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polished/reworked the entry Lie integration. But it’s still somewhat stubby.
I updated references a bit (it would be good when you put the reference link to arxiv to have the number of the article in the link name sometimes, e.g. when using the prinpouts of lab pages offline it is nice to have the reference handy, just arxiv does not help much). I do not understand
“whose origin possibly preceeds that of the previous article and which considers Banach manifold structure on the resulting ∞-groupoids”
about Henriques’ article in comparison to Getzler’s. What does it mean that it’s origin is earlier ? Getzler claims e.g. that it took him 7 years just to get a crucial improvement into a calculus of Dupont which is essential in his paper.
And Getzler does not treat arbitrary L_oo algebras, but nilpotent ones :) I’d edit the page but for the spam blocker. Perhaps, Zoran, Urs means the technique or idea from Henriques predates Getzler’s?
And Getzler does not treat arbitrary L_oo algebras, but nilpotent ones :)
No, wait before you edit, I tried to say that correctly:
Getztler notices that for all $L_\infty$-algebra, the Sullivan-like construction should be regarded as Lie integration.
It is only for the special case of nilpotent $L_\infty$-algebras that he gives a prescription to cut down the large result of the Sullivan construction to a smaller equivalent one. For the general notion of Lie-integration however, this is pretty irrelevant. It point is to give more tractable models.
I do not understand
“whose origin possibly preceeds that of the previous article and which considers Banach manifold structure on the resulting ∞-groupoids”
I saw this comment again when I reworked the entry now, but I noticed that I forgot why I had put it there. Somebody had told me about some coomplaint about false attribution of originality. But I forget the details. Maybe we should just remove it.
Urs?
Ahm, Zoran?
In 2 and 3 Zoran and David are asking you what is the meaning of strange sentence that the origin’s of Henriques paper from 2004 predate the originćs of 2002 Getzler’s paper.
And I replied:
I forgot why I had put it there. Somebody had told me about some coomplaint about false attribution of originality. But I forget the details. Maybe we should just remove it.
By the way, I added now to Lie infinity-groupoid in the section Lie group: differential coefficients a discussion of the general abstract mechanism underlying the Getzler-Henriques prescription for integration of oo-Lie algebras:
the claim is that for $G$ an $\infty$-Lie group
the object $\mathbf{\flat}_{dR} \mathbf{B}G$ is essentially given by the sheaf of flat $\mathfrak{g}$-valued forms, for $\mathfrak{g}$ the corresponding $L_\infty$-algebra; (and this is demonstrated inm the entry)
moreover the object $\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{B}G$ is $\exp(\mathfrak{g})$ (in the notation of the entry) i.e. is the Getzler-Henriques-prescription extended to simplicial presheaves in the evident way.
This is (in somewhat different notation) that old observation of mine that we should be thinking of the Getzler-Henriques prescription as forming the path oo-groupoid of the sheaf of $L_\infty$-algebra valued forms.
And indeed, this is evidently true for the evident naive model of the path $\infty$-groupoid: if for $X$ a sheaf I write
${\tilde \mathbf{\Pi}}(X) : U \mapsto Hom(U \times \Delta^\bullet_{diff}, X)$for the simplicial presheaf whose k-cells are $k$-dimensional path in $X$, then for $X = \mathbf{\flat}_{dR} \mathbf{B}G = \Omega^1_{flat}(-, \mathfrak{g})$ we have the ordinary pushout
$\array{ \mathbf{\flat}_{dR} \mathbf{B}G &\to& * \\ \downarrow && \downarrow \\ {\tilde \mathbf{\Pi}} \mathbf{\flat}_{dR} \mathbf{B}G &\to& \exp(\mathfrak{a}) }$in the category of simplicial presheaves.
So this all looks like it should look. Unfortunately, I have still to fully understand if and why the naive model $\tilde \mathbf{\Pi}$ in fact does model the correct left derived functor that defines $\mathbf{\Pi}$ in this case: because the formula for that which I have is like this $\tilde \mathbf{\Pi}$, but much thicker, with various cofibrant replacements thrown in. I still cannot show that it may be modeled by $\tilde \mathbf{\Pi}$ when applied to 0-truncated simplicial sheaves.
