Characterizations and Uniqueness of the stable motivic homotopy theory

动机同伦理论的思想是应用经典同伦理论中的构造和技术来解决代数几何和算术几何中的问题。（dfg优先计划提案第2.2.1节提供了成功示例的列表。）这里的关键对象是由Morel和Voevodsky在90年代末提出的针对给定基场k的稳定动力同伦理论SH(k)。重要的代数上同调理论如动机上同调、代数和厄米特k -理论以及代数上同调在SH(k)中得到了表征。本课题的目的是从不同的角度对稳定动同伦范畴SH(k)有更好的概念性理解，即比较它的各种描述和表征，包括唯一性问题。特别地，我们希望在Ayoub和Robalo的工作的基础上，研究SH(k)在导数和无穷范畴方面的描述。此外，我们将研究Schwede关于经典稳定同伦范畴模型唯一性的刚性定理是否允许对动机情况进行某种改进。所有这些问题的一个潜在主题是，有多少SH(k)的“高级结构”已经由它的三角结构决定，这与所选择的模型无关。描述。

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.