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@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 $\mathbf{B}G$? 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
$H$ is a reminder of “cohomology”.
and here is the (homotopy, of course) commutative diagram inducing the natural morphism $\mathbf{B}_{diff}A\to\flat_{dR}\mathbf{B}^2 A$:
$\array{ \mathbf{B}_{diff}A &\to& \flat\mathbf{B}\mathbf{E}A&\to&\flat\mathbf{B}^2 A \\ \downarrow && \downarrow&&\downarrow \\ \mathbf{B}A &\to& \mathbf{B}\mathbf{E}A&& \\ \downarrow&&&\searrow&\downarrow \\ *&\to&\to&\to&\mathbf{B}^2 A }$Jim,
Is there some (psychological?) implication to using H
$H$ is a reminder of “cohomology”.
generally Urs seems to use $H(a,b)$ for the generic hom-space between $a$ and $b$ in the $(\infty,1)$-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.
Jim,
I’ve sent you via email a pdf file of just the comments you were interested in.
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 $\omega$ defines a path $A+t\omega$ 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 $\mathfrak{inn}(\mathfrak{g})$-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
Urs seems to use $H(a,b)$ for the generic hom-space between $a$ and $b$ in the $(\infty,1)$-cat context,
More precisely, I like to call my $(\infty,1)$-topos $\mathbf{H}$ since then the hom-$\infty$-groupoid $\mathbf{H}(X,A)$ is indeed the “fat” version of the cohomology set $H(X,A)$.
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.
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
$(U,[n]) \mapsto \{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) \leftarrow CE(\mathfrak{g}) \} \,.$This describes paths in the space of flat “vertical” (not along $U$) $\mathfrak{g}$-valued forms.
Now we may thicken this to the simplicial object
$(U,[n]) \mapsto \left\{ \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(\mathfrak{g}) } \right\} \,.$This now describes paths of $\mathfrak{g}$-valued forms, i.e. paths of local $\mathfrak{g}$-connections. The $\infty$-Chern-Weil homomorphism is, up to the truncation issue, just the map that
starts with an $\infty$-Lie algebra cocycle $CE(\mathfrak{g}) \stackrel{\mu}{\leftarrow} CE(b^{k-1}\mathbb{R})$;
picks an invariant polynomial and Chern-Simons element for it to extent it to a pasting diagram
$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) &\stackrel{}{\leftarrow}& 0 \\ \uparrow && \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs,P)}{\leftarrow}& W(b^{k-1}\mathbb{R}) &\stackrel{P}{\leftarrow}& CE(b^k \mathbb{R}) }$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)
I’m trying to understand the $cosk_3(exp\mathfrak{g})\to\mathbf{B}G$. does in a first very rough approximation looks like this description below?
given a Lie algebra $\mathfrak{g}$, there is a (infinite dimensional) simplicial manifold $\int\mathfrak{g}$ integrating it, which turns out to be a Kan complex (and so we can think of as an oo-groupoid $\int\mathfrak{g$exp(\mathfrak{g})$). If $G$ is a Lie group with Lie algebra $G$, then we have natural morphism $exp\mathfrak{g}\to \mathbf{B}G$, i.e., a morphism of simplicial manifolds $\int\mathfrak{g}\to N(\mathbf{B}G)$. This is not an equivalence, and the topology of $G$ obstructs lifting maps to $\mathbf{B}G$ to maps to $exp(\mathfrak{g})$. However, if $G$ is $n$-connected, we can lift from $\mathbf{B}G$ to $cosk_{n+1}(exp\mathfrak{g})$.
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.
Domenico,
yes, that’s the idea. A good way to start thinking about this is to think about how $\mathbf{cosk}_2 \exp(\mathfrak{g})$ is equivalent to $\mathbf{B}G$, because that turns out to be a classical statement about intergration of Lie algebras.
A morphism in $exp(\mathfrak{g})$ is a $\mathfrak{g}$-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 $\mathfrak{g}$-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 G$this way, and this is still equivalent to$G$. This is given by$\mathbf{cosk}_3 \exp(\mathfrak{g})$.
