added pointer to:

- Rui Loja Fernandes, Marius Crainic,
*Lectures on Integrability of Lie Brackets*, Geometry & Topology Monographs**17**(2011) 1–107 [arxiv:math.DG/0611259, doi:10.2140/gtm.2011.17.1]

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,

- Alessandro Ardizzoni, Laiachi El Kaoutit, Paolo Saracco,
*Towards differentiation and integration between Hopf algebroids and Lie algebroids*, arXiv:1905.10288

Indeed, somehow I didn’t notice it originally.

]]>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.:…

Isn’t this a particular instance of the nerve-realization adjunction? Shouldn’t it be indicated as such?

]]>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….

]]>Thanks for catching this. Fixed now.

]]>I think something is the wrong way around. You have, in the adjunction, written $Spec$ as going from presheaves to algebras.

]]>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

$\mathcal{O} \;\colon\; \mathbf{X} \simeq \int^{n,k} y(\mathbb{R}^n \times \Delta[k]) \times \mathbf{X}(\mathbb{R}^n)_k \;\mapsto\; \int_{n,k} \underset{\mathbf{X}(\mathbb{R}^n)_k}{\prod} \Omega^\bullet_{si,vert}(\mathbb{R}^n \times \Delta^k_{mfd})$

Thanks! Pavol told me about this result a few weeks back when he visited Prague. That’s neat.

]]>Today’s reference

- Pavol Ševera, Michal Širaň,
*Integration of differential graded manifolds*, arxiv/1506.04898

insereted into Lie integration, without comments.

]]>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*.

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.

(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.

]]>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)

]]>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 ?

]]>okay, i have now fully boosted up statement and proof at Lie integration to line n-groups using Domenico’s argument.

]]>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.)

]]>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.

]]>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 $D^{n+1}$.

]]>added an argument to pass from a single $n$-form to a smooth family.

]]>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.

]]>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.)

]]>Your SVG works for me (Firefox 3.5.5 on Windows - this is work’s setup) - and looks very pretty I might add!

]]>