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

We construct a natural morphism $\rho$ from the nerve ${\text{MC}}_{\bullet }(L)=\text{MC}({\mathrm{\Omega }}_{\bullet }\widehat{\otimes }L)$ of a pronilpotent curved L${}_{\mathrm{\infty }}$-algebra $L$ to the simplicial subset ${\gamma }_{\bullet }(L)=\text{MC}({\mathrm{\Omega }}_{\bullet }\widehat{\otimes }L,s_{\bullet })$ 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${}_{\mathrm{\infty }}$-algebras to curved L${}_{\mathrm{\infty }}$-algebras. The morphism $\rho$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue ${\rho }^{\mathrm{\square }}$ of $\rho$ to identify $\rho$ with higher holonomy for semiabelian curved \Linf-algebras.