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 $\mathbf{B}G$ to” there’s a spurious $\mu$ in the upper horizontal arrow.
ii) notation $\mu \in CE(\mathfrak{g})$ 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 $\mathfrak{g}$ a Lie algebra, $Hom_{dgca}(CE(b^{k-1} \mathbb{R},CE(\mathfrak{g})$ is naturally identified with the set of $d_{CE(\mathfrak{g})}$-closed degree $k$ elements in $CE(\mathfrak{g})$, i.e., with $d_{CE(\mathfrak{g})}$-closed elements of $\wedge^k\mathfrak{g}^*$; moreover since the differential $d_{CE(\mathfrak{g})}$ is nothing but the Lie bracket of $\mathfrak{g}$, being $d_{CE(\mathfrak{g})}$-closed means to be $\mathfrak{g}$-invariant (for the (co-)adjoint action of $\mathfrak{g}$ on $\wedge^k \mathfrak{g}^*$. Also, I would recall that such a closed element in $\wedge^k \mathfrak{g}^*$, seen as a map $\wedge^k \mathfrak{g}\to \mathbb{R}$, is precisely an $L_\infty$-morphism $\mathfrak{g}\to b^{k-1}\mathbb{R}$. This adds nothing to the mathematical content, but has a nice intuitive effect: one sees that one can use $\mu$ to change a flat $\mathfrak{g}$-connection into a $b^{k-1}\mathbb{R}$-flat connection.
iii) same kind of considerations for $\langle-\rangle\in W(\mathfrak{g})$ and $cs\in W(\mathfrak{g})$: I would add a short digression to recall what is the datum of a morphism $W(b^{k-1} \mathbb{R})\to W(\mathfrak{g})$.
iv) as I have already discussed with Urs, I don’t like very much the $(A,F_A)$-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 $cs$ in place of $(cs,\langle-\rangle)$.
Hi Domenico,
thanks for this! I removed the $\mu$. I am agnostic about the $(A, F_A)$ versus $A$. If you say the latter works better for the reader, then that’s good. For $(cs, \langle-\rangle)$ versus $(cs)$ (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!
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 $W(\mathfrak{g})$: it is an additional information involving $inv(\mathfrak{g})$. so I’d like to say this explicitely. anyway I have to think better to this, so I will leave the notation $(cs,\langle-\rangle)$ 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 $\langle-\rangle$, $\mu$ and $cs$ 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 $inv(\mathfrak{g})\stackrel{\langle-\rangle}{\leftarrow}CE(b^{k}\mathbb{R})$, while leaving only $cs$ on the arrow $W(\mathfrak{g})\stackrel{cs}{\leftarrow}W(b^{k-1}\mathbb{R})$. but since the diagram would not fit neatly with “and then postcomposing with that”, I’ve left it unchangend for the time being.
at circle n-bundle with connection, the arrows $C^\infty(-;U(1))\stackrel{d_{dR}}{\to}\Omega^1(-)$ should be $C^\infty(-;U(1))\stackrel{d_{dR} log}{\to}\Omega^1(-)$. (or am I lost?)
edit: also after “So by the above definition of differential cohomology…” the curvature morphism $\mathbf{B}_{diff} U(1)\to \flat_{dR}\mathbf{B}U(1)$ seems to be written as $\mathbf{B}_{diff} U(1)\to \mathbf{B}_{dR} U(1)$. Since this could be a misunderstanding of mine, I’m not correcting it, yet.
concerning the label of these morphisms:
Yes, you are right. But I was being lazy. But also, If we think of $U(1)$ as $\mathbb{R}/\mathbb{Z}$ then writing just $d_{dR}$ is actually more true to what happens than $d log$, to my mind. But $d log$ 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.
concerning $d_{dR}$ vs. $d log$ 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.
