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
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 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