THE A-COMPLICATION OF A SPACE

W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

Abstract. Suppose that A is a pointed CW-complex. We look at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complication of B. Studying the complication leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CW_{A}(X) of an arbitrary space X.

1. Introduction

Suppose that A is a pointed CW complex. A space B is said to be A-cellular if it can be constructed from copies of A by repeatedly taking homotopy colimits. In this paper we introduce the notion of the A-complication of an A-cellular space B; this is an ordinal number, denoted κ_{A}(B), which gives the minimal number of times it is necessary to repeat the homotopy colimit construction in order to get from A to B. We then describe an explicit way to construct B in terms of functorial resolutions which depend upon information about maps A → B. The resolution process has to be iterated before it settles on the right result, and it turns out that the number of iterations required is controlled by κ_{A}(B). From this point of view the complication determines how difficult it is to build B, starting with the space A and data about the maps A → B.

Suppose now that X is an arbitrary pointed space. Let Map(A, X) denote the space of pointed maps from A to X, and say that a map X → Y is an A-equivalence if it induces a weak equivalence Map(A, X) ≃ Map(A, Y). Dror-Farjoun [5] shows that up to homotopy there is a canonical A-cellular space CW_{A}(X) together with an A-equivalence CW_{A}(X) → X; this is the A-cellular approximation to X. (If X is Acellular then CW_{A}(X) ≃ X.) The resolution process mentioned above can be applied to X as well as to CW_{A}(X). It turns out that after the

Date: August 15, 2000.

The first author was partially supported by the National Science Foundation.

The second author was partially supported by National Science Foundation Grant DMS–9802336.

2 W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

very first stage it gives the same result for these two spaces, and so, after a number of iterations controlled by κ_{A}(CW_{A} X), converges to CW_{A}(X). For this reason we extend the definition of A-complication to arbitrary spaces by setting κ_{A}(X) = κ_{A}(CW_{A} X), and we interpret this ordinal as measuring the difficulty of starting with data on maps A → X and building CW_{A}(X).

Our approach is to construct an increasing chain of categories

C_{0} ⊂ C_{1} ⊂ ⋯ ⊂ C_{α} ⊂ C_{α+1} ⋯ (1.1)

depending on A and indexed by the ordinal numbers α. (These are full subcategories of the category of pointed spaces.) The category C_{0} contains all cofibrant spaces which are homotopy equivalent to retracts of wedges of copies of A. The category C_{α}, α > 0 is defined inductively. Let C_{<α} denote the union of the categories C_{β} for β < α. Then C_{α} contains all cofibrant spaces which up to homotopy equivalence and retracts can be written as hocolim G, for some small category D and functor G : D → C_{<α}. It is clear that any A-cellular space B belongs to C_{α} for some ordinal α; the least such α is the A-complication κ_{A}(B) of B. As mentioned above, for a general X we set κ_{A}(X) = κ_{A}(CW_{A} X).

We also construct for any X a functorial sequence of spaces

F_{0}(X) → F_{1}(X) → ⋯ → F_{α}(X) → F_{α+1}(X) → · · · .

The space F_{0}(X) is equivalent to a wedge of copies of A, one for each map A → X. If α = β + 1 is a successor ordinal, then F_{α}(X) is equivalent to the realization or homotopy colimit of a resolution of X derived from iteratively applying the functor F_{β} to X. If α is a limit ordinal, then F_{α}(X) ≃ hocolim_{β<α} F_{β}(X). It is clear that F_{α}(X) ∈ C_{α}. All of the spaces F_{α}(X) map in a coherent way to X.

Our first result gives a way to interpret the functors F_{α} conceptually. Recall the definition of C_{<α} from above; if α = β + 1 is a successor ordinal, then C_{<α} is just C_{β}.

1.2. Theorem. Suppose that α > 0 is an ordinal. Then the space F_{α}(X) is the canonical approximation to X by a homotopy colimit of spaces from C_{<α} (4.4). More concretely, F_{α}(X) is a homotopy colimit, over the category of maps B → X with B ∈ C_{<α}, of the functor which assigns to the object B → X the space B itself.

Since the category C_{<α} in 1.2 is not small, part of the work in proving this theorem is to make sense of the indicated homotopy colimit. (The way in which this is done (3.11) explains why in the above statement F_{α}(X) is called “a” homotopy colimit instead of “the” homotopy colimit.) Roughly speaking, the theorem says that in spite of its awkward construction in terms of an explicit resolution, F_{α}(X) is the most natural functorial approximation to X by a homotopy colimit of A-cellular spaces of complication < α.

We next show that the functors F_{α} are invariant under A-equivalences, and that they are ultimately successful at reconstructing A-cellular spaces.

1.3. Theorem. Suppose that α > 0 is an ordinal. Then a map X → Y is an A-equivalence if and only if the induced map F_{α}(X) → F_{α}(Y) is a homotopy equivalence.

1.4. Theorem. Suppose that α and β are ordinals with α > β. Then if B belongs to C_{β}, the natural map F_{α}(B) → B is a homotopy equivalence. In particular F_{α} ∘ F_{β} ≃ F_{β}.

1.5. Remark. Now let X be an arbitrary space and let B = CW_{A}(X). Consider the commutative ladder

F_{0}(B) → ⋯ → F_{α}(B) → ⋯ B
↓ ↓ ↓
F_{0}(X) → ⋯ → F_{α}(X) → ⋯ X

By 1.3 the vertical maps indexed by α are equivalences for α > 0, and in view of 1.4 the maps F_{α}(B) → B are equivalences for α > κ_{A}(B) = κ_{A}(X). Thus the spaces F_{α}(X) give a sequence of ever more complicated approximations to CW_{A}(X), a sequence which hits the mark when α = κ_{A}(X) + 1.

The following proposition is clear from Dror-Farjoun’s construction of CW_{A}(X).

1.6. Proposition. There exists an ordinal κ, depending on A, such that κ_{A}(B) ≤ κ for any A-cellular space B.

Let κ_{A}^{max} denote the minimal such ordinal and let α = κ_{A}^{max} + 1. It follows from 1.3 and 1.4 that for any space X the space F_{α}(X) is homotopy equivalent to CW_{A}(X). In this way we have constructed CW_{A}(X) by an iterative functorial resolution process, although carrying the process to its conclusion potentially requires a transfinite reserve of patience.

