Theory and Applications of Categories, Vol. 10, No. 8, 2002, pp. 162–179.

COLOCALIZATIONS AND THEIR REALIZATIONS AS SPECTRA

FRIEDRICH W. BAUER

ABSTRACT. Every chain functor A A A_{∗} (cf. appendix B below or [2] §4), admits a 𝔏colocalization (corollary 1.3., 𝔏 a subcategory of the category on which A A A_{∗} is defined) A A A_{∗}^{𝔏} which (in contrast to the case of 𝔏-localizations, cf. [3]) in general does not allow a realization as a spectrum (even if A A A_{∗} stems from a spectrum itself). The [E, ]_{∗}- colocalization of A. K. Bousfield [6] is retrieved as a special case of a general colocalization process for chain functors.

Introduction

It is well-known that every spectrum A can be localized in the sense of A. K. Bousfield [5] with respect to another spectrum E. This amounts to the existence of a natural exact sequence of spectra

EA  [figure] A [figure: η] A_{E}  (1)

where_{E}A is E-acyclic, A_{E} E-local and η an E-isomorphism. E-local means that [B, A_{E}] = 0 for any E-acyclic spectrum B.

Every spectrum A gives rise to a chain functor A A A_{∗} (cf. appendix B or [2] for further references). Let 𝔏 ⊂ Top^{2} or 𝔏 ⊂ 𝔅 (= the Boardman category [1]) be a full subcategory (e.g. 𝔏 = {E}, being determined by a single object), then there exists a natural 𝔏localization sequence

[figure: 𝔏; A A A; ∗; A A A; ∗; η; A A A; 𝔏_{∗}]  (2)

which implies (1) as a special case.

The objective of the present paper is to establish a dual 𝔏-colocalization sequence for chain functors

[figure: A A A; {𝔏 \atop ∗}; η; A A A; ∗; 𝔏; A A A; ∗]  (3)

where ^{𝔏}A A A_{∗} is 𝔏-acyclic, η an 𝔏-isomorphism and A A A_{∗}^{𝔏} 𝔏-colocal (i.e. [A A A_{∗}^{𝔏}, B B B_{∗}] = 0 for any 𝔏-acyclic chain functor B B B_{∗}).

Theorem 1.3 and Corollary 1.4 are existence theorems for 𝔏- colocalizations. So far we have (for the assertions, not for the proofs) full duality with the case of 𝔏-localizations.

Received by the editors 2001 September 6 and, in revised form, 2002 April 3.

Transmitted by James Stasheff. Published on 2002 April 11.

2000 Mathematics Subject Classification: Primary: 55P60, 55N20; Secondary: 55U30, 55P42.

Key words and phrases: Colocalizations, chain functors, realizations of chain functors as spectra.

\begin{matrix}c \\ ◯ \end{matrix} Friedrich W. Bauer, 2002. Permission to copy for private use granted.

However, even if the chain functor A A A_{∗} originates from a spectrum A (i.e. if it can be realized as a spectrum), it turns out that in general A A A_{∗}^{𝔏} can not be realized as a spectrum, because it does not have compact carriers. So we rediscover the fact that in general there are no colocalizations on the level of spectra. In the case 𝔏 = {E} we present in §2 an explicit construction of A A A_{∗}^{E} by means of the E-colocalization of a specific (highly irregular, i.e. not realizable) chain functor ℤ_{∗}, which is taken from [3]. By enlarging the given category (e.g. the category of Boardman spectra, the same would also work for pairs of CW-spaces) by some “formal S-duals” DE, we are permitted to deal with homology theories of the form {E, }_{∗} (which are isomorphic to DE_{∗}( ) = {S^{•}, ( )∧DE}_{∗} whenever a S-dual DE exists). Although {E, }_{∗} does not have compact carriers, it turns out that (theorem 3.1.) DE-colocalization of a chain functor A A A_{∗} can be realized by a spectrum. As a result we obtain a {E, }_{∗}-colocalization of a spectrum as a spectrum and not only as a chain functor. This coincides with the results of [6].

The proof of the main theorem is accomplished in §4 and prepared by some material on chain functors in an appendix A.

In contrast to [2], [3] we do not require that homology theories and (even regular) chain functors (cf. appendix B)) automatically have compact carriers, so that {E, }_{∗} now becomes a homology theory. In our present notation a chain functor might be an irregular one (cf. Appendix B).

The results of this paper have been recorded in the expository article [4].

1. Colocalizations of chain functors:

Suppose that 𝔏 ⊂ 𝔎 ⊂ Top^{2} are categories and 𝔏 a full subcategory of 𝔎. Alternatively we may take a full subcategory of the Boardman category [1].

1.1. Definition.

1) A chain functor C C C_{∗} is 𝔏-acyclic, whenever for any (E, F) ∈ 𝔏 there exists a chain homotopy D : C_{∗}(E, F) [figure] C_{∗+1}(E, F), D : 1 ≃ 0, i.e.

dD(ζ) + D(dζ) = ζ, ζ ∈ C_{∗}(E, F).

We assume that D commutes with l : C′_{∗} ⊂ C_{∗}, i.e. D(ζ) ∈ C′_{∗+1}, ζ ∈ C′_{∗}.

2) Let [⋯, ⋯] denote the set of all homotopy classes of chain functor transformations. A chain functor A A A_{∗} is 𝔏-colocal whenever [A A A_{∗}, B B B_{∗}] = 0 for any 𝔏-acyclic chain functor B B B_{∗}.

3) A transformation ω : A A A_{∗} [figure] B B B_{∗} between chain functors is an 𝔏-isomorphism whenever H_{∗}(ω) = ω_{∗} is an isomorphism on the category 𝔏.

1.2. Remark. Observe that we do not require the naturality of the chain homotopy in 1).

The main objective of this paper is the verification of Theorem 4.2, which subsumes:

1.3. Theorem. To any 𝔏 and any (regular) chain functor A A A_{∗} there exists a natural (regular) 𝔏-colocal A A A_{∗}^{𝔏} and a 𝔏-isomorphism η : A A A_{∗}^{𝔏} [figure] A A A_{∗}.

