A TELESCOPE COMPARISON LEMMA FOR THH

M.A. MANDELL AND B. SHIPLEY

Abstract. We extend to the non-connective case a lemma of Bökstedt about the equivalence of the telescope with a more complicated homotopy colimit of symmetric spectra used in the construction of THH.

1. Introduction

In [1], Bökstedt constructed the topological Hochschild homology of a symmetric ring spectrum, before the smash product of symmetric spectra had been invented. Now that we understand the smash product of symmetric spectra, we can construct THH using the cyclic bar construction. The paper [13] compares these definitions of THH. Both of these constructions are the geometric realizations of simplicial symmetric spectra that turn out to be stably equivalent in each simplicial level. A peculiarity of the theory of symmetric spectra is that not all stable equivalences are equivalences of the underlying prespectra, and simple examples show that these two constructions of THH do not always have weakly equivalent underlying prespectra.

The problem can be understood levelwise, where the ring structure plays no role. (The multiplication is only used to define the simplicial face maps.) Understanding simplicial level zero is the key to understanding all the levels, and we concentrate on this. As traditional in working with THH, instead of thinking in terms of prespectra, we can think in terms of functors from based spaces to based spaces. Then for a symmetric spectrum T, the zeroth level of the cyclic bar construction thh_{0}(T) is just T itself, and this corresponds to the functor that sends a based space X to the space tel_{n} Ω^{n}(T_{n} ∧ X), that is, this space is the underlying infinite loop space of T ∧ X. The zeroth level of Bökstedt’s construction is a homotopy colimit. It sends X to the space

T HH_{0}(T; X) = hocolim_{n∈I} Ω^{n}(T_{n} ∧ X).

Here I is the category finite sets and injections; see Definition 2.2. There is a canonical inclusion of the telescope in this homotopy colimit and Bökstedt’s fundamental lemma for THH is that it is a weak equivalence when T is strictly connective (π_{q}T_{n} = 0 for q < n) and convergent (Definition 2.1). The purpose of this paper is to eliminate the connectivity hypothesis.

This work grew out of the authors’ previous study of topological Hochschild homology as defined for symmetric spectra and for S-modules in [3] and [13]. This material also formed the basis of a first comparison of the homotopy categories of symmetric spectra, S-modules, and several other new symmetric monoidal categories of spectra carried out in a larger project with May and Schwede. See [11] for an expository account based on this earlier outline. This first approach has since been replaced by highly structured comparisons of model categories, see [7] and [12].

1991 Mathematics Subject Classification. 55P42, 19D99.

Key words and phrases. symmetric spectra, topological Hochschild homology, infinite loop space.

Both authors partially supported by NSF Postdoctoral Fellowships.

The definition of symmetric spectra easily generalizes to other, even non-top-ologically motivated contexts. For example, symmetric spectra have also been recently used to model stable 𝔸^{1}-local homotopy theory in the sense of [14], q.v. [5]. Although the proofs do not immediately transport to this setting, one expects similar statements to hold. In fact, the impetus for resurrecting this material came from questions posed by Voevodsky that are answered here.

We work in the setting of topological symmetric spectra, but only minor modifications, such as level fibrant replacement, would be necessary to apply these results to simplicial symmetric spectra. To avoid topological technicalities, we assume that any level of a given symmetric spectrum or any given based space here is non-degenerately based.

Acknowledgments: As mentioned above, this material was developed in the context of a larger project with Peter May and Stefan Schwede. We would like to thank them for their support during this work and for encouraging us to publish this piece of work separately. In particular, we would like to thank Peter May for his editorial and mathematical guidance throughout the writing of this paper.

2. Statement of results

We first need to recall a few definitions before we state our results.

Definition 2.1. Let T be a symmetric spectrum. It has an underlying prespectrum with spaces T_{n} and structure maps σ_{n} : T_{n} ∧ S^{1} ⟶ T_{n+1}. Define the homotopy groups of T to be the homotopy groups of the prespectrum:

π_{q}T = colim π_{q+n}T_{n} ≅ π_{q}(tel_{n} Ω^{n}T_{n}).

We say that T is convergent if the adjoint structure map σ̃_{n} : T_{n} ⟶ ΩT_{n+1} is an (n + λ(n))-equivalence for some nondecreasing sequence of integers λ(n) with infinite limit. We say that T is a symmetric Ω-spectrum if the maps σ̃_{n} are weak equivalences. We say that a map f : T ⟶ T′ of symmetric spectra is a π_{∗}- isomorphism if it induces an isomorphism of homotopy groups. We say that f is a level weak equivalence if each f_{n} : T_{n} ⟶ T′_{n} is a weak equivalence. Let [X, Y] denote the set of maps X ⟶ Y in the homotopy category with respect to the level model structure [7, §6]. We say that f is a stable weak equivalence if f^{∗} : [Y, E] ⟶ [X, E] is a bijection for all symmetric Ω-spectra E.

