To start off the categorical prerequisites of the theory, this chapter reviews some of the basic elements of Quillen's theory of model categories and modern variants. We cover the main definitions and what can be done on a formal level. A more thorough discussion of the process of adding on infinitely many cells to form a cell complex, and the resulting constructions which are grouped around what is usually known as the “small object argument,” are left for the next chapter. These are the places which use the notion of locally presentable category in a fundamental way, so that treatment will start off the next chapter. For a reference to the distinction between large and small categories, the reader is therefore asked to skip ahead to the beginning of the next chapter.

Near the end of the present chapter, the notion of cartesian model category is introduced. This is crucial to the subsequent development of an iterative theory of higher categories, as has been discussed in the first part of the book. We show here how to make use of the cartesian property to gain an internal Hom, a construction which eventually leads to the (n + 1)-category nCat of n-categories.

A few words are in order about why the theory of Quillen model categories is useful for treating higher categories. It seems clear that Grothendieck was aware of the applicability of the theory of model categories to these questions, at least on an intuitive level, because “Pursuing stacks” [132] started out as a series of letters to Quillen.