Integrability of degree n Q-manifolds via the Lie functor

Establish whether the differentiation functor Lie from the category of Lie n-groupoids to the category of degree n Q-manifolds is essentially surjective; equivalently, determine whether every degree n Q-manifold (i.e., every Lie n-algebroid) integrates to a Lie n-groupoid up to Morita equivalence.

Background

The paper recalls the construction of a Lie functor that differentiates Lie n-groupoids to degree n Q-manifolds (Theorem on the Lie functor), extending the classical passage from Lie groupoids to Lie algebroids when n=1.

For n=1, integrability of Lie algebroids into Lie groupoids is well studied and obstructions are known. For higher n, only special cases are understood (e.g., string 2-groups, double Lie groupoids), and a general integrability theory matching the full scope of the Lie functor is not yet available. The authors highlight that the essential surjectivity of this functor—i.e., whether every degree n Q-manifold arises from a Lie n-groupoid—is not settled.