All known realizations of complete Lie algebras coincide

TL;DR

The paper proves that for simply connected differential graded Lie algebras, the classical Quillen geometrical realization ${\langle L\rangle_Q}$ is homotopy equivalent to a cosimplicial Lie algebra–based realization ${\langle L\rangle}$. It constructs a surjective quasi‑isomorphism $\Phi:\lambda\langle L\rangle\to L$, showing compatibility with brackets and differentials, which implies ${\langle L\rangle}$ and ${\langle L\rangle_Q}$ are homotopy equivalent; this elevates the cosimplicial realization to the universal realization in the cdgl setting and extends the Quillen equivalence. As immediate consequences, the Quillen realization is representable by ${\mathfrak{L}}_\bullet$, and the Baues–Lemaire conjecture follows in this framework, with further extensions to non‑finite type and non‑simply connected cases via enriched differential graded Lie algebras and enriched cochains. The work also clarifies the relationship between Sullivan and Quillen approaches and situates the Eckmann–Hilton dual of Sullivan’s realization in this context.

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.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Remark 2.4
• Corollary 2.5
• proof