Recall from [4, 7] that a π_{∗}-isomorphism is a stable weak equivalence, but not conversely. However, the converse does hold for stable weak equivalences between convergent spectra, and we shall only be concerned with convergent spectra here.

Definition 2.2. Let I be the category whose objects are the sets n = {1, …, n}, n ≥ 0, and whose morphisms are the injective functions. Let Σ be the subcategory of isomorphisms in I. For m < n, let i_{m,n} : m ⟶ n be the standard inclusion and observe that any map f : m ⟶ n can be written, non-uniquely, as a composite f = τ ∘ i_{m,n} for some τ ∈ Σ_{n}. Let J ⊂ I be the subcategory of objects n and standard injections i_{m,n} : m ⟶ n for m ≤ n.

Our main theorem is the following sharpening of a basic lemma of Bökstedt [1, 1.6] that eliminates its connectivity hypotheses; see also [13, 3.1.7] for a further generalization in the case X = S^{0}. Let T be the category of based spaces. A sequence of based spaces X_{n} and maps X_{n} ⟶ X_{n+1} determines a functor J ⟶ T , and there is a natural homotopy equivalence

tel_{n} X_{n} ⟶^{≃} hocolim_{J} X_{n}.

A functor defined on I restricts to a functor defined on J and thus gives an induced map of homotopy colimits.

A symmetric spectrum T gives rise to a functor I ⟶ T that sends n to Ω^{n}(T_{n} ∧ X). Using the evident natural map Ω^{n}(X)∧Y ⟶ Ω^{n}(X ∧Y), the adjoint structure map σ̃ gives

T_{m} ∧ X ⟶ (Ω^{n−m}T_{n}) ∧ X ⟶ Ω^{n−m}(T_{n} ∧ X);

the map induced by i_{m,n} is obtained from this by applying Ω^{m}. Permutations act on S^{n} and T_{n} and by conjugation on Ω^{n}(T_{n} ∧ X). This construction plays a fundamental role in the theory of symmetric spectra, see [13].

Theorem 2.3 (Bökstedt’s lemma). Let T be a convergent symmetric spectrum and X be a based space. Then the natural map

tel_{n} Ω^{n}(T_{n} ∧ X) ≃ hocolim_{J} Ω^{n}(T_{n} ∧ X) ⟶ hocolim_{I} Ω^{n}(T_{n} ∧ X)

is a weak equivalence.

A proof of Bökstedt’s original lemma appears in [6, 2.3.7]. When the symmetric spectrum T is not assumed to be connective, the homotopy groups of T_{n} ∧X do not stabilize, and the proof of Theorem 2.3 requires the construction and analysis of (−k)-connected covers; see Lemma 4.3. Bökstedt and Madsen’s argument applies on each cover, and we use standard techniques with homotopy colimits to mesh them together and deduce the result for T. For the reader’s convenience we review the definition and basic properties of homotopy colimits in Section 6.

It is a standard fact in stable homotopy theory that the map

tel_{n} Ω^{n}(T_{n} ∧ X) → Ω(tel_{n} Ω^{n}(T_{n} ∧ ΣX))

is a weak equivalence; see for example [7, 7.4.i’]. It follows that when T is convergent, the map

hocolim_{I} Ω^{n}(T_{n} ∧ X) → Ω(hocolim_{I} Ω^{n}(T_{n} ∧ ΣX))

is a weak equivalence. We obtain the following corollary of Theorem 2.3.

Corollary 2.4. If T is convergent, then, for a based space X, we have an isomorphism of homology

T_{q}(X) = π_{q}(T ∧ X) ≅ π_{q+k} hocolim_{I} Ω^{n}(T_{n} ∧ Σ^{k}X)

and an isomorphism of cohomology

T^{q}(X) = π_{−q}(F (X, T)) ≅ π_{k−q}F (Σ^{k−q}X, hocolim_{I} Ω^{n}(T_{n} ∧ S^{k})).

For S the sphere spectrum, S ∧ X is isomorphic to Σ^{∞}X, and we obtain the following corollary. Recall that QX = colim Ω^{n}Σ^{n}X. Since the maps of the colimit system are inclusions, the natural map tel_{n} Ω^{n}Σ^{n}X ⟶ QX is a weak equivalence.

Corollary 2.5. The natural map

tel_{n} Ω^{n}Σ^{n}X ≃ hocolim_{J} Ω^{n}Σ^{n}X ⟶ hocolim_{I} Ω^{n}Σ^{n}X

is a weak equivalence.

