Noncommutative localization in homotopy theory

Introduction In a sense, noncommutative localization is at the center of homotopy theory, or even more accurately, one form of it is homotopy theory. After all, Gabriel and Zisman and later Quillen observed that the homotopy category of CW-complexes can be obtained from the category of topological spaces by formally inverting the maps which are weak homotopy equivalences. More generally, the homotopy category of any Quillen model category can be built by formally inverting maps. In a slightly different direction, the process of localization with respect to a map (§2) has recently developed into a powerful tool for making homotopy-theoretic constructions; roughly speaking, localizing with respect to f involves converting an object X into a new one, L f ( X ), with the property that, as far as mapping into L f ( X ) goes, f looks like an equivalence. In this paper we will show how the Cohn noncommutative localization described in can be interpreted as an instance of localization with respect to a map (3.2). Actually, we produce a derived form of the Cohn localization, and show that the circumstances in which the Cohn localization is most useful are exactly those in which the higher derived information vanishes (3.3). Finally, we sketch how the derived Cohn localization can sometimes be computed by using a derived form of the categorical localization construction from Gabriel and Zisman (§4).