Low-Dimensional Enriched Category Theory

Supervised by Dr Simon Willerton and Dr Harsh Beohar.

The University of Sheffield is one of the leading Russell Group universities in the UK. We carry out cutting-edge research with strong links to industry. When you enrol to do a PhD with us, you will be working with world-leading academics and have access to top of the range facilities. As a PhD student you will have the opportunity to gain skills not only to conduct research, but also to take your career to the next level, whether you want to stay in academia, go into industry or the public sector, or set up your own company. You will have access to a range of training and support services to help you excel in your studies and beyond.

Project Description

Based in the Pure Mathematics group at the University of Sheffield, this PhD project explores aspects of modern category theory which seem prevalent in pure mathematics and in recent applications of category theory. On the one hand there are low dimensional structures such as monoidal categories, double categories and two- and three-categories. On the other hand there is the notion of enriched categories, which generalizes the notion of categories, and includes things like metric spaces, where instead of having a set of morphisms between each pair of elements you have a number between each pair of elements.

The context:

Category theory developed in the mid-twentieth century to help organise and relate notions in algebra, topology and geometry but it gradually became clear that it was far more encompassing and related to far more mathematics than just that. On a naive level, category theory seems to be about one-dimensional structures, namely objects and arrows between them; however, it has been clear from its outset that high-dimensional structures (such as natural transformations and monoidal products) are intrinsic parts of the theory.

In the late twentieth century, in particular with the development of topological quantum field theory and explicit connections between category theory and physics, the deep links between topological structures such as surfaces and knots and these higher dimensional categorical structures became apparent. So now category theory, which started out describing topology is actually being described by topology; diagrammatic notation such as string diagrams and surface diagrams reflect this connection in a rigorous, visual fashion.

In the twenty first century category theory continues to find new applications, in particular, in so-called ‘applied category theory’, which includes areas such as engineering, disease modelling, linguistics, probability, dynamical systems, optimization and game theory.

One pervasive structure in these areas seems to be that of the ‘symmetric monoidal enriched double category’ as typified by the double category of enriched profunctors; profunctors are a categorical notion of ‘correspondence’ which appears across many areas of mathematics.

The Project

Once you have mastered the necessary background material, the project will address questions around these structures. Diagrammatic methods are likely to play a strong role in this. Depending on your interests and strengths, you might consider applications to areas such as topological quantum field theory, algebraic geometry or optimization.

Research Environment and Career Development

You will join the Pure Mathematics group within the School of Mathematical and Physical Sciences at Sheffield where there is also strong research in algebraic topology, algebraic geometry, number theory and algebra, where there are regular seminars and working groups, including working groups run by the PhD students. We collaborate with theorists in the School of Computer Science. We are also involved in wider regional research groupings such as the Yorkshire and Manchester Category Theory seminar and the TTT topology seminar.

During your PhD you will have the opportunity to give seminars, to attend national and international conferences, and to gain teaching experience through undergraduate tutoring.

You

You should have a strong background in pure mathematics, particularly in algebra, topology, geometry or category theory.

How to apply

Interested candidates are strongly encouraged to contact the project supervisors to discuss your interest in and suitability for the project prior to submitting your application.

Please refer to the EPSRC DLA webpage for detailed information about the EPSRC DLA and how to apply.

Applicants must ensure that they use the correct application form linked to the EPSRC DLA webpage and must submit an application for October 2027 entry. The form is unique to the EPSRC DLA. It contains the list of advertised projects from which to select, and three questions specific to the DLA.

The application deadline is 17:00 GMT 13 January 2027.

The majority of our DLA studentships will be allocated to candidates who qualify for Home fee status, in accordance with EPSRC funding regulations which place strict limits on international recruitment.

Funding Notes

The award will fund the full tuition fee and a maintenance stipend at the UKRI rate (currently £21,805 per annum) for 3.5 years, as well as a research grant to support costs associated with the project.