# Towers and Fibered Products of Model StructuresAcademic research paper on "Mathematics" CC BY 0 0
Share paper
Mediterranean Journal of Mathematics
OECD Field of science
Keywords
{""}

## Academic research paper on topic "Towers and Fibered Products of Model Structures"

﻿Mediterr. J. Math.

DOI 10.1007/s00009-016-0719-3 |---

with open access at Springerlink.com I of Mathematics

CrossMark

Towers and Fibered Products of Model Structures

Javier J. Gutiérrez and Constanze Roitzheim

Abstract. 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.

Mathematics Subject Classification. 55P42, 55P60, 55S45.

Keywords. Localization, model category, Postnikov tower, homotopy fibered product, homotopy pullback.

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

••• PnX Pn-1 X —----P2X P1X P0X

and maps pn: X ^ PnX satisfying that pn = fn o pn+1 for every n > 0 and that nk (fn): nk (X) = nk (PnX) if k < n for any choice of base point of X, and (PnX) = 0 if k > n and all choices of base points.

Each of the spaces PnX can be built as a localization of X with respect to the map Sn+1 ^ *, and pn is the corresponding localization map. If X is connected, then the fiber of fn-1 is an Eilenberg-Mac Lane space

J. J. Gutierrez was supported by the NWO (SPI 61-638) and the MEC-FEDER Grants MTM2010-15831 and MTM2013-42178-P. Both authors received support from the LMS Scheme 4 Grant No. 41360.

IS? Birkhauser

Published online: 20 April 2016

K(nn(X),n) and every simplicial set X can be reconstructed as the homo-topy limit of its Postnikov tower X ~ holimn>0 PnX; see [15, Chap. VI, Theorem 3.5].

In the category of spectra, given any spectrum E, we can consider its associated homological localization functor LE which inverts the maps that induce isomorphisms in E*-homology in a universal way. Given an abelian group G, let us denote by MG the associated Moore spectrum. It is well known that any spectrum X can be built, using Bousfield's arithmetic square , as a homotopy pullback of the diagram of homological localizations

Lmzj X —> LMQX <— LMZK X.

where J and K form any partition of the set of prime numbers and ZJ are the integers localized at the set of primes J.

Furthermore, the chromatic convergence theorem [26, Theorem 7.5.7] states that a finite p-local spectrum X is the homotopy limit of its chromatic localizations LE(n)X at the prime p.

The aim of this paper is to present categorified versions of these statements in the framework of Quillen model structures. Given a diagram (left Quillen presheaf) of model categories F: Iop ^ CAT, there is an injective model structure on the category of sections associated wit F, which we can further colocalize to obtain the homotopy limit model structure. We study these model structures for towers and homotopy fibered products (homotopy pullbacks) of model categories.

First, we construct the Postnikov tower of an arbitrary combinatorial model category. As an application we show that for simplicial sets and for bounded below chain complexes these towers converge in a certain sense. Another tower model structure is the homotopy limit model structure on the left Quillen presheaf of chromatic towers Chrom(Sp), where Sp denotes here the category of p-local symmetric spectra. We show that the Quillen adjunction

const : Sp ^^ Chrom(Sp) : lim

induces a composite

Ho(Sp)fin ^t^0^ Ho(Chrom(Sp))F ^^ Ho(Sp)fin

which is isomorphic to the identity. (Here, F and fin denote suitable finiteness conditions.) This set-up is a step towards deeper insights into the structure of the stable homotopy category via viewing chromatic convergence in a cat-egorified manner.

We then move to fibered products of model categories. Using this setup, we show that the category of symmetric spectra is Quillen equivalent to the homotopy limit model structure of the left Quillen presheaf for Bousfield arithmetic squares of spectra.

As a final application we focus on a correspondence between the homo-topy fiber of a left Bousfield localization C ^ LsC and certain right Bousfield localizations. This is then used, among other examples, to understand the

layers of the Postnikov towers established earlier and to study the correspondence between stable localizations and stable colocalizations.

1. 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: Iop ^ 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: Iop ^ CAT consists of a tuple X = (Xi)iex, where each Xi is in F(i), and, for every morphism f: i ^ j in I, a morphism pf : f *Xj ^ Xi 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 \$ : (X, y) ^ (Y, y') is given by morphisms : Xi ^ Yi in F(i) such that the diagram

commutes for every morphism f: i ^ j in I.

A section (X, p) is called homotopy cartesian if for every f: i ^ j the morphism pf : f *QjXj ^ Xi is a weak equivalence in F(i), where Qj 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

(g o f )*xk^f Xi

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].

Theorem 1.2. (Barwick) Let F: Iop ^ CAT be a left Quillen presheaf such that F(i) is combinatorial for every i in I. Then there exists a combinatorial model structure on the category of sections of F, denoted by Sect(I, F) and called the injective model structure, such that a morphism of sections ^ is a weak equivalence or a cofibration if and only if is a weak equivalence or a cofibration in F(i) for every i in I, respectively. Moreover, if F(i) is left or right proper for every i &I, then so is the model structure on Sect(I, F). □

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.