as a trivial exercise in the definitions, I’m working out what the prestacks $exp(b^2\mathbb{R})$ and $exp(b^2\mathbb{R})_{diff}$ are. By dimensional reasons it seems that for $n=0,1,2,3$ we just have
$exp(b^2\mathbb{R}): (U,[n])\mapsto 0$ for $n=0,1,2$
$exp(b^2\mathbb{R}): (U,[3])\mapsto C^\infty(U)\otimes\Omega^3(\Delta^3)$
and
$exp(b^2\mathbb{R})_{diff}:(U,[n])\mapsto \Omega^3(U\times \Delta^n)$
Is this correct?
concerning $d_{dR}$ vs. $d log$ 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 $\exp(b^2 \mathbb{R})$ and $\exp(b^2 \mathbb{R})_{diff}$ are
Okay. By the way, these are actually stacks (satisfy descent) on the site $CartSp$.! That’s part of the point of using this “small” site instead of all of $Diff$. 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 $n=0,1,2,3$ we just have
$exp(b^2\mathbb{R}): (U,[n])\mapsto 0$ for $n=0,1,2$
$exp(b^2\mathbb{R}): (U,[3])\mapsto C^\infty(U)\otimes\Omega^3(\Delta^3)$
Yes. But it is important to go at least one degree higher and also look at
$exp(b^2\mathbb{R}): (U,[4])\mapsto C^\infty(U)\otimes\Omega^3_{closed}(\Delta^4)$
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 $S^3$ may be extended to a closed 3-form on $D^3$ precisely if its integral vanishes. The integral of the 3-form is the actual real number it represents under the equivalence
$\int_{\Delta^\bullet} : \exp(b^2 \mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^3 \mathbb{R} \,.$and $exp(b^2\mathbb{R})_{diff}:(U,[n])\mapsto \Omega^3(U\times \Delta^n)$
This is not quite correct: on the right we have the subset of $\Omega^3(U \times \Delta^n)$ of precisely those forms that have the property that the component of their 4-form curvature which sits in $C^\infty(U)\otimes \Omega^\bullet(\Delta^n)$ (all “legs” in the simplicial direction) vanishes.
That’s precisely what the cmmutativity of this diagram
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& CE(b^2 \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(b^2 \mathbb{R}) }$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 $\mathbf{B}^3 \mathbb{R}$-bundle (or $\mathbf{B}^3 \mathbb{R}/\mathbb{Z}$-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.
What you wrote down is the set of morphisms in the bottom of the diagram
yes, but it seems to me that for $n=0,1,2$ the upper horizontal arrow and the leftmost vertical arrow are zero, while for $n=3$ one only needs that the 3-form component in $C^\infty(U)\otimes\Omega^3(\Delta^3)$ is closed, but this is trivially true for dimensional reasons. This is no more true for higher $n$’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 $n=4$ step.
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.
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 $\mathfrak{g}$ fits the "big diagram" involving also $inv(\mathfrak{g})$, 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 $\mu$ and $cs$. 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 $\mu$ and $cs$ 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
$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) }$and then postcomposing with that. Note that the above diagram is part of a larger diagram involving the invariant polynomial $\langle-\rangle$ for $\mu$ and exhibiting the Chern-Simons element as a transgression element between these two:
$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& CE(b^{k} \mathbb{R}) }$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 $n$-gerbe with connection on $B G$.
One sees it more explicitly if we consider the “unrefined Chern-Weil” morphism
$\mathbf{B}G \to \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$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 ;-)
I find it is a beautiful fact that the Killing form of $\mathfrak{g}$ fits the “big diagram” involving also $inv(\mathfrak{g})$
By the way, I quite share your enthusiasm for that double-square diagram
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& CE(b^{k} \mathbb{R}) }$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
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes C^\infty(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& CE(b^{k} \mathbb{R}) }$if we allow the left bottom morphism to be just the gca-inclusion, not a dgca morphism.
This sounds awesome. I hope I can understand it someday.
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.
