Categorical methods in topology, algebra and computer science

范畴论旨在发现各种数学理论的共同点，并建立同时适用于数学及其在自然科学中的应用的许多不同领域的定理，而不必求助于基于集合论的传统基础。拟议的项目是集中在代数和拓扑之间的接口，但也将达到追求同伦代数，代数几何和域理论的主题。具体地说，我们希望-全面研究所谓的宽松的Eilenberg-Moore代数，这是为了使一般拓扑结构的工具，已成功地用于代数，-完善和扩展现有的“内在”理论的拓扑结构在相当任意的类别，并将其应用到上述宽松代数，- 发展一种挠率和局部化理论，该理论适用于半阿贝尔范畴和更远的范畴，以及稳定同伦理论的三角化范畴，-建立一种宽松因子分解结构理论，目的是将其应用于同伦理论，- 给一个新的治疗楚空间的范畴，在其拓扑的基础范畴，并将其纳入Sambin的方法，以正式的拓扑结构中使用的理论计算机科学。

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.