We call (A A A_{∗}^{𝔏}, η) the 𝔏-colocalization of A A A_{∗}. The analogy with 𝔏-localizations as treated in [3] can be pursued even further by introducing

𝔏A A A_{∗} = A A A_{∗} ∪_{η} coneA A A_{∗}^{𝔏}

which is 𝔏-acyclic: Suppose z = c + c_{1} ∈ A_{∗}(E) ∪_{η} cone A_{∗}^{𝔏}(E), E ∈ 𝔏, is a cycle, c ∈ A_{∗}(E)), c_{1} ∈ cone A_{∗}^{𝔏}(E). We calculate: dc_{1} = −dc ∈ (cone A_{∗}^{𝔏}) ∩ A_{∗}, hence dc_{1} ∈ η(A_{∗}^{𝔏}(E)). Because η is an 𝔏- isomorphism, dc_{1} = dc_{2}, c_{2} ∈ η(A_{∗}^{𝔏}(E)). We deduce that z ∼ z_{1} = c + c_{2} ∈ A_{∗}(E) ∪_{η} A_{∗}^{𝔏}(E) and therefore z_{1} ∼ z_{2} ∈ A_{∗}^{𝔏}(E). So z ∼ 0 ∈ ^{𝔏}A_{∗}(E). Since all chain complexes involved are free, the acyclicity of ^{𝔏}A A A_{∗} follows.

We have:

1.4. Corollary. To any 𝔏 and any chain functor A A A_{∗} there exists a natural (in the same sense as in theorem 1.3.) exact colocalization sequence

[figure: A A A; {𝔏 \atop ∗}; η; A A A; ∗; 𝔏; A A A; ∗]

with 𝔏-colocal A A A_{∗}^{𝔏}, 𝔏-acyclic ^{𝔏}A A A_{∗} which are regular, whenever A A A_{∗} is, and 𝔏-isomorphism η.

Proof. Only naturality has not yet been proven. However the construction of a^{𝔏}γ : ^{𝔏}A A A_{∗}

[figure]^{𝔏}B B B_{∗} associated with a γ : A A A_{∗} [figure] B B B_{∗}, fitting into a homology commutative diagram is immediate.

1.5. Proposition. A chain functor A A A_{∗} is 𝔏-colocal whenever the following condition is satisfied:

2′) Any 𝔏-isomorphism γ : B B B_{∗} [figure] C C C_{∗} induces an isomorphism

[A A A_{∗}, γ] : [A A A_{∗}, B B B_{∗}] [figure: ≈] [A A A_{∗}, C C C_{∗}].

Proof. Define

K K K_{∗} = C C C_{∗} ∪_{γ} coneB B B_{∗},

then we have an exact sequence of chain functors

Σ^{−1}K K K_{∗}  [figure] B B B_{∗} [figure: γ] C C C_{∗}  [figure] K K K_{∗}

where Σ^{−1}K K K_{∗} is the (formal) desuspension which is defined as for chain complexes. Now K K K_{∗} turns out to be 𝔏-acyclic and [A A A_{∗}, K K K_{∗}] = 0 is equivalent to the statement that [A A A_{∗}, γ] : [A A A_{∗},B B B_{∗}] [figure] [A A A_{∗},C C C_{∗}] is an isomorphism.

1.6. Proposition. Let ^{1}A A A_{∗} [figure: ^{1}η] A A A_{∗}, ^{2}A A A_{∗} [figure: ^{2}η] A A A_{∗} be two 𝔏-colocalizations of A A A_{∗}, then there exists a homotopy equivalence γ : ^{1}A A A_{∗} [figure]^{2}A A A_{∗}, commuting with ^{i}η, i = 1, 2.

In view of 1.5, the proof is immediate.

1.7. Remarks.

1) The formal suspensions and desuspensions used in the proof of 1.5 are not related to any excision properties of the individual (regular or irregular) chain functors involved.

2) If A A A_{∗} is a chain functor which allows a realization | A A A_{∗} | as a Boardman spectrum or alternatively A A A_{∗} is the chain functor associated with a spectrum, then it is not necessarily true that A A A_{∗}^{𝔏} can be realized as a spectrum | A A A_{∗}^{𝔏} | (cf. [2] for further reference). The reason is that A A A_{∗}^{𝔏}, while enjoying all other properties of a realizable chain functor, does not necessarily have compact support. In the next section we will encounter an explicit example even for the case that 𝔏 consists of a single object. So 𝔏-colocalizations do not always exist for (and as) spectra while they are always available as chain functors.

3) If A A A_{∗} has compact carriers, then A A A_{∗} | 𝔏 in general does not (because 𝔏 may not contain any compact subsets at all). This is also the reason that, in contrast to 𝔏-localizations in [3] (cf. in particular the proof of 4.1.), we cannot change A A A_{∗}^{𝔏} by simply taking compact carriers. The resulting chain functor would no longer be a 𝔏-colocalization of A A A_{∗}.

2. A A A_{∗}^{𝔏} for 𝔏 = {E} a single object

In this case we write A A A_{∗}^{E} instead of A A A_{∗}^{𝔏}. The purpose of this section is to give an explicit description of E-colocalization in this case. To this end we briefly recall the definition of the (irregular) chain functor ℤ ℤ ℤ_{∗} (cf. [3] §3 or appendix B):

ℤ_{n}(X) = \{ \begin{matrix}⟨z_{X}⟩ ⋯ & dz_{X} = 0, n = 0, X ≠ ∅ \\ 0 . . . . . . & otherwise \end{matrix}

ℤ_{n}(X, A) = 0, A ≠ ∅

This can easily be endowed with the structure of an irregular chain functor.

In order to describe ℤ_{∗}^{E}(X) explicitly, we refer to the existence proof of an E-colocal-ization in §4 and in appendix A.

According to the existence proof of an E-colocalization (cf. §4), the generating elements of ℤ_{∗}^{E}(X) are of the form w[ζ, f], where w denotes a word in the sense of the proof of A.1 and ζ = mz_{E}, m ∈ ℤ. Suppose A A A_{∗} is any chain functor defined on 𝔎, then we establish a new (irregular) chain functor

Ã_{n}( ) = (A_{∗}(E) ⊗ ℤ_{∗}^{E}( ))_{n} = ⨁_{p+q=n} A_{p}(E) ⊗ ℤ_{q}^{E}( ),

with the usual boundary operator. We set

Ã′_{∗}( ) = A_{∗}(E) ⊗ (ℤ_{∗}^{E})′( )

and φ, κ for Ã_{∗} are the tensor products of the identity of A_{∗}(E) with the corresponding φ, κ for ℤ_{∗}^{E}. Moreover l : Ã′_{∗}( ) ⊂ Ã_{∗}( ) is the tensor product 1 ⊗ l, correspondingly for i′ = κi. This furnishes an irregular chain functor (cf. appendix B or [3] §3).

