论文标题
完成$ l_ \ infty $ - 代数及其同义理论
Complete $L_\infty$-algebras and their homotopy theory
论文作者
论文摘要
我们分析了用于代数和代数几何变形理论的全滤光度$ L_ \ infty $ - 代数的同型$ l_ \ infty $ - 代数的模型。 We provide an explicit proof of an unpublished result of E.\ Getzler which states that the category $\hat{\mathsf{Lie}}_\infty$ of such $L_\infty$-algebras and filtration-preserving $\infty$-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory.我们方法的新应用包括明确的同型回调模型，以及对怀特海的定理的类似物：在某些温和条件下，每个过滤的$ l_ \ infty $ -quasi-ismormorphism in $ \ hat {\ mathsf {\ mathsf {lie}} _ \ iftty $具有滤液保留的均匀配合型。另外，我们证明了Simplicial Maurer--Cartan函子，该函数将KAN Simplicial集合分配给每个$ L_ \ infty $ -Algebra $ \ hat {\ Mathsf {lie}} _ \ infty $，是相应CFOS之间的精确函数。最后，我们为通过$ \ infty $ morphimism提升毛勒 - 卡丹元素的总体问题提供了阻碍理论。障碍物类别位于相应切线图的相关分级映射锥中。
We analyze a model for the homotopy theory of complete filtered $L_\infty$-algebras intended for applications in algebraic and algebro-geometric deformation theory. We provide an explicit proof of an unpublished result of E.\ Getzler which states that the category $\hat{\mathsf{Lie}}_\infty$ of such $L_\infty$-algebras and filtration-preserving $\infty$-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Novel applications of our approach include explicit models for homotopy pullbacks, and an analog of Whitehead's Theorem: under some mild conditions, every filtered $L_\infty$-quasi-isomorphism in $\hat{\mathsf{Lie}}_\infty$ has a filtration preserving homotopy inverse. Also, we show that the simplicial Maurer--Cartan functor, which assigns a Kan simplicial set to each $L_\infty$-algebra in $\hat{\mathsf{Lie}}_\infty$, is an exact functor between the respective CFOs. Finally, we provide an obstruction theory for the general problem of lifting a Maurer-Cartan element through an $\infty$-morphism. The obstruction classes reside in the associated graded mapping cone of the corresponding tangent map.
