STABILIZATION OF MODEL CATEGORIES

MARK HOVEY

Abstract. Given a sufficiently nice model category 𝒞 and a left Quillen endofunctor of it, we show how to invert G up to homotopy. That is, we construct a new model category Sp^{ℕ}(𝒞, G), an embedding 𝒞 ⟶ Sp^{ℕ}(𝒞, G), and an extension of G to a Quillen equivalence of Sp^{ℕ}(𝒞, G). In the presence of a symmetric monoidal structure on 𝒞 compatible with the model structure on G, we show how to construct a different stabilization Sp^{Σ}(𝒞, G) which is again symmetric monoidal.

Introduction

In the broadest sense, algebraic topology is the study of model categories. Of course, algebraic topologists normally concern themselves with the specific model category of topological spaces, or with model categories derived from it. Nevertheless, there is considerable value in the general study of model categories. In particular, this allows us to export the methods of algebraic topology to other branches of mathematics. This idea has been extremely important in the recent work of Voevodsky in algebraic geometry [Voe97], but this viewpoint was already apparent in [AM86].

In particular, we would like to export the methods of stable homotopy theory to other situations. Thus, we would like a way to stabilize an arbitrary model category in an appropriate sense. Given a simplicial model category 𝒞, we have a suspension functor given by smashing with the simplicial circle S^{1}, and we would like to make this functor an equivalence of categories, up to homotopy. However, this viewpoint is not general enough. Instead, we will start with a model category 𝒞 and an endofunctor G: 𝒞 ⟶ 𝒞. Of course, we should assume that G respects the model structure in some way; we will assume that G is a left Quillen functor (see Section 1 for definitions of terms used in model category theory). Then we would like a new model category G^{−1}𝒞 on which G is not an isomorphism, but a Quillen equivalence. More precisely, we would like an embedding 𝒞 ⟶ G^{−1}𝒞, which should be a left Quillen functor, and an extension of G to a left Quillen functor G: G^{−1}𝒞 ⟶ G^{−1}𝒞, which is in fact a Quillen equivalence. Let us refer to such a model category G^{−1}𝒞 as a G-stabilization, or just a stabilization of 𝒞. Such a G-stabilization will only be stable with respect to G, of course, and not necessarily with respect to the suspension. Note that a G-stabilization of a model category 𝒞 will not necessarily be unique; one could attempt to fix this by looking for an initial G-stabilization, but we do not know if such a thing exists in the most important examples. Of course, if G is already a Quillen equivalence on 𝒞, then 𝒞 is the initial G-stabilization of 𝒞. For any reasonable G-stabilization functor 𝒞 ↦ G^{−1}𝒞, the embedding 𝒞 ⟶ G^{−1}𝒞 must be a Quillen equivalence when G is already a Quillen equivalence on 𝒞.

Date: July 3, 1998.

1991 Mathematics Subject Classification. 55U35, 18G55.

Key words and phrases. model category, homotopy category, stabilization, Quillen functor, Quillen equivalence, Bousfield localization, cellular, spectra, symmetric spectra.

The goal of this paper is to construct a G-stabilization of 𝒞 under good conditions on 𝒞. As mentioned above, there may be many different G-stabilizations. We discuss two of them. The first is based on Bousfield-Friedlander spectra [BF78], and applies in considerable generality. We denote it Sp^{ℕ}(𝒞, G). The second is based on symmetric spectra [HSS98], and is useful only when there is some monoidal structure around that G preserves. Such a functor will always be of the form −∧K for some object K, and we then denote this symmetric stabilization by Sp^{Σ}(𝒞, K).

Here are the two main theorems of the paper.

Theorem A. Suppose 𝒞 is a left proper cellular model category, and suppose G is a left Quillen endofunctor of 𝒞. Then there is a left proper cellular G-stabilization Sp^{ℕ}(𝒞, G) of 𝒞 which is functorial in an appropriate sense. Furthermore, if G: 𝒞 ⟶ 𝒞 is a Quillen equivalence, then the embedding 𝒞 ⟶ Sp^{ℕ}(𝒞, G) is a Quillen equivalence.

The model category Sp^{ℕ}(𝒞, G) is defined in Definition 5.3, modulo an important technical detail proved in Theorem 6.10. We prove that Sp^{ℕ}(𝒞, G) is a Gstabilization of 𝒞 in Theorem 5.8. We prove that Sp^{ℕ}(𝒞, G) is Quillen equivalent to 𝒞 when G is already a Quillen equivalence on 𝒞 in Theorem 7.1. Finally, the functoriality of Sp^{ℕ}(𝒞, G) is considered in Proposition 7.3 and in Theorem 7.5.

The most interesting model categories, such as simplicial sets and chain complexes of abelian groups, are symmetric monoidal model categories. That is, they are closed symmetric monoidal categories and the monoidal structure is compatible with the model structure in an appropriate sense. The trouble with the naive G-stabilization Sp^{ℕ}(𝒞, G) is that Sp^{ℕ}(𝒞, G) is almost never a symmetric monoidal model category, even when 𝒞 is so and G is a module functor.

In this case, we have the following theorem.

Theorem B. Suppose 𝒞 is a left proper cellular symmetric monoidal model category in which the domains of the generating cofibrations can be taken to be cofibrant. Suppose K is a cofibrant object of 𝒞. Then there is a K-stabilization Sp^{Σ}(𝒞, K) of 𝒞 which is again a left proper cellular symmetric monoidal model category. Furthermore, there is a K-stabilization functor 𝒟 ↦ Sp^{Σ}(𝒟, K) from left proper cellular 𝒞-model categories to left proper cellular Sp^{Σ}(𝒞, K)-model categories. If − ∧ K is a Quillen equivalence on 𝒟, then the embedding 𝒟 ⟶ Sp^{Σ}(𝒟, K) is a Quillen equivalence.

This theorem is proved in Section 10 and Section 11.

In particular, if we take 𝒞 to be the category of pointed simplicial sets SSet_{∗} in Theorem B and K to be the usual simplicial circle S^{1}, then Sp^{Σ}(SSet_{∗}, S^{1}) is the category Sp^{Σ} of symmetric spectra considered in [HPS97]. We get a stabilization functor on pointed left proper cellular simplicial model categories whose output is a Sp^{Σ}-model category.

We hope that these theorems will have many applications. One application is to stable homotopy theory. Consider the p-local stable homotopy category 𝒮, and choose a particular homotopy type E. Then, by choosing a cofibrant representative for E as a symmetric spectrum, we can apply Theorem B to get a model category where smashing with E is a Quillen equivalence. At least for some choices of E, the homotopy category of this model category will be a stable homotopy category in the sense of [HPS97], in which we have forced E to be in the Picard group. Any E-acyclic spectrum therefore goes to 0 in this new category, so it is a module over the E-local stable homotopy category, but will in general be larger. We hope to investigate these categories in a future paper.

The author believes that we should not have to require that a model category be simplicial in order to stabilize it with respect to the suspension. The general principle, developed in [Hov98a] and due to Dwyer and Kan [DK80], is that there is no difference between the homotopy category of a general model category and the homotopy category of a simplicial model category. So the simplicial assumption should never be necessary. The author would like to define a stabilization functor on arbitrary pointed left proper cellular model categories, but has not been able to do so yet.

This paper is organized as follows. In the first section, we remind the reader of some of the definitions and basic facts of model category theory. In Section 2 we give the definition of the category of G-spectra 𝒞_{G}, where G is a left Quillen endofunctor of a model category 𝒞. In Section 3 we describe the projective model structure on the category of G-spectra. The stabilization functor Sp^{ℕ}(𝒞, G) is a Bousfield localization of the category 𝒞_{G}, so we describe the general theory of such localizations, due to Hirschhorn [Hir98], in Section 4. We then describe Sp^{ℕ}(𝒞, G) in Section 5. Hirschhorn does require a technical hypothesis to hold in order to construct the Bousfield localization of a model category, and we check this hypothesis for 𝒞_{G} in Section 6. We investigate the functoriality of Sp^{ℕ}(𝒞, G) in Section 7.

The second part of the paper begins in Section 8, where we show that Sp^{ℕ}(𝒞, G) is almost never a monoidal category, even when 𝒞 is symmetric monoidal and G is a module functor. We begin to remedy this problem in Section 9, where we construct the category 𝒟_{K}^{Σ} of symmetric K-spectra in 𝒟, where K is an object of a symmetric monoidal model category 𝒞 and 𝒟 is a 𝒞-model category. This section contains no model category theory; we defer that to Section 10, where we define the stabilizations Sp^{Σ}(𝒞, K) and Sp^{Σ}(𝒟, K). In Section 11, we investigate the functoriality of Sp^{Σ}(𝒟, K) and compare the two stabilizations Sp^{ℕ}(𝒟, − ∧ K) and Sp^{Σ}(𝒟, K). The final section of the paper contains some remarks concerning unfinished business. The tenor of these remarks is that the general theory of this paper will construct stabilizations, but, in the most interesting cases, will not provide enough information to work effectively with the resulting model category. These remarks deserve to be expanded into a section detailing how to get better information when enough is known about the underlying model category and the functor with respect to which one is stabilizing, but that will have to await future work.

The author would like to thank Phil Hirschhorn, Dan Kan, Brooke Shipley, and Jeff Smith for many conversations about the theory of model categories.

1. Model categories

In this section, we briefly recall some of the basic facts about model categories and Quillen functors. A gentle introduction to model categories can be found in [DS95]. The original paper is [Qui67], but the situation has changed a great deal since then. More recent references are the books [DHK], [Hir98], and [Hov98a].

1.1. Definitions. We begin with some preliminary definitions.

Definition 1.1. Suppose i: A ⟶ B and p: X ⟶ Y are maps in a category 𝒞. Then i has the left lifting property with respect to p and p has the right lifting property with respect to i if, for every commutative diagram

A ⟶^{f} X
i↓  ↓^{p}
B ⟶_{g} Y

there is a lift h: B ⟶ X such that hi = f and ph = g.

Now, given a category 𝒞, we can form the category Arr 𝒞 of arrows of 𝒞; an object of Arr 𝒞 is a morphism of 𝒞, and a morphism in Arr 𝒞 is a commutative square.

Definition 1.2. Suppose 𝒞 is a category, and let Arr 𝒞 denote the category of arrows in 𝒞.

1. A map f in 𝒞 is a retract of a map g ∈ 𝒞 if f is a retract of g as objects of Arr 𝒞.

2. A functorial factorization is an ordered pair (α, β) of functors Arr 𝒞 ⟶ Arr 𝒞 such that f = β(f) ∘ α(f) for all f ∈ Arr 𝒞. In particular, the domain of α(f) is the domain of f, the codomain of α(f) is the domain of β(f), and the codomain of β(f) is the codomain of f.

We can now define a model category. We use the definition of [Hov98a], which differs slightly from the definition used in [DHK] and [Hir98], and more significantly from the original definition of [Qui67].

Definition 1.3. Suppose 𝒞 is a category. A model structure on 𝒞 consists of three subcategories, the morphisms of which are called weak equivalences, cofibrations, and fibrations, together with two functorial factorizations (α, β) and (α′, β′), satisfying the following properties.

1. (Retracts): A retract of a weak equivalence (cofibration, fibration) is a weak equivalence (cofibration, fibration).

2. (Two out of three): If gf is defined and two out of the three maps f, g, and gf are weak equivalences, so is the third.

3. (Lifting): Define a map to be a trivial cofibration (resp. trivial fibration) if it is both a cofibration and a weak equivalence (resp. both a fibration and a weak equivalence). Then trivial cofibrations have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect to trivial fibrations.

4. For all maps f of 𝒞, α(f) is a trivial cofibration, β(f) is a fibration, α′(f) is

a cofibration, an β′(f) is a trivial fibration. A bicomplete category together with a model structure on it is called a model category.

The standard examples of model categories are the model categories of topological spaces, chain complexes over a ring, and simplicial sets. The book [Hov98a] contains descriptions of all of these. A model category, being bicomplete, always has an initial object 0 and a terminal object 1. An object X in a model category is called cofibrant if the map 0 ⟶ X is a cofibration, and is called fibrant if the map X ⟶ 1 is a fibration. By applying functorial factorization to the map 0 ⟶ X we obtain the cofibrant replacement functor Q and a natural trivial fibration QX ⟶ X. Similarly, we get a fibrant replacement functor R and a natural trivial cofibration X ⟶ RX.

A model category 𝒞 has associated to it its homotopy category Ho 𝒞, obtained by inverting the weak equivalences of 𝒞. In particular, Ho 𝒞 has the same objects as 𝒞. To describe the morphism sets, one introduces the notions of left and right homotopy. Then Ho 𝒞(X, Y) is isomorphic to the set of (left or right) homotopy classes of maps from QX to RY. See [DS95], [Qui67], or [Hov98a] for details.

The following lemma is frequently used when dealing with model categories. The proof is straightforward, but may be found in [Hov98a, Lemma 1.1.9] and [HSS98, Proposition 3.2.4].

Lemma 1.4 (The Retract Argument). Suppose we have a factorization f = pi in a category 𝒞, and suppose that f has the left lifting property with respect to p. Then f is a retract of i.

1.2. Cofibrantly generated model categories. We now define the particularly important class of cofibrantly generated model categories. See [Hov98a, Section 2.1] and [Hir98, Chapters 12-13] for more details. Some preliminary definitions are required to cope with transfinite compositions.

Definition 1.5. Suppose κ is a cardinal. A limit ordinal λ is said to be κ-filtered if every set A of ordinals less than λ such that sup A = λ has cardinality greater than κ.

The ordinal λ is κ-filtered if and only if the cofinality [Jec78, p.26] of λ is greater than κ. For any ordinal, there is a larger ordinal that is κ-filtered.

Given an ordinal λ, we will always think of λ as a category where there is one morphism from i to j if and only if i ≤ j.

Definition 1.6. Suppose 𝒞 is a cocomplete category.

1. A λ-sequence in 𝒞 is a colimit-preserving functor X: λ ⟶ 𝒞 from some ordinal λ to 𝒞. We usually denote the image of i ∈ λ by X_{i}. The map X_{0} ⟶ colim X_{i} is called the transfinite composition of X. Given a subcategory 𝒟 of 𝒞, a λ-sequence in 𝒟 is a λ-sequence X: λ ⟶ 𝒞 such that each map X_{i} ⟶ X_{i+1} is in 𝒟.

2. Given a subcategory 𝒟 of 𝒞 and a cardinal κ, an object X ∈ 𝒞 is said to be κsmall relative to 𝒟 if, for every κ-filtered ordinal λ, and for every λ-sequence Y in 𝒟, the induced map

colim_{i} 𝒞(X, Y_{i}) ⟶ 𝒞(X, colim_{i} Y_{i})

is an isomorphism.

3. Given a subcategory 𝒟 of 𝒞, an object X ∈ 𝒞 is said to be small relative to 𝒟 if X is κ-small relative to 𝒟 for some cardinal κ.

The main advantage of small objects is that one can use them to construct factorizations.

Definition 1.7. Suppose I is a set of maps in a category 𝒞. Then we define I-inj to be the class of maps with the right lifting property with respect to I, and we define I-cof to be the class of maps with the left lifting property with respect to I-inj. Note that I ⊆ I-cof.

