Topics in enriched category theory

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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.