SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES

MARK HOVEY

Abstract. We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the 𝔸^{1}-homotopy theory of Voevodsky [22]. One is based on the standard notion of spectra originated by Boardman [23]. Its input is a well-behaved model category 𝒞 and an endofunctor G, generalizing the suspension. Its output is a model category Sp^{ℕ}(𝒞, G) on which G is a Quillen equivalence. The second construction is based on symmetric spectra [11], and applies to model categories 𝒞 with a compatible monoidal structure. In this case, the functor G must be given by tensoring with a cofibrant object K. The output is again a model category Sp^{Σ}(𝒟, K) where tensoring with K is a Quillen equivalence, but now Sp^{Σ}(𝒟, K) is again a monoidal model category. We study general properties of these stabilizations; most importantly, we give a sufficient condition for these two stabilizations to be equivalent that applies both in the known case of topological spaces and in the case of A^{1}-homotopy theory.

Introduction

The object of this paper is to give two very general constructions of the passage from unstable homotopy theory to stable homotopy theory. Since homotopy theory in some form appears in many different areas of mathematics, this construction is useful beyond algebraic topology, where these methods originated. In particular, the two constructions we give apply not only to the usual passage from unstable homotopy theory of pointed topological spaces (or simplicial sets) to the stable homotopy theory of spectra, but also to the passage from the unstable 𝔸^{1}-hmootopy theory of Voevodsky [22] to the stable 𝔸^{1}-homotopy theory. This example is obviously important, and the fact that it is an example of a widely applicable theory of stabilization may come as a surprise to readers of [12], where specific properties of sheaves are used.

Suppose, then, that we are given a (Quillen) model category 𝒞 and a functor G: 𝒞 ⟶ 𝒞 that we would like to invert, analogous to the suspension. We will clearly need to require that 𝒞 be compatible with the model structure; specifically, we require G to be a left Quillen functor. We will also need some technical hypotheses on the model category 𝒞, which are complicated to state and to check, but which are satisfied in almost all interesting examples, including 𝔸^{1}-homotopy theory. It is well-known what one should do to form the category Sp^{ℕ}(𝒞, G) of spectra, as first written down for topological spaces in [1]. An object of Sp^{ℕ}(𝒞, G) is a sequence X_{n} of objects of 𝒞 together with maps GX_{n} ⟶ X_{n+1}, and a map f: X ⟶ Y is a sequence of maps f_{n}: X_{n} ⟶ Y_{n} compatible with the structure maps. There is an obvious model structure, called the projective model structure, where the weak equivalences are the maps f: X ⟶ Y such that f_{n} is a weak equivalence for all n. It is not difficult to show that this is a model structure and that there is a left Quillen functor G: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) extending G on 𝒞. But, just as in the topological case, G will not be a Quillen equivalence. So we must localize the projective model structure on Sp^{ℕ}(𝒞, G) to produce the stable model structure, with respect to which G will be a Quillen equivalence. A new feature of this paper is that we are able to construct the stable model structure with minimal hypotheses on 𝒞, using the localization results of Hirschhorn [9] (based on work of Dror-Farjoun [5]). We must pay a price for this generality, of course. That price is that stable equivalences are not stable homotopy isomorphisms, but instead are cohomology isomorphisms on all cohomology theories, just as for symmetric spectra [11]. If enough hypotheses are put on 𝒞 and G, then we show that stable equivalences are stable homotopy isomorphisms; in particular, this is the case for 𝔸^{1}-homotopy theory.

Date: April 6, 2000.

1991 Mathematics Subject Classification. 55U35, 18G55.

Key words and phrases. model category, homotopy category, stabilization, spectra, symmetric spectra.

As is well-known in algebraic topology, the category Sp^{ℕ}(𝒞, G) is not sufficient to understand the smash product. That is, if 𝒞 is a symmetric monoidal model category, and G is the functor X ↦ X ⊗ K for some cofibrant object K of 𝒞, it almost never happens that Sp^{ℕ}(𝒞, G) is symmetric monoidal. We therefore need a different construction in this case. We define a category Sp^{Σ}(𝒞, K) just as in symmetric spectra [11]. That is, an object of Sp^{Σ}(𝒞, K) is a sequence X_{n} of objects of 𝒞 with an action of the symmetric group Σ_{n} on X_{n}. In addition, we have Σ_{n}- equivariant structure maps X_{n} ⊗ K ⟶ X_{n+1}, but we must further require that the iterated structure maps X_{n} ⊗ K^{⊗p} ⟶ X_{n+p} are Σ_{n} × Σ_{p}-equivariant, where Σ_{p} acts on K^{⊗p} by permuting the tensor factors. It is once again straightforward to construct the projective model structure on Sp^{Σ}(𝒞, K). The same localization methods developed for Sp^{ℕ}(𝒞, G) apply again here to give a stable model structure on which tensoring with K is a Quillen equivalence. Once again, stable equivalences are cohomology isomorphisms on all possible cohomology theories, but this time it is very difficult to give a better description of stable equivalences even in the case of simplicial symmetric spectra (but see [21] for the best such result I know). We point out that our construction gives a different construction of the stable model category of simplicial symmetric spectra than the one appearing in [11].

We now have competing stabilizations of 𝒞 under the tensoring with K functor when 𝒞 is symmetric monoidal. Naturally, we need to prove they are the same in an appropriate sense. This was done in the topological (actually, simplicial) case in [11] by constructing a functor Sp^{ℕ}(𝒞, G) ⟶ Sp^{Σ}(𝒞, K), where K = S^{1} and G is the tensor with S^{1} functor, and proving it is a Quillen equivalence. We are unable to generalize this argument. Instead, following an idea of Hopkins, we construct a zigzag of Quillen equivalences Sp^{ℕ}(𝒞, G) ⟶ 𝒞 ⟵ Sp^{Σ}(𝒞, K). However, we need to require that the cyclic permutation map on K ⊗K ⊗K be homotopic to the identity by an explicit homotopy for our construction to work. This hypothesis holds in the topological case with K = S^{1} and in the 𝔸^{1}-local case with K equal to either the simplicial circle or the algebraic circle 𝔸^{1} − {0}. This section of the paper is by far the most delicate, and it is likely that we do not have the best possible result.

We also investigate the properties of these two stabilization constructions. There are some obvious properties one would hope for of a stabilization construction such as Sp^{ℕ}(𝒞, G) or Sp^{Σ}(𝒞, K). First of all, it should be functorial in the pair (𝒞, G). We prove this for both stabilization constructions; the most difficult point is defining what one should mean by a map from (𝒞, G) to (𝒟, H). Furthermore, it should be homotopy invariant. That is, if the map (𝒞, G) ⟶ (𝒟, H) is a Quillen equivalence, the induced map of stabilizations should also be a Quillen equivalence. We also prove this; one corollary is that the Quillen equivalence class of Sp^{Σ}(𝒞, K) depends only on the homotopy type of K. Finally, the stabilization map 𝒞 ⟶ Sp^{ℕ}(𝒞, G) should be the initial map to a model category 𝒟 with an extension of G to a Quillen equivalence. However, this last statement seems to be asking for too much, because the category of model categories is itself something like a model category. This statement is analogous to asking for an initial map in a model category from X to a fibrant object, and such things do not usually exist. The best we can do is to say that if G is already a Quillen equivalence, then the map from 𝒞 ⟶ Sp^{ℕ}(𝒞, G) is a Quillen equivalence. This gives a weak form of uniqueness, and is the basis for the comparison between Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K). See also see [19] and [18] for uniqueness results for the usual stable homotopy category.

We point out that this paper leaves some obvious questions open. We do not have a good characterization of stable equivalences or stable fibrations in either spectra or symmetric spectra, in general, and we are unable to prove that spectra or symmetric spectra are right proper. We do have such characterizations for spectra when the original model category 𝒞 is sufficiently well-behaved, and the adjoint U of G preserves sequential colimits. These hypotheses include the cases of ordinary simplicial spectra and spectra in the 𝔸^{1}-category of Voevodsky. We also prove that spectra are right proper in this situation. But we do not have a characterization of stable equivalences of symmetric spectra even with these strong assumptions. Also, we have been unable to prove that symmetric spectra satisfy the monoid axiom. Without the monoid axiom, we do not get model categories of monoids or of modules over an arbitrary monoid, though we do get a model category of modules over a cofibrant monoid. The question of whether commutative monoids form a model category is even more subtle and is not addressed in this paper. See [15] for commutative monoids in symmetric spectra of topological spaces.

There is a long history of work on stabilization, much of it not using model categories. As far as this author knows, Boardman was the first to attempt to construct a good point-set version of spectra; his work was never published (but see [23]), but it was the standard for many years. Generalizations of Boardman’s construction were given by Heller in several papers, including [6] and [7]. Heller has continued work on these lines, most recently in [8]. The review of this paper in Mathematical Reviews by Tony Elmendorf (MR98g:55021) captures the response of many algebraic topologists to Heller’s approach. I believe the central idea of Heller’s approach is that the homotopy theory associated to a model category 𝒞 is the collection of all possible homotopy categories of diagram categories ho 𝒞^{I} and all functors between them. With this definition, one can then forget one had the model category in the first place, as Heller does. Unfortunately, the resulting complexity of definition is overwhelming at present.

Of course, there has also been very successful work on stabilization by May and coauthors, the two major milestones being [14] and [4]. At first glance, May’s approach seems wedded to the topological situation, relying as it does on homeomorphisms X_{n} ⟶ ΩX_{n+1}. This is the reason we have not tried to use it in this paper. However, there has been considerable recent work showing that this approach may be more flexible than one might have expected. I have mentioned [15] above, but perhaps the most ambitious attempt to generalize S-modules has been initiated by Mark Johnson [13].

Finally, we point out that Schwede [17] has shown that the methods of Bousfield and Friedlander [1] apply to certain more general model categories. His model categories are always simplicial and proper, and he is always inverting the ordinary suspension functor. Nevertheless, the paper [17] is the first serious attempt to define a general stabilization functor of which the author is aware.

This paper is organized as follows. We begin by defining the category Sp^{ℕ}(𝒞, G) and the associated strict model structure in Section 1. Then there is the brief Section 2 recalling Hirschhorn’s approach to localization of model categories. We then construct the stable model structure modulo certain technical lemmas in Section 3. The technical lemmas we need assert that if a model category 𝒞 is left proper cellular, then so is the strict model structure on Sp^{ℕ}(𝒞, G), and therefore we can apply the localization technology of Hirschhorn. We prove these technical lemmas, and the analogous lemmas for the strict model structure on Sp^{Σ}(𝒟, K), in an Appendix. In Section 4, we study the simplifications that arise when the adjoint U of G preserves sequential colimits and 𝒞 is sufficiently well-behaved. We characterize stable equivalences as the appropriate generalization of stable homotopy isomorphisms in this case, and we show the stable model structure is right proper, giving a description of the stable fibrations as well. In Section 5, we prove the functoriality, homotopy invariance, and homotopy idempotence of the construction (𝒞, G) ↦ Sp^{ℕ}(𝒞, G). We also investigate monoidal structure.

Section 6 begins the second part of the paper, about symmetric spectra. Since we have developed all the necessary techniques in the first part, the proofs in this part are more concise. In Section 6 we discuss the category of symmetric spectra. In Section 7 we construct the projective and stable model structures on symmetric spectra, and in Section 8, we discuss some properties of symmetric spectra. This includes functoriality, homotopy invariance, and homotopy idempotence of the stable model structure. We conclude the paper in Section 9 by constructing the chain of Quillen equivalences betwee Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K), under the cyclic permutation hypothesis mentioned above. Finally, as stated previously, there is an Appendix verifying that the techniques of Hirschhorn can be applied to the projective model structures on Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K).

Obviously, considerable familiarity with model categories will be necessary to understand this paper. The original reference is [16], but a better introductory reference is [3]. More in depth references include [9], [10], and [2]. In particular, we rely heavily on the localization technology in [9].

The author would like to thank, above all, Jeff Smith and Phil Hirschhorn. It was clear to Jeff when he had the idea of symmetric spectra that they should work in general, as this paper shows that they do. But it would be impossible to make this vision work without an adequate general theory of model categories, and especially localization. Phil’s unswerving devotion to getting localization right in as much generality as possible [9] has made this paper possible. The author would also like to thank Dan Dugger, Dan Kan, Brooke Shipley, and Stefan Schwede for many conversations about the topic of this paper. And the author would like to thank Markus Spitzweck for pointing out a gap in a previous version of this paper.

