CELLULAR GENERATORS

WOJCIECH CHACHÓLSKI, PAUL-EUGENE PARENT, AND DONALD STANLEY

Abstract. The aim of this paper is two fold. On one hand, we show that the kernel \overline{C(A)} of the Bousfield periodization functor P_{A} is cellularly generated by a space B, i.e., we construct a space B such that the smallest closed class C(B) containing B is exactly \overline{C(A)}. On the other hand we show that the partial order (Spaces, ≫) is a complete lattice, where B ≫ A if B ∈ C(A). Finally, as a corollary we obtain Bousfield’s theorem which states that (Spaces, >) is a complete lattice, where B > A if B ∈ \overline{C(A)}.

1. Introduction

The main objects of our investigation are classes of unpointed spaces C which are closed under certain fundamental operations.

• We say that C is closed under weak equivalences if X ∈ C and X is weakly equivalent to Y imply that Y ∈ C.

• We say that C is closed under taking pointed homotopy colimits if the following conditions are satisfied: if, for any set I, {X_{i}}_{i∈I} consists of elements of C, then, for any choice of basepoints in X_{i}, ⋁_{I} X_{i} ∈ C; if, for i = 0, 1, 2, X_{i} ∈ C, then hocolim(X_{0} ← X_{1} → X_{2}) ∈ C[λ] (see [4]).

• We say that C is closed under extensions by fibration if, for any map X → B such that B ∈ C and over any basepoint in B the homotopy fiber F ib(X → B) ∈ C, the space X ∈ C.

Our primary interest is in closed classes which were introduced by E. Dror Farjoun [9, 10].

1.1. Definition. A class of unpointed spaces is closed if it consists of non-empty spaces, it is closed under weak equivalences, and it is closed under taking pointed homotopy colimits.

The motivation for studying such classes comes from the theory of unstable localization and colocalization functors (see [2, 10]) which are related to our key examples:

1.2. Example (Dror Farjoun, [9]). Let A be a non-empty space. Denote by C(A) the smallest closed class that contains A. The relation X ∈ C(A) is denoted by X ≫ A and X is said to be A-cellular. This defines an order on the class of all the spaces. The class C(A) can be identified with the image of the Dror-Farjoun colocalization functor CW_{A} (see [9, 5]).

1.3. Example (Bousfield, Dror Farjoun, [1, 10]). Let A be a non-empty space. Denote by \overline{C(A)} the smallest closed class that contains A and is closed under extensions by fibrations. The relation X ∈ \overline{C(A)} is denoted by X > A and X is said to be A-acyclic. This defines an order on the class of all the spaces. The class \overline{C(A)} can be identified with the kernel of the Bousfield functor P_{A} (see [5]).

In this paper we look at certain global features of the orders (Spaces, ≫) and (Spaces, >). Our work was inspired by a result of Bousfield [2] which says that (Spaces, >) is a complete lattice. Recall that an ordered class (C, >) is called a complete lattice if it satisfies the following condition.

The first author was partially supported by the NSF grant DMS-9803766.

This work has been partly supported by the Volkswagenstiftung Oberwolfach.

2000 Mathematics Subject Classification: 55Q05.

For any set J ⊂ C, there are objects sup(J) and inf(J) in C, such that: if X ∈ J, then sup(J) > X > inf(J), and for any other objects Y and Z in C for which Y > X > Z, if X ∈ J, we have Y > sup(J) and inf(J) > Z.

The main result of our paper is:

1.4. Theorem. 1. The order (Spaces, ≫) is a complete lattice.

2. For any space A there exists a space B such that \overline{C(A)} = C(B). If A is finite, then we can take B to be the wedge of representatives of the isomorphism classes of finite A-acyclic spaces.

The strategy to prove this theorem to is to look for special kind of generators of a given class C. We say that a space A is a cellular generator of C if C = C(A). We say that a set B ⊂ C strongly generates C if any space in C is weakly equivalent to the pointed homotopy colimit of a diagram F : I → Spaces_{∗}, where for i ∈ I, F (i) ∈ B. Strong generation is related to an invariant called “complication”, see [6]. Observe that if B strongly generates C, then, for any choice of basepoints in the elements of B, ⋁_{A∈B} A is a cellular generator of C.

The key step in the proof of Theorem 1.4 is a generalization of the following statement to an arbitrary space A:

1.5. Theorem. If A is finite, then a set of representatives of isomorphism classes of finite A-acyclic (cellular) spaces strongly generates the class of A-acyclic (cellular) spaces.

