All known realizations of complete Lie algebras coincide

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.