related by the Dold-Kan correspondence
Bousfield localization is a procedure that to a model category structure assigns a new one with more weak equivalences. It is a special case of a localization of model categories, corresponding to the homotopy-version of the notion of localization of categories by reflective subcategories.
The historically original example is the Bousfield localization of spectra. But the notion is much more general.
A left Bousfield localization of a model category is another model category structure on the same underlying category with the same cofibrations,
but more weak equivalences
While that’s a very simple definition, it turns out that something interesting happens to the fibrations when we keep the cofibrations fixed and increase the weak equivalences.
It follows directly that
has as fibrations a subset of fibrations of
has the same acyclic fibrations as
on the underlying categories
the identity functor preserves cofibrations and weak equivalences
the identity functor preserves fibrations and acyclic fibrations
so that this pair of functors is a Quillen adjunction
And a very special one: the category modeled by a model category is its full subcategory on fibrant-cofibrant objects. Under left Bousfield localization the fibrant-cofibrant objects of are a subcollection of those of , so that we have the full subcategory
Moreover, as we shall see, every object in is weakly equivalent in to one in : it reflects into .
Bousfield localization is a model category version of reflecting onto a reflective subcategory.
Indeed – at least when is a combinatorial simplicial model category – Bousfield localization is an example of a localization of a simplicial model category and under passage to the sub-category of fibrant-cofibrant objects this Quillen adjunction becomes the inclusion of a reflective (∞,1)-subcategory
hence of a localization of an (∞,1)-category.
Such a localization is determined by the collection of local weak equivalences in , and alternatively by the collection of -local objects in . Indeed, is the full -subcategory on the cofibrant and fibrant and -local objects of .
More in detail, the weak equivalences that are added under Bousfield localization are -local weak equivalences” for some set . We will see below why this is necessarily the case if is a cofibrantly generated model category. For the moment, we take the following to be a refined definition of left Bousfield localization.
Let be a
be a subclass of cofibrations with cofibrant domain.
We want to characterize objects in that “see elements of as weak equivalences”. Notice that
So we can “test isomorphism by homming them into objects”. This phenomenon we use now the other way round, to characterize new weak equivalences:
(-local objects and -local weak equivalences)
This is a slightly simplified version of a more general definition using derived hom spaces, where we do not have to assume that the domains and codomains of elements are and not that the local objects are fibrant.
(-local objects and -local weak equivalences)
Assume that we have fibrant and cofibrant replacement functors . Then say
an object is an -local object if for all in the morphism
conversely, say that a cofibration is an -local weak equivalence if for all -local objects the morphism is a weak equivalence.
That this second condition is indeed compatible with the first one is shown here.
We write for the collection of -local weak equivalences.
is a weak equivalence of simplicial sets. This is described in detail at enriched homs from cofibrants to fibrants.
Here this implies in particular
Every ordinary weak equivalence is also -local weak equivalence.
Therefore, for any set , we can consider the left Bousfield localization at the -local weak equivalences :
(left Bousfield localization)
The left Bousfield localization of at is, if it exists, the new model category structure on with
cofibrations are the same as before, ;
acyclic cofibrations are the cofibrations that are -local weak equivalences.
Assume that the left Bousfield localization of a given model category at a class of cofibrations with cofibrant domain exists. Then it has the following properties.
The fibrant objects in are precisely the fibrant objects in that are -local.
To see this, we modify, if necessary, the set in a convenient way without changing the class of -local weak equivalences that it defines.
Lemma We may add to any set of -local cofibrations without changing the collection of -local objects and hence without changing the collection of -local weak equivalences themselves. In particular, we may add to without changing
all generating acyclic cofibrations of , i.e. ;
for every original morphism in and for every also the canonical morphism
Proof of the Lemma
We discuss why these morphisms of the latter type are indeed -local cofibrations with cofibrant domain:
to see that is indeed a cofibration notice that for every commuting diagram
we get as components of the top morphism the left square of
and similarly the components of the bottom morphism consitute a morphism which by the commutativity of the original square is a lift of the outer diagram here. The top left triangle of this lift in turn gives a square
So this last diagram has a lift and this is adjunct to the lift of the original lifting problem that we are looking for.
Therefore is indeed a cofibration.
Notice that in these arguments we made use of
Next, again using the Quillen bifunctor property of the tensoring of over SSet we find that with cofibrant in and being cofibrant in SSet it follows that is cofibrant. Similarly for the other cases. And the coproduct of two cofibrants is cofibrant because cofibrations are preserved under pushout. Therefore is indeed a cofibrant domain of our cofibration.
With being a cofibration, we can check -locality by homming into fibrant -local objects and checking if that produces an acyclic Kan fibration.
in SSet. Here the top and the lowest morphisms are weak equivalences by the fact that is an acyclic Kan fibration by the characterization of -local cofibrations and the fact that SSet is an SSet-enriched model category. Similarly for the fibration on the left, which implies by right properness of SSet that the bottom horizontal morphism is a weak equivalence, which finally implies by 2-out-of-3 that the morphism in question is a weak equivalence.
end of the proof of the lemma
This shows that we can assume that contain the generating acyclic cofibrations and the morphism called .
