7 - An introduction to Bousfield localization

Summary

Bousfield localization encodes a wide variety of constructions in homotopy theory, analogous to localization and completion in algebra. Our goal in this chapter is to give an overview of Bousfield localization, sketch how basic results in this area are proved, and illustrate some applications of these techniques. Near the end we will give more details about how localizations are constructed using the small object argument. The underlying methods apply in many contexts, and we provide examples that exhibit a variety of behaviors.

We will begin by discussing categorical localizations. Given a collection of maps in a category, the corresponding localization of that category is formed by making these maps invertible in a universal way; this technique is often applied to discard irrelevant information and focus on a particular type of phenomenon. In certain cases, localization can be carried out internally to the category itself: this happens when there is a sufficiently ample collection of objects that already see these maps as isomorphisms. This leads naturally to the study of reflective localizations.

Bousfield localization generalizes this by taking place in a category where there are spaces of functions, rather than sets, with uniqueness only being true up to contractible choice. Bousfield codified these properties, for spaces in [54] and for spectra in [55]. The definitions are straightforward, but proving that localizations exist takes work, some of it of a set-theoretic nature.

Our presentation is close in spirit to Bousfield’s work, but the reader should go to the books of Farjoun [96] and Hirschhorn [124] for more advanced information on this material. We will focus, for the most part, on left Bousfield localization, since the techniques there are easier and this is where most of our applications lie. In [25] right Bousfield localization is discussed at greater length.

The story of localization techniques in algebraic topology probably begins with Serre classes of abelian groups [272]. After choosing a class C of abelian groups that is closed under subobjects, quotients, and extensions, Serre showed that one could effectively ignore groups in C when studying the homology and homotopy of a simply connected space X. In particular, he proved mod-C versions of the Hurewicz and Whitehead theorems, showed the equivalence between finite generation of homology and homotopy groups, determined the rational homotopy groups of spheres, and significantly reduced the technical overhead in computing the torsion in homotopy groups by allowing one to work with only one prime at a time. His techniques for computing rational homotopy groups only require rational homology groups; p-local homotopy groups only require p-local homology groups; p-completed homotopy groups only require mod-p homology groups.