Low dimensional monoidal category theory : a functorial method for constructing monoidal bicategories

Abstract

In this work, we start studying some basic concepts of classical category theory, such as categories, functors, natural transformations, products and co-products, among other important concepts, understanding its definitions and their main properties. We proceed to the theory of monoidal categories, with the objective of understanding a generalization of the product in categories and of algebraic objects within such categories. We begin this part studying properties of the neutral, the commutativity of certain diagrams and the properties of functors that preserve the monoidal structure, with the aim of being able to prove MacLane’s coherence theorem, which gives us the commutativity of a large class of diagrams, and the strictification theorem, which gives us a monoidal category equivalent to the initial one that is algebraically simpler. We finish the study of these categories by looking at additional braiding structures, symmetry and internal algebraic structures (monoids, modules, bimodules and actions in monoidal categories). Finally, we extend the study of monoidal categories to the case of low-dimensional categories to prove a theorem recently proved by Shulman (which says that a certain bicategory associated with an isofibrant monoidal double category is also monoidal through a functorial association) and then we detail the applications of this result to some scenarios.