Towers and Fibered Products of Model Structures

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products (pullbacks) of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For stable model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization.


Introduction
Localization techniques play an important role in modern homotopy theory. For several applications it is often useful to approximate a given space or spectrum by simpler ones by means of localization functors. For instance, given a simplicial set X, one can consider its Postnikov tower. This tower can be built as a sequence of fibrations and maps p n : X → P n X satisfying that p n = f n • p n+1 for every n ≥ 0 and that π k (f n ): π k (X) ∼ = π k (P n X) if k ≤ n for any choice of base point of X, and π k (P n X) = 0 if k > n and all choices of base points.
Each of the spaces P n X can be built as a localization of X with respect to the map S n+1 → * , and p n is the corresponding localization map. If X is connected, then the fiber of f n−1 is an Eilenberg-Mac Lane space

Model Structures for Sections of Quillen Presheaves
In this section we recall the injective model structure on the category of sections of diagrams of model categories. We will state the existence of this model structure in general, although we will be mainly interested in the cases of sections of towers and fibered products of model categories. Details about these model structures can be found in [4, Section 2, Application II], [6,7], [16,Section 3] and [27,Section 4].
Let I be a small category. A left Quillen presheaf on I is a presheaf of categories F : I op → CAT such that for every i in I the category F (i) has a model structure, and for every map f : i → j in I the induced functor f * : F (j) → F (i) has a right adjoint and they form a Quillen pair. Definition 1.1. A section of a left Quillen presheaf F : I op → CAT consists of a tuple X = (X i ) i∈I , where each X i is in F (i), and, for every morphism f : i → j in I, a morphism ϕ f : f * X j → X i in F (i) such that the diagram commutes for every pair of composable morphisms f : i → j and g : j → k. A morphism of sections φ : (X, ϕ) → (Y, ϕ ) is given by morphisms A section (X, ϕ) is called homotopy cartesian if for every f : i → j the morphism ϕ f : f * Q j X j → X i is a weak equivalence in F (i), where Q j denotes a cofibrant replacement functor in F (j).
Recall that a model category is left proper if pushouts of weak equivalences along cofibrations are weak equivalences, and right proper if pullbacks of weak equivalences along fibrations are weak equivalences. A model category is proper if it is left and right proper.
The category of sections admits an injective model structure, which is left or right proper, if the involved model structures are left or right proper, respectively. A proof of the following statement can be found in [4, Theorem 2.30, Propostion 2.31]. Recall that a model category is called combinatorial if it is cofibrantly generated and the underlying category is locally MJOM presentable. Foundations of the theory of combinatorial model categories may be found in [5,11,23]. The essentials of the theory of locally presentable categories can be found in [1,14,24]. Now, to model the homotopy limit of a left Quillen presheaf, we would like to construct a model structure on the category of sections whose cofibrant objects are precisely the levelwise cofibrant homotopy cartesian sections. This will be done by taking a right Bousfield localization of Sect(I, F ). The resulting model structure will be called the homotopy limit model structure.
The existence of the homotopy limit model structure when the category Sect(I, F ) is right proper was proved in [7,Theorem 3.2]. Without any properness assumptions, the homotopy limit model structure exists as a right model structure, as proved in [4,Theorem 5.25]. It follows directly from those results that if F (i) is right proper for every i in I, then we get a full model structure. For the reader's convenience we spell this out in a little more detail. Proof. Let D be the full subcategory of Sect(I, F ) consisting of the homotopy cartesian sections. Consider the functor where f runs over all morphisms of I and Arr(−) denotes the category of arrows, and let Q denote an accessible cofibrant replacement functor in Sect(I, F ).
The categories Sect(I, F ) and f : i→j Arr(F (i)) are accessible (in fact, they are locally presentable; see [1, Corollary 1.54]) and the functor Φ is an accessible functor since it preserves all colimits (as these are computed levelwise). Hence Φ is an accessible functor between accessible categories.
