Maurer-Cartan elements in the Lie models of finite simplicial complexes

TL;DR

This paper explores the relationship between differential graded Lie algebra models and finite simplicial complexes, establishing a correspondence between algebraic structures and topological components.

Contribution

It introduces a new correspondence linking the Deligne groupoid of Lie algebra models to the connected components of simplicial complexes.

Findings

Established a functorial association between simplicial complexes and Lie algebras.

Linked the homology of Lie algebras to homotopy groups of realizations.

Connected the Deligne groupoid to simplicial complex components.

Abstract

In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.