1. Spectra

In this section and throughout the paper, 𝒞 will be a model category and G: 𝒞 ⟶ 𝒞 will be a left Quillen endofunctor of 𝒞 with right adjoint U. In this section, we define the category Sp^{ℕ}(𝒞, G) of spectra and construct its strict model structure.

The following definition is a straightforward generalization of the usual notion of spectra [1].

Definition 1.1. Suppose G is a left Quillen endofunctor of a model category 𝒞. Define Sp^{ℕ}(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 Sp^{ℕ}(𝒞, G) is the Bousfield-Friedlander category of spectra [1].

Definition 1.2. Given n ≥ 0, the evaluation functor Ev_{n}: Sp^{ℕ}(𝒞, G) ⟶ 𝒞 takes X to X_{n}. The evaluation functor has a left adjoint F_{n}: 𝒞 ⟶ Sp^{ℕ}(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 Sp^{ℕ}(𝒞, G). Limits and colimits in the category of spectra are taken objectwise.

Lemma 1.3. The category of spectra is bicomplete.

Proof. Given a functor X: ℐ ⟶ Sp^{ℕ}(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 1.4. Note that the evaluation functor Ev_{n}: Sp^{ℕ}(𝒞, G) ⟶ 𝒞 preserves colimits, so should have a right adjoint R_{n}: 𝒞 ⟶ Sp^{ℕ}(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 Sp^{ℕ}(𝒞, G).

Lemma 1.5. Suppose F: 𝒞 ⟶ 𝒞 is a functor and τ: GF ⟶ F G is a natural transformation. Then there is an induced functor F: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G), called the prolongation of F.

Proof. Given X ∈ Sp^{ℕ}(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. Usually there is an obvious choice of τ.

Corollary 1.6. The functors G and U prolong to adjoint functors G and U on Sp^{ℕ}(𝒞, G).

Proof. To prolong G, take τ to be the identity in Lemma 1.5. 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 a somewhat involved exercise in adjoint functors to verify that the resulting prolongations G′ and U′ are still adjoint to each other. Like all the exercises in adjoint functors in this paper, the simplest way to proceed is to write the adjoint Y ⟶ UZ of a map f: GY ⟶ Z as the composition

Y ⟶^{η} UGY ⟶^{Uf} UZ

and to use the standard properties of the unit and counit of an adjunction.

The following remark is critically important to the understanding of our approach to spectra.

Remark 1.7. The definition we have just given of the prolongation of G to an endofunctor of Sp^{ℕ}(𝒞, G) is the only possible definition under our very general hypotheses. However, this definition does not generalize the definition of the suspension when 𝒞 is the category of pointed topological spaces and GX = X∧S^{1}. Indeed, recall from [1] that the suspension of a spectrum X in this case is defined by (X ∧ S^{1})_{n} = X_{n} ∧ S^{1}, with structure map given by

X_{n} ∧ S^{1} ∧ S^{1} ⟶^{1∧T} X_{n} ∧ S^{1} ∧ S^{1} ⟶^{σ∧1} X_{n+1} ∧ S^{1},

where T is the twist isomorphism. On the other hand, if we apply our definition of the prolongation of G above, we get a functor X ↦ X ⊗ S^{1} defined by (X ⊗ S^{1})_{n} = X_{n} ⊗ S^{1} with structure map

X_{n} ∧ S^{1} ∧ S^{1} ⟶^{σ∧1} X_{n+1} ∧ S^{1}.

Said another way, Bousfield and Friedlander choose the natural transformation τ: X ∧ S^{1} ∧ S^{1} ⟶ X ∧ S^{1} ∧ S^{1} to be 1 ∧ T, while we are taking it to be the identity. This is a crucial and subtle difference whose ramifications we will study in Section 9.

We now show that Sp^{ℕ}(𝒞, G) inherits a model structure from 𝒞, called the projective model structure. The functor G: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) will be a left Quillen functor with respect to the projective model structure, but it will not be a Quillen equivalence. Our approach to the projective model structure owes much to [1] and [11, Section 5.1]. At this point, we will slip into the standard model category terminology and notation, all of which can be found in [10], mostly in Section 2.1.

Definition 1.8. A map f ∈ Sp^{ℕ}(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 (see [10, Section 2.1] for a discussion of cofibrantly generated model categories).

Definition 1.9. Suppose 𝒞 is a cofibrantly generated model category with generating cofibrations I and generating trivial cofibrations J. Suppose G is a left Quillen endofunctor of 𝒞, and form the category of spectra Sp^{ℕ}(𝒞, G). Define sets of maps in Sp^{ℕ}(𝒞, G) by 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 Sp^{ℕ}(𝒞, G). There is a standard method for proving this, based on the small object argument [10, Theorem 2.1.14]. The first step is to show that the domains of I_{G} and J_{G} are small, in the sense of [10, Definition 2.1.3].

Proposition 1.10. 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 Sp^{ℕ}(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 1.4.

Lemma 1.11. A map in Sp^{ℕ}(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 1.12. 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 1.11, 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 1.11 completes the proof for I_{G}-cof. The proof for J_{G}-cof is similar.

Proposition 1.10 and Proposition 1.12 immediately imply the following corollary.

Corollary 1.13. 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 1.14. Suppose 𝒞 is cofibrantly generated. Then the projective cofibrations, the level fibrations, and the level equivalences define a cofibrantly generated model structure on Sp^{ℕ}(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 [1].

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 [10, Theorem 2.1.14] 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 1.12 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 [10, Lemma 1.1.9]. In particular f ∈ J_{G}-cof.

Since colimits and limits in Sp^{ℕ}(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 1.15. 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 1.12 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 1.4, 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 Sp^{ℕ}(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 prolongation of G is still a Quillen functor.

Proposition 1.16. Give Sp^{ℕ}(𝒞, G) the projective model structure. Then the prolongation G: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) of G is a Quillen functor. Furthermore, the functor F_{n}: 𝒞 ⟶ Sp^{ℕ}(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 prolongation of U to a functor U: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) preserves level fibrations and level trivial fibrations, so its left adjoint G is a Quillen functor.

2. Bousfield localization

We will define the stable model structure on Sp^{ℕ}(𝒞, G) in Section 3 as a Bousfield localization of the projective model structure on Sp^{ℕ}(𝒞, G). In this section we recall the theory of Bousfield localization of model categories from [9].

To do so, we need some preliminary remarks related to function complexes. Details can be found in [10, Chapter 5], [2], and [9, Chapter 19]. 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, Ho 𝒞 is naturally tensored and cotensored over Ho SSet, as well as enriched over it. 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 2.1. 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 [9] is the following. We will define cellular model categories, a special class of cofibrantly generated model categories, in the Appendix.

Theorem 2.2. 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 [9, Chapter 4].

Proposition 2.3. 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, because F is a Quillen equivalence on the localizations, the map QX ⟶ UL_{𝒯} F QX is a weak equivalence in L_{𝒮}𝒞 (see [10, Section 1.3.3]). 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 [10, Corollary 1.3.16].

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 [9, Section 3.6]. 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 [1]. However, L_{𝒮}𝒞 need not be right proper even if 𝒞 is, as is shown by the example of Γ-spaces in [1], where it is also shown that the expected characterization of 𝒮-local fibrations does not hold.

3. The stable model structure

Our plan now is to apply Bousfield localization to the projective model structure on Sp^{ℕ}(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 the projective model structure makes Sp^{ℕ}(𝒞, G) into a cellular model category when 𝒞 is left proper cellular. We will prove this technical result in the appendix. In this section, we will assume that Sp^{ℕ}(𝒞, G) is cellular, find a good set 𝒮 of maps to form the stable model structure as the 𝒮-localization of the projective model structure, and prove that G is a Quillen equivalence with respect to the stable model structure.

Just as in symmetric spectra [11], we want the stable equivalences to be maps which induce isomorphisms on all cohomology theories. Cohomology theories will be represented by the appropriate analogue of Ω-spectra.

Definition 3.1. 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 Sp^{ℕ}(𝒞, G) such that the 𝒮-local 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. This is true, but the proof is somewhat technical; the reader might be well-advised to skip the following proof.

Proposition 3.2. 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 [9], g̃ is a cofibrant approximation to g. By [9, Proposition 12.3.2], there is a cofibrant approximation ĩ_{β}: Ã_{β} ⟶ { \atop A_{β+1}} to i_{β} which is a pushout of g̃. This is where we need our model category to be left proper. For any fibrant object Z, the functor Map_{r}(−, Z) converts colimits to limits and cofibrations to fibrations [10, 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 [10, Lemma 5.2.6], where the dual of the version we need is proved, or [2]) 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 [10, 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 3.2 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 3.2, 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 SSet, since F_{n}QC = (LF_{n})C, where LF_{n} is the total left derived functor of F_{n}. Also, map(C, UX_{n+1}) ≅ map(F_{n+1}GQC, X). In view of this, we make the following definition.

Definition 3.3. 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 Sp^{ℕ}(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 stable model structure on Sp^{ℕ}(𝒞, G) to be the localization of the projective model structure on Sp^{ℕ}(𝒞, G) with respect to this set 𝒮. We refer to the 𝒮-local weak equivalences as stable equivalences, and to the 𝒮-local fibrations as stable fibrations.

Theorem 3.4. Suppose 𝒞 is a left proper cellular model category and G is a left Quillen endofunctor of 𝒞. Then the stably 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 3.3, 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 3.2, 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).

We would now like to claim that the stable model structure on Sp^{ℕ}(𝒞, G) that we have just defined is a generalization of the stable model structure on spectra of topological spaces or simplicial sets defined in [1]. This cannot be a trivial observation, however, both because our approach is totally different and because of Remark 1.7.

Corollary 3.5. 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 the stable model structure on Sp^{ℕ}(𝒞, G) coincides with the stable model structure on the category of Bousfield-Friedlander spectra [1].

Proof. We know already that the cofibrations are the same in the stable model structure on Sp^{ℕ}(𝒞, G) and the stable model structure of [1]. 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 the strict model category of spectra, since the cofibrations do not change under localization. Thus map(f, X) is the same in both the stable model structure on Sp^{ℕ}(𝒞, G) and in the stable model category of Bousfield and Friedlander. Since the stably fibrant objects are also the same, the corollary holds.

The purpose of the stable model structure is to make the prolongation of G into a Quillen equivalence. We begin the process of proving this with the following corollary.

Corollary 3.6. Suppose 𝒞 is a left proper cellular model category and G is a left Quillen endofunctor of 𝒞. Then the prolongation of G to a functor G: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) is a Quillen functor with respect to the stable model structure.

Proof. In view of Hirschhorn’s localization theorem 2.2, 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 3.4, 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 with respect to the stable model structure. To do so, we introduce the shift functors.

Definition 3.7. Suppose 𝒞 is a model category and G is a left Quillen endofunctor of 𝒞. Define the shift functors t: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) and s: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(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 with respect to the strict model structure on Sp^{ℕ}(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 with respect to the stable model structure as well.

We have now come to the main advantage of our approach to spectra. We can test whether a spectrum X is a U-spectrum by checking that X is level fibrant and by checking that the map X ⟶ sUX, adjoint to the structure map of X, is a level equivalence. The analogous statement is false in the Bousfield-Friedlander category [1], because the extra twist map they use (see Remark 1.7) means that there is no map of spectra X ⟶ sUX adjoint to the structure map of X! Our interpretation of this is that the Bousfield-Friedlander approach, while excellent at what it does, is probably not the right general construction. Further evidence for this is provided by the extreme simplicity of the following proof.

Theorem 3.8. 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 with respect to the stable model structures. 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 stably 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) (where we use the stable model structure). 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.

4. The finitely generated case

The reader may well object at this point that we have defined the stable model structure on Sp^{ℕ}(𝒞, G) without ever defining stable homotopy groups. This is because stable homotopy groups do not detect stable equivalences in general. The usual simplicial and topological situation is very special. The goal of this section is to put some hypotheses on 𝒞 and G so that the stable model structure on Sp^{ℕ}(𝒞, G) behaves similarly to the stable model structure on ordinary simplicial spectra. In particular, we show that, under these hypotheses, the usual Q construction gives a stable fibrant replacement functor; thus, a map f is a stable equivalence if and only if Qf is a level equivalence. This allows us to characterize Ho Sp^{ℕ}(𝒞, G)(F_{0}A, X) for well-behaved A as the usual sort of colimit colim Ho 𝒞(G^{n}A, X_{n}). It also allows us to prove that the stable model structure is right proper, so we get the expected characterization of stable fibrations.

