Chenchang Zhu
We establish a functor $Kan$ from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.
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On the local Kan structure and differentiation of simplicial manifolds
本文证明任意单纯流形在局部意义下满足 Kan 条件，从而将高阶李广群的微分技术扩展至一般单纯流形，导出了将其微分为高阶李代数胚的方法。
In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids, we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids, we use inner Kan fibrations over the interval to model bibundles of higher groupoids. In particular, we show that our definitions given by the simplicial method agree with those given by the categorification approach to actions, principal bundles, and bibundles of 2-groupoids. In addition, we use the …
Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group. In the case of the string Lie 2-algebra of Baez and Crans, this recovers the model of the string group introduced in math.QA/0504123 .
Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-group…
Lie's Third Theorem for Lie $\infty$-Algebras
本文通过局部最小模型理论，首次证明了有限型Lie ∞-代数的Lie第三定理，并给出了弦Lie 2群的新模型。
Let $Y$ be a CW-complex with a single 0-cell, let $K$ be its Kan group, a free simplicial group whose realization is a model for the space $ΩY$ of based loops on $Y$, and let $G$ be a Lie group, not necessarily connected. By means of simplicial techniques involving fundamental results of {\smc Kan's} and the standard $W$- and bar constructions, we obtain a weak $G$-equivariant homotopy equivalence from the geometric realization $|\roman{Hom}(K,G)|$ of the cosimplicial manifold $\roman{Hom}(K,G)$ of homomorphisms from $K$ to $G$ to the space $\roman{Map}^o(Y,BG)$ of based maps from $Y$ to the c…
We describe an $A_\infty$-quasi-equivalence of dg-categories between the first authors' $\mathcal{P}_{\mathcal{A}}$ ---the category of category of prefect $A^0$-modules with flat $\Z$-connection, corresponding to the de Rham dga $\mathcal{A}$ of a compact manifold $M$--- and the dg-category of \emph{infinity-local systems} on $M$ ---homotopy coherent representations of the smooth singular simplicial set of $M$, $\Pinf$. We understand this as a generalization of the Riemann--Hilbert correspondence to $\Z$-connections ($\Z$-graded superconnections in some circles). In one formulation an infinity…
Higher Vector Bundles
本文在(Kan)单纯纤维化背景下引入分裂的高阶类比，建立了同伦表示与带分裂的单纯向量丛的等价关系，作为Higher Grothendieck Correspondence的体现并扩展了VB-groupoids的早期工作。
Geometric differentiation of simplicial manifolds
本文提出了一种基于更高李叠代数的几何微分方法，通过规范形式定理和van Est同构定理，统一了同伦和正式方法，揭示了Dold-Kan对应式的单调细化机制。
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infinity algebras concentrated in degree >-n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogue…
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer-Cartan equation to closed differential forms. Following ideas of Ezra Getzler we then impose a gauge condition which cuts out a finite-dimensional simplicial submanifold. This "smaller" simplicial manifold is (the nerve of) a local Lie k-gr…
The holonomy Lie $\infty$-groupoid of a singular foliation I
本文提出通过双子浸入整合奇异叶形的有限维更高李群体，并将其1-截断为Androulidakis-Skandalis holonomy群体。
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