Groundbreaking new approach to mathematical theory

Research from the University of Leicester introduces a new approach to higher categorical structures.

New research from Dr Simona Paoli from the University of Leicester’s Department of Mathematics introduces a new approach to higher categorical structures. The new research constitutes a foundational and groundbreaking contribution to the field.

Category theory is an area of pure mathematics that studies the way in which structures arising in different areas can be described with a common language. It was originally developed to transport ideas from one branch of mathematics to another, e.g. from topology to algebra.

This unifying approach is very powerful. It allows the transfer of methods and results between different areas of mathematics and helps to gain a deeper understanding of the nature of mathematical objects.

Although very abstract, category theory has important inter-disciplinary applications; for instance to theoretical computer science, to network theory and quantum computing. These links, in the long term, potentially impact real world industrial applications.

In the last two decades, a new branch of category theory has developed – higher category theory. This cutting-edge and very active area of research has penetrated diverse fields of science: from pure mathematics to mathematical physics and, more recently, logic and computer science.

In particular, homotopy type theory uses higher categorical structures to build ‘proof assistants’: these are tools that a mathematician or a programmer can use to verify the correctness of a mathematical proof or of a computer program. Automated software verification has considerable potential in applications.

Dr Paoli said: “This monograph introduces a new approach to higher categorical structures, yielding one of their simplest model. The results of this monograph pave the way to many applications and can be used to tackle long-standing open questions in category theory and beyond.

“The theory in this monograph is described in elementary terms. It is accessible to mathematicians from a wide range of disciplines. The exposition is enhanced by intuitive explanations, diagrammatic summaries and road maps.”

The monograph will be published in the prestigious Springer series ‘Algebra and Applications’. It is the result of several years of work by the author at the University of Leicester.

The work has been supported by an EU grant and by the Centre for Australian Category Theory at Macquarie University.

• The research monograph, 'Simplicial Methods for Higher Categories: Segal-type models of weak n-categories', will appear in the Springer series 'Algebra and Applications'