Homotopy Theory and Higher Categories

这个计划的目标是研究同伦理论和形式群之间的关系，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函子是一种构造，它挑选出不稳定同伦理论的单色层的不变量。 PI提出了一系列的证明，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级范畴理论的某些模型。 更高的范畴最近被应用于拓扑场论的分类（通过霍普金斯、卢里等人的工作）。 PI将研究的模型（称为Theta-n空间），被证明是（无穷，n）-范畴理论的表示;这个项目的一个目标是证明这个猜想，并将这些结果扩展到丰富的更高范畴理论的设置。同伦理论是拓扑学的一个分支;它产生于研究空间的某些不变性质，即那些通过连续变形保持不变的性质。 研究这些性质的最有力的工具是所谓的“上同调理论”。 上同调理论由形式群理论所阐明，而形式群理论又与代数数论中的问题密切相关。 这个项目的第一个目标是了解这种关系，并有创造新的计算工具的前景。 第二个目标是理解如何使用同伦理论来阐明更高的范畴理论。

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.