3. The proof of Bökstedt’s lemma

The strategy for the proof of Theorem 2.3 is to reduce to the study of a symmetric Ω-spectrum that is closely related to T ∧ X. Let ℚ_{X}T be the symmetric spectrum with levels

ℚ_{X}T_{n} = tel_{m} Ω^{m}(T_{n+m} ∧ X).

Then ℚ_{X} is a functor of both T and X. The inclusions of the initial term in a telescope gives a natural map of symmetric spectra T ∧ X ⟶ ℚ_{X}T such that the following diagram commutes:

\xymatrix{tel Ω^{n}(T_{n} ∧ X) \ar[r] \ar[d] & tel Ω^{n}ℚ_{X}T_{n} \ar[d] \\ hocolim_{I} Ω^{n}(T_{n} ∧ X) \ar[r] & hocolim_{I} Ω^{n}ℚ_{X}T_{n}.}

Theorem 2.3 states that the left vertical arrow is a weak equivalence. To prove this, we prove that the remaining three arrows are weak equivalences. We prove the following two lemmas in the next section.

Lemma 3.1. If T is a convergent symmetric spectrum, then ℚ_{X}T is a symmetric Ω-spectrum and the natural map T ∧ X ⟶ ℚ_{X}T is a π_{∗}-isomorphism.

For q ≥ 0, the map of qth homotopy groups of spaces induced by the top arrow in the diagram is the map of qth homotopy groups of prespectra induced by T ∧X ⟶ ℚ_{X}T, so this lemma implies directly that the top arrow is a weak equivalence.

Warning 3.2. If T is not convergent, then ℚ_{X}T is not a symmetric Ω-spectrum and T ∧ X ⟶ ℚ_{X}T is not a π_{∗}-isomorphism in general, even when X = S^{0}. In fact, it is shown in [4] that the symmetric spectrum T = F_{1}(S^{1}) defined below gives a counterexample.

The next lemma shows that the bottom horizontal arrow in the diagram is a weak equivalence.

Lemma 3.3. If T is a convergent symmetric spectrum, then the natural map

hocolim_{I} Ω^{n}(T_{n} ∧ X) ⟶ hocolim_{I} Ω^{n}ℚ_{X}T_{n}

is a weak equivalence.

Finally, the following lemma allows us to exploit the fact that ℚ_{X}T is a symmetric Ω-spectrum to prove that the right vertical arrow in the diagram is a weak equivalence.

Lemma 3.4. If U is a symmetric Ω-spectrum, then the natural map

U_{0} ⟶ hocolim_{I} Ω^{n}U_{n}

is a weak equivalence.

This lemma follows from a standard result about homotopy colimits of homotopically constant diagrams, see Lemma 6.2.

By Lemmas 3.1 and 3.4, when T is convergent the inclusion

ℚ_{X}T_{0} ⟶ hocolim_{I} Ω^{n}ℚ_{X}T_{n}

is a weak equivalence. The inclusion

ℚ_{X}T_{0} ⟶ tel Ω^{n}ℚ_{X}T_{n}

is also a weak equivalence by Lemma 6.2. Since the right vertical arrow is a map under ℚ_{X}T_{0}, it too is a weak equivalence. This completes the proof of Theorem 2.3, modulo the proofs of the lemmas.

4. The proofs of two technical lemmas

The proofs of Lemmas 3.1 and 3.3 are based on the use of “strictly k-connected covers” of symmetric spectra.

Definition 4.1. A symmetric spectrum T is strictly k-connected if T_{n} is (n + k)- connected for n ≥ 0.

We prove the following lemma in the next section.

Lemma 4.2. For each integer k, there is a functor C_{k} on the category of symmetric spectra such that, for any symmetric spectrum T, C_{k}T is strictly k-connected. There are natural transformations c_{k} : C_{k} ⟶ Id and e_{k} : C_{k} ⟶ C_{k−1} such that c_{k−1} = c_{k} ∘ e_{k} and c_{k} induces an isomorphism π_{i}C_{k}T_{n} ⟶ π_{i}T_{n} for i > n + k.

The following observation is the key to the proofs of Lemmas 3.1 and 3.3.

Lemma 4.3. Let T be a convergent symmetric spectrum. For each integer k and based space X, C_{k}T ∧ X is a convergent symmetric spectrum.

Proof. Assume that the structure map σ̃ : T_{n} ⟶ ΩT_{n+1} is an (n + λ(n))-equi-valence, where λ(n) is a nondecreasing sequence with infinite limit. The structure maps of C_{k}T then have the same property. An easy diagram chase shows that the nth structure map of C_{k}T ∧ X factors as the composite

C_{k}T_{n} ∧ X ⟶^{σ̃∧id} (ΩC_{k}T_{n+1}) ∧ X ⟶ Ω(C_{k}T_{n+1} ∧ X).

