Categorical methods in topology, algebra and computer science

拓扑、代数和计算机科学中的分类方法

• 批准号: 8844-2006
• 负责人: Tholen, Walter
• 金额: $ 1.46万
• 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: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=361598
• 关键词: Categorical; methods; topology; algebra; computer

Category theory aims at detecting common aspects of various mathematical theories and at establishing theorems that are therefore simultaneously applicable to many distinct areas of mathematics and its applications in the natural sciences, without necessarily resorting to traditional foundations based on set theory. The proposed project is concentrated on the interface between algebra and topology, but will also reach out to themes pursued in homotopical algebra, algebraic geometry, and domain theory. Specifically, we wish -  to study comprehensively so-called lax Eilenberg-Moore algebras, which are designed to make structures of general topology accessible to tools that have been successfully used in algebra, - to refine and extend the existing "intrinsic" theory of topological structures in rather arbitrary categories and apply it to the afore-mentioned lax algebras, - to develop a theory of torsion and of localizations that is applicable to semi-abelian categories and beyond, as well as to the triangulated categories of stable homotopy theory, - to establish a theory of lax factorization structures with the aim of applying it in homotopy theory, - to give a new treatment of categories of Chu spaces, in terms of their topologicity over the base category, and integrate them into Sambin's approach to formal topology as used in theoretical computer science.

范畴论旨在发现各种数学理论的共同方面，并建立定理，因此，这些定理同时适用于许多不同的数学领域及其在自然科学中的应用，而不必诉诸基于集合论的传统基础。提议的项目集中在代数和拓扑之间的接口，但也将涉及到同调代数、代数几何和领域理论的主题。具体来说，我们希望——全面研究所谓的松弛Eilenberg-Moore代数，它的目的是使一般拓扑结构能够被代数中成功使用的工具所访问，——完善和扩展现有的相当任意范畴的拓扑结构的“内在”理论，并将其应用于上述松弛代数，——发展一种适用于半阿贝尔范畴及其他领域的扭转和局部化理论。对于稳定同伦理论中的三角化范畴，建立了一种松弛分解结构的理论，目的是将其应用于同伦理论；对于楚空间的范畴，根据其在基范畴上的拓扑性，给出了一种新的处理方法，并将其整合到理论计算机科学中使用的Sambin的形式拓扑方法中。