Theory and Applications of Categories, Vol. 14, No. 19, 2005, pp. 451–479.

GENERALIZED BROWN REPRESENTABILITY IN HOMOTOPY CATEGORIES

JIŘÍ ROSICKÝ

Abstract. Brown representability approximates the homotopy category of spectra by means of cohomology functors defined on finite spectra. We will show that if a model category K is suitably determined by λ-small objects then its homotopy category Ho(K) is approximated by cohomology functors defined on those λ-small objects. In the case of simplicial sets, we have λ = ω_{1}, i.e., λ-small means countable.

1. Introduction

There are two versions of Brown representability for a triangulated category T : the first one says that every cohomological functor T^{op} → Ab is representable and the second one deals with the representability of cohomological functors T_{0}^{op} → Ab defined on the full subcategory T_{0} of small objects. The first version is often called Brown representability for cohomology while the second one is called Brown representability for homology (see [10]). We will consider a whole hierarchy of Brown representabilities by asking whether every cohomological functor defined on λ-small objects (where λ is a regular cardinal) is representable. We will show that, for every combinatorial stable model category K, the triangulated category Ho(K) satisfies one of these λ-Brown representabilities. The consequence is that Ho(K) satisfies Brown representability for cohomology. It fits in similar results proved in [16], [35] and [29]. In fact, A. Neeman uses his new concept of well generated triangulated categories in his proof and we will show that every combinatorial model category K has Ho(K) well generated (cf. 6.10).

Moreover, we can extend our framework from triangulated categories to a general homotopy category Ho(K) of a model category K. Brown representabilities then deal with weakly continuous functors Ho(K_{λ})^{op} → Set and ask whether they are representable. Let us stress that this was (for λ = ω) the original setting considered by E. M. Brown [7]. Since the category of weakly continuous functors Ho(K_{λ})^{op} → Set coincides with the free completion Ind_{λ}(Ho(K_{λ})) of Ho(K_{λ}) under λ-filtered colimits, the question is whether the natural functor E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ})) is (essentially) surjective on objects. Our main result is that if K is combinatorial (in the sense of J. H. Smith) then there is always a regular cardinal λ such that E_{λ} is not only surjective on objects but also full, which means that we obtain Brown representability not only for objects but also for morphisms. In the rest of this introduction we will explain our approach in more detail.

Supported by the Ministry of Education of the Czech Republic under the project MSM 0021622409.

Received by the editors 2005-04-08 and, in revised form, 2005-11-28.

Transmitted by Ross Street. Published on 2005-12-20.

2000 Mathematics Subject Classification: 18G55, 55P99.

Key words and phrases: Quillen model category, Brown representability, triangulated category, accessible category.

\begin{matrix}c \\ ◯ \end{matrix} Jiří Rosický, 2005. Permission to copy for private use granted.

A model category K is combinatorial if it is accessible and cofibrantly generated. In very general terms, the theory of accessible categories studies those categories K which are determined by a full subcategory A consisting of “small” objects (see [33] and [1]). A typical example is the free completion Ind(A) of the category A under filtered colimits introduced by Grothendieck [3]. Categories Ind(A) where A is a small category are precisely finitely accessible categories. The most of Quillen model categories are finitely accessible and we may ask in what extent is Ho(Ind(A)) determined by Ho(A). Here, since A is not necessarily a model category, we understand Ho(A) as the full subcategory of Ho(Ind(A)). The best but very rare case is that

Ho(Ind(A)) ≅ Ind(Ho(A)) ,

which means that Ho(K) is finitely accessible as well. We will show that this happens for truncated simplicial sets SSet_{n} = Set^{Δ^{n}} where Δ_{n} is the category of ordinals {1, 2, …, n}. But the homotopy category Ho(SSet) of simplicial sets is not concrete (see [17]) and thus it cannot be accessible.

However, very often, one has the comparison functor

E : Ho(Ind(A)) → Ind(Ho(A))

and one can ask whether this functor is at least full and (essentially) surjective on objects. Non-faithfulness of this functor corresponds to the presence of phantoms in Ho(Ind(A)), i.e., of morphisms in Ho(Ind(A)) which are not determined by their restrictions on objects from Ho(A).

A sufficient condition for having E : Ho(Ind(A)) → Ind(Ho(A)) is that Ho(A) has weak finite colimits. In this case, Ind(Ho(A)) is the full subcategory of the functor category Set^{Ho(A)^{op}} consisting of functors Ho(A)^{op} → Set which are weakly left exact; they correspond to cohomological functors. Hence the essential surjectivity of E on objects precisely corresponds to the fact that every cohomological functor Ho(A)^{op} → Set is representable, i.e., to Brown representability of Ho(Ind(A)). The classical case is when Ind(A) is the category of spectra and A the category of finite spectra. Following Adams [2], the functor E : Ho(Ind(A)) → Ind(Ho(A)) is full and essentially surjective on objects.

Categories Ind(A) are important for large A as well, for instance in the dual setting of Pro(A) = (Ind(A^{op}))^{op}, i.e., Pro(A) is the free completion of A under filtered limits. Here, the functor

E : Ho(Pro(A)) → Pro(Ho(A))

is considered in [15]. Their rigidification question asks which objects belong to the image of E. In the special case of A being the category Top of topological spaces, the functor E is the comparison between the strong shape category and the shape category (see [37]).

GENERALIZED BROWN REPRESENTABILITY IN HOMOTOPY CATEGORIES 453

For every regular cardinal λ, there is the free completion Ind_{λ}(A) of A under λ-filtered colimits. While K = Ind(A) consists of filtered colimits of objects from A, Ind_{λ}(A) is the full subcategory of K consisting of λ-filtered colimits of objects from A. For any A, Ind(A) = Ind_{λ}(K_{λ}) where K_{λ} consists of filtered colimits of objects from A of size < λ (see [33], 2.3.11). In the same way as above, we get the functor

E_{λ} : Ho(Ind_{λ}(K_{λ})) → Ind_{λ}(Ho(K_{λ}))

(then E = E_{ω}). As we have mentioned, our main result says that, for any combinatorial model category K, there is a regular cardinal λ such that E_{λ} is full and essentially surjective on objects. This means that the homotopy category Ho(K) is approximated by the category Ind_{λ}(Ho(K_{λ})) of cohomological functors Ho(K_{λ})^{op} → Set defined on λ-small objects. This approximation does not distinguish morphisms f, h : K → L in Ho(K) which are λ-phantom equivalent, i.e., which have the same composition with each morphism A → K where A is λ-small.

For instance, if Ind(A) is the category of simplicial sets then λ = ω_{1}. It seems to be unknown whether E_{ω} is essentially surjective in this case, i.e., whether every weakly left exact functor H : Ho(A)^{op} → Set is representable. E. M. Brown proved this in the special case when H takes countable values (see [7]) and F. Adams [2] in the case that H takes values in the category of groups.

In the case when Ind(A) is the category Sp of spectra, our result again yields that E_{λ} is full and essentially surjective on objects for λ = ω_{1}. A consequence is that Ho(Sp) has minimal λ-filtered colimits of objects from Ho(Sp_{λ}) for λ = ω_{1} and, more generally, for each ω_{1} ◁ λ. This has been known for λ = ω (see [34]). We thus contribute to the still open problem whether Ho(Sp) has all minimal filtered colimits (see [34]).

There is well known that, for each model category K, the homotopy category Ho(K) has weak (co)limits (see, e.g., [34], [24] or [9]). They are constructed from coproducts and homotopy pushouts in the same way as colimits are constructed from coproducts and pushouts. The trivial observation about colimits is that a category D with a terminal object d^{∗} has each colimit colim D, D : D → K isomorphic to Dd^{∗}. We will show, which is non-trivial and seems to be new, that the same holds for our weak colimits in homotopy categories. This implies that, for any combinatorial model category K, there is a regular cardinal λ such that homotopy λ-filtered colimits are weak colimits, which leads to our generalized Brown representability. The proved property of weak λ-filtered colimits was called Ho(K_{λ})-priviliged in [21].

2. Basic concepts

A model structure on a category K will be understood in the sense of Hovey [22], i.e., as consisting of three classes of morphisms called weak equivalences, cofibrations and fibrations which satisfy the usual properties of Quillen [38] and, moreover, both (cofibration, trivial fibrations) and (trivial cofibrations, fibration) factorizations are functorial. Recall that trivial (co)fibrations are those (co)fibrations which are in the same time weak equivalences. The (cofibration, trivial fibration) factorization is functorial if there is a functor F : K^{→} → K and natural transformations α : dom → F and β : F → cod such that f = β_{f}α_{f} is the (cofibration, trivial fibration) factorization of f. Here K^{→} denotes the category of morphisms in K and dom : K^{→} → K (cod : K^{→} → K) assign to each morphism its (co)domain. The same for (trivial cofibration, fibration) factorization (see [39]) .

