Higher holonomy for curved L∞-algebras 1: simplicial methods

Abstract

We construct a natural morphism ρ from the nerve MC∙⁡(L)=MC⁡(Ω∙⊗^L) of a pronilpotent curved L∞-algebra L to the simplicial subset γ∙(L)=MC⁡(Ω∙⊗^L,s∙) of Maurer-Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion γ∙(L)↪MC∙⁡(L). The proof uses the extension of Berglund's homotopical perturbation theory for L∞-algebras to curved L∞-algebras. The morphism ρ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue ρ◻ of ρ to identify ρ with higher holonomy for semiabelian curved L∞-algebras. This article is part of the theme issue 'Derived Lie algebras in geometry and topology'.