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

unpointed spaces.

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

complex.

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

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