1.7. Relationship to previous work. In a sense, this paper is a result of combining Bousfield and Kan [1, XI.10] with long tower arguments from [3], turning the arrows around, and finding a more theoretical way to describe the results. To see the connection, let R = ℤ/p, and construct a sequence of categories

D_{0} ⊂ D_{1} ⊂ ⋯ ⊂ D_{α} ⊂ ⋯

4 W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

depending on R and indexed by the ordinal numbers α. The category D_{0} contains all spaces which up to weak equivalence are products of Eilenberg-MacLane spaces K(R, n), n ≥ 0 , and D_{α} for α > 0 is inductively defined to contain all spaces which up to weak equivalence and retracts are of the form can be written as holim G, for some small category D and functor G : D → D_{<α}. (Here D_{<α} = ∪_{β<α}D_{β}.) Then for any space X it is possible to construct a functorial long tower of spaces

D_{0}(X) ← D_{1}(X) ← ⋯ ← D_{α}(X) ← D_{α+1}(X) ← ⋯,

where D_{0}(X) is a topological version of RX, and D_{α}(X) for α > 0 is a right homotopy Kan extension of the inclusion J : D_{<α} → S_{∗} along the functor J itself. (In language parallel to that of 1.2, D_{α}(X) is the canonical approximation to X by a homotopy limit of spaces from D_{<α}.) Bousfield and Kan show in a simplicial setting tha D_{1}(X) = R_{∞}(X), while the results of [3] implicitly show how to construct D_{α}(X) for α > 1 in term of restricted cosimplicial resolutions, and prove that for sufficiently large α the space D_{α}(X) is the R-homology localization of X. (It is not known, however, that “sufficiently large” can be interpreted in a way which is independent of X, and it is not known whether an analogous construction of the homology localization succeeds for arbitrary homology theories.) The fact that the homology localization is not always reached at the first stage with D_{1}(X) ≃ R_{∞}X is a reflection of the fact that not all spaces are R-good. This is parallel in our setting to the fact that CW_{A}(X) is not always reached at the first stage with F_{1}(X).

1.8. Organization of the paper. Section 2 has preliminary material on homotopy colimits, and §3 identifies certain functors, called terminal functors, which preserve homotopy colimits. A device using terminal functors allows the notion of homotopy colimit to be extended to some large diagrams (3.11). Section 4 discusses left homotopy Kan extensions, which are a kind of parametrized homotopy colimit, and in §5 there is a general homotopy invariance theorem for left homotopy Kan extensions. The kind of approximation referred to in 1.2 is a special kind of homotopy Kan extension (4.4), and section 6 describes a method for computing these approximations. This method depends on the availability of a certain kind of functor F associated to the approximating category C. If such an F exists then (C, F) is called an adapted pair, and the approximation is computed by taking the homotopy colimit of a resolution constructed with the help of F. Section 7 focuses on constructing adapted pairs, and §8 uses the machinery from the previous sections to prove the main results. Finally, §9 briefly discusses a few examples: the cases in which A is a finite complex, a Moore space, or a sphere.

1.9. Notation and terminology. The symbol S_{∗} stands for the category of pointed spaces, and throughout the paper spaces are usually assumed to have a chosen basepoint. (There are some obvious exceptions, such as simplices.) Along these lines, “hocolim” is the pointed homotopy colimit (§2). The notation Y_{+} stands for Y with a disjoint basepoint added, and X ⋊ Y for X ∧ (Y_{+}). We sometimes say that f: X → Y is an equivalence (and write X ≃ Y) if f is a weak equivalence. A natural transformation between functors D → S_{∗} is called a weak equivalence if it gives a weak equivalence for each object of D. From a technical point of view we work with the model category structure on S_{∗} due to Quillen ([14], see also [7, §8]); in particular, a space is cofibrant if it is a retract of a cell complex. A weak equivalence between cofibrant spaces is a homotopy equivalence.

2. Homotopy colimits

In this section we recall the notion of (pointed) homotopy colimit, and then give two examples involving simplicial spaces and restricted simplicial spaces.

Suppose that D is a small category, and that F: D → S_{∗} is a functor. We use “hocolim F ” to denote the pointed homotopy colimit of F. This is defined as in [1, p. 328], with minor adjustments: the construction is made in the category S_{∗}, and the part of the construction due to the basepoints of the spaces involved is collapsed to the basepoint of hocolim F. See also [10, §4]. The part of the unpointed homotopy colimit which is collapsed to form the basepoint of hocolim F is homeomorphic to the geometric realization of the nerve [1, XI 2.1] of D; it follows easily that if the values of F are cofibrant and the nerve of D is contractible, then the pointed and unpointed homotopy colimits of F have the same homotopy type.

2.1. Remark. If J : C → D and F : D → S_{∗} are functors, we let J^{∗}(F) denote the composite F ∘ J. There are natural maps

hocolim J^{∗}(F) → hocolim F → colim F .

The homotopy colimit construction has the following basic invariance property.

2.2. Proposition. (cf. [1, XII 4.2]) Suppose that D is a small category, and that t : F → G is a weak equivalence between functors D → S_{∗}. Suppose in addition that for each object d of D the spaces

6 W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

F (d) and G(d) are cofibrant. Then hocolim t: hocolim F → hocolim G is an equivalence.

2.3. Remark. We will make use of hocolim F only in cases in which the values of F are cofibrant, and we write hocolim^{L}F for hocolim F^{c}, where F^{c} is some naturally associated functor with cofibrant values which maps to F by an equivalence. For instance F^{c}(x) might be the geometric realization of the singular complex of F (x). If F itself takes on cofibrant values, then hocolim^{L}F is equivalent to hocolim F.

2.4. Remark. The homotopy colimit can be viewed as the total derived functor in the sense of Quillen [9, II.7] associated to the colimit; roughly speaking, hocolim^{L} is the construction closest to colim which takes equivalences between functors C → S_{∗} to equivalences between spaces. See [7, §10] for a discussion of this in a special case. There is a slightly different construction of the homotopy colimit in [4, 2.16]. See also [2].

2.5. Remark. Given a functor F : D → S_{∗}, there exists a weak equivalence F′ → F, where F′ is a functor with cofibrant values such that colim F′ is cofibrant and the natural map hocolim F′ → colim F′ is a homotopy equivalence . See [4, §2] and [2, §16]. The functor F′ can be taken to be a cofibrant replacement for F in an appropriate Quillen model structure on the functor category.

2.6. Simplicial spaces. Let Δ^{op} be the simplicial category, so that a simplicial object in some category A is a functor X : Δ^{op} → A, or more explicitly a collection {X_{n}}_{n≥0} of objects of A together with face maps d_{i} : X_{n} → X_{n−1} and degeneracy maps s_{i} : X_{n} → X_{n+1}, 0 ≤ i ≤ n, satisfying the usual simplicial identities [12, §1]. The category Δ^{op} is the opposite of the category Δ whose objects are the finite ordered sets [n] = {0, …, n} and whose morphisms are the weakly monotone maps [12]. Let Δ_{n} be the topological n-simplex. The geometric realization of a simplicial space X is obtained from the space

⋁_{n} X_{n} ⋊ Δ_{n} (2.7)

by making the usual face and degeneracy identifications [12, §14]. If each X_{n} is cofibrant and the degeneracy maps s_{i} : X_{n} → X_{n+1} are cofibrations, then the geometric realization is equivalent to hocolim X (cf. [1, XII.3.4] or [15]).

