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
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