Characterizations and Uniqueness of the stable motivic homotopy theory

稳定动机同伦理论的特征和独特性

• 批准号: 269515708
• 负责人: Professor Dr. Jens Hornbostel
• 金额: --
• 依托单位: Arbeitsgruppe Topologie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2017-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/269515708?language=en
• 关键词: Characterizations; Uniqueness; stable; motivic; homotopy

The idea of motivic homotopy theory is to apply constructions and techniques from classical homotopy theory to solve problems in algebraic and arithmetic geometry. (A list of succesful examples is provided in section 2.2.1 of the Proposal for the DFG-Priority Program.) The key object here is the stable motivic homotopy theory SH(k) for a given base field k as invented by Morel and Voevodsky in the late 90s. Important algebraic cohomology theories such as motivic cohomology, algebraic and hermitian K-theory and algebraic cobordism are representable in SH(k). The goal of this project is to obtain a better conceptual understanding of the stable motivic homotopy category SH(k) from different points of view, that is comparing various descriptions and characterizations of it, including the question of uniqueness. In particular, we wish to study descriptions of SH(k) in terms of derivators and of infinity-categories, building upon work of Ayoub and Robalo. Moreover, we will investigate if the rigidity theorem of Schwede concerning uniqueness of models for the classical stable homotopy category allows some kind of refinement to the motivic case. An underlying theme of all these problems is the question how much of the ``higher structure'' of SH(k) is already determined by its triangulated structure, that is independent of the choosen model resp. description.

动机同伦理论的思想是应用经典同伦理论的构造和技术来解决代数和算术几何中的问题。（DFG优先计划提案第2.2.1节提供了成功实例列表。）这里的关键对象是由Morel和Voevodsky在90年代后期发明的对于给定基域k的稳定运动同伦理论SH（k）。重要的代数上同调理论，如motivic上同调，代数和埃尔米特K-理论和代数配边在SH（k）中表示。本项目的目标是从不同的角度对稳定动机同伦范畴SH（k）有一个更好的概念性理解，即比较它的各种描述和刻画，包括唯一性问题。特别是，我们希望研究SH（k）的描述的导子和无穷范畴，Ayoub和Robalo的工作的基础上。此外，我们将调查，如果刚性定理施维德关于唯一性模型的经典稳定同伦范畴允许某种细化的motivic情况。所有这些问题的一个基本主题是SH（k）的“更高结构”有多少已经由其三角结构决定的问题，这与所选择的模型无关。说明.