}[1], and the natural morphism W(𝔤)CE(𝔤)W(\mathfrak{g})\to CE(\mathfrak{g}) can be seen as the dgca counterpart of the natural L L_\infty-morphism (id,0):𝔤𝔤𝔤[1](id,0):\mathfrak{g}\to \mathfrak{g}\oplus\mathfrak{g}[1]. but then I would think of an L L_\infty-algebra strcucture on 𝔤[1]\mathfrak{g}[1] and of an L L_\infty-morphisms 𝔤𝔤[1]𝔤[1]\mathfrak{g}\oplus\mathfrak{g}[1]\to \mathfrak{g}[1] inducing CE(𝔤[1])W(𝔤)CE(\mathfrak{g}[1])\to W(\mathfrak{g}). As a vector space CE(𝔤[1])CE(\mathfrak{g}[1]) is precisely the subalgebra of W(𝔤)W(\mathfrak{g}) generated by the shifted generators, so the restriction to this part of W(𝔤)W(\mathfrak{g}) is natural from this perspective. It is not clear to me how the d Wd_W-closed condition appears, but something like this may be the case: assume the differential on CE(𝔤[1])CE(\mathfrak{g}[1]) is zero and that the dgca morphism CE(𝔤[1])W(𝔤)CE(\mathfrak{g}[1])\to W(\mathfrak{g}) takes values in the part of W(𝔤)W(\mathfrak{g}) generated by the shifted generators. Then compatibility with the differentials would force this image to consist of d Wd_W-closed elements. In other words, if everything works, inv(𝔤)inv(\mathfrak{g}) would be a handy realization of the more intrinsically defined CE(𝔤[1])CE(\mathfrak{g}[1]) as a subalgebra of W(𝔤)W(\mathfrak{g}), and this would justify the definition of inv(𝔤)inv(\mathfrak{g}) which at the present time seems a bit artificial.

    • CommentRowNumber130.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    That question you are addressiing now, Domenico, is one that I have pondered for a long time. I started asking around for this on the blog. More recently I posed this as a question here on the nForum. I still don’t have a fully satisfactory answer.

    More later..

  7. Hey, wait! maybe there’s a much simpler expanation: cosider the subalgebra 𝔤 *[1]\wedge^\bullet\mathfrak{g}^*[-1] of W(𝔤)W(\mathfrak{g}) generated by the shifted generators; the differential d Wd_W maps this algebra into itself, so ( 𝔤 *[1],d W)(\wedge^\bullet\mathfrak{g}^*[-1], d_W) is a dgca, and is CE(𝔤[1])CE(\mathfrak{g}[1]) for a certain natural L L_\infty-algebra structure on 𝔤[1]\mathfrak{g}[1]. Let me call this algebra Poly(𝔤)Poly(\mathfrak{g}) for the moment, waiting for a better name. Then Poly(𝔤)Poly(\mathfrak{g}) is not invariant polynomials, but a morphism Poly(b k1)Poly(𝔤)Poly(b^{k-1}\mathbb{R})\to Poly(\mathfrak{g}) picks up precisely a degree kk invariant polynomial! So we see that inv(𝔤)inv(\mathfrak{g}) naturally comes into play when the big diagram is involved.

    • CommentRowNumber132.
    • CommentAuthorzskoda
    • CommentTimeSep 8th 2010

    Regarding the characteristic classes, I would like to see the connection to the work of Alastair Hamilton and Andrey Lazarev on characteristic classes for strong homotopy algebras (they have versions for A-infinity, C-infinity and L-infinity algebras, though probably one could do these constructions for more general class of Koszul operads, and this is probably even well known by experts). I know that Domenico has communicated with Cattaneo about the graph homology and related issues which appear in this theory, maybe he could help (I just opened entry graph homology without any real content so far). I am very slow into getting into this as these days I work parallely very hard on certain descent problems using abstract localization theory.

  8. I guess I could actually try to give a hand with graph homology, but for me too this is a quite busy period, so my contribution there will be extremely slow and little for the time being. sorry for that. anyway, I’ll keep an eye on graph homology as the page grows.

    • CommentRowNumber134.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    Promted by the above discussion on those diagrams, I added a Summary-section oo-CW homomorphism – summary.

    • CommentRowNumber135.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010
    • (edited Sep 8th 2010)

    cosider the subalgebra 𝔤 *[1]\wedge^\bullet\mathfrak{g}^*[-1] of W(𝔤)W(\mathfrak{g}) generated by the shifted generators; the differential d Wd_W maps this algebra into itself,

    Actually, no, it does not. For instance for an ordinary Lie algebra with unshifted generators {t a}\{t^a\} and shifted generators {r a}\{r^a\} we have d Wr a=C a bct br cd_W r^a = C^a{}_{b c} t^b \wedge r^c.

    What is true is that d Wd_W stays in the kernel of the map W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}), which is the ideal generated by 𝔤 *[1]\mathfrak{g}^*[1].

    The subtlety is how to characterize in a useful intrinsic way inv(𝔤)inv(\mathfrak{g}) inside this kernel

    inv(𝔤)ker(i *)W(𝔤)i *CE(𝔤). inv(\mathfrak{g}) \hookrightarrow ker(i^*) \hookrightarrow W(\mathfrak{g}) \stackrel{i^*}{\to} CE(\mathfrak{g}) \,.

    One way is this: inv(𝔤)inv(\mathfrak{g}) are precisely the strong invariants in W(𝔤)W(\mathfrak{g}) of the canonical 𝔤\mathfrak{g}-action. Where by strong invariant I mean those element that are invariant under homotopies of diagrams of inner derivations

    CE(𝔤) [d,ι t] CE(𝔤) W(𝔤) [d W,ι t] W(𝔤). \array{ CE(\mathfrak{g}) &\stackrel{[d,\iota_t]}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{[d_W,\iota_t]}{\leftarrow}& W(\mathfrak{g}) } \,.

