Homotopy theory of complete Lie algebras and Lie models of simplicial sets

TL;DR

This paper advances Lie rational homotopy theory by extending Quillen's construction to non-simply connected spaces, establishing a model category structure, and demonstrating the preservation of homotopies and weak equivalences.

Contribution

It introduces a model category structure on complete differential graded Lie algebras and constructs explicit functors that form a Quillen pair, extending rational homotopy theory.

Findings

Realization of the model of a finite connected simplicial set is its Bousfield-Kan completion plus an external point.

The functors preserve homotopies and weak equivalences.

A basis for Lie rational homotopy theory for all spaces is established.

Abstract

In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, 'model' and 'realization', between the categories of simplicial sets and complete differential graded Lie algebras. This paper is a follow up of this work. We show that when X is a finite connected simplicial set, then the realization of the model of X is the disjoint union of the Bousfield-Kan completion of X with an external point. We also define a model category structure on the category of complete differential graded algebras making the two previous functors a Quillen pair, and we construct an explicit cylinder. In particular, these functors preserve homotopies and weak equivalences and therefore, this gives the basis for developing a Lie rational homotopy theory for all spaces.