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

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 -algebras.