Ezra Getzler: Cubical approach to higher Lie groupoids

Abstract:

Attached to any nilpotent Lie algebra \(L\) is a Kan complex \(NL\); this simplicial set, called the nerve of \(L\), is naturally isomorphic to the nerve of the associated Lie group G(L) of \(L\). In higher Lie theory, this construction generalizes: the nerve of a nilpotent differential graded Lie algebra \(L\), or more generally, nilpotent \(L_\infty\) algebra, is a certain Kan complex \(NL\). In this talk, we present a new approach to the construction of \(NL\) based on cubical instead of simplicial sets.

The construction of the cubical nerve of a nilpotent \(L_\infty\) algebra is modeled on the construction of the simplicial nerve, with the Dupont homotopy acting on differential forms on a simplex replaced by a considerably simpler homotopy acting on differential forms on a cube.

Next, we review the homotopy theory of cubical sets and the construction of an explicit functor from cubical Kan complexes to simplicial Kan complexes giving an equivalence of homotopy theories.

Our main result is that applying this functor to the cubical nerve of a nilpotent \(L_\infty\) algebra gives its simplicial nerve. The proof relies on Berglund’s homotopical perturbation theory for \(L_\infty\) algebras.

In the special case where \(L\) is a nilpotent differential graded Lie algebra concentrated in degrees \(> -2\), the nerve is a \(2\)-groupoid. Our main theorem allows us to identify this \(2\)-groupoid with the Deligne \(2\)-groupoid of \(L\).