Categorical and quantitive topology and its applications

数学研究最近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，范畴语言和理论已经发展起来，以促进这种互动。虽然这只是众多范畴理论中的一个，但这一最初的最重要的方面仍在继续努力，我们的研究建议旨在加强它在与其他数学学科相比，到目前为止只受到范畴理论的影响或帮助的领域：泛函分析和一般拓扑学。为了扩大其适用性，我们希望将一种更定量的方法扩展到一般拓扑可度量化之外，通过研究配备了距离函数的空间，该距离函数可能假定数字以外的值，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.