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

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Urtzi Buijs , Yves Félix , Aniceto Murillo , Daniel Tanré ·

YOU? · · 2017 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1702.04397 · OA: W2589021821

We show an alternative construction of the cosimplicial free complete diferential graded Lie algebra $\mathfrak{L}_\bullet=\widehat{\mathbb{L}}(s^{-1}Δ^\bullet)$ 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 $γ_\bullet(L)$, a deformation retract of the Getzler-Hinich realization $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.