2. Notation

This paper is written simplicially. The symbols Spaces and Spaces_{∗} denote the categories of unpointed and pointed simplicial sets respectively. We refer to objects of these categories as spaces. All the spaces and maps, unless clearly specified, are assumed to be unpointed.

The n-dimensional simplex is denoted by Δ[n]. Its boundary is denoted by ∂Δ[n]. The symbol Δ[n, k] denotes the horn in Δ[n]. It is a simplicial subset of ∂Δ[n] generated by all except the k-th face of Δ[n]. We refer to [8, 11] for the details of how to do homotopy theory in Spaces and Spaces_{∗}.

If K is a space, by #K we denote the cardinality of the set of all the simplices in K.

By N we denote the natural order on the set natural numbers {0, 1, … }.

Let λ be an ordinal number. By [λ] we denote a category whose objects are ordinals κ such that κ < λ, and mor_{[λ]}(κ, δ) is either empty, if κ > δ, or consists of a single element, if κ ≤ δ.

Let J be a set and λ be a limit ordinal. We say that #J is smaller than the cofinality of λ if the following condition is satisfied. If F : [λ] → Sets is a functor where, for κ < λ, colim_{[κ]}F → F (κ) is a monomorphism, then the natural map colim_{[λ]}mor_{Sets}(J, F) → mor_{Sets}(J, colim_{[λ]}F) is an isomorphism. For example the cardinality of a set is smaller than the cofinality of N if and only if this set is finite. We refer the reader to [7] for a detailed discussion of this condition.

The symbols hocolim_{I}F and hocolim_{I}^{∗}G denote respectively the homotopy colimit and the pointed homotopy colimit of F : I → Spaces and G : I → Spaces_{∗} as defined in [3].

Let F : I → Spaces be a functor. Assume that the cardinality of the set of objects of I is smaller that the cofinality of λ. If for any i ∈ I, #F (i) is also smaller than the cofinality of λ, then the same is true for #colim_{I}F and #hocolim_{I}F.

A category is called filtered if, for any two objects a and b, there are morphisms a → c and b → c, and, for any two morphisms α : a → b and β : a → c, there are morphisms α′ : b → d and β′ : c → d such that α′α = β′β. If I is filtered, then, for any diagram F : I → Spaces_{∗}, the map ∐_{i∈I} π_{n}F (i) → π_{n}(hocolim_{I}^{∗}F) is an epimorphism.

3. Approximations

To see that a set B strongly generates a class C, we need a criteria which would guarantee that a space can we expressed as the pointed homotopy colimit of a diagram whose values belong to B.

Let X be a pointed space and B be a set of unpointed spaces. Define B ↓X to be a category whose objects are pointed maps Y → X such that Y ∈ B, and a set of morphisms between α_{0} : Y_{0} → X and α_{1} : Y_{1} → X consists of cofibrations i : Y_{0} ↪ Y_{1} such that α_{1}i = α_{0}. By ω_{B} : B ↓X → Spaces_{∗} let us denote a functor that assigns to Y → X the space Y.

3.1. Proposition. Let X be a pointed space and B be a set of connected spaces. Assume:

• The category B ↓X is filtered.

• If K is a finite complex, then any map K → X can be expressed as a composition K → Y → X where Y ∈ B. Moreover any commutative diagram of form:

\xymatrix{K \ar[r] \ar@{^{(}->}[d] & Y \ar[d] \\ L \ar[r] & X} where K, L are finite, Y ∈ B, and K ↪ L is a cofibration, can be extended to a commutative diagram:

\xymatrix{K \ar[rr] \ar@{^{(}->}[dd] & & Y \ar@{^{(}->}[dl] \ar[dd] \\ & Y′ \ar[dr] \\ L \ar[rr] \ar[ur] & & X}

where Y′ ∈ B and Y ↪ Y′ is a cofibration.

Then the map hocolim_{B↓X}^{∗}(ω_{B}) → X is a weak equivalence.

Proof. Let us denote the map hocolim_{B↓X}^{∗}(ω_{B}) → X by f. As all the values of ω_{B} are connected, then so is the space hocolim_{B↓X}^{∗}(ω_{B}). Thus to show the proposition it is enough to prove that f induces an isomorphism on homotopy groups π_{∗}, for ∗ ≥ 0.

