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Contents
Context
Model category theory
model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
for ∞-groupoids
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
Modalities, Closure and Reflection
Contents
Idea
In model category-theory, the Bousfield-Friedlander theorem (Bousfield-Friedlander 78, theorem A.7, Bousfield 01, theorem 9.3) states that if an endofunctor on a model category behaves like an idempotent monad in an appropriate model category theoretic sense, then the left Bousfield localization model category structure of at the -equivalences exists.
The original proof assumed that is a right-proper model category, but it turns out that this condition is not necessary (Stanculescu 08, theorem 1.1).
Statement
Definition
Let be a proper model category. Say that a Quillen idempotent monad on is
-
an endofunctor
-
a natural transformation
such that
-
(homotopical functor) preserves weak equivalences;
-
(idempotency) for all the morphisms
and
are weak equivalences;
-
(right-properness of the localization) if in a pullback square in
we have that
-
is a fibration;
-
, , and are weak equivalences
then is a weak equivalence.
(Here the formulation of the third item follows Bousfield 01, def. 9.2. By lemma below this condition implies that is a -fibration, which is the condition required in Bousfield-Friedlander 78 (A.6)).
Definition
For a Quillen idempotent monad according to def. , say that a morphism in is
-
a -weak equivalence if is a weak equivalence;
-
a -cofibration if it is a cofibration.
-
a -fibration if it has the right lifting property against the morphisms that are both (-)cofibrations as well as -weak equivalences.
Write for equipped with these classes of morphisms.
Lemma
In the situation of def. , a morphism is an acyclic fibration in precisely if it is an acyclic fibration in .
Proof
Let be a fibration and a weak equivalence. Since preserves weak equivalences by condition 1 in def. , is also a -weak equivalence. Since -cofibrations are cofibrations, the acyclic fibration has right lifting against -cofibrations, hence in particular against against -acyclic -cofibrations, hence is a -fibration.
In the other direction, let be a -acyclic -fibration. Consider its factorization into a cofibration followed by an acyclic fibration
Now the fact that preserves weak equivalences together with two-out-of-three implies that is a -weak equivalence, hence a -acyclic -cofibration. This means by assumption that has the right lifting property against . Hence the retract argument, implies that is a retract of the acyclic fibration , and so is itself an acyclic fibration.
Lemma
In the situation of def. , if a morphism is a fibration, and are weak equivalences, then is a -fibration.
(e.g. Goerss-Jardine 96, chapter X, lemma 4.4).
Proof
We need to show that for every commuting square of the form
there exists a lifting.
To that end, first consider a factorization of the image under of this square as follows:
(This exists even without assuming functorial factorization: factor the bottom morphism, form the pullback of the resulting , observe that this is still a fibration, and then factor (through ) the universal morpism from the outer square into this pullback.)
Now consider the pullback of the right square above along the naturality square of , take this to be the right square in the following diagram
where the left square is the universal morphism into the pullback which is induced from the naturality squares of on and .
We claim that here is a weak equivalence. This implies that we find the desired lift by factoring into an acyclic cofibration followed by an acyclic fibration and then lifting consecutively as follows
To see that indeed is a weak equivalence:
Consider the diagram
Here the projections are weak equivalences as shown, because by assumption in def. the ambient model category is right proper and these projections are the pullbacks along the fibrations and of the morphisms and , respectively, where the latter are weak equivalences by assumption. Moreover is a weak equivalence, since is a -weak equivalence.
Hence now it follows by two-out-of-three (def.) that and then are weak equivalences.
Proposition
(Bousfield-Friedlander theorem)
For a Quillen idempotent monad according to def. , then , def. is a model category.
(Bousfield-Friedlander 78, theorem 8.7, Bousfield 01, theorem 9.3, Goerss-Jardine 96, chapter X lemma 4.5, lemma 4.6)
Proof
The existence of limits and colimits is guaranteed since is already assumed to be a model category. The two-out-of-three poperty for -weak equivalences is an immediate consequence of two-out-of-three for the original weak equivalences of .
Moreover, according to lemma the pair of classes equals the pair , and this is a weak factorization system by the model structure .
Hence it remains to show that is a weak factorization system. The condition holds by definition of . Once we show that every morphism factors as followed by , then the condition follows from the retract argument (and the fact that and are stable under retracts, because and are).
So we may conclude by showing the existence of factorizations:
First we consider the case of a morphism of the form . This may be factored with respect to as
Here is already a -acyclic -cofibration. Now apply to obtain
where and are weak equivalences by idempotency, and is a weak equivalence since preserves weak equivalences. Hence by two-out-of-three also is a weak equivalence. Therefore lemma gives that is a -fibration, and hence the above factorization is already as desired
Now for an arbitrary morphism , form a factorization of as above and then decompose the naturality square for on into the pullback of the resulting -fibration along :
This exhibits as the pullback of a -weak equivalence along a -fibration, and hence itself as a -weak equivalence. This way, two-out-of-three implies that is a -weak equivalence.