Proposition 1.8 (The Small Object Argument). Suppose I is a set of maps in a cocomplete category 𝒞. Suppose the domains of I are small relative to I-cof. Then there is a functorial factorization f = β(f) ∘ α(f), where α(f) is in I-cof and β(f) is in I-inj.

A proof of the small object argument can be found in [Hov98a, Section 2.1.2], in [Hir98], or in [HSS98, Lemma 3.2.11]. We point out that the map α(f) in the small object argument is constructed as the transfinite composition of pushouts of coproducts of maps of I. Furthermore, it is proved in [Hir98, Chapter 12] and [Hov98a, Section 2.1.2] that we can remove the coproducts by taking a longer transfinite composition, so that α(f) is in fact a transfinite composition of pushouts of maps of I.

We can now define a cofibrantly generated model category.

Definition 1.9. A model category 𝒞 is cofibrantly generated if there is a set I of cofibrations and a set J of trivial cofibrations such that the following properties hold.

1. The domains of I are small relative to the cofibrations.

2. The domains of J are small relative to the trivial cofibrations.

3. A map is a trivial fibration if and only if it has the right lifting property with respect to I, and a map is a fibration if and only if it has the right lifting property with respect to J.

This definition is equivalent to the definition given in [Hov98a, Section 2.1] and in [Hir98, Chapter 13].

The model categories of topological spaces, simplicial sets, and chain complexes are all cofibrantly generated [Hov98a, Chapter 2], as are most model categories in common use. The generating cofibrations in the model category of simplicial sets, for example, are the inclusions of the boundary of an n-simplex into the n-simplex.

Note that the functorial factorizations in a cofibrantly generated model category need not be the ones constructed using the small object argument, but usually they are. Also note that, in a cofibrantly generated model category, I-cof is the class of cofibrations and J-cof is the class of trivial cofibrations. It follows from the retract argument and the small object argument that every cofibration in a cofibrantly generated model category is a retract of a transfinite composition of pushouts of I.

1.3. Quillen functors. Having defined model categories, we need to define functors between them. See [Hov98a, Section 1.3] for more details.

Definition 1.10. A functor G: 𝒞 ⟶ 𝒟 between model categories is called a left Quillen functor if G is a left adjoint and G preserves cofibrations and trivial cofibrations. Dually, G is a right Quillen functor if G is a right adjoint and G preserves fibrations and trivial fibrations. The term Quillen functor will always mean a left Quillen functor. A Quillen functor G with right adjoint U is called a Quillen equivalence if, for all cofibrant X ∈ 𝒞 and fibrant Y ∈ 𝒟, a map GX ⟶ Y is a weak equivalence in 𝒟 if and only if its adjoint X ⟶ UY is a weak equivalence in 𝒞.

It is an exercise in lifting and adjointness to verify that if G is a functor between model categories with right adjoint U, then G is a left Quillen functor if and only if U is a right Quillen functor. Less obvious is Ken Brown’s lemma (see [Hov98a, Lemma 1.1.12] or [DS95, Lemma 9.9]), which implies that a left Quillen functor preserves weak equivalences between cofibrant objects, and a right Quillen functor preserves weak equivalences between fibrant objects. From this, it follows that a left Quillen functor G with right adjoint U induces a functor LG: Ho 𝒞 ⟶ Ho 𝒟 with right adjoint RU. In fact (LG)X = G(QX), and (RU)X = U(RX). This is the reason for taking the functorial factorizations as part of the model structure, so that LG and RU will be well defined.

The following lemma is very useful. We say that a functor F between model categories reflects weak equivalences between cofibrant objects if, given a map f: X ⟶ Y between cofibrant objects, if F f is a weak equivalence, then f is a weak equivalence. There is a similar definition using fibrant objects.

Lemma 1.11. Suppose F: 𝒞 ⟶ 𝒟 is a Quillen functor with right adjoint U. The following are equivalent:

(a) F is a Quillen equivalence.

(b) F reflects weak equivalences between cofibrant objects and, for every fibrant Y, the map F QUY ⟶ Y is a weak equivalence.

(c) U reflects weak equivalences between fibrant objects and, for every cofibrant X, the map X ⟶ URF X is a weak equivalence.

(d) LF is an equivalence of categories.

A proof of this lemma can be found in [Hov98a, Section 1.3], especially in Corollary 1.3.16.

1.4. Proper model categories. Many of the technical arguments in this paper and in [Hir98] require the assumption of properness, which we now define.

Definition 1.12. Suppose 𝒞 is a model category.

1. 𝒞 is left proper if, whenever we have a pushout square

\begin{matrix}A & ⟶^{f} & C \\ ^{i}↓ & & ↓ \\ B & ⟶_{g} & D \end{matrix}

in which i is a cofibration and f is a weak equivalence, the map g is also a weak equivalence.

2. 𝒞 is right proper if, whenever we have a pullback square

\begin{matrix}A & ⟶^{f} & C \\ ↓ & & ↓^{p} \\ B & ⟶_{g} & D \end{matrix}

in which p is a fibration and g is a weak equivalence, the map f is a weak equivalence.

3. 𝒞 is proper if it is both left proper and right proper.

Almost all model categories one ever runs across are proper. Simplicial sets, topological spaces, and chain complexes, for example, are proper. See [Hir98, Chapter 11] for details. On the other hand, the stable model category of Γ-spaces is not right proper [BF78], although the strict model category is.

2. Spectra

In this section and throughout the paper, 𝒞 will be a model category and G: 𝒞 ⟶ 𝒞 will be a Quillen endofunctor of 𝒞 with right adjoint U. In this section, we define the underlying category 𝒞_{G} of our stabilization Sp^{ℕ}(𝒞, G).

The following definition is a straightforward generalization of the usual notion of spectra [BF78].

Definition 2.1. Suppose G is a left Quillen endofunctor of a model category 𝒞. Define 𝒞_{G}, the category of spectra, as follows. A spectrum X is a sequence X_{0}, X_{1}, …, X_{n}, … of objects of 𝒞 together with structure maps σ: GX_{n} ⟶ X_{n+1} for all n. A map of spectra from X to Y is a collection of maps f_{n}: X_{n} ⟶ Y_{n} commuting with the structure maps, so that the diagram

GX_{n} ⟶^{σ_{X}} X_{n+1}
Gf_{n}↓  ↓^{f_{n+1}}
GY_{n} ⟶_{σ_{Y}} Y_{n+1}

is commutative for all n.

Note that if 𝒞 is either the model category of pointed simplicial sets or the model category of pointed topological spaces, and G is the suspension functor given by smashing with the circle S^{1}, then 𝒞_{G} is the Bousfield-Friedlander category of spectra [BF78].

Definition 2.2. Given n ≥ 0, the evaluation functor Ev_{n}: 𝒞_{G} ⟶ 𝒞 takes X to X_{n}. The evaluation fnctor has a left adjoint F_{n}: 𝒞 ⟶ 𝒞_{G} defined by (F_{n}X)_{m} = G^{m−n}X if m ≥ n and (F_{n}X)_{m} = 0 otherwise, where 0 is the initial object of 𝒞. The structure maps are the obvious ones.

Note that F_{0} is an full and faithful embedding of the category 𝒞 into 𝒞_{G}.

Limits and colimits in the category of spectra are taken objectwise.

Lemma 2.3. The category of spectra is bicomplete.

Proof. Given a functor X: ℐ ⟶ 𝒞_{G}, we define (colim X)_{n} = colim Ev_{n} ∘X and (lim X)_{n} = lim Ev_{n} ∘X. Since G is a left adjoint, it preserves colimits. The structure maps of the colimit are then the composites

G(colim Ev_{n} ∘X) ≅ colim(G ∘ Ev_{n} ∘X) ⟶^{colim σ∘X} colim Ev_{n+1} ∘X.

Although G does not preserve limits, there is still a natural map G(lim Y) ⟶ lim GY for any functor Y : ℐ ⟶ 𝒞. Then the structure maps of the limit are the composites

G(lim Ev_{n} ∘X) ⟶ lim(G ∘ Ev_{n} ∘X) ⟶^{lim σ∘X} lim Ev_{n+1} ∘X.

Remark 2.4. Note that the evaluation functor Ev_{n}: 𝒞_{G} ⟶ 𝒞 preserves colimits, so should have a right adjoint R_{n}: 𝒞 ⟶ 𝒞_{G}. We define (R_{n}X)_{m} = U^{n−m}X if m ≤ n, and (R_{n}X)_{m} = 1 if m > n. The structure map GU^{n−m}X ⟶ U^{n−m−1}X is adjoint to the identity map of U^{n−m}X when m < n. We leave it to the reader to verify that R_{n} is the right adjoint of Ev_{n}.

We now show that the functors G and U extend to functors on 𝒞_{G}.

Lemma 2.5. Suppose F: 𝒞 ⟶ 𝒞 is a functor and τ: GF ⟶ F G is a natural transformation. Then there is an induced functor F: 𝒞_{G} ⟶ 𝒞_{G}, called the prolongation of F.

Proof. Given X ∈ 𝒞_{G}, we define (F X)_{n} = F X_{n}, with structure maps GF X_{n} ⟶^{τ} F GX_{n} ⟶^{F σ} F X_{n+1}. Given a map f: X ⟶ Y, we define F f by (F f)_{n} = F f_{n}.

Note that the prolongation of F depends on the choice of natural transformation τ, but in practice there is always an obvious choice.

Corollary 2.6. The functors G and U prolong to functors on 𝒞_{G}, where they are still adjoint.

Proof. To prolong G, take τ to be the identity in Lemma 2.5. To prolong its right adjoint U, take τ to be the composite GUX ⟶ X ⟶ UGX of the counit and unit of the adjunction. It is an exercise in adjoint functors to verify that the prolongations of G and U are still adjoint to each other.

3. The projective model structure

Our goal is to put a model structure on the category 𝒞_{G} of G-spectra defined in the last section, so that G becomes a Quillen equivalence. In this section, we define a preliminary model structure on 𝒞_{G} called the projective model structure. This model structure could also be called the strict, or level, model structure. The extension of G to 𝒞_{G} is not a Quillen equivalence with respect to this model structure, though it is a Quillen functor. This section owes much to [BF78] and [HSS98, Section 5.1].

Definition 3.1. A map f ∈ 𝒞_{G} is a level equivalence if each map f_{n} is a weak equivalence in 𝒞. Similarly, f is a level fibration (resp.level cofibration, level trivial fibration, level trivial cofibration) if each map f_{n} is a fibration (resp. cofibration, trivial fibration, trivial cofibration) in 𝒞. The map f is a projective cofibration if f has the left lifting property with respect to every level trivial fibration.

Note that level equivalences satisfy the two out of three property, and each of the classes defined above is closed under retracts. Thus we should be able to construct a model structure using these classes. To do so, we need the small object argument, and hence we assume that 𝒞 is cofibrantly generated.

Definition 3.2. Suppose 𝒞 is a cofibrantly generated model category with generating cofibrations I and generating trivial cofibrations J. Suppose G is a Quillen endofunctor of 𝒞, so we have the category 𝒞_{G}. Define I_{G} = ⋃_{n} F_{n}I and J_{G} = ⋃_{n} F_{n}J.

The sets I_{G} and J_{G} will be the generating cofibrations and trivial cofibrations for a model structure on 𝒞_{G}. To see this, we must show that the domains of I_{G} and J_{G} are small.

Proposition 3.3. Suppose A is small relative to the cofibrations (resp. trivial cofibrations) in 𝒞, and n ≥ 0. Then F_{n}A is small relative to the level cofibrations (resp. level trivial cofibrations) in 𝒞_{G}.

Proof. The main point is that Ev_{n} commutes with colimits. We leave the remainder of the proof to the reader.

To apply this to the domains of I_{G}, we need to know that the maps of I_{G}-cof are level cofibrations. Recall the right adjoint R_{n} of Ev_{n} constructed in Remark 2.4.

Lemma 3.4. A map in 𝒞_{G} is a level cofibration (resp. level trivial cofibration) if and only if it has the left lifting property with respect to R_{n}g for all n ≥ 0 and all trivial fibrations (resp. fibrations) g in 𝒞.

Proof. By adjunction, a map f has the left lifting property with respect to R_{n}g if and only if Ev_{n} f has the left lifting property with respect to g. Since a map is a cofibration (resp. trivial cofibration) in 𝒞 if and only if it has the left lifting property with respect to all trivial fibrations (resp. fibrations), the lemma follows.

Proposition 3.5. Every map in I_{G}-cof is a level cofibration. Every map in J_{G}-cof is a level trivial cofibration.

Proof. Since G is a left Quillen functor, every map in I_{G} is a level cofibration. By Lemma 3.4, this means that R_{n}g ∈ I_{G}-inj for all n ≥ 0 and all trivial fibrations g. Since a map in I_{G}-cof has the left lifting property with respect to every map in I_{G}-inj, in particular it has the left lifting property with respect to R_{n}g. Another application of Lemma 3.4 completes the proof for I_{G}-cof. The proof for J_{G}-cof is similar.

Proposition 3.3 and Proposition 3.5 immediately imply the following corollary.

Corollary 3.6. The domains of I_{G} are small relative to I_{G}-cof. The domains of J_{G} are small relative to J_{G}-cof.

Theorem 3.7. Suppose 𝒞 is cofibrantly generated. Then the projective cofibrations, the level fibrations, and the level equivalences define a cofibrantly generated model structure on 𝒞_{G}, with generating cofibrations I_{G} and generating trivial cofibrations J_{G}. We call this the projective model structure. The projective model structure is left proper (resp. right proper, proper) if 𝒞 is left proper (resp. right proper, proper).

Note that if 𝒞 is either the model category of pointed simplicial sets or pointed topological spaces, and G is the suspension functor, the projective model structure on 𝒞_{G} is the strict model structure on the Bousfield-Friedlander category of spectra [BF78].

Proof. The retract and two out of three axioms are immediate, as is the lifting axiom for a projective cofibration and a level trivial fibration. By adjointness, a map is a level trivial fibration if and only if it is in I_{G}-inj. Hence a map is a projective cofibration if and only if it is in I_{G}-cof. The small object argument 1.8 applied to I_{G} then produces a functorial factorization into a projective cofibration followed by a level trivial fibration.

Adjointness implies that a map is a level fibration if and only if it is in J_{G}-inj. We have already seen in Proposition 3.5 that the maps in J_{G}-cof are level equivalences, and they are projective cofibrations since they have the left lifting property with respect to all level fibrations, and in particular level trivial fibrations. Hence the small object argument applied to J_{G} produces a functorial factorization into a projective cofibration and level equivalence followed by a level fibration.

Conversely, we claim that any projective cofibration and level equivalence f is in J_{G}-cof, and hence has the left lifting property with respect to level fibrations.

To see this, write f = pi where i is in J_{G}-cof and p is in J_{G}-inj. Then p is a level fibration. Since f and i are both level equivalences, so is p. Thus f has the left lifting property with respect to p, and so f is a retract of i by the retract argument 1.4. In particular f ∈ J_{G}-cof.

