论文标题
衍生的谎言$ \ infty $ - 群体和代数差异几何形状
Derived Lie $\infty$-groupoids and algebroids in higher differential geometry
论文作者
论文摘要
我们研究了使用{\ der of lie $ \ infty $ groupoids和algebroids和algebroids}在高等差异几何形状中产生的各种问题。我们的第一个研究躺在各种类别的衍生几何学对象中的$ \ infty $ groupoids中的差异几何形状，包括衍生的流形，衍生的分析空间，衍生的非派遣空间，以及派生的繁殖空间，以及派生的繁殖。我们在派生的lie $ \ infty $ groupoid类别中构造了纤维对象（CFO）结构的类别。然后，我们研究$ l _ {\ infty} $ - 代数，这是派生的lie $ \ infty $ groupoids的无限对应物。然后，我们研究派生的谎言$ \ infty $ groupoid和代数的同位代数，并研究其同拷贝代表，我们称之为$ \ infty $ pressentations。我们将$ l _ {\ infty} $的$ \ infty $ - 代表与block开发的（Quasi-）凝聚模块以及$ \ infty $ - lie $ \ iffty $ groupoids to block-lifty $ groupoids to block-like-like $ - groupepoids of block-like $ - grouptoids to Blocksmith引入的$ \ infty $ groupoids。然后，我们将这些工具应用于研究奇异叶子及其特征类别。我们为$ l _ {\ infty} $ - algebroids对构建Atiyah类。我们研究奇异的叶子及其万能。我们构建了$ _ {\ infty} $ - 用于全态奇异叶子的代数，然后我们研究椭圆形涉及的结构，并证明了$ V $ Analytial Traimantial Traimantial Cooherent Cooherent Cooherent Sheaves的DG增强。这些例子激发了我们定义{\它是完美的奇异叶子}，这是一个单一叶片的子类别，但具有更好的同源代数。接下来，我们为单一叶子构建了各种谎言$ \ infty $ groupoids。然后，我们研究堆栈和较高类固醇的叶子。最后，我们证明了叶子的$ a _ {\ infty} $ de rham定理，以及Riemann-Hilbert的foriand $ \ infty $ local-local-local System Foliated歧管的通信。
We study various problems arising in higher differential geometry using {\it derived Lie $\infty$-groupoids and algebroids}.We first study Lie $\infty$-groupoids in various categories of derived geometric objects in differential geometry, including derived manifolds, derived analytic spaces, derived noncommutative spaces, and derived Banach manifolds. We construct category of fibrant objects (CFO) structures in the category of derived Lie $\infty$-groupoids. Then we study $L_{\infty}$-algebroids which are the infinitesimal counterpart of derived Lie $\infty$-groupoids. We then study the homotopical algebras for derived Lie $\infty$-groupoids and algebroids and study their homotopy-coherent representations, which we call $\infty$-representations. We relate $\infty$-representations of $L_{\infty}$-algebroids to (quasi-) cohesive modules developed by Block, and $\infty$-representations of Lie $\infty$-groupoids to $\infty$-local system introduced by Block-Smith. Then we apply these tools in studying singular foliations and their characteristic classes. We construct Atiyah classes for $L_{\infty}$-algebroids pairs. We study singular foliations and their holonomies. We construct $Ł_{\infty}$-algebroids for holomorphic singular foliations, and then We study elliptic involutive structures and prove an dg-enhancement of $V$-analytic coherent sheaves. These examples inspire us to define {\it perfect singular foliations}, which is a subcategory of singular foliation but with better homological algebras. Next, we construct various Lie $\infty$-groupoids for singular foliations. Then we study foliations on stacks and higher groupoids. Finally, we prove an $A_{\infty}$ de Rham theorem for foliations, and Riemann-Hilbert correspondence for foliated $\infty$-local system foliated manifolds.