Finally, apply factorization in to to obtain the desired factorization
Proposition
For a Quillen idempotent monad according to def. , then a morphism in is a -fibration (def. ) precisely if
-
is a fibration;
-
the -naturality square on
exhibits a homotopy pullback in (def.), in that for any factorization of through a weak equivalence followed by a fibration , then the universally induced morphism
is weak equivalence (in ).
(e.g. Goerss-Jardine 96, chapter X, theorem 4.8)
Proof
First consider the case that is a fibration and that the square is a homotopy pullback. We need to show that then is a -fibration.
Factor as
By the proof of prop. , the morphism is also a -fibration. Hence by the existence of the -local model structure, also due to prop. , its pullback is also a -fibration
Here is a weak equivalence by assumption that the diagram exhibits a homotopy pullback. Hence it factors as
This yields the situation
As in the retract argument (prop.) this diagram exhibits as a retract (in the arrow category, rmk.) of the -fibration . Hence by the existence of the -model structure (prop. ) and by the closure properties for fibrations (prop.), also is a -fibration.
Now for the converse. Assume that is a -fibration. Since is a left Bousfield localization of (prop. ), is also a fibration. We need to show that the -naturality square on exhibits a homotopy pullback.
So factor as before, and consider the pasting composite of the factorization of the given square with the naturality squares of :
Here the top and bottom horizontal morphisms are weak (-)equivalences by the idempotency of , and is a weak equivalence since preserves weak equivalences (first and second clause in def. ). Hence by two-out-of-three also is a weak equivalence. From this, lemma gives that is a -fibration. Then is a -weak equivalence since it is the pullback of a -weak equivalence along a fibration between objects whose is a weak equivalence, via the third clause in def. . Finally two-out-of-three implies that is a -weak equivalence.
In particular, the bottom right square is a homotopy pullback (since two opposite edges are weak equivalences, by this prop.), and since the left square is a genuine pullback of a fibration, hence a homotopy pullback, the total bottom rectangle here exhibits a homotopy pullback by the pasting law for homotopy pullbacks (prop.).
Now by naturality of , that total bottom rectangle is the same as the following rectangle
where now since , as we had just established. This means again that the right square is a homotopy pullback (prop.), and since the total rectangle still is a homotopy pullback itself, by the previous remark, so is now also the left square, by the other direction of the pasting law for homotopy pullbacks (prop.).
So far this establishes that the -naturality square of is a homotopy pullback. We still need to show that also the -naturality square of is a homotopy pullback.
Factor as a cofibration followed by an acyclic fibration. Since is also a -weak equivalence, by the above, two-out-of-three for -fibrations gives that this factorization is of the form
As in the first part of the proof, but now with replaced by and using lifting in the -model structure, this yields the situation
As in the retract argument (prop.) this diagram exhibits as a retract (in the arrow category, rmk.) of .
Observe that the -naturality square of the weak equivalence is a homotopy pullback, since preserves weak equivalences (first clause of def. ) and since a square with two weak equivalences on opposite sides is a homotopy pullback (prop.). It follows that also the -naturality square of is a homotopy pullback, by the pasting law for homotopy pullbacks (prop.).
In conclusion, we have exhibited as a retract (in the arrow category, rmk.) of a morphism whose -naturality square is a homotopy pullback. By naturality of , this means that the whole -naturality square of is a retract (in the category of commuting squares in ) of a homotopy pullback square. This means that it is itself a homotopy pullback square (prop.).
Examples
Stable model structure on sequential spectra
The Bousfield-Friedlander model structure on sequential spectra (in any proper, pointed simplicial model category), modelling stable homotopy theory, arises via the Bousfield-Friedlander theorem from localizing the strict model structure transferred from the model structure on sequences (in the classical model structure on simplicial sets/on topological spaces) at being the spectrification endofunctor.
(For pre-spectra in the classical model structure on simplicial sets, spectrification is readily defined, more generally one needs to prooceed as in Schwede 97, section 2.1.)
References
The theorem is due to
- Aldridge Bousfield, Eric Friedlander, section A.3 of Homotopy theory of -spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math. 658, Springer, Berlin, 1978, pp. 80–130. (pdf)
and in improved form due to
- Aldridge Bousfield, section 9 of On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391–2426 (AMS, jstor)
The right-properness condition is shown to be unnecessary in
Textbook accounts include