    That statement is already pretty good. But not good enough, because it is still a statement in a model for some intrinsic oo-statement. This is the infintiesimal analog of the statement that for GG an oo-group and EG\mathbf{E}G its universal principal oo-bundle (a notion that is the hallmark of being in a model!) we have that BG=(EG)/G\mathbf{B}G = (\mathbf{E}G)/G.

    What I don’t quite fully understand is what abstract intrinsic concept this quotient is actually a model for. In some sense we are looking at an extension of the fiber sequence

    G dRBGBGBG G \to \mathbf{\flat}_{dR}\mathbf{B}G \to \mathbf{\flat} \mathbf{B}G \to \mathbf{B}G

    one step further to the right.

    I have a guess what should go there. Namely the unit

    BG dRΠ dRBG \mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{\Pi}_{dR} \mathbf{B}G

    of the \infty-adjunction (Π dR dR)(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR}). I wasted two months or so thinking about this. But I still don’t have this under control.

    • CommentRowNumber136.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    Er, I didn’t say the last bit well. But too tired now. More tomorrow.

    • CommentRowNumber137.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2010

    Regarding the characteristic classes, I would like to see the connection to the work of Alastair Hamilton and Andrey Lazarev on characteristic classes for strong homotopy algebras

    I haven’t read that. Or is this the kind of thing we talked about by email recently? Otherwise, can you give me a rough idea of what it is these people are considering?

    • CommentRowNumber138.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 9th 2010
    • (edited Sep 9th 2010)

    Actually, no, it does not.

    Oh, right! in the formula for d Wd_W on the shifted generators the leftost shift acts as a derivation! I got confused on that! :(

    • CommentRowNumber139.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2010

    acts as a derivation! I got confused on that! :(

    Yes, right, that’s a pitfall with the notation. Would you know a way to improve the notation to make this clearer? I had wondered about this before, but couldn’t come up with a nice solution.

    • CommentRowNumber140.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    This is in response to 105 - 119
    #105 and #108 and several others

    My problem with H(X,A) as signifying cohomology is that
    calling it cohomology implies A is fixed and H( -, A) is a functor

    Hom(X,A) in a specified cat would mean morphisms X --> A

    but it is crucial to distinguish that Hom(X,A) from Hom(X,A)/~
    where the ~ gives homotopy classes


    #107 domenico_fiorenza

    when you want to map a connection to a closed differntial form such that..
    I find this a little misleading OF COuRSE you had in mind e.g. map to the curvature
    but it would help to say so

    pseudoconnection as the local datum - now that at last makes sense
    assuming you mean
    the local datum which would be a connenction if...

    #109 thicken is over used - try for some terminology that distinguishes various kinds
    you would NOT want to call an extension of groups as a thickening of one of them or would you?

    #112 Urs

    if cosk_2exp(g) is equivalent to BG why invoke it?

    #113 domenico_fiorenza

    in ii) I was at last able to see that you are talking about elements in certain
    Hom_dgca as \emph{picking out a particular cochain} - say so!


    #115 Urs

    If a cs-element is NOT a connection, why refer to its curvature?
    don't cut too many corners in your expo
    • CommentRowNumber141.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    #123

    It's probably too late but I wish you wouldn't use Lie integration to mean `anti-differentiation' - i.e. asserting the existence of or finding a group of whihc the Lie alg is what's given

    sometimes you really do mean integration in the sense of integrating something
    • CommentRowNumber142.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    #125
    the invariant polynomial as heh (greatly generalized) `curvature' of something - I can live with that but
    I don't see any Lie or other integration involved!

    #126 same comment in re: the beautiful diagram
    what would waving the wand over it produce

    #127 as to its lack of being appreciated, a picture may be worth a thousand words
    but a diagram could use a lot of words!

    a minor source of non-comprehension: the use on inv(g) rahter than the base of W(g)
    a major source: the right hand column which `just' picks out particular cochains/forms

    #129 W(g) is shorthand for CE(inn(g)) - no, CE(inn(g)) is a fancy name for the beautifully simple W(g)

    also notice the potential typos lurking in inn(g) vs inv(g)
    invariance should be part of a result - not adefintion'
    '
    #130 what question are you referring to?

    #131 picks up precisely - YES

    #132 by Zoran - yes, characteristic classes for strong homotopy algebras is related how to characteristic classes for bundles?

    That brings me up to #137
    IF NO ONE ELSE WEIGHS IN, I'LL DO A RECENT ARXIV SEARCH
    • CommentRowNumber143.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    #137
    and how about Characteristic classes of Q-bundles
    cf. Kotov and Strobl back in 2007 and i assume sequels
    • CommentRowNumber144.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2010

    Here a list of quick replies to Jim:

    My problem with H(X,A) as signifying cohomology is that calling it cohomology implies A is fixed and H( -, A) is a functor

    Yes, and that’s even true. Also for XX fixed, H(X,)H(X,-) is a functor. That’s because H(,)\mathbf{H}(-,-) is actually the hom-functor in an \infty-topos.

    but it is crucial to distinguish that Hom(X,A) from Hom(X,A)/~ where the ~ gives homotopy classes

    Yes, that’s why we write boldface H(X,A)\mathbf{H}(X,A) for the \infty-groupoid of cocycles and H(X,A)H(X,A) for its set of connected components.

    pseudoconnection as the local datum - now that at last makes sense

    By the way, yesterday I startedpseudo-connection.

    if cosk_2exp(g) is equivalent to BG why invoke it?

    Because it’s the natural functorial operation to achieve what it is supposed to achieve. It’s a main part of the whole game being played here that there may be many equivalent incarnations of a given abstract entity.