That’s been a stumbling block for me for quite some time now.
am working on the entry Lie integration
Here is what I did so far:
moved the discussion of references from the introduction to a References-section at the end and polished slightly.
created a Definition-section with two subsections:
first is the Sullivan-Hinich-Getztler integration to a “bare oo-groupoid” (no smooth structure),
second is the integration to $\infty$-Lie groupoids.
I have now expanded the (currently) three Examples-sections at Lie integration:
integration of Lie algebras to Lie groups
integration to line/circle Lie n-groups;
integration of Lie 2-algebras to Lie 2-groups.
There is considerably more to be said. But I am running out of steam.
did some polishing of the exposition at Lie integration, following a list of comments by Jim Stasheff
I tried to polish the discussion of forms on simplices that have sitting instants a bit. But it is still not really good.
didn’t find much time over the weekend, but tried to work a bit on material related to Lie integration.
I have now typed a detailed proof of the claim that for the Lie $n$-algebra$b^n \mathbb{R}$ its Lie integration to a smooth $n$-group is indeed $\mathbf{B}^n \mathbb{R}$.
That’s in the section Integration to line n-groups.
I had gone through the trouble of preparing an SVG graphics, displayed there, that is supposed to illustrate the idea of how to identify smooth forms with sitting instants on the $n$-simplex with smooth forms on the $n$-ball. This was the first time I used the SVG editor, but we haven’t become friends yet. (And by the way: the version linked to from within the nLab edit pages does not work for me (Firefox on Win). It appears with duplicated menu items that don’t react properly to mouse clicks. I used the corresponding version found elsewhere on the web.)
Your SVG works for me (Firefox 3.5.5 on Windows - this is work’s setup) - and looks very pretty I might add!
Your SVG works for me (Firefox 3.5.5 on Windows - this is work’s setup)
Oh, for me, too. What I meant as that the editor didn’t work! When I open an nLab page, hit “edit”, click on a point in the edit pane such that the “create an SVG”-button appears, then click on that button, what I get is a window that tries to display the SVG editor properly but fails, and which does not accept any mouse click input. But the same SVG editor can be found elsewhere on the web, and that works for me.
and looks very pretty I might add!
Okay. I was very dissatisfied, as it is has a large amount of free hand drawing and I didn’t really have the nerve for that. I didn’t figure out how to create a copy of some element, for instance. And Is there a way to invoke a grid such as to facilitate drawing well-aligned lines?
(Just asking. If the answer is: no, but you can use any of the one hundred other SVG editors out there, that’s fine with me.)
I have further polished the proof at integration to line n-group (now the use of symbols might even be consistent…) and have tried – following a suggestion by Domenico – to indicate better how we are essentially just invoking the de Rham theorem but need to be careful to do it properly in smooth families.
added an argument to pass from a single $n$-form to a smooth family.
Thanks, Domenico. That looks of course like a more elegant/powerful/better argument. Hm, I still need to think about this. But maybe not tonight. But this should also give the result nicely for extension of closed $(k \lt n)$-forms from $S^n$ to $D^{n+1}$.
yes, it should: the argument picks a solution of $d A=\omega$ once it is known that $\omega$ is exact. So on a $n$-sphere it applies to any closed $(0\lt k\lt n)$-form, and to closed $n$-forms whose integral over $S^n$ vanishes.
Okay, good. That then gives a nice elegant proof that $\exp(b^{n-1}\mathbb{R}) \simeq \mathbf{B}^n \mathbb{R}$ also without the truncation. (One can show it with my original style of argument, too, but by far not as elegantly.)
okay, i have now fully boosted up statement and proof at Lie integration to line n-groups using Domenico’s argument.
I have now typed a detailed proof of the claim that for the Lie $n$-algebra$b^n \mathbb{R}$ its Lie integration to a smooth $n$-group is indeed $\mathbf{B}^n \mathbb{R}$.
I always expected that this should be a corollary of a relation between two model structures, responsible for two nonabelian cohomology theories – for Lie groups and Lie algebras; or corresponding infinity categories. Is this too far from the present understanding ?
Right, I think it is a little subtle:
consider the non-smooth case, the $\infty$-topos just over bare dgc-algebras. There is then the kind of inclusion of dg-algebras discussed at function algebras on infinity-stacks which gives a right Quillen functor (roughly)
$Spec : dgcAlg^{op} \to [dgcAlg,sSet]_{proj}$that is given by the “Lie integration”-formula
$Spec A : (Spec U, [k]) \mapsto dgcAlg(A , U \otimes \Omega^\bullet_{pl}(\Delta^k)) \,,$where on the right we have the standard polynomial differential forms on the simplex.