The first map is an (n + λ(n))-equivalence and for n + k > 1 the second map is a (2n + 2k + 1)-equivalence, by Lemma 4.4 below. The composite is therefore an (n + min(n + 2k + 1, λ(n)))-equivalence.

Lemma 4.4. If Y is r-connected, then the canonical map (ΩY) ∧ X ⟶ Ω(Y ∧ X) is a (2r − 1)-equivalence.

Proof. This follows from the Freudenthal suspension theorem applied to ΩY ∧ X and to ΩY, together with the factorization of the cited map as the composite

ΩY ∧ X ⟶^{η} ΩΣ(ΩY ∧ X) ≅ Ω(ΣΩY ∧ X) ⟶^{Ω(ε∧id)} Ω(Y ∧ X),

where η and ε are the unit and counit of the (Σ, Ω)-adjunction.

We need the following general result about homotopy colimits over I to prove Lemmas 3.1 and 3.3.

Proposition 4.5. Let F and F′ be functors I ⟶ T and let ϕ : F ⟶ F′ be a natural transformation. Assume that ϕ(n) : F (n) ⟶ F′(n) is a λ(n)-equivalence, where λ(n) ≤ λ(n + 1) and lim_{n} λ(n) = ∞. Then the induced map

ϕ_{∗} : hocolim_{I} F ⟶ hocolim_{I} F′

is a weak equivalence.

The proof of this proposition is delayed to section 6 where we discuss homotopy colimits.

The following simple observation is used several times in the following proofs of Lemmas 3.1 and 3.3.

Lemma 4.6. For any sequence of based spaces Y_{n} and based maps Y_{n} ⟶ Y_{n+1}, the evident maps ΩY_{n} ⟶ Ω tel_{n} Y_{n} induce a weak equivalence

tel_{n} ΩY_{n} ⟶ Ω tel_{n} Y_{n}.

Proofs of Lemmas 3.1 and 3.3. We are given a convergent symmetric spectrum T. We first show that both lemmas hold when T is replaced by C_{k}T for any fixed integer k. We show that ℚ_{X}C_{k}T is a symmetric Ω-spectrum, and the map

C_{k}T ∧ X ⟶ ℚ_{X}C_{k}T (4.7)

is a π_{∗}-isomorphism.

The group π_{j}ℚ_{X}C_{k}T_{n} is the colimit of the sequence

\xymatrix{π_{j}(C_{k}T_{n} ∧ X) \ar[r] & π_{j+1}(C_{k}T_{n+1} ∧ X) \ar[r] & π_{j+2}(C_{k}T_{n+2} ∧ X) \ar[r] & · · · .}

We can calculate the maps π_{j}ℚ_{X}C_{k}T_{n} ⟶ π_{j+1}ℚ_{X}C_{k}T_{n+1} induced by the structure maps ℚ_{X}C_{k}T_{n} ⟶ Ω(ℚ_{X}C_{k}T_{n+1}) as maps of colimits arising from the diagrams

\xymatrix{π_{j}(C_{k}T_{n} ∧ X) \ar[r] \ar[d] & π_{j+1}(C_{k}T_{n+1} ∧ X) \ar[r] \ar[d] & ⋯ \\ π_{j+1}(C_{k}T_{n+1} ∧ X) \ar[r] & π_{j+2}(C_{k}T_{n+2} ∧ X) \ar[r] & · · · .}

Since C_{k}T ∧X is convergent, the vertical maps in each such diagram are eventually isomorphisms and so induce isomorphisms of the colimits. Thus each structure map ℚ_{X}C_{k}T_{n} ⟶ Ωℚ_{X}C_{k}T_{n+1} is a weak equivalence. The map on homotopy groups induced by (4.7) is the map

colim_{n} π_{j+n}(C_{k}T_{n} ∧ X) ⟶ colim_{m,n} π_{j+n+m}(C_{k}T_{n+m} ∧ X)

obtained by mapping the terms in the source to the terms with m = 0 in the target. Extending the diagram above vertically and arguing similarly, we see that this map is an isomorphism. We should note here that the subtlety mentioned in Warning 3.2 appears because the vertical maps here differ from the horizontal maps up to isomorphism, see [4, 5.6.3]. But this difficulty is avoided here since C_{k}T ∧ X is convergent.

