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

Abstract

We construct a natural morphism $\ddot{I}$ from the nerve ${\text{MC}}_{\bullet }(L)=\text{MC}(\hat{I}{\text{c◯}}_{\bullet }\hat{\otimes }L)$ of a pronilpotent curved L${}_{\infty }$-algebra $L$ to the simplicial subset ${\hat{I}}_{\bullet }^3(L)=\text{MC}(\hat{I}{\text{c◯}}_{\bullet }\hat{\otimes }L,s_{\bullet })$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion ${\hat{I}}_{\bullet }^3(L)\hookrightarrow {\text{MC}}_{\bullet }(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_{\infty }$-algebras to curved L${}_{\infty }$-algebras. The morphism $\ddot{I}$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $\ddot{I}{}^{\square }$ of $\ddot{I}$ to identify $\ddot{I}$ with higher holonomy for semiabelian curved \Linf-algebras.

18 pages; final version, to appear in Philosophical Transactions of the Royal Society A