Bousfield localization


In category theory, a branch of mathematics, a Bousfield localization of a model category 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.

Model category structure of the Bousfield localization

Given a class C of morphisms in a model category M the left Bousfield localization is a new model structure on the same category as before. Its equivalences, cofibrations and fibrations, respectively, are
and
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 for any C-local object W. An object W is called C-local if it is fibrant and
is a weak equivalence for all maps in C. The notation is, for a general model category a certain simplicial set whose set of path components agrees with morphisms in the homotopy category of M:
If M is a simplicial model category, 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.

Existence

The left Bousfield localization model structure, as described above, is known to exist in various situations, provided that C is a set:
Combinatoriality and cellularity of a model category guarantee, in particular, a strong control over the cofibrations of M.
Similarly, the right Bousfield localization exists if M is right proper and cellular or combinatorial and C is a set.

Universal property

The localization of an category C with respect to a class W of morphisms satisfies the following universal property:
The Bousfield localization is the appropriate analogous notion for model categories, keeping in mind that isomorphisms in ordinary category theory are replaced by weak equivalences. That is, the Bousfield localization is such that

Localization and completion of a spectrum

Localization and completion of a spectrum at a prime number p are both examples of Bousfield localization, resulting in a local spectrum. For example, localizing the sphere spectrum S at p, one obtains a local sphere.

Stable model structure on spectra

The stable homotopy category is the homotopy category of spectra, endowed with the stable model structure. The stable model structure is obtained as a left Bousfield localization of the level model structure on spectra, whose weak equivalences are those maps which are weak equivalences in all levels.

Morita model structure on dg categories

Morita model structure on the category of small dg categories is Bousfield localization of the standard model structure.