Most of the results in this section do not depend on the existence of the stable model structure on Sp^{ℕ}(𝒞, G), so we do not usually need to assume 𝒞 is left proper cellular.

In order to state our hypothesis on 𝒞, we make the following definition.

Definition 4.1. An object A of a category 𝒞 is called finitely presented if the functor 𝒞(A, −) preserves direct limits of sequences X_{0} ⟶ X_{1} ⟶ … ⟶ X_{n} ⟶ . . .. A cofibrantly generated model category 𝒞 is said to be finitely generated if the domains and codomains of the generating cofibrations and the generating trivial cofibrations are finitely presented.

This definition differs slightly from other definitions. In particular, an object A is usually said to be finitely presented if 𝒞(A, −) preserves all directed (or, equivalently, filtered) colimits. We are trying to assume the minimum necessary. Finitely generated model categories were introduced in [10, Section 7.4], but in that definition we assumed only that 𝒞(A, −) preserves (transfinitely long) direct limits of sequences of cofibrations. The author would now prefer to call the model categories of [10, Section 7.4] compactly generated. Thus, the model category of simplicial sets is finitely generated, but the model category on topological spaces is only compactly generated. Since we will only be working with finitely generated model categories in this section, our results will not apply to topological spaces. As we go along, we will indicate where our results fail for compactly generated model categories, and a possible way to amend them in the compactly generated case.

Now, given a spectrum X, there is an obvious candidate for a stable fibrant replacement of X.

Definition 4.2. Suppose G is a left Quillen endofunctor of a model category 𝒞 with right adjoint U. Define R: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒞, G) to be the functor sU, where s is the shift functor. Then we have a natural map ι_{X}: X ⟶ RX, and we define

R^{∞}X = colim(X ⟶^{ι_{X}} RX ⟶^{Rι_{X}} R^{2}X ⟶^{R^{2}ι_{X}} … ⟶^{R^{n−1}ι_{X}} R^{n}X ⟶^{R^{n}ι_{X}} …).

Let j_{X}: X ⟶ R^{∞}X denote the obvious natural transformation.

The following lemma, though elementary, is crucial.

Lemma 4.3. The maps ι_{RX}, Rι_{X}: RX ⟶ R^{2}X coincide.

Proof. The map ι_{RX} is the adjoint of the structure map

GUX_{n+1} ⟶^{ε} X_{n+1} ⟶^{η} UGX_{n+1} ⟶^{Uσ} UX_{n+2}

of RX, where ε denotes the counit of the adjunction, η denotes the unit, and σ denotes the structure map of X. Thus ι_{RX} is the composite

UX_{n+1} ⟶^{η} UGUX_{n+1} ⟶^{Uε} UX_{n+1} ⟶^{Uη} U^{2}GX_{n+1} ⟶^{U^{2}σ} U^{2}X_{n+2}.

Since Uε ∘ η is the identity, it follows that ι_{RX} = Rι_{X}.

We stress that Lemma 4.3 fails for symmetric spectra, and it is the major reason we must work with finitely generated model categories rather than compactly generated model categories. Indeed, in the compactly generated case, R^{∞} is not a good functor, since maps out of one of the domains of the generating cofibrations will not preserve the colimit that defines R^{∞}. The obvious thing to try is to replace the functor R by a functor W, obtained by factoring X ⟶ RX into a projective cofibration X ⟶ W X followed by a level trivial fibration W X ⟶ RX. The difficulty with this plan is that we do not see how to prove Lemma 4.3 for W. An alternative plan would be to use the mapping cylinder X ⟶ W′X on X ⟶ RX; this might make Lemma 4.3 easier to prove, but the map X ⟶ W′X will not be a cofibration. The map X ⟶ W′X may, however, be good enough for the required smallness properties to hold. It is a closed inclusion if 𝒞 is topological spaces, for example. The author knows of no good general theorem in the compactly generated case.

This lemma leads immediately to the following proposition.

Proposition 4.4. Suppose G is a left Quillen endofunctor of a model category 𝒞, and suppose that its right adjoint U preserves sequential colimits. Then the map ι_{R^{∞}X}: R^{∞}X ⟶ R(R^{∞}X) is an isomorphism. In particular, if X is level fibrant, 𝒞 is finitely generated, and U preserves sequential colimits, then R^{∞}X is a Uspectrum.

Proof. The map ι_{R^{∞}X} is the colimit of the vertical maps in the diagram below.

X ⟶^{ι_{X}} RX ⟶^{Rι_{X}} R^{2}X ⟶^{R^{2}ι_{X}} … ⟶^{R^{n−1}ι_{X}} R^{n}X ⟶^{R^{n}ι_{X}} …
ι_{X} ↓  ^{ι_{RX}} ↓  ^{ι_{R^{2}X}} ↓  ^{ι_{R^{n}X}} ↓
RX ⟶_{Rι_{X}} R^{2}X ⟶_{R^{2}ι_{X}} R^{3}X ⟶_{R^{3}ι_{X}} … ⟶_{R^{n}ι_{X}} R^{n+1}X ⟶_{R^{n+1}ι_{X}} …

Since the vertical and horizontal maps coincide, the result follows. For the second statement, we note that if X is level fibrant, each R^{n}X is level fibrant since R is a right Quillen functor (with respect to the projective model structure). Then we use the finitely generated condition to show that sequential colimits of fibrant objects in 𝒞 are again fibrant, following [10, Lemma 7.4.1]. Thus R^{∞}X is level fibrant, and hence a U-spectrum.

Proposition 4.5. Suppose G is a left Quillen endofunctor of a model category 𝒞 with right adjoint U. If 𝒞 is finitely generated, and X is a U-spectrum, then the map j_{X}: X ⟶ R^{∞}X is a level equivalence.

Proof. By assumption, the map ι_{X}: X ⟶ RX is a level equivalence between level fibrant objects. Since R is a right Quillen functor, R^{n}ι_{X} is a level equivalence as well. Then the method of [10, Corollary 7.4.2] completes the proof.

Proposition 4.5 gives us a slightly better method of detecting stable equivalences.

Corollary 4.6. Suppose G is a left Quillen endofunctor of a model category 𝒞 with right adjoint U. Suppose 𝒞 is finitely generated and U preserves sequential colimits. Then a map f: A ⟶ B is a stable equivalence in Sp^{ℕ}(𝒞, G) if and only if map(f, X) is a weak equivalence for all level fibrant spectra X such that ι_{X}: X ⟶ RX is an isomorphism.

Proof. By definition, f is a stable equivalence if and only if map(f, Y) is a weak equivalence for all U-spectra Y. But we have a level equivalence Y ⟶ R^{∞}Y by Proposition 4.5, and so it suffices to know that map(f, R^{∞}Y) is a weak equivalence for all U-spectra Y. But, by Proposition 4.4, ι_{R^{∞}Y} is an isomorphism.

This corollary, in turn, allows us to prove that R^{∞} detects stable equivalences. The following theorem is similar to [11, Theorem 3.1.11].

Theorem 4.7. Suppose G is a left Quillen endofunctor of a model category 𝒞 with right adjoint U. Suppose that 𝒞 is finitely generated and sequential colimits in 𝒞 preserve finite products. Suppose also that U preserves sequential colimits. If f: A ⟶ B is a map in Sp^{ℕ}(𝒞, G) such that R^{∞}f is a level equivalence, then f is a stable equivalence.

Proof. Suppose X is a U-spectrum such that the map ι_{X}: X ⟶ RX is an isomorphism. We will show that map(f, X) as a retract of map(R^{∞}f, R^{∞}X); this will obviously complete the proof. We first note that there is a natural map map(R^{∞}C, R^{∞}X) ⟶ map(C, X) obtained by precomposition with C ⟶ R^{∞}C and postcomposition with k: R^{∞}X ⟶ X. Here k is the inverse of the map j_{X}: X ⟶ R^{∞}X, which is an isomorphism since ι_{X} is so. On the other hand, we claim that there is also a natural map map(C, X) ⟶ map(R^{∞}C, R^{∞}X) obtained by applying the total right derived functor of R^{∞}. This is not obvious, since we are asserting that the total right functor of R^{∞} preserves the enrichment of the projective homotopy category of Sp^{ℕ}(𝒞, G) over the homotopy category of simplicial sets, even though R^{∞} is not a right Quillen functor. Nevertheless, if we assume that this natural map exists, it follows easily that the composite map(C, X) ⟶ map(R^{∞}C, R^{∞}X) ⟶ map(C, X) is the identity (in the homotopy category if simplicial sets), and therefore that map(f, X) is a retract of map(R^{∞}f, R^{∞}X).

It remains to show that the total right derived functor of R^{∞} preserves the enriched structure. We first point out that R^{∞} preserves level fibrations and level trivial fibrations, because 𝒞 is finitely generated, using the argument of [10, Lemma 7.4.1]. It follows from Ken Brown’s lemma [10, Lemma 1.1.12] that R^{∞} preserves level equivalences between level fibrant objects. Because sequential colimits in 𝒞 preserve finite products, R^{∞} also preserves finite products.

We claim that, for any functor H on a model category 𝒟 that preserves fibrations, weak equivalences between fibrant objects, and finite products, the total right derived functor of H preserves the enriched structure over Ho SSet. Indeed, analysis of the definition of this enrichment [10, Chapter 5] shows that it suffices to check that such a functor H preserves simplicial frames on a fibrant object Y. A simplicial frame on Y is a factorization ℓ_{•}Y ⟶^{α} Y_{∗} ⟶^{β} r_{•}Y in the diagram category of simplicial objects in 𝒟, where ℓ_{•}Y is the constant simplicial diagram on Y, the nth space of r_{•}Y is the product Y^{n+1}, α is a level equivalence, and β is a level fibration. We further require that β is an isomorphism in degree 0. Since H preserves products, weak equivalences between fibrant objects, and fibrations, it follows that HY_{∗} is a simplicial frame on HY.

Corollary 4.8. Suppose G is a left Quillen endofunctor of a model category 𝒞 with right adjoint U. Suppose that 𝒞 is finitely generated and sequential colimits in 𝒞 preserve finite products. Suppose also that U preserves sequential colimits. Then j_{A}: A ⟶ R^{∞}A is a stable equivalence for all A ∈ Sp^{ℕ}(𝒞, G).

Proof. One can easily check that R^{∞}j_{A} is an isomorphism, using Proposition 4.4.

Finally, we get the desired characterization of stable equivalences.

Theorem 4.9. Suppose G is a left Quillen endofunctor of a model category 𝒞 with right adjoint U. Suppose that 𝒞 is finitely generated and sequential colimits in 𝒞 preserve finite products. Suppose as well that U preserves sequential colimits. Let L′ denote a fibrant replacement functor in the projective model structure on Sp^{ℕ}(𝒞, G). Then, for all A ∈ Sp^{ℕ}(𝒞, G), the map A ⟶ R^{∞}L′A is a stable equivalence into a U-spectrum. Also, a map f: A ⟶ B is a stable equivalence if and only if R^{∞}L′f is a level equivalence.

Proof. The first statement follows immediately from Proposition 4.4 and Corollary 4.8. By the first statement, if f is a stable equivalence, so is R^{∞}L′f. Since R^{∞}L′f is a map between U-spectra, it is a stable equivalence if and only if it is a level equivalence. The converse follows from Theorem 4.7.

Since we did not need the existence of the stable model structure to prove Theorem 4.9, one can imagine attempting to construct it from the functor R^{∞}L′. This is, of course, the original approach of Bousfield-Friedlander [1], and this approach has been generalized by Schwede [17]. Also, if one has some way to detect level equivalences in 𝒞, say using appropriate generalizations of homotopy groups, Theorem 4.9 implies that stable equivalences in Sp^{ℕ}(𝒞, G) are detected by the appropriate generalizations of stable homotopy groups. One can see these generalizations in the following corollary as well.

Corollary 4.10. Suppose 𝒞 is a pointed, left proper, cellular, finitely generated model category where sequential colimits preserve finite products. Suppose G: 𝒞 ⟶ 𝒞 is a left Quillen functor whose right adjoint U commutes with sequential colimits. Finally, suppose A is a finitely presented cofibrant object of 𝒞 that has a finitely presented cylinder object A × I. Then

Ho Sp^{ℕ}(𝒞, G)(F_{k}A, Y) = colim_{m} Ho 𝒞(A, U^{m}Y_{k+m}).

for all level fibrant Y ∈ Sp^{ℕ}(𝒞, G).

Here we are using the stable model structure to form Ho Sp^{ℕ}(𝒞, G), of course.

