Bousfield Localization
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In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, 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. Bousfield localization is named after
Aldridge Bousfield Aldridge Knight Bousfield (April 5, 1941 – October 4, 2020), known as "Pete", was an American mathematician working in algebraic topology, known for the concept of Bousfield localization. Work and life Bousfield obtained both his undergrad ...
, 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 The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all ma ...
, 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 In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given c ...
with respect to the cofibrations in ''M'' which are also ''C''-local equivalences. In this definition, a ''C''-local equivalence is a map f\colon X \to Y which, roughly speaking, does not make a difference when mapping to a ''C''-local object. More precisely, f^* \colon \operatorname (Y, W) \to \operatorname (X, W) 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 :s^* \colon \operatorname (B, W) \to \operatorname (A, W) is a weak equivalence for ''all'' maps s\colon A \to B in ''C''. The notation \operatorname(-, -) 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'': :\pi_0 (\operatorname(X, Y)) = \operatorname_(X, Y). 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).


Existence

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. 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 C ^/math> of an (ordinary) category ''C'' with respect to a class ''W'' of morphisms satisfies the following universal property: * There is a functor C \to C ^/math> which sends all morphisms in ''W'' to isomorphisms. * Any functor C \to D 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 L_C M is such that * There is a left Quillen functor M \to L_C M whose left derived functor sends all morphisms in ''C'' to weak equivalences. * Any left Quillen functor M \to N whose left derived functor sends ''C'' to weak equivalences factors uniquely through M\to L_C M.


Examples


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 In stable homotopy theory, a branch of mathematics, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''0, i.e., a set of two points. Explicitly, the ''n''th space in the sphere spectrum ...
''S'' at ''p'', one obtains a
local sphere In stable homotopy theory, a branch of mathematics, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''0, i.e., a set of two points. Explicitly, the ''n''th space in the sphere spectrum ...
S_.


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.


Morita model structure on dg categories

Morita model structure on the category of small dg categories is Bousfield localization of the standard model structure (the one for which the weak equivalences are the quasi-equivalences).


See also

* Localization of a topological space


References

{{Reflist * Hirschhorn, ''Model Categories and Their Localizations'', AMS 2002
Absence of Maps Between p-local and q-local spectra


External links


Bousfield localization in nlab
*J. Lurie
Lecture 20
in Chromatic Homotopy Theory (252x). Category theory Homotopy theory