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 $CartSp$.

]]>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?

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.

]]>