Localization

Abstract

Localization is more a way of life than any specific collection of results. For example, under this rubric one can include Bousfield localization with respect to a homology 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 Bous-field factorization, 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 several contexts.