Van Est integration in higher Lie theory

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Lie algebroids and their higher version the Lie $n$-algebroids are differential graded manifolds and they describe infinitesimal symmetries. The integration of these structures to structures encoding global symmetries -- that is, to Lie groupoids and their higher versions -- is a problem that has interested many mathematicians in the last decades. In the early 2000's, Crainic and Fernandes proposed in a seminal paper a path-integration of Lie algebroids. This beautiful method has a lot of analytical difficulties. Now that the adjoint representation (up to homotopy) of a Lie algebroid is well-understood, we propose to study the integration problem from a new and more algebraic point of view, extending van Est's method for integrating Lie algebras.