2.8. Restricted simplicial spaces. We need a variation of the above. Let Λ^{op} be the restricted simplicial category; this is the subcategory of Δ^{op} whose morphisms are the composites of face maps d_{i}. This is the opposite of the category Λ whose objects are the finite ordered sets [n] and whose morphisms are the monotone injections. A restricted simplicial object in a category A [3, 3.13] is a functor X : Λ^{op} → A, or, more explicitly, a collection {X_{n}}_{n≥0} of objects of A together with maps d_{i} : X_{n} → X_{n−1}, 0 ≤ i ≤ n, such that d_{i}d_{j} = d_{j−1}d_{i} for i < j. This is a “simplicial object without degeneracy maps.” The geometric realization of a restricted simplicial space X is obtained from the onepoint union 2.7 by making the identifications (d_{i}x) ⋊ y ∼ x ⋊ (d^{i}y), where x ∈ X_{n}, y ∈ Δ_{n−1}, and d^{i} : Δ_{n−1} → Δ_{n} is the i’th face inclusion. In this case the geometric realization is homeomorphic to hocolim X.

3. Terminal functors

In this section we point out that a functor captures homotopy colimits (3.1) if and only if it is terminal in an appropriate sense (3.4). We then check that the condition of being terminal is satisfied in some key examples involving (restricted) simplicial categories. We also use the notion of a terminal functor to extend the construction of homotopy colimits to certain diagrams indexed by large categories (3.11).

3.1. Capturing homotopy colimits. We will say that a functor J : C → D between small categories captures (pointed) homotopy colimits if for every F : D → S_{∗} the natural map hocolim^{L}J^{∗}(F) → hocolim^{L}F is an equivalence. There is a similar sense in which J can be said to capture unpointed homotopy colimits.

Given J : C → D and an object d of D, let d ↓ J denote the category whose objects are pairs (c, f) where c is an object of C and f : d → J(c) is a map in D; a morphism (c, f) → (c′, f′) is a map g : c → c′ in C such that J(g)f = f′.

3.2. Definition. A functor J : C → D is said to be terminal if for each object d of D the nerve of the category d ↓ J is contractible.

3.3. Remark. An object d_{0} of D is usually said to be terminal if for any object d of D the set Hom(d, d_{0}) has exactly one point. It is easy to see that d_{0} is terminal in this sense if and only if the inclusion J : {d_{0}} → D of the trivial subcategory with object d_{0} is terminal in the sense of 3.2.

In general, in the situation of 3.2 let Hom(d, J) denote the functor C → Sets which sends c ∈ C to Hom(d, J(c)). It is easy to check that the geometric realization of the nerve of d ↓ J is homeomorphic to the unpointed homotopy colimit of Hom(d, J), and so J is terminal if and only if for each object d of D the unpointed homotopy colimit of Hom(d, J) is equivalent to a single point.

8 W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

3.4. Proposition. Suppose that J : C → D is a functor between small categories. Then J captures pointed or unpointed homotopy colimits if and only if J is terminal.

Proof. The “if” half of the statement appears in [6, 9.4], at least in a simplicial form in the unpointed case. The proof in the pointed case is identical. See also [10, 4.4] and [2]. The “only if” part is proved in the pointed case by looking at functors F_{d} : D → S_{∗} given by F_{d}(x) = Hom_{D}(d, x)_{+}. The homotopy colimit of F_{d} is equivalent to S^{0}, while the homotopy colimit of J^{∗}(F_{d}) is obtained up to homeomorphism by adding a disjoint basepoint to the geometric realization of the nerve of d ↓ J.

3.5. Example. If d_{0} is a terminal object of D, then (3.2, 3.4) for any F : D → S_{∗} the natural map hocolim^{L}F → F (d_{0}) is an equivalence.

3.6. Contractions. An augmentation for a (restricted) simplicial object X in the category A is an object X_{−1} of A together with a map d_{0} : X_{0} → X_{−1} such that d_{0}d_{0} = d_{0}d_{1} : X_{1} → X_{−1}. A left contraction for such an augmented X is a collection of maps s_{−1} : X_{n} → X_{n+1}, n ≥ −1, such that d_{0}s_{−1} = 1 and d_{j}s_{−1} = s_{−1}d_{j−1} for j > 0; if X is simplicial (as opposed to restricted simplicial) there are also identities s_{j}s_{−1} = s_{−1}s_{j−1} for j > −1. A right contraction for X is a collection of maps s_{n+1} : X_{n} → X_{n+1} such that d_{n+1}s_{n+1} = 1 and d_{j}s_{n+1} = s_{n}d_{j} for j < n+1; in the simplicial case there are also identities s_{j}s_{n+1} = s_{n+2}s_{j} for j ≤ n + 1. See [8, 6.1], [3, 3.16], and [13] for similar formulations. Let _{l}Δ^{op} and _{l}Λ^{op} be the extensions of Δ^{op} and Λ^{op} respectively which correspond to left contractions, and_{r}Δ^{op} and_{r}Λ^{op} the extensions which corresponding to right contractions. All of these categories have objects [n] for n ≥ −1; the category _{l}Λ^{op}, for instance, has morphisms d_{i} : [n + 1] → [n], s_{−1} : [n] → [n + 1] together with the identities required so that an augmented restricted simplicial object with a left contraction amounts to a functor cΛ^{op} → S_{∗}. There are commutative squares of categories and functors

(3.7)  Λ^{op} ⟶ Δ^{op} ↓  ↓ _{l}Λ^{op} ⟶ _{l}Δ^{op}  Λ^{op} ⟶ Δ^{op} ↓  ↓ _{r}Λ^{op} ⟶ _{r}Δ^{op} .

3.8. Proposition. All of the functors in display 3.7 are terminal.

Proof. We first deal with the left-hand square. The upper functor is terminal by [3, 3.17], the one on the left by [3, p. 196], the lower one by 3.5 and the fact that both of the categories involved have [−1] as a terminal object, and the one on the right by an easy argument depending on 3.4 and the fact that the other three functors are terminal.

The right-hand square differs from the left-hand square by the automorphism of Δ^{op} which in dimension n interchanges d_{i} and d_{n−i}.

3.9. Example. If X is a (restricted) simplicial space which admits both an augmentation by a space X_{−1} and an associated left or right contraction, then the map hocolim^{L}X → X_{−1} induced by the augmentation is an equivalence. This follows from observing that the categories _{l}Δ^{op}, lΛ^{op}, _{r}Δ^{op} _{r}Λ^{op} all have [−1] as a terminal object, and then combining 3.8 with 3.5.

3.10. Example. Consider a functor J : Δ → D or Λ → D, and let d be an object of D. In these cases Hom(d, J) (see 3.3) is a (restricted) simplicial set; combining 3.3 with 2.6 and 2.8 shows that J is terminal if and only if for each d in D this restricted simplicial set has a contractible unpointed homotopy colimit. By 3.9, this will be the case if for each d the unique augmentation Hom(d, J) → ∗ admits either a left or a right contraction. (Note that along the lines of 2.8, the unpointed homotopy colimit of a restricted simplicial set can be easily described in terms of a geometrical realization.)

3.11. Homotopy colimits over large categories. We will say that a category is large if the totality of its objects does not form a set, although the collection of morphisms between any two objects does form a set; for example, the category S_{∗} is large. Suppose that D is a large category, and that F : D → S_{∗} is a functor. There are clearly settheoretic difficulties with forming hocolim^{L}F (or even colim F). However, if D is terminally small in the sense that there exists a small category C and a terminal functor J : C → D, we will follow [1, XI.10.2] and say that hocolim^{L}J^{∗}(F) is a homotopy colimit for F. If the values of F are cofibrant, then hocolim J^{∗}(F) is equivalent to hocolim^{L}J^{∗}(F), and so hocolim J^{∗}(F) is also a homotopy colimit for F.