A model category is a complete and cocomplete category together with a model structure. In a model category K, the classes of weak equivalences, cofibrations and fibrations will be denoted by W, C and F, resp. Then C_{0} = C ∩ W and F_{0} = F ∩ W denote trivial cofibrations and trivial fibrations, resp. We have

F_{0} = C^{□} , F = C_{0}^{□} , C = ^{□}F_{0} and C_{0} = ^{□}F

where C^{□} denotes the class of all morphisms having the right lifting property w.r.t. each morphism from C and^{□}F denotes the class of all morphisms having the left lifting property w.r.t. each morphism of F. K is called cofibrantly generated if there are sets of morphisms I and J such that F_{0} = I^{□} and F = J^{□}. If K is locally presentable then C is the closure of I under pushouts, transfinite compositions and retracts in comma-categories K ↓ K and, analogously, C_{0} is this closure of J.

An object K of a model category K is called cofibrant if the unique morphism 0 → K from an initial object is a cofibration and K is called fibrant if the unique morphism K → 1 to a terminal object is a fibration. Let K_{c}, K_{f} or K_{cf} denote the full subcategories of K consisting of objects which are cofibrant, fibrant or both cofibrant and fibrant resp. We get the cofibrant replacement functor R_{c} : K → K and the fibrant replacement functor R_{f} : K → K. We will denote by R = R_{f}R_{c} their composition and call it the replacement functor. The codomain restriction of the replacement functors are R_{c} : K → K_{c}, R_{f} : K → K_{f} and R : K → K_{cf}.

Let K be a model category and K an object of K. Recall that a cylinder object C(K) for K is given by a (cofibration, weak equivalence) factorization

∇ : K ⨿ K ⟶^{γ_{K}} C(K) ⟶^{σ_{K}} K

of the codiagonal ∇. Morphisms f, g : K → L are left homotopic if there is a morphism h : C(K) → L with

f = hγ_{1K} and g = hγ_{2K}

where γ_{1K} = γ_{K}i_{1} and γ_{2K} = γ_{K}i_{2} with i_{1}, i_{2} : K → K ⨿K being the coproduct injections. In fact, cylinder objects form a part of the cylinder functor C : K → K and γ_{1}, γ_{2} : Id → C are natural transformations.

On K_{cf}, left homotopy ∼ is an equivalence relation compatible with compositions, it does not depend on a choice of a cylinder object and we get the quotient

Q : K_{cf} → K_{cf}/∼ .

The composition

P : K ⟶^{R} K_{cf} ⟶^{Q} K_{cf}/∼

is, up to equivalence, the projection of K to the homotopy category Ho(K) = K[W^{−1}] (see [22]). In what follows, we will often identify K_{cf}/∼ with Ho(K).

A category K is called λ-accessible, where λ is a regular cardinal, provided that

(1) K has λ-filtered colimits,

(2) K has a set A of λ-presentable objects such that every object

of K is a λ-filtered colimit of objects from A. Here, an object K of a category K is called λ-presentable if its hom-functor hom(K, −) : K → Set preserves λ-filtered colimits; Set is the category of sets. A category is called accessible if it is λ-accessible for some regular cardinal λ. The theory of accessible categories was created in [33] and for its presentation one can consult [1]. We will need to know that λ-accessible categories are precisely categories Ind_{λ}(A) where A is a small category. If idempotents split in A then A precisely consists of λ-presentable objects in Ind(A). In what follows, we will denote by K_{λ} the full subcategory of K consisting of λ-presentable objects.

A locally λ-presentable category is defined as a cocomplete λ-accessible category and it is always complete. Locally λ-presentable categories are precisely categories Ind_{λ}(A) where the category A has λ-small colimits, i.e., colimits of diagrams D : D → A where D has less then λ morphisms. In general, the category Ind_{λ}(A) can be shown to be the full subcategory of the functor category Set^{A^{op}} consisting of λ-filtered colimits H of hom-functors hom(A, −) with A in A. In the case that A has λ-small colimits this is equivalent to the fact that H : A^{op} → Set preserves λ-small limits. More generally, if A has weak λ-small colimits then Ind_{λ}(A) precisely consists of left λ-covering functors (see [26] 3.2). Let us recall that a weak colimit of a diagram D : D → A is a cocone from D such that any other cocone from D factorizes through it but not necessarily uniquely. If X is a category with weak λ-small limits then a functor H : X → Set is left λ-covering if, for each λ-small diagram D : D → X and its weak limit X, the canonical mapping H(X) → lim HD is surjective (see [8] for λ = ω). A left λ-covering functor preserves all λ-small limits which exist in X. Moreover, a functor H : X → Set is left λ-covering iff it is weakly λ-continuous, i.e., iff it preserves weak λ-small limits. This immediately follows from [8], Proposition 20 and the fact that surjective mappings in Set split. A functor H is called weakly continuous if it preserves weak limits. Hence a weakly continuous functor H : X → Set preserves all existing limits.

A functor F : K → L is called λ-accessible if K and L are λ-accessible categories and F preserves λ-filtered colimits. An important subclass of λ-accessible functors are those functors which also preserve λ-presentable objects. In the case that idempotents split in B, those functors are precisely functors Ind_{λ}(G) where G : A → B is a functor. The uniformization theorem of Makkai and Paré says that for each λ-accessible functor F there are arbitrarily large regular cardinals μ such that F is μ-accessible and preserves μ-presentable objects (see [1] 2.19). In fact, one can take λ ◁ μ where ◁ is the set theoretical relation between regular cardinals corresponding to the fact that every λ-accessible category is μ-accessible (in contrast to [1] and [33], we accept λ◁λ). For every λ there are arbitrarily large regular cardinals μ such that λ ◁ μ. For instance, ω ◁ μ for every regular cardinal μ.

3. Combinatorial model categories

We will follow J. H. Smith and call a model category K λ-combinatorial if K is locally λ-presentable and both cofibrations and trivial cofibrations are cofibrantly generated by sets I and J resp. of morphisms having λ-presentable domains and codomains. Then both trivial fibrations and fibrations are closed in K^{→} under λ-filtered colimits. K will be called combinatorial if it is λ-combinatorial for some regular cardinal λ.

The following result is due to J. H. Smith and is presented in [12], 7.1 and 7.2. We just add a little bit more detail to the proof.

3.1. Proposition. [Smith] Let K be a combinatorial model category. Then the functors K^{→} → K giving (cofibration, trivial fibration) and (trivial cofibration, fibration) factorizations are accessible.

Proof. This means that there is a regular cardinal λ such that K (and hence K^{→} are locally λ-presentable and the (cofibration, trivial fibration) factorization A → C → B of a morphism A → B preserves λ-filtered colimits; the same for the (trivial cofibration, fibration) factorization. There is a regular cardinal λ such that K is locally λ-presentable and domains and codomains of morphisms from the generating set I of cofibrations are λ-presentable. For every morphism f : A → B form a colimit F_{0}f of the diagram

[figure: A; X; u; h; Y]

consisting of all spans (u, h) with h : X → Y in I such that there is v : Y → B with vh = fu. Let α_{0f} : A → F_{0}f denote the component of the colimit cocone (the other components are Y → F_{0}f and they make all squares

\xymatrix{A \ar[r]^{α_{0f}} & F_{0}f \\ X \ar[u]^{u} \ar[r]_{h} & Y \ar[u]}

to commute). Let β_{0f} : F_{0}f → B be the morphism induced by f and v’s. Then F_{0} : K^{→} → K is clearly λ-accessible. Let F_{i}f, α_{if} and β_{if}, i ≤ λ, be given by the following transfinite induction: F_{i+1}f = F_{0}β_{if}, α_{i+1,f} = α_{0,β_{if}} α_{if}, β_{i+1,f} = β_{0,β_{if}} and the limit step is given by taking colimits. Then all functors F_{i} : K^{→} → K, i ≤ λ are λ-accessible and F_{λ} yields the desired (cofibration, trivial fibration) factorization.

3.2. Remark. Following the uniformization theorem ([1] Remark 2.19), there is a regular cardinal μ such that the functors from 3.1 are μ-accessible and preserve μ-presentable objects. This means that the factorizations A → C → B of a morphism A → B have C μ-presentable whenever A and B are μ-presentable. This point is also well explained in [12].

3.3. Notation. Let K be a locally presentable model category. Consider the following conditions

(G_{λ}^{1}) the functor F : K^{→} → K giving the (cofibration, trivial fibration) factorization is λ-accessible and preserves λ-presentable objects,

(G_{λ}^{2}) the replacement functor R : K → K (being the composition R = R_{f}R_{c} of the cofibrant and the fibrant replacement functors) is λ-accessible and preserves λpresentable objects, and

(G_{λ}^{3}) weak equivalences are closed under λ-filtered colimits in K^{→}.

3.4. Remark. (1) (G_{λ}^{1}) implies that the functor R_{c} is λ-accessible and preserves λ-presentable objects. Thus (G_{λ}^{2}) only adds that R_{f} is λ-accessible and preserves λ-presentable objects. (G_{λ}^{1}) also implies

(G_{λ}^{4}) the cylinder functor C : K → K is λ-accessible and preserves λ-presentable objects.

(2) Following [1] 2.11 and 2.20, if K satisfies (G_{λ}^{i}) and λ ◁ μ then K satisfies (G_{μ}^{i}) for i = 1, 2, 3, 4. In particular, if K satisfies (G_{ω}^{i}) then it satisfies (G_{λ}^{i}) for any regular cardinal λ (see [1] 2.13 (1)).

