All known realizations of complete Lie algebras coincide

TL;DR

This paper proves that all known realizations of complete Lie algebras are homotopy equivalent, establishing a unique realization functor and providing new insights into their structure and applications.

Contribution

It demonstrates that various realization functors for complete differential graded Lie algebras are homotopy equivalent, unifying their understanding and proving the Baues-Lemaire conjecture.

Findings

All realization functors are homotopy equivalent.

The Quillen realization is representable.

Provides an elementary proof of the Baues-Lemaire conjecture.

Abstract

We prove that for any reduced differential graded Lie algebra L, the classical Quillen geometrical realization $\langle L\rangle_Q$ is homotopy equivalent to the realization $\langle L\rangle= Hom_{\bf cdgl}(\mathfrak{L}_\bullet, L)$ constructed via the cosimplicial free complete differential graded Lie algebra $\mathfrak{L}_\bullet$. As the latter is a deformation retract of the Deligne-Getzler-Hinich realization MC${}_\bullet(L)$ we deduce that, up to homotopy, there is only one realization functor for complete differential graded Lie algebras. Immediate consequences include an elementary proof of the Baues-Lemaire conjecture and the description of the Quillen realization as a representable functor.