Proof. We have Ho Sp^{ℕ}(𝒞, G)(F_{k}A, Y) = Sp^{ℕ}(𝒞, G)(F_{k}A, R^{∞}Y)/ ∼, by Theorem 4.9, where ∼ denotes the left homotopy relation. We can use the cylinder object F_{k}(A × I) as the source for our left homotopies. Then adjointness implies that Sp^{ℕ}(𝒞, G)(F_{k}A, R^{∞}Y)/ ∼= 𝒞(A, Ev_{k} R^{∞}Y)/ ∼. Since A and A×I are finitely presented, we get the required result.

By assuming slightly more about 𝒞, we can also characterize the stable fibrations.

Corollary 4.11. Suppose 𝒞 is a pointed, proper, cellular, finitely generated model category such that sequential colimits preserve pullbacks. Suppose G: 𝒞 ⟶ 𝒞 is a left Quillen functor whose right adjoint U commutes with sequential colimits. Then the stable model structure on Sp^{ℕ}(𝒞, G) is proper. In particular, a map f: X ⟶ Y is a stable fibration if and only if f is a level fibration and the diagram

X ⟶ R^{∞}L′X
f↓  ↓^{Lf}
Y ⟶ R^{∞}L′Y

is a homotopy pullback square in the projective model structure, where L′ is a fibrant replacement functor in the projective model structure.

Proof. We wil actually show that, if p: X ⟶ Y is a level fibration and f: B ⟶ Y is a stable equivalence, the pullback B ×_{A} Y ⟶ X is a stable equivalence. The first step is to use the right properness of the projective model structure on Sp^{ℕ}(𝒞, G) to reduce to the case where B and Y are level fibrant. Indeed, let Y′ = L′Y, B′ = L′B, and f′ = L′f. Then factor the composite X ⟶ Y ⟶ Y′ into a projective trivial cofibration X′ ⟶ Y′ followed by a level fibration p′: X′ ⟶ Y′. Then we have the commutative diagram below,

B ⟶^{f} Y ⟵^{p} X
↓  ↓  ↓
B′ ⟶_{f′} Y′ ⟵_{p′′} X′

where the vertical maps are level equivalences. Then Proposition 12.2.4 and Corollary 12.2.8 of [9], which depend on the projective model structure being right proper, imply that the induced map B ×_{Y} X ⟶ B′ ×_{Y′} X′ is a level equivalence. Hence B ×_{Y} X ⟶ X is a stable equivalence if and only if B′ ×_{Y′} X′ ⟶ X′ is a stable equivalence, and so we can assume B and X are level fibrant.

Now let S denote the pullback square below.

B ×_{Y} X ⟶ X
↓  ↓^{p}
B ⟶_{f} Y

Then R^{n}S is a pullback square for all n, and there are maps R^{n}S ⟶^{R^{n}ι_{S}} R^{n+1}S. Since pullbacks commute with sequential colimits, R^{∞}S is a pullback square. Furthermore, R^{∞}p is a fibration, since each R^{n}p is a fibration and 𝒞 is finitely generated (see [10, Lemma 7.4.1]). Since f is a stable equivalence between level fibrant spectra, R^{∞}f is a level equivalence by Theorem 4.9. So, since the projective model structure is right proper, the map R^{∞}(B ×_{Y} X ⟶ X) is a level equivalence, and thus B ×_{Y} X ⟶ X is a stable equivalence.

The characterization of stable fibrations then follows from [9, Proposition 3.6.8].

5. Properties of the stabilization functor

In this section we explore some of the properties of the correspondence (𝒞, G) ↦ Sp^{ℕ}(𝒞, G), where, throughout this section, we mean the stable model structure on 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. This important fact is as close as we can get to proving that Sp^{ℕ}(𝒞, G) is the initial, up to homotopy, stabilization of 𝒞 with respect to G. We also 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). In particular, this means, for example, that the Bousfield-Friedlander category of spectra of simplicial sets does not depend, up to Quillen equivalence, on which model of the circle S^{1} one chooses. We conclude the section by pointing out that our stabilization condition preserves some monoidal structure. For example, if 𝒞 is a simplicial model category, and G is a simplicial functor, then Sp^{ℕ}(𝒞, G) is a gain a simplicial model category, and the extension of G is again a simplicial functor. However, if 𝒞 is monoidal, and G is a monoidal functor, Sp^{ℕ}(𝒞, G) will almost never be a monoidal category; this is the reason we need the symmetric spectra introduced in the next section.

Theorem 5.1. 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, where Sp^{ℕ}(𝒞, G) has the stable model structure.

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 [10, Corollary 1.3.16]. 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 [10, Corollary 1.3.16], 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 stably fibrant replacement functor in Sp^{ℕ}(𝒞, G). Let R′ denote a fibrant replacement functor in the strict model structure on Sp^{ℕ}(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 U-spectrum, this map is in fact a level equivalence. Hence the map X ⟶ (L_{𝒮}F_{0}X)_{0} is a weak equivalence, as required.

It remains to prove 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.

In particular, this theorem means that the passage (𝒞, G) ↦ (Sp^{ℕ}(𝒞, G), G) is idempotent, up to Quillen equivalence. This suggests that we are doing some kind of fibrant replacement of (𝒞, G), but the author knows no way of making this precise.

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

Definition 5.2. 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 (Φ, τ): (𝒞, G) ⟶ (𝒟, H) is a left Quillen functor Φ: 𝒞 ⟶ 𝒟 and a natural transformation τ: ΦG ⟶ HΦ 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 5.3. Suppose (Φ, τ): (𝒞, G) ⟶ (𝒟, H) is a map of pairs. Then there is an induced map of pairs (Sp^{ℕ(Φ)}, Sp^{ℕ(τ)}): (Sp^{ℕ}(𝒞, G), G) ⟶ (Sp^{ℕ}(𝒟, H), H) such that Sp^{ℕ(Φ)} ∘ F_{n} = F_{n}Φ. This induced map of pairs is compatible with composition and identities.

Proof. Suppose G has right adjoint U, H has right adjoint V, and Φ has right adjoint Γ. The natural transformation τ induces a dual natural transformation Dτ: ΓV ⟶ UΓ. Define Sp^{ℕ(Γ)} : Sp^{ℕ}(𝒟, H) ⟶ Sp^{ℕ}(𝒞, G) by (Sp^{ℕ(Γ)}Y)_{n} = ΓY_{n}, with structure maps adjoint to the composite

ΓY_{n} ⟶^{Γσ̃} ΓV Y_{n+1} ⟶^{Dτ} UΓY_{n+1}

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

∐_{p+q+r=n} H^{p}ΦG^{q}X_{r} ⇉ ∐_{p+q=n} H^{p}ΦX_{q}

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

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

The functor Sp^{ℕ(Γ)} clearly preserves level fibrations and level trivial fibrations, so Sp^{ℕ(Φ)} is a left Quillen functor with respect to the projective model structures. Also, since Ev_{n} Sp^{ℕ(Γ)} = Γ Ev_{n}, we have F_{n}Φ = Sp^{ℕ(Φ)}F_{n}. To show that Sp^{ℕ(Φ)} is a left Quillen functor with respect to the stable model structures, we must show that Sp^{ℕ(Φ)}s_{n}^{A} is a stable equivalence for all cofibrant A, by Theorem 2.2. Using the fact that Sp^{ℕ(Φ)}F_{n} = F_{n}Φ and the fact that τ_{A} is a weak equivalence, we reduce to showing that s_{n}ΦA is a stable equivalence in Sp^{ℕ}(𝒟, H). This follows from Theorem 3.4, so Sp^{ℕ(Φ)} is a left Quillen functor with respect to the stable model structures. the map Φs_{n}^{A} is a stable equivalence. Thus Φ: Sp^{ℕ}(𝒞, G) ⟶ Sp^{ℕ}(𝒟, H) is a left Quillen functor with respect to the stable model structures.

We define Sp^{ℕ(τ)} by defining its adjoint DSp^{ℕ(τ)} : Sp^{ℕ(Γ)}V ⟶ USp^{ℕ(Γ)}. Indeed, Dτ 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 induces 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 equivalence on all fibrant objects of 𝒟. Thus DSp^{ℕ(τ)} will be a level equivalence on all level fibrant objects of Sp^{ℕ}(𝒟, H), so Sp^{ℕ(τ)} is a level equivalence on all cofibrant objects of Sp^{ℕ}(𝒞, G). We leave it to the reader to check compatibility of (Sp^{ℕ(Φ)}, Sp^{ℕ(τ)}) with compositions and identities.

Proposition 5.3 and Theorem 5.1 give us a weak universal property of Sp^{ℕ}(𝒞, G).

Corollary 5.4. Suppose (Φ, τ): (𝒞, G) ⟶ (𝒟, H) is a map of pairs such that H is a Quillen equivalence. Then there is a functor Φ̃: Ho Sp^{ℕ}(𝒞, G) ⟶ 𝒟 such that Φ̃ ∘ LF_{0} = LΦ: Ho 𝒞 ⟶ Ho 𝒟.

This corollary is trying to say that (Sp^{ℕ}(𝒞, G), G) is homotopy initial among maps of pairs (𝒞, G) ⟶ (𝒟, H) where H is a Quillen equivalence. Though the statement of the corollary is the best statement of this concept we have been able to find, we suspect there is a better one.

Proof. By Proposition 5.3 there is a map of pairs

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

induced by (Φ, τ). By Theorem 5.1, F_{0} : 𝒟 ⟶ Sp^{ℕ}(𝒟, H) is a Quillen equivalence. Define Φ̃ to be the composite R Ev_{0} ∘LSp^{ℕ(Φ)}.

We have now shown that the correspondence (𝒞, G) ↦ (Sp^{ℕ}(𝒞, G), G) is functorial. We would like to know that it is homotopy invariant. In particular, we would like to know that (Sp^{ℕ(Φ)}, Sp^{ℕ(τ)}) is a Quillen equivalence of pairs when Φ is a Quillen equivalence. Our proof of this seems to require some hypotheses.

Theorem 5.5. Suppose (Φ, τ): (𝒞, G) ⟶ (𝒟, H) is a map of pairs such that Φ 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 (Sp^{ℕ(Φ)}, Sp^{ℕ(τ)}): (Sp^{ℕ}(𝒞, G), G) ⟶ (Sp^{ℕ}(𝒟, H), H), the Quillen functor Sp^{ℕ(Φ)} is a Quillen equivalence.

Proof. We will first show that Sp^{ℕ(Φ)} is a Quillen equivalence on the projective model structures. Use the same notation as in the proof of Proposition 5.3, so that Γ denotes the right adjoint of Φ. Then, since Φ is a Quillen equivalence, Γ reflects weak equivalences between fibrant objects, by [10, Corollary 1.3.16]. It follows that Sp^{ℕ(Γ)} reflects level equivalences between level fibrant objects. Hence to show that Sp^{ℕ(Φ)} is a Quillen equivalence of the projective model structures, it suffices to show that X ⟶ Sp^{ℕ(Γ)}RSp^{ℕ(Φ)}X is a level equivalence for all cofibrant X, where R is a fibrant replacement functor in the projective model structure on Sp^{ℕ}(𝒟, H). Thus, we need only show that X_{n} ⟶ ΓR(Sp^{ℕ(Φ)}X)_{n} is a weak equivalence for all n and all cofibrant X, where now R is a fibrant replacement functor in 𝒟. Since X_{n} is cofibrant and Φ is a Quillen equivalence, it suffices to show that ΦX_{n} ⟶ (Sp^{ℕ(Φ)}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, ΦX_{n}^{β} ⟶ (Sp^{ℕ(Φ)}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 [9, Proposition 17.9.12], since each of the maps ΦX_{n}^{β} ⟶ ΦX_{n}^{β+1} and each of the maps (Sp^{ℕ(Φ)}X^{β})_{n} ⟶ (Sp^{ℕ(Φ)}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

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

and another pushout diagram

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

in 𝒟. Note that Φ(F_{m}C)_{n} = ΦG^{n−m}C, where we interpret G^{n−m}C to be the initial object if n < m. Similarly, (Sp^{ℕ(Φ)}F_{m}C)_{n} = (F_{m}ΦC)_{n} = H^{n−m}ΦC. 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 [10, 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 Sp^{ℕ(Φ)} is a Quillen equivalence with respect to the projective model structures. In view of Proposition 2.3, to show that Sp^{ℕ(Φ)} is a Quillen equivalence with respect to the stable model structures, we need to show that if Y is level fibrant in Sp^{ℕ}(𝒟, H) and Sp^{ℕ(Γ)}Y is a U-spectrum, then Y is a V - spectrum. To see this, note that, since Sp^{ℕ(Γ)}Y is a U-spectrum, the natural map ΓY_{n} ⟶ UΓY_{n+1} is a weak equivalence for all n. There is a natural transformation Dτ: ΓV ⟶ UΓ dual to τ. Furthermore, (Dτ)_{X} is a weak equivalence for all fibrant X, as we have seen in the proof of Proposition 5.3. Thus, the natural map ΓY_{n} ⟶ ΓV Y_{n+1} is a weak equivalence for all n. Since Γ reflects weak equivalences between fibrant objects, it follows that Y is a V -spectrum, as required.

