In category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences.
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
- 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 enriched over simplicial sets) 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 (such as, say, simplicial sets or topological spaces), then "map" above can be taken to be the (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:
- 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.
Combinatoriality and cellularity of a model category guarantee, in particular, a strong control over the cofibrations of M.
The right Bousfield localization exists if M is right proper and cellular.
The localization of an (ordinary) category C with respect to a class W of morphisms satisfies the following universal property:
- There is a functor which sends all morphisms in W to isomorphisms.
- Any functor that sends W to isomorphisms in D factors uniquely over the previously mentioned functor.
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 (left) Bousfield localization is such that
- There is a (left Quillen) functor which sends all morphisms in C to weak equivalences in the localized model structure (i.e., to C-local equivalences, or, equivalently, to isomorphisms in the homotopy category ).
- Any (left Quillen) functor that sends C to weak equivalences factors uniquely over the previously mentioned functor.
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 (in the sense of model categories) of spectra, endowed with the stable model structure. The stable model structure is obtained as a left Bousfield localization of the level (or projective) model structure on spectra, whose weak equivalences (fibrations) are those maps which are weak equivalences (fibrations, respectively) in all levels.
- Aldridge Bousfield, The localization of spectra with respect to homology, Topology vol 18 (1979)
- Aldridge Bousfield, The localization of spaces with respect to homology, Topology vol. 14 (1975)
- M. Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra 165 (2001), section 3
- Hirschhorn, Model Categories and Their Localizations, AMS 2002