Observe that any pointed map α : K → X, where K is finite, factors through f. To see this let K → Y → X be a factorization of α given by the assumption. Let Y → hocolim_{B↓X}^{∗}(ω_{B}) be a map induced by Y → X in B ↓X. It is clear that the composition K → Y → hocolim_{B↓X}^{∗}(ω_{B})→X equals α.

An element in π_{∗}X is represented by a pointed map α : sd^{k}∂Δ[n] → X where sd^{k}∂Δ[n] denotes the k-th barycentric subdivision of ∂Δ[n]. As sd^{k}∂Δ[n] is finite the above remark shows that α factors through f. It follows that f induces an epimorphism on π_{∗}. In particular X is connected.

Since B ↓ X is filtered any element of π_{∗}hocolim_{B↓X}^{∗}(ω_{B}) can be represented by a pointed map α : sd^{k}∂Δ[n] → Y, for some Y → X in B ↓ X. Assume that α represents an element in the kernel of π_{∗}f. This means that after possibly further subdividing there is a commutative diagram:

\xymatrix{sd^{l}∂Δ[n] \ar@{^{(}->}[d] \ar[r] & sd^{k}∂Δ[n] \ar[r] & Y \ar[d] \\ L \ar[rr] &  & X}

where L is finite and contractible, for example the cone on sd^{l}∂Δ[n]. By the assumptions we can then find Y′ ∈ B, for which the above diagram can be extended to:

\xymatrix{sd^{l}∂Δ[n] \ar@{^{(}->}[d] \ar[r] & sd^{k}∂Δ[n] \ar[r] & Y \ar@{^{(}->}[dl] \ar[d] \\ L \ar[r] & Y′ \ar[r] & X}

It follows that α represents a trivial class in the homotopy groups of hocolim_{B↓X}^{∗}(ω_{B}) and thus π_{∗}f is also a monomorphism.

4. “Small” C-spaces

In order to find strong generators of closed classes we are going to consider spaces for which a relative Quillen’s small object argument holds. Let C be a class of spaces and λ be a limit ordinal. Note that the collection of isomorphism classes of spaces in C, whose cardinality is smaller than the cofinality of λ, forms a set. We denote this set by C_{≪λ}. Sometimes, by slight abuse of language, we say that X belongs to C_{≪λ}, if X is isomorphic to a space in C whose cardinality is smaller than the cofinality of λ. For example C_{≪N} consists of finite complexes that belong to C.

4.1. Definition. Let C be a class of spaces and λ be a limit ordinal. A space X is called λ-C-small if, for any map K → X such that #K is smaller than the cofinality of λ, there exists a factorization K → Y → X where Y ∈ C_{≪λ}.

In the case C is closed under weak equivalences we can always arrange so that the map K → Y, in the above definition, is a cofibration. To do that just take Y′ = colim(K × Δ[1] ↩ K → Y).

The above definition is motivated by Proposition 3.1:

4.2. Lemma. Let C be a closed class and λ be a limit ordinal. Assume that C consists of connected spaces. If X is λ-C-small then, for any choice of a basepoint in X, the set C_{≪λ} satisfies the assumptions of Proposition 3.1.

Proof. If A → X and B → X are elements of C_{≪λ} ↓ X, then so is A ∨ B → X. Moreover, as C is closed, for any morphisms A ↪ B and A ↪ C in C_{≪λ} ↓ X, the space D = colim(B ↩ A ↪ C) belongs to C_{≪λ}. This shows that C_{≪λ} ↓ X is filtered.

Consider now a commutative diagram:

\xymatrix{K \ar[r] \ar@{^{(}->}[d] & Y \ar[d] \\ L \ar[r] & X} where K, L are finite, Y ∈ C_{≪λ}, and K ↪ L is a cofibration.

The push-out Z = colim(L ↩ K → Y) clearly has cardinality smaller than the cofinality of λ. Since X is λ-C-small, the induced map Z → X can be factored as Z ↪ Y′ → X, where Y′ ∈ C_{≪λ}. This shows that the second condition of Proposition 3.1 is also satisfied.

4.3. Proposition. Let C be a closed class and λ be a limit ordinal whose cofinality is bigger than N or λ = N. Then the following is a closed class:

C[λ] = {X | there exists X →^{∼} X′ such that X′ is λ-C-small }

4.4. Proposition. Let λ be a limit ordinal whose cofinality is bigger than N. If {C_{i}}_{i∈I} is a family of closed classes such that #I is smaller than the cofinality of λ, then:

( ⋂_{i∈I} C_{i})[λ] = ⋂_{i∈I} (C_{i}[λ])

Propositions 4.3 and 4.4 will be proved at the end of the section.

