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.