2.1. Claim. For any ζ ∈ A_{∗}(E) we have a chain homotopy ζ ≃ l κ(ζ).

Proof. The relations φκ ≃ 1, j_{#}φ ≃ l imply lκ ≃ j_{#} : A_{∗}(E) [figure] A_{∗}(E, ∅) = A_{∗}(E). Let ζ ⊗ mw[z_{E}, f] ∈ Ã_{∗}(X, Y), f : E [figure] (X, Y), w a word as in the proof of A.1, be given, then we find

η̃: Ã_{∗}(X, Y) [figure] A_{∗}(X, Y)

by setting

η̃(ζ ⊗ mw[z_{E}, f]) = mwf_{#}(ζ), m ∈ ℤ

whenever this is defined and mwκ(ζ), when w(ζ) is not defined, which happens whenever ζ ∉ A′_{∗}(E), w [z_{E}, 1] ∈ (ℤ_{∗}^{E})′(X, Y) and w is of the form w = w_{1} φf, with continuous f. This furnishes a mapping of irregular chain functors.

2.2. Proposition.

1) η̃ is an E-isomorphism;

2) let B B B_{∗} be E-acyclic, then for any γ : Ã Ã Ã_{∗} [figure] B B B_{∗} one has

γ ≃ 0.

Proof. .

1) Let ζ ⊗ mw[z_{E}, f] ∈ Ã_{∗}(E) be an element, we can assume that f = 1_{E}, moreover w is a word w : E [figure] E. By inspecting all possible words of this kind in A.1, we deduce that

2.3. Claim. [(**)] The assignment

ζ ⊗ mw[z_{E}, 1] ↦ wζ ⊗ m[z_{E}, 1]

induces an E-isomorphism.

Proof of (**). The inverse (up to homology) to

η̃: Ã_{∗}(E) [figure] A_{∗}(E)

is

η̄(ζ) = ζ ⊗ [z_{E}, 1].

We calculate:

η̄η̃(ζ ⊗ mw[z_{E}, 1]) = mwζ ⊗ [z_{E}, 1]
η̃η̄(ζ) = η̃(ζ ⊗ [z_{E}, 1]) = ζ.

Due to (**) this implies that η̃ is an E-isomorphism.

2) Let z ∈ Z_{n}(Ã_{∗}(X, Y)) be a cycle, then there exists because of proposition A.1 B) a cycle

z ∼ z′ = ∑_{i=1}^{m} ζ_{i} ⊗ [z_{E}, f_{i}]. We collect all elements with the same f_{i}, obtaining z′ = ∑ f_{i#}z_{i},

observing that all z_{i} are cycles themselves. Therefore γz_{i} = dx_{i} is bounding. Hence γf_{i#}z_{i} = f_{i#}γz_{i} = df_{i#}x_{i} and consequently

γz′ = d ∑ f_{i#}x_{i}

is bounding. As a result we confirm that any γz is bounding which completes the proof of 2).

2.4. Theorem. There is a homotopy equivalence of (irregular) chain functors

γ : A_{∗}(E) ⊗ ℤ_{∗}^{E}( ) ≃ A A A_{∗}^{E}( ),

which is compatible with η̃, η.

Proof. This follows from 1.6 and 2.2.

2.5. Remark. A_{∗}(E)⊗ℤ_{∗}^{E} does not have compact carriers unless E is compact, because a continuous f : E [figure]X does not necessarily factor through some f_{1} : E [figure] K ⊂ X, K compact. As a result [z_{E}, 1_{E}] ∈ ℤ_{∗}^{E}(E) does not have a counterimage in some ℤ_{∗}^{E}(K), K ⊂ E, K compact. This implies that ℤ_{∗}^{E}( ) and consequently that A_{∗}(E)⊗ℤ_{∗}^{E}( ) does not necessarily have compact carriers.

3. [E, ]_{∗}-colocalization:

Among the homology theories defined on the Boardman category 𝔅 (cf. [1]), respectively on a category of pairs of CW-spaces 𝔎, we encounter those of the type [E, ]_{∗} (E ∈ 𝔅) respectively

{E, }_{∗} = colim_{k}[Σ^{∗+k}E, Σ^{k}( )], E ∈ 𝔎.

They satisfy all conditions of a generalized homology theory with the exception of the compact carrier condition. According to a theorem of A. Neeman ([7], theorem 2.1.) a homology theory h_{∗}( ) is isomorphic to a homology theory {E, }_{∗}, for some E ∈ 𝔅 if and only if h_{∗} preserves products, i. e. whenever

h_{∗}( Π_{ι∈J}X_{ι}) ≈ Π_{ι∈J}h_{∗}(X_{ι}).

If E is an object which admits an S-dual (e.g. a finite CW-spectrum), we have

{E, }_{∗} = DE_{∗}( ).

So, by an abuse of notation, we call such a homology theory DE_{∗} also, even when the object DE is not defined within the original category.

The natural transformations γ : DE_{∗} [figure]DE′_{∗} are in one-to-one correspondence with stable homotopy classes of mappings γ̂ : E′ [figure]E, while the transformations γ : [E, ]_{∗} [figure]F_{∗} are in one-to-one correspondence with stable mappings γ̂ : S^{•}

[figure] E ∧ F. In the same way we associate γ : F_{∗} [figure] [E, ]_{∗} with γ̂ : F ∧ E [figure] S^{•}. If DE is not defined as an object in the original category, we employ this strategy as a definition of new morphisms, so that we can enlarge the original categories 𝔅 or 𝔎 by these new objects and morphisms, denoting the new category by \overline{𝔅} respectively \overline{𝔎}. The [E, ]_{∗}-colocalization is nothing but the DE-colocalization in \overline{𝔎}.

Let A be a spectrum, A A A_{∗} the associated chain functor with A ≃| A A A_{∗} |, then we define

A_{∗}(DE) = (DE)_{∗}(A).

Using the terminology of §4, the elements of Ã_{∗}(X, Y) are classes of pairs [ζ, f̄], ζ ∈ A_{∗}(DE), f̄: DE [figure] X ∪CY. By the previous identification, we discover the associated f : S^{•} [figure] (X ∪ CY) ∧ E which factors over a finite subcomplex K ⊂ (X ∪ CY) ∧ E, whose cells are ∧-products of finitely many cells σ ∧ ε, σ in X ∪ CY, ε in E. As a result, f factors over X′ ∧ E, X′ ⊂ X ∪ CY, a finite, hence compact, subcomplex. This confirms that Ã_{∗} admits compact carriers.