As usual, we say that given a set of morphisms and an object that has the extension property with respect to if every diagram
has a lift.
We claim now that the the objects of that have the extension property with respect to our set are precisely the fibrant and -local objects. The argument proceeds along the same lines as the proof of the above lemma.
In one direction, if that has the extension property with respect to it has it in particular with respect to the generating acyclic cofibrations and hence is fibrant.
And it in particular has the extension property with respect to . Observe that by the pushout definition of a morphism
consists of two component maps and such that
And in terms of this a lift
consists of a lift
Conversely, if is fibrant and -local, then for all in the map in is an acyclic Kan fibration hence in particular its underlying map of sets is a surjection, so has the extension property.
Now every fibrant object in has the extension property with respect to hence in particular with respect to , so is -local and fibrant in .
Conversely, if it is -local and fibrant in ; then, as mentioned before, for all the map is an acyclic Kan fibration in SSet so that in particular is a surjection, which means that has the extension property with respect to all and is hence fibrant in .
We had remarked already in the previous argument that objects with the extension property relative to , i.e. objects whose morphism to the terminal object is in , are fibrant as well as -local in .
Therefore is in particular a fibrant approximation functor in and is the weak equivalence
in relating an object to its fibrant approximation.
The -local weak equivalences between -local fibrant objects are precisely the original weak equivalences between these objects.
Consider the full subcategory of the homotopy category of on the -local objects. The image of an -local weak equivalence in there satisfies for every object in there that is an isomorphism. By the Yoneda lemma this implies that is an isomorphism in . Since that is a full subcategory, it follows that is also an isomorphism in . But that means precisely that it is a weak equivalence in .
We have considered two definitions of left Bousfield localization: in the first we just required that cofibrations are kept and weak equivalences are increased. In the second we more specifically took the weak equivalences to be -local weak equivalences.
We now show that every localization in the first sense is indeed of the second kind if we demand that both the original and the localized category are left proper, cofibrantly generated simplicial model categories.
In the context of left proper, cofibrantly generated simplicial model categories,
for a left Bousfield localization of (i.e. a structure with the same cofibrations as and more weak equivalences), there is a set such that
We show that choosing to be the set of generating acyclic cofibrations does the trick.
First, the cofibrations of and coincide. Moreover, the acyclic cofibrations of contain all the acyclic cofibrations of because
It remains to show that, conversely, every acyclic cofibration in is an acyclic cofibration in .
Choose a cofibrant replacement for and
Then by 2-out-of-3 and since the morphism is still an acyclic cofibration on . Again by 2-out-of-3 and since , it is sufficient to show that is an acyclic cofibration in .
To show that it is an acyclic cofibration in it suffices to show that for every fibrant object the morphism
is a trivial fibration. Either by assumption or by the characterization of S-local cofibrations this is the case if is -local and fibrant in . The first statement is one of the direct consequences of the definition of and the second follows because .
Let and be categories for which left Bousfield localization exists, and let
be a Quillen equivalence. Then for every small set there is an induced Quillen equivalence of left Bousfield localizations
This is due to Hirschhorn.
If the left Bousfield localization exists, i.e. is indeed a model category with the above definitions of cofibrations and weak equivalences, then it is indeed a localization of a model category in that there is a left Quillen functor
(i.e. preserves cofibrations and trivial cofibrations and has a right adjoint)
such that the total left derived functor
takes the images of in to isomorphisms
and every other left Quillen functor with this property factors by a unique left Quillen functor through .
Moreover, the identity functor on the underlying category is a Quillen adjunction
(and is itself a localization functor).
The first part is theorem 3.3.19 in ModLoc . The second part is prop 3.3.4, which follows directly from the following proposition.
We discuss the existence of left Bousfield localization in the context of combinatorial model categories. A similar existence result is available in the slightly more general context of cellular model categories, but for the combinatorial case a somewhat better theory is available.
By the corollary to Dugger’s theorem on presentations for combinatorial model categories we have that every combinatorial model category is Quillen equivalent to a left proper simplicial combinatorial model category.
Therefore there is little loss in assuming this extra structure, which the following statement of the theorem does.
If is a
then the left Bousfield localization does exist as a combinatorial model category.
Moreover, it satisfies the following conditions:
We follow Bar for the proof that the assumptions of Smith’s recognition theorem are satisfied and follow HTT, prop. A.3.7.3 for the characterization of the fibrant objects. The details are spelled out in the following subsections.
The proof we give is self-contained, except that it builds on the following notions and facts.
A cardinal number is regular if it is not the cardinality of a union of sets of size .
This means that a morphism from a -compact object into an object that is a -directed colimit over component objects always lifts to one of these component objects.
An object is a small object if it is -compact for some .
A locally small but possibly non-small category is an accessible category if it has a small sub-set of generating -compact objects such that every other object is a -directed colimit over such generators.
If such a category has all small colimits, it is called a locally presentable category.
In particular, in a locally presentable category the small object argument for factoring of morphisms applies with respect to every set of morphisms.