Theorem 1.3. Let F: Iop ^ CAT be a left Quillen presheaf such that F(i) is right proper and combinatorial for every i in I. Then there exists a combinatorial model structure on the category of sections of F, called the homotopy limit model structure, with the same fibrations as Sect(I, F) and whose cofibrant objects are the sections that are cofibrant in Sect(I, F) and homotopy cartesian.

Proof. Let D be the full subcategory of Sect(I, F) consisting of the homotopy cartesian sections. Consider the functor

\$: Sect(I,F) —> Arr(F(i)) f : i^j

defined as \$((Xi)iei) = J}-. y-, 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 -. Arr(F(i)) are accessible (in fact, they are locally presentable; see [1, Corollary 1.54]) and the functor \$ is an accessible functor since it preserves all colimits (as these are computed levelwise). Hence \$ 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, -. weq(F(i)) is an accessible and accessibly embedded subcategory of -. Arr(F(i)).

By [1, Remark 2.50], the preimage (\$ o Q) (nf • 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 A such that every object of D is a A-filtered colimit (and hence a homotopy colimit if we choose A 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 Rk Sect(I, F). (We can perform this right Bousfield localization because every F(i) and hence Sect(I, F) are right proper.) The fact that the cofibrant objects of this new model structure are precisely the levelwise cofibrant ho-motopy cartesian sections follows from [19, Theorem 5.1.5]. □

2. Towers of Model Categories

Let N be the category 0 ^ 1 ^ 2 ^ •••. A tower of model categories is a left Quillen presheaf F: Nop ^ CAT. The objects of the category of sections are then sequences X0, Xi,..., Xn,..., where each Xi is an object of F(i), together with morphisms : f *Xi+i ^ Xi in F(i) for every i > 0, where f: i ^ i +1 is the unique morphism from i to i + 1 in N. A morphism between two sections : X, ^ Y, consists of morphisms \$i: Xi ^ Yi in F(i) such that the diagram

f *Xi+i-Xi

f & f * Yi+i-* Yi

commutes for every i > 0.

Proposition 2.1. Let F: Nop ^ CAT be a tower of model categories, where F(i) is a combinatorial model category for every i > 0. There exists a combinatorial model structure on the category of sections, denoted by Sect(N, F), where a map is a weak equivalence or a cofibration if and only if for every i > 0 the map is a weak equivalence or a cofibration in F(i), respectively. The fibrations are the maps : X, ^ Y, such that is a fibration in F(0) and

Xi+i -> Yi+i xf„Yi f*Xi

is a fibration in F(i + 1) for every i > 0, where f* denotes the right adjoint to f *. The fibrant objects are those sections X, such that Xi is fibrant in F(i) and the morphism

Xi+i -► f*Xi

is a fibration in F(i + 1) for every i > 0.

Proof. The existence of the required model structure follows from Theorem 1.2. The description of the fibrations follows from [16, Theorem 3.1]. □

Proposition 2.2. Let F : Nop ^ CAT be a tower of model categories, where each F(i) is combinatorial and right proper for every i > 0. Then there is a model structure Tow(F) on the category of sections of F with the following properties:

(i) A morphism is a fibration in Tow(F) if and only is a fibration in Sect(N, F).

(ii) A section X, is cofibrant in Tow(F) if and only if Xi is cofibrant in F(i) and the morphism f *Xi+i ^ Xi is a weak equivalence in F(i) for every i > 0.

(iii) A morphism between cofibrant sections is a weak equivalence in Tow(F) if and only if ^i is a weak equivalence in F(i) for every i > 0.

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). □

2.1. Postnikov Sections of Model Structures

Let C be a left proper combinatorial model category and n > 0. The model structure PnC of n-types in C is the left Bousfield localization of C with respect to the set of morphisms Ic □fn,. Here Ic is the set of generating cofibrations of C, fn : Sn+1 ^ Dn+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 PmC ^ PnC. Thus we have a tower of model categories P,C : Nop ^ CAT. The objects X, of the category of sections are sequences

• • • —> Xn —> • • • —> X2 —> Xi —> Xo

of morphisms in C, and its morphisms f, : X, ^ Y, are given by commutative ladders

Xo-Xi -Xn

* Yn-^----^ Yo-^ Yi -^ Yn.

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 fn is a weak equivalence or a cofibration in PnC, respectively. The fibrations are the maps f,: X, ^ Y, such that f0 is a fibration in P0C and

Xn -> Yn XYn-! Xn-1

is a fibration in PnC for every n > 1. The fibrant objects can be characterized as follows:

Lemma 2.3. Let X. be a section of P.C. The following are equivalent:

(i) X. is fibrant in Sect(N, P. C).

(ii) Xo is fibrant in PoC and X„+i ^ Xn is a fibration in Pn+iC for all n > 0.

(iii) Xn is fibrant in PnC and Xn+i ^ Xn is a fibration in C for all n > 0.

Proof. This follows because a fibration in PnC is also a fibration in Pn+1C as well as a fibration in C. □

If the model structures for n-types PnC 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:

(i) A morphism f. is a fibration in Post(C) if and only if f. is a fibration in Sect(N, P.C).