3.5. Proposition. Let K be a combinatorial model category. Then there is a regular cardinal λ such that K satisfies the conditions (G_{λ}^{i}) for i = 1, 2, 3.

Proof. Let K be a combinatorial model category. It immediately follows from 3.4 that there are arbitrarily large regular cardinals λ such that the conditions (G_{λ}^{1}) and (G_{λ}^{2}) are satisfied. Following [12] 7.3, there is λ such that (G_{λ}^{3}) holds. This proves the theorem.

Combinatorial model categories form a very broad class. We will discuss the conditions (G_{λ}^{i}) in a couple of examples.

3.6. Examples. (i) The model category SSet of simplicial sets is ω-combinatorial and satisfies (G_{ω}^{1}), (G_{ω_{1}}^{2}) and (G_{ω}^{3}). The first and the third statements are clear and the second one follows from the fact that finitely presentable simplicial sets have ω_{1}-presentable fibrant replacements. This observation can be found in [27], Section 5, as well.

The same is true for the model category SSet_{∗} of pointed simplicial sets.

(ii) The category Sp of spectra with the strict model category structure (in the sense of [5]) is ω-combinatorial (see [40] A.3). We will show that it satisfies the conditions (G_{ω_{1}}^{1}), (G_{ω_{1}}^{2}) and (G_{ω}^{3}).

Let us recall that a spectrum X is a sequence (X_{n})_{n=0}^{∞} of pointed simplicial sets equipped with morphisms σ_{n}^{X} : ΣX_{n} → X_{n+1} where Σ is the suspension functor. This means that ΣX_{n} = S^{1} ∧ X_{n} where S^{1} ∧ − is the smash product functor, i.e., a left adjoint to

−^{S^{1}} = hom(S^{1}, −) : SSet_{∗} → SSet_{∗}.

A spectrum X is ω_{1}-presentable iff all X_{n}, n ≥ 0, are ω_{1}-presentable in SSet_{∗}. The strict model structure on Sp has level equivalences as weak equivalences and level fibrations as fibrations. This means that f : X → Y is a weak equivalence (fibration) iff all f_{n} : X_{n} → Y_{n} are weak equivalences (fibrations) in SSet_{∗}. A morphism f : X → Y is a (trivial) cofibration iff f_{0} : X_{0} → Y_{0} is a (trivial) cofibration and all induced morphisms t_{n} : Z_{n} → Y_{n}, n ≥ 1, from pushouts are (trivial) cofibrations

\xymatrix{ΣX_{n−1} \ar[r]^{σ_{n−1}^{X}} \ar[d]_{Σf_{n−1}} & X_{n} \ar[d] \ar[ddr]^{f_{n}} \\ ΣY_{n−1} \ar[r] \ar[drr]_{σ_{n−1}^{Y}} & Z_{n} \ar[dr]^{t_{n}} \\  &  & Y_{n}}

(see [5], [25] or [23]). Then a (cofibration, trivial fibration) factorization X ⟶^{g} Z ⟶^{h} Y of a morphism f : X → Y is made as follows.

One starts with a (cofibration, trivial fibration) factorization

f_{0} : X_{0} ⟶^{g_{0}} Z_{0} ⟶^{h_{0}} Y_{0}

in SSet_{∗}. Then one takes a (cofibration, trivial fibration) factorization

t : Z′_{1} ⟶^{u} Z_{1} ⟶^{h_{1}} Y_{1}

of the induced morphism from a pushout

\xymatrix{ΣX_{0} \ar[r]^{σ_{0}^{X}} \ar[d]_{Σg_{0}} & X_{1} \ar[d]_{q} \ar[ddr]^{f_{1}} \\ ΣZ_{0} \ar[r]^{p} \ar[drr]_{σ_{0}^{Y} ·Σh_{0}} & Z′_{1} \ar[dr]_{t} \\  &  & Y_{1}}

and puts σ_{1}^{Z} = up and g_{1} = uq. This yields

f_{1} : X_{1} ⟶^{g_{1}} Z_{1} ⟶^{h_{1}} Y_{1}

and one continues the procedure. Analogously, one constructs a (trivial cofibration, fibration) factorization. It is now easy to see that the strict model structure on Sp satisfies (G_{ω_{1}}^{1}) and (G_{ω_{1}}^{2}). (G_{ω}^{3}) is follows from (i).

(iii) The model category Sp of spectra with the stable Bousfield-Friedlander model category structure (see [5]) is ω-combinatorial (see [40] A.3). We will show that it satisfies the conditions (G_{ω_{1}}^{1}), (G_{ω_{1}}^{2}) and (G_{ω}^{3}) too.

The stable model structure is defined as a Bousfield localization of the strict model structure, i.e., by adding a set of new weak equivalences. Cofibrations and trivial fibrations remain unchanged, which means that the condition (G_{ω_{1}}^{1}) is satisfied following (ii). Weak equivalences are closed under filtered colimits in Sp^{→}, which follows from their characterization (see, e.g., [25] 4.2.2). Stably fibrant spectra are those strictly fibrant spectra X for which the adjoint transposes σ̃_{h}^{X} : X_{n} → X_{n+1}^{S^{1}} of structure morphisms are weak equivalences. For checking the condition (G_{ω_{1}}^{2}), it suffices to show that the stable fibrant replacement functor R_{f} preserves ω_{1}-presentable objects. But this follows from [5] or [23]: consider the functor

Θ : Sp → Sp

such that

(ΘX)_{n} = X_{n+1}^{S^{1}}

and

σ_{n}^{ΘX} = (σ̃_{n}^{X})^{S^{1}} .

Let Θ^{∞}X be a colimit of the chain

X ⟶^{ι_{X}} ΘX ⟶^{Θι_{X}} Θ^{2}X → …

where ι_{X_{n}} = σ̃_{n}^{X} : X_{n} → X_{n+1}^{S^{1}}. Then Θ^{∞} is stably fibrant (see [23] 4.6). Then a stable fibrant replacement of X is defined by a (cofibrant, trivial fibrant) factorization of ι_{X}^{∞}

X ⟶ R_{f}X ⟶ Θ^{∞}X .

It is easy to see that R_{f}X is ω_{1}-presentable whenever X is ω_{1}-presentable.

4. Weak colimits

There is well known that the homotopy category of any model category K has products, coproducts, weak limits and weak colimits. We will recall their constructions.

4.1. Remark. (i) Let K_{i}, i ∈ I be a set of objects of K. Without any loss of generality, we may assume that they are in K_{cf}. Then their product in K

p_{i} : K → K_{i}

is fibrant and let

q_{K} : R_{c}K → K

be its cofibrant replacement. Then R_{c}K ∈ K_{cf} and

Q(p_{i}q_{K}) : QR_{c}K → QK_{i}

is a product in Ho(K). Recall that Q : K_{cf} → Ho(K) = K_{cf}/∼ is the quotient functor. In fact, consider morphisms

Qf_{i} : QL → QK_{i} , i ∈ I

in K_{cf}/∼. Let f : L → K be the induced morphism and g : L → R_{c}K be given by the lifting property:

\xymatrix{0 \ar[r] \ar[d] & R_{c}K \ar[d]^{q_{K}} \\ L \ar@{-->}[ur]_{g} \ar[r]_{f} & K}

We have Q(p_{i}q_{K}g) = Qf_{i} for each i ∈ I. The unicity of g follows from the facts that Qq_{K} is an isomorphism and that left homotopies h_{i} from p_{i}f to p_{i}f′, i ∈ I, lift to the left homotopy from f to f′.

Since K^{op} is a model category and

Ho(K^{op}) = (Ho(K))^{op} ,

Ho(K) has coproducts.

(ii) In order to show that Ho(K) has weak colimits, it suffices to prove that it has weak pushouts. In fact, a weak coequalizer

[figure: A; f; g; B; h; D]

is given by a weak pushout

\xymatrix{B \ar[r]^{h} & D \\ A ⨿ B \ar[u]^{(f,id_{B})} \ar[r]_{(g,id_{B})} & B \ar[u]_{h}}

and weak colimits are constructed using coproducts and weak coequalizers in the same way as colimits are constructed by coproducts and coequalizers.

Let

\xymatrix{B \\ A \ar[u]^{f} \ar[r]_{g} & D}

be a diagram in K. Consider a pushout

\xymatrix{B_{1} \ar[r]^{\overline{g}} & E \\ A \ar[u]^{f_{1}} \ar[r]_{g_{1}} & D_{1} \ar[u]_{\overline{f}}}

in K where f = f_{2}f_{1} and g = g_{2}g_{1} are (cofibration, trivial fibration) factorizations. Then

\xymatrix{P B_{1} \ar[r]^{P \overline{g}} & P E \\ P A \ar[u]^{P f_{1}} \ar[r]_{P g_{1}} & P D \ar[u]_{P \overline{f}}}

is a weak pushout in Ho(K) which is called the homotopy pushout of the starting diagram. Recall that P : K → Ho(K) is the canonical functor.

Following [9], we will call the resulting weak colimits in Ho(K) standard. By duality, Ho(K) has weak limits.

