Homotopy Theory and Higher Categories

本文的目的是研究同伦理论与形式群的关系，以及同伦理论在范畴论中的应用。PI计划研究Morava e理论的幂运算，将这些运算的代数与Bousfield-Kuhn函子联系起来，Bousfield-Kuhn函子是一种挑选不稳定同伦理论单色层不变量的构造。PI提出证明一系列猜想，这将导致一些新的计算技术，也将在不稳定同伦理论和Morava e理论中的上同调运算代数之间建立有趣的联系。此外，PI将研究基于同伦理论思想的高范畴论的某些模型。最近（通过Hopkins， Lurie和其他人的工作），更高的分类被应用于拓扑场理论的分类。PI将研究的模型（称为Theta-n空间）被推测为（无穷大，n）类别理论的表示；本课题的目标是证明这一猜想，并将这些结果推广到丰富的高等范畴论的设置中。同伦理论是拓扑学的一个分支；它起源于对空间某些不变性质的研究，即那些连续变形所留下的不变性质。研究这些性质最有力的工具是所谓的“上同调理论”。上同调理论是由形式群理论阐释的，而形式群理论又与代数数论中的问题密切相关。该项目的第一个目标是理解这种关系，并展望创造新的计算工具。第二个目标是了解如何使用同伦理论来阐明高等范畴理论。

The goal of this proposal is to study the relationship between homotopy theory and formal groups, and applications of homotopy theory to category theory. The PI plans to study power operations for Morava E-theory, relating the algebra of such operations to the Bousfield-Kuhn functor, a construction which picks out invariants of a single chromatic layer of unstable homotopy theory. The PI proposes to prove a a series of conjectures, which would lead to some new computational techniques, and would also draw interesting connections between unstable homotopy theory and the algebra of cohomology operations in Morava E-theory. In addition, the PI will study certain models for higher category theory based on the ideas of homotopy theory. Higher categories have recently been applied (through work of Hopkins, Lurie and others) to the classification of topological field theories. The models the PI will study (called Theta-n spaces), are conjectured to be presentations for the theory of (infinity,n)-categories; a goal of this project is to prove this conjecture, and to extend these results to the setting of enriched higher category theory.Homotopy theory is a branch of topology; it arose as the study of certain invariant properties of spaces, namely those left unchanged by continuous deformations. The most powerful tools for studying such properties are what are called "cohomology theories". Cohomology theories are illuminated by the theory of formal groups, which in turn are closely related to problems in algebraic number theory. The first goal of this project is to understand this relationship, with the prospect of creating new computational tools. A second goal is to understand how to use homotopy theory to shed light on higher category theory.