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 $L_\infty$-algebra $b^{k-1}\mathbb{R}$ has trivial differential and brackets, every polynomial in the shifted generators in $W(b^{k-1}\mathbb{R})$ is $D_W$-closed, and so is an invariant polynomial. This says that $inv(b^{k-1}\mathbb{R})=CE(b^k\mathbb{R})$ and the big diagram above can be rewritten in the following more coherent form:
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }$Also, I see from this I’m not completely satisfied by our current definition of invariant polynomials. Namely $W(\mathfrak{g})$ is a shorthand notation for $CE(inn(\mathfrak{g})$. as a graded vector space, $inn(\mathfrak{g})=\mathfrak{g}\oplus\mathfrak{g}[1]$, and the natural morphism $W(\mathfrak{g})\to CE(\mathfrak{g})$ can be seen as the dgca counterpart of the natural $L_\infty$-morphism $(id,0):\mathfrak{g}\to \mathfrak{g}\oplus\mathfrak{g}[1]$. but then I would think of an $L_\infty$-algebra strcucture on $\mathfrak{g}[1]$ and of an $L_\infty$-morphisms $\mathfrak{g}\oplus\mathfrak{g}[1]\to \mathfrak{g}[1]$ inducing $CE(\mathfrak{g}[1])\to W(\mathfrak{g})$. As a vector space $CE(\mathfrak{g}[1])$ is precisely the subalgebra of $W(\mathfrak{g})$ generated by the shifted generators, so the restriction to this part of $W(\mathfrak{g})$ is natural from this perspective. It is not clear to me how the $d_W$-closed condition appears, but something like this may be the case: assume the differential on $CE(\mathfrak{g}[1])$ is zero and that the dgca morphism $CE(\mathfrak{g}[1])\to W(\mathfrak{g})$ takes values in the part of $W(\mathfrak{g})$ generated by the shifted generators. Then compatibility with the differentials would force this image to consist of $d_W$-closed elements. In other words, if everything works, $inv(\mathfrak{g})$ would be a handy realization of the more intrinsically defined $CE(\mathfrak{g}[1])$ as a subalgebra of $W(\mathfrak{g})$, and this would justify the definition of $inv(\mathfrak{g})$ which at the present time seems a bit artificial.
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..
Hey, wait! maybe there’s a much simpler expanation: cosider the subalgebra $\wedge^\bullet\mathfrak{g}^*[-1]$ of $W(\mathfrak{g})$ generated by the shifted generators; the differential $d_W$ maps this algebra into itself, so $(\wedge^\bullet\mathfrak{g}^*[-1], d_W)$ is a dgca, and is $CE(\mathfrak{g}[1])$ for a certain natural $L_\infty$-algebra structure on $\mathfrak{g}[1]$. Let me call this algebra $Poly(\mathfrak{g})$ for the moment, waiting for a better name. Then $Poly(\mathfrak{g})$ is not invariant polynomials, but a morphism $Poly(b^{k-1}\mathbb{R})\to Poly(\mathfrak{g})$ picks up precisely a degree $k$ invariant polynomial! So we see that $inv(\mathfrak{g})$ naturally comes into play when the big diagram is involved.
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.
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.
Promted by the above discussion on those diagrams, I added a Summary-section oo-CW homomorphism – summary.
cosider the subalgebra $\wedge^\bullet\mathfrak{g}^*[-1]$ of $W(\mathfrak{g})$ generated by the shifted generators; the differential $d_W$ maps this algebra into itself,
Actually, no, it does not. For instance for an ordinary Lie algebra with unshifted generators $\{t^a\}$ and shifted generators $\{r^a\}$ we have $d_W r^a = C^a{}_{b c} t^b \wedge r^c$.
What is true is that $d_W$ stays in the kernel of the map $W(\mathfrak{g}) \to CE(\mathfrak{g})$, which is the ideal generated by $\mathfrak{g}^*[1]$.
The subtlety is how to characterize in a useful intrinsic way $inv(\mathfrak{g})$ inside this kernel
$inv(\mathfrak{g}) \hookrightarrow ker(i^*) \hookrightarrow W(\mathfrak{g}) \stackrel{i^*}{\to} CE(\mathfrak{g}) \,.$One way is this: $inv(\mathfrak{g})$ are precisely the strong invariants in $W(\mathfrak{g})$ of the canonical $\mathfrak{g}$-action. Where by strong invariant I mean those element that are invariant under homotopies of diagrams of inner derivations
$\array{ CE(\mathfrak{g}) &\stackrel{[d,\iota_t]}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{[d_W,\iota_t]}{\leftarrow}& W(\mathfrak{g}) } \,.$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 $G$ an oo-group and $\mathbf{E}G$ its universal principal oo-bundle (a notion that is the hallmark of being in a model!) we have that $\mathbf{B}G = (\mathbf{E}G)/G$.