According to the proof of 4.1 all generating elements of A_{∗}^{DE}(X, Y) are of the form w[ζ, f̄], with w being a word as in the proof of A.1. Hence A A A_{∗}^{DE} has compact carriers. All other properties of a chain functor are readily verified as in the appendix. As a result we have:

3.1. Theorem. Let A be a Boardman spectrum, then the chain functor A A A_{∗}^{DE} can be realized as a spectrum | A A A_{∗}^{DE} |= A A A^{[E, ]_{∗}}.

3.2. Remark. In [6] A. K. Bousfield introduced a [E, ]_{∗}-colocalization which agrees with the present one where defined.

4. Proof of the colocalization theorem:

Suppose 𝔏 ⊂ 𝔎 is a subcategory of a category of pairs of topological spaces or of pairs of spectra in the Boardman category. We need the following:

4.1. Theorem. For each chain functor L L L_{∗} : 𝔎 [figure] ch, there exists a chain functor L̄ L L_{∗} on 𝔎 and a transformation of chain functors τ : L L L_{∗} | 𝔏 [figure] L̄ L L_{∗} | 𝔏 (defined on 𝔏) such that for any chain functor K K K_{∗} on 𝔎 and any transformation α : L L L_{∗} | 𝔏 [figure] K K K_{∗} | 𝔏, there exists a ᾱ : L̄ L L_{∗} [figure] K K K_{∗} satisfying (ᾱ | 𝔏)τ = α. Regularity of L L L_{∗} implies that of L̄ L L_{∗}.

Proof. In a first step we neglect the specific properties of a chain functor, dealing merely with functors L_{∗}, K_{∗} going into the category of free chain complexes, equipped with subfunctors L′_{∗}, K′_{∗} and natural inclusions l : L′_{∗} ⊂ L_{∗}, i′ : L_{∗}(A) ⊂ L′_{∗}(X, A) respectively for K_{∗}. Having extended L_{∗} | 𝔏 over 𝔎 to a L̃_{∗}, we are able to apply A.1.

I) Let (ζ_{i}, f_{i}) be pairs, ζ_{i} ∈ L_{∗}(E_{i}, F_{i}), f_{i} : (E_{i}, F_{i}) [figure] (X, A) continuous, then we establish an equivalence relation

(ζ_{1}, f_{1}) ∼ (ζ_{2}, f_{2})

which is generated by the relationship

(ζ_{1}, f_{1}g) ∼ (g_{#}ζ_{1}, f_{1}), g ∈ 𝔏.

Let L̃_{∗}(X, A) (L̃′_{∗}(X, A)) be the free abelian group generated by equivalence classes [ζ, f] (ζ ∈ L′_{∗}(E, F)) of such pairs together with the following relations:

−[ζ, f] = [−ζ, f]

[ζ_{1}, f] + [ζ_{2}, f] = [ζ_{1} + ζ_{2}, f].

The resulting groups are still free abelian.

Suppose h : (X, A) [figure] (Y, B) ∈ 𝔎, then we set

h_{#}[ζ, f] = [ζ, hf]  (1)

d[ζ, f] = [dζ, f].  (2)
This furnishes a functor L̃_{∗} : 𝔎  [figure] ch (L̃′_{∗} : 𝔎  [figure] ch) together with inclusions l, i′,

defined in the following way:

The inclusion l[ζ, f] = [l(ζ), f] is obvious. Suppose [ζ, f] ∈ L̃_{n}(F), f : F [figure] A, then f determines a mapping f̃ as the composition

(F, F) [figure: (f,f)] (A, A) ⊂ (X, A)

allowing us to set i′[ζ, f] = [i′(ζ), f̃]. Since i′ for L L L_{∗} is by definition natural (in (X, A)) this equips L̃_{∗} with the required i′.

Suppose ζ ∈ L_{n}(X, A), then there exists by definition a natural chain homotopy χ(ζ), commuting with l and i′, satisfying

dχ(ζ) + χ(dζ) = i_{0#}(ζ) − i_{1#}(ζ).

By setting

[ζ, f] = [χ(ζ), f × 1],

we find a chain homotopy for L̃_{∗}

dχ[ζ, f] = [χ(ζ), f] + χ[dζ, f] = i_{0#}[ζ, f] − i_{1#}[ζ, f].

We have a transformation

τ : L_{∗} | 𝔏 [figure] L̃_{∗} | 𝔏
ζ [figure] [ζ, 1_{E,F)}], ζ ∈ L_{∗}(E, F).

Suppose α is given, then

α̃ : L̃_{∗} [figure] K_{∗}
[ζ, f] [figure] K_{∗}(f)(α(ζ))

is defined on 𝔎 and satisfies

(α̃ | 𝔏)τ = α.

Both τ and α̃ commute with the natural chain homotopies inherent in L̃ L L_{∗}, K K K_{∗} and L L L_{∗}. Suppose [ζ, f] ∈ L̃_{∗}(X, X), f : (E, F) [figure] (X, X), then we find f′ : (E, E) [figure] (X, X) and a ξ ∈ L_{∗+1}(E, E) such that dξ = g_{#}ζ, g : (E, F) ⊂ [E, E). As a result

d[ξ, f′] = [ζ, f].

This ensures that L̃_{∗}(X, X) is always acyclic. All other properties of a partial chain functor (cf. appendix A) are immediately verified.

We still have to ensure that for the derived homology of L̃_{∗} an excision axiom holds: Let (ζ, f), f : (E, F) [figure] (X, A), ζ ∈ Z_{∗}(L_{∗}(E, F)) be a pair and suppose Ū ⊂ Int_{X}A, then we define V = f^{−1}(U) and observe that V̄ ⊂ Int_{E}F. By restricting f we obtain a map f′ : (E \ V, F \ V)) [figure](X \ U, A \ U). Let i : (X \ U, A \ U) ⊂ (X, A), i′ : (E \ V, F \ V)) ⊂ (E, F) be the inclusions, then there exists a ζ′ ∈ Z_{∗}(L_{∗}(E \ V, F \ V)) such that i′_{#}ζ′ ∼ ζ. Hence

i_{#}([ζ′, f′]) ∼ [ζ, f].

So i_{#} is epic.

