Algebraic applications of the homotopy theory of homotopy theories

这一建议涉及同伦理论的同伦理论的各种模型在代数和拓扑中更具体的情况下的应用。更具体地说，我们想把传统上在模型范畴和派生范畴的背景下提出的问题转化为完备的西格尔空间的背景。这种方法在派生范畴发挥作用的情况下似乎特别有希望，因为我们可以在传递到派生范畴时丢失的完整西格尔空间中保留高阶信息。在这一点上，我们的主要应用是推广Toen对与某些类型的稳定模型范畴相关的派生Hall代数的定义。通过在稳定完备的西格尔空间上定义一个导霍尔代数，我们希望得到一个与某些李代数产生的霍尔代数和量子包络代数相关的结果的推广.这项工作将采用传统的代数拓扑学中的技巧，并将它们应用于代数中的问题.特别是，我们希望回答表示论中的一个悬而未决的问题，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中的结构提供一个新的视角。

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.