Higher Lie Theory

The construction of the infinity-groupoid associated to homotopy Lie algebras due to Getzler, after Hinich’s works, can actually be presented in a more simple and more powerful way. While Getzler defines it intrinsically as the gauge (kernel) of the Dupont’s contraction, I will use the seminal approach of Kan to adjoint functors to/from simplicial sets. This will provide me with a new way to integrate homotopy Lie algebras which shows how Getzler’s function can be represented by a universal object, that it admits a left adjoint, that it is actually functorial with respect to infinity-morphisms. This approach will allow me to fully describe the sets of horn fillers: this will settle the Kan property in a canonical way which will make this object and algebraic infinity-groupoid, that is with given horn fillers. This change of paradigm will make us leave algebraic topology and enter algebra. In this way, we can perform explicit and algorithmic computations. For instance, the first horn filler gives the celebrated Baker—Campbell—Hausdorff formula. The higher horn fillers introduce for the first time higher BCH formulas. This also gives us homotopy Lie algebra models in rational homotopy theory.

This is a project with Daniel Robert-Nicoud.