4.5. Corollary. Let C be a closed class and λ be a limit ordinal whose cofinality is bigger than N or λ = N. Then C_{≪λ} strongly generates C[λ] and, for any choice of basepoints in the elements of C_{≪λ}, C[λ] = C(⋁_{A∈C_{≪λ}} A).

Proof. It is clear that C_{≪λ} ⊂ C[λ].

If C contains a non-connected space, then C consists of all the spaces (as C is closed). In particular all discrete spaces belong to C. In this case the corollary is clear since any λ-C-small space can be expressed as the pointed homotopy colimit of a diagram whose values are discrete spaces of cardinality smaller than the cofinality of λ.

If C consists of connected spaces, the corollary follows from Lemma 4.2

We can now prove the “cellular” part of Theorem 1.5:

4.6. Corollary. Let A be a space. Assume that either λ = N, if A is finite, or λ is a limit ordinal whose cofinality is bigger than #A, if #A is infinite. Then C(A) = C(A)[λ] and C(A) is strongly generated by the set C(A)_{≪λ}.

Proof. The inclusion C(A)[λ] ⊂ C(A) is obvious. Since the cardinality of A is smaller than the cofinality of λ, it is also clear that A ∈ C(A)[λ]. As C(A)[λ] is closed, we have C(A) ⊂ C(A)[λ].

As a corollary we also get part 1 of Theorem 1.4:

4.7. Corollary. The order (Spaces, ≫) is a complete lattice.

Proof. Let J be a set of spaces. For any choice of basepoints in the spaces of J, inf(J) = ⋁_{A∈J} A.

Note that sup(J) is represented by a cellular generator of ⋂_{A∈J} C(A). To show the existence of such a generator choose a limit ordinal λ whose cofinality is bigger than the cardinalities of spaces in J, #J, and N. According to Proposition 4.4 and Corollary 4.6 we have equalities:

( ⋂_{A∈J} C(A))[λ] = ⋂_{A∈J} (C(A)[λ]) = ⋂_{A∈J} C(A)

As any class of the form C[λ] has a cellular generator (see Corollary 4.5), then so does ⋂_{A∈J} C(A).

To prove propositions 4.3 and 4.4 we start with showing that, in the case C is closed, λ-C-small spaces are preserved by many operations.

4.8. Lemma. Let C be a closed class and λ be a limit ordinal whose cofinality is bigger than N or λ = N.

1. Let X → Y → X be maps whose composition is the identity. If Y is λ-C-small, then so is X.

2. If E ↠^{∼} B is an acyclic fibration, then E is λ-C-small if and only if B is.

3. Let I be an arbitrary set and X = colim(X_{0} ← ∐_{I} Δ[n, k] ↪ ∐_{I} Δ[n]). If X_{0} is λ-C-small, then so is X.

4. Let X = colim(X_{0} ↩ X_{1} ↪ X_{2}), where the indicated arrows are cofibrations. If, for all i, X_{i} is λ-C-small, then so is X.

5. Let X = colim(X_{0} ↪ X_{1} ↪ X_{2} ↪ ⋯), where the indicated arrows are cofibrations. If, for all i, X_{i} is λ-C-small, then so is X.

6. Let I be an arbitrary set. If, for all i ∈ I, X_{i} is λ-C-small, then so is ⋁_{i∈I} X_{i}.

7. If X is λ-C-small, then any map X → Y can be expressed as a composition X ↪^{∼} X′ ↠ Y, where X′ is λ-C-small and the indicated maps are an acyclic cofibration and a fibration.

8. Let X →^{∼} Y be a weak equivalence. If X is λ-C-small, then so is Y.

9. Let X →^{∼} Y be a weak equivalence. If Y is λ-C-small and X is fibrant, then X is also λ-C-small.

10. If X ∈ C[λ] and X is fibrant, then X is λ-C-small.

Proof of 1. This follows directly from the definitions.

Proof of 2. This part follows easily from the fact that any map K → B factors as K → E ↠^{∼} B.

Proof of 3. Let K → X be a map where #K is smaller than the cofinality of λ. Note that there is J ⊂ I for which #J is smaller than the cofinality of λ and K → X factors through X′ = colim(X_{0} ← ∐_{J} Δ[n, k] ↪ ∐_{J} Δ[n]). Note that the cardinalities of ∐_{J} Δ[n, k] and ∐_{J} Δ[n] are smaller that the cofinality of λ.