    It’s probably too late but I wish you wouldn’t use Lie integration to mean ‘anti-differentiation’ - i.e. asserting the existence of or finding a group of whihc the Lie alg is what’s given sometimes you really do mean integration in the sense of integrating something

    In which way is “anti-differentiation” not a synonym for “integration”? And in fact, the ordinary integration of a differential form is precisely a special case of “Lie integration”, namely the weak equivalence Δ :exp(b k)B k+1\int_{\Delta^\bullet} : \exp(b^k \mathbb{R}) \to \mathbf{B}^{k+1}\mathbb{R}.

    in re: the beautiful diagram what would waving the wand over it produce

    It produces from the local form data Cech-Deligne cocycles. For the first nontrivial case, this is discussed in detail at Chern-Simons circle 3-bundle.

    what question are you referring to?

    The intrinsic, \infty-categorical characterization of inv(𝔤)inv(\mathfrak{g})

    and how about Characteristic classes of Q-bundles

    A “QQ-bundle” is what here we are calling an infinity-Lie algebroid 𝔞\mathfrak{a}. So its characteristic classes are what here we are calling the invariant polynomials inv(𝔞)inv(\mathfrak{a}).

    • CommentRowNumber145.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 9th 2010

    @Jim,

    pseudoconnection as the local datum - now that at last makes sense assuming you mean the local datum which would be a connenction if…

    Yes! And I’m starting to think that as a connection can be seen as a certain sort of functor, a pseudoconnection may actually be a pseudofunctor - we have compatibility up to some terms, which themselves satisfy some equation. I’m thinking even of the homotopy coherent functors from your paper with James Wirth, but in a slightly different guise.

    • CommentRowNumber146.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    #145 David

    pseudoconnection may actually be a pseudofunctor - we have compatibility up to some terms, which themselves satisfy some equation. I'm thinking even of the homotopy coherent functors from your paper with James Wirth, but in a slightly different guise.

    that would be fine
    but in other places a pseudoconnection seems to be a collection with no constraints at all
    and indeed homotopy coherence as in Wirth or only up to a point (cd homotopy reps of Abad et al)
    fits just fine
    • CommentRowNumber147.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    #134 by Urs

    You write as if the invariant polynomial came first, giving rise in the transgressive manner
    to a cocycle in CE(g)

    although G and BG are in some sense equivalent
    surely G comes first, at least historically and in accessibility to a new audience

    I've already inveighed against treating inv(g) as the base of W(g)
    would you ever write a principal bundle as G \to P \to H(B)?

    in re: H(X,A) it's very dangerous to rely on a change in font as simple as plain to bold
    to distinguish cochains for cohomology

    you sem to have missed my point that Hom(X,A)/~ can be regarded as cohomology for fixed A
    or homotopy (cf homtopy groups) for fixed X

    If you won't settle for [X,A] to mean homotopy classe of maps X \to A,
    how about the exisiting usuage \pi(X,A) which doesn't play favorites
    • CommentRowNumber148.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    #139 \sigma as a derivation
    the notation is ok as long as you use words such as
    the derivation \sigma
    or
    \sigma extended as a derivation
    but if you want the notation to do it
    how about: let d_\sigma denote \sigma extended as a derivation
    • CommentRowNumber149.
    • CommentAuthorzskoda
    • CommentTimeSep 9th 2010
    • (edited Sep 9th 2010)
    #137 Right, the thing from Hamilton-Lazarev we talked via internet. I will spend few hours today on that topic and write some comments in nlab to start, and maybe I open a new thread in nForum.
    • CommentRowNumber150.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 9th 2010
    • (edited Sep 9th 2010)

    @Urs

    The intrinsic, oo-categorical characterization of inv(𝔤)inv(\mathfrak{g})

    Indeed that seems to be a crucial issue here. For the time being I will content myself of thinking that since inv(𝔤)inv(\mathfrak{g}) is intended to model the de Rahm algebra of BGBG, it can be seen as the subalgebra of Ω (EG)\Omega^\bullet(EG) consisting of horizontal equivariant forms. This suggests teh two algebraic conditios we presently have: using W(𝔤)W(\mathfrak{g}) as a model for Ω (EG)\Omega^\bullet(EG), horizontality should correspond to being a polynomial in the shifted generators, while equivariance should be encoded in being d Wd_W-closed. This is not an intrinsic description, but I find it motivates the definition we currently have, so in case of no objections, I would add this to the idea section in invariant polynomial.

    by the way, I’m goinf to add the big diagram to Chern-Simons circle 3-bundle, as discussed yesterday.

    @Jim,

    you had in mind e.g. map to the curvature but it would help to say so

    yes. there I intended to stress that one has to see the whole problem: not only obtaining a closed differential form out of a connection, but doing so in such a way that the cohomology class one gets does not depend on the particular connection chosen.

    in ii) I was at last able to see that you are talking about elements in certain Hom_dgca as picking out a particular cochain - say so!

    yes. In a sense it is the old story that giving an element in a vector space VV is the same thing as giving a morphism 𝕂V\mathbb{K}\to V, just a bit refined to the dgca context. You’re right, we shoud say this more clearly somewhere.

    W(g) is shorthand for CE(inn(g)) - no, CE(inn(g)) is a fancy name for the beautifully simple W(g)

    I can see your point..

    • CommentRowNumber151.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2010

    so in case of no objections

    No objections from me. Thanks for taking care of that!

    (I think a discussion to that extent was already there in a entry in a previous version. Apparently it got overwritten.)

    • CommentRowNumber152.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    • (edited Sep 9th 2010)
    #150 Urs wrote:

    For the time being I will content myself of thinking that since inv(
    • CommentRowNumber153.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    #150 and 152

    I tried to edit my incomplete #152 but no can do

    so instead here is 153


    Urs wrote:
    For the time being I will content myself of thinking that since inv(
    • CommentRowNumber154.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 9th 2010
    and even that doesn't work
    something wrong in that begining partial sentence blocks any additions

    so look back to Urs in #150 to make sense of this

    intended to --- but not defined as

    de Rham *cohomology* algebra - NOT form algebra

    note that Chern Weil identifies the basic forms in EG which happen to be cocycles
    and hence also can be identified with cohomology classes

    It would be worth looking at this stuff in the context of H Cartan's mathfrak g -algebras

    what kind of intrinsic would you like with that?
    • CommentRowNumber155.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    de Rham cohomology algebra - NOT form algebra

    Isn’t BGB G, for GG a compact Lie group a formal space? I thought so, and so in my mind a model for Ω (BG)\Omega^\bullet(B G) was the same as a model for H (BG)H^\bullet(B G) (as dg-algebras). But I see that since the differential in inv(𝔤)inv(\mathfrak{g}) is trivial, our current definition of inv(𝔤)inv(\mathfrak{g}) aims to pick H (BG)H^\bullet(B G) up to isomorphisms, and not up to quasi-isomorphism.

