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| 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.
| | Hello, I'm Kristi, a 17 year old from Reykjavik, Iceland.<br>My hobbies include (but are not limited to) Petal collecting and pressing, Rock climbing and watching Supernatural.<br><br>Take a look at my weblog; get help for your health ([http://www.glyconutrientsinfo.com www.glyconutrientsinfo.com]) |
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| Bousfield localization is named after [[Aldridge Bousfield]], who first introduced this technique in the context of localization of topological spaces and spectra.<ref>Aldridge Bousfield, ''[http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf The localization of spectra with respect to homology]'', Topology vol 18 (1979)</ref><ref>Aldridge Bousfield, ''The localization of spaces with respect to homology'', Topology vol. 14 (1975)</ref>
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| ==Model category structure of the Bousfield localization==
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| Given a [[class (mathematics)|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
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| * the ''C''-local equivalences
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| * the original cofibrations of ''M''
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| and (necessarily, since cofibrations and weak equivalences determine the fibrations)
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| * the maps having the right [[lifting property]] with respect to the cofibrations in ''M'' which are also ''C''-local equivalences.
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| In this definition, a ''C''-local equivalence is a map <math>f: X \to Y</math> which, roughly speaking, does not make a difference when mapping to a ''C''-local object. More precisely, <math>f^* : map (Y, W) \to map (X, W)</math>
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| is required to be a weak equivalence (of [[simplicial set]]s) for any ''C''-local object ''W''. An object ''W'' is called ''C''-local if its is fibrant (in ''M'') and
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| :<math>s^* : map (B, W) \to map (A, W)</math>
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| is a weak equivalence for ''all'' maps <math>f: A \to B</math> in ''C''. The notation <math>map(-, -)</math> is, for a general model category (not necessarily [[enriched category|enriched]] over simplicial sets) a certain simplicial set whose set of [[path component]]s agrees with morphisms in the [[homotopy category]] of ''M'':
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| :<math>\pi_0 (map(X, Y)) = Hom_{Ho(M)}(X, Y).</math>
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| 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''.
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| This description does not make any claim about the existence of this model structure, for which see below.
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| Dually, there is a notion of ''right Bousfield localization'', whose definition is obtained by replacing cofibrations by fibrations (and reversing directions of all arrows).
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| ==Existence==
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| The left Bousfield localization model structure, as described above, is known to exist in various situations:
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| * ''M'' is left proper (i.e., the [[pushout]] of a weak equivalence along a cofibration is again a weak equivalence) and combinatorial
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| * ''M'' is left proper and cellular.
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| Combinatoriality and cellularity of a model category guarantee, in particular, a strong control over the cofibrations of ''M''.
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| The right Bousfield localization exists if ''M'' is right proper and cellular.
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| ==Universal property==
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| The [[localization of a category|localization]] <math>C[W^{-1}]</math> of an (ordinary) category ''C'' with respect to a class ''W'' of morphisms satisfies the following universal property:
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| * There is a functor <math>C \to C[W^{-1}]</math> which sends all morphisms in ''W'' to isomorphisms.
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| * Any functor <math>C \to D</math> that sends ''W'' to isomorphisms in ''D'' factors uniquely over the previously mentioned functor.
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| 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 <math>L_C M</math> is such that
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| * There is a ([[Quillen adjunction|left Quillen]]) functor <math>M \to L_C M</math> 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 <math>Ho(L_C M)</math>).
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| * Any (left Quillen) functor <math>M \to N</math> that sends ''C'' to weak equivalences factors uniquely over the previously mentioned functor.
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| ==Examples==
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| ===Localization and completion of a spectrum===
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| 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]] <math>S_{(p)}</math>.
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| ===Stable model structure on spectra===
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| 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.<ref>M. Hovey, ''[http://www.math.uiuc.edu/K-theory/0402/ Spectra and symmetric spectra in general model categories]'', Journal of Pure and Applied Algebra 165 (2001), section 3</ref>
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| == See also ==
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| *[[Localization of a topological space]]
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| == References ==
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| <references />
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| * Hirschhorn, ''Model Categories and Their Localizations'', AMS 2002
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| *http://mathoverflow.net/questions/87174/absence-of-maps-between-p-local-and-q-local-spectra
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| == External links ==
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| *[http://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories Bousfield localization in nlab].
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| *J. Lurie, [http://www.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf Lecture 20] in Chromatic Homotopy Theory (252x).
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| [[Category:Category theory]]
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| [[Category:Homotopy theory]]
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Hello, I'm Kristi, a 17 year old from Reykjavik, Iceland.
My hobbies include (but are not limited to) Petal collecting and pressing, Rock climbing and watching Supernatural.
Take a look at my weblog; get help for your health (www.glyconutrientsinfo.com)