Since colimits and limits in 𝒞_{G} are taken levelwise, and since every projective cofibration is in particular a level cofibration, the statements about properness are immediate.

We also characterize the projective cofibrations. We denote the pushout of two maps A ⟶ B and A ⟶ C by B ⨿_{A} C.

Proposition 3.8. A map i: A ⟶ B is a projective (trivial) cofibration if and only if the induced maps A_{0} ⟶ B_{0} and A_{n} ⨿_{GA_{n−1}} GB_{n−1} ⟶ B_{n} for n ≥ 1 are (trivial) cofibrations.

Proof. We only prove the cofibration case, leaving the similar trivial cofibration case to the reader. First suppose i: A ⟶ B is a projective cofibration. We have already seen in Proposition 3.5 that A_{0} ⟶ B_{0} is a cofibration. Suppose p: X ⟶ Y is a trivial fibration in 𝒞, and suppose we have a commutative diagram

A_{n} ⨿_{GA_{n−1}} GB_{n−1} ⟶ X
↓  ↓^{p}
B_{n}  ⟶ Y

We must construct a lift in this diagram. By adjointness, it suffices to construct a lift in the induced diagram

A ⟶  R_{n}X
i↓  ↓
B ⟶ R_{n}Y ×_{R_{n−1}UY} R_{n−1}UX

where R_{n} is the right adjoint of Ev_{n}. Using the description of R_{n} given in Remark 2.4, one can easily check that the map R_{n}X ⟶ R_{n}Y ×_{R_{n−1}UY} R_{n−1}UX is a level trivial fibration, so a lift exists.

Conversely, suppose that the map i satisfies the conditions in the statement of the proposition. Suppose p: X ⟶ Y is a level trivial fibration in 𝒞_{G}, and suppose the diagram

A ⟶^{f} X
i↓  ↓^{p}
B ⟶_{g} Y

commutes. We construct a lift h_{n}: B_{n} ⟶ X_{n}, compatible with the structure maps, by induction on n. There is no difficulty defining h_{0}, since i_{0} has the left lifting property with respect to the trivial fibration p_{0}. Suppose we have defined h_{j} for j < n. Then by lifting in the induced diagram

A_{n} ⨿_{GA_{n−1}} GB_{n−1} ⟶^{(f_{n},σ∘Gh_{n−1})} X_{n}
↓  ↓^{p_{n}}
B_{n}  ⟶_{g_{n}}  Y_{n}

we find the required map h_{n}: B_{n} ⟶ X_{n}.

Finally, we point out that the extension G is still a Quillen functor.

Proposition 3.9. Give 𝒞_{G} the projective model structure. Then the extension of G to an endofunctor of 𝒞_{G} is a Quillen functor. Furthermore, the functor F_{n}: 𝒞 ⟶ 𝒞_{G} is a Quillen functor.

Proof. The functor Ev_{n} obviously takes level fibrations to fibrations and level trivial fibrations to trivial fibrations. Hence Ev_{n} is a right Quillen functor, and so its left adjoint F_{n} is a left Quillen functor. Similarly, the functor U: 𝒞_{G} ⟶ 𝒞_{G} preserves level fibrations and level trivial fibrations, so its left adjoint G is a Quillen functor.

4. Bousfield localization

We will define the stabilization Sp^{ℕ}(𝒞, G) in Section 5 as a Bousfield localization of the projective model structure on 𝒞_{G}. In this section we recall the theory of Bousfield localization of model categories from [Hir98].

To do so, we need some preliminary remarks related to function complexes. Details can be found in [Hov98a, Chapter 5], [DHK], and [Hir98, Chapter 17]. Given an object A in a model category 𝒞, there is a functorial cosimplicial resolution of A induced by the functorial factorizations of 𝒞. By mapping out of this cosimplicial resolution we get a simplicial set Map_{ℓ}(A, X). Similarly, there is a functorial simplicial resolution of X, and by mapping into it we get a simplicial set Map_{r}(A, X). One should think of these as replacements for the simplicial structure present in a simplicial model category. Let us define the homotopy function complex map(A, X) = Map_{r}(QA, RX). Then map(A, X) is canonically isomorphic in the homotopy category of simplicial sets to Map_{ℓ}(QA, RX), and makes Ho 𝒞 an enriched category over the homotopy category Ho SSet of simplicial sets. In fact, this functor is part of a functorial closed action of Ho SSet on Ho 𝒞. In particular, if G is an arbitrary left Quillen functor between model categories with right adjoint U, we have map((LG)X, Y) ≅ map(X, (RU)Y) in Ho SSet, where (LG)X = GQX is the total left derived functor of G and (RU)Y = URY is the total right derived functor of U.

Definition 4.1. Suppose we have a set S of maps in a model category 𝒞.

1. A S-local object of 𝒞 is a fibrant object W such that, for every f: A ⟶ B in S, the induced map map(B, W) ⟶ map(A, W) is a weak equivalence of simplicial sets.

2. A S-local equivalence is a map g: A ⟶ B in 𝒞 such that the induced map map(B, W) ⟶ map(A, W) is a weak equivalence of simplicial sets for all S-local objects W.

Note that S-local equivalences between S-local objects are in fact weak equivalences.

Then the main theorem of [Hir98] is the following. We will define cellular model categories, a special class of cofibrantly generated model categories, in Section 6.

Theorem 4.2. Suppose S is a set of maps in a left proper cellular model category 𝒞. Then there is a left proper cellular model structure on 𝒞 where the weak equivalences are the S-local equivalences and the cofibrations remain unchanged. The S-local objects are the fibrant objects in this model structure. We denote this new model category by L_{S}𝒞 and refer to it as the Bousfield localization of 𝒞 with respect to S. Left Quillen functors from L_{S}𝒞 to 𝒟 are in one to one correspondence with left Quillen functors F: 𝒞 ⟶ 𝒟 such that F (Qf) is a weak equivalence for all f ∈ S.

We will also need the following fact about localizations, which is implicit in [Hir98, Chapter 4].

Proposition 4.3. Suppose 𝒞 and 𝒟 are left proper cellular model categories, S is a set of maps in 𝒞, and T is a set of maps in 𝒟. Suppose F: 𝒞 ⟶ 𝒟 is a Quillen equivalence with right adjoint U, and suppose F (Qf) is a T -local equivalence for all f ∈ S. Then F induces a Quillen equivalence F: L_{S}𝒞 ⟶ L_{T} 𝒟 if and only if, for every S-local X ∈ 𝒞, there is a T -local Y in 𝒟 such that X is weakly equivalent in 𝒞 to UY. This condition will hold if, for all fibrant Y in 𝒟 such that UY is S-local, Y is T -local.

Proof. Suppose first that F does induce a Quillen equivalence on the localizations, and suppose that X is S-local. Then QX is also S-local. Let L_{T} denote a fibrant replacement functor in L_{T} 𝒟. Then, by Lemma 1.11, QX ⟶ UL_{T} F QX is a weak equivalence in L_{S}𝒞. But both QX and UL_{T} F QX are S-local, so QX ⟶ UL_{T} F QX is a weak equivalence in 𝒞. Hence X is weakly equivalent in 𝒞 to UY, where Y is the T -local object L_{T} F QX.

Before proving the converse, note that, since F is a Quillen equivalence before localizing, the map F QUX ⟶ X is a weak equivalence for all fibrant X. Since the functor Q does not change upon localization, and every T -local object of 𝒟 is in particular fibrant, this condition still holds after localization. Thus F is a Quillen equivalence after localization if and only if F reflects local equivalences between cofibrant objects, by Lemma 1.11.

Suppose f: A ⟶ B is a map between cofibrant objects such that F f is a T -local equivalence. We must show that map(f, X) is a weak equivalence for all S-local X. Adjointness implies that map(f, UY) is a weak equivalence for all T -local Y, and our condition then guarantees that this is enough to conclude that map(f, X) is a weak equivalence for all S-local X.

We still need to prove the last statement of the proposition. So suppose X is S-local. Then QX is also S-local, and, in 𝒞, we have a weak equivalence QX ⟶ URF QX. Our assumption then guarantees that Y = RF QX is T -local, and X is in indeed weakly equivalent to UY.

The fibrations in L_{S}𝒞 are not completely understood [Hir98, Section 6.1]. The S-local fibrations between S-local fibrant objects are just the usual fibrations. In case both 𝒞 and L_{S}𝒞 are right proper, there is a characterization of the S-local fibrations in terms of homotopy pullbacks analogous to the characterization of stable fibrations of spectra in [BF78]. However, L_{S}𝒞 need not be right proper even if 𝒞 is, as is shown by the example of Γ-spaces in [BF78], where it is also shown that the expected characterization of S-local fibrations does not hold.

5. The stable model structure

Our plan now is to apply Bousfield localization to the projective model structure on 𝒞_{G} to obtain a model structure with respect to which G is a Quillen equivalence. In order to do this, we will have to prove that 𝒞_{G} is a cellular model category when 𝒞 is left proper cellular. We will do this in the next section. In this section, we will assume that 𝒞_{G} is cellular, find a good set S of maps to form the localization L_{S}𝒞_{G}, and prove that G is a Quillen equivalence on Sp^{ℕ}(𝒞, G) = L_{S}𝒞_{G}.

Definition 5.1. A spectrum X is a U-spectrum if X is level fibrant and the adjoint X_{n} \underrightarrow{σ̃} UX_{n+1} of the structure map is a weak equivalence for all n.

Of course, if 𝒞 is the category of pointed simplicial sets or pointed topological spaces, and G is the suspension functor, U-spectra are just Ω-spectra. We will find a set S of maps of 𝒞_{G} such that the S-local fibrant objects are the U-spectra. To do so, note that if map(A, X_{n}) ⟶ map(A, UX_{n+1}) is a weak equivalence of simplicial sets for all cofibrant A in 𝒞, then X_{n} ⟶ UX_{n+1} will be a weak equivalence as required. Since 𝒞 is cofibrantly generated, we should not need all cofibrant A, but only those A related to the generating cofibrations.

Proposition 5.2. Suppose 𝒞 is a left proper cofibrantly generated model category with generating cofibrations I, and f: X ⟶ Y is a map. Then f is a weak equivalence if and only if map(C, X) ⟶ map(C, Y) is a weak equivalence for all domains and codomains C of maps of I.

Proof. The only if half is clear. Since every cofibrant object is a retract of a cell complex (i.e. an object A such that the map 0 ⟶ A is a transfinite composition of pushouts of maps of I), it suffices to show that map(A, f), or, equivalently, Map_{r}(A, Rf), is a weak equivalence for all cell complexes A. Given a cell complex A, there is an ordinal λ and a λ-sequence

0 = A_{0} ⟶ A_{1} ⟶ … ⟶ A_{β} ⟶ …

with colimit A_{λ} = A. We will show by transfinite induction on β that Map_{r}(A_{β}, Rf) is a weak equivalence for all β ≤ λ. Since A_{0} = 0, getting started is easy. For the successor ordinal case of the induction, suppose map(A_{β}, f) is a weak equivalence. We have a pushout square

C ⟶ A_{β}
g↓  ↓^{i_{β}}
D ⟶ A_{β+1}

where g is a map of I. Factor the composite QC ⟶ C ⟶ D into a cofibration g̃: QC ⟶ D̃ followed by a trivial fibration D̃ ⟶ D. In the terminology of [Hir98], g̃ is a cofibrant approximation to g. By [Hir98, Proposition 11.3.2], there is a cofibrant approximation ĩ_{β}: Ã_{β} ⟶ { \atop A_{β+1}} to i_{β} which is a pushout of g̃. For any fibrant object Z, the functor Map_{r}(−, Z) converts colimits to limits and cofibrations to fibrations [Hov98a, Corollary 5.4.4]. Hence we have two pullback squares of fibrant simplicial sets

Map_{r}({ \atop A_{β+1}}, RZ) ⟶ Map_{r}(D̃, RZ)
↓  ↓
Map_{r}(Ã_{β}, RZ) ⟶ Map_{r}(QC, RZ)

where Z = X and Z = Y, respectively. Here the vertical maps are fibrations. There is a map from the square with Z = X to the square with Z = Y induced by f. By hypothesis, this map is a weak equivalence on every corner except possibly the upper left. But then Dan Kan’s cube lemma (see [Hov98a, Lemma 5.2.6], where the dual of the version we need is proved, or [DHK]) implies that the map on the upper left corner is also a weak equivalence, and hence that Map_{r}(A_{β+1}, Rf) is a weak equivalence.

Now suppose β is a limit ordinal and Map_{r}(A_{γ}, Rf) is a weak equivalence for all γ < β. Then, for Z = X or Z = Y, the simplicial sets Map_{r}(A_{γ}, RZ) define a limit-preserving functor β^{op} ⟶ SSet such that each map Map_{r}(A_{γ+1}, RZ) ⟶ Map_{r}(A_{γ}, RZ) is a fibration of fibrant simplicial sets. There is a natural transformation from the functor with Z = X to the functor with Z = Y, and by hypothesis this map is a weak equivalence at every stage. It follows that it is a weak equivalence on the inverse limits, as one can see in different ways. The simplest is probably to note that the inverse limit is a right Quillen functor [Hov98a, Corollary 5.1.6]. Thus Map_{r}(A_{β}, Rf) is a weak equivalence, as required. This completes the transfinite induction and the proof.

Note that the left properness assumption in Proposition 5.2 is unnecessary when the domains of the generating cofibrations are themselves cofibrant, since there is then no need to apply cofibrant approximation.

In view of Proposition 5.2, we need to choose our set S so as to make

map(C, X_{n}) ⟶ map(C, UX_{n+1})

a weak equivalence for all S-local objects X and all domains and codomains C of the generating cofibrations I. Adjointness implies that, if X is level fibrant, map(C, X_{n}) ≅ map(F_{n}QC, X) in Ho 𝒞_{G}, since F_{n}QC = (LF_{n})C, where LF_{n} is the total left derived functor of F_{n}. Similarly, map(C, UX_{n+1}) ≅ map(F_{n+1}GQC, X). In view of this, we make the following definition.

Definition 5.3. Suppose 𝒞 is a left proper cellular model category with generating cofibrations I, and G is a left Quillen endofunctor of 𝒞. Define the set S of maps

in 𝒞_{G} as {F_{n+1}GQC ⟶^{s_{n}^{QC}} F_{n}QC}, as C runs through the set of domains and codomains of the maps of I and n runs through the non-negative integers. Here the map s_{n}^{QC} is adjoint to the identity map of GQC. Define the model category Sp^{ℕ}(𝒞, G) = L_{S}𝒞_{G} to be the category 𝒞_{G} given the S-local model structure. We refer to this model structure as the stable model structure on the category of spectra. We refer to the S-local weak equivalences as stable equivalences, and to the S-local fibrations as stable fibrations.

Theorem 5.4. Suppose 𝒞 is a left proper cellular model category and G is a left Quillen endofunctor of 𝒞. Then the fibrant objects in Sp^{ℕ}(𝒞, G) are the U-spectra.

Furthermore, for all cofibrant A ∈ 𝒞 and for all n ≥ 0, the map F_{n+1}GA ⟶^{s_{n}^{A}} F_{n}A is a stable equivalence.

