An inductive approach to ω-categories and their computads

Abstract

Given an algebraic structure, a basic question is to determine the data from which it can be freely generated. In the case of ω-categories, this data can be either globular sets, or the computads of Street and Batanin. In this thesis, we give an inductive description to computads and the free weak ω-categories they generate. We then generalise this approach to other higher structures. The first part of the thesis is concerned with strict ω-categories and globular pasting diagrams. We give an inductive description of the set of Batanin trees, parametrising globular pasting diagrams, as well as a recursive definition of globular pasting diagrams using the wedge sum and the suspension of globular sets. We then give a new direct proof that globular pasting diagrams familially represent the free strict ω-category monad on globular sets, giving in particular a structurally recursive description of the monad multiplication. The second part of the thesis introduces weak ω-categories and their computads. First, computads are defined mutually inductively with the underlying globular set of the free ω-category they generate. This globular set is defined inductively via a pair of constructors, and not as a pushout in an already-known category of ω-categories. This yields a new understanding of computads, and allows a new definition of ω-categories that avoids the technology of globular operads. This new description permits direct proofs of important results via structural induction, and we use this to give new proofs that every ω-category is equivalent to a free one, and that the category of computads with generator-preserving maps is a presheaf topos. We also use this description to construct the hom and suspension of an ω-category, and to prove that the homs of a computad are computads. We then show that our definition of ω-category agrees with that of Batanin and Leinster. In the final part of the thesis, we generalise our inductive approach to other higher structures. We introduce a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe how computads give rise to algebras. Motivated by work of Batanin, we show that computads with generator-preserving morphisms form a presheaf category, and describe a forgetful functor from algebras to computads. Algebras free on a computad turn out to be the cofibrant objects for a certain cofibrantly generated weak factorisation system, and the adjunction above induces the universal cofibrant replacement in the sense of Garner, for this weak factorisation system. Finally, we conclude by explaining how many-sorted structures and algebraic semi-simplicial Kan complexes are algebras of such signatures, and by proposing a notion of weak multiple category.