For Lemma 3.3, since C_{k}T ∧X is convergent the colimit, colim π_{i+k}(C_{k}T_{n+k}∧X), is attained at π_{i}(C_{k}T_{n} ∧ X) for i ≤ n + λ(n). This directly implies the hypotheses required by Proposition 4.5 to show that

(4.8) hocolim_{I} Ω^{n}(C_{k}T_{n} ∧ X) ⟶ hocolim_{I} Ω^{n}ℚ_{X}C_{k}T_{n}

is a weak equivalence.

We now deduce that the lemmas hold for T. We first show that ℚ_{X}T is a symmetric Ω-spectrum. Using the maps C_{−k}T ⟶ C_{−(k+1)}T, we define T′ = tel_{k} C_{−k}T. The compatible maps C_{−k}T ⟶ T induce a level weak equivalence of symmetric spectra c : T′ ⟶ T. Observe that T′ ∧ X ≅ tel_{k}(C_{−k}T ∧ X). Similarly, using the induced maps ℚ_{X}C_{−k}T ⟶ ℚ_{X}C_{−(k+1)}T, we define ℚ′_{X}T = tel_{k} ℚ_{X}C_{−k}T. The compatible maps ℚ_{X}C_{−k}T ⟶ ℚ_{X}T induce a map of symmetric spectra d : ℚ′_{X}T ⟶ ℚ_{X}T such that the following diagram commutes:

\xymatrix{T′ ∧ X \ar[r] \ar[d]_{c∧id} & ℚ′_{X}T \ar[d]^{d} \\ T ∧ X \ar[r] & ℚ_{X}T.}

The top horizontal arrow is a π_{∗}-isomorphism since it is obtained by passage to telescopes from the π_{∗}-isomorphisms (4.7). The map c∧id is a level weak equivalence since c is a level weak equivalence. Using Lemma 4.6, we see that ℚ′_{X}T is a symmetric Ω-spectrum since each ℚ_{X}C_{k}T is a symmetric Ω-spectrum. Thus it suffices to show that d is a level weak equivalence to conclude both that ℚ_{X}T is a symmetric Ω-spectrum and that the bottom horizontal arrow in the diagram is a π_{∗}- isomorphism, giving Lemma 3.1. On passage to telescopes, the weak equivalences

tel_{k} Ω^{m}(C_{−k}T_{n+m} ∧ X) ⟶ Ω^{m} tel_{k} C_{−k}T_{n+m} ∧ X

given by Lemma 4.6 induce a weak equivalence

ℚ′_{X}T ≅ tel_{m} tel_{k} Ω^{m}(C_{−k}T_{n+m} ∧ X)
⟶ tel_{m} Ω^{m}(tel_{k} C_{−k}T_{n+m} ∧ X) = ℚ_{X}T′.

This is the nth term of a level weak equivalence b : ℚ′_{X}T ⟶ ℚ_{X}T′. It is easily seen that d factors as the composite

ℚ′_{X}T ⟶^{b} ℚ_{X}T′ ⟶^{ℚ_{X}c} ℚ_{X}T.

Since c is a level weak equivalence, so is ℚ_{X}c. Therefore d is a level weak equivalence. This completes the proof of Lemma 3.1.

Finally, to complete the proof of Lemma 3.3, we observe that the level weak equivalences c and d induce the maps \overline{c} and \overline{d} displayed in the following commutative diagram:

\xymatrix{tel_{k} hocolim_{I} Ω^{n}(C_{k}T_{n} ∧ X) \ar[r] \ar[d]_{≅} & tel_{k} hocolim_{I} Ω^{n}ℚ_{X}C_{k}T_{n} \ar[d]^{≅} \\ hocolim_{I} tel_{k} Ω^{n}(C_{k}T_{n} ∧ X) \ar[r] \ar[d]_{≃} & hocolim_{I} tel_{k} Ω^{n}ℚ_{X}C_{k}T_{n} \ar[d]^{≃} \\ hocolim_{I} Ω^{n}T′_{n} ∧ X) \ar[r] \ar[d]_{\overline{c}} & hocolim_{I} Ω^{n}(ℚ′_{X}T_{n}) \ar[d]^{\overline{d}} \\ hocolim_{I} Ω^{n}(T_{n} ∧ X) \ar[r] & hocolim_{I} Ω^{n}ℚ_{X}T_{n}.}

The top horizontal arrow is a weak equivalence because it is the telescope of the weak equivalences (4.8). The vertical arrows labelled ≅ are homeomorphisms obtained by commuting tel_{k} with hocolim_{I}. The vertical arrows labelled ≃ are weak equivalences induced by weak equivalences of Lemma 4.6. The maps \overline{c} and \overline{d} are weak equivalences since c and d are level weak equivalences. Therefore the bottom horizontal arrow is a weak equivalence, which is the conclusion of Lemma 3.3.