(ii) A section X. is cofibrant if and only if Xn is cofibrant in C and Xn+1 ^ Xn is a weak equivalence in PnC for every n > 0.

(iii) A morphism f. between cofibrant sections is a weak equivalence if and only if fn is a weak equivalence in PnC for every n > 0.

For every n > 0 the identity functors form a Quillen pair id : C ^ PnC : id, since PnC is a left Bousfield localization of C. This extends to a Quillen pair

id : CNP ^ Sect(N, P.C) : id,

where C^P denotes the category of Nop-indexed diagrams with the injective model structure. Indeed weak equivalences and cofibrations in CNjP are defined levelwise and every weak equivalence in C is a weak equivalence in PnC for all n > 0. Hence, there is a Quillen pair

const id id

C ; ' Cjn.% ; Sect(N, P.C) , " Post(C),

lim J id id

where const denotes the constant diagram functor.

Lemma 2.4. The adjunction const : C ^ Post(C) : lim is a Quillen pair.

Proof. By [19, Proposition 8.5.4(2)], it is enough to check that the left adjoint preserves trivial cofibrations and cofibrations between cofibrant objects. If f is a trivial cofibration in C then const(f) is a trivial cofibration in Sect(N, P.C). But since Post(C) is a right Bousfield localization of Sect(N,P.C) it has the same trivial cofibrations. Hence const(f) is a trivial cofibration in Post(C).

Let f: X ^ Y be a cofibration between cofibrant objets in C. Then const(f) is a cofibration between cofibrant objects in Sect(N,P.C). But const(X) and const(Y) are both cofibrant in Post(C) by Proposition 2.2. Hence const(f) is a cofibration in Post(C) if and only if it is a cofibration in Sect(N,P.C) (see [19, Proposition 3.3.16(2)]). □

Let sSet* denote the category of pointed simplicial sets with the Kan-Quillen model structure. Then the model structure Post(sSet*) exists, since Pn sSet is right proper for every n > 0; see [10, Theorem 9.9].

Theorem 2.5.

equivalence.

The Quillen pair const : sSet* ^ Post(sSet*) : lim is a Quillen

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 Xn is fibrant in Pn sSet* and Xn+i ^ Xn is a fibration in sSet* and a weak equivalence in Pn 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. ^ Xn is a weak equivalence in Pn sSet* for every n > 0. First note that since the category = • • • ^ n + 3 ^ n + 2 ^ n +1 is homotopy left cofinal in Nop we have that lim X. is weakly equivalent to limN°p X. for every n (see [19, Theorem 19.6.13]). Hence it is enough to check that the map limN°p X. ^ Xn is a weak equivalence in Pn sSet* for all n > 0. For every n > 0 we have a map of towers

■ Xn

where each vertical map is a weak equivalence in Pn+i sSet*. Using the Milnor exact sequence (see [15, Chap. VI, Proposition 2.15]) we get a morphism of short exact sequences

0-s- ni+iX, -s- ni (limN°Pi X,)-s- niX, -^ o

0-s- limjN°p^ ni+iXn+i -s- ni(limN°Pi X„+i) -s- niXn+i-o.

For 0 < i < n the left and right vertical morphisms are isomorphisms; hence the map limN°p X. ^ Xn+i is a weak equivalence in Pn sSet*. Therefore, the map

limN^ X. —> Xn+i —> Xn is a weak equivalence in Pn sSet for n > 0. □

Corollary 2.6. Let X ^ Y be a map in Post(sSet*). Then X ^ Y is a weak equivalence if and only if lim X ^ lim Y is a weak equivalence in sSet*, where X and Y denote a fibrant replacement of X and Y, respectively.

Proof. We have the following commutative diagram:

const(lim X) ~ > X < ~ X

const(lim Y)-Y -Y.

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. □

2.2. 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,

X ~ holimn LnX,

where Ln 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 homo-topy 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  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 LnC 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 Ic□Se^), where Ic is the set of generating cofibrations of C and SE(n) the generating acyclic cofibrations of LE(n) Sp = Ln Sp. (The square denotes the pushout-product.) This defines a left Quillen presheaf

L.C: Nop —> CAT . By Proposition 2.1 we get the following:

Proposition 2.7. There is a left proper, combinatorial and stable model structure on the category of sections Sect(N,L.C), such that a map is a weak equivalence or a cofibration if and only if each

is a weak equivalence or a cofibration in LnC, respectively. A map fn : Xn ^ Yn is a fibration if and only if fo is a fibration in LqC and

Note that the resulting model structure is stable as each LnC 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] as this right localization is stable in the sense of [2, Definition 5.3]. As left localization with respect to E(n) is also stable in the sense of [2, Definition 4.2], LnC is both left and right proper if C is; see [2, Propositions 4.6 and 4.7]. Hence, Proposition 2.2 implies the following result:

Proposition 2.8. Let C be a proper, combinatorial and stable model category. There is a model structure Chrom(C) on Sect(N, L,C) with the following properties:

(i) A morphism is a fibration in Chrom(C ) if and only if it is a fibration in Sect(N, L.C).

(ii) An object X. is cofibrant in Chrom(C) if and only if all the Xn are cofibrant in C and Xn+i ^ Xn 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.

