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

Abstract

We show an alternative construction of the cosimplicial free complete diferential graded Lie algebra ${\mathfrak{L}}_{\bullet }=\hat{\mathbb{L}}(s^{-1}{\Delta }^{\bullet })$ 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 $⟨L⟩={\text{Hom}}_{\text{cdgl}}({\mathfrak{L}}_{\bullet },L)$ is isomorphic to its nerve ${\gamma }_{\bullet }(L)$, a deformation retract of the Getzler-Hinich realization $\text{MC}({\mathcal{A}}_{\bullet }\hat{\otimes }L)$.

19 pages

References in corpus (3)

• Representing the deformation $\infty$-groupoid
• Homotopy theory of complete Lie algebras and Lie models of simplicial sets
• Maurer-Cartan elements and homotopical perturbation theory

Cited by in corpus (4)

• Homotopy theory of complete Lie algebras and Lie models of simplicial sets
• Descent of Deligne-Getzler $\infty$-groupoids
• A model structure for the Goldman-Millson theorem
• Spatial realization of a Lie algebra and Bar construction of a group