5. Postnikov towers and connective covers of symmetric spectra

We construct the strict coverings promised in Lemma 4.2 from the functorial strict Postnikov towers given by the following lemma.

Lemma 5.1. For each integer k, there is a functor P_{k} on the category of symmetric spectra and a natural transformation ξ_{k} : Id ⟶ P_{k} such that for every symmetric spectrum T and every n, the map

ξ_{k}(n) : T_{n} ⟶ P_{k}T_{n}

is a (k+n+1)-equivalence and π_{i}P_{k}T_{n} = 0 for i > k+n. Further, there are natural transformations χ_{k} : P_{k} ⟶ P_{k−1} such that ξ_{k−1} = χ_{k} ∘ ξ_{k}.

Proof of Lemma 4.2. We define the symmetric spectrum C_{k}T to be the (levelwise) homotopy fiber of the map ξ_{k}. For each n in Σ, we define c_{k} : C_{k}T_{n} ⟶ T_{n} to be the left vertical arrow in the following pullback diagram:

\xymatrix{C_{k}T_{n} \ar[d]_{c_{k}} \ar[r] & map_{T} (I, P_{k}T_{n}) \ar[d] \\ T_{n} \ar[r]_{ξ_{k}(n)} & P_{k}T_{n}.}

It is clear from the functoriality of our Postnikov sections that C_{k} defines a functor on symmetric spectra and that c_{k} is natural. Since ξ_{k}(n) is a (k+n+1)-equivalence, C_{k}T is strictly k-connected. Since π_{i}P_{k}T_{n} = 0 for i > k + n, c_{k}(n) induces an isomorphism on homotopy groups in degrees k + n + 1 and above. The maps e_{k} are induced by the maps χ_{k}, and the equation c_{k−1} = c_{k} ∘ e_{k} follows from the corresponding equation ξ_{k−1} = χ_{k} ∘ ξ_{k}.

We will prove Lemma 5.1 by a localization technique analogous to one that allows a direct construction of functorial Postnikov towers in the category of spaces. We will need the left adjoint F_{n} to the nth space functor from symmetric spectra to spaces. Thus we have homeomorphisms of hom spaces

map_{Sp^{Σ}}(F_{n}X, T) ≅ map_{T} (X, T_{n}).

As noted in [7, 4.2], F_{n} is given explicitly by

(F_{n}X)_{m} = Σ_{m+} ∧_{Σ_{m−n}} X ∧ S^{m−n},

where we interpret S^{m−n} as ∗ for m < n.

We construct P_{k} using the “small objects argument” from model category theory. For each n, define a functor L_{k,n} on the category of symmetric spectra and a natural transformation η : Id ⟶ L_{k,n} as follows. For a symmetric spectrum T, let D_{k,n}T be the set of maps of symmetric spectra F_{n}S^{k+n+1} ⟶ T. Define L_{k,n}T and η by the following pushout diagram in the category of symmetric spectra:

\xymatrix{⋁_{f∈D_{k,n}T} F_{n}S^{k+n+1} \ar[r]^{∨f} \ar[d]_{∨F_{n}i} & T \ar[d]^{η} \\ ⋁_{f∈D_{k,n}T} F_{n}CS^{k+n+1} \ar[r] & L_{k,n}T.}

Here i denotes the inclusion of the sphere in the cone.

When m < n, the space F_{n}S_{m}^{n+k+1} is a point and so η(m) : T_{m} ⟶ L_{k,n}T_{m} is an isomorphism. When m ≥ n, the space F_{n}S_{m}^{n+k+1} is a wedge of S^{m+k+1}’s, and so the map η(m) : T_{m} ⟶ L_{k,n}T_{m} induces an isomorphism on homotopy groups in degrees 0 to m + k inclusive. Moreover, by adjunction, η(n) induces the zero map π_{n+k+1}T_{n} ⟶ π_{n+k+1}L_{k,n}T_{n}. Let N_{k,n} be the telescope of iterates of η:

N_{k,n} = tel(Id ⟶^{η} L_{k,n} ⟶^{η} L_{k,n} ∘ L_{k,n} ⟶^{η} L_{k,n} ∘ L_{k,n} ∘ L_{k,n} ⟶^{η} ⋯).

Then π_{n+k+1}N_{k,n}T_{n} = 0 and there is a natural transformation ν_{k,n} : Id ⟶ N_{k,n} such that ν_{k,n}(m) : T_{m} ⟶ N_{k,n}T_{m} induces an isomorphism on homotopy groups in degrees 0 to m + k inclusive.