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
$G \to \mathbf{\flat}_{dR}\mathbf{B}G \to \mathbf{\flat} \mathbf{B}G \to \mathbf{B}G$one step further to the right.
I have a guess what should go there. Namely the unit
$\mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{\Pi}_{dR} \mathbf{B}G$of the $\infty$-adjunction $(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})$. I wasted two months or so thinking about this. But I still don’t have this under control.
Er, I didn’t say the last bit well. But too tired now. More tomorrow.
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?
Actually, no, it does not.
Oh, right! in the formula for $d_W$ on the shifted generators the leftost shift acts as a derivation! I got confused on that! :(
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.
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 $X$ fixed, $H(X,-)$ is a functor. That’s because $\mathbf{H}(-,-)$ 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 $\mathbf{H}(X,A)$ for the $\infty$-groupoid of cocycles and $H(X,A)$ 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 $\int_{\Delta^\bullet} : \exp(b^k \mathbb{R}) \to \mathbf{B}^{k+1}\mathbb{R}$.
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 $inv(\mathfrak{g})$
and how about Characteristic classes of Q-bundles
A “$Q$-bundle” is what here we are calling an infinity-Lie algebroid $\mathfrak{a}$. So its characteristic classes are what here we are calling the invariant polynomials $inv(\mathfrak{a})$.
@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.
@Urs
The intrinsic, oo-categorical characterization of $inv(\mathfrak{g})$
Indeed that seems to be a crucial issue here. For the time being I will content myself of thinking that since $inv(\mathfrak{g})$ is intended to model the de Rahm algebra of $BG$, it can be seen as the subalgebra of $\Omega^\bullet(EG)$ consisting of horizontal equivariant forms. This suggests teh two algebraic conditios we presently have: using $W(\mathfrak{g})$ as a model for $\Omega^\bullet(EG)$, horizontality should correspond to being a polynomial in the shifted generators, while equivariance should be encoded in being $d_W$-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 $V$ is the same thing as giving a morphism $\mathbb{K}\to V$, 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..
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.)
de Rham cohomology algebra - NOT form algebra
Isn’t $B G$, for $G$ a compact Lie group a formal space? I thought so, and so in my mind a model for $\Omega^\bullet(B G)$ was the same as a model for $H^\bullet(B G)$ (as dg-algebras). But I see that since the differential in $inv(\mathfrak{g})$ is trivial, our current definition of $inv(\mathfrak{g})$ aims to pick $H^\bullet(B G)$ 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 $B G$? I guess that’s what we should try to reproduce, since in the sequence $inv(\mathfrak{g})\to W(\mathfrak{g})\to CE(\mathfrak{g})$ 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.
Hi Domenico,
that’s spot-on.
Yes, $B G$ is a formal space, so $\Omega^\bullet(B G)$ is isomorphic to $inv(\mathfrak{g})$ 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 $\mu$ one can use any element in $W(\mathfrak{g})$ of the form $d_{W(\mathfrak{g})} \hat \mu$ for $\hat \mu|_{CE(\mathfrak{g})} = \mu$ 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 $ker(i^* : W(\mathfrak{g}) \to CE(\mathfrak{g}))$.
(Because that kernel is in fact a homotopy kernel, because $i^*$ is a fibration).
Indeed, which is classically a Sullivan model for BG?
But a semi-free dg-algebra with vanishing differential is (trivially) a Sullivan algebra! So $inv(\mathfrak{g}) = H^\bullet(B G)$ is.
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 $\mathfrak{g}$-connections are sent to $b^n \mathbb{R}$-connections.