Proof. By definition, X is S-local if and only if X is level fibrant and

map(F_{n}QC, X) ⟶ map(F_{n+1}GQC, X)

is a weak equivalence for all n ≥ 0 and all domains and codomains C of maps of I. By the comments preceding Definition 5.3, this is equivalent to requiring that X be level fibrant and that the map map(C, X_{n}) ⟶ map(C, UX_{n+1}) be a weak equivalence for all n ≥ 0 and all domains and codomains C of maps of I. By Proposition 5.2, this is equivalent to requiring that X be a U-spectrum.

Now, by definition, s_{n}^{A} is a stable equivalence if and only if map(s_{n}^{A}, X) is a weak equivalence for all U-spectra X. But by adjointness, map(s_{n}^{A}, X) can be identified with map(A, X_{n}) ⟶ map(A, UX_{n+1}). Since X_{n} ⟶ UX_{n+1} is a weak equivalence between fibrant objects, so is map(s_{n}^{A}, X).

Corollary 5.5. If 𝒞 is either the category of pointed simplicial sets or pointed topological spaces, and G is the suspension functor given by smashing with S^{1}, then Sp^{ℕ}(𝒞, G) coincides with the stable model structure on the category of Bousfield-Friedlander spectra [BF78].

Proof. We know already that the cofibrations are the same in Sp^{ℕ}(𝒞, G) and the stable model structure of [BF78]. We will show that the weak equivalences are the same. In any model category at all, a map f is a weak equivalence if and only if map(f, X) is a weak equivalence of simplicial sets for all fibrant X. Construction of map(f, X) requires replacing f by a cofibrant approximation f′ and building cosimplicial resolutions of the domain and codomain of f′. In the case at hand, we can do the cofibrant replacement and build the cosimplicial resolutions in 𝒞_{G}, the strict model category of spectra, since the cofibrations do not change under localization. Thus map(f, X) is the same in both Sp^{ℕ}(𝒞, G) and in the stable model category of spectra. Since the fibrant objects are also the same, the corollary holds.

Corollary 5.6. Suppose 𝒞 is a left proper cellular model category and G is a left Quillen endofunctor of 𝒞. Then the functor G: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) is a Quillen functor.

Proof. In view of Hirschhorn’s localization theorem 4.2, we must show that G(Qf) is a stable equivalence for all f ∈ S. Since the domains and codomains of the maps of S are already cofibrant, it is equivalent to show that Gf is a stable equivalence for all f ∈ S. Since GF_{n} = F_{n}G, we have Gs_{n}^{A} = s_{n}^{GA}. In view of Theorem 5.4, this map is a weak equivalence whenever A, and hence GA, is cofibrant. Taking A = QC, where C is a domain or codomain of a map of I, completes the proof.

We will now show that G is in fact a Quillen equivalence on Sp^{ℕ}(𝒞, G). To do so, we introduce the shift functors.

Definition 5.7. Suppose 𝒞 is a model category and G is a left Quillen endofunctor of 𝒞. Define the shift functors t: 𝒞_{G} ⟶ 𝒞_{G} and s: 𝒞_{G} ⟶ 𝒞_{G} by (sX)_{n} = X_{n+1} and (tX)_{n} = X_{n−1}, (tX)_{0} = 0, with the evident structure maps. Note that t is left adjoint to s.

It is clear that s preserves level equivalences and level fibrations, so t is a left Quillen functor on 𝒞_{G} and s is a right Quillen functor. Also, sU = Us, so tG = Gt. Similarly, Ev_{n} s = Ev_{n+1}, so tF_{n} = F_{n+1}. It follows that ts_{n}^{A} = s_{n+1}^{A}, so that t is a Quillen functor on Sp^{ℕ}(𝒞, G) as well as on 𝒞_{G}.

The main advantage of the shift functor is that there is a natural map X ⟶ sUX, adjoint to the structure map of X, and X is a U-spectrum if and only if X is level fibrant and this map is a level equivalence.

Theorem 5.8. Suppose 𝒞 is a left proper cellular model category and G is a left Quillen endofunctor of 𝒞. Then the functors G: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) and t: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) are Quillen equivalences. Furthermore, Rs is naturally isomorphic to LG, and RU is naturally isomorphic to Lt.

Proof. There is a a natural map X ⟶ sUX which is a weak equivalence when X is a fibrant object of Sp^{ℕ}(𝒞, G). This means that the total right derived functor R(sU) is naturally isomorphic to the identity functor on Ho Sp^{ℕ}(𝒞, G). On the other hand, R(sU) is naturally isomorphic to Rs ∘ RU and also to RU ∘ Rs, since s and U commute with each other. Thus the natural isomorphism from the identity to R(sU) gives rise to an natural isomorphism 1 ⟶ Rs∘RU and a natural isomorphism RU ∘Rs ⟶ 1, displaying RU and Rs as adjoint equivalences of categories. It follows that U and s are both Quillen equivalences, as required. Since RU is adjoint to LG and Rs is adjoint to Lt, we must also have RU naturally isomorphic to Lt and Rs naturally isomorphic to LG.

6. Cellular model categories

In this section we define cellular model categories and show that 𝒞_{G} is cellular when 𝒞 is so. This is necessary to be sure that the Bousfield localization used in the last section does in fact exist. The definitions in this section are taken from [Hir98].

A cellular model category is a special kind of cofibrantly generated model category. Three additional hypotheses are needed.

Definition 6.1. A model category 𝒞 is cellular if there is a set of cofibrations I and a set of trivial cofibrations J making 𝒞 into a cofibrantly generated model category and also satisfying the following conditions.

1. The domains and codomains of I are compact relative to I.

2. The domains of J are small relative to the cofibrations.

3. Cofibrations are effective monomorphisms.

The first and third hypotheses above require considerable explanation, which we will provide below. We first point out that the second hypothesis will hold in 𝒞_{G} when it holds in 𝒞.

Lemma 6.2. Suppose 𝒞 is a cofibrantly generated model category with generating cofibrations I and generating trivial cofibrations J, and G is a left Quillen endofunctor of 𝒞. If the domains of J are small relative to the cofibrations in 𝒞, then the domains of the generating trivial cofibrations J_{G} of 𝒞_{G} are small relative to the cofibrations in 𝒞_{G}.

Proof. This follows immediately from the definition of J_{G}, Proposition 3.3, and Proposition 3.5.

We now discuss the third hypothesis.

Definition 6.3. Suppose 𝒞 is a category. A map f: X ⟶ Y is an effective monomorphism if f is the equalizer of the two obvious maps Y ⇉ Y ⨿_{X} Y.

In [Hir98, Chapter 15], it is only assumed that inclusions of relatve cell complexes (i.e. transfinite compositions of pushouts of I) are effective monomorphisms. But it is equivalent to require that all cofibrations be monomorphisms, in view of the following lemma.

Lemma 6.4. Suppose f: A ⟶ B is a retract of g: C ⟶ D, and g is an effective monomorphism. Then f is an effective monomorphism.

Proof. Let f′: A′ ⟶ B denote the equalizer of the obvious pair of maps B ⇉ B ⨿_{A} B. Since f does equalize these two maps, there is a map α: A ⟶ A′ such that f′α = f. We will show that α is an isomorphism.

Since f is a retract of g, we have a commutative diagram

A ⟶^{f} B
i↓  ↓^{j}
C ⟶^{g} D
r↓  ↓^{s}
A ⟶_{f} B

where the vertical composites are identity maps. It follows easily that the maps B ⇉ B ⨿_{A} B are retracts of the corresponding maps D ⇉ D ⨿_{C} D. Since g is an effective monomorphism, we therefore have a map i′: A′ ⟶ C such that i′α = i. Furthermore, we have jf′ = gi′.

Let β: A′ ⟶ A be the composite ri′. Then we have βα = ri′α = ri = 1_{A}. To check that αβ = 1_{A′}, it suffices to show that f′αβ = f′, since f′, being an equalizer, is in particular a monomorphism. But we have f′αβ = fri′ = sgi′ = sjf′ = f′, as required.

Proposition 6.5. Suppose 𝒞 is a cofibrantly generated model category and G is a left Quillen endofunctor of 𝒞. If cofibrations are effective monomorphisms in 𝒞, then level cofibrations, and in particular projective cofibrations, are effective monomorphisms in 𝒞_{G}.

Proof. This is immediate, since limits in 𝒞_{G} are taken levelwise.

We must now define compactness. This will involve some preliminary definitions. These definitions are extremely technical; the reader is advised to keep the example of CW-complexes in mind.

Definition 6.6. Suppose I is a set of maps in a cocomplete category. A relative I-cell complex is a map which can be written as the transfinite composition of pushouts of coproducts of maps of I. That is, given a relative I-cell complex f, there is an ordinal λ and a λ-sequence X: λ ⟶ 𝒞 and a collection {(T^{β}, e^{β}, h^{β})_{β<λ}} satisfying the following properties.

1. f is isomorphic to the transfinite composition of X.

2. Each T^{β} is a set.

3. Each e^{β} is a function e^{β}: T^{β} ⟶ I.

4. Given β < λ and i ∈ T^{β}, if e_{i}^{β}: C_{i} ⟶ D_{i} is the image of i under e^{β}, then h_{i}^{β} is a map h_{i}^{β}: C_{i} ⟶ X_{β}.

5. Each X_{β+1} is the pushout in the diagram

∐_{T^{β}} C_{i} ⟶ ∐_{T^{β}} D_{i}
∐ h_{i}^{β} ↓  ↓
X_{β} ⟶ X_{β+1}

The ordinal λ together with the λ-sequence X and the collection {(T^{β}, e^{β}, h^{β})_{β<λ}} is called a presentation of f. The set ∐_{β} T_{β} is the set of cells of f, and given a cell e, its presentation ordinal is the ordinal β such that e ∈ T^{β}. The presentation ordinal of f is λ.

We also need to define subcomplexes of relative I-cell complexes.

Definition 6.7. Suppoe 𝒞 is a cocomplete category and I is a set of maps in 𝒞. Given a presentation λ, X: λ ⟶ 𝒞, and {(T^{β}, e^{β}, h^{β})_{β<λ}} of a map f as a relative I-cell complex, a subcomplex of f (or really of the presentation of f), is a collection {(T̃^{β}, e^{β}, h̃^{β})_{β<λ}} such that the following properties hold.

1. Every T̃^{β} is a subset of T^{β}, and e^{β} is the restriction of e^{β} to T̃^{β}.

2. There is a λ-sequence X̃: λ ⟶ 𝒞 such that X̃_{0} = X_{0} and a map of λ-sequences X̃ ⟶ X such that, for every β < λ and i ∈ T̃^{β}, the map h̃_{i}^{β}: C_{i} ⟶ X̃_{β} is a factorization of h_{i}^{β}: C_{i} ⟶ X_{β} through the map X̃_{β} ⟶ X_{β}.

3. Every X_{β+1} is the pushout in the diagram

∐T̃^{β} C_{i} ⟶ ∐T̃^{β} D_{i}
∐ h̃_{i}^{β} ↓  ↓
X̃_{β} ⟶ X̃_{β+1}

Given a subcomplex of f, the size of that subcomplex is the cardinality of its set of cells ∐_{β<λ} T̃^{β}. Usually, I will be a set of cofibrations in a model category where the cofibrations are essential monomorphisms. This condition guarantees that a subcomplex is uniquely determined by its set of cells [Hir98, Proposition 12.5.9]. Of course, every set of cells does not give rise to a subcomplex.

We can now define compactness.

Definition 6.8. Suppose 𝒞 is a cocomplete category and I is a set of maps in 𝒞.

1. Given a cardinal κ, an object X is κ-compact relative to I if, for every relative I-cell complex f: Y ⟶ Z and for every presentation of f, every map X ⟶ Z factors through a subcomplex of size at most κ.

2. An object X is compact relative to I if X is κ-compact relative to I for some cardinal κ.

The following proposition is adapted from an argument of Phil Hirschhorn’s.

Proposition 6.9. Suppose 𝒞 is a cellular model category with generating cofibrations I, and G is a left Quillen endofunctor of 𝒞. Let A be a domain or codomain of I. Then F_{n}A is compact relative to I_{G} in 𝒞_{G}.

Proof. Throughout this proof we will use Proposition 6.5, which guarantees that subcomplexes in 𝒞_{G} are determined by their cells. Choose an infinite cardinal γ such that the domains and codomains of I are all γ-compact relative to I. When dealing with relative I-cell complexes, we can assume that we have a presentation as a transfinite composition of pushouts of maps of I, rather than as a transfinite composition of pushouts of coproducts of maps of I, using [Hov98a, Lemma 2.1.13] or [Hir98, Section 12.2]. A similar comment holds for relative I_{G}-cell complexes. We will proceed by transfinite induction on β, where the induction hypothesis is that for every presented relative I_{G}-cell complex f: X ⟶ Y whose presentation ordinal is ≤ β, and for every map F_{n}A ⟶^{f} Y where n is an integer and A is a domain or codomain of I, f factors through a subcomplex with at most γ I_{G}-cells. Getting the induction started is easy. For the induction step, suppose the induction hypothesis holds for all ordinals α < β, and suppose we have a presentation

X = X^{0} ⟶ X^{1} ⟶ … ⟶ X^{α} ⟶ … X^{β} = Y

of f: X ⟶ Y as a transfinite composition of pushouts of maps of I_{G}. Then the boundary of each I_{G}-cell of this presentation is represented by a map F_{m}C ⟶ X^{α}, for some α < β, some m ≥ 0, and some domain C of a map of I. This map factors through a subcomplex with at most γ I_{G}-cells, by induction. It follows that the I_{G}-cell itself is contained in a subcomplex of at most γ I_{G}-cells, since we can just attach the interior of the I_{G}-cell to the given subcomplex.

Now suppose we have an arbitrary map F_{n}A ⟶ Y, where A is a domain or codomain of a map of I. Such a map is determined by a map A ⟶ Y_{n} in 𝒞. The map f_{n}: X_{n} ⟶ Y_{n} is the transfinite composition of the cofibrations X_{n}^{α} ⟶ X_{n}^{α+1}. For each α, there is an m and a map h of I such that X_{n}^{α} ⟶ X_{n}^{α+1} is the pushout of G^{m}h, where we interpret G^{m}h as the identity map if m is negative. Apply [Hir98, Lemma 12.4.19] to write the β-sequence X_{n}^{α} as a retract of a β-sequence

X_{n} = Z^{0} ⟶ Z^{1} ⟶ … ⟶ Z^{α} ⟶ … Z^{β} = Z

where each map Z^{α} ⟶ Z^{α+1} is a relative I-cell complex. We denote the retraction by r: Z ⟶ Y_{n}, noting that the restriction of r to Z^{α} factors (uniquely) through X_{n}^{α}. We can think of the entire map X_{n} ⟶ Z as a relative I-cell complex, each cell e of which appears in the relative I-cell complex Z^{t(e)} ⟶ Z^{t(e)+1} for some unique ordinal t(e), and so has associated to it the I_{G}-cell c(e) of f used to form X^{t(e)} ⟶ X^{t(e)+1}. The composite A ⟶ Y_{n} ⟶ Z then factors through a subcomplex V with at most γ I-cells. The proof will be completed if we can find a subcomplex W of Y with at most γ I_{G}-cells such that the restriction of r to V factors through W_{n}.

