In this chapter we consider the general theory of localization which is due independently to Dror Farjoun and to A.K. Bousfield. The theory is founded on the homotopy theoretic consequences of inverting a specific map µ of spaces. Those spaces for which the mapping space dual of µ is an equivalence are called local. In turn, the local spaces define a set of maps called local equivalences. The localization of a space X is defined to be a universal local space which is locally equivalent to X.

Localizations always exist. It is a nice fact that the localizations of simply connected spaces are also simply connected. This makes it possible to restrict the theory to simply connected spaces which is what we do in this chapter.

For simply connected spaces, the Dror Farjoun–Bousfield theory specializes to the classical example of localization of spaces at a subset of primes S. The complementary set of primes is inverted. We begin by inverting the maps M → * for all Moore spaces M with one nonzero first homology group isomorphic to ℤ/qℤ where q is a prime not in S. In this case, the equivalences are maps which induce an isomorphism of homology localized at S.

Localization of spaces first occurs in the works of Daniel Quillen, of Dennis Sullivan, and of A.K. Bousfield and D.M. Kan.