If we use instead a more general element in $ker(i^*)$, then the CW homomorphism will in general send genuine $\mathfrak{g}$-connections to $b^n \mathbb{R}$-pseudoconnections. Which are however guaranteed to be equivalent to some genuine $b^n \mathbb{R}$-connection.
But a semi-free dg-algebra with vanishing differential is (trivially) a Sullivan algebra! So $inv(\mathfrak{g})=H^\bullet(BG)$ is.
yes, I see I expessed myself very badly. what I meant was “how can we cook up a Sullivan model for $B G$ starting with $\mathfrak{g}$?” from this point of view, going from $\mathfrak{g}$ to $G$ and then to $B G$ 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 $B G$ is a formal space.
If we use instead a more general element in $ker(i^*)$, then the CW homomorphism will in general send genuine $\mathfrak{g}$-connections to $b^n\mathbb{R}$-pseudoconnections
I should see this better. How does exactly the invariant polynomial manifest itself in the CW construction at Chern-Simons circle 3-bundle?
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
$curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$is modeled by a certain truncation and quotienting of the span of simplicial presheaves
$\{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(b^{n-1} \mathbb{R}) \} \leftarrow \left\{ \array{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\} \to \left\{ \array{ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\}$This means that for $\mathbf{B}G \to \mathbf{B}^n U(1)$ a characteristic class arising by Lie integration of a cocycle $\mu \in CE(\mathfrak{g})$
$\{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(\mathfrak{g}) \} \to \{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(b^{n-1}) \}$the total composite morphism
$\mathbf{B}G \to \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$is modeld by the zig-zag
$\{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(\mathfrak{g}) \} \to \{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) \leftarrow CE(b^{n-1} \mathbb{R}) \} \leftarrow \left\{ \array{ C^\infty(-)\otimes \Omega^\bullet(\Delta^-) &\leftarrow& CE(b^{n-1} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\} \to \left\{ \array{ \Omega^\bullet(-)\otimes \Omega \bullet(\Delta^-) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\} \,.$For computing with this, we want to “iron out the zag in the zig-zag”. We have a zig-zag of the form
$\array{ && C &\to& D \\ && \downarrow^{\mathrlap{\simeq}} \\ A &\to& B }$and want to complete it to a commuting diagram
$\array{ \hat A &\to& C &\to& D \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ A &\to& B }$because that will mean that then the outer span
$\array{ \hat A &\to& C &\to& D \\ \downarrow^{\mathrlap{\simeq}} \\ A }$models the same $\infty$-morphism.
To be continued in the next comment.
Here is the continuation of the above comment.
Recall that we want to find an object $\hat A$ that makes a zig-zag of morphisms just a span of morphisms. (We want to “compose anafunctors”.) We see that we can find $\hat A$ in the case at hand by completing the diagram
$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ && \uparrow \\ && W(b^{n-1}\mathbb{R}) }$in any way such that the bottom left bit is contractible (so that $\hat A \to A$ indeed is a weak equivalence). An obvious solution starts with completing it as
$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) && W(b^{n-1}\mathbb{R}) } \,.$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 $\hat \mu \in W(\mathfrak{g})$ of degree $n$ such that its restriction to $CE(\mathfrak{g})$ is $\mu$:
$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(\hat \mu, d_{W(\mathfrak{g})} \hat \mu)}{\leftarrow}& W(b^{n-1}\mathbb{R}) } \,.$This alone is sufficient to model the morphism $\mathbf{B}G \to \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$, postcomposition with which is the Chern-Weil homomorphism
$\mathbf{H}(X,\mathbf{B}G) \to \mathbf{H}_{dR}^{n+1}(X) \,.$The above construction models this by a span
$\array{ [CartSp^{op}, sSet](C(U), \mathbf{B}G_{diff}) &\to& [CartSp^{op}, sSet](C(U), \mathbf{B}^n \mathbb{R}_{diff}) &\to& [CartSp^{op}, sSet](C(U), \mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} \\ [CartSp^{op}, sSet](C(U), \mathbf{B}G) }$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 $[CartSp^{op}, sSet](C(U), \mathbf{B}G_{diff})$ which are pseudo-connections that are in fact genuine connections.