Lemma 2.9. Let f : X, ^ Y, be a weak equivalence in Chrom(Sp). Then

holim X, —> holim Y, is a weak equivalence of spectra.

Proof. Let f : X, ^ Y,, be a weak equivalence in Chrom(Sp). This implies that

Ho(Chrom(Sp))(const(A),X.) —» Ho(Chrom(Sp))(const(A),Y.)

is an isomorphism for all cofibrant A G 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 G 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

Xn+1 -► Yn+1 X;

is a fibration in Ln+iC for all n > 1.

[A, holim X.] —> [A, holim Y,]

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 holim-isomorphisms.

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).

Theorem 2.12. The Quillen adjunction const : Sp ^ Chrom(Sp) : lim induces an adjunction

Ho(Sp)fin — Ho(Chrom(Sp))F

and the derived unit is a weak equivalence.

Proof. First, we notice that the derived adjunction

Lconst : Ho(Sp) Ho(Chrom(Sp)) : R lim = holim

Lconst : Ho(Sp)/in — Ho(Chrom(Sp))F : Rlim = holim .

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

X —> (holim(const(X)) = holimn LnX)

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

const(holimY,) —> Y,

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)

has a constant set of generators but also that those generators are finitary, that is, the homotopy limit of each generator is finite.

2.3. Convergence of Towers

Let C be a left proper combinatorial model structure such that the model structures PnC 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

const : C ^^ Post(C) : lim. The following terminology appears in [4, Definition 5.35].

Definition 2.13. The model category C is hypercomplete if the derived left adjoint of the previous Quillen adjunction is full and faithful, that is, if the composite

Ho(C) Ho(Post(C)) Ho(C)

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 = Ls sSet*. In general, this model category will not be hypercomplete. Let X be fibrant in Ls 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(Ls sSet*), we obtain a tower

(const(Y))fib = (----► Yn ^ m-i ^----► Yo)

such that all the Y are S-local, Y is Pj-local for all i and Yn ^ Yn-1 is a weak equivalence in Pn-1Ls sSet*. However, this is not a fibrant replacement of const(Y) in Post(sSet*), unless Ls commutes with all the localizations Pn. In this case, a Postnikov tower in Ls sSet* is also a Postnikov tower in sSet*, and hypercompleteness holds. This would be the case for Ls = LMR 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 ^ limn PnX is a weak equivalence. This is equivalent to saying that for all i,

n (X) ^ n (lim PnX)

is an isomorphism of groups. But we have also seen that n (lim PnX) =lim ni(PnX)

as well as

MPnX) = { ni(0X) if ^ n'

0 if i > n.

Putting this together we get that, indeed, ni(limnPnX) = ^¿(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 ma^(-, —). Then, for a cofibrant X, the map X ^ holimn PnX is a weak equivalence in C if and only if

mapc(G, X) —> mapc(G, holimn PnX) = holimn mapc(G, PnX)

is a weak equivalence in sSet for all G G G, where the equality holds by [19, Theorem 19.4.4(2)].

So from this we can see that if we had mapc(G, PnX) = Pn mapc(G, X) for all G in G, then we would get the desired weak equivalence because again

ni mapc(G, PnX) = ni (Pn mapc (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

mapc(G, PnX) = Pn mapc(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 Chb(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, Pn Chb(Z) is the left Bousfield localization of Chb(Z) at

Wk = Ichb(zp{fk : Sk+1 Dk+2}.

The generating cofibrations of the projective model structure of Chb(Z) are the inclusions

Ichb(z) = {S^1 —^ Dn I n > 1},

where Sn_1 is the chain complex which only contains Z in degree n — 1 and is zero in all other degrees, and Dn is Z in degrees n and n — 1 with the identity differential and zero everywhere else. We can thus work out that

Wk = {Sn+k+1 —» Dn+k+2 | n > 0}.

This means that a chain complex is a fc-type if and only if its homology vanishes in degrees k +1 and above. The localization M —> PkM is simply truncation above degree k.

Let Hom(M, N) denote the mapping chain complex for M, N in Ch&(Z). that is,

Hom(M, N)k ^Homz(Mi,Nj+fc)

with differential (df)(x) = d(f(x)) + ( — 1)k+1/(d(x)); see for example [20, Chap. 4.2]. We note that

nj(mapCh6(z)(M,N)) = Hj(Hom(M, N))

because

nj(mapCht(z)(M,N)) = [ST, mapCht(z)(M, N)Ut* = [M ,NHt(z) = [M[i],N]Cht(z) = [M ® Z[i],N]Chb(z) = [Z[i], Hom(M,N)]Chb(z) = Hj(Hom(M, N)).

So Chb(Z) is hypercomplete if Hom(G, PnN) is quasi-isomorphic to Pn Hom( G, N) for all G in HQ. For bounded below chain complexes, a set that detects weak equivalences can be taken to be

HQ = {Sj = Z[i] | i > 0}.

We have the following diagram of short exact sequences:

Extz(Hi(M),Hi+1 (N))-> Hi(Hom(M, N))-> Homz(Hi(M),Hi(N))

Extz(Hi(M),Hi+1(PnN)) Ht(Hom(M,PnN)) —^Homz(Hi(M),Ht(PnN)).

