Comparing algebraic and non-algebraic foundations of $n$-category theory

Abstract

Many definitions of weak n-category have been proposed. It has been widely observed that each of these definitions is of one of two types: algebraic definitions, in which composites and coherence cells are explicitly specified, and non-algebraic definitions, in which a coherent choice of composites and constraint cells is merely required to exist. Relatively few comparisons have been made between definitions, and most of those that have concern the relationship between definitions of just one type. The aim of this thesis is to establish more comparisons, including a comparison between an algebraic definition and a non-algebraic definition. The thesis is divided into two parts. Part 1 concerns the relationships between three algebraic definitions of weak n-category: those of Penon and Batanin, and Leinster's variant of Batanin's definition. A correspondence between the structures used to define composition and coherence in the definitions of Batanin and Leinster has long been suspected; we make this precise for the first time. We use this correspondence to prove several coherence theorems that apply to all three definitions, and to take the first steps towards describing the relationship between the weak n-categories of Batanin and Leinster. In Part 2 we take the first step towards a comparison between Penon's definition of weak n-category and a non-algebraic definition, Simpson's variant of Tamsamani's definition, in the form of a nerve construction. As a prototype for this nerve construction, we recall a nerve construction for bicategories proposed by Leinster, and prove that the nerve of a bicategory given by this construction is a Tamsamani--Simpson weak 2-category. We then define our nerve functor for Penon weak n-categories. We prove that the nerve of a Penon weak 2-category is a Tamsamani--Simpson weak 2-category, and conjecture that this result holds for higher n.