Let [ζ′, f′] be a given cycle and suppose d[ξ, f] = [dξ, f] = i_{#}[ζ′, f′], then excision for L L L_{∗} enables us to find [ξ′, f′] satisfying d[ξ′, f′] = [ζ′, f′], so that i_{#} is monic.

II) This provides us with a L̃_{∗} : 𝔎 [figure] ch (L̃′_{∗} : 𝔎 [figure] ch) satisfying all prerequisites of A.1. We obtain a chain functor L̄ L L_{∗} = ¯ L̃ L L_{∗} by applying A.1.

There is an inclusion L_{∗} ⊂ L̄_{∗} commuting with the operators φ, κ, δ, δ_{i} etc., and therefore a transformation of chain functors τ : L L L_{∗} | 𝔏 [figure]L̄ L L_{∗} | 𝔏. Let α be a transformation as in the theorem. Assume that ζ̄ ∈ L̄_{∗} is any element, then either ζ̄ ∈ L̃_{∗} allowing us to set ᾱ(ζ̄) = α̃(ζ̄) or ζ̄ ∉ L̃_{∗}. In this case ζ̄ = Σw_{i}ζ̃_{i}, ζ̃_{i} ∈ L̃_{∗} in a unique way, where w_{i} is a word in the sense of the proof of A.1, so that we are entitled to set

ᾱ(ζ̄) = Σw_{i}α̃(ζ̃_{i}).

This furnishes a transformation ᾱ : L̄ L L_{∗} [figure] K K K_{∗} satisfying (ᾱ | 𝔏)τ = α.

This completes the proof of the theorem for regular L L L_{∗}. The changes in the irregular case are, in view of A.5, immediate.

4.2. Theorem. Let A A A_{∗} be any chain functor, 𝔏 as before, then there exists a natural (in A A A_{∗}) 𝔏-colocalization

η : A A A_{∗}^{𝔏} [figure] A A A_{∗}

(cf. definition 1.1.2)). More precisely we have:

1) η is an 𝔏-isomorphism;

2) if B B B_{∗} is 𝔏 acyclic, then [A A A_{∗}^{𝔏}, B B B_{∗}] = 0;

3) Let γ : A A A_{∗} [figure]B B B_{∗} be a transformation of chain functors then there exists in a canonical way a γ^{𝔏} : A A A_{∗}^{𝔏} [figure] B B B_{∗}^{𝔏} such that the diagram

[figure: B B B_{∗}^{𝔏}; B B B_{∗}; η; ^{A A A}{𝔏 \atop ∗}; B B B_{∗}^{𝔏}; γ^{𝔏}; ^{A A A}{𝔏 \atop ∗}; ^{A A A}^{∗}; η; ^{A A A}^{∗}; B B B_{∗}; γ]  (3)

is commutative up to homology. If A A A_{∗} is regular, then A A A_{∗}^{𝔏} is also regular.

Proof. Let L L L_{∗} = A A A_{∗}, α : A A A_{∗} | 𝔏 [figure]A A A_{∗} | 𝔏 be the identity, then we have ᾱ : Ā_{∗} [figure] A A A_{∗} such that (ᾱ | 𝔏_{∗}) is an isomorphism. We set

A A A_{∗}^{𝔏} = Ā A A_{∗}

and

ᾱ = η.

Suppose B B B_{∗} is 𝔏-acyclic, z ∈ Ā_{∗}(X, Y) a cycle, then A.1 B) implies that z ∼ z′ ∈ Z_{∗}(Ã_{∗}(X, Y)). Now z′ = ∑[ζ_{i}, f_{i}] and, because of (2), we can assume that there is a representation of z′ as a sum with all [ζ_{i}, f_{i}] being cycles. Let ω : A A A_{∗}^{𝔏} [figure]B B B_{∗} be a mapping. We deduce, because B B B_{∗} is 𝔏-acyclic, that there exists a ξ_{i} in B B B_{∗} such that dξ_{i} = ω[ζ_{i}, 1], hence d ∑ f_{i#}ξ_{i} = ∑ f_{i#}ω[ζ_{i}, 1] = ω ∑[ζ_{i}, f_{i}] = ω(z′). Thus ω(z) is bounding. Since B B B_{∗} is a free chain complex, this implies ω ≃ 0.

Concerning naturality, we consider γ : A A A_{∗} [figure] B B B_{∗} and construct in a functorial way γ^{𝔏} : A A A_{∗}^{𝔏} [figure] B B B_{∗}^{𝔏}. First, we set

γ^{𝔏}[ζ, f] = [γ ζ, f].

For free generators, we define

γ^{𝔏}w[ζ, f] = w[γ ζ, f].

This furnishes a transformation γ^{𝔏} : A A A_{∗}^{𝔏} [figure]B B B_{∗}^{𝔏} in a canonical way. In order to investigate the homology commutativity of (3), let z ∈ Z_{n}(A_{∗}^{𝔏}(X, Y)) be a cycle, then there exists a cycle z ∼ z′ = ∑[ζ_{i}, f_{i}]. Since

γ η(z′) = ∑ f_{i#}γ ζ_{i} = η γ^{𝔏}(z′),

the assertion follows.

A. Appendix: Auxiliary material for the proof of the main theorem

This section is devoted to some material on chain functors which is needed in §4 to establish Theorem 4.1 and especially Theorem 4.2, which is the existence theorem for colocalizations. For further background, e.g. concerning explicit definitions of chain functors, regularity, etc., we refer to [2] for further references or to the next section (appendix B), where, for the convenience of the reader, the definitions are recorded.

Let C_{∗} : 𝔎 [figure] ch be a functor into the category of free chain complexes, l : C′_{∗} ⊂ C_{∗} a subfunctor, i′ : C_{∗}(A) ⊂ C′_{∗}(X, A) a natural inclusion (all going into direct summands, cf. [2] for further references), then C C C_{∗} need not be a chain functor: There may be no φ : C′_{∗}(X, A) [figure]C_{∗}(X), κ : C_{∗}(X) [figure]C′_{∗}(X, A) and no chain homotopies j_{#} φ ≃ l, φ κ ≃ 1 together with all the other relations which are required for a chain functor and which are eventually not available.

We list now all these properties and conditions, by expressing them as specific operations on C C C_{∗}:

1) There are possibly non-natural chain mappings φ, κ.

