On nLab’s page on Lie’s three theorems, Lie’s theorem 2 says that the functor $Lie: \textsf{Local LieGp} \rightarrow \textsf{Finite Dim LieAlg}$ is fully faithfull, and Lie’s theorem 3 says this functor is essentially surjective, so there is an equivalence of categories. But Lie’s theorem 1 is purely local. They say:
Here one lacks a good notion of differentiable manifold for extending this to a global result.
What good notion of differentiable manifold is lacking?
Is our notion of differentiable manifolds too general or too strict? Is the category $\textsf{DiffMan}$ not ‘nice’ enough? I hope someone can explain to me what they mean.