Lie theory for nilpotent $L_\infty$-algebras


The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor $\gamma$ from $L_\infty$-algebras concentrated in degree $>-n$ to $n$-groupoids. (We actually construct the nerve of the $n$-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont’s proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula.

In the case of abelian $L_\infty$ algebras (i.e., chain complexes), the functor $\gamma$ is the Dold-Kan simplicial set.