A combinatorial model category is a locally presentable category that is equipped with a cofibrantly generated model category structure. So in particular there is a set of generating (acyclic) cofibrations that map between small objects.
(If we’d require smallness of the domains of the generating cofibrations only with respect to these cofibrations themselves, we have instead the notion of cellular model category.)
Smith’s recognition theorem says that a locally presentable category has a combinatorial model category structure already if it has weak equivalences and generating cofibrations satisfying a simple condition and if weak equivalences form an accessible subcategory of the arrow category. This means that only two thirds of the data for a generic combinatorial model category needs to be checked and greatly facilitates checking model category structures.
By the axioms of an enriched model category it follows that the functor
Similarly, the collection of morphisms such that for all -local objects is a weak equivalence is called the collection of -local weak equivalences.
Sometimes ordinary (co)limits in a model category are already also homotopy colimits:
Since these are the two operations under which is closed, this facilitates finding this closure given that by the above the elements of are characterized by their images under for -local .
The following proof uses the small object argument several times. In particular, at one point it is applied relative to the collection of morphisms at which we localize. It is at this point that we need that assumption that is indeed a (small) set, and not a proper class.
For the small object argument itself, this requirement comes from the fact that it involves colimits indexed by . These won’t in general exist if is not a set.
The collection of -local weak equivalences , however, won’t be a small set in general even if is. But for Smith’s recognition theorem to apply we need to check that the full subcategory of on is, while not small, accessible.
To establish this we need two properties of accessible categories: the inverse image of an accessible subcategory under a functor is accessible, and the collections of fibrations, weak equivalences and acyclic fibrations in a combinatorial model category are accessible.
Beginning of the proof of the existence of the left Bousfield localization of a left proper combinatorial simplicial model category at a set of morphisms.
check that the weak equivalences form an accessibly embedded accessible subcategory.
For the first item choose with any set of generating cofibrations of , that exists by assumption on . Then .
It remains to demonstrate closure of under pushout and transfinite composition.
for an -local morphism – a morphism in – and for
of -groupoids. where the bottom morphism is a weak equivalence by assumption of -locality of and . But then also the top horizontal morphism is a weak equivalence for all -local and therefore is in .
Similarly for transfinite composition colimits.
Therefore, indeed, is closed under pushouts and transfinite composition.
By the general properties of accessible categories for that it is sufficient to exhibit as the inverse image of under functor of the accessible category spanned by ordinary weak equivalences in .
That functor we take to be the -local fibrant replacement functor from above
By one of the above propositions, -local weak equivalence between -local objects are precisely the ordinary weak equivalences. This means that the inverse image under of the weak equivalences in are all -local weak equivalences
Therefore this is an accessible category.
End of the proof of the existence of the left Bousfield localization of a left proper combinatorial simplicial model category at a set of morphisms.
Every combinatorial localization of is already of the form for a set of just cofibrations.
We demonstrate that does the trick.
If one assumes large cardinal axioms then the existence of Bousfield localization follows much more generally.
Vopěnka's principle implies the statement:
This is theorem 2.3 in (RosickyTholen).
The above statement is generalized to the context of enriched model category theory by the following result:
a small set
(all with respect to a fixed Grothendieck universe).
This is (Barwick, theorem 4.46).
the category of symmetric spectra in a pointed left-proper cellular model category
and so on
the over category for any object .
As described at presentable (∞,1)-category, an (∞,1)-category is presentable precisely if, as an simplicially enriched category, it arises as the full subcategory of fibrant-cofibrant objects of a combinatorial simplicial model category.
The proof of this proceeds via Bousfield localization, and effectively exhibits Bousfield localization as the procedure that models localization of an (∞,1)-category when -categories are modeled by model categories.
For notice that
a presentable -category is one arising as the localization
in terms of the simplicial model category the prescription for localization as an (∞,1)-category and passing to the subcategory of fibrant-cofibrant objects of the Bousfield localization is literally the same: in both cases one passes to the full subcategory on the -local objects.
Moreover, by Dugger’s theorem on combinatorial model categories every combinatorial simplicial model category arises this way.
This is the argument of HTT, prop A.3.7.6.
This gives a good conceptual interpretation of Bousfield localization, since the localization of an (∞,1)-category is nothing but an adjunction
that exhibits as a reflective (∞,1)-subcategory of .
So we find the diagram
Localization of -presheaf categories
Here denotes passing to the full simplicially enriched subcategory on the fibrant-cofibrant objects, regarding that as an (∞,1)-category. (If one wants to regard that as a quasi-category, then also involves taking the homotopy coherent nerve of this simplicially enriched category.)
There is also a notion of Bousfield localization of triangulated categories.
Under suitable conditions it should be true that for a model category whose homotopy category is a triangulated category the homotopy category of a left Bousfield localization of is the left Bousfield localization of . See this answer on MO.
Bousfield localization appears as definition 3.3.1 in
for left proper cellular model categories.
In proposition A.3.7.3 of
A detailed discussion of Bousfield localization in the general context of enriched model category theory is in
The relation to Vopěnka's principle is discussed in