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 $\mathfrak{L}\bullet=\widehat{\mathbb{L}}(s^{-1}\Delta^\bullet)$ 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 $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.