    And now that I write it, maybe it is precisely this fact that makes the definition non-intrinsic. Indeed, which is classically a Sullivan model for BGB G? I guess that’s what we should try to reproduce, since in the sequence inv(𝔤)W(𝔤)CE(𝔤)inv(\mathfrak{g})\to W(\mathfrak{g})\to CE(\mathfrak{g}) the two rightmost terms are Sullivan algebras, and so I would find it natural to have a Sullivan algebra also at the beginning of the sequence.

    • CommentRowNumber156.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    Hi Domenico,

    that’s spot-on.

    Yes, BGB G is a formal space, so Ω (BG)\Omega^\bullet(B G) is isomorphic to inv(𝔤)inv(\mathfrak{g}) in the homotopy category of \infty-Lie algebroids (i.e. of duals of dg-algebras). (We have this at formal dg-algebra).

    I have tried to ask around if there is anything known about formality or else of classifying spaces of higher Lie groups, but so far I didn’t find any answers.

    At some point I thought answering this question is pressing. Now I no longer think it is pressing, just that it would be useful. Because, as we discussed by email, for the purposesof oo-Chern-Weil theory for a given \infty-Lie algebra cocycle μ\mu one can use any element in W(𝔤)W(\mathfrak{g}) of the form d W(𝔤)μ^d_{W(\mathfrak{g})} \hat \mu for μ^| CE(𝔤)=μ\hat \mu|_{CE(\mathfrak{g})} = \mu in place of the corresponding invariant polynomial. The only thing that will change is that the curvarture characteristic forms induced by this are not globally defined forms, but more general sheaf cohomology cocycles with values in the de Rham complex. But intrinsically this does not matter, because all these are of course equivalent to globally defined forms (if we are over a manifold). So from this perspective, an invariant polynomial is just a particularly nice representative of something that is intrinsically defined, namely an element of ker(i *:W(𝔤)CE(𝔤))ker(i^* : W(\mathfrak{g}) \to CE(\mathfrak{g})).

    (Because that kernel is in fact a homotopy kernel, because i *i^* is a fibration).

    • CommentRowNumber157.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    Indeed, which is classically a Sullivan model for BG?

    But a semi-free dg-algebra with vanishing differential is (trivially) a Sullivan algebra! So inv(𝔤)=H (BG)inv(\mathfrak{g}) = H^\bullet(B G) is.

    • CommentRowNumber158.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010

    I wrote:

    The only thing that will change […]

    To say this in a more pronounced way:

    using an invariant polynomial in the \infty-Chern-Weil homomorphisms ensures that 𝔤\mathfrak{g}-connections are sent to b nb^n \mathbb{R}-connections.

    If we use instead a more general element in ker(i *)ker(i^*), then the CW homomorphism will in general send genuine 𝔤\mathfrak{g}-connections to b nb^n \mathbb{R}-pseudoconnections. Which are however guaranteed to be equivalent to some genuine b nb^n \mathbb{R}-connection.

    • CommentRowNumber159.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    But a semi-free dg-algebra with vanishing differential is (trivially) a Sullivan algebra! So inv(𝔤)=H (BG)inv(\mathfrak{g})=H^\bullet(BG) is.

    yes, I see I expessed myself very badly. what I meant was “how can we cook up a Sullivan model for BGB G starting with 𝔤\mathfrak{g}?” from this point of view, going from 𝔤\mathfrak{g} to GG and then to BGB G and taking cohomology would not be a good strategy, since adopting it presumes that we already know somehow that in the end we’ll find a polynomial algebra and that BGB G is a formal space.

    If we use instead a more general element in ker(i *)ker(i^*), then the CW homomorphism will in general send genuine 𝔤\mathfrak{g}-connections to b nb^n\mathbb{R}-pseudoconnections

    I should see this better. How does exactly the invariant polynomial manifest itself in the CW construction at Chern-Simons circle 3-bundle?

    • CommentRowNumber160.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    How does exactly the invariant polynomial manifest itself in the CW construction

    Here is the explanation. You might want to read this with a pen and a piece of paper in hand to draw the diagrams of diagrams of morphisms of dg-algebras that are involved in a nice fashion. Here in this comment I can only give a non-nice typetessing of these.

    Also, the software forces me to break this up into pieces. So here is the first piece:

    The starting point is the theorem that the morphism

    curv:B nU(1) dRB n+1U(1) curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)

    is modeled by a certain truncation and quotienting of the span of simplicial presheaves

    {C ()Ω (Δ )CE(b n1)}{C ()Ω (Δ ) CE(b n1) Ω ()Ω(Δ ) W(b n1)}{Ω ()Ω(Δ ) W(b n1)} \{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(b^{n-1} \mathbb{R}) \} \leftarrow \left\{ \array{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\} \to \left\{ \array{ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\}

    This means that for BGB nU(1)\mathbf{B}G \to \mathbf{B}^n U(1) a characteristic class arising by Lie integration of a cocycle μCE(𝔤)\mu \in CE(\mathfrak{g})

    {C ()Ω (Δ )CE(𝔤)}{C ()Ω (Δ )CE(b n1)} \{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(\mathfrak{g}) \} \to \{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(b^{n-1}) \}

    the total composite morphism

    BGB nU(1) dRB n+1U(1) \mathbf{B}G \to \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)

    is modeld by the zig-zag

    {C ()Ω (Δ )CE(𝔤)}{C ()Ω (Δ )CE(b n1)}{C ()Ω (Δ ) CE(b n1) Ω ()Ω(Δ ) W(b n1)}{Ω ()Ω(Δ ) W(b n1)}. \{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(\mathfrak{g}) \} \to \{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(b^{n-1} \mathbb{R}) \} \leftarrow \left\{ \array{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\} \to \left\{ \array{ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\} \,.