Let N_{k} be the telescope of the maps ν_{k,n}:

N_{k} = tel(Id ⟶^{ν_{k,0}} N_{k,0} ⟶^{ν_{k,1}} N_{k,1} ∘ N_{k,0} ⟶^{ν_{k,2}} N_{k,2} ∘ N_{k,1} ∘ N_{k,0} ⟶^{ν_{k,3}} ⋯).

Then π_{n+k+1}N_{k}T_{n} = 0 for all n and there is a natural transformation ν_{k} : Id ⟶ N_{k} such that ν_{k}(n) : T_{n} ⟶ N_{k}T_{n} induces an isomorphism on homotopy groups in degrees 0 to n + k inclusive for all n.

Since the map ν_{k} is the inclusion of the initial object in a telescope, each map ν_{k}(n) : T_{n} ⟶ N_{k}T_{n} is a cofibration. Let P_{k} be the colimit of the levelwise cofibrations ν_{j} for j > k:

P_{k} = colim(N_{k} ⟶^{ν_{k+1}} N_{k+1} ∘ N_{k} ⟶^{ν_{k+2}} N_{k+2} ∘ N_{k+1} ∘ N_{k} ⟶^{ν_{k+3}} ⋯).

There is a natural transformation ξ_{k} : Id ⟶ P_{k} induced by ν_{k}. It follows from the properties of N_{k} that each space P_{k}T_{n} has zero homotopy groups above degree n + k and that each map ξ_{k}(n) is a (k + n + 1)-equivalence. There are natural transformations χ_{k} : P_{k} ⟶ P_{k−1} induced by the following map of diagrams:

\xymatrix{ & N_{k} \ar[r]^{ν_{k+1}} \ar[d]_{N_{k}ν_{k−1}} & N_{k+1} ∘ N_{k} \ar[r]^{ν_{k+2}} \ar[d]_{N_{k+1}N_{k}ν_{k−1}} & ⋯ \\ N_{k−1} \ar[r]_{ν_{k}} & N_{k} ∘ N_{k−1} \ar[r]_{ν_{k+1}} & N_{k+1} ∘ N_{k} ∘ N_{k−1} \ar[r]_{ν_{k+2}} & · · · .}

The diagram commutes because of the naturality of the horizontal maps. We see that χ_{k} ∘ ξ_{k} = ξ_{k−1} since the following diagram commutes:

\xymatrix{Id \ar[r]^{ν_{k}} \ar[d]_{ν_{k−1}} & N_{k} \ar[d]^{N_{k}ν_{k−1}} \\ N_{k−1} \ar[r]_{ν_{k}} & N_{k} ∘ N_{k−1}.}

The functors P_{k} and natural transformations ξ_{k} and χ_{k} have the properties specified in Lemma 5.1.

6. Homotopy colimits of spaces and symmetric spectra

We first describe what we have used about homotopy colimits of spaces.

Let C be a small discrete category, thought of as an indexing category. For a functor F : C ⟶ T , the homotopy colimit of F is defined in terms of the one-sided categorical bar construction B(∗, −, −) as

hocolim_{C} F = B(∗, C, F).

See [3, X.3] or [10, §12]. The same definition is given in different language in [2, XII§2]. This is the geometric realization of a simplicial space, and the inclusion of its subspace of zero simplices gives a map

⋁_{c∈ Ob C} F (c) ⟶ hocolim_{C} F.

Homotopy colimits are functorial in F: a natural transformation ϕ : F ⟶ F′ of functors C ⟶ T induces a map

ϕ_{∗} : hocolim_{C} F ⟶ hocolim_{C} F′.

The following main property of homotopy colimits, that they are homotopy invariant, can be proven as in [9, A.4].

Lemma 6.1. If ϕ is an levelwise n-equivalence, weak equivalence, or homotopy equivalence, then ϕ_{∗} is an n-equivalence, weak equivalence, or homotopy equivalence.

We have used the following easy consequence.

Lemma 6.2. Let F : C ⟶ T be a functor such that F (α) is a weak equivalence for every morphism α in C and let C have an initial object \underline{0}. Then the inclusion

F (\underline{0}) ⟶ hocolim_{C} F

is a weak equivalence.

Proof. Let E be the constant functor that takes the value F (\underline{0}). By inspection of definitions, hocolim_{C} E is homeomorphic to F (\underline{0}) ∧ BC_{+}, and BC is contractible since J has an initial object. Therefore the inclusion F (\underline{0}) ⟶ hocolim_{C} E is a homotopy equivalence. Applying F to the unique map from \underline{0} to each object of C, we obtain a natural transformation ϕ : E ⟶ F. By the previous lemma, the induced map ϕ_{∗} : hocolim_{C} E ⟶ hocolim_{C} F is a weak equivalence. Clearly ϕ_{∗} restricts to the inclusion on F (\underline{0}).

