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.

The usefulness and richness of 2-polygraphs is confirmed by the large number and variety of categories they present. In order to show that a given polygraph is a presentation of a given category, one can either tackle the issue directly, by using rewriting tools, or take a modular approach, by combining already known presentations: this is the route taken in the present chapter. Three significant applications are given. First addressed is the presentation of limits and colimits by means of given presentations of the base categories, and precisely shown is how to systematically build presentations of products, coproducts, and pushouts. Next, it is shown how to add formal inverses to some morphisms of a category at the level of presentations. Finally, distributive laws are investigated in relation to factorization systems on categories. A notion of composition along a distributive law between two small categories sharing the same set of objects is introduced, and it is shown how to derive a presentation of this composite from presentations of the components.

The present chapter concentrates on some useful families of n-polygraphs based on familiar shapes: cylinders, cubes, and simplices. These families are crucial in the development of a homotopy theory of ω-categories. Two methods for generating these families are explained. The first one is based on a direct definition of the cylinder polygraph of a polygraph. The second is based on Steiner’s theory of augmented directed complexes, which is a very powerful tool to build polygraphs using chain complexes. In particular, it allows to define a tensor product for polygraphs (or even ω-categories) from which the cylinder polygraph, as well as a join operation, can be recovered.