By pulling back K → X′ along the appropriate maps we can form the following commutative diagram where the squares on the right are pull-backs:

[figure: K; =; colim (; K_{0}; K_{1}; K_{2}; ); X′; =; colim (; X_{0}; ∐_{J} Δ[n, k]; ∐_{J} Δ[n]; )]

Since #K and #J are smaller than the cofinality of λ, the same is true for #K_{i}, and thus also for for the cardinality of P = colim(K_{0} ← K_{1} → ∐_{J} Δ[n, k]). Let P → X_{0} be the map induced by the above diagram. By the assumption we can express P → X_{0} as P → Y → X_{0}, where Y ∈ C_{≪λ}. To get the desired factorization of K → X take Y′ = colim(Y ← ∐_{J} Δ[n, k] ↪ ∐_{J} Δ[n]). This space has cardinality smaller than the cofinality of λ and is weakly equivalent to Y, thus belongs to C_{≪λ}.

Proof of 4, 5, and 6. As the proofs are analogous we show only part 4. Let us point out however that part 5 requires the assumption that either λ = N or the cofinality of λ is bigger than N.

Let K → X be a map where #K is smaller than the cofinality of λ. By pulling back K → X along the appropriate maps, we can form the following commutative diagram, where the squares on the right are pull-backs and the indicated arrows are cofibrations:

[figure: K; =; colim (; K_{0}; K_{1}; K_{2}; ); X; =; colim (; X_{0}; X_{1}; X_{2}; )]

As K_{i} ⊂ K, #K_{i} is smaller than the cofinality of λ. Let K_{1} → Y_{1} → X_{1} be a factorization of K_{1} → X_{1} where Y_{1} ∈ C_{≪λ}. Define \overline{Y_{i}} = colim(Y_{1} ← K_{1} ↪ K_{i}). There are natural maps \overline{Y_{i}} → X_{i}. As the cardinalities of \overline{Y_{i}} are smaller than the cofinality of λ, we can find appropriate factorizations \overline{Y_{i}} ↪ Y_{i} → X_{i}. The space Y = colim(Y_{0} ↩ Y_{1} ↪ Y_{2}) gives the desired factorization of K → X.

Proof of 7. This follows from parts 3 and 5 and the standard way of factoring a map in Spaces into an acyclic cofibration followed by a fibration (the acyclic cofibrations are generated by the inclusions of horns into simplexes).

Proof of 8. According to part 7 we can can factor X →^{∼} Y as X ↪^{∼} X′ ↠^{∼} Y where X′ is λ-C-small. Part 8 follows now from part 2.

Proof of 9. Factor X →^{∼} Y by an acyclic cofibration followed by an acyclic fibration X ↪^{∼} Y′ ↠^{∼} Y. According to part 2, Y′ is λ-C-small. As X is fibrant, it is a retract of Y′. Part 9 follows now from part 1.

Proof of Proposition 4.3. In view of Lemma 4.8, to prove the proposition it is enough to show that C[λ] is closed under weak equivalences.

Let X →^{∼} Y be a weak equivalence. We need to prove that X ∈ C[λ] if and only if Y ∈ C[λ]. Assume that X ∈ C[λ]. By possibly factoring a given map into a cofibration followed by an cyclic fibration, using part 2 of Lemma 4.8, we can assume that there is an acyclic cofibration X ↪^{∼} X′, where X′ is λ-C-small. Let Y′ = colim(X′ ↩^{∼} X →^{∼} Y). As X′ is λ-C-small, part 8 of Lemma 4.8 implies that so is Y′. It follows that Y ∈ C[λ].

Assume that Y ∈ C[λ]. Let Y →^{∼} Y′ be a weak equivalence where Y′ is λ-C-small. The composition X →^{∼} Y →^{∼} Y′ gives the desired weak equivalence.

Proof of Proposition 4.4. The inclusion (⋂_{i∈I} C_{i})[λ] ⊂ ⋂_{i∈I}(C_{i}[λ]) is clear since, for any i ∈ I, (⋂_{i∈I} C_{i})[λ] ⊂ C_{i}[λ].