2) There are (non-natural) homomorphisms

δ_{1} : C_{∗}(X) [figure] C_{∗+1}(X)
δ_{2} : C′_{∗}(X, A) [figure] C′_{∗+1}(X, A)

satisfying

\{ \begin{matrix}dδ_{1}(ζ) + δ_{1}(dζ) = & φ κ(ζ) − ζ \\ dδ_{2}(ζ) + δ_{2}(dζ) = & j_{#} φ(ζ) − l(ζ). \end{matrix}  (1)

The existence of δ_{i} takes care of the above mentioned chain homotopies. The existence of δ_{i} (as well as that of δ) follow for a chain functor, because C_{∗}( ) is assumed to be free.

3) There is a homomorphism

η : Z_{∗}(C_{∗}(X, A)) [figure] C′_{∗}(X, A) ⊕ C_{∗+1}(X, A) ⊕ C_{∗−1}(A)
η(ζ) = (η_{1}(ζ), η_{2}(ζ), η_{3}(ζ))

satisfying

ζ + d η_{2}(ζ) = l η_{1}(ζ) + q_{#}δs_{#}η_{4}(ζ),
q : (A, A) ⊂ (X, A), s : A ⊂ (A, A)

where δ stems from 1). This expresses the fact that every cycle z ∈ C_{∗}(X, A) is homologous to a cycle of the form

l z′ + q_{#}a, z′ ∈ C′_{∗}(X, A), a ∈ C_{∗}(A, A)

dz′ ∈ im i′, dz′ = i′ da.

4) There exists a homomorphism

β : B_{∗}(X, A) [figure] C_{∗}(A)

B_{∗}( ) = bounding cycles, satisfying

i′ β(ζ) = dη_{1} (ζ);

5)

λ : j_{#}^{−1} B_{∗}(X, A) [figure] C′_{∗}(X, A) ⊕ C_{∗}(A)
λ(ζ) = (λ_{1}(ζ), λ_{2}(ζ)),

satisfying

κ ζ = dλ_{1}(ζ) + i′ λ_{2}(ζ).

Here 4), respectively 5), are translations of

ker ψ ⊂ ker ¯∂

and

ker j_{∗} ⊂ ker p_{∗} κ_{∗},

p : C′_{∗}(X, A) [figure] C′′_{∗}(X, A) = C′(X, A)/im i′, j_{∗}, p_{∗} denoting the induced mappings for homology.

There are no relations between all these operations and continuous mappings except those already mentioned and the fact that i′ = κ i is natural (i.e. commutes with continuous mappings).

By an abuse of notation, we write e.g. δ : (X, X) [figure](X, X), η_{2} : (X, A) [figure] (X, A) and do not distinguish between these symbols and the original homomorphisms. This allows us to combine them with continuous maps as if they constitute new morphisms which induce the original homomorphisms.

We now form words w by using these operators or continuous mappings, where we assume of course that the compositions (like φκ, κδ_{1}, but not necessarily φδ_{1}) make sense.

Suppose that (C_{∗}, C′_{∗}, l, i′) is in the following sense partially a (free) chain functor (or simply: a partial-chain-functor):

p1) C_{∗}(X, X) is always acyclic; all inclusions j induce monomorphisms C_{∗}(j) onto direct summands, the same holds for i′ and l; the homology groups of C_{∗} satisfy an excision property; there are natural and chain homotopies D, commuting with i′ and l, satisfying, for each ζ ∈ L_{∗}(X, A),

dD(ζ) + D(dζ) = i_{0#}(ζ) − i_{1#}(ζ);

p2) if ζ ∈ C_{∗} and v is any of the operators 1) - 6) (i.e. φ, κ, δ_{i}, η_{i}, β, λ, ), then either none or all those vζ which formally make sense, are defined; p3) if κζ is defined, then φ κ ζ is also defined; p4) let ζ = ∑ a_{i} ζ_{i}, a_{i} ≠ 0, ζ_{i} a base element (with respect to a basis of C_{∗}) be given and vζ is defined, then all vζ_{i} are defined. A.1. Proposition. There exists a chain functor C̄ C C_{∗} = {C̄_{∗}, C̄′_{∗} , i′, l, φ, κ}, C_{∗} ⊂ C̄_{∗}, C′_{∗} ⊂ C̄′_{∗} such that A) all operations 2) - 7) of C̄_{∗} agree with those of C_{∗} whenever both are defined. B) The inclusion C_{∗}(X, A) ⊂ C̄_{∗}(X, A) induces an isomorphism of homology groups. Proof. Let ^{1}C_{∗} ⊃ C_{∗} be a partial chain functor satisfying p1) - p4) and in addition the following condition: p5) ^{1}ζ ∈ ^{1}C_{∗} [figure]^{1}ζ = ∑ w_{i}ζ_{i}, where w_{i} denotes an operation v as in p2) or a word of the form φ κ and ζ_{i} ∈ C_{∗}. Such an extension is called strongly admissible and denoted by 1C_{∗} ≥ C_{∗}. More generally we have for C_{∗} ≤ ^{1}C_{∗} ⊂ ^{2}C_{∗} the possibility to detect ^{2}C_{∗} as a strongly admissible extension ^{1}C_{∗} ≤ ^{2}C_{∗} by the same procedure. We observe: A.2. Claim. [(*)] If ^{1}C_{∗} ≥ ^{2}C_{∗} is a strongly admissible extension, then the inclusion induces an isomorphism of homology groups:

H_{∗}(^{1}C_{∗}(X, A)) ≈ H_{∗}((^{2}C_{∗}(X, A)).

Proof. The conditions p1) - p5) guarantee that the extension of ^{2}C_{∗} to ^{1}C_{∗} neither produces new cycles nor does it convert non-bounding into bounding cycles.

A chain of strongly admissible extensions

1C_{∗} ≤ ⋯ ≤ ^{2}C_{∗}

is called an admissible extension. Let 𝔖 be the set of all admissible extensions with the ≤ relation as partially ordering. A.3. Claim. [(**)] 𝔖 is inductive. Proof. Let 𝔗 ⊂ 𝔖 be a tower, then

⋃_{C̃_{∗}∈𝔗} C̃_{∗}

is an upper bound for all C̃_{∗} ∈ 𝔗.

The proposition is now reduced to the verification of: A.4. Claim. [(***)] A maximal C̄_{∗} is a chain functor satisfying B). Proof. Suppose there exists a ζ ∈ C̄_{∗} such that vζ for some v as before is not defined. Without loss of generality, we can assume that ζ is a base element. Considering the free abelian group generated by vζ for all v as in p2), p3), we achieve an admissible extension of C̄_{∗} contrary to the assumption that C̄_{∗} is maximal. So, with every ζ ∈ C̄_{∗} and every letter v, we conclude that vζ ∈ C̄_{∗}, guaranteeing that C̄_{∗} is a chain functor. By construction of 𝔖 and in view of (*), the homologies of C̄_{∗} and C_{∗} are isomorphic.

