Categorical and quantitive topology and its applications

最近数学研究的许多进展都是通过将一个领域的思想和方法引入另一个领域而取得的。从20世纪40年代开始，范畴语言和理论的发展促进了这种互动。虽然这只是范畴论的众多方面之一，但它仍在继续努力，与其他数学学科相比，我们的研究计划旨在加强范畴论在一些领域的发展，这些领域迄今为止只受到范畴论的适度影响或帮助：泛函分析和一般拓扑。为了扩大其适用性，我们希望通过研究配备距离函数的空间来扩展一种更定量的方法来超越度量性，该距离函数可以假设数值以外的值，例如概率函数。

Much of the recent progress in mathematical research has been achieved by the import of ideas and methods of one area to another. From its beginnings in the 1940s, categorical language and theory have been developed to facilitate such interaction. Although being only one of many, this original overarching aspect of category theory continues to strive, and our research proposal aims to reinforce it in areas which, by comparison to other disciplines of mathematics, have so far only moderately been influenced or aided by category theory: functional analysis and general topology. In order to broaden its applicability, we want to extend a more quantitative approach to general topology beyond metrizability, by investigating spaces equipped with a distance function that may assume values other than numbers, such as probability functions.