Let X ∈ ⋂_{i∈I}(C_{i}[λ]). This means that, for any i ∈ I, there is an acyclic cofibration X ↪^{∼} X_{i} where X_{i} is λ-C_{i}-small. Define Z := ⋃_{X} X_{i}. Observe that the inclusions X_{i} ↪ ⋃_{X} X_{i} are weak equivalences. Part 8 of Lemma 4.8 implies that, for all i ∈ I, Z is λ-C_{i}-small. As X → Z is also a weak equivalence, to show that X ∈ (⋂_{i∈I} C_{i})[λ] it is enough to prove that Z is λ-(⋂_{i∈I} C_{i})-small.

Let K → Z be a map, where #K is smaller than the cofinality of λ. We need to prove that this map factors as K → Y → Z, where #Y is smaller than the cofinality of λ and Y ∈ ⋂_{i∈I} C_{i}. We will construct Y as the “telescope” of a diagram indexed by a certain ordinal number. Let us choose a well ordering on the set I and denote by e its smallest element. Define κ to be the following order on I × N. Set (i, m) < (j, n) if m < n in N or m = n and i < j in I. Our strategy is to define, by induction, a diagram F : [κ] → Spaces together with a natural transformation F → Z such that:

• the map K → Z factors as K → F (e, 0) → Z;

• for any (i, n) ∈ κ, #F (i, n) is smaller than the cofinality of λ;

• for any (i, n) ∈ κ, F (i, n) ∈ C_{i}. Assume that we have such a functor. Observe that the imposed assumptions on I imply that #hocolim_{[κ]}F is smaller than the cofinality of λ. Moreover, for any i ∈ I, hocolim_{[κ]}F ∈ C_{i} since hocolim_{[κ]}F ≃ hocolim_{n∈{0,1,... }}F (i, n). This shows that K → hocolim_{[κ]}F → Z is the desired factorization.

As Z is λ-C_{e} small we can find a factorization K → Y → Z, where #Y is smaller that the cofinality of λ and Y ∈ C_{e}. Define F (e, 0) := Y and F (e, 0) → Z to be the map Y → Z.

Assume that we have defined F for all (i, m) < (j, n). Consider the map hocolim_{[j,n]}F → Z. As #hocolim_{[j,n]}F is smaller than the cofinality of λ, we can factor this map as hocolim_{[j,n]}F → Y → Z, where Y ∈ C_{j} and #Y is smaller than the cofinality of λ. Define F (j, n) := Y. It is clear that F satisfies the above requirements.

5. Parameterized cellular spaces

To prove part 2 of Theorem 1.4 we need to discuss parameterized families of cellular spaces. Let f : E → B be a map. For any simplex σ : Δ[n] → B, define df(σ) := lim(Δ[n] →^{σ} B ←^{f} E).

5.1. Definition. Let C be a class of spaces. A map f : E → B is called a relative C-complex if for any simplex σ : Δ[n] → B, df(σ) ∈ C.

Let C be a closed class. Since C consists of non-empty spaces a relative C-complex f : E → B is always an onto map.

A key property of closed classes is given by:

5.2. Theorem ([4]). Let C be a closed class. Let H, G : I → Spaces be diagrams and Ψ : H → G be a natural transformation. If for any i ∈ I, Ψ_{i} : H(i) → G(i) is a relative C-complex, then for any basepoint in hocolim_{I}G, the homotopy fiber of hocolim_{I}Ψ : hocolim_{I}H → hocolim_{I}G is in C.

The above implies:

5.3. Corollary. Let C be a closed class.

1. If f : E → B is a relative C-complex, then for any basepoint in B, the homotopy fiber of f is in C.

2. Let H, G : I → Spaces be diagrams and H → G be a natural transformation. If for any i ∈ I, H(i) → G(i) is a relative C-complex and hocolim_{I}G is contractible, then hocolim_{I}H ∈ C.

Relative complexes enjoy the following properties:

5.4. Proposition. Let C be a closed class and λ be a limit ordinal.

1. If E → B is a relative C_{≪λ}-complex, then, for any map h : X → B, so is the pull-back h^{∗}E → X.

2. Consider the following commutative diagram where the indicated arrows are cofibrations:

[figure: E; f; =; colim (; E_{0}; f_{0}; E_{1}; f_{1}; E_{2}; f_{2}; ); B; =; colim (; B_{0}; B_{1}; B_{2}; )]

If f_{i} are relative C_{≪λ}-complexes, then so is f.