Using the 5-lemma we can read off that Hj(Hom(M, PnN)) = 0 for i > n as desired and that

Hi(Hom(M,PnN)) = Hj(Hom(M, N))

for i < n — 1, but unless ExtZ(Hn(M), Hn+i(N)) = 0 we do not get that

Hn (Hom( M, Pn N )) = Hn(Hom(M, N)).

Note that in general it is not true that Hom(M, PnN) ~ Pn Hom(M, N). However, as we only require the case M = Sj, we have that

Hom(Sj, N) = N[n],

where N[n] is the n-fold suspension of N. Thus,

Hom(G, PnN) = Pn Hom(G, N)

for all G in hG, so Chb(Z) is hypercomplete as expected.

Remark 2.15. Another important example of a tower of model structures occurring in nature is given by the Taylor tower of Goodwillie calculus, where for every n one considers the n-excisive model structure on the category of small endofunctors of simplicial sets; see [8, Section 4]. We do not discuss this example in this paper, and detailed relations to the aforementioned references could be a topic for future research.

3. Homotopy Fibered Products of Model Categories

Let I be the small category

1 ^ 0 2.

A pullback diagram of model categories is a left Quillen presheaf F: lop ^ CAT. The objects X• of the category of sections are given by three objects Xq, X\ and X2 in F(0), F(1) and F(2), respectively, together with morphisms

a*Xi —> Xo <— {3*X2

in F(0). A morphism fa,: X, ^ Y, consists of morphisms fa: Xi ^ Yi in F(i) for i = 0,1, 2, such that the diagram

- Xo ^-ß*X2

■ Yo*-ß*Y>

commutes.

Proposition 3.1. Let F: Iop ^ CAT be a pullback diagram of model categories such that F(i) is a combinatorial model category for every i in I. Then there exists a combinatorial model structure on the category of sections Sect(I, F), where a map fa, is a weak equivalence or a cofibration if and only if fa is a weak equivalence or cofibration in F(i) for every i in I. The fibrations are the maps fa, : X, ^ Y, such that fo is a fibration in F(0) and

Xi —> Yi Xaty0 a*Xo and X2 —> Y2 Xp„y0 Xo

are fibrations in F(1) and F(2), respectively. In particular, X, is fibrant if Xi is fibrant in F(i) and

Xi —> a*X0 and X2 —> 3*X0

are fibrations in F(1) and F(2), respectively .

Proof. The existence of the required model structure follows from Theorem 1.2. The description of the fibrations follows from [16, Theorem 3.1]. □

Proposition 3.2. Let F: Iop ^ CAT be a pullback diagram of model categories such that F(i) is combinatorial and right proper for every i in I. Then there is a model structure Fibpr(F) on the category of sections of F, called the homotopy fibered product model structure, with the following properties:

(i) A morphism fa, is a fibration in Fibpr(F) if and only if fa, is a fibration in Sect(I, F).

(ii) A section X, is cofibrant in Fibpr(F) if and only if Xi is cofibrant in F(i) for every i in I and the morphisms a*Xi ^ Xo and 3*X2 ^ Xo are weak equivalences in F(0).

(iii) A morphism fa, between cofibrant sections is a weak equivalence if and only if fai is a weak equivalence in F(i) for every i in I.

Proof. The existence of the model structure Fibpr(F ) follows from Theorem 1.3 applied to the left Quillen presheaf F. The characterization of the weak equivalences between cofibrant objects follows since Fibpr(F) is a right Bousfield localization of Sect(I, F). □

3.1. Bousfield arithmetic Squares of Homological Localizations

Let C be a left proper combinatorial stable model category and E any spectrum. The model structure LEC is the left Bousfield localization of C with respect to the set Ic □Se. Here Ic is the set of generating cofibrations of C, the set SE consists of the generating trivial cofibrations of the homological localization LE Sp, and □ is the pushout-product defined via the action Cx Sp —» C. This model structure is an example of a left Bousfield localization along a Quillen bifunctor, as studied in .

Now, let J and K be a partition of the set of prime numbers. By Zj we denote the J-local integers, and by MG the Moore spectrum of the group G. Consider the model structures LMzjC, LMzk C and LMqC . Since, for every set of primes P, every MZP-equivalence is an MQ-equivalence, the identities LMzjC — LMqC and LMzkC — LMqC are left Quillen functors.

Thus we have a pullback diagram of model categories L,C : Iop — CAT, where I =1 ^ 0 — 2 and L0C = LMqC, L1C = LMzjC and L2C = LMzkC.

If C is a left proper combinatorial stable model category, then by Proposition 3.1 the model structure Sect(I, L• C) exists, and it is also a stable model structure because each of the involved model categories is stable.

Moreover, if in addition the model structures LMzjC, LMzk C and LMqC are right proper, then by Proposition 3.2 the model structure Fibpr(L.C); which we denote by Bou(C), also exists. The model structure Bou(C) is also stable, since it is a right Bousfield localization with respect to a set of stable objects; see [2, Proposition 5.6].