This completes the proof of A.1.

The following corollary is immediate:

A.5. Corollary. Suppose that i′ : C_{∗}(A)  [figure]C′_{∗}(X, A), l : C′_{∗}  [figure]C_{∗} as well

as j_{#} = C_{∗}(j) for inclusions j are not necessarily inclusions, then there still exists an irregular C_{∗} satisfying A) and B).

B. Appendix: Chain functors and associated homology theories

In this appendix, we present without proofs for the convenience of the reader some material about the definition of and the motivation for chain functors. Concerning details we refer to [2].

It would be advantageous to define a homology theory h_{∗}( ) as the derived homology of a functor

C_{∗} : 𝔎 [figure] ch ch ch,

𝔎 = the category on which h_{∗} is defined (e.g. a subcategory of the category of all pairs of topological spaces, or pairs of spectra or pairs of CW spaces or CW spectra together with the appropriate morphisms), ch ch ch = category of chain complexes (i.e. C_{∗} = {C_{n}, d_{n}, n ∈ ℤ, d^{2} = 0} ∈ ch ch ch). Let (X, A) ∈ 𝔎 be a pair, then one would like to have an exact sequence

0  [figure] C_{∗}(A) [figure: i_{#}] C_{∗}(X) [figure: j_{#}] C_{∗}(X, A)  [figure] 0  (1)

such that the associated boundary ¯∂ : H_{n}(C_{∗}(X, A)) [figure]H_{n−1}(C_{∗}(A)) corresponds to the boundary ∂ : h_{n}(X, A) [figure]h_{n−1}(A) via the isomorphism h_{∗}( ) ≈ H_{∗}(C_{∗}( )). In accordance with [2] we call a homology with this property flat. Due to a result of R. O. Burdick, P. E. Conner and E. E. Floyd (cf. [4] for further reference), this implies for 𝔎 = category of CW pairs that h_{∗}( ) is a sum of ordinary homology theories, i.e. of those satisfying a dimension axiom, although not necessarily in dimension 0.

We call a functor C_{∗} together with a short exact sequence (1) determining the boundary operator, a chain theory for h_{∗}. The non-existence of such a chain theory gives rise to the theory of chain functors.

A chain functor C C C_{∗} = {C_{∗}, C′_{∗}, l, i′, κ, φ} is

1) a pair of functors C_{∗}, C′_{∗} : 𝔎 [figure] ch ch ch, natural inclusions i′ : C_{∗}(A) ⊂ C′_{∗}(X, A), and l : C′_{∗}(X, A) ⊂ C_{∗}(X, A,)

2) possibly non-natural chain mappings

φ : C′_{∗}(X, A) [figure] C_{∗}(X)
κ : C_{∗}(X) [figure] C′_{∗}(X, A),

chain homotopies φκ ≃ 1, j_{#} φ ≃ l (j : X ⊂ (X, A)), as well as an identity

κ i_{#} = i′ i : A ⊂ X.

3) All inclusions k : (X, A) ⊂ (Y, B) are supposed to induce monomorphisms. All C_{∗}(X, X) are acyclic.

The exact sequence (1) is replaced by the sequence

0  [figure] C_{∗}(A) [figure: i′] C′_{∗}(X, A) [figure: p] C′_{∗}(X, A)/im i′  [figure] 0  (2)

and there exists a homomorphism

ψ : H_{∗}(C′_{∗}(X, A)/im i′)  [figure] H_{∗}(C_{∗}(X, A))  (3)
[z′] [figure] [l(z′) + q_{#}ā]

where z′ ∈ C′_{∗}(X, A), dz′ ∈ im i′, q : (A, A) ⊂ (X, A), ā ∈ C_{∗}(A, A), dā = −dz′. It is assumed that ψ is epic.

Since C_{∗}(A, A) is acyclic, dz′ ∈ im i′, such that an ā exists and [l(z′) + q_{#}(ā)] is independent of the choice of ā. This assumption implies that each cycle z ∈ C_{∗}(X, A) is homologous to a cycle of the form l(z′)+q_{#}(ā), with z′ being a relative cycle, the analogue of a classical relative cycle z ∈ C_{∗}(X) with dz ∈ im i_{#}, whenever (1) holds, i.e. whenever we are dealing with a chain theory.

4) We assume

ker ψ ⊂ ker ¯∂,  (4)

¯∂ : H_{n}(C′_{∗}(X, A)/im i′) [figure]H_{n−1}(C_{∗}(A)) being the boundary induced by the exact sequence (2). Moreover, we assume

ker j_{∗} ⊂ ker p_{∗} κ_{∗},  (5)

with e.g. κ_{∗} denoting the mapping induced by κ for the homology groups.

5) Homotopies H : (X, A) × I [figure](Y, B) induce chain homotopies D(H) : C_{∗}(X, A) [figure] C_{∗+1}(Y, B) naturally and compatible with i′ and l. These are almost all the ingredients of a chain functor we need. The derived (or

associated) homology of a chain functor

h_{∗}(X, A) = H_{∗}(C_{∗}(X, A))

respectively for the induced mappings, is endowed with a boundary operator

∂ : H_{n}(C_{∗}(X, A)) [figure] H_{n−1}(C_{∗}(A)),

determined by ¯∂ as follows: We seek to ζ ∈ H_{n}(C_{∗}(X, A)) a representative l(z′) + q_{#}(ā) and set

∂ζ := ¯∂[z′] = [i′^{−1} d z′].  (6)

This turns out to be independent of the choices involved.

This h_{∗}( ) satisfies all properties of a homology theory with the exception of an excision axiom. Therefore it is convenient to add:

6) Let p : (X, A) [figure](X′, A′) be an excision map (of some kind, e.g. p : (X, A) [figure] (X/A, ⋆),) then p_{∗} = H_{∗}(C_{∗}(p)) is required to be an isomorphism. Then it turns out that H_{∗}(C_{∗}( )) = h_{∗}( ) becomes a homology theory. Moreover

it turns out that under very general conditions on 𝔎 every homology theory h_{∗}( ) is isomorphic to the derived homology theory of some chain functor.