    For computing with this, we want to “iron out the zag in the zig-zag”. We have a zig-zag of the form

    C D A B \array{ && C &\to& D \\ && \downarrow^{\mathrlap{\simeq}} \\ A &\to& B }

    and want to complete it to a commuting diagram

    A^ C D A B \array{ \hat A &\to& C &\to& D \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ A &\to& B }

    because that will mean that then the outer span

    A^ C D A \array{ \hat A &\to& C &\to& D \\ \downarrow^{\mathrlap{\simeq}} \\ A }

    models the same \infty-morphism.

    To be continued in the next comment.

    • CommentRowNumber161.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    Here is the continuation of the above comment.

    Recall that we want to find an object A^\hat A that makes a zig-zag of morphisms just a span of morphisms. (We want to “compose anafunctors”.) We see that we can find A^\hat A in the case at hand by completing the diagram

    CE(𝔤) μ CE(b n1) W(b n1) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ && \uparrow \\ && W(b^{n-1}\mathbb{R}) }

    in any way such that the bottom left bit is contractible (so that A^A\hat A \to A indeed is a weak equivalence). An obvious solution starts with completing it as

    CE(𝔤) μ CE(b n1) W(𝔤) W(b n1). \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) && W(b^{n-1}\mathbb{R}) } \,.

    For producing the span that irons out our zig-zag, we can take any bottom morphism here that makes this diagram commute. This amounts to choosing any element μ^W(𝔤)\hat \mu \in W(\mathfrak{g}) of degree nn such that its restriction to CE(𝔤)CE(\mathfrak{g}) is μ\mu:

    CE(𝔤) μ CE(b n1) W(𝔤) (μ^,d W(𝔤)μ^) W(b n1). \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(\hat \mu, d_{W(\mathfrak{g})} \hat \mu)}{\leftarrow}& W(b^{n-1}\mathbb{R}) } \,.

    This alone is sufficient to model the morphism BGB nU(1) dRB n+1U(1)\mathbf{B}G \to \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1), postcomposition with which is the Chern-Weil homomorphism

    H(X,BG)H dR n+1(X). \mathbf{H}(X,\mathbf{B}G) \to \mathbf{H}_{dR}^{n+1}(X) \,.

    The above construction models this by a span

    [CartSp op,sSet](C(U),BG diff) [CartSp op,sSet](C(U),B n diff) [CartSp op,sSet](C(U), dRB n+1) [CartSp op,sSet](C(U),BG) \array{ [CartSp^{op}, sSet](C(U), \mathbf{B}G_{diff}) &\to& [CartSp^{op}, sSet](C(U), \mathbf{B}^n \mathbb{R}_{diff}) &\to& [CartSp^{op}, sSet](C(U), \mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} \\ [CartSp^{op}, sSet](C(U), \mathbf{B}G) }

    Here on the right we have the hypercohomology with coefficients in the de Rham complex. its cohomology is of course ordinary de Rham cohomology.

    We know that on the left we can always find representatives of cohomology classes in [CartSp op,sSet](C(U),BG diff)[CartSp^{op}, sSet](C(U), \mathbf{B}G_{diff}) which are pseudo-connections that are in fact genuine connections.

    So we can ask if we can find choice of μ^\hat \mu such that with the above model for this morphism genuine connections are sent to genuine connection, so that chasing them through this span lands us in cocycles in [CartSp op,sSet](C(U), dRB n+1)[CartSp^{op}, sSet](C(U), \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}) that are globally defined forms.

    This is precisely the case if the composite dg-algebra morphism

    Ω (U)Ω (Δ k)(A,F A)W(𝔤)(μ^,d W(𝔤)μ^)W(b n1)d W(𝔤)μ^CE(b n) \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) \stackrel{(A,F_A)}{\leftarrow} W(\mathfrak{g}) \stackrel{(\hat \mu, d_{W(\mathfrak{g})} \hat \mu)}{\leftarrow} W(b^{n-1}\mathbb{R}) \stackrel{d_{W(\mathfrak{g})} \hat \mu}{\leftarrow} CE(b^n \mathbb{R})

    is a n+1n+1-form on U×Δ kU \times \Delta^k with no leg along the simplex, i.e. if it lands in Ω (U)C (Δ k)\Omega^\bullet(U) \otimes C^\infty(\Delta^k).

    If we assume that AA is a genuine connection on a GG-bundle, then by definition the F AF_A do have this property. So we can ensure the above condition if we manage to find μ^\hat \mu such that d W(𝔤)μ^d_{W(\mathfrak{g})} \hat \mu ends up sitting just in the shifted copy, i.e. in 𝔤 *[1]\wedge^\bullet \mathfrak{g}^*[1].

    This is precisely the case if d W(𝔤)μ^W(𝔤)d_{W(\mathfrak{g})} \hat \mu \in W(\mathfrak{g}) is an invariant polynomial in the standard sense. So this is precisely the good choice of span AA^CA \leftarrow \hat A \to C that ensures that nice representatives (genuine connections) are sent to nice representatives (genuine connections).