3.12. Remark. The argument of [1, XI.10.3] shows that the homotopy type of hocolim^{L}J^{∗}(F) does not depend upon the choice of terminal functor J.

4. Homotopy Kan extensions and approximations

Here we use objectwise homotopy colimits to define the homotopy left Kan extension of a functor F : C → S_{∗} over a functor J : C → D. Taking the homotopy left Kan extension of a suitable functor over itself (4.4) leads to the notion of approximation.

10 W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

4.1. Homotopy Kan extensions. Suppose that J : C → D is a functor between small categories, and that F is a functor from C to some category A with colimits (e.g.. Sets). For each object d of D, let J ↓ d denote the category whose objects are pairs (c, f), where c is an object of C and f : J(c) → d is a map in D; a morphism (c, f) → (c′, f′) is a map g : c → c′ in C such that f′ ∘ (F (g)) = f. If J is the inclusion of a subcategory, we sometimes write C ↓ d and refer to this as the category of objects of C over d [11, II.6]. We let F_{d}: J ↓ d → A be given by F_{d}(c, f) = F (c). The left Kan extension of F along J, denoted J_{∗}F, is the functor D → A with

(J_{∗}F)(d) = colim F_{d}

See [11, X.4]. The construction J_{∗}, considered as an operation on functors C → A, is left adjoint to J^{∗} [11, X.3]. If A is the category S_{∗}, the homotopy left Kan extension J_{h∗}F is defined by the analogous formula

(J_{h∗}F)(d) = hocolim F_{d} .

There is also a variant which is better-behaved if the values of F are not cofibrant:

(J_{L∗}F)(d) = hocolim^{L}F_{d} .

The construction J_{L∗} can be viewed as the total derived functor in the sense of Quillen [9, II.7] associated to J_{∗}; roughly speaking, J_{L∗} is the construction closest to J_{∗} which takes weak equivalences between functors C → S_{∗} to weak equivalences between functors D → S_{∗}. See [10, §5] for a topological discussion, or [9, p. 439] for a simplicial treatment which includes a description [9, IX.1.4] of the model categories involved in applying Quillen’s machinery. There is a more general discussion in [2, II.16].

4.2. Transitivity. Suppose that J : C → D and K : D → E are functors between small categories, and that F is a functor from C to say, Sets. The left Kan extension construction then satisfies the transitivity relation (KJ)_{∗}F = K_{∗}(J_{∗}F); this follows from the identity (KJ)^{∗} = J^{∗}K^{∗} and the fact that the left adjoint of a composite is the composite of the left adjoints. The homotopy left Kan extension construction has a similar property. If F takes values in S_{∗}, then

(KJ)_{L∗}F ≃ K_{L∗}(J_{L∗}F) .

See [6, 9.8] or [10, 5.5].

4.3. Large categories. If J : C → D is a functor between large categories and F : C → S_{∗} is a functor, we can follow 3.11 and speak, under appropriate conditions, of a homotopy left Kan extension of F along J. In particular, we do this below when for each object d of D there is a terminal functor J_{d} : Λ^{op} → J ↓ d which is constructed in a way which is natural in d. In this case the functor which sends d to hocolim(F_{d} ∘ J_{d}) is a homotopy left Kan extension of F.

4.4. Approximation. Suppose that J : C → S_{∗} is the inclusion of a full subcategory; we can think of J in two ways, either as a map of index categories, or as a diagram of spaces indexed by C. If there exists a left homotopy Kan extension of J (in the second sense) along J (in the first sense), we denote this extension α_{C}(X); the natural map α_{C}(X) → X is the canonical approximation to X by a homotopy colimit of spaces from C (cf. [11, X.6]).

4.5. Remark. There is one case in which it is easy to compute α_{C}(X), namely, if J : C → S_{∗} is the inclusion of a full subcategory and X ∈ C. In this case C ↓ X has the identity map of X a terminal object, and so it follows from 3.5 and 3.12 that α_{C}(X) → X is a weak equivalence.

5. Invariance of homotopy Kan extensions

Suppose that C is a small full subcategory of S_{∗}, and recall that α_{C}(X) → X is the canonical approximation to X by a homotopy colimit of spaces from C (4.4). In this section we look at conditions which will guarantee that α_{C}(X) is homotopy invariant as a functor of X. The main result is the following one.

5.1. Proposition. Suppose that C is a small full subcategory of S_{∗} which has the following two properties:

1. each object of C is cofibrant, and

2. if A is an object of C, then each of the products A ⋊ Δ_{n}, n ≥ 0 also belongs to C.

Then α_{C} is homotopy invariant. More generally, α_{C}(X) → α_{C}(Y) is an equivalence whenever X → Y is an A-equivalence for each object A of C.

5.2. Examples. It may seem paradoxical that a construction like α_{C}(X) described in terms of homotopy colimits might not be homotopy invariant. But consider the following example. Let C be any small full subcategory of S_{∗} which contains only discrete spaces, but is large enough to contain both S^{0} and the one-point space ∗. We will show that α_{C}(X) is equivalent to X^{δ} (this is X with the discrete topology). In particular, α_{C}(X) is not homotopy invariant as a functor of X. Let D ⊂ C be the subcategory which contains only S^{0} with its identity map. We have functors

D ⟶^{I}C ⟶^{J} S_{∗} and also K = J ∘ I .

Let Id be the identity functor on S_{∗}. Essentially by inspection, K_{L∗}(K) is weakly equivalent to Id_{+}^{δ}; restricting this observation to C and using the fact that X = X^{δ} for X ∈ C gives I_{L∗}(K) ≃ J_{+}. By transitivity of homotopy Kan extensions (4.2),

J_{L∗}(J_{+}) ≃ J_{L∗}(I_{L∗}K) ≃ K_{L∗}(K) ≃ Id_{+}^{δ} . (5.3)

Let cS^{0} be the constant functor with value S^{0} (on whatever category is in question), and let c∗ be the constant functor with value the one-point space. There is a map of homotopy pushout squares

\begin{matrix}J_{L∗}(cS^{0}) & ⟶ & J_{L∗}(c∗) \\ ↓ & & ↓ \\ J_{L∗}(J_{+}) & ⟶ & J_{L∗}(J) \end{matrix}  ⟶  \begin{matrix}cS^{0} & ⟶ & c∗ \\ ↓ & & ↓ \\ Id_{+}^{δ} & ⟶ & Id^{δ} \end{matrix} .

Since J_{L∗}(c∗) is trivially the constant functor with value a point, to show that J_{L∗}(J) is equivalent to Id^{δ} it is enough to show that J_{L∗}(cS^{0}) is weakly equivalent to the constant functor on S_{∗} with value S^{0}. By inspection, the value of J_{L∗}(cS^{0}) on a space X is the disjoint union of a basepoint with the nerve of the category J ↓ X. This nerve is contractible, because J ↓ X has ∗ → X as an initial object.

