Algebraic applications of the homotopy theory of homotopy theories

本文讨论了同伦理论中的同伦理论的各种模型在代数和拓扑学中更为具体的应用。更具体地说，我们想把传统上在模型类别和派生类别的设置中提出的问题转化为完整的西格尔空间。这种方法似乎特别适用于派生范畴发挥作用的情况，因为我们可以在完整的Segal空间中保留更多的高阶信息，而这些信息在传递给派生范畴时丢失了。在这一点上，我们的主要应用是推广Toen对与某些稳定模型范畴相关的派生霍尔代数的定义。通过定义一个稳定完备西格尔空间的派生霍尔代数，我们希望对由某些李代数产生的有关霍尔代数和量子包络代数的结果进行推广。这项工作将采用代数拓扑中传统使用的技术，并将其应用于代数中出现的问题。特别是，我们希望回答表征理论中的一个开放问题，这是一个尚未使用这些方法的领域。这种方法有望为代数和拓扑学的构造提供新的视角。

This proposal concerns the application of various models for the homotopy theory of homotopy theories to more concrete situations in algebra and topology. More specifically, we would like to take questions traditionally posed in the settings of model categories and derived categories and translate them into the context of complete Segal spaces. This approach seems particularly promising for situations in which the derived category plays a role, in that we can retain much higher-order information in a complete Segal space that is lost when passing to the derived category. Our primary application at this point is generalizing Toen's definition of derived Hall algebras associated to certain kinds of stable model categories. By defining a derived Hall algebra for a stable complete Segal space, we hope to gain insight into extending a result relating Hall algebras and quantum enveloping algebras arising from certain Lie algebras.This work will take techniques traditionally used in algebraic topology and apply them to questions arising in algebra. In particular, we hope to answer an open problem in representation theory, an area in which these methods have not yet been used. This approach is expected to provide a new perspective on constructions in algebra as well as in topology.