Each F (i) is combinatorial for every i in I, and hence by [23, Corollary A.2.6.6] the subcategory of weak equivalences weq(F (i)) is an accessible and accessibly embedded subcategory of Arr(F (i)). Therefore, f : i→j weq(F (i)) is an accessible and accessibly embedded subcategory of f : i→j Arr(F (i)).
Vol. 13 (2016) Towers and Fibered Products of Model Structures 3867 By [1, Remark 2.50], the preimage (Φ • Q) −1 ( f : i→j weq(F (i))) is an accessible and accessibly embedded subcategory of Sect(I, F ). But this preimage is precisely D. Now, since D is accessible there exists a set K of objects and a regular cardinal λ such that every object of D is a λ-filtered colimit (and hence a homotopy colimit if we choose λ big enough; see [11,Proposition 7.3]) of objects in K. Moreover, since D is accessibly embedded this homotopy colimit lies in D.
The homotopy limit model structure is then the right Bousfield localization R K Sect(I, F ). (

Towers of Model Categories
Let N be the category 0 → 1 → 2 → · · · . A tower of model categories is a left Quillen presheaf F : N op → CAT. The objects of the category of sections are then sequences X 0 , X 1 , . . . , X n , . . ., where each X i is an object of F (i), together with morphisms Proof. The existence of the model structure Tow(F ) follows from Theorem 1.3 applied to the left Quillen presheaf F . The characterization of the weak equivalences between cofibrant objects follows since Tow(F ) is a right Bousfield localization of Sect(N, F ).

Postnikov Sections of Model Structures
Let C be a left proper combinatorial model category and n ≥ 0. The model structure P n C of n-types in C is the left Bousfield localization of C with respect to the set of morphisms I C f n . Here I C is the set of generating cofibrations of C, f n : S n+1 → D n+2 is the inclusion of simplicial sets from the (n + 1)sphere to the (n + 2)-disk, and denotes the pushout-product of morphisms constructed using the action of simplicial sets on C coming from the existence of framings; see [20,Section 5.4]. A longer account about model structures for n-types can be found in [18,Section 3]. For every n < m the identity is a left Quillen functor P m C → P n C. Thus we have a tower of model categories P • C : N op → CAT. The objects X • of the category of sections are sequences of morphisms in C, and its morphisms f • : X • → Y • are given by commutative ladders By Proposition 2.1, if C is a left proper combinatorial model category, then there exists a left proper combinatorial model structure on the category of sections Sect(N, P • C), where a map f • is a weak equivalence or a cofibration if for every n ≥ 0 the map f n is a weak equivalence or a cofibration in P n C, respectively. The fibrations are the maps f • : is a fibration in P n C for every n ≥ 1. The fibrant objects can be characterized as follows: Proof. This follows because a fibration in P n C is also a fibration in P n+1 C as well as a fibration in C.
If the model structures for n-types P n C are right proper for every n ≥ 0, then by Proposition 2.2 the model structure Tow(P • C) exists and will be denoted by Post(C). It has the following properties: only if f n is a weak equivalence in P n C for every n ≥ 0. For every n ≥ 0 the identity functors form a Quillen pair id : C P n C : id, since P n C is a left Bousfield localization of C. This extends to a Quillen pair id : inj denotes the category of N op -indexed diagrams with the injective model structure. Indeed weak equivalences and cofibrations in C N op inj are defined levelwise and every weak equivalence in C is a weak equivalence in P n C for all n ≥ 0. Hence, there is a Quillen pair Proof. By [20,Proposition 1.3.13] it suffices to check that the derived unit and counit are weak equivalences. Let X be a fibrant simplicial set. Then const(X) is cofibrant in Post(sSet * ), since const is a left Quillen functor. Let be a fibrant replacement of const(X) in Post(sSet * ). Hence we have that X n is fibrant in P n sSet * and X n+1 → X n is a fibration in sSet * and a weak equivalence in P n sSet * for all n ≥ 0. By [15, Chap. VI, Theorem 3.5], the map X → lim X • is a weak equivalence. Now, let X • be any fibrant and cofibrant object in Post(sSet * ). We have to see that the map const(lim X • ) → X • is a weak equivalence in Post(sSet * ). This is equivalent to seeing that the map lim X • → X n is a weak equivalence in P n sSet * for every n ≥ 