So we can ask if we can find choice of $\hat \mu$ 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 $[CartSp^{op}, sSet](C(U), \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R})$ that are globally defined forms.
This is precisely the case if the composite dg-algebra morphism
$\Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) \stackrel{(A,F_A)}{\leftarrow} W(\mathfrak{g}) \stackrel{(\hat \mu, d_{W(\mathfrak{g})} \hat \mu)}{\leftarrow} W(b^{n-1}\mathbb{R}) \stackrel{d_{W(\mathfrak{g})} \hat \mu}{\leftarrow} CE(b^n \mathbb{R})$is a $n+1$-form on $U \times \Delta^k$ with no leg along the simplex, i.e. if it lands in $\Omega^\bullet(U) \otimes C^\infty(\Delta^k)$.
If we assume that $A$ is a genuine connection on a $G$-bundle, then by definition the $F_A$ do have this property. So we can ensure the above condition if we manage to find $\hat \mu$ such that $d_{W(\mathfrak{g})} \hat \mu$ ends up sitting just in the shifted copy, i.e. in $\wedge^\bullet \mathfrak{g}^*[1]$.
This is precisely the case if $d_{W(\mathfrak{g})} \hat \mu \in W(\mathfrak{g})$ is an invariant polynomial in the standard sense. So this is precisely the good choice of span $A \leftarrow \hat A \to C$ that ensures that nice representatives (genuine connections) are sent to nice representatives (genuine connections).
that “inv(” seems to be causing problems…
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
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: $x + y = 9$
@Urs, #161
This is the part I don’t clearly see:
This is precisely the case if the composite dg-algebra morphism
$\Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) \stackrel{(A,F_A)}{\leftarrow} W(\mathfrak{g}) \stackrel{(\hat \mu, d_{W(\mathfrak{g})} \hat \mu)}{\leftarrow} W(b^{n-1}\mathbb{R}) \stackrel{d_{W(\mathfrak{g})} \hat \mu}{\leftarrow} CE(b^n \mathbb{R})$is a $n+1$-form on $U \times \Delta^k$ with no leg along the simplex, i.e. if it lands in $\Omega^\bullet(U) \otimes C^\infty(\Delta^k)$.
If we assume that $A$ is a genuine connection on a $G$-bundle, then by definition the $F_A$ do have this property. So we can ensure the above condition if we manage to find $\hat \mu$ such that $d_{W(\mathfrak{g})} \hat \mu$ ends up sitting just in the shifted copy, i.e. in $\wedge^\bullet \mathfrak{g}^*[1]$.
Indeed, deleting the spurious $d_{W(\mathfrak{g})} \hat \mu$ 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
$\Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{\hat \mu}{\leftarrow} W(b^{n-1}\mathbb{R}) \leftarrow CE(b^n \mathbb{R})$is a $n+1$-form on $U \times \Delta^k$ with no leg along the simplex, i.e. if it lands in $\Omega^\bullet(U) \otimes C^\infty(\Delta^k)$.
which is fine. But then I’m unable to make any sense of the line that follows:
If we assume that $A$ is a genuine connection on a $G$-bundle, then by definition the $F_A$ do have this property. So we can ensure the above condition if we manage to find $\hat \mu$ such that $d_{W(\mathfrak{g})} \hat \mu$ ends up sitting just in the shifted copy, i.e. in $\wedge^\bullet \mathfrak{g}^*[1]$.
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]
Domenico,
so if $inv(\mathfrak{g})$ is quasi-isomorphic to $ker(i ^* : W(\mathfrak{g}) \to CE(\mathfrak{g}))$ then that gives another explanation for my long comment above, at least as long $\mathfrak{g}$ is an ordinary Lie algebra.