    • CommentRowNumber162.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    #155 and #156
    yes, but... for such G, G and BG are formal, in fact, intrinsically formal.
    The Sullivan minimal models are respectively free graded commutative algebras
    on odd dim generators for G and even dim generators for BG, related by transgression
    but these are NOT the generators of CE(g + g[1])

    N.B. CE(g) is NOT a MINIMAL model for G - some people say Sullivan model meaning minimal,
    others don't distinguish

    that's what's been bothering me for so long about
    inv(
    • CommentRowNumber163.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 10th 2010

    that “inv(” seems to be causing problems…

    • CommentRowNumber164.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    #156 more

    Urs wrote: btw, how do you copy parts of previous to appear in a blue box?
    Ω•(BG) is isomorphic to inv(
  9. I guess the source of the problems is the gothic g in inv(g): Jim can you try replacing it with a simple “g”?

    to produce this just write a >\gt at the beginning of the line

    • CommentRowNumber166.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    #158 and 159
    Since b^n R is being used to pick out particular closed forms/cocycles,
    why are more general elements of ker only pseudo-connections?

    given a minimal Sullivan model for G, i.e. S(V) where S means free graded comm and V is
    a graded vector space concentrated in odd degree so S(V) is the tensor product of S(x_i)
    where x_i is odd, use that separately S(s x_i) is a model for the corresponding classifying space
    then reassemble to get S(sV) as candidate
    then `guess' the twisting cochain to make S(x_i)\otimes S(s x_i) acyclic
    • CommentRowNumber167.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    how do you copy parts of previous to appear in a blue box?

    This is achieved by beginning the paragraph with a greater-than sympol followed by a space. Typing

      > Hello world.
    

    produces

    Hello world.

    The yellow box, by the way, is obtained by beginning a paragraph with 6 spaces in a row. Inside such a box you can also display all extra symbols without them being interpreted as code. For instance you can do

      This is how to display math: enclose it in dollar signs: $x + y = 9$
    
    • CommentRowNumber168.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    #165
    aha! I was using cut and paste to write inv(g)
    so to continue my comments about what was bothering me
    namely that
    inv(g) \to W(g) \to CE(g) is NOT a fibration in the strong sense
    though inv(g) is quasiiso to the fibre
  10. @Urs, #161

    This is the part I don’t clearly see:

    This is precisely the case if the composite dg-algebra morphism

    Ω (U)Ω (Δ k)(A,F A)W(𝔤)(μ^,d W(𝔤)μ^)W(b n1)d W(𝔤)μ^CE(b n) \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) \stackrel{(A,F_A)}{\leftarrow} W(\mathfrak{g}) \stackrel{(\hat \mu, d_{W(\mathfrak{g})} \hat \mu)}{\leftarrow} W(b^{n-1}\mathbb{R}) \stackrel{d_{W(\mathfrak{g})} \hat \mu}{\leftarrow} CE(b^n \mathbb{R})

    is a n+1n+1-form on U×Δ kU \times \Delta^k with no leg along the simplex, i.e. if it lands in Ω (U)C (Δ k)\Omega^\bullet(U) \otimes C^\infty(\Delta^k).

    If we assume that AA is a genuine connection on a GG-bundle, then by definition the F AF_A do have this property. So we can ensure the above condition if we manage to find μ^\hat \mu such that d W(𝔤)μ^d_{W(\mathfrak{g})} \hat \mu ends up sitting just in the shifted copy, i.e. in 𝔤 *[1]\wedge^\bullet \mathfrak{g}^*[1].

    Indeed, deleting the spurious d W(𝔤)μ^d_{W(\mathfrak{g})} \hat \mu on the rightmost arrow, and erasing the curvature terms from the notation for morphisms stemming out of Weil algebras, the above is rewritten as:

    This is precisely the case if the composite dg-algebra morphism

    Ω (U)Ω (Δ k)AW(𝔤)μ^W(b n1)CE(b n) \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{\hat \mu}{\leftarrow} W(b^{n-1}\mathbb{R}) \leftarrow CE(b^n \mathbb{R})

    is a n+1n+1-form on U×Δ kU \times \Delta^k with no leg along the simplex, i.e. if it lands in Ω (U)C (Δ k)\Omega^\bullet(U) \otimes C^\infty(\Delta^k).

    which is fine. But then I’m unable to make any sense of the line that follows:

    If we assume that AA is a genuine connection on a GG-bundle, then by definition the F AF_A do have this property. So we can ensure the above condition if we manage to find μ^\hat \mu such that d W(𝔤)μ^d_{W(\mathfrak{g})} \hat \mu ends up sitting just in the shifted copy, i.e. in 𝔤 *[1]\wedge^\bullet \mathfrak{g}^*[1].

    • CommentRowNumber170.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    though inv(g) is quasiiso to the fibre

    I asked you this long time ago, and you said then that you didn’t know. Where is this written up?

    [edit: beind-the-scenes email exchange clarified this, Jim means by the fiber CE(g[1]) for g[1] regarded as an abelian dg-Lie algebra, which is different from what I am having in mind here]

    • CommentRowNumber171.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010

    Domenico,

    so if inv(𝔤)inv(\mathfrak{g}) is quasi-isomorphic to ker(i *:W(𝔤)CE(𝔤))ker(i ^* : W(\mathfrak{g}) \to CE(\mathfrak{g})) then that gives another explanation for my long comment above, at least as long 𝔤\mathfrak{g} is an ordinary Lie algebra.

    • CommentRowNumber172.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    Urs,

    if inv(𝔤)inv(\mathfrak{g}) is quasi-isomorphic to ker(i *:W(𝔤)CE(𝔤))ker(i^*:W(\mathfrak{g})\to CE(\mathfrak{g})), nothing else needs to be said! :)

    (I mean: then it is a nice model for the intrinsic object)

    • CommentRowNumber173.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    So we should try to figure out whether this is true for all L L_\infty-algebras. Dang it, I have asked myself this question for ages. If this turns out to have a simple proof, I’ll have to sit ashamed in the corner for a day or so ;-)

    • CommentRowNumber174.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    on further thought, knowing H(G), it is easy to compute H(BG)
    cf. Cartan constructions
    then observe the result is intrinsically formal
    • CommentRowNumber175.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    @Urs #170

    > though inv(g) is quasiiso to the fibre



    I asked you this long time ago, and you said then that you didn't know. Where is this written up?

