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

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.