Summary of Part 1
Part 1 contains our discussion of localization of model categories. Throughout
Part 1 we freely use the results of Part 2, which is our reference for techniques of
homotopy theory in model categories, and which logically precedes Part 1.
In Chapters 1 and 2 we discuss localization in a category of spaces. We work
in parallel in four different categories:
• The category of (unpointed) topological spaces.
• The category of pointed topological spaces.
• The category of (unpointed) simplicial sets.
• The category of pointed simplicial sets.
Given a map / , in Chapter 1 we discuss /-local spaces, /-local equivalences, and
/-localizations of spaces. We construct an /-localization functor, as well as a con-
tinuous version of the /-localization functor. We discuss commuting localizations
with the total singular complex and geometric realization functors, and compare
localizations in a category of pointed spaces with localizations in a category of
In Chapter 2 we establish a model category structure on the category of spaces
in which the weak equivalences are the /-local equivalences and the fibrant ob-
jects are the /-local spaces. This requires a careful analysis of the cell complexes
constructed by the /-localization functor defined in Chapter 1, and the main argu-
ment involves studying the cardinality of the set of cells in the localization of a cell
In Chapter 3 we define left and right localizations of a model category M with
respect to a class C of maps in M. A left localization of M with respect to C is a
left Quillen functor defined on M that is initial among those that take the images
in the homotopy category of the elements of C into isomorphisms. A right locali-
zation is the analogous notion for right Quillen functors. We also define Bousfield
localizations, which are localizations obtained by constructing a new model cate-
gory structure on the original underlying category. (The localization of Chapter 2
is a left Bousfield localization.) We discuss local objects, local equivalences, and
localization functors in this more general context.
Chapter 4 contains our main existence results for left localizations. We show
that if M is a left proper cellular model category, then the left Bousfield localization
of 3V C with respect to an arbitrary set S of maps in M exists. The proof requires
that we first define an 5-localization functor for objects of M, and then carefully
analyze the cardinality of the set of cells in the localization of a cell complex.
Chapter 5 contains our main existence results for right localizations. If M is
a model category and K is a set of objects of M, then a map / : X — » Y in M is
defined to be a K-colocal equivalence if for every object A in K the induced map of