Categorical and quantitive topology and its applications

分类和定量拓扑及其应用

• 批准号: RGPIN-2017-05715
• 负责人: Tholen, Walter
• 金额: $ 1.17万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675715
• 关键词: Categorical; quantitive; topology; its; applications

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. ***A particular type of such spaces should allow for a categorical unification of novel results in functional analysis that are based on "approximate" notions, defined to miss the standard one only by a preset margin. In Banach space theory, a prominent classical example is the so-called Gurarii space, which is separable and has an approximate extension property with respect to isometric embeddings of finite-dimensional spaces. We have recently established a categorical existence proof which, when applied to metric spaces, yields also the existence of the so-called Urysohn universal space. Our first goal is to crystallize the categorical principles that would allow us to give not only an existence but also a unified uniqueness proof for these two classical spaces, and then exploit them for other types of objects, especially for spaces with additional properties or structures of interest in topology or functional analysis.***In a metric space (infinite distances permitted) one has a numerically defined closest distance from a point x to any set A of points of the space. So-called approach spaces come with an axiomatized point-set-distance function. They faithfully encompass, as categories, both metric and topological spaces, i.e., degenerate approach spaces where the distance from x to A may only be 0 or infinite (x lies in or outside the closure of A). For novel applications, we study spaces where the "distances" may not only be binary or numerical, but take values in a well-structured lattice. Moreover, by considering distances from points to features of the space in question other than its subsets, such as ultrafilters, we employ algebraic methods to study special types of spaces, such as the ones permitting the satisfactory formation of their function spaces. As a thesis project we seek to find situations when the totality of such spaces behaves dually to another category, usually of an algebraic type, which will then allow for the transfer of information between both types. Another thesis project will study "connected" objects and transformations in these and even broader contexts.*****

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。*一种特定类型的这种空间应该允许泛函分析中的新结果的分类统一，这些新结果建立在“近似”概念的基础上，定义为仅通过预设的差值与标准结果不符。在Banach空间理论中，一个突出的经典例子是所谓的Gurarii空间，它是可分的，并且关于有限维空间的等距嵌入具有近似扩张性质。我们最近建立了一个绝对存在证明，当它应用于度量空间时，也给出了所谓的Urysohn泛空间的存在。我们的第一个目标是明确分类原则，使我们不仅可以给出这两个经典空间的存在而且统一的唯一性证明，然后将它们用于其他类型的对象，特别是对于具有拓扑学或泛函分析中感兴趣的额外性质或结构的空间。*在度量空间(允许无限距离)中，一个人有一个数值定义的从点x到空间中任何一组点A的最近距离。所谓的逼近空间具有公理化的点集距离函数。它们忠实地包含度量空间和拓扑空间作为范畴，即退化逼近空间，其中从x到A的距离可能只有0或无穷远(x位于A的闭包之内或之外)。对于新的应用，我们研究的空间中的“距离”不仅可以是二进制或数值，而且可以在结构良好的格子中取值。此外，通过考虑点到所讨论空间的子集以外的特征的距离，例如超滤子，我们使用代数方法来研究特殊类型的空间，例如那些允许满意形成其函数空间的空间。作为一个论文项目，我们试图找出这样的空间的全体表现为另一个范畴的对偶行为的情况，通常是代数类型的，这将允许在两种类型之间传递信息。另一个论文项目将在这些甚至更广泛的背景下研究“连接的”对象和变换。