Homotopy colimits are also functorial in C. Given indexing categories C and K and a functor f : C ⟶ K, we obtain an induced map

f_{∗} : hocolim_{C} F ∘ f ⟶ hocolim_{K} F.

We have used the following result relating natural transformations to homotopies.

Lemma 6.3. Let η : f ⟶ g be a natural transformation between functors f, g : C ⟶ K. Then the following diagram is naturally homotopy commutative:

\xymatrix{hocolim_{C} F ∘ f \ar[rr]^{(F η)_{∗}} \ar[dr]_{f_{∗}} &  & hocolim_{J} F ∘ g \ar[dl]^{g_{∗}} \\  & hocolim_{K} F}

Proof. Let I be the category with two objects [0] and [1] and one non-identity morphism [0] ⟶ [1]. Then BI ≅ [0, 1]. It is standard that η determines and is determined by a functor \overline{η} : C × I ⟶ K that restricts to f on C × \{[0]\} and to g on C × {[1]}. Let π : C × I ⟶ C be the projection. Then η also determines a natural transformation η̃ : f ∘ π ⟶ \overline{η}. For an object c of C, η̃(c, [0]) = id : f(c) ⟶ f(c) and η̃(c, [1]) = η : f(c) ⟶ g(c). The required natural homotopy h : f_{∗} ≃ g_{∗} ∘(F η)_{∗} is the composite

(hocolim_{C} F ∘ f) × BI  ≅  hocolim_{C×I} F ∘ f ∘ π
⟶^{(F η̃)_{∗}} hocolim_{C×I} F ∘ \overline{η}
⟶^{\overline{η}_{∗}} hocolim_{K} F,

where the first isomorphism is a direct inspection of definitions.

We now turn to the proof of Proposition 4.5 which depends on the expression of homotopy colimits over I as homotopy colimits over certain subcategories of I. Let I_{n} be the full subcategory of I consisting of the objects m for m ≥ n and let f_{n} be the inclusion of I_{n} in I.

Lemma 6.4. Let F : I ⟶ T be a functor. For each n, the map

(f_{n})_{∗} : hocolim_{I_{n}}(F ∘ f_{n}) ⟶ hocolim_{I} F

induced by f_{n} is a homotopy equivalence.

Proof. Define a functor g_{n} : I ⟶ I_{n} by concatenation with n; explicitly, g_{n}(m) = m + n and g_{n}(i) = i ⊔ id_{n} for an injection i : m ⟶ m′. Let z_{n} : 0 ⟶ n be the unique map, define ζ(m) = id_{m} ⊔z_{n} : m ⟶ m + n, and define η(m) = ζ(m) for m ≥ n. Then ζ is a natural transformation Id_{I} ⟶ f_{n} ∘ g_{n}, η is a natural transformation Id_{I_{n}} ⟶ g_{n} ∘ f_{n}, and f_{n}η = ζf_{n} : f_{n} ⟶ f_{n} ∘ g_{n} ∘ f_{n}. Via two applications of Lemma 6.3 and a little diagram chasing, we find that the composite

\xymatrix{hocolim_{I} F \ar[r]^{(F ζ)_{∗}} & hocolim_{I} F ∘ f_{n} ∘ g_{n} \ar[r]^{(g_{n})_{∗}} & hocolim_{I_{n}} F ∘ f_{n}}

is a homotopy inverse to (f_{n})_{∗}.

Proof of Proposition 4.5. It suffices to show that ϕ_{∗} is an N-equivalence for each N > 0. Choose n such that λ(n) > N. Then for every object m in I_{n} the map ϕ : F (m) ⟶ F′(m) is an N-equivalence. By Lemma 6.1, this implies that

ϕ_{∗} : hocolim_{I_{n}} F ∘ f_{n} ⟶ hocolim_{I_{n}} F′ ∘ f_{n}

is an N-equivalence. The proposition now follows from Lemma 6.4.

For a based space X, we define F ∧ X by setting (F ∧ X)(c) = F (c) ∧ X and have a natural homeomorphism

hocolim_{C}(F ∧ X) ≅ (hocolim_{C} F) ∧ X.

We define homotopy colimits of symmetric spectra levelwise. A functor F : C ⟶ Sp^{Σ} restricts to give functors F (n) : C ⟶ T for each n ≥ 0, and

(hocolim_{C} F)(n) = hocolim_{C} F (n),

and since homotopy colimits commute with the smash product hocolim_{C} F is a symmetric spectrum. All of the statements above remain valid for homotopy colimits of symmetric spectra.

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Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139

E-mail address: mandell@math.mit.edu

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637

E-mail address: bshipley@math.uchicago.edu