The situation changes drastically if C is enlarged to contain spaces which are homotopy equivalent to S^{0} but not discrete. In fact, let C be any small full subcategory of S_{∗} which contains the spaces (Δ_{n})_{+} for n ≥ 0 as well as the one-point space ∗. We will sketch a modification of the above argument which shows that α_{C}(X) → X is an equivalence for every space X and so, in particular, that α_{C} is homotopy invariant. Consider the functor I : Δ → C with I([n]) = (Δ_{n})_{+}. We have functors

Δ ⟶^{I}C ⟶^{J} S_{∗} and also K = J ∘ I .

There is a natural transformation K → cS^{0} which is a weak equivalence on each object of Δ, and so there are weak equivalences of functors

I_{L∗}(K) ≃ I_{L∗}(cS^{0}) ≃ I_{h∗}(cS^{0})

where the last equivalence comes from the fact that S^{0} is cofibrant. Looking at 2.6 shows that for any space X, (I_{h∗}(cS^{0}))(X) is obtained by adding a disjoint basepoint to the geometric realization of the singular complex of X, and it follows that I_{L∗}(K) ≃ J_{+}. Similarly, K_{L∗}(K) ≃ Id_{+}. Transitivity of homotopy Kan extension gives as in 5.3 that J_{L∗}(J_{+}) ≃ Id_{+}, and the disjoint basepoints are disposed of as before.

Proof of 5.1. We will use some machinery from [10]. If D is a small category, F is a contravariant functor D → S_{∗} (a right D-module) and G is a covariant functor D → S_{∗} (a left D-module), we write F ⊗_{D}^{h} G for what in the notation of [10, §3] would be described as the quotient B(Y, D, X)/B(∗, D, ∗); this reduced bar construction is the pointed homotopy coend of F and G. This has a homotopy invariance property analogous to that of the homotopy colimit; in fact, hocolim G is exactly S^{0} ⊗_{D}^{h} G. There is also a variant F ⊗_{D}^{L} G, which is defined as F^{c} ⊗_{D}^{h} G^{c} (2.3) and is better-behaved if the values of F and G are not cofibrant. If J : D → E and F : D → S_{∗} are functors, the homotopy Kan extension J_{h∗}(F) can be identified as E_{+}⊗_{D}^{h}F, where for short E_{+} denotes the (E, D)-bimodule which assigns to a pair of objects (e, d) the pointed set Hom_{E}(J(d), e)_{+}.

Consider the functors

C ⟶^{I}C × Δ ⟶^{J} C ⟶^{K} S_{∗} ,

where I(A) = (A, [0]), J(A, [n]) = A ⋊ Δ_{n}, and K is the inclusion functor. The composite KJI is naturally equivalent to K. Let L denote the composite KJ. The diagram gives maps

(KJI)_{h∗}(K) = K_{h∗}(K) → L_{h∗}(L) → K_{h∗}(K)

such that the composite is the identity map. This shows that the approximation functor α_{C} = K_{h∗}(K) is a retract of L_{h∗}(L), and so in order to complete the proof it is enough to show that L_{h∗}(L) has the appropriate invariance property. The functor L is weakly equivalent to the functor L′ given by L′(A, [n]) = A, so in fact it is enough to treat L_{h∗}(L′). Let P : C × Δ → C be the projection functor given by P (A, [n]) = A. Since L′ = P^{∗}(K), we are faced with the functor which in hocoend notation is

(S_{∗})_{+} ⊗_{C×Δ}^{h} K

where K is implicitly considered a left (C × Δ)-module via P. Any functor F : C → S_{∗} is equivalent to C⊗_{C}^{h} F (see [10, 3.1(5)]; homotopy Kan extending over the identity functor leaves functors unchanged up to homotopy). After substituting C ⊗_{C}^{h} K for K and reparenthesizing [10, 3.1(3)], we are left with

((S_{∗})_{+} ⊗_{C×Δ}^{h} C) ⊗_{C}^{h} K .

Recall that this expression represents a functor which depends upon a space X. The way in which X enters the picture is through the functor

14 W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

on the left, which can be decoded as the homotopy left Kan extension, over the projection map P : (C × Δ)^{op} → C^{op} of the functor H_{X} which assigns to a pair (A, [n]) the pointed set Hom_{S_{∗}}(A ⋊ Δ_{n}, X)_{+}. Taking a homotopy left Kan extension over a projection map such as this is a particularly simple process; the value P_{h∗}(H_{X}) on an object A is just the homotopy colimit of the original functor over the category P^{−1}(A) ≅ Δ^{op}. (Compare [10, §6]. The simplest way to see this is to calculate that the inclusion P^{−1}(A) → A ↓ P is terminal, by noticing that each one of its under categories has a terminal object and hence a contractible nerve.) Let Map(A, X) denote the space of pointed maps from A to X. The conclusion, via 2.6 and inspection, is that the functor P_{h∗}(H_{X}) assigns to an object A of C the geometric realization of the singular complex of Map(A, X)_{+}. The proposition now follows from the fact that an A-equivalence X → Y induces a weak equivalence Map(A, X)_{+} → Map(A, Y)_{+}.

For later use we record the following result, which is proved by the above observations and is implicit in [10, §6]; see also [2, III.24].

5.4. Proposition. Suppose that C and D are small categories, and that F: : C × D → S_{∗} is a functor. For each d ∈ D let F_{d} : C → S_{∗} be given by F_{d}(x) = F (x, d), and let G : D → S_{∗} be given by G(d) = hocolim^{L}F_{d}. Then there is a natural weak equivalence

hocolim^{L}F ≃ hocolim^{L}G .

We will use this in the case in which α is a limit ordinal and D is the ordered set (category), also denoted α, of all ordinals below α. A functor F : α → S_{∗} is just a directed system (X_{β})_{β<α}. In that case we will need the following observation.

5.5. Proposition. Let α be a limit ordinal and (X_{β})_{β<α} a directed system of spaces. Suppose that there is some β_{0} < α such that for all γ > β_{0} the map X_{β_{0}} → X_{γ} is a weak equivalence. Then hocolim_{β<α}^{L} X_{β} is naturally weakly equivalent to X_{β}.

Proof. Interpret (X_{β})_{β<α} as a functor F : α → S_{∗}. Let [β_{0}, α) denote the category of all ordinals γ with β_{0} ≤ γ < α. It is easy to check that the inclusion J : [β_{0}, α) → α is terminal (3.2), and so hocolim^{L}F ≃ hocolim^{L}J^{∗}F (3.4). Let F′ be the constant functor on [β_{0}, α) which assigns to each γ some cofibrant space weakly equivalent to F (β_{0}); it is clear that there is a natural transformation F′ → J^{∗}(F) which is a weak equivalence on each object, so that hocolim J^{∗}(F) ≃ hocolim F′ (2.2). Finally, hocolim F′ is homeomorphic to F′(β_{0}) ⋊ N, where N is the geometric realization of the nerve of the category [β_{0}, α); we refer to [1, XII.3.5] (in the special case in which the functor Y in question is constant and has value the one-point space) for the fact that N is contractible.

