The primary aim of this work is an intrinsic homotopy theory of strict ω-categories. We establish a model structure on ωCat, the category of strict ω-categories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on n-categories for arbitrary n ∈ N, as specialisations of the ω-categorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.

Polygraphs generalize to 2-categories the usual notion of equational theory, by describing them as quotients, modulo equations, of freely generated 2-categories on a given set of generators. In order to work with morphisms modulo the equations, it is often convenient to orient the equations into a confluent rewriting system. In the case of a terminating system, confluence can be checked by showing that critical pairs are joinable. However, the computation of the critical pairs is more complicated for polygraphs than for term rewriting systems: in particular, two left members of a rule don’t necessarily have a finite number of unifiers. We advocate here that a more general notion of rewriting system should be considered instead, and introduce an operad of compact contexts in a 2-category, in which two rules have a finite number of unifiers. A concrete representation of contexts is proposed, as well as an unification algorithm for these.

We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZM-modules. 1

The pasting theorem showed that pasting schemes are useful in studying free !-categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !-categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !-category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !-categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...

Abstract. We adapt the work of Power [14] to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in the 2-category of k-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack [9], proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov [10]. 1.

Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2-naturality and Gray’s tensor product of 2-categories. It generalises the existing more specific notions of distributive

Polygraphs generalize to n-categories the usual notion of equational theory, thus allowing one to describe a category by the means of generators and relations. When the relations are oriented, such a presentation can be considered as a rewriting system and one might wonder whether the rewriting system is confluent and terminating in order to provide a notion of canonical representative of morphisms modulo equations (the normal forms of the morphisms). In term rewriting systems, confluence is often proved by computing the critical pairs, which are in finite number, and showing that they are joinable. We extend here this methodology to polygraphs presenting 2-categories. This task is not straightforward because a finite polygraph might admit an infinite number of critical pairs. This leads us to introduce the multicategory of contexts of the free compact 2-category generated by a 2-category, in which we can embed the original 2-category generated by the polygraph and compute a finite number

§1 Types, shapes and occurrences p. 3 §2 Factorization and geometry p. 13 §3 Cuts in partial orders, and planar arrangements p. 32

Abstract. The primary purpose of this work is to characterise strict ω-categories as simplicial sets with structure. We prove the Street-Roberts conjecture which states that they are exactly the “complicial sets ” defined and named by John Roberts in his handwritten notes of that title [26].2 VERITY

Abstract: We adapt the work of Power [13] to describe general, not-necessarily composable, notnecessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in k-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack [9], proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov [10].