Derived Lie $\infty$-groupoids and algebroids in higher differential geometry

Abstract

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 $\L_{\infty}$-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.