CHAPTER 1

Model categories

In this first chapter, we discuss the basic theory of model categories. It very

often happens that one would like to consider certain maps in a category to be

isomorphisms when they are not. For example, these maps could be homology

isomorphisms of some kind, or homotopy equivalences, or birational equivalences

of algebraic varieties. One can always invert these "weak equivalences" formally,

but there is a foundational problem with doing so, since the class of maps between

two objects in the localized category may not be a set. Also, it is very difficult to

understand the maps in the resulting localized category. In a model category, there

are weak equivalences, but there are also other classes of maps called cofibrations

and fibrations. This extra structure allows one to get precise control of the maps

in the category obtained by formally inverting the weak equivalences.

Model categories were introduced by Quillen in [Qui67], as an abstraction

of the usual situation in topological spaces. This is where the terminology came

from as well. Quillen's definitions have been modified over the years, by Quillen

himself in [Qui69] and, more recently, by Dwyer, Hirschhorn, and Kan [DHK].

We modify their definition slightly to require that the functorial factorizations be

part of the structure. The reader may object that there is now more than one

different definition of a model category. That is true, but the differences are slight:

in practice, a structure that satisfies one definition satisfies them all. We present

our definition and some of the basic facts about model categories in Section 1.1.

At this point, the reader would certainly like some interesting examples of

model categories. However, that will have to wait until the next chapter. The

axioms for a model category are very powerful. This means that one can prove

many theorems about model categories, but it also means that it is hard to check

that any particular category is a model category. We need to develop some theory

first, before we can construct the many examples that appear in Chapter 2.

In Section 1.2 we present Quillen's results about the homotopy category of a

model category. This is the category obtained from a model category by inverting

the weak equivalences. The material in this section is standard, as the approach of

Quillen has not been improved upon.

In Section 1.3 we study Quillen functors and their derived functors. The most

obvious requirement to make on a functor between model categories is that it

preserve cofibrations, fibrations, and weak equivalences. This requirement is much

too stringent, however. Instead, we only require that a Quillen functor preserve

half of the model structure: either cofibrations and trivial cofibrations, or fibrations

and trivial fibrations, where a trivial cofibration is both a cofibration and a weak

equivalence, and similarly for trivial fibrations. This gives us left and right Quillen

functors, and could give us two different categories of model categories. However,

in practice functors of model categories come in adjoint pairs. We therefore define

http://dx.doi.org/10.1090/surv/063/01