Take W to be a subcomplex of Y containing the I_{G}-cells c(e) as e runs through the cells of V. Then W_{n} contains r(e) for every cell of V, so W_{n} contains rV, as required. Furthermore, since each I_{G}-cell c(e) lies in a subcomplex with no more that γ I_{G}-cells, and V has no more than γ cells, there is a choice for W which has no more than γ^{2} = γ cells. This completes the induction step and the proof.

Altogether then, we have the following theorem.

Theorem 6.10. Suppose 𝒞 is a left proper cellular model category, and G is a left Quillen endofunctor on 𝒞. Then the category 𝒞_{G} of G-spectra is also a left proper cellular model category.

7. Properties of the stabilization functor

In this section we explore some of the properties of the correspondence (𝒞, G) ↦ Sp^{ℕ}(𝒞, G). We begin by showing that, if G is already a Quillen equivalence, then the embedding 𝒞 ⟶ Sp^{ℕ}(𝒞, G) is a Quillen equivalence. We show that Sp^{ℕ}(𝒞, G) is functorial in the pair (𝒞, G), with a suitable definition of maps of pairs. Under mild hypotheses, we show that Sp^{ℕ}(𝒞, G) preserves Quillen equivalences in the pair (𝒞, G).

Theorem 7.1. Suppose 𝒞 is a left proper cellular model category, and suppose G is a left Quillen endofunctor of 𝒞 that is a Quillen equivalence. Then F_{0} : 𝒞 ⟶ Sp^{ℕ}(𝒞, G) is a Quillen equivalence.

Proof. We first point out that the right adjoint Ev_{0} : Sp^{ℕ}(𝒞, G) ⟶ 𝒞 reflects weak equivalences between fibrant objects. Indeed, suppose X and Y are U-spectra, and f: X ⟶ Y is a map such that Ev_{0} f = f_{0} is a weak equivalence. Then, because X and Y are U-spectra, U^{n}f_{n} is a weak equivalence for all n. Since G is a Quillen equivalence, U reflects weak equivalences between fibrant objects by Lemma 1.11. Thus f_{n} is a weak equivalence for all n, and so f is a level equivalence and hence a stable equivalence, as required.

In view of Lemma 1.11, to complete the proof it suffices to show that X ⟶ (L_{S}F_{0}X)_{0} is a weak equivalence for all cofibrant X ∈ 𝒞, where L_{S} denotes a fibrant replacement functor in Sp^{ℕ}(𝒞, G). Let R′ denote a fibrant replacement functor in 𝒞_{G}. Then X ⟶ (R′F_{0}X)_{0} is certainly a weak equivalence. We claim that R′F_{0}X is already a U-spectrum. Suppose for the moment that this is true; then by lifting we can construct a stable equivalence R′F_{0}X ⟶ L_{S}F_{0}X, and since R′F_{0}X is a Uspectrum, this map is in fact a level equivalence. Hence the map X ⟶ (L_{S}F_{0}X)_{0} is a weak equivalence, as required.

We have reduced ourselves to showing that R′F_{0}X is a U-spectrum. Since G is a Quillen equivalence, the map (F_{0}X)_{n} = G^{n}X ⟶ URG^{n+1}X = UR(F_{0}X)_{n+1} is a weak equivalence. By lifting, we can factor the weak equivalence (F_{0}X)_{n+1} ⟶ (R′F_{0}X)_{n+1} through the trivial cofibration (F_{0}X)_{n+1} ⟶ R(F_{0}X)_{n+1}. This implies that the map (F_{0}X)_{n} ⟶ U(R′F_{0}X)_{n+1} is a weak equivalence, and hence that R′F_{0}X is a U-spectrum, as required.

We now examine the functoriality of Sp^{ℕ}(𝒞, G).

Definition 7.2. Suppose 𝒞 and 𝒟 are left proper cellular model categories, G is a left Quillen endofunctor of 𝒞, and H is a left Quillen endofunctor of 𝒟. A map of pairs (F, τ): (𝒞, G) ⟶ (𝒟, H) is a left Quillen functor F: 𝒞 ⟶ 𝒟 and a natural transformation τ: F G ⟶ HF such that τ_{A} is a weak equivalence for all cofibrant A ∈ 𝒞

Note that there is an obvious associative and unital composition of maps of pairs.

Proposition 7.3. Suppose (F, τ): (𝒞, G) ⟶ (𝒟, H) is a map of pairs. Then there is an induced map of pairs (F′, τ′) = Sp^{ℕ}(F, τ): (Sp^{ℕ}(𝒞, G), G) ⟶ (Sp^{ℕ}(𝒟, H), H) such that F′ ∘ F_{n} = F_{n}F. This induced map of pairs is compatible with composition and identities.

Proof. Suppose G has right adjoint U, H has right adjoint V, and F has right adjoint W. The natural transformation τ induces a dual natural transformation Dτ: W V ⟶ UW. Define W_{G}: 𝒟_{H} ⟶ 𝒞_{G} by (W_{G}Y)_{n} = W Y_{n}, with structure maps adjoint to the composite

W Y_{n} ⟶^{W σ̃} W V Y_{n+1} ⟶^{Dτ} UW Y_{n+1}

where σ̃ is adjoint to the structure map of Y. The functor W_{G} is analogous to restriction in the theory of group representations, and we must now define the analog to induction F_{G}. To do so, first note that τ defines natural transformations τ^{q}: F G^{q} ⟶ H^{q}F for all q, by iteration. Define (F_{G}X)_{n} to be the coequalizer of the two maps

∐_{p+q+r=n} H^{p}F G^{q}X_{r} ⇉ ∐_{p+q=n} H^{p}F X_{q}

where the top map is induced by H^{p}F G^{q}X_{r} ⟶ H^{p}F X_{q+r} and the bottom map is

induced by H^{p}F G^{q}X_{r} ⟶^{H^{p}τ^{q}} H^{p+q}F X_{r}. To define the structure map of F_{G}X, note that the coequalizer diagram for H(F_{G}X)_{n} is just the subdiagram of the coequalizer diagram for (F_{G}X)_{n+1} consisting of all terms with a positive power of H. The inclusion of diagrams induces the desired structure map H(F_{G}X)_{n} ⟶ (F_{G}X)_{n+1}. We leave to the reader the exercise in adjointness required to prove that F_{G} is left adjoint to W_{G}.

The functor W_{G} clearly preserves level fibrations and level trivial fibrations, so F_{G}: 𝒞_{G} ⟶ 𝒟_{H} is a left Quillen functor. Also, since Ev_{n} W_{G} = W Ev_{n}, we have F_{n}F = F_{G}F_{n}. In particular, if A is cofibrant in 𝒞, then F_{G}s_{n}^{A} = F_{n+1}F GA ⟶ F_{n}F A. Since τ is a weak equivalence on A, F_{n+1}F GA is level equivalent to F_{n+1}HF A. Since F A is cofibrant, the map s_{n}^{F A} is a stable equivalence in 𝒟_{H} by Theorem 5.4, so the map F_{G}s_{n}^{A} is a stable equivalence in 𝒟_{H}. Thus F′ = F_{G}: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒟, H) is a left Quillen functor, by Theorem 4.2.

We define τ′ = τ_{G}: F_{G}G ⟶ HF_{G} by defining its adjoint Dτ_{G}: W_{G}X ⟶ UW_{G}. Indeed, Dτ_{G} is just the prolongation of the adjoint Dτ of τ. Since τ is a weak equivalence on all cofibrant objects of 𝒞, Dτ is a weak equivalence on all fibrant objects of 𝒟. To see this, note that τ induce a natural isomorphism in the homotopy category. Adjointness implies that Dτ also induces a natural isomorphism in the homotopy category, and it follows that Dτ is a weak equivlance on all fibrant objects of 𝒟. Thus Dτ_{G} will be a level equivalence on all level fibrant objects of 𝒟_{H},so τ_{G} is a level equivalence on all cofibrant objects of 𝒞_{G}. We leave it to the reader to check compatibility of Sp^{ℕ}(F, τ) with compositions and identities.

Proposition 7.3 and Theorem 7.1 give us a weak universal property of Sp^{ℕ}(𝒞, G).

Corollary 7.4. Suppose (F, τ): (𝒞, G) ⟶ (𝒟, H) is a map of pairs such that H is a Quillen equivalence. Then there is a functor F̃: Ho Sp^{ℕ}(𝒞, G) ⟶ 𝒟 such that F̃ ∘ LF_{0} = LF: Ho 𝒞 ⟶ Ho 𝒟.

Proof. By Proposition 7.3 there is a map of pairs

(F′, τ′): (Sp^{ℕ}(𝒞, G), G) ⟶ (Sp^{ℕ}(𝒟, H), H)

induced by (F, τ). By Theorem 7.1, F_{0} : 𝒟 ⟶ Sp^{ℕ}(𝒟, H) is a Quillen equivalence. Define F̃ to be the composite R Ev_{0} ∘LF′.

We would also like to know that Sp^{ℕ}(F, τ) is a Quillen equivalence of pairs when F is a Quillen equivalence.

Theorem 7.5. Suppose (F, τ): (𝒞, G) ⟶ (𝒟, H) is a map of pairs such that F is a Quillen equivalence. Suppose as well that either the domains of the generating cofibrations for 𝒞 can be taken to be cofibrant, or that τ_{X} is a weak equivalence for all X. Then, in the induced map of pairs (F′, τ′): (Sp^{ℕ}(𝒞, G), G) ⟶ Sp^{ℕ}(𝒟, H), the Quillen functor F′ is a Quillen equivalence.

Proof. We will first show that F_{G} is a Quillen equivalence before Bousfield localization. Use the same notation as in the proof of Proposition 7.3, so that W denotes the right adjoint of F. Then, since F is a Quillen equivalence, W reflects weak equivalences between fibrant objects, by Lemma 1.11. It follows that W_{G} reflects level equivalences between level fibrant objects. Hence to show that F_{G}: 𝒞_{G} ⟶ 𝒟_{H} is a Quillen equivalence, it suffices to show that X ⟶ W_{G}RF_{G}X is a level equivalence for all cofibrant X. Thus, it suffices to show that X_{n} ⟶ W R(F_{G}X)_{n} is a weak equivalence for all n and all cofibrant X. Since X_{n} is cofibrant and F is a Quillen equivalence, it suffices to show that F X_{n} ⟶ (F_{G}X)_{n} is a weak equivalence for all n and all cofibrant X. In fact, we can assume that X is an I_{G}-cell complex.

Write X as the colimit of a λ-sequence

0 = X^{0} ⟶ X^{1} ⟶ X^{2} ⟶ … ⟶ X^{β} ⟶ … ⟶ X^{λ} = X

where each map X^{β} ⟶ X^{β+1} is a pushout of a map of I_{G}. We will prove that, for all β ≤ λ, F X_{n}^{β} ⟶ (F_{G}X^{β})_{n} is a weak equivalence for all n, by transfinite induction on β. Getting started is easy. The limit ordinal part of the induction follows from [Hir98, Proposition 18.4.1], since each of the maps F X_{n}^{β} ⟶ F X_{n}^{β+1} and each of the maps (F_{G}X^{β})_{n} ⟶ (F_{G}X^{β+1})_{n} is a cofibration of cofibrant objects.

For the succesor ordinal part of the induction, suppose X^{β} ⟶ X^{β+1} is a pushout of the map F_{m}C ⟶^{F_{m}f} F_{m}D of I_{G}. Then we have a pushout diagram

F (F_{m}C)_{n} ⟶ F (F_{m}D)_{n}
↓  ↓
F X_{n}^{β} ⟶ F X_{n}^{β+1}

and another pushout diagram

(F_{G}F_{m}C)_{n} ⟶ (F_{G}F_{m}D)_{n}
↓  ↓
(F_{G}X^{β})_{n} ⟶ (F_{G}X^{β+1})_{n}

in 𝒟. Note that F (F_{m}C)_{n} = F G^{n−m}C, where we interpret G^{n−m}C to be the initial object if n < m. Similarly, (F_{G}F_{m}C)_{n} = (F_{m}F C)_{n} = H^{n−m}F C. Thus the natural transformation τ induces a map from the first of these pushout squares to the second. If C (and hence also D) is cofibrant, then this map of pushout squares is a weak equivalence at both the upper left and upper right corners. Or, if τ is a weak equivalence for all X, then again this map is a weak equivalence at both the upper left and upper right corners. It is also a weak equivalence at the lower left corner, by the induction hypothesis. Since the top horizontal map is a cofibration in 𝒟, Dan Kan’s cube lemma [Hov98a, Lemma 5.2.6] implies that the map is a weak equivalence on the lower right corner. This completes the induction.

We have now proved that F_{G}: 𝒞_{G} ⟶ 𝒟_{H} is a Quillen equivalence. In view of Proposition 4.3, to show that F′ = F_{G}: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒟, H) is a Quillen equivalence, we need to show that if Y is level fibrant in 𝒟_{H} and W Y is a Uspectrum, then Y is a V -spectrum. To see this, note that, since W Y is a Uspectrum, the natural map W Y_{n} ⟶ UW Y_{n+1} is a weak equivalence for all n. There is a natural transformation Dτ: W V ⟶ UW dual to τ. Furthermore, (Dτ)_{X} is a weak equivalence for all fibrant X, as we have seen in the proof of Proposition 7.3. Thus, the natural map W Y_{n} ⟶ W V Y_{n+1} is a weak equivalence for all n. Since W reflects weak equivalences between fibrant objects, it follows that Y is a V -spectrum, as required.

As an example of Theorem 7.5, suppose we take a pointed simplicial set K weakly equivalent to S^{1}. Then there is a weak equivalence K ⟶ RS^{1}, where R is the fibrant replacement functor. This induces a natural transformation of left Quillen functors τ: K ∧ − ⟶ RS^{1} ∧ −. In Theorem 7.5, take 𝒟 = 𝒞 equal to the model category of pointed simplicial sets, take F to be the identity, and take τ to be this natural transformation. Then we get a Quillen equivalence between the stable model categories of spectra obtained by inverting K and inverting RS^{1}. Therefore, the choice of simplicial circle does not matter, up to Quillen equivalence, for Bousfield-Friedlander spectra.

8. Monoidal model categories and modules over them

Most model categories that arise in nature have some extra structure, Often they are simplicial model categories, for example. In this section, we remind the reader of the definition of a monoidal model category (analogous to a ring) and of the definition of a module over a monoidal model category (analogous to a module). We then show that, if 𝒟 is a left proper cellular model category which is a module over the monoidal model category 𝒞, and if G is a left Quillen endofunctor of 𝒟 which is also a 𝒞-module functor, then, under mild hypotheses, Sp^{ℕ}(𝒟, G) is also a module over 𝒞. In particular, the stabilization of a left proper cellular simplicial model category is still a simplicial model category.

However, the stabilization of a monoidal model category is not itself a monoidal model category, in general. We address this problem in the following sections.