As an example of Theorem 5.5, 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 5.5, take 𝒟 = 𝒞 equal to the model category of pointed simplicial sets, take Φ 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.

We now investigate to what extent the correspondence that takes (𝒞, G) to the stable model structure on Sp^{ℕ}(𝒞, G) preserves monoidal structure. We begin by assuming that 𝒞 is a 𝒟-model category, for some symmetric monoidal model category 𝒟. This means that 𝒟 is a symmetric monoidal category with a compatible model structure. Since we will need to work with this compatibility, we remind the reader of the precise definition (see also [10, Chapter 4]).

Definition 5.6. Suppose 𝒟 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

The compatibility condition we require is then that, if f and g are cofibrations, then so is f □ g, and furthermore, if one of f and g is a trivial cofibration, so is f □ g. We must also require that, if S is the unit of ⊗ and QS ⟶ S is a cofibrant approximation, then QS ⊗ X ⟶ X is still a weak equivalence.

Then, by saying that 𝒞 is a 𝒟-model category, we mean that 𝒞 is tensored, cotensored, and enriched over 𝒟, compatibly with the model structure. This compatibility is precisely analogous to the compatibility above.

We then have the following theorem.

Theorem 5.7. Let 𝒟 be a cofibrantly generated monoidal model category, and suppose the domains of the generating cofibrations are cofibrant. Suppose 𝒞 is a left proper cellular 𝒟-model category, and that G is a left 𝒟-Quillen endofunctor of 𝒟. This means that G(X ⊗ K) ≅ GX ⊗ K, coherently, for X ∈ 𝒞 and K ∈ 𝒟. Then Sp^{ℕ}(𝒞, G), with the stable model structure, is again a 𝒟-model category, and the extension of G is a 𝒞-Quillen self-equivalence of Sp^{ℕ}(𝒞, G).

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

Proof. We define the action of 𝒟 on Sp^{ℕ}(𝒞, G) levelwise. That is, given X ∈ Sp^{ℕ}(𝒞, G) and K ∈ 𝒟, 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 makes Sp^{ℕ}(𝒞, G) tensored over 𝒟. Similarly, 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}. This makes Sp^{ℕ}(𝒞, G) cotensored over 𝒟. Finally, given X and Y in Sp^{ℕ}(𝒞, G), define Map(X, Y) ∈ 𝒟 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 preserves the 𝒟 action, and the second map is induced by the structure map of X. This functor makes Sp^{ℕ}(𝒞, G) enriched over 𝒟.

We must now check that these structures are compatible with the model structure. We begin with the projective model structure on Sp^{ℕ}(𝒞, G). One can easily check that F_{n}f □ g = F_{n}(f □ g). Thus, if f is one of the generating cofibrations of the projective model structure on Sp^{ℕ}(𝒞, G), and g is a cofibration in 𝒟, then f □ g is a cofibration in Sp^{ℕ}(𝒞, G). It follows that f □ g is a cofibration for f an arbitrary cofibration of Sp^{ℕ}(𝒞, G) (see [20, Lemma 2.3] and [11, Corollary 5.3.5]). A similar argument shows that f □ g is a projective trivial cofibration in Sp^{ℕ}(𝒞, G) if either f is a projective cofibration in Sp^{ℕ}(𝒞, G) or g is a trivial cofibration in 𝒟. Finally, if QS ⟶ S is a cofibrant approximation to the unit S in 𝒟, and X is cofibrant in Sp^{ℕ}(𝒞, G), then each X_{n} is cofibrant in 𝒞, so the map X ⊗ QS ⟶ X is a level equivalence as required. Thus Sp^{ℕ}(𝒞, G) with its projective model structure is a 𝒟-model category.

To show that Sp^{ℕ}(𝒞, G) with its stable model structure is also a 𝒟-model category, we need to show that, if f is a stable trivial cofibration and g is a cofibration in 𝒟, then f □ g is a stable equivalence. It suffices to check this for g: K ⟶ L one of the generating trivial cofibrations of 𝒟. In this case, by hypothesis, K and L are cofibrant in 𝒟. Thus the functor − ⊗ K is a Quillen functor with respect to the projective model structure on Sp^{ℕ}(𝒞, 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 preserves the 𝒟-action. In view of Theorem 3.4, the map s_{n}^{QC⊗K} is a stable equivalence. Theorem 2.2 then implies that −⊗K is a Quillen functor with respect to the stable model structure on Sp^{ℕ}(𝒞, 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.

Remark 5.8. Suppose that the functor G is actually given by GX = X ⊗ K for some cofibrant object K of 𝒟. We then have two different ways of tensoring with K on Sp^{ℕ}(𝒞, G). The first way is the extension of G to a Quillen equivalence of Sp^{ℕ}(𝒞, G). Recall from Remark 1.7 that this functor, which we denote by X ↦ X\overline{⊗}K, does not use the twist map. On the other hand, we also have the functor X ↦ X ⊗ K that is part of the 𝒟-action on Sp^{ℕ}(𝒞, G) constructed in Theorem 5.7. This functor does use the twist map as part of its structure map; indeed, in order to construct the isomorphism G(X ⊗ K) ≅ GX ⊗ K we need to permute the two different copies of K. Therefore, we do not know that X ↦ X ⊗ K is a Quillen equivalence, even though X ↦ X\overline{⊗}K is. We will have to deal with this point more thoroughly in Section 9, when we compare Sp^{ℕ}(𝒞, G) with symmetric spectra.

Theorem 5.7 gives us a functorial stabilization. We first simplify the notation. Suppose K is a cofibrant object of a symmetric monoidal model category 𝒟. Then G = − ⊗ K is a left Quillen functor on any 𝒟-model category 𝒞. In this case, we denote Sp^{ℕ}(𝒞, G) by Sp^{ℕ}(𝒞, K).

Corollary 5.9. Suppose K is a cofibrant object of a cofibrantly generated symmetric monoidal model category 𝒟 where the domains of the generating cofibrations can be taken to be cofibrant. Then the correspondence 𝒞 ↦ Sp^{ℕ}(𝒞, K) defines an endofunctor of the category of left proper cellular 𝒟-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 𝒞 ↦ Sp^{ℕ}(𝒞, K) is actually a 2-functor. See [10] for a description of this point of view on model categories.

Proof. Given a left proper cellular 𝒟-model category 𝒞, we have seen in Theorem 5.7 that Sp^{ℕ}(𝒞, K) is a 𝒟-model category. Just as in Lemma 1.5, a 𝒟-Quillen functor H: 𝒞 ⟶ 𝒞′ induces a functor H: Sp^{ℕ}(𝒞, K) ⟶ Sp^{ℕ}(𝒞′, K), as does its right adjoint V. Since H is defined levelwise, it preserves the action of 𝒟. It is easy to check that V preserves level fibrations and level trivial fibrations, so that H is a 𝒞-Quillen functor with respect to the projective model structures. Furthermore, we have Hs_{n}^{QC} = s_{n}^{HQC}, so Theorem 3.4 and Theorem 2.2 imply that H is a 𝒞-Quillen functor with respect to the stable model structures as well.

We now point out that, if 𝒟 is a symmetric monoidal model category, and G = − ⊗ K for some cofibrant object K, the category Sp^{ℕ}(𝒟, G) is almost never itself monoidal, though, as we have seen, it has an action of 𝒟. To see this, consider the category 𝒟^{ℕ} of sequences from 𝒟. An object of 𝒟^{ℕ} is a sequence X_{n} of objects of 𝒟, and a map f: X ⟶ Y is a sequence of maps f_{n}: X_{n} ⟶ Y_{n}. Then 𝒟^{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 preserves the 𝒟-action.

Lemma 5.10. Suppose 𝒟 is a symmetric monoidal model category and G is a left 𝒟-Quillen functor. Then Sp^{ℕ}(𝒟, 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. The important point is that the monoid T is almost never commutative, and therefore Sp^{ℕ}(𝒟, G) can not be a symmetric monoidal category with unit T. Indeed, let K = GS ∈ 𝒟, so that G = − ⊗ K. Then T is commutative if and only if the commutativity isomorphism on K ∧ K is the identity. This happens only very rarely.

6. Symmetric spectra

We have just seen that the stabilization functor Sp^{ℕ}(𝒞, G) is not good enough in case 𝒟 is a symmetric monoidal model category and G is a 𝒟-Quillen functor, 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}(𝒟, 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 [10, Section 4.1].

Through most of this section, then, 𝒟 will be a bicomplete closed symmetric monoidal category with unit S, and K will be an object of 𝒟, The category 𝒞 will be a bicomplete category enriched, tensored, and cotensored over 𝒟. Note that any 𝒟-functor G on 𝒟 is of the form G(X) = X ∧ K for K = GS, so we will only consider 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 [11]. The main idea of symmetric spectra is that the commutativity isomorphism of 𝒟 makes K^{∧n} into a Σ_{n}-object of 𝒟, 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 [11, Definition 2.1.1].

Definition 6.1. Let Σ = ∐_{n≥0} Σ_{n} be the category whose objects are 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 𝒟 is a closed symmetric monoidal category, so is 𝒟^{Σ}, as explained in [11, 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 𝒟. 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, …), where 0 is the initial object of 𝒞. To define the closed structure, we first define Hom_{Σ_{n}}(X, Y) for X, Y ∈ 𝒟^{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 𝒞 is enriched, tensored, and cotensored over 𝒟, then 𝒞^{Σ} is enriched, tensored, and cotensored over 𝒟^{Σ}. Indeed, the same definition as above works to define the tensor structure. The cotensor structure is defined as follows. First we define Hom_{Σ_{n}}(K, X) for X ∈ 𝒞^{Σ_{n}} and K ∈ 𝒟^{Σ_{n}} as an appropriate equalizer. Then, for X ∈ 𝒞^{Σ} and K ∈ 𝒟^{Σ}, we define X_{k}^{K} = ∏_{n} Hom_{Σ_{n}}(K_{n}, X_{n+k}). The enrichment Map(X, Y) is defined similarly. In the same way, if 𝒞 is an enriched monoidal category over 𝒟, then 𝒞^{Σ} is an enriched monoidal category over 𝒟^{Σ}.

Consider the free commutative monoid Sym(K) on the object (0, K, 0, …, 0, …) of 𝒟^{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 [11, Section 4.4].

Definition 6.2. Suppose 𝒟 is a symmetric monoidal model category, 𝒞 is a 𝒟model category, and K is an object of 𝒟. The category of symmetric spectra Sp^{Σ}(𝒞, K) is the category of modules in 𝒞^{Σ} over the commutative monoid Sym(K) in 𝒟^{Σ}. That is, a symmetric spectrum is a sequence of objects X_{n} ∈ 𝒞^{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 Sp^{Σ}(𝒟, K) 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 [11]). We denote the monoidal structure by X ∧ Y = X ⊗_{Sym(K)} Y, and the closed structure by Hom_{Sym(K)}(X, Y). Similarly, Sp^{Σ}(𝒞, K) is bicomplete, enriched, tensored, and cotensored over Sp^{Σ}(𝒟, K) with the tensor structure denoted X ∧Y again, and, if 𝒞 is a 𝒟-monoidal model category, then Sp^{Σ}(𝒞, K) will be a monoidal category enriched over Sp^{Σ}(𝒟, K).

Of course, if we take 𝒞 = SSet_{∗} and K = S^{1}, we recover the definition of symmetric spectra given in [11], except that we are using right modules instead of left modules.