Lemma 3.3. The adjunction const : C ^ Bou(C) : lim is a Quillen pair. Proof. The proof is the same as the one for Lemma 2.4. □

Note that for any spectrum E, the model structure LE Sp is right proper [2, Proposition 4.7]; hence the model structure Bou(Sp) exists.

Theorem 3.4. Let C be a proper and combinatorial stable model category. The Quillen pair const : C ^ Bou(C) : lim is a Quillen equivalence.

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 and cofibrant object in C. We need to show that

X —» lim(const(X)fib)

is a weak equivalence in C, where (—)fib denotes a fibrant replacement in Bou(C). The constant diagram const(X) is cofibrant in Bou(C) since const is a left Quillen functor. Let

LMzj X —> LMqX <— LMzK X

be a fibrant replacement of const(X) in Bou(C). We have that LMzk X, Lmzj X and LmqX are fibrant in Lmzk C, LmzjC and LmqC , respectively, and the two maps are fibrations in C and weak equivalences in LMqC. Furthermore, the three localisations are smashing in Sp, so by [3, Lemma 6.7]

Lmzk X = X A MZk , LmqX = X A MQ and Lmzj X = X A MZj . By [9, Proposition 2.10] we have that

lim(MZK —» MQ <— MZJ) = S, where S denotes the sphere spectrum. Thus, the map

X -> lim(LMzkX -^ LMQX <- LMzJX)

= X A lim(MZK —» MQ <— MZJ )

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

const(lim X.) —> X,

is a weak equivalence in Bou(C). This is equivalent to saying that the map lim X, ^ Xi is a weak equivalence in LMzjC, lim X, ^ X2 is a weak equivalence in LMzkC and limX, ^ Xi2 is a weak equivalence in LMqC.

Note that if A ^ B is a weak equivalence in LMqC , A is fibrant in LmzkC and B is fibrant in LmqC, then A ^ B is a weak equivalence in LmzjC. To see this, let A ^ LmzjA be a fibrant replacement of A in LmzjC. We are going to use [3, Lemma 6.7] again, which says that the weak equivalences in LMzj C are morphisms f in C such that f A MZJ 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 LmqC, it is so in LmzjC. Thus, there is a lifting

Lm zj A.

The left arrow is a weak equivalence in LMzj C and hence a weak equivalence in LMqC. Therefore, the dotted arrow is a weak equivalence in LMqC between fibrant objects in LmqC. (Observe that LmzjA is fibrant in LmzjC and LMzkC and hence in LMqC.) Thus, it is a weak equivalence in C. This completes the proof of the claim since weak equivalences in C are weak equivalences in LmzjC.

Since X, is fibrant and cofibrant, we have that in the pullback diagram

lim X,^^ X2

X\-X12

Xi, X2 and X12 are fibrant in LM j C, LMzK C and LMqC , respectively, and the right and bottom arrows are weak equivalences in LMqC and fibrations in LMzk C and LMzj C, respectively. By the previous observation and right properness of the model structures involved, the map f1: lim X, ^ X1 is a weak equivalence in LMzj , and f2 : lim X, ^ X2 is a weak equivalence in LMzk C, respectively. Thus, the map lim X, ^ X12 is also a weak equivalence in MQ, 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

LnX-s- LK(n)X

Ln-1X-Ln-1 LK(n)X;

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 LK(n)Ln-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

Ln-1 Sp —> Ln-1LK(n) Sp <— ¿K(n) Sp. A fibrant and cofibrant diagram

1 -> X0 <- X2

would have to satisfy that X1 is E(n — 1)-local and f1 is an Ln-1LK(n) localization. By the universal property of localizations, this means that f1 factors over Ln-1LK(n)X1 ^ X0. However, as X1 is E(n — 1)-local and thus K(n)-acyclic, this map (and thus f1) is trivial. Thus we cannot reconstruct a pullback square like the above from this model structure.

3.2. 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 ^ LsC is a left Quillen functor and thus we have a pullback diagram of model categories LSC: Iop ^ CAT,

where I =1 ^ 0 ^ 2, and LfC = LSC, Lf C = * and LfC = C. (Here * denotes the category with one object and one identity morphism with the trivial model structure.)

A section of Lf C is a diagram * ^ Y ^ X in C where * denotes the zero object. There is an adjunction

const : C Sect(l, Lf C ) : ev2,

where const(X) = X ^- X) and ev2(* ^ Y ^ X) = X. We will denote Fibpr(Lf ) by Fib(Lf ) and we will call it the homotopy fiber of the Quillen pair C ^ Lf 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 CkC and the localized model structure LsC 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 CkC and Lf C are compatible, then it follows from the definitions that * —s- Y <— X is cofibrant in Fib(Lf C) if and only if both X and Y are K-colocal and cofibrant in C. If *^Y ^ X is moreover fibrant in Fib(Lf 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.

Theorem 3.8. 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. If CkC and LsC are compatible, then the adjunction

const : CkFib(Lf C) : ev2

is a Quillen equivalence.

