XIV

INTRODUCTION

left proper (see Definition 13.1.1), then for a map to have the right lifting property

with respect to all cofibrations that are S-local equivalences, it is sufficient that it

have the right lifting property with respect to all inclusions of cell complexes that

are S-local equivalences (see Proposition 13.2.1). In order to make the cardinality

argument, though, we need to assume that maps between cell complexes in M are

sufficiently well behaved; this leads us to the definition of a cellular model category

(see Definition 12.1.1).

A cellular model category is a cofibrantly generated model category with addi-

tional properties that ensure that

• the intersection of a pair of subcomplexes (see Definition 12.2.5) of a cell

complex exists (see Theorem 12.2.6),

• there is a cardinal a (called the size of the cells of M; see Definition 12.3.3)

such that if X is a cell complex of size r, then any map from X to a

cell complex factors through a subcomplex of size at most O~T (see Theo-

rem 12.3.1), and

• if X is a cell complex, then there is a cardinal n such that if Y is a cell

complex of size v (y 2), then the set M(X, Y) has cardinal at most v^

(see Proposition 12.5.1).

Fortunately, these properties follow from a rather minimal set of conditions on the

model category M (see Definition 12.1.1), satisfied by almost all model categories

that come up in practice.

Left localization and right localization. There are two types of morphisms

of model categories: Left Quillen functors and right Quillen functors (see Defini-

tion 8.5.2). The localizations that we have been discussing are all left localizations,

because the functor from the original model category to the localized model cate-

gory is a left Quillen functor that is initial among left Quillen functors whose total

left derived functor takes the images of the designated maps into isomorphisms in

the homotopy category (see Definition 3.1.1). There is an analogous notion of right

localization.

Given a CW-complex A, Dror Farjoun [20, 21, 23, 24] defines a map of

topological spaces / : X — Y to be an A-cellular equivalence if the induced map

of function spaces /* : Map(A, X) — Map(A, Y) is a weak equivalence. He also

defines the class of A-cellular spaces to be the smallest class of cofibrant spaces that

contains A and is closed under weak equivalences and homotopy colimits. We show

in Theorem 5.1.1 and Theorem 5.1.6 that this is an example of a right localization,

i.e., that there is a model category structure in which the weak equivalences are

the A-cellular equivalences and in which the cofibrant objects are the A-cellular

spaces. In fact, we do this for an arbitrary right proper cellular model category

(see Theorem 5.1.1 and Theorem 5.1.6).

The situation here is not as satisfying as it is for left localizations, though.

The left localizations that we construct for left proper cellular model categories

are again left proper cellular model categories (see Theorem 4.1.1), but the right

localizations that we construct for right proper cellular model categories need not

even be cofibrantly generated if not every object of the model category is fibrant.

However, if every object is fibrant, then a right localization will again be right

proper cellular with every object fibrant; see Theorem 5.1.1.