Title:

Comparing algebraic and nonalgebraic foundations of ncategory theory

Many definitions of weak ncategory 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 nonalgebraic 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 nonalgebraic definition. The thesis is divided into two parts. Part 1 concerns the relationships between three algebraic definitions of weak ncategory: 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, and we make this precise for the first time. We use this correspondence to prove several coherence theorems that apply to all three definitions, and also to take the first steps towards describing the relationship between the weak ncategories of Batanin and Leinster. In Part 2 we take the first step towards a comparison between Penon's definition of weak ncategory and a nonalgebraic 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 TamsamaniSimpson weak 2category. We then define our nerve functor for Penon weak ncategories. We prove that the nerve of a Penon weak 2category is a TamsamaniSimpson weak 2category, and conjecture that this result holds for higher n.
