  We present a novel approach to the problem of integrating homotopy Lie
algebras by representing the Maurer-Cartan space functor with a universal
cosimplicial object. This recovers Getzler's original functor but allows us to
prove the existence of additional, previously unknown, structures and
properties. Namely, we introduce a well-behaved left adjoint functor, we
establish functoriality with respect to infinity-morphisms, and we construct a
coherent hierarchy of higher Baker-Campbell-Hausdorff formulas. Thanks to these
tools, we are able to establish the most important results of higher Lie
theory. We use the recent developments of the operadic calculus, which leads us
to explicit tree-wise formulas at all stage. We conclude by applying this
theory to rational homotopy theory: the left adjoint functor is shown to
provide us with homotopy Lie algebra models for topological spaces which
faithfully capture their rational homotopy type.
