External traced monoidal categories

Abstract

String diagrams provide a novel technique for modern developments in category theory, and in particular, higher category theory. Many results about category theory can only be properly stated in the setting of 2-categories. We analyse the approach of using profunctors, which are a categorification of relations, to study category theory itself. Specifically, we examine the structure of ∗-autonomous categories and various other categories with duals, and develop a new characterisation of traced monoidal categories with respect to Prof: the 2-category of categories, profunctors, and natural transformations. We discuss at length the topic of diagrammatic methods, and advocate for a fancy kind of string diagram called an internal string diagram.