3. Let δ be an ordinal number smaller than the cofinality of λ. Let Ψ : H → G be a natural transformation between functors H, G : [δ] → Spaces. Assume that for any l < δ, colim_{i<l}H(i) ↪ H(l) and colim_{i<l}G(i) ↪ G(l) are cofibrations and Ψ_{l} : H_{l} → G_{l} is a relative C_{≪λ}-complex. Then colim_{[δ]}Ψ : colim_{[δ]}H → colim_{[δ]}G is also a relative C_{≪λ}-complex.

Proof. Part 1 is clear.

As the proofs of parts 2 and 3 are analogous we show only part 2. Let us consider a simplex σ : Δ[n] → B. We have to prove that df(σ) ∈ C_{≪λ}.

Denote by K the image of the map σ. It is a finite subspace of B. As f_{i} are relative C_{≪λ}- complexes, the subspace f^{−1}(K) ⊂ E has cardinality smaller than the cofinality of λ. It follows that so does df(σ).

Let us consider the pull-backs:

K_{i} = lim(Δ[n] →^{σ} B ← B_{i}) X_{i} = lim(K_{i} → B_{i} ⟵^{f_{i}} E_{i})

These spaces fit into the following commutative diagram where the indicated arrows are cofibrations:

[figure: df(σ); =; colim (; X_{0}; X_{1}; X_{2}; ); Δ[n]; =; colim (; K_{0}; K_{1}; K_{2}; )]

As f_{i} is a relative C-complex, then so is X_{i} → K_{i}, by part 1. We can therefore use part 2 of Corollary 5.3 to conclude that df(σ) belongs to C.

6. Relatively “small”-C-spaces

6.1. Theorem. Let λ be a limit ordinal whose cofinality is bigger than N or λ = N. If C is a closed class which is closed under extensions by fibrations, then so is C[λ].

As a corollary we get both the second part of Theorem 1.4 and Theorem 1.5.

6.2. Corollary. Let A be a space. Assume that either λ = N, if A is finite, or the cofinality of λ is bigger than #A, if A is infinite. Then \overline{C(A)} = \overline{C(A)}[λ]. Moreover \overline{C(A)} is strongly generated by the set \overline{C(A)}_{≪λ} and, for any choice of basepoints in the elements of \overline{C(A)}_{≪λ}, \overline{C(A)} = C( ⋁_{X∈\overline{C(A)}_{≪λ}} X).

Proof. The inclusion \overline{C(A)}[λ] ⊂ \overline{C(A)} is clear. Since #A is smaller than the cofinality of λ, A ∈ \overline{C(A)}[λ]. As \overline{C(A)}[λ] is a closed class which is closed under extensions by fibrations (see Theorem 6.1), \overline{C(A)} ⊂ \overline{C(A)}[λ]. The rest of the corollary follow from Corollary 4.5.

6.3. Corollary (Bousfield [2]). The order (Spaces, >) is a complete lattice.

Proof. Let J be a set of spaces. For any choice of basepoints in the spaces of J, inf(J) = ⋁_{A∈J} A.

Note that sup(J) is represented by a cellular generator of the class ⋂_{A∈J} \overline{C(A)}. Since, by Corollary 6.2, any class of the form \overline{C(A)} has a cellular generator, the existence of sup(J) follows from Corollary 4.7.

To prove Theorem 6.1 we need to introduce a relative notion of λ-C-smallness.

6.4. Definition. Let C be a class of spaces and λ be a limit ordinal. A map p : X → B is called λ-C-small if, for any α : K → X such that #K is smaller than the cofinality of λ, there exists a relative C_{≪λ}-complex f : E → B and a commutative diagram:

[figure: K; α; E; f; X; p; B]

6.5. Remark. Observe that if C is closed under weak equivalences, we can always arrange so that the map K → E, in the above definition, is a cofibration. If not let E′ = colim(E ← K ↪ K ×Δ[1]) and E′ → E be induced by id_{E} and K × Δ[1] →^{pr_{K}} K → E. By a direct verification one can show that the composition E′ → E →^{f} B is also a relative C_{≪λ}-complex

6.6. Lemma. Let C be a closed class and λ be a limit ordinal whose cofinality is bigger than N or λ = N. Let p : X ↠ B be a fibration. If #B is smaller than the cofinality of λ and, for any vertex b ∈ B, p^{−1}(b) is a λ-C-small space, then p is a λ-C-small map.

