Bousfield localization

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.

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 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 (necessarily, since cofibrations and weak equivalences determine the fibrations)

In this definition, a C-local equivalence is a map ${\displaystyle f:X\to Y}$ which, roughly speaking, does not make a difference when mapping to a C-local object. More precisely, ${\displaystyle f^{*}:map(Y,W)\to map(X,W)}$ 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

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