We will assume the reader is familiar with the basics of monoidal categories. Given a monoidal category 𝒞, we will always denote the unit by S^{0} (to aviod conflict with the set S used to form Sp^{ℕ}(𝒞, G)), and the monoidal product by ∧. Our monoidal categories will always be closed, which means that both functors X ∧ − and − ∧ X have right adjoints, which are as natural as possible.

Definition 8.1. Suppose 𝒞 is a monoidal category. Given maps f: A ⟶ B and g: C ⟶ D, we define the pushout product f □ g of f and g to be the map f □ g: (A ∧ D) ⨿_{A∧C} (B ∧ C) ⟶ B ∧ D induced by the commutative square

A ∧ C ⟶^{f∧1} B ∧ C
1∧g↓  ↓^{1∧g}
A ∧ D ⟶_{f∧1} B ∧ D

Definition 8.2. Suppose 𝒞 is both a model category and a closed monoidal category. Then 𝒞 is called a monoidal model category if both of the following conditions hold:

(a) If f and g are cofibrations, so is f □ g. Moreover, if either f or g is a weak equivalence as well, so is f □ g.

(b) If QS^{0} ⟶ S^{0} is a weak equivalence, where QS^{0} is cofibrant, then the induced maps QS^{0} ∧ X ⟶ S^{0} ∧ X ≅ X and X ∧ QS^{0} ⟶ X are weak equivalences for all cofibrant X.

The second condition is automatic in case the unit is cofibrant, but is necessary in general to be sure that Ho 𝒞 is a monoidal category. Note that, if X is cofibrant in a monoidal model category 𝒞, the functors X ∧ − and − ∧ X are left Quillen functors. Monoidal model categories are studied in [Hov98a], [SS97], and [Hov98b].

We have a similar definition of a module over a monoidal model category. Again we assume that the reader is familiar with the notion of a module over a monoidal category, and we point out that our modules will always be closed. That is, if 𝒞 is our monoidal category and 𝒟 is our (right) 𝒞-module, the functor X ∧ − has a right adjoint for all X ∈ 𝒟, and the functor −∧K has a right adjoint for all K ∈ 𝒞, and these adjoints are as natural as possible.

Note that, if 𝒟 is a module over 𝒞, we can define the pushout product f □ g for f ∈ 𝒟 and g ∈ 𝒞.

Definition 8.3. Suppose 𝒟 is a closed module over the monoidal model category 𝒞. Then 𝒟 is defined to be a 𝒞-model category if the following properties hold.

(a) If f is a cofibration in 𝒟 and g is a cofibration in 𝒞, then f □g is a cofibration in 𝒟. Moreover, if either f or g is a weak equivalence as well, so is f □ g.

(b) If QS^{0} ⟶ S^{0} is a weak equivalence in 𝒞, where QS^{0} is cofibrant, then the induced map X ∧ QS^{0} ⟶ X ∧ S^{0} ≅ X is a weak equivalence for all cofibrant X ∈ 𝒟.

Once again, the second condition is automatic if S^{0} is cofibrant in 𝒞, but not otherwise. Note that if 𝒞 is the category of simplicial sets, then a 𝒞-model category is just a simplicial model category.

A left Quillen functor F: 𝒟 ⟶ 𝒟′ between 𝒞-model categories is called a 𝒞-Quillen functor if there is a coherent natural isomorphism F X ∧ K ⟶ F (X ∧ K). In case 𝒞 is the category of simplicial sets, then a left 𝒞-Quillen functor is just a left simplicial Quillen functor.

We now come to the main result of this section.

Theorem 8.4. Suppose 𝒞 is a cofibrantly generated monoidal model category, where the domains of the generating cofibrations are cofibrant. Suppose 𝒟 is a left proper cellular 𝒞-model category, and that G is a left 𝒞-Quillen endofunctor of 𝒟. Then Sp^{ℕ}(𝒟, G) is again a 𝒞-model category, and the extension of G to an endofunctor on Sp^{ℕ}(𝒟, G) is 𝒞-Quillen equivalence.

Of course, the Quillen functors F_{n}: 𝒟 ⟶ Sp^{ℕ}(𝒟, G) will be 𝒞-Quillen functors as well.

Proof. We define the action of 𝒞 on 𝒟_{G} levelwise. That is, given X ∈ 𝒟_{G} and K ∈ 𝒞, we define (X ∧ K)_{n} = X_{n} ∧ K. The structure map is given by

G(X_{n} ∧ K) ≅ GX_{n} ∧ K ⟶ X_{n+1} ∧ K.

One can easily verify that this defines an associative unital action of 𝒞 on 𝒟_{G}. To see that the action is closed, define (X^{K})_{n} = X_{n}^{K}, with structure maps G(X_{n}^{K}) ⟶ X_{n+1}^{K} adjoint to the composite

G(X_{n}^{K}) ∧ K ≅ G(X_{n}^{K} ∧ K) ⟶^{G(ev)} GX_{n} ⟶ X_{n+1}

where ev: X_{n}^{K} ∧ K is the evaluation map, adjoint to the identity of X_{n}^{K}. Now, given X and Y in 𝒟_{G}, define Map(X, Y) ∈ 𝒞 to be the equalizer of the two maps

α, β: ∏_{n} Map(X_{n}, Y_{n}) ⟶ ∏_{n} Map(X_{n}, UY_{n+1})

where α is the product of the maps Map(X_{n}, Y_{n}) ⟶ Map(X_{n}, UY_{n+1}) induced by the adjoint of the structure map of Y, and β is the product of the maps

Map(X_{n+1}, Y_{n+1}) ⟶ Map(UX_{n+1}, UY_{n+1}) ⟶ Map(X_{n}, UY_{n+1}).

Here the first map exists since G is a 𝒞-Quillen functor, and the second map is induced by the structure map of X. These functors define the closed action of the monoidal category 𝒞 on 𝒟_{G}.

We first note that this action makes 𝒟_{G} into a 𝒞-model category. Indeed, one can easily check that F_{n}f □ g = F_{n}(f □ g). Thus, if f is one of the generating cofibrations of 𝒟_{G}, and g is a cofibration in 𝒞, then f □ g is a cofibration in 𝒟_{G}. It follows that f □ g is a cofibration for f an arbitrary cofibration of 𝒟_{G} (see [SS97, Lemma 2.3] and [HSS98, Corollary 5.3.5]). A similar argument shows that f □ g is a trivial cofibration in 𝒟_{G} if either f is a trivial cofibration in 𝒟_{G} or g is a trivial cofibration in 𝒞. If QS^{0} is a cofibrant replacement for S^{0} in 𝒞, and X is cofibrant in 𝒟_{G}, then each X_{n} is cofibrant in 𝒟, so the map X ∧QS^{0} ⟶ X is a level equivalence as required. Thus 𝒟_{G} is a 𝒟-model category.

To complete the proof that Sp^{ℕ}(𝒟, G) is a 𝒞-model category, we need to show that, if f is a stable trivial cofibration and g is a cofibration in 𝒞, then f□g is a stable equivalence. It suffices to check this for g: K ⟶ L one of the generating trivial cofibrations of 𝒞. In this case, by hypothesis, K and L are cofibrant in 𝒞. Thus the functor − ∧ K is a Quillen functor on 𝒟_{G}, and similarly for L. Furthermore, if s_{n}^{QC}: F_{n+1}GQC ⟶ F_{n}QC is an element of the set S, then s_{n}^{QC} ∧ K ≅ s_{n}^{QC∧K}, since G is a 𝒞-Quillen functor. In view of Theorem 5.4, the map s_{n}^{QC∧K} is a stable equivalence. Theorem 4.2 them implies that − ∧ K is a Quillen functor on Sp^{ℕ}(𝒟, G), and similarly for − ∧ L. Thus, if f is a stable trivial cofibration, so are f ∧ K and f ∧ L. It follows from the two out of three property that f □ g is a stable equivalence, as required.

Theorem 8.4 gives us a functorial stabilization.

Corollary 8.5. Suppose K is a cofibrant object of a cofibrantly generated symmetric monoidal model category 𝒞 where the domains and codomains of the generating cofibrations can be taken to be cofibrant. Given a left proper cellular 𝒞-model category 𝒟, define G to be the 𝒞-Quillen functor given by − ∧ K. Then the correspondence 𝒟 ↦ Sp^{ℕ}(𝒟, G) defines an endofunctor of the category of left proper cellular 𝒞-model categories.

Note that the “category” of left proper cellular 𝒟-model categories is not really a category, because the Hom-sets need not be sets. It is really a 2-category, and the correspondence 𝒟 ↦ Sp^{ℕ}(𝒟, G) is actually a 2-functor. See [Hov98a] for a description of this point of view on model categories.

Proof. Since 𝒞 is symmetric, − ∧ K is a 𝒞-Quillen functor. In fact, any 𝒞-Quillen functor defined on 𝒞 must be of the form − ∧ K for K the image of S^{0}. Given a left proper cellular 𝒞-model category 𝒟, we have seen in Theorem 8.4 that Sp^{ℕ}(𝒟, G) is a 𝒞-model category. Just as in Lemma 2.5, a 𝒞-Quillen functor H: 𝒟 ⟶ 𝒟′ induces a functor H: 𝒟_{G} ⟶ 𝒟′_{G}, as does the right adjoint V of H. Both H and V are defined levelwise, and are 𝒞-functors. It is easy to check that V preserves level fibrations and level trivial fibrations, so that H is a 𝒞-Quillen functor before localization. Furthermore, we have Hs_{n}^{QC} = s_{n}^{HQC}, so Theorem 5.4 and Theorem 4.2 imply that H is a 𝒞-Quillen functor after localization as well.

In particular, if we take 𝒞 to be pointed simplicial sets and K to be S^{1}, then we get a functorial stabilization of pointed left proper cellular simplicial model categories.

However, this stabilization is not as well-behaved as one would like. Suppose 𝒞 is a symmetric monoidal model category. Given a left 𝒞-Quillen functor G on 𝒞, one would like Sp^{ℕ}(𝒞, G) to be itself a symmetric monoidal model category. Then, given a 𝒞-model category 𝒟, one would like Sp^{ℕ}(𝒟, G) to be a Sp^{ℕ}(𝒞, G)-model category. Unfortunately, this is false. Indeed, consider the category 𝒞^{ℕ} of sequences from 𝒞. An object of 𝒞^{ℕ} is a sequence X_{n} of objects of 𝒞, and a map f: X ⟶ Y is a sequence of maps f_{n}: X_{n} ⟶ Y_{n}. Then 𝒞^{ℕ} is a symmetric monoidal category, where we define (X ∧ Y)_{n} = ∐_{p+q=n} X_{p} ∧ Y_{q}. The functor G defines a monoid T = (S^{0}, GS^{0}, G^{2}S^{0}, …, G^{n}S^{0}, …) in this category, using the fact that G is a 𝒞-module functor.

Lemma 8.6. Suppose 𝒞 is a symmetric monoidal model category and G is a left 𝒞-Quillen functor. Then 𝒞_{G} is the category of left modules over the monoid T = (S^{0}, GS^{0}, …, G^{n}S^{0}, …).

We leave the proof of this lemma to the reader. Note that the monoid T is not commutative in general, and therefore 𝒞_{G} can not be a symmetric monoidal category with unit T. Indeed, let K = GS ∈ 𝒞. Then T is commutative if and only if the commutativity isomorphism on K ∧K is the identity. This happens only very rarely.

For this reason, we need an alternative stabilization construction when G is a 𝒞-Quillen functor. This will be the goal of the rest of the paper.

9. Symmetric spectra

We have just seen that the stabilization functor Sp^{ℕ}(𝒞, G) is not good enough in case 𝒞 is a symmetric monoidal model category and G is a 𝒞-model category, because Sp^{ℕ}(𝒞, G) is not usually itself a symmetric monoidal model category. In this section, we begin the construction of a better stabilization functor Sp^{Σ}(𝒞, K) for this case. We will concentrate on the category theory in this section, leaving the model structures for the next section. The terms used for the algebra of symmetric monoidal categories and modules over them are all defined in [Hov98a, Section 4.1].

Through most of this section, then, 𝒞 will be a bicomplete closed symmetric monoidal category, 𝒟 will be a bicomplete (right) closed 𝒞-module category, and K will be an object of 𝒞. Note that any 𝒞-functor G on 𝒞 is of the form G(X) = X ∧K for some K, namely K = GS^{0}, so there is minimal loss of generality in considering only such functors. Because of this, we will drop the letter G from our notations and replace it with K.

This section is based on the symmetric spectra and sequences of [HSS98]. The main idea of symmetric spectra is that the commutativity isomorphism of 𝒞 makes K^{∧n} into a Σ_{n}-object of 𝒞, where Σ_{n} is the symmetric group on n letters. We must keep track of this action if we expect to get a symmetric monoidal category of K-spectra.

The following definition is [HSS98, Definition 2.1.1].

Definition 9.1. The category Σ = ∐_{n≥0} Σ_{n} has as objects the sets \overline{n} = \{1, 2, …, n\} for n ≥ 0, where \overline{0} = ∅. The morphisms of Σ are the isomorphisms of \overline{n}. Given a category 𝒞, a symmetric sequence in 𝒞 is a functor Σ ⟶ 𝒞. The category of symmetric sequences is the functor category 𝒞^{Σ}.

A symmetric sequence in 𝒞 is a sequence X_{0}, X_{1}, …, X_{n}, … of objects of 𝒞 with an action of Σ_{n} on X_{n}. It is sometimes more useful to consider a symmetric sequence as a functor from the category of finite sets and isomorphisms to 𝒞; since the category Σ is a skeleton of the category of finite sets and isomorphisms, there is no difficulty in doing so.

As a functor category, the category of symmetric sequences in 𝒞 is bicomplete if 𝒞 is so; limits and colimits are taken objectwise. Furthermore, if 𝒞 is a closed symmetric monoidal category, so is 𝒞^{Σ}, as explained in [HSS98, Section 2.1]. Recall that the monoidal structure is given by

(X ⊗ Y)(C) = ∐_{A∪B=C,A∩B=∅} X(A) ∧ Y (B)

where we think of X, Y, and X ⊗ Y as functors from finite sets to 𝒞. Equivalently, though less canonically, we have

(X ⊗ Y)_{n} = ∐_{p+q=n} Σ_{n} ×_{Σ_{p}×Σ_{q}} (X_{p} ∧ Y_{q}).

The unit of the monoidal structure is the symmetric sequence (S^{0}, 0, …, 0, …), where 0 is the initial object of 𝒞. To define the closed structure, we first define Hom_{Σ_{n}}(X, Y) for X, Y ∈ 𝒞^{Σ_{n}} in the usual way, as an equalizer of the two obvious maps Hom(X, Y) ⟶ Hom(X × Σ_{n}, Y). The closed structure is then given by

Hom(X, Y)_{k} = ∏_{n} Hom_{Σ_{n}}(X_{n}, Y_{n+k}).

