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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711774
• 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.

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，范畴语言和理论已经发展到促进这种相互作用。虽然只是众多中的一个，范畴论的这个原始的总体方面继续努力，我们的研究计划旨在加强它的领域，通过比较其他学科的数学，迄今为止只有适度的影响或帮助范畴论：功能分析和一般拓扑。为了扩大它的适用性，我们想扩展一个更定量的方法，一般拓扑超越度量化，通过调查配备了距离函数，可以假设值以外的数字，如概率函数的空间。 一种特殊类型的空间应该允许一个明确的统一的新的结果，在功能分析的基础上，“近似”的概念，定义错过标准的一个只有一个预设的利润。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，并且对于有限维空间的等距嵌入具有近似的扩张性质。我们最近建立了一个范畴存在的证明，当应用到度量空间，也产生存在的所谓Urysohn普遍空间。我们的第一个目标是明确的范畴原则，使我们不仅可以给这两个经典空间的存在性，但也统一的唯一性证明，然后利用它们的其他类型的对象，特别是空间的额外的属性或结构的拓扑或功能分析的兴趣。 在度量空间（允许无限距离）中，从点x到空间中任何点的集合A都有一个数值定义的最近距离。所谓的接近空间带有公理化的点集距离函数。它们忠实地包含，作为类别，度量和拓扑空间，即，退化逼近空间，其中从x到A的距离可能仅为0或无限（x位于A的闭包内或外）。 对于新的应用，我们研究的空间中的“距离”可能不仅是二进制或数值，但在一个结构良好的晶格值。此外，通过考虑从点到所讨论的空间的特征的距离，而不是它的子集，例如超滤子，我们采用代数方法来研究特殊类型的空间，例如允许令人满意地形成其函数空间的空间。作为一个论文项目，我们试图找到这样的空间的总体行为时，对偶的另一个类别，通常是一个代数类型，这将允许两种类型之间的信息传输的情况。另一个论文项目将研究这些甚至更广泛的上下文中的“连接”对象和转换。