0. First note that since the category N op >n = · · · → n + 3 → n + 2 → n + 1 is homotopy left cofinal in N op we have that lim X • is weakly equivalent to lim N op >n X • for every n (see [19,Theorem 19.6.13]). Hence it is enough to check that the map lim N op >n X • → X n is a weak equivalence in P n sSet * for all n ≥ 0. For every n ≥ 0 we have a map of towers where each vertical map is a weak equivalence in P n+1 sSet * . Using the Milnor exact sequence (see [15, Chap. VI, Proposition 2.15]) we get a morphism of short exact sequences For 0 ≤ i < n the left and right vertical morphisms are isomorphisms; hence the map lim N op >n X • → X n+1 is a weak equivalence in P n sSet * . Therefore, the map Proof. We have the following commutative diagram: The horizontal arrows are weak equivalences because they are either a fibrant replacement or because the Quillen pair const and lim is a Quillen equivalence. So f is a weak equivalence if and only if g is a weak equivalence. But since const preserves and reflects weak equivalences between cofibrant objects (because it is the left adjoint of a Quillen equivalence), it follows that g is a weak equivalence if and only if lim X → lim Y is a weak equivalence.

Chromatic Towers of Localizations
We can also use the homotopy limit model structure on towers of categories to obtain a categorified version of yet another classical result. The chromatic convergence theorem states that for a finite p-local spectrum X, where L n denotes left localization at the chromatic homology theory E(n) at a fixed prime p; see [26,Theorem 7.5.7]. The prime p is traditionally omitted from notation. We will see that the Quillen adjunction between spectra and the left Quillen presheaf of chromatic localizations of spectra induces an adjunction between the homotopy category of finite spectra and the homotopy category of chromatic towers subject to a suitable finiteness condition.
The chromatic convergence theorem then shows that the derived unit of this adjunction is a weak equivalence. By Sp in this section we always mean the category of p-local spectra symmetric spectra [21] and the prime p will be fixed throughout the section.
Recall from [20, Section 6.1] that the homotopy category of a pointed model category supports a suspension functor with a right adjoint loop functor defined via framings. A model category is called stable if it is pointed and the suspension and loop operators are inverse equivalences on the homotopy category. Every combinatorial stable model category admits an enrichment over the category of symmetric spectra via stable frames; see [12,22].
Let C be a proper and combinatorial stable model category. Given a prime p, we define L n C to be the left Bousfield localization of C with respect to the E(n)-equivalences, where E(n) is considered at the prime p. By this, we mean Bousfield localisation at the set Note that the resulting model structure is stable as each L n C is stable. We then perform a right Bousfield localization to obtain the homotopy limit model structure. Note that this again results in a stable model category [2,Proposition 5.6]  (

i) A morphism is a fibration in Chrom(C) if and only if it is a fibration in
if and only if all the X n are cofibrant in C and X n+1 → X n is an E(n)-equivalence for each n.
The following is useful to justify the name "homotopy limit model structure". Recall that Sp denotes here the category of p-local spectra. Proof. Let f : X • → Y • be a weak equivalence in Chrom(Sp). This implies that is an isomorphism for all cofibrant A ∈ Sp. By Lemma 2.4, (const, lim) is a Quillen pair, so the above is equivalent to the claim that is an isomorphism for all cofibrant A ∈ C, where the square brackets denote morphisms in the stable homotopy category. But as the class of all cofibrant spectra detects isomorphisms in the stable homotopy category, this is equivalent to holim X • −→ holim Y • being a weak equivalence of spectra as desired.
Remark 2.10. It is important to note that we do not know if the converse is true. Looking at the proof of this lemma, we see that the following are equivalent: (i) There is a set of objects of the form const(G) in Chrom(Sp) that detect weak equivalences. (ii) The weak equivalences in Chrom(Sp) are precisely the holimisomorphisms. Unfortunately, it is not known from the definition of the homotopy limit model structure whether any of those equivalent conditions hold.