    I must not have understood the question - rephrase it for me with full details: given .....
    prove inv(g) is quasiiso to the fibre

    and then with given... changed to ?? an arbitrary L_\infty algebra?

    how much of the following do you agree we know:

    CE(g[1]) --> W(g) --> CE(g) is a fibration of dgcas

    IF H(CE(g)) is a free gca, then so is H(CE(g[1]) )

    therefore we have a gca map and quism H(CE(g[1]) ) --> (CE(g[1])

    now we have to go inside to recognize H(CE(g[1]) ) as invariant polynomials



    ditto for the question
    • CommentRowNumber176.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    I must not have understood the question

    Yeah, I just checked my email to remind me. That was January 2008. Re-reading it now, I see that you probably intended to tell me that it is a homotopy fiber sequence. But I understood that you were asking me if that was my question. Then I asked you to tell me how we would check this. And then you said you didn’t know and forwarded that question to Johannes Huebschmann. Who also didn’t give the answer!

    I suppose it must have been a huge misunderstanding between us three. I am also thinking now this will have an easy answer and that I will hate myself for not having thought this through in the L \L_\infty-case years ago.

    how much of the following do you agree we know:

    CE(g[1]) –> W(g) –> CE(g) is a fibration of dgcas

    Let’s just get on the same page with terminology. W(g)CE(g)W(g) \to CE(g) is degreewise surjective, hence a fibration in the standard model structure on dgcas. Is that the meaning of “fibration” we are both talking about?

    Okay, and then what do you mean by CE(g[1])CE(g[1])? Is this that supposed to be the kernel of the map W(g)CE(G)W(g) \to CE(G)? In that case I would call this the ideal generated by g *[1]g^*[1] in W(g)W(g). Is that what you mean by CE(g[1])CE(g[1])?

    • CommentRowNumber177.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    > ditto for the question

    as supposed to continue: about Linfty algebras
    • CommentRowNumber178.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010

    now we have to go inside to recognize H(CE(g[1]) ) as invariant polynomials

    That’s the crucial step.

    • CommentRowNumber179.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    @Urs #176

    > Let's just get on the same page with terminology.
    W(g)→CE(g) is degreewise surjective, hence a fibration in the standard model structure on dgcas. Is that the meaning of "fibration" we are both talking about?

    I hope this > works - if not, tell me how else to cut and paste

    ah, that's the confusion - I NEVER think in terms of model structures if you mean the Quillenesque
    i.e. closed model cat

    I'm in rational homotopy theory following Sullivan

    a fibration is a twisted tensor product

    Okay, and then what do you mean by
    CE(g[1]). Is this that supposed to be the kernel of the map
    W(g)→CE(G)? In that case I would call this the ideal generated by
    g *[1] in
    W(g). Is that what you mean by
    CE(g[1])?

    almost - NOT the ideal but the sub dgca
    it will be the homotopy kernel or quasiiso to it but not the gca kernel
    but by CE(g[1]) i mean it literally regardeing g[1] as an abelian Lie algebra
    • CommentRowNumber180.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 10th 2010
    Notice in the classical case inv(g) means invariant under G for suitable G
    I think that may be a historical accident since G-invariant polys were already known?
    for general Linfty algebra L, invariance under L is straightforward, but what's G?
    • CommentRowNumber181.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    mmm.. and what about considering morphisms of Lie algebras 𝔤 1𝔤0\mathfrak{g}_{-1}\to \mathfrak{g}0 (the source of the morphism is “in degree -1”) as the starting point ? so that

    (𝔤0)𝔤[1] (\mathfrak{g}\to 0)\mapsto \mathfrak{g}[1]

    (𝔤id𝔤)inn(𝔤) (\mathfrak{g}\stackrel{id}\to \mathfrak{g})\mapsto inn(\mathfrak{g})

    (0𝔤)𝔤 (0\to\mathfrak{g})\mapsto \mathfrak{g}

    this way the obvious commutative diagram

    0 𝔤 𝔤 id 𝔤 𝔤 0 \array{ 0 &\to& \mathfrak{g}&\to&\mathfrak{g} \\ \downarrow && \downarrow_{id} &&\downarrow \\ \mathfrak{g} &\to& \mathfrak{g}&\to&0 }

    would naturally induce the desired 𝔤inn(𝔤)𝔤[1]\mathfrak{g}\to inn(\mathfrak{g})\to \mathfrak{g}[1].

    with Marco Manetti we considered a similar question in math.QA/0601312, but there the starting point was a pair of morphisms 𝔤 0𝔤 1\mathfrak{g}_0\stackrel{\to}{\to}\mathfrak{g}_1, seen as a tiny bit of a semicosimplicial object in dglas, whereas here we would rather be considering an augmented semicosimplicial object.

    • CommentRowNumber182.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    Domenico,

    concerning this remark:

    if […(this is the cae)..] nothing else needs to be said!

    It seems to me that one would still need the lengthy discussion that i posted above. But we’d be guaranteed that, in the words of this discussion, the nice choice of span would always exist.

  11. It seems to me that one would still need the lengthy discussion that i posted above

    yes, sure. I meant that one had no more to wonder why to use that definition for inv(𝔤)inv(\mathfrak{g}): the simple answer is: because it is an extremely nice model for ker(i *)ker(i^*).

    • CommentRowNumber184.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    I think we have been taking this to private email for the moment.

    But for the record:

    Jim was talking about “fiber” in a different sense than Domenico and I was, and I think it is easy to see that the quasi-iso that seemed to materialize in above comments cannot exist. But maybe one can say something about useful isos on generators of cohomology rings.

    • CommentRowNumber185.
    • CommentAuthorEric
    • CommentTimeSep 10th 2010

    how do you copy parts of previous to appear in a blue box?

    Jim, a good trick to see (and copy) what others have done is to click the “Source” link on the top right of each comment.

    This lets you even copy and paste the latex commands, diagrams, etc. Very useful. One of my favorite technical features of the n-Forum. I wish the n-Cafe had this feature.

