There is a canonical and efficient way to extend a convergent presentation of a category by a 2-polygraph into a coherent one. Precisely, the 3-cells used in this extension procedure are in one-to-one correspondence with the confluence diagrams of critical branchings in the polygraph. Now, if the polygraph is finite, so is the set of its critical branchings, and therefore the set of 3-cells generating coherence can be taken to be finite. In such a situation, the polygraph is said to have finite derivation type, or FDT. The relevance of this concept, introduced by Squier, lies in the following invariance property: if a category admits a finite presentation having finite derivation type, then all finite presentations of also have FDT. This invariance will prove essential to show that some finitely presented categories do not admit convergent presentations. Using these conditions, Squier managed to produce an explicit example of a finitely presented monoid, with decidable word problem, but having no finite convergent presentation. This provides a negative answer to the question of universality of finite convergent rewriting.

This chapter presents techniques for proving the termination of 3-polygraphs. A first method is based on a certain type of well-founded orders called reduction orders. Attention then turns to functorial interpretations: these amount to construct a functor from the underlying category to another category which already bears a reduction order. This covers quite a few useful examples. To address more complex cases, a powerful technique, due to Guiraud, is presented, based on the construction of a derivation from the polygraph. Here, termination is obtained by specifying quantities on 2-cells which decrease during rewriting, based on information propagated by the 2-cells themselves.

The study of universal algebra, that is, the description of algebraic structures by means of symbolic expressions subject to equations, dates back to the end of the 19th century. It was motivated by the large number of fundamental mathematical structures fitting into this framework: groups, rings, lattices, and so on. From the 1970s on, the algorithmic aspect became prominent and led to the notion of term rewriting system. This chapter briefly revisits these ideas from a polygraphic viewpoint, introducing only what is strictly necessary for understanding. Term rewriting systems are introduced as presentations of Lawvere theories, which are particular cartesian categories. It is shown that a term rewriting system can also be described by a 3-polygraph in which variables are handled explicitly, i.e., by taking into account their duplication and erasure. Finally, a precise meaning is given to the statement that term rewriting systems are "cartesian polygraphs".

This appendix provides an explicit description of the free n-category generated by an n-polygraph. This section is mostly inspired of the work of Makkai. A formal definition of the syntax of n-categories is first provided, describing morphisms in an (n+1)-category freely generated by an n-polygraph, allowing reasoning by induction on its terms to prove results on free categories. It turns out that this syntax for n-categories, which corresponds to the one used throughout the book, is very "redundant", in the sense that there are many ways to express a composite of cells which will give rise to the same result, and is sometimes not very practical for this reason. An alternative syntax, which suffers less from these problems, is provided by restricting compositions. Finally, a brief mention of the word problem for free n-categories is made.

This appendix lists some examples of presentations of classical monoids and categories by a 2-polygraph.

This chapter is dedicated to the definition of 2-polygraphs, which are a 2-dimensional generalization of 1-polygraphs. Before introducing this notion, a refined viewpoint over 1-polygraphs is given. Instead of merely focusing on the set presented by a 1-polygraph as a set of equivalence classes of generators modulo the relations, the free category generated by the polygraph is now considered. The notion of 2-polygraph naturally appears as soon as arbitrary, non necessarily free, small categories are considered. In order to present such a category, one starts with a polygraph such that the 1-generators generate the morphisms of the category, but now it must be taken into account the relations induced by the category among the morphisms of the free category generated the resulting 1-polygraph. These relations will be generated by a set of 2-generators, consisting in certain pairs of morphisms intended to be equalized in the category. Following the same pattern, it will be explained that a 2-polygraph can also be seen as a system of generators for a free 2-category, thus preparing the study of 3-polygraphs. The variant where a (2,1)-category is freely generated is also examined.

This chapter introduces in full generality the central concept of this book, namely the notion of polygraph. Given an n-category, a cellular extension of it consist in attaching cells of dimension n+1 between certain pairs of parallel n-cells. This operation freely generates an (n+1)-category. Polygraphs are then obtained by starting with a set, considered as a 0-category and inductively repeating the above process in all dimensions. The construction yields a fundamental triangle of adjunctions between omega-categories, polygraphs, and globular sets. A brief description of (n,p)-polygraphs, that is, the notion of polygraph adapted to (n,p)-categories, concludes the chapter.

This chapter introduces the notion of acyclic extension of a 2-category, which consists of the additional data of 3-generators "filling all the spheres". This leads to the notion of coherent presentation of a category, which consists of a 2-polygraph presenting the category together with an acyclic extension of the free (2,1)-category on the polygraph. Coherent presentations are then constructed from convergent ones, and the appropriate notion of Tietze transformation between coherent presentations is studied: this allows formulation of a coherent variant of the Knuth-Bendix completion procedure, but also a reduction procedure, which can be used to obtain smaller coherent presentations. Finally, coherent presentations of algebras are studied, thereby defining the proper notion of coherent extension for linear polygraphs.

The purpose of this chapter is to introduce the notion of a polygraphic resolution of an ω-category. This notion was introduced by Métayer to define a homology theory for ω-categories, that is now known as the polygraphic homology. It was then showed by himself and Lafont that this homology recovers the classical homology of monoids for ω-categories coming from monoids. It is now known by work of Lafont, Métayer, and Worytkiewicz that these polygraphic resolutions are resolutions in the sense of a model category structure on ω-categories, the so-called folk model structure. Every ω-category is shown to admit such a resolution, and the relationship between two resolutions of the same ω-category is examined.

This appendix is a quick introduction to locally presentable categories. This notion is in some sense a formalization of what is an algebraic structure. When category theory is restricted to locally presentable categories, many things get simpler. In particular, there are characterizations of adjoint functors purely in terms of preservation of limits and colimits. Locally presentable categories also play an important role in the theory of model categories through the concept of combinatorial model categories. There are many ways to define locally presentable categories. The appendix begins by presenting the concept using sketches, which encode the syntax of an algebraic structure. These sketches are used several times in the body of the book. The intrinsic categorical characterization is then provided, introducing several notions that are important for the theory of model categories. Finally, the syntactic characterization is discussed.

Among the many existing notions of higher categories, the notion of strict globular n-category is, in some sense, the most basic one. In this chapter, the essential definitions and notations are set. Starting with a description of the basic "shapes", that is, the presheaf category of globular sets, family of operations endowing a globular set with a structure of ω-category is defined. Then, it is proven that the category of strict ω-categories is exactly the category of algebras of the monad induced by the forgetful functor from ω-categories to globular sets. Finally, important subcategories of ω-categories, obtained by requiring cells to be invertible above a given dimension, are defined.

The notion of polygraph introduced so far is a particular case of a general construction due to Batanin. In fact, any finitary monad on globular sets yields a appropriate notion of polygraph. The original motivation was the study of weak ω-categories seen as algebras of such a monad. Another example, of particular relevance to this book, is the case of linear polygraphs presented in the last section.

This chapter discusses 1-polygraphs, which are simply directed graphs, thought of here as abstract rewriting systems: they consist of vertices, which represent the objects of interest, and arrows, which indicate that one object can be rewritten into another. After formally introducing those, it will be shown that they provide a notion of presentation for sets, by generators and relations. Of course presentations of sets are of little interest in themselves, but merely used here as a gentle introduction to some of the main concepts discussed in this work: in particular, the notion of Tietze transformations is introduced, which generates the equivalence between two presentations of the same set. In this context, an important question consists in deciding when two objects are equivalent, i.e., represent the same element of the presented set. In order to address it, the theory of abstract rewriting systems is developed.