Definition 6.3. Given n ≥ 0, the evaluation functor Ev_{n}: Sp^{Σ}(𝒞, K) ⟶ 𝒞 takes X to X_{n}. the evaluation functor has a left adjoint F_{n}: 𝒞 ⟶ Sp^{Σ}(𝒞, K), 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 = Sym(K). Also, if X ∈ 𝒞 and Y ∈ 𝒟, there is a natural isomorphism F_{n}X ∧ F_{m}Y ≅ F_{n+m}(X ⊗Y), just as in [11, Proposition 2.2.6]. In particular, F_{0} : 𝒟 ⟶ Sp^{Σ}(𝒟, K) is a (symmetric) monoidal functor, and so Sp^{Σ}(𝒞, K) is naturally enriched, tensored, and cotensored over 𝒟. In fact, this structure is very simple. Indeed, if X ∈ Sp^{Σ}(𝒞, K) and L ∈ 𝒟, X ⊗ L = X ⊗_{Sym(K)} F_{0}L is just the symmetric sequence whose nth term is X_{n} ⊗ L. The structure map is the composite

X_{n} ⊗ L ⊗ K ⟶^{1⊗τ} X_{n} ⊗ K ⊗ L ⟶ X_{n+1} ⊗ L.

Note the presence of the twist map; this is required even when L = K to get a symmetric spectrum, unlike the case of ordinary spectra. Similarly, X^{L} = Hom_{Sym(K)}(F_{0}L, X) is the symmetric sequence whose nth term is X_{n}^{L}, with the twist map again appearing as part of the structure map.

Remark 6.4. Just as in the spectrum case, the functor Ev_{n} has a right adjoint R_{n}: 𝒞 ⟶ Sp^{Σ}(𝒞, K). 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}}.

7. Model structures on symmetric spectra

Throughout this section, 𝒟 will denote a left proper cellular symmetric monoidal model category, 𝒞 will denote a left proper cellular 𝒟-model category, and K will denote a cofibrant object of 𝒟. In this section, we discuss the projective and stable model structures on the category Sp^{Σ}(𝒞, K) 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 7.1. A map f ∈ Sp^{Σ}(𝒞, K) 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.

Then, just as in Definition 1.9, if we denote the generating cofibrations of 𝒞 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 1.10–1.13 with the same proofs. This gives us the projective model structure.

Theorem 7.2. The projective cofibrations, the level fibrations, and the level equivalences define a left proper cellular model structure on Sp^{Σ}(𝒞, K).

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 cellularity of the projective model structure is proved in the Appendix.

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 7.3. The category Sp^{Σ}(𝒟, K), with the projective model structure, is a symmetric monoidal model category. The category Sp^{Σ}(𝒞, K), with its projective model structure, is a Sp^{Σ}(𝒟, K)-model category.

Proof. We first show that the pushout product f □ g is a (trivial) cofibration when f is a cofibration in Sp^{Σ}(𝒞, K), and g is a cofibration in Sp^{Σ}(𝒟, K) (and one of them is a level equivalence). As explained in [10, 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 denote a cofibrant replacement for the unit S in 𝒟. Then F_{0}QS is a cofibrant replacement for F_{0}S = Sym(K) in Sp^{Σ}(𝒟, K). Indeed, F_{0}QS is cofibrant, and Ev_{n} F_{0}QS is just QS ⊗ K^{⊗n}. Since K is cofibrant and 𝒟 is a monoidal model category, the map F_{0}QS ⟶ F_{0}S is a level equivalence. Now, if X is cofibrant in Sp^{Σ}(𝒞, K), then each X_{n} is cofibrant. Hence the map X_{n} ⊗ QS ⟶ X_{n} is a weak equivalence for all n, and so the map X ∧ F_{0}QS ⟶ X is a level equivalence, as required.

We point out here that one can show that the projective model structure on Sp^{Σ}(𝒟, K) satisfies the monoid axiom of [20], assuming that 𝒟 itself does so. This means there is a projective model structure on the category of monoids in Sp^{Σ}(𝒟, K) and on the category of modules over any monoid. We do not include the proofs of these statement since we have not been able to prove the analogous statements for the stable model structure.

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

Definition 7.4. Define the symmetric spectrum \overline{Sym(K)} in Sp^{Σ}(𝒟, K) 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 ∈ Sp^{Σ}(𝒞, K) 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 𝒞. There is a model structure on Σ_{n}-objects of 𝒞 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 𝒞, then the set of generating cofibrations of 𝒞^{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 7.5. A map f: X ⟶ Y in Sp^{Σ}(𝒞, K) 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 𝒞^{Σ_{n}} for all n.

Proof. We only prove the cofibration case, as the trivial cofibration case is analogous. 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 1.15. 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 7.6. A symmetric spectrum X ∈ Sp^{Σ}(𝒞, K) 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 7.7. Define the set of maps 𝒮 in Sp^{Σ}(𝒞, K) 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 𝒞, 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 the stable model structure on Sp^{Σ}(𝒞, K) to be the Bousfield localization with respect to 𝒮 of the projective model structure on Sp^{Σ}(𝒞, K). 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 3.4, and has the same proof.

Theorem 7.8. The stably fibrant symmetric spectra are the Ω-spectra. Further-

more, for all cofibrant A ∈ 𝒞 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 3.5, this theorem implies that, when 𝒞 = SSet_{∗} or Top_{∗}, Sp^{Σ}(𝒟, K) is the same as the stable model category on (simplicial or topological) symmetric spectra discussed in [11].

The analog of Corollary 3.6 also holds, with the same proof, so that tensoring with K is a Quillen endofunctor of Sp^{Σ}(𝒞, K). Of course, we want this functor to be a Quillen equivalence. As in Definition 3.7, we prove this by introducing the shift functors.

Definition 7.9. Define the right shift functor s: Sp^{Σ}(𝒞, K) ⟶ Sp^{Σ}(𝒞, K) by sX = Hom_{Sym(K)}(F_{1}S, X). Thus (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}(𝒞, K) ⟶ Sp^{Σ}(𝒞, K) by tX = X ⊗_{Sym(K)} F_{1}S, so that t is left adjoint of s. We have (tX)_{n} = Σ_{n} ×_{Σ_{n−1}} X_{n−1}, with the induced structure maps.

Note that adjointness gives natural isomorphisms

(sX)^{K} ≅ Hom_{Sym(K)}(F_{1}K, X) ≅ s(X^{K}).

There is a map F_{1}K ⟶ F_{0}S which is the identity in degree 1. By adjointness, this map induces a map

X = Hom_{Sym(K)}(F_{0}S, X) ⟶ Hom_{Sym(K)}(F_{1}K, X) = (sX)^{K}.

X is an Ω-spectrum if and only if this map is a level equivalence and X is level fibrant. Therefore, the same method used to prove Theorem 3.8 also proves the following theorem.

Theorem 7.10. The functors X ↦ X ⊗ K and t are Quillen equivalences on Sp^{Σ}(𝒞, K). Furthermore, Rs is naturally isomorphic to L(− ⊗ K) and R((−)^{K}) is naturally isomorphic to Lt.

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

Theorem 7.11. Suppose that the domains of the generating cofibrations of both 𝒞 and 𝒟 are cofibrant. Then the category Sp^{Σ}(𝒟, K) is a symmetric monoidal model category, and the category Sp^{Σ}(𝒞, K) is a Sp^{Σ}(𝒟, K)-model category.

Proof. We prove this theorem in the same way as Theorem 5.7. Since the cofibrations in the stable model structure are the same as the cofibrations in the projective model structure, 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}(𝒞, K) and g is a stable trivial cofibration in Sp^{Σ}(𝒟, K); the argument for f a stable trivial cofibration and g a generating cofibration in Sp^{Σ}(𝒟, K) is the same. Then, by hypothesis, A and B are cofibrant in 𝒞. We claim that F_{n}A ∧ − is a Quillen functor Sp^{Σ}(𝒟, K) to Sp^{Σ}(𝒞, K) with their stable model structures, and similarly for F_{n}B ∧−. Indeed, in view of Theorem 2.2, 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 7.8 implies that this map is a stable equivalence, as required.

Thus, both functors F_{n}A ∧ − and F_{n}B ∧ − are Quillen functors in the stable model structures. A two out of three argument, as in Theorem 5.7, then shows that f □ g is a stable equivalence, as required.

Note that the functor F_{0} : 𝒟 ⟶ Sp^{Σ}(𝒟, K) is a symmetric monoidal Quillen functor, so of course Sp^{Σ}(𝒞, K) is a 𝒟-model category as well, under the hypotheses of Theorem 7.11. In fact, we only need the domains of the generating cofibrations of 𝒟 to be cofibrant to conclude that Sp^{Σ}(𝒞, K) is a 𝒟-model category, using the argument of Theorem 7.11.

As we mentioned above, we do not know if the stable model structure satisfies the monoid axiom. Given a particular monoid R, one could attempt to localize the projective model structure on R-modules to obtain a stable model structure. However, for this to work one would need to know that the projective model structure is cellular, and the author does not see how to prove this. This plan will certainly fail for the category of monoids, since the projective model structure on monoids will not be left proper in general.

We also point out that it may be possible to prove some of the results of Section 4 for symmetric spectra. All of those results cannot hold, since stable homotopy isomorphisms do not coincide with stable equivalences even for symmetric spectra of simplicial sets. Nevertheless, in that case, every stable homotopy isomorphism is a stable equivalence [11, Theorem 3.1.11], and there is a replacement for the functor R^{∞} constructed in [21]. We do not know if these results hold for symmetric spectra over a general well-behaved finitely generated model category.

8. Properties of symmetric spectra

In this section, we point out that the arguments of Section 5 also apply to symmetric spectra. In particular, if smashing with K is already a Quillen equivalence on 𝒞, then F_{0} : 𝒞 ⟶ Sp^{Σ}(𝒞, K) is a Quillen equivalence. This means that, under mild hypotheses, the homotopy category of 𝒞 is enriched, tensored, and cotensored over Ho Sp^{Σ}(𝒟, K). We also show symmetric spectra are functorial in an appropriate sense. In particular, we show that the Quillen equivalence class of Sp^{Σ}(𝒞, K) is an invariant of the homotopy type of K.

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

The proof of the following theorem is the same as the proof of Theorem 5.1.

Theorem 8.1. Suppose smashing with K is a Quillen equivalence on 𝒞. Then F_{0} : 𝒞 ⟶ Sp^{Σ}(𝒞, K) is a Quillen equivalence.

Corollary 8.2. 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 𝒞. Then Ho 𝒟 is enriched, tensored, and cotensored over Ho Sp^{Σ}(𝒟, K).

Proof. Note that Ho Sp^{Σ}(𝒞, K) is certainly enriched, tensored, and cotensored over Ho Sp^{Σ}(𝒟, K). Now use the equivalence of categories LF_{0} : Ho 𝒞 ⟶ Ho Sp^{Σ}(𝒞, K) to transport this structure back to 𝒞.

Recall that the homotopy category of any model category is naturally enriched, tensored, and cotensored over Ho SSet [10, Chapter 5]. This corollary is the first step to the assertion that the homotopy category of any stable (with respect to the suspension) model category is naturally enriched, tensored, and cotensored over the homotopy category of (simplicial) symmetric spectra. See [19] for further results along these lines.

Symmetric spectra are functorial in a natural way.

Theorem 8.3. Suppose the domains of the generating cofibrations of 𝒟, 𝒞, and the left proper cellular 𝒞-model category 𝒞′ are cofibrant. Then any 𝒟-Quillen functor Φ: 𝒞 ⟶ 𝒞′ extends naturally to a Sp^{Σ}(𝒟, K)-Quillen functor

Sp^{Σ}(Φ): Sp^{Σ}(𝒞, K) ⟶ Sp^{Σ}(𝒞′, K).

Furthermore, if Φ is a Quillen equivalence, so is Sp^{Σ}(Φ).

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

ΦX_{n} ⊗ K ≅ Φ(X_{n} ⊗ K) ⟶ ΦX_{n+1}.

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

Φ(ΓY_{n} ⊗ K) ≅ ΦΓY_{n} ⊗ K ⟶ Y_{n} ⊗ K ⟶ Y_{n+1}.

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

It is clear that Sp^{Σ}(Γ) preserves level fibrations and level equivalences, so Sp^{Σ}(Φ) is a Quillen functor with respect to the projective model structure. In view of Theorem 2.2, to see that Sp^{Σ}(Φ) defines a Quillen functor with respect to the stable model structures, it suffices to show that Sp^{Σ}(Φ)(s_{n}^{QC}) is a stable equivalence for all domains and codomains of C of the generating cofibrations of 𝒞. But one can readily verify that Sp^{Σ}(Φ)(s_{n}^{QC}) = s_{n}^{ΦQC}, which is a stable equivalence as required. Thus Sp^{Σ}(Φ) is a Quillen functor with respect to the stable model structures.

