Monoidal categories in, and linking, geometry and algebra

TL;DR

This paper explores the theory and applications of monoidal categories, linking geometry and algebra, with implications for knot theory, representation theory, and low-dimensional topology.

Contribution

It provides an overview of monoidal categories, illustrating their connections to knot theory, representation theory, and potential uses in physics and manifold invariants.

Findings

Link between knot theory and monoidal categories via string notation

Insights into representation theory through monoidal category structures

Proposals for new invariants of low-dimensional manifolds

Abstract

This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. The fourth and final section looks at ways in which monoidal categories are, and might, be used for new invariants of low-dimensional manifolds and for the field theory of theoretical physics.