(iii) Consider a diagram D : D → K, its colimit (δ̄_{d} : Dd → \overline{K}) and (δ_{d} : Dd → K) such that (P δ_{d} : P Dd → P K) is a standard weak colimit of P D. There is the comparison morphism p : K → \overline{K} such that P (k)δ_{d} = P (\overline{δ}_{d}) for each d ∈ D. It suffices to find this morphism for a pushout diagram

\xymatrix{B \ar[r]^{g′} & \overline{E} \\ A \ar[u]^{f} \ar[r]_{g} & D \ar[u]_{f′}}

But it is given by p\overline{g} = g′f_{2} and p\overline{f} = f′g_{2}; we use the notation from (ii).

(iv) There is another construction of weak pushouts in Ho(K). Consider

\xymatrix{B \\ A \ar[u]^{f} \ar[r]_{g} & D}

in K_{cf}. Form the double mapping cylinder of f, g, i.e., the colimit

\xymatrix{B \ar[rr]^{\overline{g}} &  & E \\ A \ar[u]^{f} \ar[r]^{γ_{1A}} & C(A) \ar@{-->}[ur]^{t} \\  & A \ar[u]_{γ_{2A}} \ar[r]_{g} & D \ar[uu]_{\overline{f}}}

of the diagram

\xymatrix{B \\ A \ar[u]^{f} \ar[r]^{γ_{1A}} & C(A) \\  & A \ar[u]_{γ_{2A}} \ar[r]_{g} & D}

where C(A) is the cylinder object. Then

\xymatrix{QB \ar[r]^{P \overline{g}} & P E \\ QA \ar[u]^{Qf} \ar[r]_{Qg} & QD \ar[u]_{P \overline{f}}}

is a weak pushout in Ho(K) (cf. [28]).

We will show that homotopy pushouts and double mapping cylinders are naturally weakly equivalent in SSet. The double mapping cylinder is given by the pushouts

\xymatrix{B \ar[r]^{j_{f}} & E_{1} \ar[r]^{^{\overline{g}^{2}}} & E \\ A \ar[u]^{f} \ar[r]_{γ_{1A}} & C(A) \ar[u]_{f′} \\  & A \ar[u]_{γ_{2A}} \ar[r]_{g} & D \ar[uu]_{\overline{f}}}

The left (square) pushout is called the mapping cylinder of f. Since γ_{1A} is a trivial cofibration, j_{f} is a trivial cofibration too. Since fσ_{A}γ_{1A} = f, there is a unique morphism q_{f} : E_{1} → B with q_{f}j_{f} = id_{B} and q_{f}f′ = fσ_{A}. Thus q_{f} is a weak equivalence. Since, in SSet, f′γ_{2A} is a cofibration,

q_{f}(f′γ_{2A}) = fσ_{A}γ_{2A} = f

is a (cofibration, weak equivalence) factorization of f. Hence, following [20] 13.3.4 and 13.3.8, the right (rectangle) pushout is naturally weakly equivalent to the homotopy pushout of f and g.

(v) Another, and very important, colimit construction in model categories are homotopy colimits (see, e.g., [6], [13], [20]). Both coproducts and homotopy pushouts described above are instances of this concept. While weak colimits correspond to homotopy commutative diagrams, homotopy colimits corrspond to homotopy coherent ones. So, one cannot expect that homotopy colimits are weak colimits. There is a construction of homotopy colimits using coproducts and homotopy pushouts which is presented in [32] 5.1.1 in the context of Segal categories. It extends the usual construction to higher homotopies and witnesses the fact that homotopy colimits are weak colimits. There is always a morphism wcolim D → hocolim D from the standard weak colimit to the homotopy colimit for each diagram D : D → K.

Homotopy colimits tend to have homotopy cofinality properties. For example, if D has a terminal object d^{∗} then the natural morphism Dd^{∗} → hocolim D is a weak equivalence (see [20] 19.6.8(1) or [6] 3.1(iii)). We will show that standard weak colimits have this property in each model category K. As the author knows, this is a new result which will be used to prove the Brown representability property for morphisms (see Theorem 5.7). It also has an impact to a relation between homotopy filtered colimits and homotopy colimits (see Remark 4.4).

4.2. Proposition. Let K be a model category, P : K → Ho(K) the canonical functor, D a small category having a terminal object d^{∗}, D : D → K_{c} a functor and (P δ_{d} : P Dd → P K) the standard weak colimit of P D. Then δ_{d^{∗}} is a weak equivalence.

Proof. Consider the construction of K using coproducts and weak coequalizers

\xymatrix{Dd \ar[d]_{u_{e}} \ar[dr]^{v_{d}} \\ ∐_{e:d→d′} Dd \ar[r]^{f} \ar[r]_{g} & ∐_{d} Dd \ar[r]^{h} & K \\ Dd \ar[u]^{u_{e}} \ar[r]_{De} & Dd′ \ar[u]_{v_{d′}}}

where u_{e} : Dd → C, e : d → d′ in D and v_{d} : Dd → ∐_{d} Dd, d in D are coproduct injections

and h is a standard weak coequalizer of f and g.

I. Assume at first that K = SSet. This means that h is given by the double mapping cylinder (see 4.1 (iv))

\xymatrix{B \ar[rr]^{h} &  & K \\ A ⨿ B \ar[u]^{(f,id_{B})} \ar[r]^{γ_{1,A⨿B}} & C(A ⨿ B) \ar[ur]_{t} \\  & A ⨿ B \ar[u]_{γ_{2,A⨿B}} \ar[r]_{(g,id_{B})} & B \ar[uu]_{h′}}

where A = ∐_{e} Dd and B = ∐_{d} Dd. We have δ_{d} = hv_{d} for d in D. Since the cylinder functor

C(−) = Δ_{1}×− preserves colimits, we have C(A⨿B) = C(A)⨿C(B) and γ_{i,A⨿B} = γ_{iA}⨿γ_{iB} for i = 1, 2. It is easy to see that tC(j_{2}) : C(B) → K is a monomorphism and thus a cofibration; here

\xymatrix{A \ar[r]_{j_{1}} & A ⨿ B & B \ar[l]^{j_{2}}}

are coproduct injections. Since σ_{B} is a weak equivalence, the morphism w in the pushout below is a weak equivalence (see [20] 18.1.2):

\xymatrix{B \ar[r]^{\overline{t}} & L \\ C(B) \ar[u]^{σ_{B}} \ar[r]_{tC(j_{2})} & K \ar[u]_{w}}

Clearly, L appears as a colimit

\xymatrix{B \ar[rr]^{wh} &  & L \\ A ⨿ B \ar[u]^{(f,id_{B})} \ar[r]^{γ_{1,A}⨿id_{B}} & C(A) ⨿ B \ar[ur]_{(wt_{1},wh)} \\  & A ⨿ B \ar[u]_{γ_{2,A}⨿id_{B}} \ar[r]_{(g,id_{B})} & B \ar[uu]_{wh}}

where t = (t_{1}, t_{2}) : C(A) ⨿ C(B) → K. Let e_{d} : d → d^{∗} denote the unique morphism from d to the terminal object d^{∗} in D. Since

(De_{d})_{e}σ_{A}γ_{1A} = (De_{d})_{e} = (De_{d})_{d}f

and

(De_{d})_{e}σ_{A}γ_{2A} = (De_{d})_{e} = (D(e_{d′}e))_{e} = (De_{d})_{d}g ,

we get a unique morphisms p : L → Dd^{∗} such that pwh = (De_{d})_{d} and pwt_{1} = (De_{d})_{e:d→d′} · σ_{A}. For the morphism whv_{d^{∗}} : Dd^{∗} → L we have p(whv_{d^{∗}}) = id_{Dd^{∗}}. It suffices to find a left homotopy q : C(L) → L such that qγ_{1L} = id_{L} and qγ_{2L} = (whv_{d^{∗}})p. In this case, P (whv_{d^{∗}}p) is an isomorphism and thus wδ_{d^{∗}} = whv_{d^{∗}} is a weak equivalence. Therefore δ_{d^{∗}} is a weak equivalence.