6. How to compute approximations

Suppose that C is a full subcategory of S_{∗} (not necessarily small) such that all of the objects of C are cofibrant. In this section we describe a technique for computing the homotopy approximations (4.4) α_{C}(X) → X. The technique depends on the existence of a functor F on S_{∗} which has a special relationship to C (6.2), and α_{C}(X) is then constructed as the homotopy colimit or realization of a resolution of X derived from this functor. At the end we give a criterion for the homotopy approximations we construct to be homotopy invariant as functors of X.

6.1. Adapted pairs. An augmented functor on S_{∗} is a functor F: S_{∗} → S_{∗} together with a natural transformation from F to the identity functor. We write this natural transformation ϵ_{F} (X): F (X) → X, or just ϵ(X) if F is understood.

6.2. Definition. Suppose that C is a full subcategory of S_{∗} such that all of the objects of C are cofibrant. An augmented functor F on S_{∗} is said to be adapted to C if

1. F takes values in C, and

2. for each space B in C, the map ϵ(B): F (B) → B has a section

(i.e. a right inverse). If F is adapted to C, then (C, F) is called an adapted pair.

6.3. Remark. A morphism (C, F) → (C′, F′) of adapted pairs is an inclusion C ⊂ C′ together with a natural transformation F → F′ which respects the augmentations of the two functors involved. The functor F in an adapted pair (C, F) is not in any sense determined by C; for instance, if (C, F) is an adapted pair, so is (C, F^{n}) for any n > 1.

Now suppose that (C, F) is an adapted pair and that X is a space. Associated to F is a restricted simplicial space FX (see 2.8) with (FX)_{n} = F^{n+1}(X) and

((FX)_{n} ⟶^{d_{i}} (FX)_{n−1}) = (F^{n+1}X ⟶^{F^{i}ϵ(F^{n−i}X)} F^{n}X) .

The restricted simplicial identities amount to diagrams which commute because ϵ is a natural transformation. This object is augmented by the map ϵ(X) : (FX)_{0} = F X → X, and so can be interpreted as a kind of resolution of X by values of F. We define a functor F̂ by taking the homotopy colimit or realization of the resolution (2.8):

F̂(X) = hocolim FX .

The augmentation of FX induces a natural map F̂(X) → X, so that F̂ is itself an augmented functor.

Since F takes values in C, the resolution together with its augmentation can be considered as a functor FX : Λ^{op} → C ↓ X (4.1). Our constructions depend on the following property of this functor.

6.4. Proposition. If (C, F) is an adapted pair, then for any X ∈ S_{∗} the resolution functor

FX : Λ^{op} → C ↓ X

is terminal (3.2).

The proof of this is below. In particular, 6.4 implies that the categories C ↓ X are terminally small (3.11). The following result is immediate (4.4).

6.5. Proposition. If (C, F) is an adapted pair, then the natural map F̂(X) → X is an approximation map α_{C}(X) → X, i.e., F̂ is a left homotopy Kan extension of the inclusion functor J : C → S_{∗} over itself.

6.6. Remark. In particular, if (C, F) is an adapted pair and X ∈ C, then (4.5) the map F̂(X) → X is an equivalence.

Proof of 6.4. Write Hom_{X} for the set of morphisms in the category C ↓ X. To verify that FX is terminal in C ↓ X, it is enough to check that for each object B → X of this category the restricted simplicial set Hom_{X}(B, FX) admits a suitable left contraction (3.6). Since F is adapted to C, we can choose a section s : B → F (B) of the augmentation F (B) → B, and define the contraction by letting s_{−1} : Hom_{X}(B, (FX)_{n}) → Hom_{X}(B, (FX)_{n+1}) send a map f : B → F^{n+1}(X) to the composite

B ⟶^{s} F B ⟶^{F (f)} F^{n+2}(X) .

It is routine to verify the necessary identities.

6.7. Invariance properties of approximations. We now study when the construction α_{C}(X) = F̂(X) described above is homotopy invariant as a functor of X. In fact under fairly mild conditions the construction satisfies an even stronger invariance property.

6.8. Proposition. Suppose that (C, F) is an adapted pair. Assume that that the category C is closed under the functors B ↦ B ⋊ Δ_{n}, n ≥ 0. Then if X → Y is a map of spaces which is a B-equivalence for every object B of C, the induced map F̂(X) → F̂(Y) is an equivalence.

Proof. This almost follows from a combination of 6.5 and 5.1, but there is a smallness hypothesis in 5.1 which is not necessarily satisfied here. Suppose that X → Y is a map of the indicated type. Let D be the smallest full subcategory of C which contains the spaces F^{n}X, F^{n}Y, n ≥ 1, and is closed under the constructions B ↦ B ⋊ Δ_{n}, n ≥ 0. It is clear that D is in fact a small category. The argument of 6.4 shows that the functors FX : Λ^{op} → D ↓ X and FY : Λ^{op} → D ↓ Y are terminal, so that F̂(X) and F̂(Y) compute respectively α_{D}(X) and α_{D}(Y). By 5.1, these approximations are equivalent.

7. Building adapted pairs

In this section we start with an adapted pair (C, F) in which C is closed under equivalences (7.4) and construct a directed system (6.3)

(C, F) = (C_{0}, F_{0}) → (C_{1}, F_{1}) → ⋯ → (C_{α}, F_{α}) → ⋯

of adapted pairs, indexed by the ordinal numbers. Although this is a general process, we use notation reminiscent of §1 because it is the techniques of this section that yield the objects described in §1. It is useful to start with some definitions.

7.1. Definition. Suppose that C is a full subcategory of S_{∗} such that all of the objects of C are cofibrant. The category Hocolim(C) is then the category of all spaces Y for which there exists a small category D and functor G : D → C with Y = hocolim G.

7.2. Definition. Suppose that G is an augmented functor. Then G^{fib} is the augmented functor constructed by factoring each natural map G(X) → X in some fixed functorial way into a composite G(X) → G^{fib}(X) → X, where G(X) → G^{fib}(X) is a cofibration and a weak equivalence, and G^{fib}(X) → X is a fibration.

The categories C_{α} are defined inductively. To start with, C_{0} is the category C. Suppose that α > 0, and that C_{β} has been defined for all β < α, such that C_{γ} ⊂ C_{β} if γ < β. Let C_{<α} denote the increasing union ∪_{β<α}C_{β}. Then C_{α} is defined to be the full subcategory of S_{∗} whose objects consist of all cofibrant spaces which up to homotopy are retracts of objects of Hocolim(C_{<α}).

