LOCALIZING MODEL CATEGORY STRUCTURE S xi
more complete description of the contents of Part 2, see the summary on page 103
and the introductions to the individual chapters. For a description of Part 1, which
discusses localizing model category structures, see below, as well as the summary
on page 3.
Prerequisites. The category of simplicial sets plays a central role in the homo-
topy theory of a model category, even for model categories unrelated to simplicial
sets. This is because a homotopy function complex between objects in a model
category is a simplicial set (see Chapter 17). Thus, we assume that the reader has
some familiarity with the homotopy theory of simplicial sets. For readers without
the necessary background, we recommend the works by Curtis [18], Goerss and
Jardine [39], and May [49].
Localizing model category structures
Localizing a model category with respect to a class of maps does not mean
making the maps into isomorphisms; instead, it means making the images of those
maps in the homotopy category into isomorphisms (see Definition 3.1.1). Since the
image of a map in the homotopy category is an isomorphism if and only if the
map is a weak equivalence (see Theorem 8.3.10), localizing a model category with
respect to a class of maps means making maps into weak equivalences.
Localized model category structures originated in Bousfield's work on local-
ization with respect to homology ([8]). Given a homology theory /i*, Bousfield
established a model category structure on the category of simplicial sets in which
the weak equivalences were the maps that induced isomorphisms of all homology
groups. A space (i.e., a simplicial set) W was defined to be local with respect
to h* if it was a Kan complex such that every map / : X Y that induced
isomorphisms /*: h*X w h*Y of homology groups also induced an isomorphism
/*: 7r(Y, W) ~ TT(X, W) of the sets of homotopy classes of maps to W. In Bous-
field's model category structure, a space was fibrant if and only if it was local with
respect to /i*.
The problem that led to Bousfield's model category structure was that of con-
structing a localization functor for a homology theory. That is, given a homology
theory /i*, the problem was to define for each space X a local space L/^X and a
natural homology equivalence X L/^X. There had been a number of partial
solutions to this problem (perhaps the most complete being that of Bousfield and
Kan [14]), but each of these was valid only for some special class of spaces, and
only for certain homology theories. In [8], Bousfield constructed a functorial /i*-
localization for an arbitrary homology theory /i* and for every simplicial set. In
Bousfield's model category structure, a fibrant approximation to a space (i.e., a
weak equivalence from a space to a fibrant space) was exactly a localization of that
space with respect to h*.
Some years later, Bousfield [9, 10, 11, 12] and Dror Farjoun [20, 22, 24]
independently considered the notion of localizing spaces with respect to an arbitrary
map, with a definition of "local" slightly different from that used in [8]: Given a
map of spaces / : A B, a space W was defined to be f-local if / induced a weak
equivalence of mapping spaces /*: Map(5, W) = Map(^4, W) (rather than just a
bijection on components, i.e., an isomorphism of the sets of homotopy classes of
maps), and a map g: X Y was defined to be an f-local equivalence if for every f-
local space W the induced map of mapping spaces g* : Map(Y, W) Map(X, W)
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