If 𝒟 is a closed module over 𝒞, then 𝒟^{Σ} is a closed module over 𝒞^{Σ}. Indeed, the same definition works to define the action. The closed structures are defined analogously. That is, we first define Hom_{Σ_{n}}(K, X) for X ∈ 𝒟^{Σ_{n}} and K ∈ 𝒞^{Σ_{n}} as an appropriate equalizer. Then, for X ∈ 𝒟^{Σ} and K ∈ 𝒞^{Σ}, we define X_{k}^{K} = ∏_{n} Hom_{Σ_{n}}(K_{n}, X_{n+k}). The other adjoint Map(X, Y) is defined similarly. In the same way, if 𝒟 is a closed algebra over 𝒞, then 𝒟^{Σ} is a closed algebra over 𝒞^{Σ}.

Consider the free commutative monoid Sym(K) on the object (0, K, 0, …, 0, …) of 𝒞^{Σ}. One can easily check that Sym(K) is the symmetric sequence (S^{0}, K, K ∧ K, …, K^{∧n}, …) where Σ_{n} acts on K^{∧n} by the commutativity isomorphism, as in [HSS98, Section 4.4].

Definition 9.2. Suppose 𝒞 is a symmetric monoidal model category, 𝒟 is a 𝒞model category, and K is an object of 𝒞. The category of symmetric spectra 𝒟_{K}^{Σ}, is the category of modules in 𝒟^{Σ} over the commutative monoid Sym(K) in 𝒞^{Σ}. That is, a symmetric spectrum is a sequence of objects X_{n} ∈ 𝒟^{Σ_{n}} and Σ_{n}-equivariant maps X_{n} ∧ K ⟶ X_{n+1}, such that the composite

X_{n} ∧ K^{∧p} ⟶ X_{n+1} ∧ K^{∧p−1} ⟶ … ⟶ X_{n+p}

is Σ_{n} × Σ_{p}-equivariant for all n, p ≥ 0. A map of symmetric spectra is a collection of Σ_{n}-equivariant maps X_{n} ⟶ Y_{n} compatible with the structure maps of X and Y.

Because Sym(K) is a commutative monoid, the category 𝒞_{K}^{Σ} is a bicomplete closed symmetric monoidal category, with Sym(K) itself as the unit (see Lemma 2.2.2 and Lemma 2.2.8 of [HSS98]). We denote the monoidal structure by X ∧ Y = X ⊗_{Sym(K)} Y, and the closed structure by Hom_{Sym(K)}(X, Y). Similarly, 𝒟_{K}^{Σ} is a bicomplete closed module over 𝒞_{K}^{Σ} with action denoted X ∧ Y again, and, if 𝒟 is a 𝒞-monoidal model category, then 𝒟_{K}^{Σ} will be a closed algebra over 𝒞_{K}^{Σ}.

Of course, if we take 𝒞 = SSet_{∗} and K = S^{1}, we recover the definition of symmetric spectra given in [HSS98], except that we are using right modules instead of left modules.

Definition 9.3. Given n ≥ 0, the evaluation functor Ev_{n}: 𝒟_{K}^{Σ} ⟶ 𝒟 takes X to X_{n}. the evaluation functor has a left adjoint F_{n}: 𝒟 ⟶ 𝒟_{K}^{Σ}, defined by F_{n}X = F̃_{n}X ⊗ Sym(K), where F̃_{n}X is the symmetric sequence (0, …, 0, Σ_{n} × X, 0, …).

Note that F_{0}X = (X, X ∧ K, …, X ∧ K^{∧n}, …), and in particular F_{0}S^{0} = Sym(K). Also, if X ∈ 𝒟 and Y ∈ 𝒞, there is a natural isomorphism F_{n}X ∧ F_{m}Y ≅ F_{n+m}(X ∧ Y), just as in [HSS98, Proposition 2.2.6]. In particular, F_{0} : 𝒞 ⟶ 𝒞_{K}^{Σ} is a (symmetric) monoidal functor, and so 𝒟_{K}^{Σ} is naturally a closed 𝒞-module. In fact, this structure is very simple. Indeed, if X ∈ 𝒟_{K}^{Σ} and L ∈ 𝒞, X ∧ L is just the symmetric sequence whose nth term is X_{n} ∧ L, with the evident structure maps. Similarly, X^{L} is the symmetric sequence whose nth term is X_{n}^{L}.

Remark 9.4. Just as in the spectrum case, the functor Ev_{n} has a right adjoint R_{n}: 𝒟 ⟶ 𝒟_{K}^{Σ}. Indeed, R_{n}X = Hom(Sym(K), R̃_{n}L), where R̃_{n}L is the symmetric sequence concentrated in degree n whose nth term is X^{Σ_{n}}.

10. Model structures on symmetric spectra

Throughout this section, 𝒞 will denote a left proper cellular symmetric monoidal model category, 𝒟 will denote a left proper cellular 𝒞-model category, and K will denote a cofibrant object of 𝒞. In this section, we discuss the projective and stable model structures on the category 𝒟_{K}^{Σ} of symmetric spectra. The results in this section are very similar to the corresponding results for spectra, so we will leave most of the proofs to the reader.

Definition 10.1. A map f ∈ 𝒟_{K}^{Σ} is a level equivalence if each map f_{n} is a weak equivalence in 𝒟. Similarly, f is a level fibration (resp. level cofibration, level trivial fibration, level trivial cofibration) if each map f_{n} is a fibration (resp. cofibration, trivial fibration, trivial cofibration) in 𝒟. The map f is a projective cofibration if f has the left lifting property with respect to every level trivial fibration.

Then, just as in Definition 3.2, if we denote the generating cofibrations of 𝒟 by I and the generating trivial cofibrations by J, we define I_{K} = ⋃_{n} F_{n}I and J_{K} = ⋃_{n} F_{n}J.

We have analogues of 3.3–3.6 with the same proofs. This gives us the projective model structure.

Theorem 10.2. The projective cofibrations, the level fibrations, and the level equivalences define a left proper cellular model structure on 𝒟_{K}^{Σ}.

The set I_{K} is the set of generating cofibrations of the projective model structure, and J_{K} is the set of generating trivial cofibrations. The fact that the projective model structure is cellular depends on the analogue of Proposition 6.9, whose proof is virtually the same.

Note that Ev_{n} takes level (trivial) fibrations to (trivial) fibrations, so Ev_{n} is a right Quillen functor and F_{n} is a left Quillen functor.

Theorem 10.3. With respect to the projective model structures, the category 𝒞_{K}^{Σ} is a symmetric monoidal model category, and the category 𝒟_{K}^{Σ} is a 𝒞_{K}^{Σ}-model category.

Proof. We first show that the pushout product f □ g is a (trivial) cofibration when f is a cofibration in 𝒟_{K}^{Σ}, and g is a cofibration in 𝒞_{K}^{Σ} (and one of them is trivial). As explained in [Hov98a, Corollary 2.5]. we may as well assume that f and g belong to the sets of generating cofibrations or generating trivial cofibrations. In either case, we have f = F_{m}f′ and g = F_{n}g′. But then f □ g = F_{m+n}(f′ □ g′). Since F_{m+n} is a Quillen functor, the result follows.

Now let QS^{0} denote a cofibrant replacement for S^{0} in 𝒞. Then F_{0}QS^{0} is a cofibrant replacement for F_{0}S^{0} = Sym(K) in 𝒞_{K}^{Σ}. Indeed, F_{0}QS^{0} is cofibrant, and Ev_{n} F_{0}QS^{0} is just QS^{0} ∧ K^{∧n}. Since K is cofibrant and 𝒞 is a monoidal model category, the map F_{0}QS^{0} ⟶ F_{0}S^{0} is a level equivalence. Now, if X is cofibrant in 𝒟_{K}^{Σ}, then each X_{n} is cofibrant. Hence the map X_{n} ∧ QS^{0} ⟶ X_{n} is a weak equivalence for all n, and so the map X ∧ F_{0}QS^{0} ⟶ X is a level equivalence, as required.

The projective cofibrations are more complicated than they are in the case of ordinary spectra.

Definition 10.4. Define the symmetric spectrum \overline{Sym(K)} in 𝒞_{K}^{Σ} to be 0 in degree 0 and K^{∧n} in degree n, for n > 0, with the obvious structure maps. Define the nth latching space L_{n}X of X ∈ 𝒟_{K}^{Σ} by L_{n}X = Ev_{n}(X ∧ \overline{Sym(K)}).

The obvious map i: \overline{Sym(K)} ⟶ Sym(K) induces a Σ_{n}-equivariant natural transformation L_{n}X ⟶ X.

Note that the latching space is a Σ_{n}-object of 𝒟. There is a model structure on Σ_{n}-objects of 𝒟 where the fibrations and weak equivalences are the underlying ones. This model structure is cofibrantly generated: if I is the set of generating cofibrations of 𝒟, then the set of generating cofibrations of 𝒟^{Σ_{n}} is the set Σ_{n} × I. Here, for an object A, Σ_{n} × A is the coproduct of n! copies of A, given the obvious Σ_{n}-structure.

Proposition 10.5. A map f: X ⟶ Y in 𝒟_{K}^{Σ} is a projective (trivial) cofibration if and only if the induced map Ev_{n}(f □ i): X_{n} ⨿_{L_{n}X} L_{n}Y ⟶ Y_{n} is a (trivial) cofibration in 𝒟^{Σ_{n}} for all n.

Proof. The trivial cofibration case is analogous to the cofibration case. If each map X_{n} ⨿_{L_{n}X} L_{n}Y ⟶ Y_{n} is a cofibration, then we can show that f is a projective cofibration by showing f has the left lifting property with respect to level trivial fibrations. Indeed, we construct a lift by induction, just as in the proof of Proposition 3.8. To prove the converse, it suffices to show that Ev_{n}(f □i) is a Σ_{n}-cofibration for f ∈ I_{K}, since Ev_{n} is itself a left Quillen functor. Then we can write f = F_{m}g, and we find that Ev_{n}(f □i) is an isomorphism when n ≠ m, and is the map Σ_{m} ×g when n = m. This is a Σ_{m}-cofibration, as required.

We must now localize the projective model structure to obtain the stable model structure.

Definition 10.6. A symmetric spectrum X ∈ 𝒟_{K}^{Σ} is an Ω-spectrum if X is level fibrant and the adjoint X_{n} ⟶ X_{n+1}^{K} of the structure map is a weak equivalence for all n.

Just as with Bousfield-Friedlander spectra, we would like the Ω-spectra to be the fibrant objects in the stable model structure. We invert the same maps we did in that case.

Definition 10.7. Define the set of maps S in 𝒟_{K}^{Σ} to be {F_{n+1}(QC ∧ K) ⟶^{s_{n}^{QC}} F_{n}QC} as C runs through the domains and codomains of the generating cofibrations of 𝒟, and n ≥ 0. The map s_{n}^{QC} is adjoint to the map QC ∧ K ⟶ Ev_{n+1} F_{n}QC = Σ_{n+1}×(QC∧K) corresponding to the identity of Σ_{n+1}. Define Sp^{Σ}(𝒟, K) to be the Bousfield localization L_{S}𝒟_{K}^{Σ}. We refer to this model structure as the stable model structure on the category of symmetric spectra. The S-local weak equivalences are called the stable equivalences, and the S-local fibrations are called the stable fibrations.

The following theorem is then analogous to Theorem 5.4, and has the same proof.

Theorem 10.8. The stably fibrant symmetric spectra are the Ω-spectra. Further-

more, for all cofibrant A ∈ 𝒟 and for all n ≥ 0, the map F_{n+1}(A ∧ K) ⟶^{s_{n}^{A}} F_{n}A is a stable equivalence.

Just as in Corollary 5.5, this theorem implies that, when 𝒞 = SSet_{∗} or Top_{∗}, Sp^{Σ}(𝒞, K) is the same as the stable model category on (simplicial or topological) symmetric spectra discussed in [HSS98].

The analog of Corollary 5.6 also holds, with the same proof, so that smashing with K is a Quillen endofunctor of Sp^{Σ}(𝒟, K). Of course, we want this functor to be a Quillen equivalence. As in Definition 5.7, we prove this by introducing the shift functors.

Definition 10.9. Define the right shift functor s: Sp^{Σ}(𝒟, K) ⟶ Sp^{Σ}(𝒟, K) by (sX)_{n} = X_{n+1}, where the Σ_{n}-action on X_{n+1} is induced by the usual inclusion Σ_{n} ⟶ Σ_{n+1}. The structure maps of sX are the same as the structure maps of X. Define the left shift functor t: Sp^{Σ}(𝒟, K) ⟶ Sp^{Σ}(𝒟, K) to be the left adjoint of s. We have (tX)_{n} = Σ_{n} ×_{Σ_{n−1}} X_{n−1}, with the induced structure maps.

Then the same argument as in Theorem 5.8 implies the following theorem.

Theorem 10.10. The functors X ↦ X ∧ K and t are Quillen equivalences on Sp^{Σ}(𝒟, K). Furthermore, Rs is naturally isomorphic to L(− ∧ K) and R((−)^{K}) is naturally isomorphic to Lt.

We have now shown that Sp^{Σ}(𝒟, K) is a K-stabilization of 𝒟. However, for this to be of any use, we must show that Sp^{Σ}(𝒞, K) is a symmetric monoidal model category.

Theorem 10.11. Suppose that the domains of the generating cofibrations of both 𝒞 and 𝒟 are cofibrant. Then the category Sp^{Σ}(𝒞, K) is a symmetric monoidal model category, and the category Sp^{Σ}(𝒟, K) is a Sp^{Σ}(𝒞, K)-model category.

Proof. We prove this theorem in the same way as Theorem 8.4. Since the cofibrations in L_{S}𝒟_{K}^{Σ} are the same as the cofibrations in 𝒟_{K}^{Σ}, the only thing to check is that f □ g is a stable equivalence when f and g are cofibrations and one of them is a stable equivalence. We may as well assume that f: F_{n}A ⟶ F_{n}B is a generating cofibration in Sp^{Σ}(𝒟, K) and g is a stable trivial cofibration in Sp^{Σ}(𝒞, K); the argument for f a stable trivial cofibration and g a generating cofibration is the same. Then, by hypothesis, A and B are cofibrant in 𝒟. We claim that F_{n}A ∧ − is a Quillen functor from Sp^{Σ}(𝒞, K) to Sp^{Σ}(𝒟, K), and similarly for F_{n}B. Indeed, in view of Theorem 4.2, it suffices to show that F_{n}A ∧ s_{m}^{QC} is a stable equivalence for all m ≥ 0 and all domains or codomains C of the generating cofibrations of 𝒞. But one can easily check that F_{n}A ∧ s_{m}^{QC} = s_{n+m}(A ∧ QC). Then Theorem 10.8 implies that this map is a stable equivalence, as required.

Thus, both functors F_{n}A ∧ − and F_{n}B ∧ − are Quillen functors from Sp^{Σ}(𝒞, K) to Sp^{Σ}(𝒟, K). A two out of three argument, as in Theorem 8.4, then shows that f □ g is a stable equivalence, as required.

Note that we only require the domains of the generating cofibrations of 𝒞 to be cofibrant in order to conclude that Sp^{Σ}(𝒟, K) is a 𝒞-model category.

11. Properties of symmetric spectra

In this section, we point out that the arguments of Section 7 also apply to symmetric spectra. In particular, if smashing with K is already a Quillen equivalence on 𝒟, then F_{0} : 𝒟 ⟶ Sp^{Σ}(𝒟, K) is a Quillen equivalence, and so, under mild hypotheses, the homotopy category of 𝒟 is a closed module over Ho Sp^{Σ}(𝒞, K). We also show that symmetric spectra are Quillen equivalent to ordinary spectra.