We can now turn to the main result of this subsection. For this, we need to specify our finiteness conditions. Recall that a p-local spectrum is called finite if it is in the full subcategory of the stable homotopy category Ho(Sp) which contains the sphere spectrum and is closed under exact triangles and retracts. We denote this full subcategory by Ho(Sp) fin . Definition 2.11. We call a diagram X • in Chrom(Sp) finitary if holim X • is a finite spectrum. By Ho(Chrom(Sp)) F we denote the full subcategory of the finitary diagrams in the homotopy category of Chrom(Sp). By definition, the homotopy limit of each finitary diagram is assumed to be a finite spectrum. On the other side, holim(Lconst(X)) X is exactly the chromatic convergence theorem for finite spectra. The derived unit of the above adjunction is a weak equivalence. For a cofibrant spectrum is again the chromatic convergence theorem.
We would really like to show that the above adjunction is an equivalence of categories, that is, that the counit is a weak equivalence, meaning that is a weak equivalence for Y • a fibrant and cofibrant finitary diagram in Chrom(Sp). However, to show this we would need to know that the weak equivalences in Chrom(Sp) are exactly the holim-isomorphisms; see Remark 2.10. Furthermore, we would not just have to know that Chrom(Sp) MJOM has a constant set of generators but also that those generators are finitary, that is, the homotopy limit of each generator is finite.

Convergence of Towers
Let C be a left proper combinatorial model structure such that the model structures P n C of n-types (see Sect. 2.1) are right proper, and hence the model structure Post(C) exists. In this section we are going to take a closer look at what it means for a tower in Post(C) to converge. Recall that we have a Quillen adjunction The following terminology appears in [4,Definition 5.35]. is isomorphic to the identity.
We have seen in Sect. 2.1 that this is true for C = sSet * . We have also seen in Theorem 2.12 that, under a finiteness assumption, the chromatic tower of spectra Chrom(Sp) is hypercomplete in this sense. We can also consider the case of left Bousfield localizations of sSet * , that is, C = L S sSet * . In general, this model category will not be hypercomplete. Let X be fibrant in L S sSet * , that is, fibrant as a simplicial set and S-local. If we take a fibrant replacement of the constant tower const(Y) in Post(L S sSet * ), we obtain a tower such that all the Y i are S-local, Y i is P i -local for all i and Y n → Y n−1 is a weak equivalence in P n−1 L S sSet * . However, this is not a fibrant replacement of const(Y) in Post(sSet * ), unless L S commutes with all the localizations P n . In this case, a Postnikov tower in L S sSet * is also a Postnikov tower in sSet * , and hypercompleteness holds. This would be the case for L S = L MR for R a subring of the rational numbers Q, but it cannot be expected in general.
Let us recapture the classical case to get a more general insight into hypercompleteness. For X in sSet * we know that X → lim n P n X is a weak equivalence. This is equivalent to saying that for all i, is an isomorphism of groups. But we have also seen that π i (lim n P n X) = lim n π i (P n X) as well as Putting this together we get that, indeed, π i (lim n P n X) ∼ = π i (X) for all i. This is a special case of the following. A set of homotopy generators for a model category C consists of a small full subcategory G such that every object of C is weakly equivalent to a filtered homotopy colimit of objects of G and that by [11,Proposition 4.7] every combinatorial model category has a set of homotopy generators that can be chosen to be cofibrant. Let C be a proper combinatorial model category with a set of homotopy generators G and homotopy function complex map C (−, −). Then, for a cofibrant X, the map X → holim n P n X is a weak equivalence in C if and only if is a weak equivalence in sSet for all G ∈ G, where the equality holds by [19,Theorem 19.4.4(2)]. So from this we can see that if we had map C (G, P n X) ∼ = P n map C (G, X) for all G in G, then we would get the desired weak equivalence because again π i map C (G, P n X) = π i (P n map C (G, X)).
