∞-Lie theory (higher geometry)
There is an obvious functor
which sends every Lie group to its Lie algebra and every homomorphism of Lie groups to the corresponding homomorphism of Lie algebras.
Lie’s three theorems can be understood as establishing salient properties of this functor. More exactly, Lie’s theorems provide a foundation establishing an equivalence between local Lie groups and Lie algebras; subsequent work by Elie Cartan and others extended the theorems to give information on (global) Lie groups via the functor $Lie$.
Lie’s first theorem is purely local; see the Encyclopedia of Math for a statement. (Here one lacks a good notion of differentiable manifold for extending this to a global result.)
Lie II Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak{g} = Lie(G)$ and $\mathfrak{h} = Lie(H)$, such that $G$ is simply connected. If $f : \mathfrak{g} \to \mathfrak{h}$ is a morphism of Lie algebras, then there is a unique morphism $F : G \to H$ of Lie groups lifting $f$, i.e. such that $f = Lie(F)$.
Lie III (Cartan-Lie theorem) The functor $Lie$ is essentially surjective on objects: for every finite dimensional real Lie algebra $\mathfrak{g}$ there is a real Lie group $G$ such that $\mathfrak{g} \cong Lie(G)$. Moreover, there exists such $G$ which is simply connected.
In his third theorem, Lie proved only the existence of of a local Lie group, but not the global existence (nor simply connected choice) which were established a few decades later by Elie Cartan. Hence the full theorem is properly called the Cartan-Lie theorem. From an nPOV, the third Lie theorem establishes the essential surjectivity of the functor $Lie$ from the category of local Lie groups to the category of finite dimensional real Lie algebras, and similarly the second Lie theorem establishes that this functor is fully faithful (so the two together establish that this functor is an equivalence). The historically incorrect naming of the Cartan-Lie theorem as the “third Lie theorem” is largely due to the influence of a book based on lectures of Jean-Pierre Serre (Lie algebras and Lie groups, W.A. Benjamin, 1965).
Let $LieGroups_{simpl}$ be the full subcategory of $LieGroups$ consisting of simply connected Lie groups. Then the above implies that restricted to $LieGroups_{simpl}$, the functor $Lie$ becomes an equivalence of categories.
The horizontal categorification of Lie’s theorems for Lie groups leads to analogous statements for Lie groupoids. In other words, there are analogous properties for the differentiation functor
from Lie groupoids to Lie algebroids.
In the case of Lie groupoids, the condition of a group being simply connected which plays an important role in the above statements is generalized to the condition that source fibers of the Lie groupoid (the preimages $s^{-1}(x)$ of the source map $s : C_1 \to C_0$ at every object $x \in C_0$ of the Lie groupoid $C$) are simply connected. One says
Lie II for Lie groupoids now reads exactly as Lie II for Lie groups, with “simply connected” replaced by “source simply connected”.
Lie II for Lie groupoids was proven in
and
Lie III for Lie groupoids does not hold in direct generalization:
by the general mechanism of Lie integration the space of morphisms of the source simply-connected topological groupoid $G$ integrating a Lie algebroid $\mathfrak{g}$ is a quotient space. This quotient may fail to be a manifold due to singularities.
The precise conditions under which Lie algebroids do have Lie groupoids integrating them were found in
A comprehensive review is in
A review of Lie theory of Lie groupoids in on pages 3-5 of
and in the introduction of
This failure of Lie III for Lie groupoids, i.e. for internal groupoids in Diff seems to suggest that the category of manifolds is not the natural home for general Lie theory. More concretely, it seems to suggest that Lie theory ought to be practiced internal to some category of generalized smooth spaces.
One such choice is given by replacing manifolds by differentiable stacks.
The generalization of Lie’s theorems from Lie groups to to stacky Lie groupoids is discussed in