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

Abstract: We construct a natural morphism $\rho$ from the nerve $$\text{MC}<em>\bullet(L) = \text{MC}(\Omega</em>\bullet \widehat{\otimes} L)$$ of a pronilpotent curved L${}<em>\infty$-algebra $L$ to the simplicial subset $$\gamma</em>\bullet(L) = \text{MC}(\Omega_\bullet \widehat{\otimes} L,s_\bullet)$$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $$\gamma_\bullet(L) \hookrightarrow \text{MC}<em>\bullet(L)$$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}</em>\infty$-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 $\rho<sup>\square$ of $\rho$ to identify $\rho$ with higher holonomy for semiabelian curved \Linf-algebras.