Let F_{0} = F. Suppose that α > 0, and that we have already constructed a directed system {F_{β}}_{β<α} of augmented functors, such that F_{β} takes values in C_{β}. For any space X there is a functor

ρ_{α}^{X} : α × Λ^{op} → C_{<α} ↓ X (7.3)

given by ρ_{X}(β, [n]) = (F_{β}X)_{n} = F_{β}^{n+1}(X). This functor is built in such a way that it is a combination of all of the F_{β}–resolutions of X for β < α (§6). Let G_{α}(X) = hocolim ρ_{α}^{X}; it is clear that G_{α} is an augmented functor, and we define F_{α} inductively by letting F_{α} = G_{α}^{fib} (7.2). By construction the functor F_{α} takes values in C_{α}. Notice that if β < α then the restriction of ρ_{α}^{X} to β × Λ^{op} is exactly ρ_{β}^{X}; we thus get compatible maps

G_{β}(X) = hocolim ρ_{β}^{X} → hocolim ρ_{α}^{X} = G_{α}(X), 0 < β < α

which, by the functoriality of the factorization process used in 7.2, pass to compatible maps

F_{β}(X) = G_{β}^{fib}(X) → G_{α}^{fib}(X) = F_{α}(X), 0 < β < α.

We leave it to the reader to construct F_{0} → F_{1}.

The main result of this section is the following one. Say that the category C is closed under equivalences if whenever A and B are cofibrant spaces which are homotopy equivalent, then both are in C if either one is.

7.4. Proposition. Suppose that (C, F) is an adapted pair and that C is closed under equivalences. Then each of the pairs (C_{α}, F_{α}) constructed above is also an adapted pair. Moreover, for α > 0 the natural map F_{α}(X) → X is the canonical approximation to X by a homotopy colimit of spaces from C_{<α} (4.4).

7.5. Remark. The construction above may seem elaborate, but the functors F_{α} are much less complicated than they look; the objects ρ_{α}^{X} appear only as a device for obtaining the correct maps F_{β} → F_{α}. In fact, if α = β + 1 is a successor ordinal then F_{α}(X) ≃ F̂_{β}(X); if α is a limit ordinal, the F_{α}(X) ≃ hocolim_{β<α} F_{β}(X).

To see this, suppose first that α = β + 1 is a successor ordinal. The category α then has the element β as a terminal object (maximal element), and it follows easily from 5.4 or direct calculation that the inclusion functor

Λ^{op} → α × Λ^{op}, [n] ↦ (β, [n]) (7.6)

is terminal (3.2). The restriction of ρ_{α}^{X} to Λ^{op} over this inclusion functor is the resolution functor F_{β}X, and so by 3.4 the resulting maps

F̂_{β}(X) = hocolim F_{β}X → hocolim ρ_{α}^{X} = G_{α}(X) → F_{α}(X)

are equivalences. If α is a limit ordinal then, by 5.4, hocolim ρ_{α}^{X} is equivalent to hocolim_{β<α} F̂_{β}(X). Since as above F̂_{β}(X) is equivalent to F_{β+1}(X), we obtain an equivalence

F_{α}(X) ≃ hocolim_{β<α} F_{β+1}(X) ≃ hocolim_{β<α} F_{β}(X) .

Proof of 7.4. We prove both statements by induction. By assumption, (C, F) = (C_{0}, F_{0}) is an adapted pair. Suppose that α > 0 and that for each β < α (C_{β}, F_{β}) is an adapted pair; suppose as well that if 0 < β < α then F_{β} is the indicated approximation functor. We prove first that F_{α} has the desired approximation property. By definition (4.4), this will be the case if for each X the functor ρ_{α}^{X} (7.3) is terminal. Suppose that α = β + 1 is a successor ordinal. Then the composite of ρ_{α}^{X} with the inclusion functor 7.6 is terminal (see 6.4), and, as mentioned above, the inclusion functor itself is terminal. By 3.4, ρ_{α}^{X} is also terminal. Suppose on the other hand that α is a limit ordinal; by 5.4 and 5.5, in order to show that ρ_{α}^{X} is terminal it is enough to show that for each B ∈ C_{<α} ↓ X and all sufficiently large ordinals γ < α, the restricted simplicial set Hom_{X}(B, F_{γ}X) has a contractible realization (see the proof of 6.4 for the notation here). But this is clear: if B ∈ C_{β}, it is sufficient to choose γ ≥ β, and then the desired contraction is exhibited in the proof of 6.4.

What remains is to prove that F_{α} is adapted to C_{α}, or in other words to verify 6.2(2). Note that for any space B ∈ C_{<α} the natural map F_{α}(B) → B is a weak equivalence; this follows (4.5) from the approximation property proved above. Suppose that X ∈ Hocolim(C_{<α}). Given 2.5 and the fact that C_{<α} is closed under equivalences, X is homotopy equivalent to an appropriate Y, where Y is cofibrant and Y = colim G for some G : D → C_{<α} with hocolim G ≃ colim G. Consider the commutative diagram

hocolim F_{α} ∘ G ⟶ F_{α}(Y)
↓  ↓^{ϵ_{Fα}(Y)}
hocolim G ⟶^{≃} Y

in which the vertical maps are induced by the augmentation of F_{α}. The left vertical map is an equivalence because of the invariance property of hocolim (2.2) and the observation above that F_{α}(B) ≃ B for each B ∈ C_{<α} (6.6). It follows that the map ϵ_{F_{α}}(Y) has a section up to homotopy. Since the map is a fibration and the base is cofibrant, the homotopy lifting property implies that this homotopy section can be deformed into a genuine section s_{Y} of ϵ_{F_{α}}(Y). Any X in C_{α} is up to homotopy a retract of such a Y. Given X, choose maps u : X → Y and v : Y → X such that vu is homotopic to the identity map of X. The composite

X ⟶^{u} Y ⟶^{s_{Y}} F_{α}(Y) ⟶^{F_{α}(v)} F_{α}(X)

is then a section up to homotopy of ϵ_{F_{α}}(X). As above, this can be deformed into a genuine section.

8. The main results

Here we prove the results described in §1. The first step is to construct the chain 1.1 of categories. In fact we will construct a chain

(C_{0}, F_{0}) → (C_{1}, F_{1}) → ⋯ → (C_{α}, F_{α}) ⋯

of adapted pairs (6.2). Recall that A is a fixed cofibrant object of S_{∗}. The category C_{0} is defined to be the full subcategory of S_{∗} containing all cofibrant spaces B which are homotopy equivalent to wedges of copies of A. Let G be the augmented functor which assigns to X the wedge ∨_{f}A, where the wedge is taken over the set of all maps A → X; the augmentation G(X) → X is the obvious one which on the wedge summand of G(X) corresponding to the map f is the map f itself. The functor F_{0} is defined to be G^{fib} (7.2). It is clear that (C_{0}, F_{0}) satisfies the conditions of 6.2.

The pairs (C_{α}, F_{α}) for α > 0 are now obtained by the method of §7. Theorem 1.2 follows from 7.4, and Theorem 1.4 from 7.4 and 4.5.

Proof of 1.3. We first assume that X → Y is an A-equivalence, and show by induction on α that F_{α}(X) ≃ F_{α}(Y) for α > 0. If α = β + 1 is a successor ordinal this is a consequence of 7.5 and 6.8; note that by construction each space B in C_{β} is A-cellular, so that X → Y is a fortiori a B-equivalence. If α is a limit ordinal the result follows from 2.2 and the description of F_{α} in 7.5.