We have a colimit

\xymatrix{C(B) \ar[rr]^{C(wh)} &  & C(L) \\ C(A) ⨿ C(B) \ar[u]^{(C(f),id_{CB})} \ar[r]^{C(γ_{1,A}⨿id)} & CC(A) ⨿ C(B) \ar[ur]_{(C(wt_{1}),C(wh))} \\  & C(A) ⨿ C(B) \ar[u]_{C(γ_{2,A}⨿id)} \ar[r]_{(C(g),id_{B})} & C(B) \ar[uu]_{C(wh)}}

The desired morphism q : C(L) → L will be uniquely determined by the composition \overline{q} = qC(wt_{1}) : CC(A) → L, i.e., by the morphisms

\overline{q}_{e} = \overline{q}CC(u_{e}) : CC(Dd) → L

where e : d → d′ is in D. The simplicial set CC(Dd) = Δ_{1} × Δ_{1} × Dd has points (ik, x) where i, k = 0, 1 and x is a point of Dd. Then \overline{q}_{e}(0k, x) = wh(x) and \overline{q}_{e}(1k, x) = whv_{d^{∗}}D(e_{d})(x). Edges (= 1-simplices) of Δ_{1} × Δ_{1} × Dd are (s_{1}, s) where s_{1} is an edge of Δ_{1} × Δ_{1} and s is an edge of Dd. Now,

\overline{q}_{e}(s_{1}, s) = \begin{cases}wh(s) & for s_{1} = (00, 01) \\ whv_{d^{∗}}D(e_{d})(s) & for s_{1} = (10, 11) \\ wt_{1}C(u_{e_{d}})(s) & otherwise. \end{cases}

Recall that Δ_{1} × Δ_{1} has the following edges

\xymatrix{10 \ar[r] & 11 \\ 00 \ar[u] \ar[r] \ar[ur] & 01 \ar[u]}

Analogously, we define \overline{q}_{e} on n-simplices for n > 1. II. Assume now that K is the functor category SSet^{X} (where X is a small category) with the Bousfield-Kan model category structure. This means that (trivial) cofibrations in K are generated by F_{X}(C) (F_{X}(C_{0})) where

F_{X} : SSet → K

is left adjoint to the evaluation ev_{X} : K → SSet at X ∈ X (i.e., ev_{X}(A) = A(X)). Hence any generating cofibration F_{X}(f), f : K → L is a pointwise cofibration because

(F_{X}(f))_{Y} = ∐_{X (X,Y)} f : ∐_{X (X,Y)} K → ∐_{X (X,Y)} L .

Since the evaluation functors ev_{X} preserves colimits, every cofibration φ in K is a pointwise cofibration (i.e., φ_{X}, X ∈ X are cofibrations). Since homotopy pushouts are homotopy invariant in SSet (see [20] 13.3.4), they do not depend on a choice of a (cofibration, trivial fibration) factorization. Consequently, the homotopy pushouts, and thus the standard weak colimits, in K are pointwise in the sense that (δ_{X} : (Dd)(X) → K(X)) is a standard weak colimit for each X ∈ X provided that (δ : Dd → K) is a standard weak colimit in K. Since weak equivalences in K are precisely pointwise weak equivalences, the claim follows from I. III. Let K be an arbitrary model category. Following [11], there is a left Quillen functor H : SSet^{D^{op}} → K such that HY = D where Y : D → SSet^{D^{op}} is given by taking the discrete simplicial presheaves. Since H is left Quillen, it preserves colimits, (trivial) cofibrations and weak equivalences between cofibrant objects (see [22]). Since discrete simplicial presheaves are cofibrant, H preserves the standard weak colimit (δ_{d} : Y Dd → K)_{d} of Y D : D → SSet^{D^{op}}. Since cofibrant objects are closed under coproducts and homotopy pushouts, K is cofibrant. Hence H(δ_{d^{∗}}) is a weak equivalence, which proves the claim.

4.3. Proposition. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{1}) and (G_{λ}^{3}). Let D : D → K_{c} a λ-filtered diagram where card D ≥ λ. Then the comparison morphism from 4.1(iii) is a weak equivalence.

Proof. Since D is λ-filtered and card D ≥ λ, D is a λ-directed union of subcategories E such that card E < λ and E has a terminal object d_{E}. Let D_{E} : E → K_{c} be the domain restriction of D. Following 4.2 the comparison morphism p_{E} : K_{E} → \overline{K}_{E} for D_{E} is a weak equivalence for each E (because p_{E} = δ_{d_{E}} in this case). Following (G_{λ}^{1}), the formation of homotopy pushouts preserves λ-filtered colimits. Therefore K = colim_{E}K_{E} and, of course,

\overline{K} = colim_{E}\overline{K}_{E}. Consequently, p = colim_{E} p_{E} and, following (G_{λ}^{3}), p is a weak equivalence.

4.4. Remark. (1) We have not needed the full strength of (G_{λ}^{1}) – it suffices to assume that F is λ-accessible.

(2) Under (G_{λ}^{3}), homotopy λ-filtered colimits are the same as λ-filtered colimits (in the sense that the comparison morphism from 4.1 (v) is a weak equivalence), see, e.g., [12], the proof of 4.7. The consequence is that, in locally λ-presentable model categories satisfying (G_{λ}^{1}) and (G_{λ}^{3}), standard weak λ-filtered colimits and homotopy λ-filtered colimits of diagrams D : D → K with card D ≥ λ coincide.

5. Brown model categories

Given a small, full subcategory A of a category K, the canonical functor

E_{A} : K → Set^{A^{op}}

assigns to each object K the restriction

E_{A}K = hom(−, K)/A^{op}

of its hom-functor hom(−, K) : K^{op} → Set to A^{op} (see [1] 1.25). This functor is (a) A-full and (b) A-faithful in the sense that

(a) for every f : E_{A}A → E_{A}K with A in A there is f′ : A → K such that E_{A}f′ = f and

(b) E_{A}f = E_{A}g for f, g : A → K with A in A implies f = g. Let K be a locally λ-presentable model category and denote by Ho(K_{λ}) the full subcategory P (K_{λ}) of Ho(K) consisting of P -images of λ-presentable objects in K in the canonical functor P : K → Ho(K). Let E_{λ} denote the canonical functor E_{Ho(K_{λ})}.

5.1. Proposition. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{2}) and (G_{λ}^{4}). Then the composition

E_{λ}P : K → Set^{Ho(K_{λ})^{op}}

preserves λ-filtered colimits.

Proof. Consider a λ-filtered diagram D : D → K and its colimit (k_{d} : Dd → K) in K. Since R preserves λ-filtered colimits, (Rk_{d} : RDd → RK) is a λ-filtered colimit. Let X ∈ K_{λ} and f : P X → P K be a morphism in Ho(K). Then f = Q\overline{f} for \overline{f} : RX → RK and, since R preserves λ-presentable objects, RX is λ-presentable in K. Thus \overline{f} = R(k_{d})\overline{g} for some \overline{g} : RX → RDd, d ∈ D.

Assume that Q(R(k_{d})\overline{g}_{1}) = Q(R(k_{d})\overline{g}_{2}) for \overline{g}_{1}, \overline{g}_{2} : RX → RDd. Then R(k_{d})\overline{g}_{1} and R(k_{d})\overline{g}_{2} are homotopy equivalent and thus R(k_{d})\overline{g}_{i} = hγ_{iR(X)} for i = 1, 2. Since CRX is λ-presentable as well, there is e : d → d′ in D and \overline{h} : CRX → RDd′ such that R(k_{d′})\overline{h} = h and \overline{h}γ_{iR(X)} = RD(e)\overline{g}_{i}. Therefore Q(RD(e)\overline{g}_{1}) = Q(RD(e)\overline{g}_{2}). We have proved that

( hom(P X, P (k_{d})) : hom(P X, P Dd) → hom(P X, P K))_{d∈D}

is a λ-filtered colimit in Set. Consequently,

(E_{λ}P (k_{d}) : E_{λ}P Dd → E_{λ}P K)

is a λ-filtered colimit in Set^{Ho(K_{λ})^{op}}.

Let P_{λ} : K_{λ} → Ho(K_{λ}) denote the domain and codomain restriction of the canonical functor P : K → Ho(K). We get the induced functor

Ind_{λ} P_{λ} : K = Ind_{λ} K_{λ} → Ind_{λ} Ho(K_{λ}) .

5.2. Corollary. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{2}) and (G_{λ}^{4}). Then E_{λ}P ≅ Ind_{λ} P_{λ}.

Remark.This means that E_{λ} factorizes through the inclusion

Ind_{λ} Ho(K_{λ}) ⊆ Set^{Ho(K_{λ})^{op}}

and that the codomain restriction of E_{λ}, which we denote E_{λ} as well, makes the composition E_{λ}P isomorphic to Ind_{λ} P_{λ}.

Proof. Since both E_{λ}P and Ind_{λ} P_{λ} have the same domain restriction on K_{λ}, the result follows from 5.1.

For λ < μ we get a unique functor

F_{λμ} : Ind_{μ}(Ho(K_{μ})) → Ind_{λ}(Ho(K_{λ}))

which preserves μ-filtered colimits and whose domain restriction on Ho(K_{μ}) coincides with that of E_{λ}.

5.3. Corollary. Let K be a locally λ-presentable model category satisfying the conditions (G_{μ}^{2}) and (G_{μ}^{4}) for a regular cardinal λ < μ. Then F_{λμ}E_{μ} ≅ E_{λ}.

Proof. Following 5.2, we have E_{μ} ≅ Ind_{μ}(P_{μ}) and thus the functors F_{λμ}E_{μ}P ≅ F_{λμ} Ind_{μ}(P_{μ}) and E_{λ}P have the isomorphic domain restrictions on K_{μ}. We will show that the functor E_{λ}P preserves μ-filtered colimits. Since F_{λμ} Ind_{μ}(P_{μ}) has the same property, we will obtain that F_{λμ}E_{μ}P ≅ E_{λ}P and thus F_{λμ}E_{μ} ≅ E_{λ}.

The functor E_{λ}P preserves μ-filtered colimits iff for every object A in K_{λ} the functor

hom(P A, P −) : K → Set

preserves μ-filtered colimits. Since K_{λ} ⊆ K_{μ}, this follows from 5.1.

