Universal Monoidal Categories with Duals

Abstract

String diagrams form a diagrammatic notation used in many domains. To understand the simplest diagrams that express some properties, we can look at universal categories. These satisfy universal properties and can be described by presentations in terms of generators and relations. In this thesis, we examine some examples of universal categories, namely the (oriented) Temperley-Lieb and (oriented) Brauer categories. These are respectively the free linear monoidal category on a self-dual object or pair of dual objects, and the free linear symmetric monoidal category on a symmetrically self-dual object or pair of dual objects. Then, to make precise the connection between presentations and universal properties, we exhibit an adjoint functor from a category of generators and relations to the category of linear monoidal categories. We also suggest a general recipe to find a presentation of the category satisfying a specific universal property. Our main goal is to better understand the links between string diagrams, representation theory, generators and relations, and universal properties.