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 CkC. Then f is a trivial cofibration in C and, therefore, const(f ) is a trivial cofibration in Sect(l, Lf C) and thus a trivial cofibration in Fib(Lf C).

Now let f : X ^ Y be a cofibration between cofibrant objects in CkC. Then f is a cofibration between cofibrant objects in C and hence const(f ) is also a cofibration between cofibrant objects in Sect(l,LfC). But const(X) and const(Y) are cofibrant in Fib(LfC), since CkC and LsC are compatible and, therefore, the maps X and *^Y are S-local equivalences. Hence const(f ) is a cofibration in Fib(LfC), 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

X be a cofibrant object in CkC. Then we can construct a fibrant replacement for const(X) in Fib(LfC) as follows:

*-> LsX^-X

where the map X ^ LsX is a trivial cofibration in LsC and X ^ X' ^ LsX is a factorization in C of the previous map as a trivial cofibration followed by a fibration. Indeed, the map between the two sections is a trivial cofibration in Fib(LSC) since it is a levelwise trivial cofibration, and * ^ LsX ^ X' is fibrant in Fib(LSC) since LsX is fibrant in LsC, X' is fibrant in C and X' ^ LsX is a fibration in C.

Therefore, the map X ^ ev2(const(X)) ^ ev2(R(const(X))), where R denotes fibrant replacement in Fib(LSC), is precisely the map X ^ X', which is a weak equivalence in CkC since it was already a weak equivalence in C.

Finally, let Y ^ X be a fibrant and cofibrant section in Fib(LSC). We need to check that the composite

const(g(ev2(* ^Y ^ X))) —> const(ev2(* ^Y ^ X)) —> Y ^ X)

is a weak equivalence in Fib(LSC). But ev2 (* ^ Y ^ X) = X is already cofibrant in CkC, by Remark 3.7. Therefore, we need to show that the map of sections

*-> X^=X

*-^ Y^-X

is a weak equivalence in Fib(LfC). Since both sections are cofibrant, it is enough to see that the map in the middle is a weak equivalence in LsC, which follows again from Remark 3.7. □

3.3. Postnikov Sections and Connective Covers of Simplicial Sets

We can use this setup to describe the "layers" of Postnikov towers. Let sSet* denote the category of pointed simplicial sets. Consider the model structure Pk sSet* = Ls sSet* for fc-types, that is, the left Bousfield localization of sSet* with respect to the set of inclusions S = ^ Dk+2}. If K =

|^fc+i}, then the right Bousfield localization Ck sSet* = Ck sSet* is the model structure for fc-connective covers, and Pk sSet* and Ck sSet* are compatible, since for every X there is a fiber sequence

Ck X —> X —> Pk X,

where CkX denotes the fcth connective cover of X. By Theorem 3.8 the model categories Ck sSet* and Fib(LS sSet*) are Quillen equivalent.

Let 5 = ^ Dn+2} and K = as before, and let C be a

proper combinatorial model category. Then we define LsC as the left Bous-field localization of C with respect to the set Ic □S and CkC as the right Bousfield localization of C with respect to Gc <8>K. Here Ic is the set of generating cofibrations of C, Gc is a set of homotopy generators, ® denotes the simplicial action given by a framing and □ the pushout product. A fuller account of localized model structures along Quillen bifunctors can be found in . In general, LsC and CkC are not necessarily compatible, so Theorem 3.8 will not hold in this case for arbitrary C. However, examples where compatibility holds include the category of chain complexes Chb(fl) and stable localizations; see Sect. 3.5.

We can also consider Fib(LSPk+1 sSet*). Since for every X we have a fibration

K(nk+iX, k +1) Pk+iX PkX,

the model structures CkPk+1 sSet* and PkPk+1 sSet* = Pk sSet* are compatible. Hence Theorem 3.8 directly implies

Corollary 3.9. The model structures CkPk+1 sSet* and Fib(LSPk+1 sSet*) are Quillen equivalent. □

This means that we can view Ck Pk+1 sSet* as the kth layer of the Postnikov tower model structure. Note that Ho(CkPk+1 sSet*) is equivalent to the category of abelian groups for k > 1.

3.4. Nullifications and Cellularizations of Spectra

Let Sp be a suitable model structure for the category of spectra, for instance, symmetric spectra and let S be a single map E ^ *. Then Ls Sp = PE Sp is called the E-nullification of Sp and CE Sp is called the E-cellularization of Sp. As follows from [17, Theorem 3.6] we have the following compatibility between localized and colocalized model structures:

(i) If the induced map Ho(Sp)(S^1 E, CEX) ^ Ho(Sp)(S^1E, X) is injec-tive for every X, then CE Sp and PE Sp are compatible.

(ii) If the induced map Ho(Sp)(E, X) ^ Ho(Sp)(E, P^EX) is the zero map for every X, then CE Sp and P^E Sp are compatible.

3.5. Stable Localizations and Colocalizations

Let C be a proper combinatorial stable model category and let Gsp denote a set of cofibrant homotopy generators for the model category of symmetric spectra Sp. Recall that a set of homotopy generators for a model category C consists of a small full subcategory Gc such that every object of C is weakly equivalent to a filtered homotopy colimit of objects of Gc and that by [11, Proposition 4.7] every combinatorial model category has a set of homotopy generators that can be chosen to be cofibrant.