5.4. Theorem. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{1}), (G_{λ}^{2}) and (G_{λ}^{4}). Then the functor

E_{λ} : Ho(K) → Ind_{λ} Ho(K_{λ})

is essentially surjective on objects.

Proof. Consider X in Ind_{λ} Ho(K_{λ}) and express it as a canonical λ-filtered colimit (δ_{d} : Dd → X) of objects from Ho(K_{λ}). This means that D = Ho(K_{λ}) ↓ X and D : Ho(K_{λ}) ↓ X → Ho(K_{λ}) is the projection. Since E_{λ} is Ho(K_{λ})-full and faithful, D lifts along E_{λ}, i.e., D = E_{λ}\overline{D} for \overline{D} : D → Ho(K). Let (\overline{δ}_{d} : \overline{D}d → K) be a standard weak colimit of \overline{D}. Moreover, \overline{D} is the canonical diagram of K w.r.t. Ho(K_{λ}). There is \overline{K} in K_{cf} such that P \overline{K} = K. Let \overline{\overline{D}} : \overline{D} = K_{λ} ↓ \overline{K} → K_{λ} be the canonical diagrams of \overline{K} w.r.t. K_{λ} and \overline{P} : \overline{D} → D the functor induced by P : K → Ho(K), i.e., \overline{P} (f) = P (f) for each f : A → \overline{K}, A ∈ K_{λ}. Since R preserves λ-presentable objects, \overline{P} is surjective on objects.

We will show that \overline{P} is essentially full in the sense that for every h : f → g in D there is \overline{h} : \overline{f} → \overline{g} in \overline{D} such that there is a commutative triangle

\xymatrix{f = \overline{P} \overline{f} \ar[r]^{h} \ar[dr]_{_{\overline{P} \overline{h}}} & g \\  & _{\overline{P} \overline{g}} \ar[u]_{v}}

for some isomorphism v in D. Indeed, we have a homotopy commutative triangle

\xymatrix{A \ar[r]_{∼}^{\overline{f}} \ar[d]_{h′} & \overline{K} \\ B \ar[ur]_{g′}}

where f = \overline{P} (\overline{f}), g = \overline{P} (g′), h = P (h′) and A, B ∈ K_{λ} ∩ K_{cf}. Let

A ⟶^{\overline{h}} Z ⟶^{\overline{v}} B

be the (cofibration, trivial fibration) factorization of h′. Following (G_{λ}^{1}), we have Z ∈ K_{λ} ∩ K_{cf}. Since g′\overline{v}\overline{h}∼\overline{f} and A, B, \overline{K} ∈ K_{cf}, there is \overline{g} : Z → \overline{K} such that \overline{g}∼g′\overline{v} and \overline{g}\overline{h} = \overline{f}. Here, we use the homotopy extension property of cofibrations ([23] 7.3.12 (1)) and the fact that right and left homotopy coincide on K_{cf} ([22]). Thus \overline{h} : \overline{f} → \overline{g} is in \overline{D}, v = P (\overline{v}) is an isomorphism and

v\overline{P} (\overline{h}) = P (\overline{v} \overline{h}) = P (h′) = h .

Since \overline{P} is essentially full and surjective on objects, it is essentially cofinal in the sense that, given h_{1} : f → \overline{P} g_{1} and h_{2} : f → \overline{P} g_{2}, there is a commutative diagram

\xymatrix{f = \overline{P} \overline{f} \ar[rr]^{h_{1}} \ar[drr]^{\overline{P} (\overline{h}_{1})} \ar[ddr]_{\overline{P} (\overline{h}_{2})} \ar[dd]_{h_{2}} &  & \overline{P} g_{1} \ar[d]^{v_{1}^{−1}} \\  &  & \overline{P} (\overline{g}_{1}) \ar[d]^{\overline{P} (u_{1})} \\ \overline{P} (g_{2}) \ar[r]_{v_{2}^{−1}} & \overline{P} (\overline{g}_{2}) \ar[r]_{\overline{P} (u_{2})} & \overline{P} (\overline{g})}

The existence of morphisms u_{1} and u_{2} follows from \overline{D} being λ-filtered. Consequently, following 5.2,

E_{λ}P \overline{K} ≅ (Ind_{λ} P_{λ})(\overline{K}) = colim Y P \overline{\overline{D}} = colim Y \overline{D} \overline{P}
≅ colim Y \overline{D} = colim E_{λ}\overline{D} = colim D = X .

5.5. Definition. A locally λ-presentable model category K will be called λ-Brown on objects, where λ is a regular cardinal, provided that the codomain restriction E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ})) of E_{λ} is essentially surjective on objects.

K will be called λ-Brown on morphisms provided that E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ})) is full.

K will be called λ-Brown when it is both λ-Brown on objects and on morphisms. It will be called Brown if it is λ-Brown for some regular cardinal λ.

5.6. Remark. (i) Theorem 5.4 says that, under (G_{λ}^{1}), (G_{λ}^{2}) and (G_{λ}^{4}), a locally λpresentable model category K is λ-Brown on objects. To get an analogous result for K being λ-Brown on morphisms, we will use the Proposition 4.3. Probably, there is an alternative way of using [12] to get that each object of Ho(K) is a canonical homotopy λ-filtered colimit of objects from Ho(K_{λ}). Moreover, since λ is large enough to make the canonical diagram to contain all information about higher homotopies, the canonical λ-filtered colimits are weak colimits (cf. [12]).

(ii) Whenever a compactly generated triangulated category is ω-Brown on morphisms then it is ω-Brown on objects (see [4], 11.8). In our setting, we have such a result provided that K is a locally finitely presentable model category satisfying the conditions (G_{ω}^{2}) and (G_{ω}^{4}). Then, by 5.2, we have E_{ω}P ≅ Ind_{ω} P_{ω}. Now, if K is ω-Brown on morphisms then the functor Ind_{ω} P_{ω} is full. Since each object of Ind_{ω}(K_{ω}) can be obtained by iterative taking of colimits of smooth chains (see [1]) and P_{ω} is essentially surjective on objects, Ind_{ω} P_{ω} is essentially surjective on objects as well. Hence K is ω-Brown on objects. This argument does not work for λ > ω because, in the proof, we need colimits of chains of cofinality ω. Thus, due to the condition (G_{ω}^{2}), this result is of a limited importance.

5.7. Theorem. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{1}), (G_{λ}^{2}) and (G_{λ}^{3}). Then the functor

E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ}))

is full, i.e., K is λ-Brown on morphisms.

Proof. Consider a morphism f : E_{λ}P \overline{K} → E_{λ}P L and express \overline{K} as a λ-filtered colimit (\overline{δ}_{d} : Dd → \overline{K}), D : D → K_{λ} of λ-presentable objects. Since E_{λ} is Ho(K_{λ})-full, we may assume that \overline{K} is not λ-presentable, i.e., that card D ≥ λ. Let (δ_{d} : P Dd → P K) be a standard weak colimit of P D. Following 4.3, we may assume that P \overline{K} = P K and P (\overline{δ}_{d}) = δ_{d} for each d in D. Following 5.1, (E_{λ}δ_{d} : E_{λ}P Dd → E_{λ}P \overline{K}) is a λ-filtered colimit. Since E_{λ} is Ho(K_{λ})-full and faithful, there is a compatible cocone (f_{d} : P Dd → P L)_{d∈D} such that E_{λ}(f_{d}) = fE_{λ}(δ_{d}) for each d ∈ D. Since (δ_{d} : P Dd → P \overline{K}) is a weak colimit, there is g : P \overline{K} → P L with gδ_{d} = f_{d} for each d ∈ D. Hence

E_{λ}(g)E_{λ}(δ_{d}) = E_{λ}(f_{d}) = fE_{λ}(δ_{d})

for each d ∈ D and thus E_{λ}(g) = f.

5.8. Corollary. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{1}), (G_{λ}^{2}) and (G_{λ}^{3}). Then K is λ-Brown.

5.9. Corollary. Every combinatorial model category is Brown.

5.10. Corollary. The model categories SSet and Sp (with the stable model structure) are ω_{1}-Brown.

Proof. It follows from 5.8 and 3.6.

5.11. Remark. (1) In fact, both SSet and Sp are λ-Brown for every ω_{1} ◁ λ. We do not know whether SSet is ω-Brown (see the Introduction). Following [2], Sp is also ω-Brown.

(2) If K is a locally finitely presentable model category such that Ho(K) is a stable homotopy category in the sense of [24] then K is ω-Brown in our sense iff Ho(K) is Brown in the sense of [24].

(3) Let K be a λ-Brown model category and D : D → Ho(K_{λ}) be a λ-filtered diagram. Let (k_{d} : E_{λ}Dd → K)_{d∈D} be a colimit of E_{λ}D in Ind_{λ} Ho(K_{λ})). Since K is λ-Brown, we have K = E_{λ}\overline{K} and k_{d} = E_{λ}\overline{k}_{d} for each d ∈ D. Since E_{λ} is Ho(K_{λ})-faithful, (\overline{k}_{d} : Dd → \overline{K}) is a cone in Ho(K). Let l_{d} : Dd → L be another cone. There is a unique t : K → E_{λ}L such that tk_{d} = E_{λ}l_{d} for each d ∈ D. Since K is λ-Brown, we have t = E_{λ}\overline{t} where \overline{t} : \overline{K} → L. Since E_{λ} is Ho(K_{λ})-faithful, t\overline{k}_{d} = l_{d} for each d ∈ D. Hence \overline{k}_{d} : Dd → \overline{K} is a weak colimit. We will call this weak colimit minimal.