Throughout this section, 𝒞 will denote a left proper cellular symmetric monoidal model category, 𝒟 will denote a left proper cellular 𝒞-model category, and K will denote a cofibrant object of 𝒞.

The proof of the following theorem is the same as the proof of Theorem 7.1.

Theorem 11.1. Suppose smashing with K is a Quillen equivalence on 𝒟. Then F_{0} : 𝒟 ⟶ Sp^{Σ}(𝒟, K) is a Quillen equivalence.

Corollary 11.2. Suppose that the domains of the generating cofibrations of both 𝒞 and 𝒟 are cofibrant, and suppose that smashing with K is a Quillen equivalence on 𝒟. Then the Ho 𝒞-module structure on Ho 𝒟 extends to a closed Ho Sp^{Σ}(𝒞, K)- module structure.

Proof. Certainly Ho Sp^{Σ}(𝒟, K) is a closed Ho Sp^{Σ}(𝒞, K)-module. Now use the equivalence of categories LF_{0} : Ho 𝒟 ⟶ Ho Sp^{Σ}(𝒟, K) to transport this structure back to 𝒟.

Recall that the homotopy category of any model category is naturally a closed Ho SSet-module [Hov98a, Chapter 5]. This corollary is the first step to the assertion that any stable (with respect to the suspension) model category is naturally a closed module over the homotopy category of (simplicial) symmetric spectra.

Symmetric spectra are functorial in a natural way.

Theorem 11.3. Suppose the domains of the generating cofibrations of 𝒞, 𝒟, and the left proper cellular 𝒞-model category ℰ are cofibrant. Then any 𝒞-Quillen functor F: 𝒟 ⟶ ℰ extends naturally to a Sp^{Σ}(𝒞, K)-Quillen functor Sp^{Σ}(F): Sp^{Σ}(𝒟, K) ⟶ Sp^{Σ}(ℰ, K). Furthermore, if F is a Quillen equivalence, so is Sp^{Σ}(F).

Proof. The functor F induces a 𝒞^{Σ}-module functor 𝒟^{Σ} ⟶ ℰ^{Σ}, which takes the symmetric sequence (X_{n}) to the symmetric sequence (F X_{n}). It follows that F induces a Sp^{Σ}(𝒞, K)-functor Sp^{Σ}(F): 𝒟_{K}^{Σ} ⟶ ℰ_{K}^{Σ}, that takes the symmetric spectrum (X_{n}) to the symmetric spectrum (F X_{n}), with structure maps

F X_{n} ∧ K ≅ F (X_{n} ∧ K) ⟶ F X_{n+1}.

Let U denote the right adjoint of F. Then the right adjoint of Sp^{Σ}(F) is Sp^{Σ}(U), which takes the symmetric spectrum (Y_{n}) to the symmetric spectrum (UY_{n}), with structure maps adjoint to the composite

F (UY_{n} ∧ K) ≅ F UY_{n} ∧ K ⟶ Y_{n} ∧ K ⟶ Y_{n+1}.

Since Ev_{n} Sp^{Σ}(U) = U Ev_{n}, it follows that Sp^{Σ}(F)F_{n} = F_{n}F.

It is clear that Sp^{Σ}(U) preserves level fibrations and level equivalences, so Sp^{Σ}(F) is a Quillen functor with respect to the projective model structure. In view of Theorem 4.2, to see that Sp^{Σ}(F) defines a Quillen functor with respect to the stable model structures, it suffices to show that Sp^{Σ}(F)(s_{n}^{QC}) is a stable equivalence for all domains and codomains of C of the generating cofibrations of 𝒟. But one can readily verify that Sp^{Σ}(F)(s_{n}^{QC}) = s_{n}^{F QC}, which is a stable equivalence as required. Thus Sp^{Σ}(F) is a Quillen functor as required.

If F is a Quillen equivalence, one can easily check that Sp^{Σ}(F) is a Quillen equivalence with respect to the projective model structure. To see that it is still a Quillen equivalence with respect to the stable model structures, we need only show that Sp^{Σ}(U) reflects stably fibrant objects, in view of Proposition 4.3. But, if X is level fibrant and Sp^{Σ}(U)(X) is an Ω-spectrum, this means that the map UX_{n} ⟶ (UX_{n+1})^{K} ≅ U(X_{n+1}^{K}) is a weak equivalence for all n. Since U reflects weak equivalences between fibrant objects, this means that X is an Ω-spectrum, as required.

Symmetric spectra are also functorial, in a limited sense, in the cofibrant object K.

Theorem 11.4. Suppose f: K ⟶ K′ is a weak equivalence of cofibrant objects of 𝒞, and suppose the domains of the generating cofibrations of 𝒞 and 𝒟 are cofibrant. Then f induces a Quillen equivalence Sp^{Σ}(𝒟, f): Sp^{Σ}(𝒟, K) ⟶ Sp^{Σ}(𝒟, K′) which is natural with respect to 𝒞-Quillen functors of 𝒟.

Proof. The map f induces a map of commutative monoids Sym(K) ⟶ Sym(K′). This induces the usual induction and restriction adjunction

Sp^{Σ}(𝒟, f): Sp^{Σ}(𝒟, K) ⟶ Sp^{Σ}(𝒟, K′).

That is, if X is in Sp^{Σ}(𝒟, K), then Sp^{Σ}(𝒟, f)(X) = X ⊗_{Sym(K)} Sym(K′). Restriction obviously preserves level fibrations and level equivalences, so is a Quillen functor with respect to the projective model structure. One can easily check that Sp^{Σ}(𝒟, f) ∘ F_{n} = F_{n}. It follows that Sp^{Σ}(𝒟, f)(s_{n}^{QC}) is the map

F_{n+1}(QC ∧ K) ⟶ F_{n}QC

in Sp^{Σ}(𝒟, K′). The weak equivalence QC ∧ K ⟶ QC ∧ K′ induces a level equivalence F_{n+1}(QC ∧ K) ⟶ F_{n+1}(QC ∧ K′). Since the map F_{n+1}(QC ∧ K′) ⟶ F_{n}QC is a stable equivalence, so is the given map. Thus induction is a Quillen functor with respect to the stable model structures.

We now prove that induction is a Quillen equivalence between the projective model structures. The proof of this is similar to the proof of Theorem 7.5. That is, restriction certainly reflects level equivalences between level fibrant objects. It therefore suffices to show that the map X ⟶ X ⊗_{Sym(K)} Sym(K′) is a level equivalence for all cofibrant X. The argument of Theorem 7.5 will prove this without difficulty.

To complete the proof, we need only check that restriction reflects stably fibrant objects. This follows from the fact that the map Z^{K′} ⟶ Z^{K} is a weak equivalence for all fibrant Z.

In particular, it does not matter, up to Quillen equivalence, what model of the simplicial circle one takes in forming the symmetric spectra of [HSS98].

Finally, we show that symmetric spectra and spectra are related by a chain of Quillen equivalences. This method of proof was suggested to the author by Mike Hopkins as a way of comparing symmetric spectra and the S-modules of [EKMM97].

Proposition 11.5. Suppose the domains of the generating cofibrations of 𝒞 are cofibrant. Denote the stable category of spectra Sp^{ℕ}(𝒟, − ∧ K) by Sp^{ℕ}(𝒟, K). Then there is a 𝒞-model category ℰ and 𝒞-Quillen equivalences Sp^{Σ}(𝒟, K) ⟶ ℰ ⟵ Sp^{ℕ}(𝒟, K).

Proof. We take ℰ = Sp^{ℕ}(Sp^{Σ}(𝒟, K), K), which makes sense since Sp^{Σ}(𝒟, K) is a 𝒞-model category by the comment following the proof of Theorem 10.11. We claim that ℰ is isomorphic as a model category with Sp^{Σ}(Sp^{ℕ}(𝒟, K), K), which again makes sense by Theorem 8.4. This is mostly a matter of unwinding definitions. An object of ℰ is a set {Y_{m,n}} of objects of 𝒟, where m, n ≥ 0. There is an action of Σ_{n} on Y_{m,n}, and there are Σ_{n}-equivariant maps Y_{m,n} ∧ K ⟶^{σ_{1}} Y_{m+1,n}

and Y_{m,n} ∧ K ⟶^{σ_{2}} Y_{m,n+1}. In addition, the composite Y_{m,n} ∧ K^{∧p} ⟶ Y_{m,n+p} is

Σ_{n} × Σ_{p}-equivariant, and there is a compatibility between σ_{1} and σ_{2}, expressed in the commutativity of the following diagram.

Y_{m,n} ∧ K ∧ K ⟶^{1∧T} Y_{m,n} ∧ K ∧ K ⟶^{σ_{2}∧1} Y_{m,n+1} ∧ K
σ_{1}∧1↓  ↓^{σ_{1}}
Y_{m+1,n} ∧ K  Y_{m+1,n} ∧ K ⟶_{σ_{2}} > Y_{m+1,n+1}

An object {Z_{m,n}} of the category Sp^{Σ}(Sp^{ℕ}(𝒟, K), K) has the same description if we switch m and n. There is then an isomorphism of categories between ℰ and Sp^{Σ}(Sp^{ℕ}(𝒟, K), K) which simply switches m and n. The model structures are also the same. Indeed, they are both the localization of the evident bigraded projective model structure with respect to the maps F_{m,n+1}(QC ∧ K) ⟶ F_{m,n}QC and F_{m+1,n}(QC ∧ K) ⟶ F_{m,n}QC, where F_{m,n} is left adjoint to the evaluation functor Ev_{m,n}.

Therefore, the functors F_{0} : Sp^{Σ}(𝒟, K) ⟶ ℰ and F_{0} : Sp^{ℕ}(𝒟, K) ⟶ ℰ are 𝒞-Quillen equivalences, by Theorem 7.1 and Theorem 11.1.

12. Concluding remarks

In this final section, we address some of the questions left unanswered by the body of the paper.

Firstly, it would be nice to have a theory of stabilization functors. The author views the stabilization of a model category as something like a fibrant replacement functor in a model category. Suppose we are given two fibrant replacement functors R and R′, so that there are natural weak equivalences X ⟶ RX and X ⟶ R′X, and both RX and R′X are fibrant. Then there is not any obvious relationship between R and R′, and fibrant replacement functors are certainly not unique. Nevertheless, we can choose a fibrant replacement functor R′′, using the functorial factorization, such that the natural transformation X ⟶ R′′X is a trivial cofibration. We can then use lifting to construct weak equivalences R′′X ⟶ R′X and R′′X ⟶ RX. These weak equivalences may not be natural, but given rise to natural isomorphisms in the homotopy category. A fancy way to say this, the author believes, is that the category of fibrant replacement functors is weakly contractible. This is the only sort of uniqueness one could hope for in the theory of stabilization functors: that the category of stabilization functors is weakly contractible. But the author has no idea whether this is true, as he does not have any idea how to replace the lifting properties used above.

The best general argument known to the author is the argument of Proposition 11.5, which shows that two commuting stabilization functors are related by a chain of Quillen equivalences. However, there is a Quillen functor Sp^{ℕ}(𝒟, K) ⟶ Sp^{Σ}(𝒟, K) left adjoint to the forgetful functor. Its description is very much like the analogous Quillen functor in [HSS98, Section 4.4]. We do not know if this functor is a Quillen equivalence. The argument used in [HSS98] relies good smallness properties of simplicial sets that may not be present in general.

Another problem with the methods of this paper is that the stable model structure on Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K) is not as well-described as one would like. In particular, we do not have an explicit set of generating trivial cofibrations and we do not have an explicit desciption of the fibrations. Nor do we understand the stable equivalences as well as we would like. Even with Sp^{ℕ}(𝒞, G), it is not clear that stable equivalences are the analog of stable homotopy isomorphisms. This is a fundamental drawback of the Bousfield localization method.

In certain cases, however, it is likely that one can say more. The necessary assumptions are likely to be the following. First, we should assume that 𝒞 is a finitely generated model category, in the sense of [Hov98a, Section 2.1]. We should also assume that the right adjoint U of the Quillen endofunctor G preserves transfinite compositions of cofibrations. That is, if X: λ ⟶ 𝒞 is a λ-sequence of cofibrations, we assume that the natural map colim UX_{α} ⟶ U(colim X_{α}) is an isomorphism. Finally, we need to assume that U preserves cofibrations, in some weak sense. More precisely, we require that U preserve a class of maps larger than the cofibrations, with respect to which the domains and codomains of the generating cofibrations and generating trivial cofibrations are small. These conditions hold, for example, in the category of compactly generated topological spaces (or topological symmetric spectra) when UX = X^{K} for some compact space K, where the class of maps in question is the class of closed inclusions.

These conditions will guarantee that transfinite compositions of cofibrations of stably fibrant spectra, in either Sp^{ℕ}(𝒞, G) or Sp^{Σ}(𝒞, K), are again stably fibrant. Analysis of [Hov98a, Section 7.4] shows that this is the key condition one requires in order to conclude that the homotopy category has a set of small weak generators.

These conditions are probably also the right ones needed to describe the stable equivalences in Sp^{ℕ}(𝒞, G) as the analog of stable homotopy isomorphisms, and to imitate the explicit construction of a generating set of trivial cofibrations, as in [HSS98]. This would in turn allow the expected characterization of the stable fibrations in terms of homotopy pullbacks. Proving that the stable model structures are right proper, however, may be more subtle.

Finally, we have not addressed the monoid axiom [SS97] at all in this paper. Recall that this is the axiom needed to ensure that there are model categories of monoids and of modules over them in a symmetric monoidal model category. It is not difficult to show that the projective model structure on Sp^{Σ}(𝒞, K) satisfies the monoid axiom, but we do not know how to approach the monoid axiom on the stable model structure.

References

[AM86] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer-Verlag, Berlin, 1986, Reprint of the 1969 original.

[BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II (M. G. Barratt and M. E. Mahowald, eds.), Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130.

[DHK] W. G. Dwyer, P. S. Hirschhorn, and D. M. Kan, Model categories and general abstract homotopy theory, in preparation.

[DK80] W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427–440.

[DS95] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbook of algebraic topology (Amsterdam), North-Holland, Amsterdam, 1995, pp. 73–126.

[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, xii+249 pp.

[Hir98] P. S. Hirschhorn, Localization, cellularization, and homotopy colimits, preprint, 1998.

[Hov98a] Mark Hovey, Model categories, to appear in Mathematical Surveys and Monographs, Amer. Math. Soc., 1998.

[Hov98b] Mark Hovey, Monoidal model categories, preprint, 1998.

[HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114.

[HSS98] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, preprint, 1998.

[Jec78] Thomas Jech, Set theory, Pure and applied mathematics, vol. 79, Academic Press, New York, 1978.

[Qui67] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967.

[SS97] Stefan Schwede and Brooke Shipley, Algebras and modules in monoidal model categories, preprint, 1997.

[Voe97] Vladimir Voevodsky, The Milnor conjecture, preprint, 1997.

Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA

   E-mail address: hovey@member.ams.org