Lie's third theorem via graded geometry

Question

Lie's third theorem : Given any finite dimensional Lie algebra $\mathfrak{g}$, there exists a Lie group $G$ whose Lie algebra is equal to $\mathfrak{g}$.

In one of the talks, speaker mentions that this is easier to believe if one think of this in terms of differential graded manifolds.

Think of $\mathfrak{g}$ as a dg-manifold $\mathfrak{g}[1]$. Then, "the associated fundamental group" is the candidate for the Lie group.

Here, paths are to be seen as morphisms of dg-manifolds $T[1]I\rightarrow \mathfrak{g}[1]$, and homotopies are to be seen as morphisms of dg-manifolds $T[1](I\times I)\rightarrow \mathfrak{g}[1]$.

Can some one point me to a reference where this is discussed in detail.

Answer 1

This is the Duistermaat–Kolk construction of a simply connected Lie group that integrates the given Lie algebra $\def\g{{\frak g}}\g$.

The starting observation is that for any simply connected Lie group $G$ the canonical morphism of group objects in diffeological spaces (or smooth sets) $$\def\Hom{\mathop{\rm Hom}}\Hom([0,1],G)_0/\Hom([0,1]^2,G)_0→G$$ that evaluates at 1 is an isomorphism. Here $\Hom$ denotes the internal hom in sheaves of sets on smooth manifolds (or, equivalently, diffeological spaces), equipped with the group structure induced from $G$, and $\Hom([0,1],G)_0$ denotes the subobject of $\Hom([0,1],G)$ consisting of paths that evaluate to $1∈G$ on 0, i.e., paths $[0,1]→G$ that start at $1∈G$. Likewise, $\Hom([0,1]^2,G)_0$ denotes the subobject of $\Hom([0,1]^2,G)$ comprising homotopies that do not move the endpoints, and the initial endpoint stays fixed at $1∈G$. That is, paths that are homotopic relative endpoints are identified.

A key observation to make now is that both ingredients of the quotient can be recovered from their differentials (or derivatives). That is, the above isomorphism can rewritten as $$\Hom([0,1],\g)/\Hom([0,1]^2,\g^2)_{0,\flat}→G.$$ To ensure that a smooth map $ω\colon[0,1]^2→\g^2$ is the derivative of a homotopy described above, we require that the endpoints stays fixed (encoded by the vanishing of the corresponding partial derivatives, depicted by the subscript $0$) and the curvature form $\def\d{{\rm d}}\d\,ω+[ω,ω]/2$ vanishes, depicted by the subscript $\flat$.

The advantage of this construction is that it is manifestly functorial, easy to describe, and readily generalizes to Lie ∞-algebras and Lie ∞-algebroids.

Answer 2

A sketch of how to use graded geometry to integrate dg manifolds and, in particular, Lie algebroids appeared first in the legendary paper https://arxiv.org/abs/math/0105080

More details appeared later in https://arxiv.org/abs/1506.04898

Roughly speaking, in order to integrate a given Lie algebroid one can produce an infinite dimensional simplicial manifold given at level $n$ by all the Lie algebroid maps from the tangent bundle of the $n$-dimensional simplex to your fixed Lie algebroid.