Every object of Ho(K) is a minimal λ-filtered colimit of objects from Ho(K_{λ}).

(4) K being λ-Brown can be viewed as a weak λ-accessibility of Ho(K) because Ho(K) is λ-accessible with Ho(K)_{λ} = Ho(K_{λ}) iff E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ})) is an equivalence.

5.12. Definition. Let K be a model category. Morphisms f, g : K → L in Ho(K) will be called λ-phantom equivalent if E_{λ}f = E_{λ}g.

This means that f, g : K → L are λ-phantom equivalent iff fh = gh for each morphism h : A → K with A ∈ Ho(K_{λ}).

5.13. Proposition. Let K be a λ-Brown model category. Then for each object K in Ho(K) there exists a weakly initial λ-phantom equivalent pair f, g : K → L.

Proof. Express K as a minimal weak λ-filtered colimit (k_{d} : Dd → K)_{d∈D} of objects from Ho(K_{λ}), take the induced morphism p : ∐_{d∈D}Dd → K and its weak cokernel pair f, g

\xymatrix{∐ Dd \ar[r]^{p} & K \ar[r]^{f} \ar[r]_{g} & L .}

Since the starting weak colimit is minimal, E_{λ}p is an epimorphism in Ind_{λ}(Ho(K_{λ})). Thus f and g are λ-phantom equivalent.

Let f′, g′ : K → L′ be a λ-phantom equivalent. Then f′p = g′p and thus the pair f′, g′ factorizes through f, g. Thus f, g is a weakly initial λ-phantom equivalent pair.

5.14. Examples. We will show that the homotopy categories

Ho(SSet_{n})

are finitely accessible for each n = 1, 2, …, i.e., that E_{ω} is an equivalence in this case. Recall that SSet_{n} = Set^{Δ^{n}} where Δ_{n} is the category of ordinals {1, 2, …, n}. The model category structure is the truncation of that on simplicial sets, i.e., cofibrations are monomorphisms and trivial cofibrations are generated by the horn inclusions

j_{m} : Δ_{m}^{k} → Δ_{m} 0 < k ≤ m ≤ n.

Here, Δ_{m} = Y_{n}(m + 1) where Y_{n} : Δ_{n} → SSet_{n} is the Yoneda embedding for m < n and Δ_{n} is Y_{n}(n + 1) without the (n + 1)-dimensional simplex {0, 1, …, n}.

For example SSet_{1} = Set and trivial cofibrations are generated by j_{1} : 1 → 2. Then weak equivalences are precisely mappings between non-empty sets and Ho(SSet_{1}) is the category 2; all non-empty sets are weakly equivalent. SSet_{2} is the category of oriented multigraphs with loops. Trivial cofibrations are generated by the embedding j_{1} of

•0

to

\xymatrix{0• \ar[r] & • 1}

(degenerated loops are not depicted), by the embedding j_{2} of

\xymatrix{ & •1 \\ 0• \ar[ur] \ar[dr] \\  & •2}

to

\xymatrix{ & •1 \ar[dd] \\ 0• \ar[ur] \ar[dr] \\  & •2}

and their orientation variants. This makes all connected multigraphs weakly equivalent and Ho(SSet_{2}) is equivalent to Set; the cardinality of a set corresponds to the number of connected components.

In the case of SSet_{3}, 1-connected objects cease to be weakly equivalent and their contribution to Ho(SSet_{3}) are trees (with a single root) of height ≤ 2. For example,

•  \begin{matrix}• \\ • \end{matrix}  [figure: •; •; •]

correspond to

[figure: •; •; •; •]

(degenerated loops are not depicted). Therefore Ho(SSet_{3}) is equivalent to the category of forests of height ≤ 2. Analogously Ho(SSet_{n}) is equivalent to the category of forests of height ≤ n. Hence it is finitely accessible.

Let us add that SSet_{2} is a natural model category of oriented multigraphs with loops (cf. [31]) and that the symmetric variants Set^{F_{n}^{op}}, where F_{n} is the category of cardinals {1, …, n}, are Quillen equivalent to SSet_{n} and left-determined by monomorphisms in the sense of [39].

6. Strongly Brown model categories

6.1. Proposition. Let K be a locally λ-presentable model category satisfying the conditions (G_{μ}^{2}) and (G_{μ}^{4}) for a regular cardinal λ < μ. Then E_{μ} reflects isomorphisms provided that E_{λ} reflects isomorphisms.

Proof. It follows from 5.3.

6.2. Definition. A λ-Brown model category K will be called strongly λ-Brown if, in addition, E_{λ} also reflects isomorphisms.

K will be called strongly Brown if it is strongly λ-Brown for some regular cardinal λ.

6.3. Remark. (1) For strongly λ-Brown model categories, any minimal λ-filtered colimit k_{d} : Dd → K has the property that each endomorphism t : K → K satisfying tk_{d} = k_{d} for each d ∈ D is an isomorphism. Thus k_{d} : Dd → K is a minimal colimit in the sense of [24]. Minimal colimits are determined uniquely up to an isomorphism. Another possible terminology, going back to [19], is a stable weak colimit.

(2) Following [24], 5.8 and 6.1, Sp is strongly λ-Brown for each ω_{1} ◁ λ. Therefore, following (3), Ho(Sp) has minimal λ-filtered colimits of objects from Ho(Sp_{λ}) for each ω_{1} ◁ λ. This is known for λ = ω (see [34] and [24]) and there is still an open problem whether Ho(Sp) has all minimal filtered colimits.

M. Hovey [22] introduced the concept of a pre-triangulated category (distinct from that used in [35]) and showed that the homotopy category of every pointed model category is pre-triangulated in his sense. He calls a pointed model category K stable if Ho(K) is triangulated. In particular, K is stable provided that Ho(K) is a stable homotopy category in the sense of [24].

6.4. Proposition. Every combinatorial stable model category is strongly Brown.

Proof. We know that K is Brown (see 5.9) Following [22] 7.3.1, every combinatorial pointed model category K has a set G of weak generators. Let Σ^{∗} = {Σ^{n}Z|Z ∈ G, n ∈ Z}. Following [35] 6.2.9, there is a regular cardinal λ such that E_{λ} reflects isomorphisms.

6.5. Proposition. Let K be a strongly λ-Brown model category. Then the functor E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ})) preserves (existing) λ-filtered colimits of objects from Ho(K_{λ}).

Proof. Let D : D → Ho(K_{λ}) be a λ-filtered diagram, (k_{d} : Dd → K) its colimit and (\overline{k}_{d} : Dd → \overline{K}) its minimal colimit. We get morphisms f : K → \overline{K} and g : \overline{K} → K such that fk_{d} = \overline{k}_{d} and g\overline{k}_{d} = k_{d} for each d in D. We have gf = id_{K} and E_{λ}(fg) = id_{E_{λ}\overline{K}}. Therefore E_{λ}f = (E_{λ}g)^{−1} and thus f is an isomorphism. Therefore λ-filtered colimits and minimal λ-filtered colimits of objects from Ho(K_{λ}) coincide and the latter are sent by E_{λ} to λ-filtered colimits.

6.6. Corollary. Let K be a locally λ-presentable model category. Then Ho(K) is λaccessible iff K is strongly λ-Brown and Ho(K) has λ-filtered colimits.

6.7. Remark. If K is a strongly λ-Brown category then the functor E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ})) is essentially injective on objects in the sense that E_{λ}K ≅ E_{λ}L implies that K ≅ L. This means that the isomorphism classification is the same in Ho(K) and in Ind_{λ}(Ho(K_{λ})).

6.8. Theorem. Let K be a locally λ-presentable model category such that E_{λ} is essentially injective on objects and (G_{μ}^{1}), (G_{μ}^{2}) and (G_{μ}^{4}) hold for some λ ◁ μ. Then every weakly continuous functor H : Ho(K)^{op} → Set is representable.

Proof. Let H : Ho(K)^{op} → Set be weakly continuous. Then each domain restriction H_{α} : Ho(K_{α})^{op} → Set is left α-covering. Following 3.4 and 5.4, E_{α} is essentially surjective on objects for each μ ◁ α. Thus, for each μ ◁ α, we have A_{α} in Ho(K) such that H_{α} ≅ hom(−, A_{α}). Therefore, following 5.3, we have

E_{μ}(A_{α}) ≅ F_{μα}E_{α}(A_{α}) ≅ F_{μα}(H_{α}) ≅ H_{μ} ≅ E_{μ}(A_{μ}).

Since E_{λ} is essentially injective on objects, E_{μ} has the same property (following 5.3) and thus A_{α} ≅ A_{μ} for each μ ◁ α. This implies that H ≅ hom(−, A_{μ}).

