The infinity Quillen functor, Maurer-Cartan elements and DGL realizations

TL;DR

This paper introduces a new construction of a cosimplicial free complete differential graded Lie algebra and proves its geometric realization is isomorphic to the nerve of any complete DGL, linking it to the Getzler-Hinich realization.

Contribution

It provides an alternative construction of the cosimplicial free complete DGL and establishes an isomorphism between its geometric realization and the nerve, connecting different models of DGL realizations.

Findings

New Lie bracket formula for Lie polynomials on tensor algebra.

Geometric realization of a complete DGL is isomorphic to its nerve.

Nerve is a deformation retract of the Getzler-Hinich realization.

Abstract

We show an alternative construction of the cosimplicial free complete diferential graded Lie algebra $\mathfrak{L}_\bullet=\widehat{\mathbb{L}}(s^{-1}\Delta^\bullet)$ based on a new Lie bracket formulae for Lie polynomials on a general tensor algebra. Based on it,we prove that for any complete differential graded Lie algebra $L$, its geometrical realization $\langle L\rangle=\text{Hom}_{\text{cdgl}}(\mathfrak{L}_\bullet,L)$ is isomorphic to its nerve $\gamma_\bullet(L)$, a deformation retract of the Getzler-Hinich realization $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.