Homotopy 'groups' of Lie algebroids and obstruction of integration

Lie algebroids are certain degree 1 super manifolds. We propose a definition of fibration in the category of Lie algebroids, and prove the long exact sequence of homotopy groups associated to a fibration. On the other hand, certain Lie algebroids, unlike (finite dimensional) Lie algebras which can always integrate to Lie groups, can not integrate to Lie groupoids. The obstructions are classified by Crainic and Fernandes. It turns out that the long exact sequence of homotopy groups associated to a suitable fibration encodes the obstruction of integration. This is a joint work (DOI:10.1016/j.aim.2010.10.006)
with Olivier Brahic.