We could also reformulate this statement by not using the full set of generators G, since we are only making use of the fact that they detect weak equivalences.
Proposition 2.14. Let hG be a set in C that detects weak equivalences. If map C (G, P n X) ∼ = P n map C (G, X) for every G in hG, then C is hypercomplete.
We can follow this through with a non-simplicial example, bounded chain complexes of Z-modules Ch b (Z). Let us briefly recall Postnikov sections of chain complexes, which are discussed in detail in [18,Section 3.4]. As mentioned in Sect. 2.1, P n Ch b (Z) is the left Bousfield localization of Ch b (Z) at The generating cofibrations of the projective model structure of Ch b (Z) are the inclusions where S n−1 is the chain complex which only contains Z in degree n − 1 and is zero in all other degrees, and D n is Z in degrees n and n − 1 with the identity differential and zero everywhere else. We can thus work out that This means that a chain complex is a k-type if and only if its homology vanishes in degrees k + 1 and above. The localization M −→ P k M is simply truncation above degree k. MJOM Let Hom(M, N ) denote the mapping chain complex for M , N in Ch b (Z), that is, ; see for example [20,Chap. 4.2]. We note that (Hom(M, N )).
So Ch b (Z) is hypercomplete if Hom(G, P n N ) is quasi-isomorphic to P n Hom(G, N ) for all G in hG. For bounded below chain complexes, a set that detects weak equivalences can be taken to be We have the following diagram of short exact sequences:  Hi(PnN )).
Using the 5-lemma we can read off that H i (Hom(M, P n N )) = 0 for i > n as desired and that

Homotopy Fibered Products of Model Categories
Let I be the small category A pullback diagram of model categories is a left Quillen presheaf F : I op → CAT. The objects X • of the category of sections are given by three objects X 0 , X 1 and X 2 in F (0), F (1) and F (2), respectively, together with morphisms are fibrations in F (1) and F (2), respectively. In particular, X • is fibrant if X i is fibrant in F (i) and By [9, Proposition 2.10] we have that where S denotes the sphere spectrum. Thus, the map is a weak equivalence. The last equality follows because homotopy pullbacks commute with the action of spectra coming from framings, since in stable categories they are equivalent to homotopy pushouts. Now, let X • be any fibrant and cofibrant object in Bou(C). We have to see that the map is a weak equivalence in Bou(C). This is equivalent to saying that the map lim X • → X 1 is a weak equivalence in L M ZJ C, lim X • → X 2 is a weak equivalence in L M ZK C and lim X • → X 12 is a weak equivalence in L M Q C.
Note that if A → B is a weak equivalence in L M Q C, A is fibrant in L M ZK C and B is fibrant in L M Q C, then A → B is a weak equivalence in L M ZJ C. To see this, let A → L M ZJ A be a fibrant replacement of A in L M ZJ C. We are going to use [3, Lemma 6.7] again, which says that the weak equivalences in L M ZJ C are morphisms f in C such that f ∧ M Z J is a weak equivalence in C. This makes the following argument the same as it would be for C = Sp.
Since B is fibrant in L M Q C, it is so in L M ZJ C. Thus, there is a lifting ; ; The left arrow is a weak equivalence in L M ZJ C and hence a weak equivalence in L M Q C. Therefore, the dotted arrow is a weak equivalence in L M Q C between fibrant objects in L M Q C. (Observe that L M ZJ A is fibrant in L M ZJ C and L M ZK C and hence in L M Q C.) Thus, it is a weak equivalence in C. This completes the proof of the claim since weak equivalences in C are weak equivalences in L M ZJ C.
/ / X 12 X 1 , X 2 and X 12 are fibrant in L M ZJ C, L M ZK C and L M Q C, respectively, and the right and bottom arrows are weak equivalences in L M Q C and fibrations in L M ZK C and L M ZJ C, respectively. By the previous observation and right properness of the model structures involved, the map f 1 : lim X • → X 1 is a weak equivalence in L M ZJ , and f 2 : lim X • → X 2 is a weak equivalence in L M ZK C, respectively. Thus, the map lim X • → X 12 is also a weak equivalence in M Q, which means that const(lim X • ) −→ X • is an objectwise weak equivalence, and thus a weak equivalence in Bou(C) as claimed.