If Φ is a Quillen equivalence, one can easily check that Sp^{Σ}(Φ) 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}(Γ) reflects stably fibrant objects, in view of Proposition 2.3. But, if X is level fibrant and Sp^{Σ}(Γ)(X) is an Ω-spectrum, this means that the map ΓX_{n} ⟶ (ΓX_{n+1})^{K} ≅ Γ(X_{n+1}^{K}) is a weak equivalence for all n. Since Γ 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 8.4. Suppose f: K ⟶ K′ is a weak equivalence of cofibrant objects of 𝒟, and suppose the domains of the generating cofibrations of 𝒟 and 𝒞 are cofibrant. Then f induces a Quillen equivalence Sp^{Σ}(𝒞, f): Sp^{Σ}(𝒞, K) ⟶ Sp^{Σ}(𝒞, K′) which is natural with respect to 𝒟-Quillen functors of 𝒞.

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

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

That is, if X is in Sp^{Σ}(𝒞, K), then Sp^{Σ}(𝒞, 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}(𝒞, f)∘ F_{n} = F_{n}. It follows that Sp^{Σ}(𝒞, f)(s_{n}^{QC}) is the map

F_{n+1}(QC ⊗ K) ⟶ F_{n}QC

in Sp^{Σ}(𝒞, 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 5.5. 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 5.5 will prove this without difficulty.

To show that induction is a Quillen equivalence between the stable model structures, 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 [11].

9. Comparison of spectra and symmetric spectra

In this section, suppose 𝒟 is a left proper cellular symmetric monoidal model category, K is a cofibrant object of 𝒟, and 𝒞 is a left proper cellular 𝒟-model category. Let G denote the left Quillen endofunctor GX = X ⊗ K of 𝒞. Then we have two different stabilizations of 𝒞, namely Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K). The object of this section is to compare them. We show that Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K) are related by a chain of Quillen equivalences whenever the cyclic permutation self-map of K ⊗ K ⊗ K is homotopic to the identity.

This is not the ideal theorem; one might hope for a direct Quillen equivalence rather than a zigzag of Quillen equivalences, and one might hope for weaker hypotheses, or even no hypotheses. However, some hypotheses are necessary, as pointed out to the author by Jeff Smith. Indeed, the category Ho Sp^{Σ}(𝒟, K) is symmetric monoidal, and therefore Ho Sp^{Σ}(𝒟, K)(F_{0}S, F_{0}S), the self-maps of the unit, form a commutative monoid. If we have a chain of Quillen equivalences between Sp^{ℕ}(𝒟, K) and Sp^{Σ}(𝒟, K) that preserves the functors F_{0}, then Ho Sp^{ℕ}(𝒟, K)(F_{0}S, F_{0}S) would also have to be a commutative monoid. With sufficiently many hypotheses on 𝒟 and K, for example if 𝒟 is the category of simplicial sets and K is a finite simplicial set, we have seen in in Section 4 that this mapping set is the colimit colim Ho 𝒟(K^{⊗n}, K^{⊗n}). There are certainly examples where this monoid is not commutative; for example K could be the mod p Moore space, and then homology calculations show this colimit is not commutative. In fact, this monoid will be commutative if and only if the cyclic permutation map of K ⊗K ⊗K becomes the identity in Ho 𝒟 after tensoring with sufficiently many copies of K. Hence we need some hypothesis on the cyclic permutation map.

The heart of our argument is the following theorem. The argument of this theorem can be summarized by saying that commuting stabilization functors are equivalent, and as such, was suggested to the author by Mike Hopkins in a different context. Recall that there are two different ways to tensor with K on Sp^{ℕ}(𝒞, G); the functor X ↦ X\overline{⊗}K that is a Quillen equivalence but does not involve the twist map, and the functor X ↦ X ⊗ K that may not be a Quillen equivalence but does involve the twist map.

Theorem 9.1. Suppose that the functor X ↦ X ⊗ K is a Quillen equivalence of Sp^{ℕ}(𝒞, G), and also that the domains of the generating cofibrations of 𝒟 are cofibrant. Then there is a 𝒟-model category ℰ together with 𝒟-Quillen equivalences Sp^{Σ}(𝒞, K) ⟶ ℰ ⟵ Sp^{ℕ}(𝒞, G).

Proof. We take ℰ = Sp^{ℕ}(Sp^{Σ}(𝒞, K), K), where the functor G: Sp^{Σ}(𝒞, K) ⟶ Sp^{Σ}(𝒞, K) used to form ℰ is defined by GX = X ⊗ K and is part of the 𝒟model structure of Sp^{Σ}(𝒞, K) (see the comment following Theorem 7.11). This means that the structure map of G involves the twist map. By Theorem 8.1, F_{0} : Sp^{Σ}(𝒞, K) ⟶ ℰ is a 𝒟-Quillen equivalence. On the other hand, consider Sp^{Σ}(Sp^{ℕ}(𝒞, G), K), where now we use the action of 𝒟 on Sp^{ℕ}(𝒞, G) that comes from Theorem 5.7. As pointed out in Remark 5.8, this means that we are using the functor X ↦ X ⊗ K to form Sp^{Σ}(Sp^{ℕ}(𝒞, G), K), not the functor X ↦ X\overline{⊗}K. By hypothesis, this functor is already a Quillen equivalence, so Theorem 5.1 implies that F_{0} : Sp^{ℕ}(𝒞, G) ⟶ Sp^{Σ}(Sp^{ℕ}(𝒞, G), K) is a 𝒟-Quillen equivalence.

It remains to prove that ℰ and Sp^{Σ}(Sp^{ℕ}(𝒞, G), K) are isomorphic as model categories. This is mostly a matter of unwinding definitions. An object of ℰ is a set {Y_{m,n}} of objects of 𝒞, 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 ⟶^{ν} Y_{m+1,n} and Y_{m,n} ⊗ K ⟶^{ρ} 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 ν and ρ, expressed in the commutativity of the following diagram.

Y_{m,n} ⊗ K ⊗ K ⟶^{1⊗T} Y_{m,n} ⊗ K ⊗ K ⟶^{ρ⊗1} Y_{m,n+1} ⊗ K
ν⊗1↓  ↓^{ν}
Y_{m+1,n} ⊗ K  Y_{m+1,n} ⊗ K ⟶_{ρ} Y_{m+1,n+1}

An object {Z_{m,n}} of the category Sp^{Σ}(Sp^{ℕ}(𝒞, G), 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}(𝒞, G), 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}.

Theorem 9.1 indicates that we should try to prove that − ⊗ K is a Quillen equivalence of Sp^{ℕ}(𝒞, G). The only reasonable way to do this is by comparing this functor to −\overline{⊗}K, which we know is a Quillen equivalence. The basic idea of the proof is to compare X ⊗ K ⊗ K to X\overline{⊗}K\overline{⊗}K. Both of these spectra have the same spaces, and their structure maps differ precisely by the cyclic permutation self-map of K ⊗ K ⊗ K. So if we knew that this map were the identity, they would be the same spectra. The hope is then that, if we know that the cyclic permutation map is homotopic to the identity, these two spectra are equivalent in Ho Sp^{ℕ}(𝒞, G). One can in fact construct a map of spectra X\overline{⊗}K\overline{⊗}K ⟶ R(X ⊗ K ⊗ K), where R is a level fibrant replacement functor and X is cofibrant, by inductively modifying the identity map. Unfortunately, the author does not know how to do this modification in a natural way, so is unable to prove that the derived functors L(X ⊗ K ⊗ K) and L(X\overline{⊗}K\overline{⊗}K) are equivalent in this way.

Instead, we will follow a suggestion of Dan Dugger and construct a new functor F on cofibrant objects of Sp^{ℕ}(𝒞, G) and natural level equivalences F X ⟶ X\overline{⊗}K\overline{⊗}K and F X ⟶ X ⊗ K ⊗ K, for cofibrant X. It will follow immediately that L(X\overline{⊗}K\overline{⊗}K) is naturally equivalent to L(X ⊗ K ⊗ K), and so that X ↦ X ⊗ K is a Quillen equivalence on Sp^{ℕ}(𝒞, G). Unfortunately, to make this construction we will need to make some unpleasant assumptions that ought to be unnecessary.

Definition 9.2. Given a symmetric monoidal model category 𝒟 whose unit S is cofibrant, a unit interval in 𝒟 is a cylinder object I for S such that there exists a map H_{I}: I ⊗ I ⟶ I satisfying H_{I} ∘ (1 ⊗ i_{0}) = H_{I} ∘ (i_{0} ⊗ 1) = i_{0}π and H_{I} ∘ (1 ⊗ i_{1}) is the identity. Here i_{0}, i_{1} : S ⟶ I and π: I ⟶ S are the structure maps of I. Given a cofibrant object K of 𝒟, we say that K is symmetric if there is a unit interval I and a homotopy

H: K ⊗ K ⊗ K ⊗ I ⟶ K ⊗ K ⊗ K

from the cyclic permutation to the identity.

Note that [0, 1] is a unit interval in the usual model structure on compactly generated topological spaces, and Δ[1] is a unit interval in the category of topological spaces. Indeed, the required map H_{1} : Δ[1] × Δ[1] takes both of the nondegenerate 2-simplices 011 × 001 and 001 × 011 to 001. Similarly, the standard unit interval chain complex of abelian groups is a unit interval in the projective model structure on chain complexes. Also, any symmetric monoidal left Quillen functor preserves unit intervals. It follows, for example, that the unstable A^{1}-model category of Voevodksy has a unit interval.

Our goal, then, is to prove the following theorem.

Theorem 9.3. Suppose 𝒟 is a symmetric monoidal model category with cofibrant unit S, and 𝒞 is a left proper cellular 𝒟-model category. Suppose that K is a cofibrant object of 𝒟, and that either K is itself symmetric or the domains of the generating cofibrations of 𝒞 are cofibrant and K is weakly equivalent to a symmetric object of 𝒟. Then the functor X ↦ X ⊗ K is a Quillen equivalence of Sp^{ℕ}(𝒞, G).

This theorem is certainly not the best one can do. For example, by considering the analogous functors with more than three tensor factors of K, it should be possible to show that the same theorem holds if there is a left homotopy between some even permutation of K^{⊗n} and the identity. Also, it seems clear that one should only need the cyclic permutation, or more generally some even permutation, to be equal to the identity in Ho 𝒟. That is, we should not need a specific left homotopy. But the author does not know how to remove this hypothesis.

In any case, we have the following corollary.

Corollary 9.4. Suppose 𝒟 is a left proper cellular symmetric monoidal model category whose unit S is cofibrant, and whose generating cofibrations can be taken to have cofibrant domains. Suppose 𝒞 is a left proper 𝒟-model category. Suppose K is a cofibrant object of 𝒟, and either that K is itself symmetric, or else that the domains of the generating cofibrations of 𝒞 are cofibrant and K is weakly equivalent to a symmetric object of 𝒟. Then there is a 𝒟-model category ℰ and 𝒟-Quillen equivalences

Sp^{Σ}(𝒞, K) ⟶ ℰ ⟵ Sp^{ℕ}(𝒞, G).

We will prove Theorem 9.3 in a series of lemmas.

Lemma 9.5. Suppose 𝒟 is a symmetric monoidal model category whose unit S is cofibrant. Suppose we have a square

A ⟶^{f} X
r↓  ↓^{s}
B ⟶_{g} Y

in a 𝒟-model category 𝒞, where A and B are cofibrant, and a left homotopy H: A⊗ I ⟶ Y from gr to sf, for some unit interval I. Then there is an object B′ of 𝒞, a weak equivalence B′ ⟶^{q} B, a commutative square

A ⟶^{f} X
r′↓  ↓^{s}
B′ ⟶_{g′} Y

such that qr′ = r, and a left homotopy H′: B′ ⊗ I ⟶ Y between gq and g′. Furthermore, this construction is natural in an appropriate sense.

Naturality means that, if we have a map of such homotopy commutative squares that preserves the homotopies, then we get a map of the resulting commutative squares that preserves the maps q and H′. The precise statement is complicated, and we leave it to the reader.

Proof. We let B′ be the mapping cylinder of r. That is, we take B′ to be the pushout in the diagram below.