The fibrant objects in $dgcAlg^{op}$ are cofibrant in $dgcAlg$ hence in particular semi-free hence may be thought of as CE-algebras of $L_\infty$-algebras.
So that’s good. And we can generalize this general situation to a smooth setup, as Herman Stel has done in his thesis (master thesis Stel (schreiber)).
However, what I am still unfortunately lacking is an understanding of the relation of the Spec-functor as obtained there, to the formula for smooth Lie integration with smooth forms on the simplex. I can guess how it should all be related, but I cannot prove it yet. Maybe it’s easy and I am just being dense, of course.
But for the present purpose I like to adopt a slightly different perspective anyway. From general abstract reasoning in cohesive $\infty$-toposes, one finds that “exponentiated $\infty$-Lie algebras” are objects that are sent by $\Pi$ to the point. One can show that smooth Lie integration produces such objects in $Smooth \infty Grpd$ and then just take it as a machine that acts as a source for examples of some objects. Then you can work backwards and check if that machine sends an $L_\infty$-algebra to the smooth $\infty$-groupoid that you would expect to be associated it, and the currentl proofs at Lie integration serve to confirm that.
There is yet another issue: even in the non-smooth case the Spec functor is not necessarily doing what you would think it does. That’s because its right derived functor involves fibrant replacement. Now $b^{n-1}\mathbb{R}$ is fibrant in $dgcAlg^{op}$, for for instance a general non-abelian Lie algebra $\mathfrak{g}$ is not. It’s not clear that $Spec CE(\mathfrak{g})$ is indeed equivalent to what we are calling here $\exp(\mathfrak{g})$, because it is $\exp( P \mathfrak{g})$ for some fibrant replacement $P\mathfrak{g}$.
I think there are different ways here in which $L_\infty$-algebras map into $\infty$-stacks, and it is important so keep them sorted out.
For instance there is yet another way where we don’t apply $\exp(-)$ or $Spec$ but realized the $L_\infty$-algebras directly as “infinitesimal $\infty$-groupoids”. After some back and forth I am now thinking that this is described by the discussion at cohesive oo-topos – Infinitesimal cohesion.
(sorry, I am writing all this in a bit of a haste, this really deserves to be discussed in more detail)
(sorry, I am writing all this in a bit of a haste
It is still a fantastic answer!
lacking is an understanding of the relation of the Spec-functor as obtained there, to the formula for smooth Lie integration with smooth forms on the simplex. I can guess how it should all be related, but I cannot prove it yet.
It looks very optimistic conceptually being squeezed to somewhat lower level technical question! Thanks for sharing the state of the art.
I have worked a bit more on the Idea section at Lie integration, expanded it, tried to make it read more smoothly, and added more pointers to the references.
I have worked on further polishing and streamlining the entry. Have collected all the discussion of $\mathfrak{a}$-valued differential forms on simplices into a new subsection Higher dimensional paths in an infinity-Lie algebroid.
Today’s reference
insereted into Lie integration, without comments.
Thanks! Pavol told me about this result a few weeks back when he visited Prague. That’s neat.
