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

TL;DR

This paper constructs a natural morphism linking the nerve of a curved L-infinity algebra to a subset satisfying a gauge condition, extending homotopical perturbation theory and relating to higher holonomy concepts.

Contribution

It introduces a morphism connecting the nerve of curved L-infinity algebras to Maurer--Cartan elements under a gauge condition, extending perturbation theory to curved cases.

Findings

The morphism equals the identity on its image.

The morphism generalizes holonomy for nilpotent Lie algebras.

Extension of perturbation theory to curved L-infinity algebras.

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.