Urs,
if $inv(\mathfrak{g})$ is quasi-isomorphic to $ker(i^*:W(\mathfrak{g})\to CE(\mathfrak{g}))$, nothing else needs to be said! :)
(I mean: then it is a nice model for the intrinsic object)
So we should try to figure out whether this is true for all $L_\infty$-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 ;-)
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 $\L_\infty$-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. $W(g) \to CE(g)$ 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 $CE(g[1])$? Is this that supposed to be the kernel of the map $W(g) \to CE(G)$? In that case I would call this the ideal generated by $g^*[1]$ in $W(g)$. Is that what you mean by $CE(g[1])$?
now we have to go inside to recognize H(CE(g[1]) ) as invariant polynomials
That’s the crucial step.
mmm.. and what about considering morphisms of Lie algebras $\mathfrak{g}_{-1}\to \mathfrak{g}0$ (the source of the morphism is “in degree -1”) as the starting point ? so that
$(\mathfrak{g}\to 0)\mapsto \mathfrak{g}[1]$
$(\mathfrak{g}\stackrel{id}\to \mathfrak{g})\mapsto inn(\mathfrak{g})$
$(0\to\mathfrak{g})\mapsto \mathfrak{g}$
this way the obvious commutative diagram
$\array{ 0 &\to& \mathfrak{g}&\to&\mathfrak{g} \\ \downarrow && \downarrow_{id} &&\downarrow \\ \mathfrak{g} &\to& \mathfrak{g}&\to&0 }$would naturally induce the desired $\mathfrak{g}\to inn(\mathfrak{g})\to \mathfrak{g}[1]$.
with Marco Manetti we considered a similar question in math.QA/0601312, but there the starting point was a pair of morphisms $\mathfrak{g}_0\stackrel{\to}{\to}\mathfrak{g}_1$, seen as a tiny bit of a semicosimplicial object in dglas, whereas here we would rather be considering an augmented semicosimplicial object.
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.
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 $inv(\mathfrak{g})$: the simple answer is: because it is an extremely nice model for $ker(i^*)$.
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.
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.
Concerning what I write in #169, if the big diagram in #126 where
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }$instead, then everything would be clear to me.
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
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U) \otimes C^\infty(\Delta^n) &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }$where now the left bottom morphism is just the gca-inclusion, not a dgca-morphism.
I’d rather have
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ (\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed} &\stackrel{F_A}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }$where now $(\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed}$ inside $\Omega^\bullet(U)\otimes\Omega^\bullet(\Delta^n))$ plays exactly the same role of $inv(\mathfrak{g})$ inside $W(\mathfrak{g})$.
(by the way, $(\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed}$ is nothing but $\Omega^\bullet(U)_{closed}$, but I prfer writing it as $(\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed}$ in the big diagram)
[edit: removed after some private email conversation, as being all caused by a stupid misunderstanding of notation, see comment below now]
[edit: removed after some personal email conversation, see comment below]
Sorry, i wasn’t concentrating and then the new $F_A$-notation led me astray.
We need to write $\langle F_A \rangle : inv(\mathfrak{g}) \to etc$, with the angular brackets. And then, yes, the diagram, for genuine $\infty$-connections, is
$\array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{k-1} \mathbb{R}) \\ \uparrow && \uparrow && \uparrow \\ (\Omega^\bullet(U) \otimes C^\infty(\Delta^n))_{closed} &\stackrel{\langle F_A \rangle}{\leftarrow}& inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^{k-1} \mathbb{R}) }$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.
Jim, we are just talking about the formal dual of a general Weil algebra.
@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.
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.
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.
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
$\mathbf{B}G \to \mathbf{B}^n A$of $\infty$-Lie groups, and then produces from this a characteristic class with coefficients $\mathbf{\flat}_{dR} \mathbf{B}^{n+1} A$ the differential forms with values in $\mathbf{B}^{n+1} A$.
Now, an $A$-valued differential form sends infinitesimal paths in some base to infinitesimal paths in $A$. If $A$ 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 $A = \mathbb{R}/\mathbb{Z}$ the real line divided by a discrete group, the corresponding differential forms are forms with values in the real numbers.
But for $A = Disc \mathbb{Z}/2$ 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 $\mathbb{Z}/2$ 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 $O$–principal bundle under an SW class is is a $\mathbf{B}^n \mathbb{Z}/2$-principal $(n+1)$-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.