A set of maps S in a stable model category is said to be stable if the class of S-local objects is closed under suspension. Let S be a stable set of morphisms in C and let K = cof(S) be the set of cofibers of the elements of S. Then we have that cof(S ® GSp) = cof(S) ® GSp = K ® GSp, where

® denotes the action of Sp on C. Hence, by [2, Proposition 10.3] it follows that Ls<g)gSpC and Cfc®gSpC are compatible. Therefore, Theorem 3.8 readily implies the following fact:

Corollary 3.10. The model categories Cfc@,gSpC and Fib(LS®GspC) are Quillen equivalent. □

Acknowledgments

J.J. Gutierrez would like to thank Dimitri Ara for many useful conversations on some of the topics of this paper, and Ieke Moerdijk for suggesting the idea of studying towers of localized model structures. C. Roitzheim would like to thank David Barnes for motivating discussions and the Radboud Universiteit Nijmegen for their hospitality. Both authors thank the referee and the associate editor for very useful comments that helped improving the contents and presentation of the paper.

References

 Adamek, J., Rosicky, J.: Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, vol. 189. Cambridge University Press, Cambridge (1994)

 Barnes, D., Roitzheim, C.: Stable left and right Bousfield localisations. Glasg. Math. J. 56(1), 13-42 (2014)

 Barnes, D., Roitzheim, C.: Homological localisation of model categories. Appl. Categ. Struct. 23(3), 487-505 (2015)

 Barwick, C.: On left and right model categories and left and right Bousfield localizations. Homol. Homotopy Appl. 12(2), 245-320 (2010)

 Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129(3), 447-475 (2000)

 Bergner, J: Homotopy fiber products of homotopy theories. Isr. J. Math. 185, 389-411 (2011)

 Bergner, J.: Homotopy limits of model categories and more general homotopy theories. Bull. Lond. Math. Soc. 44(2), 311-322 (2013)

 Biedermann, G., Chorny, B., Rondigs, O.: Calculus of functors and model categories. Adv. Math. 214, 92-115 (2007)

 Bousfield, A.K.: The localization of spectra with respect to homology. Topology 18(4), 257-281 (1979)

 Bousfield, A.K.: On the telescopic homotopy theory of spaces. Trans. Am. Math. Soc. 353(6), 2391-2426 (2001)

Dugger, D.: Combinatorial model categories have presentations. Adv. Math. 164, 177-201 (2001)

Dugger, D: Spectral enrichments of model categories. Homol. Homotopy Appl. 8(1), 1-30 (2006)

Dwyer, W.G.: Localizations, Axiomatic, Enriched and Motivic Homotopy Theory, 3-28, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 131. Kluwer Academic Publications, Dordrecht (2004)

Gabriel, P., Ulmer, F.: Lokal prasentierbare Kategorien. Lecture Notes in Mathematics, vol. 221. Springer, Berlin (1971)

Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Progress in Mathematics, vol. 174. Birkhauser, Basel (1999)

Greenlees, J.P.C., Shipley, B.: Homotopy theory of modules over diagrams of rings. Proc. Am. Math. Soc. Ser. B 1, 89-104 (2014)

Gutierrez, J.J.: Cellularization of structures in stable homotopy categories. Math. Proc. Camb. Philos. Soc. 153, 399-418 (2012) Gutierrez, J.J., Roitzheim, C.: Bousfield Localisations Along Quillen Bifunctors (2014) arXiv:1411.0500

Hirschhorn, P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence (2003)

Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999)

Hovey, M., Shipley, B., Smith, J.H.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149-208 (2000)

Lenhardt, F.: Stable frames in model categories. J. Pure Appl. Algebra 216(5), 1080-1091 (2012)

Lurie, J.: Higher Topos Theory. Annals of Mathematical Studies, vol. 170. Princeton University Press, Princeton (2009)

Makkai, M., Pare, R.: Accessible Categories: The Foundations of Categorical Model Theory. Contemporary Mathematics, vol. 104. American Mathematical Society, Providence (1989)

Ravenel, D.C.: Localization with respect to certain periodic homology theories. Am. J. Math. 106(2), 351-414 (1984)

Ravenel, D.C.: Nilpotence and Periodicity in Stable Homotopy Theory. Annals of Mathematical Studies, vol. 128. Princeton University Press, Princeton (1992) Toen, B.: Derived Hall algebras. Duke Math. J. 135(3), 587-615 (2006)

Javier J. Gutierrez

Institute for Mathematics, Astrophysics and Particle Physics

Heyendaalseweg 135

6525 AJ Nijmegen

The Netherlands

e-mail: j.gutierrez@math.ru.nl

URL: http://www.math.ru.nl/gutierrez

Constanze Roitzheim School of Mathematics Statistics and Actuarial Science University of Kent Canterbury, Kent CT2 7NF UK

e-mail: c.roitzheim@kent.ac.uk

URL: http://www.kent.ac.uk/smsas/personal/csrr

Received: July 20, 2015. Revised: March 4, 2016. Accepted: March 30, 2016.