Topics in enriched category theory

丰富范畴论的主题

• 批准号: 2745681
• 金额: --
• 依托单位: University of Sheffield
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2745681
• 关键词: Topics; enriched; category; theory

Topics in enriched category theoryCategory theory grew as a foundational language for many part of pure mathematics and was capable of expressing similarities and connections between different areas, in particular between geometric and algebraic areas of mathematics. At its simplest a category consists of things called objects and for each pair of objects, say A and B, a set of things going from A to B, where these things could be an enormous variety of things such as functions or paths, with the proviso that if you have something going from A to B and something going from B to C then you can combine them to form something going from A to C. These seemingly simple idea gives a very rich theory which manifest itself in lots of areas including logic and theoretical computer science. More recently, it was realised then a generalized notion, that of "enriched" category was even more useful and pervasive. For instance, the supervisor on this project has been recently working on the idea that many structures used in the area of optimization fit as neatly into enriched category theory as structures in topology do.It is very common in practice to find enriched categories combining together to form categorical structure themselves, for instance enriched profuctor double categories are particularly pervasive in many areas. The project will look deeper into such structures.Aims and objectivesGuided by both abstract principals and wide-ranging examples, the student will deepen the knowledge of the low dimensional categorical structures formed by enriched categories.Potential applications and benefitsThis is blue sky research that is looking at deep structures that pervade areas of mathematics, so it is not easy to say what potential applications are, however, it is worth mentioning that work in a related area, "magnitude", has gone on to inform measurement of biodiversity and to help measure effects in machine learning.

主题丰富的范畴理论范畴理论成长为一个基础语言的许多部分纯数学和能够表达相似性和不同领域之间的联系，特别是几何和代数领域的数学。在最简单的情况下，一个范畴由称为对象的东西组成，对于每一对对象，比如A和B，一组从A到B的东西，这些东西可以是各种各样的东西，比如函数或路径，前提是，如果你有从A到B的东西和从B到C的东西，那么你可以将它们联合收割机接收机，形成从A到C的东西。这些看似简单的思想给出了一个非常丰富的理论，体现在许多领域，包括逻辑和理论计算机科学。最近，人们意识到一个广义的概念，即“丰富”类别甚至更有用和普遍。例如，该项目的主管最近一直在研究这样一种想法，即在优化领域中使用的许多结构与拓扑学中的结构一样整齐地符合丰富范畴理论。在实践中，发现丰富范畴组合在一起形成范畴结构本身是非常常见的，例如丰富的profuctor双范畴在许多领域中特别普遍。该项目将深入研究这些结构。目的和目标在抽象原则和广泛的例子的指导下，学生将加深对丰富类别形成的低维类别结构的认识。潜在的应用和好处这是一个蓝天研究，正在寻找渗透数学领域的深层结构，所以很难说潜在的应用是什么，但是，值得一提的是，在一个相关领域“规模”中的工作已经开展，为生物多样性的衡量提供了信息，并帮助衡量机器学习的效果。