6.9. Remark. The property that every weakly continuous functor H : Ho(K)^{op} → Set is representable is called the Brown representability for cohomology, while the property that every left covering functor H : Ho(K_{ω})^{op} → Set is representable is called the Brown representability for homology, see [10]. The consequence of 6.8 is that, for every strongly Brown model category K, Ho(K) satisfies the Brown representability for cohomology. This also follows from [21] 1.3.

Brown representability theorems for cohomology for triangulated categories are considered in [16], [35] and [29]. A. Neeman [35] introduced the concept of a well generated triangulated category. These categories naturally generalize compactly generated ones and they still satisfy the Brown representability for cohomology (see [35] 8.4.2). In [36] he shows that, for any Grothendieck abelian category K, the derived category D(K) is well generated. Since D(K) is the homotopy category of the model category of (unbounded) chain complexes Ch(K) on K and this model category is combinatorial, his result follows from our 6.10. Neeman’s result was generalized by H. Krause [29] to perfectly generated triangulated categories; in [30] he compares his perfect generation with the well generation of Neeman.

6.10. Proposition. Let K be a combinatorial stable model category. Then Ho(K) is well generated.

Proof. Following [22], Ho(K) has a set A of weak generators. This means that hom(Σ^{n}A, X) = 0 for all A ∈ A and all n ∈ ℤ implies that X ≅ 0. Following 3.5 and 3.4 (2), there is a regular cardinal λ such that K is locally λ-presentable and satisfies the conditions (G_{λ}^{i}) for i = 1, 2, 3 and A ⊆ Ho(K_{λ}).

We will find a regular cardinal λ ◁ μ such that Ho(K_{μ}) generates Ho(K) in the sense of [35] 8.1.1. Since Ho(K_{μ}) weakly generates Ho(K) it remains to show that Ho(K_{μ}) is closed under suspension and desuspension. This is the same as ΣX, ΩX ∈ K_{μ} for each X ∈ K_{μ}. But, since Σ is left adjoint to Ω, this follows from [1] 2.23 and 2.19.

It remains to show that Ho(K_{μ}) is μ-perfect in the sense of [35] 3.3.1. Recall that, following 3.4 (2), K satisfies the conditions (G_{μ}^{i}) for i = 1, 2, 3, as well. Consider a morphism f : A → ∐_{i∈I} K_{i} where card I < μ. Without a loss of generality, we may

assume that f = P \overline{f} for \overline{f} : \overline{A} → ∐_{i∈I} \overline{K}_{i}; of course, A = P \overline{A}, K_{i} = P \overline{K}_{i} for i ∈ I and

P : K → Ho(K) is the canonical functor. Since K is locally μ-presentable, each \overline{K}_{i}, i ∈ I, is a μ-filtered colimit

(k_{j}^{i} : \overline{A}_{ij} → \overline{K}_{i})_{j∈J_{i}}

of μ-presentable objects \overline{A}_{ij} . Hence all μ-small subcoproducts X = ∐_{i∈I} \overline{A}_{ij_{i}} are μ-

presentable and ∐_{i∈I} \overline{K}_{i} is their μ-filtered colimit. Thus \overline{f} factorizes through some sub-

coproduct X and therefore f factorizes through P X = ∐_{i∈I} P \overline{A}_{ij_{i}}. This yields [35] 3.3.1.2.

If f = 0 then we can assume that \overline{f} = 0 and, for a factorization

\overline{f} : \overline{A} ⟶^{g} X ⟶^{h} ∐_{i∈I} \overline{K}_{i},

there is a subcoproduct morphism u : X → X′ such that ug = 0 and h′u = h where h′ : X′ → ∐_{i∈I} \overline{K}_{i}. But this is the condition [35] 3.3.1.3. Hence Ho(K_{μ}) is μ-perfect.

6.11. Proposition. Let K be a locally λ-presentable model category such that E_{λ} is essentially injective on objects and (G_{μ}^{1}), (G_{μ}^{2}) and (G_{μ}^{4}) hold for some λ ◁ μ. Then idempotents split in Ho(K).

Proof. Let f : K → K be an idempotent in Ho(K). Then hom(−, K) is an idempotent in the category of all small functors (i.e., small colimits of representable functors) Ho(K)^{op} → Set. Let H : Ho(K)^{op} → Set be its splitting. Then H is weakly continuous and thus it is representable following 6.8. Its representing object splits f.

6.12. Remark. Since idempotents do not split in Ho(SSet) (see [18], [14]), no E_{λ} : Ho(SSet) → Ind_{λ}(K_{λ})) is essentially injective on objects.

References

[1] J. Adámek and J. Rosický, Locally Presentable and Accessible Categories, Cambridge University Press 1994.

[2] F. Adams, A variant of E. H. Brown representability theorem, Topology 10 (1971), 185–198.

[3] M. Artin, A. Grothendieck and J. L. Verdier, Théorie des topos et cohomologie étale des schemas, Lecture Notes in Math. 269, Springer-Verlag 1972.

[4] A. Beligiannis, Relative homological algebra and purity in triangulated categories, J. Alg. 227 (2000), 268–361.

[5] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, In: Geometric applications of homotopy theory (ed. M. G. Barratt and M. E. Mahowald), Lecture Notes in Math. 658 (1978), 80–130.

[6] A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer-Verlag 1972.

[7] E. M. Brown, Abstract homotopy theory, Trans. Amer. Math. Soc. 119 (1965), 79–85.

[8] A. Carboni and E. M. Vitale, Regular and exact completions, J. Pure Appl. Algebra 125 (1998), 79–117.

[9] J. D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), 284–339.

[10] J. D. Christensen, B. Keller and A. Neeman, Failure of Brown representability in derived categories, Topology 40 (2001), 1339–1361.

[11] D. Dugger, Universal homotopy theories, Adv. Math. 164 (2001), 144–176.

[12] D. Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001), 177–201.

[13] W. G. Dwyer, P. S. Hirschhorn, D. M. Kan and J. H. Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, AMS 2004.

[14] J. Dydak, A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR’s, Bull. Acad. Polon. Sci., Sér. Math. Astron. Phys. 25 (1977), 55–62.

[15] D. A. Edwards and H. M. Hastings, Čech and Steenrod Homotopy Theories with Applications to Geometric Topology, Lecture Notes in Math. 542, Springer-Verlag 1976.

[16] J. Franke, On the Brown representability theorem for triangulated categories, Topology 40 (2001), 667–680.

[17] P. Freyd, Homotopy is not concrete, Lecture Notes in Math. 168 (1970), 25–34.

[18] P. Freyd and A. Heller, Splitting homotopy idempotents, J. Pure Appl. Alg. 89 (1993), 93–106.

[19] D. Harris, The Wallman compactification as a functor, Gen. Top. Appl. 1 (1971), 273–281.

478  JIŘÍ ROSICKÝ

[20] P. S. Hirschhorn, Model Categories and Their Localizations, Amer. Math. Soc. 2003.

[21] A. Heller, On the representability of homotopy functors, J. London Math. Soc. 23 (1981), 551–562.

[22] M. Hovey, Model Categories, AMS 1999.

[23] M. Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Alg. 165 (2001), 63–127.

[24] M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic Stable Homotopy Theory, Mem. Amer. Math. Soc. 610 (1997).

[25] M. Hovey, B. Shipley and J. H. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.

[26] H. Hu, Flat functors and free exact completions, J. Austral. Math. Soc. Ser. A, 60 (1996), 143–156.

[27] D. C. Isaksen, Strict model structures for pro-categories, In: Progr. Math. 215, Birkhäuser 2004, 179–198.

[28] K. H. Kamps and T. Porter, Abstract and Simple Homotopy Theory, World Scientific 1997.

[29] H. Krause, A Brown representability via coherent functors, Topology 41, (2002), 853–861.

[30] H. Krause, On Neeman’s well generated triangulated categories, Doc. Math. 6, (2001), 121–126.

[31] A. Kurz and J. Rosický, Weak factorizations, fractions and homotopies, Appl. Categ. Structures 13 (2005), 141–160.

[32] J. Lurie, On ∞-topoi, arXiv:math.CT/0306109

[33] M. Makkai and R. Paré, Accessible categories: the foundation of categorical model theory, Contemp. Math. 104, AMS 1989.

[34] H. R. Margolis, Spectra and the Steenrod Algebra, North-Holland 1983.

[35] A. Neeman, Triangulated Categories, Princeton Univ. Press 2001.

[36] A. Neeman, On the derived category of sheaves on a manifold, Doc. Math. 6 (2001), 483–488.

[37] T. Porter, Proper homotopy theory, In: Handbook of Algebraic Topology (edited by I. M. James), North-Holland 1995, 127–168.

[38] D. G. Quillen, Homotopical Algebra, Lecture Notes in Math. 43, Springer-Verlag [39] J. Rosický and W. Tholen, Left-determined model categories and universal homotopy

theories, Trans. Amer. Math. Soc. 355 (2003), 3611–3623.

[40] S. Schwede, Stable homotopy of algebraic theories, Topology 40 (2001), 1–41.

Department of Mathematics Masaryk University 602 00 Brno, Czech Republic

Email: rosicky@math.muni.cz

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