Topics in enriched category theory

丰富范畴论的主题范畴论发展成为纯数学许多部分的基础语言，能够表达不同领域之间的相似性和联系，特别是数学的几何和代数领域之间。最简单的范畴是由对象组成的对于每一对对象，比如a和B，都是一组从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，附带条件是，如果你有从A到B的东西和从B到C的东西，那么你可以把它们组合成从A到C的东西。这些看似简单的想法给出了一个非常丰富的理论，在很多领域都表现出来，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”类别更加有用和普遍。例如，这个项目的主管最近一直在研究这样一个想法，即优化领域中使用的许多结构与拓扑学中的结构一样适合丰富的范畴论。在实践中，富范畴自身组合形成范畴结构的现象十分普遍，如富生产者双范畴在许多领域尤为普遍。该项目将更深入地研究这些结构。目的和目标在抽象原则和广泛例子的指导下，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项蓝天研究，它关注的是数学领域的深层结构，因此很难说出潜在的应用是什么，然而，值得一提的是，在一个相关领域，“量级”的工作已经开始为生物多样性的测量提供信息，并帮助测量机器学习的影响。

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.