    • CommentRowNumber186.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    Concerning what I write in #169, if the big diagram in #126 where

    C (U)Ω (Δ n) A vert CE(𝔤) μ CE(b k1) Ω (U)Ω (Δ n) A W(𝔤) cs W(b k1) Ω (U) F A inv(𝔤) inv(b k1) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }

    instead, then everything would be clear to me.

    • CommentRowNumber187.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010

    Right, so a genuine connection (generally: genuine \infty-connection) is characterized, in this language, precisely by the fact that its curvature forms have “no legs along the simplicial direction”.

    So the diagram we want is

    C (U)Ω (Δ n) A vert CE(𝔤) μ CE(b k1) Ω (U)Ω (Δ n) A W(𝔤) cs W(b k1) Ω (U)C (Δ n) F A inv(𝔤) inv(b k1) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U) \otimes C^\infty(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }

    where now the left bottom morphism is just the gca-inclusion, not a dgca-morphism.

    • CommentRowNumber188.
    • CommentAuthordomenico_fiorenza
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    I’d rather have

    C (U)Ω (Δ n) A vert CE(𝔤) μ CE(b k1) Ω (U)Ω (Δ n) A W(𝔤) cs W(b k1) (Ω (U)C (Δ n)) closed F A inv(𝔤) inv(b k1) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ (\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed} &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }

    where now (Ω (U)C (Δ n)) closed(\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed} inside Ω (U)Ω (Δ n))\Omega^\bullet(U)\otimes\Omega^\bullet(\Delta^n)) plays exactly the same role of inv(𝔤)inv(\mathfrak{g}) inside W(𝔤)W(\mathfrak{g}).

    (by the way, (Ω (U)C (Δ n)) closed(\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed} is nothing but Ω (U) closed\Omega^\bullet(U)_{closed}, but I prfer writing it as (Ω (U)C (Δ n)) closed(\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed} in the big diagram)

    • CommentRowNumber189.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    [edit: removed after some private email conversation, as being all caused by a stupid misunderstanding of notation, see comment below now]

    • CommentRowNumber190.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    [edit: removed after some personal email conversation, see comment below]

    • CommentRowNumber191.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2010
    • (edited Sep 10th 2010)

    Sorry, i wasn’t concentrating and then the new F AF_A-notation led me astray.

    We need to write F A:inv(𝔤)etc\langle F_A \rangle : inv(\mathfrak{g}) \to etc, with the angular brackets. And then, yes, the diagram, for genuine \infty-connections, is

    C (U)Ω (Δ n) A vert CE(𝔤) μ CE(b k1) Ω (U)Ω (Δ n) A W(𝔤) cs W(b k1) (Ω (U)C (Δ n)) closed F A inv(𝔤) inv(b k1) \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ (\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed} &\stackrel{\langle F_A \rangle}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }
    • CommentRowNumber192.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 11th 2010
    g --> inn(g) --> g{1]
    do you mean something more than g acting on g[1] by the adjoint actions shifted?
    • CommentRowNumber193.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2010

    do you mean something more than g acting on g[1] by the adjoint actions shifted?

    For an ordinary Lie algebra, nothing else is meant.

    • CommentRowNumber194.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 11th 2010
    and for an Linfty algebra?
    • CommentRowNumber195.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2010

    Jim, we are just talking about the formal dual of a general Weil algebra.

    • CommentRowNumber196.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 13th 2010

    @Jim,

    I hope this > works - if not, tell me how else to cut and paste

    to get the

      >
    

    to work, you need to select ’Markdown’ below the comment box.

    • CommentRowNumber197.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2010

    I am working on finalizing some things. Today I went through the Motivation-section at infinity-Chern-Weil theory and polished it a bit more and added missing references.

    • CommentRowNumber198.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 19th 2011

    Can \infty-Chern-Weil theory provide an answer to this MO question:

    I am wondering if there is a more general or abstract framework that allows one to define the Stiefel-Whitney classes in the spirit of Chern-Weil.

    • CommentRowNumber199.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2011
    • (edited Jan 19th 2011)

    Can ∞-Chern-Weil theory provide an answer to this MO question:

    I am wondering if there is a more general or abstract framework that allows one to define the Stiefel-Whitney classes in the spirit of Chern-Weil.

    At least not in any immediate way. The reason is this:

    \infty-Chern-Weil theory (and ordinary Chern-Weil theory in special cases) reads in a characteristic class given by a morphism

    BGB nA \mathbf{B}G \to \mathbf{B}^n A

    of \infty-Lie groups, and then produces from this a characteristic class with coefficients dRB n+1A\mathbf{\flat}_{dR} \mathbf{B}^{n+1} A the differential forms with values in B n+1A\mathbf{B}^{n+1} A.

    Now, an AA-valued differential form sends infinitesimal paths in some base to infinitesimal paths in AA. If AA is a discrete group then the only infinitesimal paths inside it are constants and hence there are no nontrivial differential forms with values in a discrete group.

    So for instance for A=/A = \mathbb{R}/\mathbb{Z} the real line divided by a discrete group, the corresponding differential forms are forms with values in the real numbers.

    But for A=Disc/2A = Disc \mathbb{Z}/2 just a discrete group itself, the corresponding differential forms are trivial.

    This is what the person asking the question alludes to: SW-classes have coefficients in /2\mathbb{Z}/2 and this necessarily vanishes in de Rham cohomology, where Chern-Weil theory could say something about it.

    Now, if understood correctly, Chern-Weil theory does not actually forget any information about SW classes, because the refined CW homomorphisms remembers not just the differential forms, but also the bundle that they are curvature forms on. So in the refined CW homomorphism the image of an OO–principal bundle under an SW class is is a B n/2\mathbf{B}^n \mathbb{Z}/2-principal (n+1)(n+1)-bundle representing that class, with an \infty-connection that represents that class in de Rham cohomology. But that connection is necessarily flat and the image of the class in de Rham cohomology contains no information.

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