Remark 3.5. There is a higher chromatic version of the objectwise statement.
Here Sp denotes the category of p-local spectra. There is a homotopy fiber square see [13,Section 3.9]. However, we cannot apply the methods of this section to get a result analogously to Theorem 3.4. This is due to the fact that L K(n) L n−1 Sp is trivial as a model category. (By [25, Theorem 2.1], a spectrum is E(n − 1)-local if and only if it is K(i)-local for 1 ≤ i ≤ n − 1. But the K(n)-localization of a K(m)-local spectrum is trivial for n = m.) Consider the homotopy fibered product model structure on A fibrant and cofibrant diagram would have to satisfy that X 1 is E(n − 1)-local and f 1 is an L n−1 L K(n) localization. By the universal property of localizations, this means that f 1 factors over L n−1 L K(n) X 1 → X 0 . However, as X 1 is E(n − 1)-local and thus K(n)-acyclic, this map (and thus f 1 ) is trivial. Thus we cannot reconstruct a pullback square like the above from this model structure.

Homotopy Fibers of Localized Model Categories
We will use the homotopy fibered product model structure to describe the homotopy fiber of Bousfield localizations. We can then use this to describe the layers of a Postnikov tower, among other examples. Let C be a left proper pointed combinatorial model category and let S be a set of morphisms in C. The identity C → L S C is a left Quillen functor and thus we have a pullback diagram of model categories L S • C : I op → CAT, where I = 1 ← 0 → 2, and L S 0 C = L S C, L S 1 C = * and L S 2 C = C. (Here * denotes the category with one object and one identity morphism with the trivial model structure.) A section of L S • C is a diagram * → Y ← X in C where * denotes the zero object. There is an adjunction where const(X) = ( * → X 1 ← X) and ev 2 ( * → Y ← X) = X. We will denote Fibpr(L S • ) by Fib(L S • ) and we will call it the homotopy fiber of the Quillen pair C L S C.
Definition 3.6. Let C be a proper pointed combinatorial model category and let K be a set of objects and S be a set of morphisms in C. We say that the colocalized model structure C K C and the localized model structure L S C are compatible when for every object X in C, X is K-colocal if and only if X is cofibrant in C and the map * → X is an S-local equivalence.
The stable case is discussed in detail in [2,Section 10] where such model structures are called "orthogonal"; see also Sect. 3.5.
Remark 3.7. Note that if C K C and L S C are compatible, then it follows from the definitions that * → Y ← X is cofibrant in Fib(L S • C) if and only if both X and Y are K-colocal and cofibrant in C. If * → Y ← X is moreover fibrant in Fib(L S • C), then Y is weakly contractible since Y is S-local and * → Y is an S-equivalence and X → Y is a fibration in C. Proof. We will first show that the adjunction is a Quillen pair. By [19,Propostion 8.5.4(2)], it is enough to check that the left adjoint preserves trivial cofibrations and sends cofibrations between cofibrant objects to cofibrations.
Let f be a trivial cofibration in C K C. Then f is a trivial cofibration in C and, therefore, const(f ) is a trivial cofibration in Sect(I, L S • C) and thus a trivial cofibration in Fib(L S • C). Now let f : X → Y be a cofibration between cofibrant objects in C K C. Then f is a cofibration between cofibrant objects in C and hence const(f ) is also a cofibration between cofibrant objects in Sect(I, L S • C). But const(X) and const(Y ) are cofibrant in Fib(L S • C), since C K C and L S C are compatible and, therefore, the maps * → X and * → Y are S-local equivalences. Hence const(f ) is a cofibration in Fib(L S • C), by [19, Proposition 3.3.16 (2)]. To prove that it is a Quillen equivalence, it suffices to show that the derived unit and counit are weak equivalences; see [20, Proposition 1.3.13]. Let