无穷大局部系统的Riemann-Hilbert对应
A Riemann--Hilbert correspondence for infinity local systems
摘要：
本文描述了第一作者的$\mathcal{P}_{\mathcal{A}}$ --具有平坦$\Z$ -联络的完美$A^0$ -模范畴的范畴与$M$上的\emph{infinity-local systems}--光滑奇异单纯集的同伦相干表示$\Pinf$的dg-范畴之间的$A_\infty$ -拟等价.我们把这理解为Riemann-Hilbert对应到$\Z$ -联络（在某些圈子里是$\Z$ -分次超联络）的推广。在一个公式中，无穷-局部系统是单纯集$π_{\infty}M$之间的单纯映射和借助于单纯神经和Dold-Kan对上链复形的dg-范畴的重新包装。这个理论的关键使用Igusa的概念更高的holonomy运输的$\Z$ -联系，这是一个衍生的陈的主要思想广义holonomy。
英文摘要：
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-local system is simplicial map between the simplicial sets $π_{\infty}M$ and a repackaging of the dg-category of cochain complexes by virtue of the simplicial nerve and Dold-Kan. This theory makes crucial use of Igusa's notion of higher holonomy transport for $\Z$-connections which is a derivative of Chen's main idea of generalized holonomy.
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