Lie theory of Lie algebroids via higher structures in differentiable geometry
Higher structures such as differentiable stacks, simplicial objects, Lie groupoids, gerbes have been recently introduced in differential geometry. They are useful, for example, to integrate Lie algebroids. Lie algebroids are a mixture of Lie algebras and manifolds, or it could be understood as a degree-1 super-manifold with a degree-1 vector field Q with Q2 = 0. Unlike a (finite dimensional) Lie algebra which always has a (simply connected) Lie group corresponding to, a Lie algebroid may not have a corresponding Lie groupoid. But there is a one-to-one correspondence between Lie algebroids and stacky Lie groupoids (or Lie 2-groupoids). Lie’s second theorem also holds for these universal objects