Bousfield localization

In , a branch of mathematics, a (left) Bousfield localization of a replaces the model structure with another model structure with the same cofibrations but with more weak equivalences. Bousfield localization is named after Aldridge Bousfield, who first introduced this technique in the context of localization of topological spaces and spectra. Given a class C of morphisms in a M the left Bousfield localization is a new model structure on the same category as before. Its equivalences, cofibrations and fibrations, respectively, are the C-local equivalences the original cofibrations of M and (necessarily, since cofibrations and weak equivalences determine the fibrations) the maps having the right lifting property with respect to the cofibrations in M which are also C-local equivalences. In this definition, a C-local equivalence is a map which, roughly speaking, does not make a difference when mapping to a C-local object. More precisely, is required to be a weak equivalence (of simplicial sets) for any C-local object W. An object W is called C-local if it is fibrant (in M) and is a weak equivalence for all maps in C. The notation is, for a general model category (not necessarily over simplicial sets) a certain simplicial set whose set of path components agrees with morphisms in the of M: If M is a simplicial model category (such as, say, simplicial sets or topological spaces), then "map" above can be taken to be the derived simplicial mapping space of M. This description does not make any claim about the existence of this model structure, for which see below. Dually, there is a notion of right Bousfield localization, whose definition is obtained by replacing cofibrations by fibrations (and reversing directions of all arrows). The left Bousfield localization model structure, as described above, is known to exist in various situations, provided that C is a set: M is left proper (i.e., the pushout of a weak equivalence along a cofibration is again a weak equivalence) and combinatorial M is left proper and cellular.