added statement of Vincent Braunack-Mayer’s result that higher Lie integration as defined in FSS 12 is right Quillen as a functor to smooth $\infty$-groupoids (here):
There is a Quillen adjunction
$dgcAlg^{op}_{\mathbb{R}, \geq 0, proj} \; \underoverset {\underset{ Spec }{\longrightarrow}} {\overset{ \mathcal{O} }{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\bot_{Qu}\phantom{A}} \; [CartSp^{op},sSet_{Qu}]_{proj,loc}$between
the projective local model structure on simplicial presheaves over CartSp, regarded as a site via the good open cover coverage (i.e. presenting smooth ∞-groupoids);
the opposite projective model structure on connective dgc-algebras over the real numbers
given by nerve and realization with respect to the functor of smooth differential forms on simplices $CartSp \times \Delta \overset{\Omega^\bullet_{vert,si}}{\longrightarrow} dgcAlg_{\mathbb{R}, conn}^{op}$ from this Def.:
the right adjoint $Spec$ sends a dgc-algebra $A \in dgcAlg_{\mathbb{R},\geq 0}$ to the simplicial presheaf which in degree $k$ is the set of dg-algebra-homomorphism form $A$ into the dgc-algebras of smooth differential forms on simplices $\Omega^\bullet_{si,vert}(-)$ (this Def.):
$Spec(A) \;\colon\; \mathbb{R}^n \times \Delta[k] \;\mapsto\; Hom_{dgcAlg_{\mathbb{R}}} \left( A , \Omega^\bullet_{si, vert}(\mathbb{R}^n \times \Delta^k_{mfd}) \right)$the left adjoint $\mathcal{O}$ is the Yoneda extension of the functor $\Omega^\bullet_{vert,si} \;\colon\; CartSp \times \Delta \to dgcAlg_{\mathbb{R},conn}^{op}$ assigning dgc-algebras of smooth differential forms on simplices from this Def.,
hence which acts on a simplicial presheaf $\mathbf{X} \in [CartSp^{op}, sSet] \simeq [\CartSp^{op} \times \Delta^{op}, Set]$, expanded via the co-Yoneda lemma as a coend of representables, as
$\mathcal{O} \;\colon\; \mathbf{X} \simeq \int^{n,k} y(\mathbb{R}^n \times \Delta[k]) \times \mathbf{X}(\mathbb{R}^n)_k \;\mapsto\; \int_{n,k} \underset{\mathbf{X}(\mathbb{R}^n)_k}{\prod} \Omega^\bullet_{si,vert}(\mathbb{R}^n \times \Delta^k_{mfd})$I think something is the wrong way around. You have, in the adjunction, written $Spec$ as going from presheaves to algebras.
Thanks for catching this. Fixed now.
The big question now is whether this Quillen adjunction exhibits R-cohomology localization, as in section 3 of function algebras on infinity-stacks. It must come at least close….
Isn’t this a particular instance of the nerve-realization adjunction? Shouldn’t it be indicated as such?
Yes, it does say so:
…given by nerve and realization with respect to the functor of smooth differential forms on simplices $CartSp \times \Delta \overset{\Omega^\bullet_{vert,si}}{\longrightarrow} dgcAlg_{\mathbb{R}, conn}^{op}$ from this Def.:…
Indeed, somehow I didn’t notice it originally.
Integration from Lie algebroids to groupoids is also studied in the dual language and generality of integration of Lie-Reinhart algebras and commutative Hopf algebroids,
added pointer to:
Some questions I didn’t find answers to in Cech cocycles or $L_{\infty}-$algebra connections or on this page (maybe for lack of a thorough search):
Do we know if $\mathrm{exp}_{\Delta}(\mathfrak{g})$ satisfies homotopy descent over cartesian spaces?
Given a Lie $\infty-$group $G$, and its tangent $L_{\infty}$ algebra $\g$, we have a canonical map $\mathrm{exp}_{\Delta}(\mathfrak{g})\to \mathbf{B}G$. Can we say when this map is a weak equivalence? We know that it is when $G$ is a simply connected Lie group for example (or the other cases listed on this page). What can be said about general Lie $n-$groups and Lie $\infty-$groups?
On the second question:
For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, it will be the 1-truncation $\tau_1 \exp_\Delta(\mathfrak{g})$ which is weakly equivalent to $\mathbf{B}G$.
The higher truncations of $\exp_\Delta(\mathfrak{g})$ will pick up higher stacky homotopy groups from the ordinary homotopy groups of $G$.
For simply connected Lie groups we can equivalently take $\tau_2 \exp_\Delta(\mathfrak{g})$. This is modeled by $cosk_3 \exp_\Delta(\mathfrak{g})$ and this is for instance made use of in constructing the stacky refinement of $\tfrac{1}{2} \mathbf{p}_1 \,:\, \mathbf{B} Spin(n) \to \mathbf{B}^3 U(1)$ as a map of simplicial presheaves out of $cosk_3 \exp_\Delta(\mathfrak{g})$.
Regarding the first question:
This is a good question, which, I am afraid, I had never really discussed. But one can make some progress using the recognition Lemma for local fibrancy over $CartSp$ which we more recently we proved with Dmitri Pavlov, recorded on pp. 134 in our “Equivariant Principal $\infty$-bundles”. This gives that $cosk_2 \exp_\Delta(\mathfrak{g})$, being isomorphic to the $\overline{W}(-)$ of the sheaf of groups $G$, is locally fibrant on $CartSp$.
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