Higher holonomy for curved L${}_\infty$-algebras 1: simplicial methods

Abstract

We construct a natural morphism $\rho$ from the nerve $\text{MC}_\bullet(L) = \text{MC}(\Omega_\bullet \widehat{\otimes} L)$ of a pronilpotent curved L${}_\infty$-algebra $L$ to the simplicial subset $\gamma_\bullet(L) = \text{MC}(\Omega_\bullet \widehat{\otimes} L,s_\bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $\gamma_\bullet(L) \hookrightarrow \text{MC}_\bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_\infty$-algebras to curved L${}_\infty$-algebras. The morphism $\rho$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $\rho^\square$ of $\rho$ to identify $\rho$ with higher holonomy for semiabelian curved \Linf-algebras.