Suppose on the other hand that F_{α}(X) ≃ F_{α}(Y) for some α > 0. An induction using the result of the paragraph above shows that F_{α}^{n}(X) ≃ F_{α}^{n}(Y) for n ≥ 1, and hence by 2.2 and 7.5 that F_{α+1}(X) ≃ F_{α+1}(Y). Further induction along these lines leads to the conclusion that F_{β}(X) ≃ F_{β}(Y) for all β ≥ α. Choosing β sufficiently large (see 1.5, which depends only on the “if” part of 2.2) gives CW_{A}(X) ≃ CW_{A}(Y), and hence that X → Y is an A-equivalence.

9. Examples

Here we look at some particular choices for the space A: finite complexes, Moore spaces, and spheres.

9.1. Finite complexes. It is probably not surprising that if A is a finite complex there cannot be spaces of large A-complication.

9.2. Proposition. Suppose that A is a finite complex. Then any X has κ_{A}(X) ≤ ω, where ω is the first infinite ordinal. Moreover, F_{ω}(X) is equivalent to CW_{A}(X).

9.3. Remark. In general, κ_{A}(X) ≤ ω would only imply F_{ω+1}(X) ≃ CW_{A}(X) (1.4).

Proof of 9.2. To prove the result, it is enough to show that the natural map F_{ω}(X) → X is an A-equivalence. Say that a space K is a finite Acellular complex if it is built from a point by attaching a finite number of copies of A and its suspensions. In order to complete the proof it is enough to show that in any commutative diagram

K ⟶ F_{ω}(X)
↓  ↓^{ϵ(X)}
L ⟶ X

in which K and L are finite A-cellular complexes, there is a lift L → F_{ω}(X) which makes the diagram commute up to homotopy. The finiteness assumption on A implies that K is a finite complex in the ordinary sense, and so the map A → F_{ω}(X) = hocolim_{n} F_{n}(X) actually factors through F_{n_{0}}(X) for some n_{0}. Consider the map of squares

\begin{matrix}F_{ω}(K) & ⟶ & F_{ω}F_{n_{0}}(X) \\ ↓ & & ↓ \\ F_{ω}(L) & ⟶ & F_{ω}(X) \end{matrix} ⟶ \begin{matrix}K & ⟶ & F_{n_{0}}(X) \\ ↓ & & ↓ \\ L & ⟶ & X \end{matrix}

induced by the augmentation of the functor F_{ω}. Since K, L, and F_{n_{0}}(X) all lie in C_{N} for sufficiently large N, the map of squares is an equivalence except at the lower right-hand corner (1.4). In particular, the map L ≃ F_{ω}(L) → F_{ω}(X) gives up to homotopy the required lift.

9.4. The Moore space. We next show that in some cases the bound from 9.2 is sharp.

9.5. Proposition. Suppose that A is a Moore space M(ℤ/p, n), n ≥ 1. Then κ_{A}^{max} = ω, i.e.,there exists spaces X with κ_{A}(X) = ω.

Proof. It is elementary to see that M(ℤ/p^{∞}, n + 1) is an A-cellular space. By 9.2, κ_{A}(M(ℤ/p^{∞}, n + 1)) ≤ ω. We will show that the complication of this space actually equals ω, by showing with an induction on n that if B is a space with κ_{A}(B) = n for some integer n, then

22 W. CHACHÓLSKI, W. G. DWYER, AND M. INTERMONT

each reduced integral homology group of B is killed by some power of p which depends only on the integer n and the degree of the homology group. This is clear if κ_{A}(B) = 0. Inductively, any space B of complication n can be expressed as a homotopy colimit of spaces of complication (n − 1). There is an associated first quadrant homology spectral sequence [1, XII.5.6] which in the limit gives each homology group of B up to a finite filtration. Inspecting the E^{2}-page shows that the filtration quotients Gr_{j} H̃_{k}(B), 0 ≤ j ≤ k, are subquotients of groups of the form ⊕_{α}H̃_{k−j}(Y_{α,j}), where {Y_{α,j}} is a collection of spaces of complication (n − 1). The result follows immediately.

9.6. The sphere. Suppose that A is a sphere S^{n}; in this case a space is A-cellular if and only if it is (n − 1)-connected. Stover [16] shows by an explicit construction that every (n − 1) connected space is weakly equivalent to the homotopy colimit of a functor whose values are equivalent to wedges of copies of S^{k}, k ≥ n. It is immediate that such a wedge has A-complication ≤ 1; it follows that for any space X, κ_{A}(X) ≤ 2.

With more work it is probably possible to do better. We briefly indicate how to accomplish this for A = S^{1}. Any 1-connected space X is weakly equivalent to the geometric realization of the classifying space of a simplicial group GX with the property that each constituent (GX)_{n} is a free group. For instance, take GX to be the Kan loop group [12, §26] of the singular complex of X. One can take the geometric realization of the classifying space of each of the constituent groups (GX)_{n}; by functoriality these form a simplicial space Y. Of course each space Y_{n} is equivalent to a wedge of circles. It is not hard to see that X is equivalent to the homotopy colimit or realization of Y, and so κ_{A}(X) ≤ 1.

References

1. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 304.

2. W. Chacholski and J. Scherer, Homotopy theory of diagrams, Memoirs of the AMS, to appear.

3. E. Dror Farjoun and W. G. Dwyer, A long homology localization tower, Comment. Math. Helv. 52 (1977), no. 2, 185–210.

4. E. Dror Farjoun, Homotopy and homology of diagrams of spaces, Algebraic topology (Seattle, Wash., 1985), Springer, Berlin, 1987, pp. 93–134.

5. , Cellular spaces, null spaces and homotopy localization, Springer-Verlag, Berlin, 1996.

6. W. G. Dwyer and D. M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23 (1984), no. 2, 139–155.

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

8. W. Dwyer, H. Miller, and J. Neisendorfer, Fibrewise completion and unstable Adams spectral sequences, Israel J. Math. 66 (1989), no. 1-3, 160–178.

9. P. G. Goerss and J. F. Jardine, Simplicial homotopy theory, Birkhäuser Verlag, Basel, 1999.

10. J. Hollender and R. M. Vogt, Modules of topological spaces, applications to homotopy limits and E_{∞} structures, Arch. Math. (Basel) 59 (1992), no. 2, 115–129.

11. S. Mac Lane, Categories for the working mathematician, second ed., Springer-Verlag, New York, 1998.

12. J. P. May, Simplicial objects in algebraic topology, University of Chicago Press, Chicago, IL, 1992, Reprint of the 1967 original.

13. J.-P. Meyer, Cosimplicial homotopies, Proc. Amer. Math. Soc. 108 (1990), no. 1, 9–17.

14. D. G. Quillen, Homotopical algebra, Springer-Verlag, Berlin, 1967, Lecture Notes in Mathematics, No. 43.

15. G. Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.

16. C. R. Stover, A van Kampen spectral sequence for higher homotopy groups, Topology 29 (1990), no. 1, 9–26.

Yale University, New Haven, CT 06520 USA

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

University of Notre Dame, Notre Dame IN 46556 USA

E-mail address: dwyer.1@nd.edu

Kalamazoo College, Kalamazoo MI, 49006 USA

E-mail address: intermon@kzoo.edu