Localization is more a way of life than any one specific result. For example, under this rubric one can include Bousfield localization with respect to a ho- mology theory, localization with respect to a map as pioneered by Bousfield, Dror-Farjoun and elaborated on by many others, and even the formation of the stable homotopy category. We will touch on all three of these subjects, but we also have another purpose. There is a body of extremely useful techniques that we will explore and expand on. These have come to be known as Bousfield factor- ization, which is a kind of “trivial cofibration-fibration” factorization necessary for producing localizations, and the Bousfield-Smith cardinality argument. This latter technique arises when one is confronted with a situation where a fibration is defined to be a map which has the right lifting property with respect to some class of maps. However, for certain arguments one needs to know it is sufficient to check that the map has the right lifting property with respect to a set of maps. We explain both Bousfield factorization and the cardinality argument and explore the implications in a variety of contexts. In particular, we explore localization in diagram categories, with an eye towards producing a model for the stable homotopy category, and we produce a simplicial model category structure on categories of diagrams that will be useful in a later discussion of homotopy inverse limits.
The concept of localization probably has its roots in the notion of a Serre class of abelian groups and the Whitehead Theorem mod a Serre class [86, §9.6]. This result is still useful and prevalent — so prevalent, in fact, that it is often used without reference. The idea of localizing a space with respect to a homology theory appeared in Sullivan’s work on the Adams conjecture [85], where there is an explicit localization of a simply connected space with respect to ordinary homology with Z[1/p] coefficients. Bousfield and Kan [14] gave the first cate- gorical definition of localization with respect to homology theory and provided a localization for nilpotent spaces with respect to H∗(·, R), where R = Fp for some prime p or R a subring of the rationals. Their technique was the R-completion of space, recapitulated in Section 3 below for the case R = Fp. It was Bousfield himself who introduced model category theoretic techniques to provide the lo- calization of any space with respect to an arbitrary homology theory. His paper [8] has been enormously influential, as much for the methods as for the results, and it’s hard to overestimate its impact. For example, the concept of localization with respect to a map and the construction of its existence, which appears in the work for Dror-Farjoun [22] and [23] is directly influenced by Bousfield’s ideas. About the time Dror-Farjoun’s papers were first circulating, a whole group of people began to explore these ideas, both in homotopy theory and in relatedalgebraic subjects. The paper by Cascuberta [17] is a useful survey. One should also mention the important paper of Bousfield [11], which uses similar techniques for its basic constructions. The longest and most general work in this vein, a work that includes an exposition of the localization model category in an arbi- trary cellular model category is that of Hirschhorn [42], available at this writing over the Internet. The notion of a cellular model category is one way of axioma- tizing the structure necessary to make Bousfield’s arguments work, based on the concept of cell complexes and inclusions of sub-complexes. We give, in Section 4, a slightly different list of hypotheses along these lines. Both systems of axioms work in any example that we know. We emphasize, however, that Bousfield’s ideas had influence outside of the area of homotopy localization. For example, Jeff Smith realized very early on that one could use these constructions to put a model category structure on the category of small diagrams of simplicial sets, so that homotopy inverse limits can be computed as total right derived functors of inverse limit. This never made it into print, and we go through the arguments in Section 5. Beyond this, there is the second author’s work on the homotopy theory of simplicial presheaves [46], see also [38] as well as Joyal’s result for simplicial sheaves [53]. In the context of the present discussion that work can be interpreted as follows: the category of presheaves on a Groethendieck site is a category of diagrams and there is a closed model category structure obtained by localizing with respect to a class of cofibrations determined by the topology of the underlying site. END
nLab page on Localization 2