Algebraic applications of the homotopy theory of homotopy theories

这一建议涉及应用各种模型的同伦理论同伦理论更具体的情况下，代数和拓扑。 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，并把它们转换到完整的西格尔空间的背景下。这种方法似乎特别有前途的情况下，派生类别发挥作用，因为我们可以保留更高阶的信息在一个完整的西格尔空间时，失去了传递到派生类别。 我们在这一点上的主要应用是推广Toen的定义导出霍尔代数与某些种类的稳定模型类别。 通过在稳定的完备Segal空间上定义一个导出的Hall代数，我们希望能够深入理解推广一个关于Hall代数和由某些李代数产生的量子包络代数的结果，这项工作将采用代数拓扑学中传统使用的技术，并将其应用于代数中出现的问题。 特别是，我们希望回答一个开放的问题，在表示论，这些方法尚未被使用的一个领域。 这种方法预计将提供一个新的角度在代数以及拓扑结构。

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.