Proof. We first prove the lemma in the case when B is contractible. Let α : K → X be a map, where #K is smaller than the cofinality of λ. Since in this case p^{−1}(b) ↪ X is a weak equivalence, by part 8 of Lemma 4.8, X is λ-C-small. We can find therefore a factorization K →^{g} Y →^{h} X of α where Y ∈ C_{≪λ}. Consider now the following commutative diagram:

\xymatrix{ & Y \ar[r]^{h} \ar[d]|{id×p h} & X \ar@{->>}[d]^{p} \\ K \ar[ur]^{g} \ar[r]^{g×p α} & Y × B \ar[r]^{pr_{B}} \ar[ur]^{s} & B}

where s is any lifting in the above square. Such a lifting does exists since id × p h : Y → Y × B is an acyclic cofibration. It is clear that pr_{B} : Y × B → B is the desired relative C_{≪λ}-complex.

For a general B, we prove the lemma by induction on the cell decomposition of B. Let B = colim(B_{0} ← ∂Δ[n] ↪ Δ[n]). Let α : K → X be a map, where #K is smaller than the cofinality of λ. By pulling-back α and p along the appropriate maps we get the following commutative diagram where the squares on the right are pull-backs and the indicated arrows are respectively fibrations and cofibrations :

[figure: K; α; =; colim (; K_{0}; α_{0}; K_{1}; α_{1}; K_{2}; α_{2}; ); X; p; =; colim (; X_{0}; p_{0}; X_{1}; p_{1}; X_{2}; p_{2}; ); B; =; colim (; B_{0}; ∂Δ[n]; Δ[n]; )]

By the inductive assumption, the map α_{0} : K_{0} → X_{0} yields an appropriate relative C_{≪λ}-complex g_{0} : E_{0} → B_{0}. Define E_{1} = lim(E_{0} →^{g_{0}} B_{0} ← ∂Δ[n]). Let g_{1} : E_{1} → ∂Δ[n] be the induced map. According to part 1 of Proposition 5.4 g_{1} is also a relative C_{≪λ}-complex. As ∂Δ[n] is finite, it follows that #E_{1} is smaller than the cofinality of λ. Let P = colim(E_{1} ← K_{1} ↪ K_{2}). It is clear that #P is also smaller than the cofinality of λ. This space fits into a commutative diagram:

[figure: K_{2}; α_{2}; P; X_{2}; p_{2}; Δ[n]]

Since Δ[n] is contractible, P → X_{2} yields an appropriate relative C_{≪λ}-complex g_{2} : E_{2} → Δ[n] where P ↪ E_{2} is a cofibration (see Remark 6.5). Define g : E → B to be:

[figure: E; g; :=; colim (; E_{0}; g_{0}; E_{1}; g_{1}; E_{2}; g_{2}; ); B; =; colim (; B_{0}; ∂Δ[n]; Δ[n]; )]

According to part 2 of Proposition 5.4, g is a relative C_{≪λ}-complex. This concludes the lemma in the case when B is finite. The case when B is infinite follows from part 3 of Proposition 5.4.

Proof of Theorem 6.1. Let p : X → B be a map such that B ∈ C[λ] and, for any choice of a basepoint in B, F ib(X → B) ∈ C[λ]. We have to show that X ∈ C[λ].

Since C[λ] is closed under weak equivalences, we can arrange so that p is a fibration and B is fibrant. It follows from part 10 of Lemma 4.8 that spaces B and p^{−1}(b), for any vertex b in B, are λ-C-small. To prove the theorem it is enough to show that X is also λ-C-small. Let α : K → X be a map where #K is smaller than the cofinality of λ. We can factor the composition p α : K → B into K → B′ → B, where B′ ∈ C_{≪λ}. Let us pull-back p along B′ → B and form the following commutative diagram:

[figure: K; α′; α; X′; p′; X; p; B′; B]

The fibration p′ : X′ → B′ satisfies the assumptions of Lemma 6.6. Let g : E → B′ be the induced relative C_{λ}-complex for α′ : K → X′. It is clear that E ∈ C as C is closed under extensions by fibrations. It is also clear that #E is smaller than the cofinality of λ.

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Yale University, Department of Mathematics, 10 Hillhouse Avenue, P.O. Box 208283, New Haven, Connecticut 06520-8283, USA

E-mail address: chachols@math.yale.edu

Université catholique de Louvain, Département de mathméatiques, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgique

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University of Alberta, Department of Mathematical Sciences, 632 Central Academic Building, Edmonton, Alberta, T6G 2G1, Canada

E-mail address: stanley@math.ualberta.ca