Let λ : C_{∗} [figure]L_{∗}, λ′ : C′_{∗} [figure]L′_{∗} be natural transformations, where C C C_{∗}, L L L_{∗} are chain functors, compatible with i′, l and the natural homotopies of 5), then we call λ : C_{∗} C_{∗} C_{∗}

[figure] L_{∗} L_{∗} L_{∗} a transformation of chain functors. Such a transformation induces obviously a transformation λ_{∗} : H_{∗}(C_{∗} C_{∗} C_{∗}) [figure] H_{∗}(L_{∗} L_{∗} L_{∗}) of the derived homology (i.e. λ_{∗} commutes also with the boundary ∂ as defined in (6)). This furnishes a category ℭ𝔥 of chain functors. Aweak equivalence in ℭ𝔥 is a λ : C_{∗} C_{∗} C_{∗} [figure] L_{∗} L_{∗} L_{∗} which has a homotopy inverse. Incidentally there are also fibrations and cofibrations, endowing ℭ𝔥 with the essential features of a closed model category .

One could require that λ also commutes with φ and κ, however it turns out that up to homotopy this does not matter ([2] proposition 4.5.). Furthermore we can introduce the homotopy category ℭ𝔥_{h} with chain homotopy classes of transformations of chain functors ℭ𝔥_{h}(⋯, ⋯) = [⋯, ⋯] as morphisms.

In order to establish all this, it becomes sometimes necessary to assume that a chain functor C_{∗} C_{∗} C_{∗} satisfies:

7) All chain complexes C_{∗}(X, A) are free. However this is not a severe restriction as the following lemma ensures:

B.1. Lemma. To any chain functor C_{∗} C_{∗} C_{∗} there exists a canonically defined chain functor L_{∗} L_{∗} L_{∗} and a transformation of chain functors λ : L_{∗} L_{∗} L_{∗} [figure]C_{∗} C_{∗} C_{∗} compatible with φ and κ, inducing an isomorphism of homology, such that:

L1) All L_{∗}(X, A) have natural bases bbb in all dimensions;

L2) b ∈ bbb  [figure] db ∈ bbb; b ∈ bbb  [figure] i′(b) ∈ bbb, l(b) ∈ bbb, whenever this is defined and makes

sense;

L3) For every homology class ζ ∈ H_{∗}(C_{∗}(X, A)) there exists a basic b ∈ (λ_{∗})^{−1}ζ.

Proof. Consider the free abelian group F (C_{n}(X, A)) determined by the elements of C_{∗}(X, A) and convert this into a chain complex F_{∗}(X, A) in an obvious way. To each a ∈ C_{n}(X, A) corresponds a basic ā ∈ F (C_{n}(X, A)). Let i : M_{∗} ⊂ F_{∗} be the subcomplex generated by all elements of the form ∑ m_{i} ā_{i} − \overline{∑ m_{i} a_{i}} and define

L_{∗}(X, A) = F_{∗}(X, A) ∪_{i} cone M_{∗}(X, A).

This furnishes evidently a functor into the category of chain complexes. We set

λ(∑ m_{i} ā_{i}) = ∑ m_{i} a_{i},

and λ | M_{∗} = 0.

Moreover ∑ m_{i} ā_{i} ∈ L′_{∗} whenever all a_{i} ∈ C′_{∗}, respectively for the elements of cone M_{∗}. This implies that L2) holds. One can immediately equip L_{∗} L_{∗} L_{∗} and λ with the structure of a chain functor, respectively of a transformation between chain functors.

Every cycle z ∈ Z_{n}(C_{∗}(X, A)) is of the form λ(z̄) = z, hence λ_{∗} is epic. Any cycle z̃ ∈ Z_{n}(L_{∗}(X, A)) is homologous to a z̄, for some z ∈ Z_{n}(C_{∗}(X, A)): Suppose z̃ = Σm_{i}\overline{a}_{i} + c where c ∈ cone M_{∗}, then we have z̃ = \overline{a} + c_{1} where c_{1} ∈ cone M, hence d\overline{a} = \overline{da} ∈ cone M_{∗}, implying that d\overline{a} = \overline{d}a = 0. So \overline{a} and c_{1} are cycles, and since c_{1} is bounding in cone M_{∗}, we conclude that z̃ ∼ \overline{a}. If z = dx, then z̄ = dx̄ and λ_{∗} is therefore monic.

This completes the proof of the lemma.

Finally we repeat the definition of an irregular chain functor (cf. [2] definition 4.1.) for more details or [3] §3 for an example): {C_{∗}, C′_{∗}, φ, κ, i′, l} satisfies all conditions of a chain functor, but we do no longer require a) that all inclusions induce isomorphisms; b) nor that i′, l are necessarily monomorphisms; c) nor any excision properties. Whenever we talk about a regular chain functor, we mean that it is not irregular.

The role of the unnatural mappings φ and κ seems at the first glance to be a little mysterious. A chain functor K K K_{∗} is called flat whenever φ, κ and the chain homotopies φκ ≃ 1, j_{#} φ ≃ l are natural. In the beginning, we introduced the concept of a flat homology theory.

B.2. Theorem. The following conditions for a homology theory are equivalent:

1) h_{∗} is flat;

2) there exists a flat chain functor associated with h_{∗}.

B.3. Corollary. For a homology theory defined on the category of CW spaces, conditions 1), 2) are equivalent to

3) h_{∗} is the direct sum of ordinary homology theories.

References

[1] F. Adams, Stable homotopy and generalized homology. The Univ. of Chicago Press, Chicago and London (1974).

[2] F. W. Bauer, Chain functors with isomorphic homology. Homology, Homotopy and Applications 3 (2001), 37–53.

[3] F. W. Bauer, Tensor products of spectra and localizations. Homology, Homotopy and Applications 3 (2001), 55–85.

[4] F. W. Bauer, The Boardman category of spectra, chain complexes and (co-)localiza-tions. Homology, Homotopy and Applications 1 (1999), 95–116.

[5] A. K. Bousfield, The localization of spectra with respect to homology. Topology 18 (1979), 257–281.

[6] A. K. Bousfield, The Boolean algebra of spectra. Com. Math. Helvetici 54 (1979) 368–377.

[7] A. Neeman, Brown representability for the dual. Invent. Math. 133 (1998), 97–105.

Fachbereich Mathematik Johann Wolfgang Goethe Universität 60054 Frankfurt a.M. Robert-Mayer Str. 8-10 Germany

Email: f.w.bauer@mathematik.uni-frankfurt.de

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