Comparing algebraic and non-algebraic foundations of n-category theory by Thomas Peter Cottrell. A thesis submitted for the degree of Doctor of Philosophy. School of Mathematics and Statistics The University of Sheffield February 2014. 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 def- initions, in which composites and coherence cells are explicitly specified, and non-algebraic definitions, in which a coherent choice of composites and con- straint cells is merely required to exist. Relatively few comparisons have been made between definitions, and most of those that have concern the relation- ship 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 I 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 be- tween 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 de- scribing the relationship between the weak n-categories of Batanin and Leinster. In Part II we take the first step towards a comparison between Penon's def- inition 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. i ii Contents Introduction 1 Notation and terminology . .7 I Algebraic definitions of weak n-category 11 1 Penon weak n-categories 13 1.1 Definition of Penon weak n-categories . 13 1.2 Construction of Penon's left adjoint . 17 1.2.1 Left adjoint to V ....................... 19 1.2.2 Free contraction structure . 21 1.2.3 Free magma structure . 29 1.2.4 Interleaving the contraction and magma structures . 41 2 Operadic definitions of weak n-category 45 2.1 Globular operads . 45 2.2 Batanin weak n-categories . 52 2.3 Initial object in OCS ........................ 55 2.4 Leinster weak n-categories . 73 2.5 Coherence for algebras for n-globular operads . 84 3 Comparisons between algebraic definitions of weak n-category 93 3.1 The operad for Penon weak n-categories . 93 3.2 Towards a comparison between B-Alg and L-Alg ........ 106 3.2.1 Comparison functors between B-Alg and L-Alg ..... 107 3.2.2 Left adjoint to u∗ ....................... 110 3.2.3 The relationship between u∗ and v∗ ............ 123 II A multisimplicial nerve construction for Penon weak n-categories 131 4 Tamsamani{Simpson weak n-categories 133 4.1 Nerves of categories . 133 4.2 Tamsamani{Simpson weak n-categories . 137 iii iv Contents 5 Nerves of Penon weak 2-categories 143 5.1 Leinster's nerve construction for bicategories . 143 5.2 The nerve construction for n =2.................. 154 5.3 The Segal condition . 162 6 Nerves of Penon weak n-categories 183 6.1 The nerve construction for general n ................ 183 6.2 Directions for further investigation . 189 Introduction An n-category is a higher-dimensional categorical structure in which, as well as objects and morphisms between those objects, we have morphisms between mor- phisms (\2-morphisms"), morphisms between 2-morphisms (\3-morphisms"), and so on up to n-morphisms for some fixed natural number n. Such structures arise in areas as diverse as homotopy theory, computer science, and theoretical physics, as well as category theory itself. The case of strict n-categories, in which composition of morphisms is strictly associative and unital, is well-understood, but for many applications it is not sufficiently general, since many naturally occurring \composition-like" operations satisfy associativity and unitality only up to some kind of higher-dimensional isomorphism or equivalence. Thus, a no- tion of weak n-category is required. The theory of weak n-categories has grown rapidly over the past two decades; many different definitions of weak n-category have been proposed, using a wide variety of approaches, but the relationships between these definitions are not yet well understood, with few comparisons having been made. It has been widely observed (see [Lei02]) that each of these definitions belongs to one of two groups, called \algebraic" and \non-algebraic". The distinction between algebraic and non-algebraic definitions lies in the way in which composites and coherence cells are treated, and is often described as follows: in an algebraic definition composites and coherence cells are explic- itly specified; in a non-algebraic definition a suitable choice of composites and coherence cells is required to exist, but is not specified and is not necessarily unique. However, the difference is more deeply ingrained in the approaches used than this description would suggest. Algebraic definitions draw upon tech- niques from universal algebra, such as the theories of monads and operads, whereas non-algebraic definitions use topological techniques, such as homotopy theory and model category theory, and are closely related to the more algebraic notions of topological space, such as Kan complexes. Thus when making com- parisons between definitions that belong to just one group, there are pre-existing techniques that can be used, but making a comparison between an algebraic def- inition and a non-algebraic definition is more of a challenge. The way in which the algebraic and non-algebraic approaches fit into the bigger picture of the relationship between algebra and topology is well-illustrated in the following diagram, by Leinster [Lei10]: 1 2 Introduction Leinster used this diagram to illustrate the homotopy hypothesis of Grothen- dieck, an important application of the theory of weak n-categories. Very roughly, this hypothesis states that \!-groupoids should be the same as spaces" (and, in the n-dimensional case, \n-groupoids should be the same as n-types"). To state it formally we need to choose a notion of weak n-category and a notion of space, with the \strongest" statement of the hypothesis arising when we use an algebraic definition of weak n-category and a non-algebraic definition of space. A statement of the hypothesis using a non-algebraic notion of weak n-category or an algebraic notion of space is less strong since it connects concepts that are more similar to one another, but it should be easier to prove for the same reason. Understanding the relationship between algebraic and non-algebraic definitions of weak n-category would thus represent a significant step towards proving the strongest version of the homotopy hypothesis. The case of weak n-groupoids, weak n-categories in which all morphisms (including higher morphisms) are invertible up to some higher cell, is of particular interest not only in the case of the homotopy hypothesis, but also in the study of Homotopy Type Theory [UFP13]. We now discuss the various definitions of weak n-category that have been proposed. The earliest algebraic definitions were made for specific values of n, and took a very direct approach; these include the classical definitions of bicategory [B´en67]and tricategory (first defined in [GPS95], and made fully algebraic in [Gur06]), and Trimble's definition of tetracategory [Tri95]. The definition of tetracategory is long, and highlights the fact that it is not practical to use the classical approach for higher values of n. Thus, in algebraic definitions of weak n-category for a general value of n, more abstract methods are used. Algebraic definitions for a general value of n include Penon's definition [Pen99], which uses a weakened version of the free strict n-category monad, Batanin's definition ([Bat98], with variants [Lei04b, Ber02, Lei02, Cis07, Gar10, BG11, Che11a]), which uses globular operads, and the definitions of Trimble ([Tri99], first published in [Lei02], with variants [CG07, Che11a]) and May [May01], which use weakened forms of enrichment. Non-algebraic definitions take the form \a weak n-category consists of some underlying data (usually a presheaf) satisfying a certain condition". This con- dition ensures that a coherent choice of composition structure can be made from the underlying data, but the exact composition structure is not explicitly spec- ified. Various types of underlying data are used in non-algebraic definitions, in- Introduction 3 cluding simplicial sets [Str87], multisimplicial sets [Tam99, Sim97], cellular sets [Joy97], opetopic and multitopic sets [BD98, HMP00, HMP01, HMP02, Lei98b], and !-hypergraphs [HMT99, MT00]. Many non-algebraic definitions deal with the case of (1; n)-categories, a type of higher category in which there is no maximum dimension of cell, but in which all cells of dimension greater than n are equivalences. These definitions include quasi-categories (also known as weak Kan complexes) [BV73, Vog73, Joy02, Lur09], Segal n-categories [HS98, Sim12], and complete Segal spaces [Rez01]. Although there is a large number of different definitions, relatively few com- parisons have been made between them, and most of the comparisons that have been made are either exclusively between algebraic definitions, or exclusively between non-algebraic definitions. In the case of algebraic definitions, Batanin has made a comparison between his definition and that of Penon [Bat02], show- ing that his definition is weaker than Penon's and conjecturing some sort of weak equivalence between the two; Cheng has shown that a generalisation of Trimble's definition is an instance of a variant of Batanin's [Che11a].