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

${\displaystyle s^{*}\colon \operatorname {map} (B,W)\to \operatorname {map} (A,W)}$

is a weak equivalence for all maps ${\displaystyle s\colon A\to B}$ in C. The notation ${\displaystyle \operatorname {map} (-,-)}$ 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:

${\displaystyle \pi _{0}(\operatorname {map} (X,Y))=\operatorname {Hom} _{Ho(M)}(X,Y).}$

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).

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.

The localization ${\displaystyle C[W^{-1}]}$ of an (ordinary) category C with respect to a class W of morphisms satisfies the following universal property:

• There is a functor ${\displaystyle C\to C[W^{-1}]}$ which sends all morphisms in W to isomorphisms.
• Any functor ${\displaystyle C\to D}$ 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 ${\displaystyle L_{C}M}$ is such that

• There is a left Quillen functor ${\displaystyle M\to L_{C}M}$ whose left derived functor sends all morphisms in C to weak equivalences.
• Any left Quillen functor ${\displaystyle M\to N}$ whose left derived functor sends C to weak equivalences factors uniquely through ${\displaystyle M\to L_{C}M}$.

• Localization of a topological space

• Hirschhorn, Model Categories and Their Localizations, AMS 2002
• Absence of Maps Between p-local and q-local spectra

• Bousfield localization in nlab.
• J. Lurie, Lecture 20 in Chromatic Homotopy Theory (252x).