论文标题
所有已知的完全谎言代数的实现重合
All known realizations of complete Lie algebras coincide
论文作者
论文摘要
我们证明，对于任何降低分级的差异为代数，经典的Quillen几何实现$ \ langle l \ rangle_q $是同质的，等于$ \ langle l \ langle l \ rangle = hom _ hom _ {\ bf cdgl}（\ bf cdgl}（\ bf cdgl}（\ mathfrak）分级为lie代数$ \ mathfrak {l} _ \ bullet $。由于后者是Deligne-Getzler-Hinich实现MC $ {} _ \ bullet（l）$的变形缩回，因此，我们推断出它，直到同型，只有一个实现函数来实现完整的差异级别的lie代数。直接的后果包括Baues-Lemaire猜想的基本证据，以及将Quillen实现为代表函数的描述。
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.