A ⟶^{i_{0}} A ⊗ I
r↓  ↓^{h}
B ⟶_{j} B′

The map r′ is then the composite hi_{1}, and the map g′ is the map that is g on B and H on A ⊗ I. It follows that g′r′ = Hi_{1} = sf, as required. The map q: B′ ⟶ B is defined to be the identity on B and the composite A⊗I ⟶^{π} A ⟶^{r} B on A⊗I. Since j is a trivial cofibration (as a pushout of i_{0}), it follows that q is a weak equivalence, and it is clear that qr′ = r. We must now construct the homotopy H′. First note that B′ is cofibrant, since B is so and j is a trivial cofibration, and so B′ ⊗ I is a cylinder object for B′. In fact, B′ ⊗ I is the pushout of A ⊗ I ⊗ I and B ⊗ I over A ⊗ I. Define H′ to be the constant homotopy B ⊗ I ⟶^{π} B ⟶^{g} Y on B ⊗ I and the homotopy

A ⊗ I ⊗ I ⟶^{1⊗H_{I}} A ⊗ I ⟶^{H} Y

on A ⊗ I ⊗ I. The fact that H_{I} ∘ (i_{0} ⊗ 1) = i_{0}π guarantees that H′ is well-defined, and the other conditions on H_{I} guarantee that H′ is a left homotopy from qg to g′. We leave the naturality of this construction to the reader.

We also need the following lemma about the behavior of unit intervals.

Lemma 9.6. Suppose 𝒟 is a symmetric monoidal model category whose unit S is cofibrant. Let I and I′ be unit intervals, and define J by the pushout diagram below.

S ⟶^{i_{0}} I′
i_{1}↓  ↓^{j_{1}}
I ⟶_{j_{0}} J

Then J is a unit interval.

Proof. The reader is well-advised to draw a picture in the topological or simplicial case, from which the proof should be clear. We think of J as the interval whose left half is I and whose right half is I′. In particular, J is a cylinder object for S, where i′_{0} = j_{0}i_{0} and i′_{1} = j_{1}i_{1}. Then, because the tensor product preserves pushouts, we can think of J ⊗ J as a square consisting of a copy of I ⊗ I in the lower left, a copy of I ⊗ I′ in the upper left, a copy of I′ ⊗ I is the lower right, and a copy of I′ ⊗ I′ in the upper right. We define the necessary map G: J ⊗J ⟶ J by defining G on each subsquare. On the lower left, we use the composite I ⊗ I ⟶^{H} I ⟶^{j_{0}} J, where H is the homotopy making I into a unit interval. Similarly, on the upper right square, we use the composite j_{1}H′. On the upper left square we use the constant homotopy j_{0}(1 ⊗ π), and on the lower right square we use the constant homotopy j_{0}(π ⊗ 1). We leave it to the reader to check that this makes J into a unit interval.

The importance of these two lemmas for spectra is indicated in the following consequence.

Lemma 9.7. Suppose 𝒟 is a left proper cellular symmetric monoidal model category with a unit interval I, whose unit S is cofibrant. Let K be a cofibrant object of 𝒟, and let 𝒞 be a left proper cellular 𝒟-model category. Suppose A, B ∈ Sp^{ℕ}(𝒞, G), where A is cofibrant, and we have maps f_{n}: A_{n} ⟶ B_{n} for all n and a homotopy H_{n}: A_{n} ⊗ K ⊗ I ⟶ B_{n+1} from f_{n+1}σ_{A} to σ_{B}(f_{n} ⊗ 1), where σ_{(−)} is the structure

map of the spectrum (−). Then there is a spectrum C, a level equivalence C ⟶^{h} A, and a map of spectra C ⟶^{g} B such that g_{n} is homotopic to f_{n}h_{n}. Furthermore, this construction is natural in an appropriate sense.

Once again, the naturality involves the homotopies H_{n} as well as the maps f_{n}. We leave the precise statement to the reader.

Proof. We define C_{n}, h_{n}, g_{n} and a homotopy H′_{n} : C_{n} ⊗ I_{n} ⟶ B_{n} from g_{n} to f_{n}h_{n}, where I_{n} is a unit interval, inductively on n, using Lemma 9.5. To get started, we take C_{0} = A_{0}, h_{0} to be the identity, g_{0} to be f_{0}, and H′_{0} to be the constant homotopy (with I_{0} = I). For the inductive step, we apply Lemma 9.5 to the diagram

C_{n} ⊗ K ⟶^{g_{n}⊗1} B_{n} ⊗ K
σ(h_{n}⊗1)↓  ↓^{σ}
A_{n+1} ⟶_{f_{n+1}} B_{n+1}

and the homotopy obtained as follows. We have a homotopy

C_{n} ⊗ K ⊗ I_{n} ⟶^{1⊗T} C_{n} ⊗ I_{n} ⊗ K ⟶^{H′_{n}⊗1} B_{n} ⊗ K ⟶^{σ} B_{n+1}

from σ(f_{n} ⊗1)(h_{n} ⊗1) to σ(g_{n} ⊗1). On the other hand, we also have the homotopy H_{n}(h_{n} ⊗ 1) from f_{n+1}σ(h_{n} ⊗ 1) to σ(f_{n} ⊗ 1)(h_{n} ⊗ 1). We can amalgamate these to get a homotopy G_{n}: C_{n} ⊗ K ⊗ I_{n+1} ⟶ B_{n+1} from f_{n+1}σ(h_{n} ⊗ 1) to σ(g_{n} ⊗ 1), and I_{n+1} is still a unit interval by Lemma 9.6. Hence Lemma 9.5 gives us an object C_{n+1}, a map σ: C_{n} ⊗ K ⟶ C_{n+1}, and a map g_{n+1} : C_{n+1} ⟶ B_{n+1} such that g_{n+1}σ = σ(g_{n} ⊗ 1). It also gives us a map h_{n+1} : C_{n+1} ⟶ A_{n+1} such that h_{n+1}σ = σ(h_{n} ⊗ 1) and a homotopy H′_{n+1} : C_{n+1} ⊗ I_{n+1} ⟶ B_{n+1} from f_{n+1}h_{n+1} to g_{n+1}. This completes the induction step and the proof (we leave naturality to the reader).

With this lemma in had we can now give the proof of Theorem 9.3.

Proof of Theorem 9.3. We first reduce to the case where K is itself symmetric. So suppose the generating cofibrations of 𝒞 have cofibrant domains, and suppose K′ is symmetric and weakly equivalent to K; this means there are weak equivalences K ⟶ RK ⟶ RK′ ⟵ K′, where R denotes a fibrant replacement functor. This means the total left derived functors X ↦ X ⊗^{L} K and X ↦ X ⊗^{L} K′ are naturally isomorphic on the homotopy category of any 𝒟-model category. In particular, it suffices to show that X ↦ X ⊗ K′ is a Quillen equivalence on Sp^{ℕ}(𝒞, K). On the other hand, by Theorem 5.5, there are 𝒟-Quillen equivalences

Sp^{ℕ}(𝒞, K) ⟶ Sp^{ℕ}(𝒞, RK) ⟶ Sp^{ℕ}(𝒞, RK′) ⟵ Sp^{ℕ}(𝒞, K′).

It therefore suffices to show that X ↦ X⊗K′ is a Quillen equivalence of Sp^{ℕ}(𝒞, K′); that is, we can assume that K itself is symmetric.

Let H denote the given homotopy from the cyclic permutation to the identity of K ⊗ K ⊗ K. Let X be a cofibrant spectrum, let σ̃ denote the structure map of X\overline{⊗}K\overline{⊗}K, and let σ denote the structure map of X ⊗ K ⊗ K. These two structure maps differ by the cyclic permutation, and therefore we are in the situation of Lemma 9.7, with A = X\overline{⊗}K\overline{⊗}K, B = X ⊗ K ⊗ K, f_{n} equal to the identity map, and H_{n} = (σ_{X} ⊗ 1 ⊗ 1)(X_{n+1} ⊗ H). It follows that we get a functor F defined on cofibrant objects of Sp^{ℕ}(𝒞, G) and natural level equivalences F X ⟶^{h} X\overline{⊗}K\overline{⊗}K and F X ⟶^{g} X ⊗ K ⊗ K, where the latter map is a level equivalence since g_{n} is homotopic to h_{n}. Thus the total left derived functors of (−)\overline{⊗}K\overline{⊗}K and (−)⊗K⊗K are naturally isomorphic. Since we know already that (−)\overline{⊗}K\overline{⊗}K is a Quillen equivalence, so is (−) ⊗ K ⊗ K, and hence so is (−) ⊗ K.

Appendix A. Cellular model categories

In this section we define cellular model categories and show that the projective model structures on Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K) are cellular when 𝒞 is so. This is necessary to be sure that the Bousfield localizations used in the paper do in fact exist. The definitions in this section are taken from [9]. Throughout this section, then, G will be a left Quillen endofunctor of 𝒞; when we refer to Sp^{Σ}(𝒞, K), we will be thinking of 𝒞 as a 𝒟-model category, where 𝒟 is some symmetric monoidal model category, and of K as a cofibrant object of 𝒟.

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

Definition A.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 hypothesis above requires considerable explanation, which we will provide below. We first point out that the second hypothesis will hold in the projective model structure on Sp^{ℕ}(𝒞, G) or Sp^{Σ}(𝒞, K) when it holds in 𝒞.

Lemma A.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 the projective model structure on Sp^{ℕ}(𝒞, G) (Sp^{Σ}(𝒞, K)) are small relative to the cofibrations in Sp^{ℕ}(𝒞, G) (Sp^{Σ}(𝒞, K)).

Proof. For Sp^{ℕ}(𝒞, G), this follows immediately from the definition of J_{G}, Proposition 1.10, and Proposition 1.12. The proof for Sp^{Σ}(𝒞, K) is similar.

We now discuss the third hypothesis.

Definition A.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.

Proposition A.4. 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 Sp^{ℕ}(𝒞, G) and in Sp^{Σ}(𝒞, K).

Proof. This is immediate, since limits in Sp^{ℕ}(𝒞, G) and Sp^{Σ}(𝒞, K) 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 A.5. 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 A.6. 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 [9, Proposition 12.5.9]. Of course, every set of cells does not give rise to a subcomplex.

We can now define compactness.

Definition A.7. 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 A.8. 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 Sp^{ℕ}(𝒞, G) or Sp^{Σ}(𝒞, K).

Proof. We will prove the proposition only for Sp^{ℕ}(𝒞, G), as the Sp^{Σ}(𝒞, K) case is similar. Throughout this proof we will use Proposition A.4, 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 [10, Lemma 2.1.13] or [9, 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 [9, 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 A.9. Suppose 𝒞 is a left proper cellular model category, and G is a left Quillen endofunctor on 𝒞. Then the category Sp^{ℕ}(𝒞, G) of G-spectra and the category Sp^{Σ}(𝒞, K) of symmetric spectra, with the projective model structures, are left proper cellular model categories.

References

[1] 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.

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

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

[4] 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.

[5] Emmanuel Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Springer-Verlag, Berlin, 1996.

[6] Alex Heller, Stable homotopy categories, Bull. Amer. Math. Soc. 74 (1968), 28–63.

[7] , Completions in abstract homotopy theory, Trans. Amer. Math. Soc. 147 (1970), 573–602.

[8] , Stable homotopy theories and stabilization, J. Pure Appl. Algebra 115 (1997), no. 2, 113–130.

[9] Philip S. Hirschhorn, Localization of model categories, preprint, available at

http://www-math.mit.edu/∼ psh, 2/22/2000.

[10] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1998.

[11] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208.

[12] J. F. Jardine, A^{1}-local symmetric spectra, preprint, 1999.

[13] Mark W. Johnson, Enriched sheaf theory as a framework for stable homotopy theory, Ph.D. thesis, University of Washington, 1999.

[14] L. G. Lewis, Jr., J. P. May, and M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, 1986.

[15] M. Mandell, J. P. May, B. Shipley, and S. Schwede, Model categories of diagram spectra, to appear in Proc. London Math. Soc., 1999.

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

[17] Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Algebra 120 (1997), no. 1, 77–104.

[18] Stefan Schwede, The stable homotopy category has a unique model at the prime 2, preprint, 2000.

[19] Stefan Schwede and Brooke Shipley, A uniqueness theorem for stable homotopy theory, preprint, 16pp., 2000.

[20] Stefan Schwede and Brooke E. Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491–511.

[21] Brooke Shipley, Symmetric spectra and topological Hochschild homology, K-Theory 19 (2000), 155–183.

[22] Vladimir Voevodsky, A^{1}-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), vol. 1998, pp. 579–604 (electronic).

[23] R. Vogt, Boardman’s stable homotopy category, Aarhus Univ. Lect. Notes, vol. 21, Aarhus University, 1970.

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

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