摘要翻译：
在这篇笔记中，我们提出了一个正在进行的工作，其主要目的是建立一个分类版本的sheaf理论。本文提出了一个导出范畴束的概念，它是模束的复形概念的范畴化版本，以及它的准相干和完美版本。我们还解释了如何利用派生代数几何和高等范畴理论的思想来构造这些范畴束的Chern特征，这是具有循环同调值的完美复形的Chern特征的范畴化版本。我们的构造以一种基本的方式使用了格式X的导出循环空间，该格式是一个函数理论与X的循环同调密切相关的导出格式。这一工作可以看作是定义椭圆对象的代数类比及其特征类的一种尝试。本文是一项正在进行的工作的概述，细节将在其他地方出现。
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英文标题：
《A note on Chern character, loop spaces and derived algebraic geometry》
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作者：
B. To\"en, G. Vezzosi
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Algebraic Topology        代数拓扑
分类描述：Homotopy theory, homological algebra, algebraic treatments of manifolds
同伦理论，同调代数，流形的代数处理
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英文摘要：
  In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.
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PDF链接：
https://arxiv.org/pdf/0804.1274