Chapter VII Cosimplicial spaces

One of the most commonly used tools in homotopy theory is the homotopy spectral sequence of a cosimplicial space. This first appeared in the work of Bousfield and Kan [14] and has been further analyzed by Bousfield [10]. Two of the standard examples include the Bousfield-Kan spectral sequence — an unstable Adams spectral sequence that arose before the general example [7] — and the spectral sequence for computing the homotopy groups of the homotopy inverse limit of a diagram of pointed spaces. One of the main purposes of this chapter is to define and discuss this kind of spectral sequence. We will do much more however; for example, we will give a detailed analysis of the total tower of a cosimplicial space, which is the basic object from which the spectral sequence is derived.

We begin with a discussion of skeleta in a very general context. This is the purpose of Section 1, and the main results are Propositions 1.8 and 1.13, which discuss to what extent a simplicial object in a category C with enough colimits can be built by “attaching cells”. One application is a characterization of cofibrations in the kind of model category considered in Section II.5. See Example 1.14.

If the category C happens to itself be a closed model category, then the category sC inherits a closed model category structure in which a morphism X → Y in sC is a weak equivalence in and only if X_{n} → Y_{n} is a weak equivalence for all n ≥ 0. This is the Reedy model category structure on sC and one of the main results here is that there is a geometric realization functor

| · | : sC ⟶ C

which preserves weak equivalences between Reedy cofibrant objects. See Proposition 3.6. The Reedy model category structure is discussed in detail in Section 2.

In Section 4, this theory is specialized to the case of cosimplicial spaces cS; that is, to the case of the opposite category to the category of simplicial objects in S^{op}. The resulting model category structure on cS — probably the original example of a Reedy model category structure — is the one discussed by Bousfield and Kan, and it has some unusual features. In particular, one can give a simple characterization of cofibrations. This is done in Proposition 4.16.

If x ∈ cS, the total space of Tot(X) of X is the geometric realization of X; that is,

Tot(·) = | · |^{op} : cS = (sS)^{op} → (S^{op})^{op}.

As a result, if X ∈ cS is fibrant, Tot(X) is the inverse limit of a tower of fibrations. If X is pointed, the tower is pointed, and there is spectral sequence in homotopy groups. This is the spectral sequence of the cosimplicial space. There are several issues: defining the spectral sequence in low degrees in homotopy, where one might not have groups, or abelian groups; the question of whether the spectral sequence converges to π_{∗} Tot(X) or not; and what the E_{2} term is. This all occupies Sections 5, 6, and 7. The main examples are discussed at the end of Section 7.

Finally, there is a possibility that Tot(X) might be empty — it’s an inverse limit, and inverse limits can be empty. To analyze this question, and to address the calculation of π_{0} Tot(X) in general, Bousfield has developed an extensive obstruction theory in the reference cited above. We give a small, but very useful, example of this theory in Section 8. The main results are Propositions 8.3 and 8.4.

1. Decomposition of simplicial objects.

Let C be a category with all limits and colimits. The purpose of this section is to analyze how simplicial objects are constructed out of smaller components. We will use this inductive argument in later sections.

We begin with skeleta. The category sC is the functor category CΔ Δ Δ^{op}. Let i_{n} : Δ Δ Δ_{n} ⊆ Δ Δ Δ be the inclusion of the full subcategory with objects k, k ≤ n, and let s_{n}C = CΔ Δ Δ{op \atop n}. There is a restriction function i_{n∗} : sC → s_{n}C which simply forgets the k-simplices, k > n. This restriction functor has a left adjoint given by

(1.1)  (i_{n}^{∗}X)_{m} = \underrightarrow{lim}_{m→k} X_{k} = \underrightarrow{lim}_{m↓Δ Δ Δ_{n}} X_{k}

and the colimit is over morphisms m → k in Δ Δ Δ with k ≤ n. This is an example of a left Kan extension. Since every morphism m → k in Δ Δ Δ can be factored uniquely as

m ⟶^{ϕ} k′ ⟶^{ψ} k

where ϕ is a surjection and ψ is one-to-one, the surjections m → k, k ≤ n, are cofinal in the category of all morphisms used to define the colimit of (1.1) and we get

(1.2)  (i_{n}^{∗}X)_{m} = \underrightarrow{lim}_{m↓Δ Δ Δ_{n}^{+}} X_{k}

where Δ Δ Δ_{+} ⊆ Δ Δ Δ is the subcategory with the same objects but only surjections as morphisms and Δ Δ Δ_{n}^{+} = Δ Δ Δ_{n} ∩ Δ Δ Δ_{+}.

If X ∈ sC we define the n^{th} skeleton of X by the formula

(1.3)  sk_{n} X = i_{n}^{∗}i_{n∗}X.

There are natural maps sk_{m} X → sk_{n} X, m ≤ n, and sk_{n} X → X. In light of (1.2), (sk_{n} X)_{m} = X_{m} for m ≤ n It is clear that there is a natural isomorphism

\underrightarrow{lim}_{n} sk_{n} X ⟶^{≅} X.

Here is an example. Because C has limits and colimits, sC has a canonical structure as a simplicial category. (See Section II.2). In particular if X ∈ sC and K ∈ S, then

(X ⊗ K)_{n} = ⨆_{K_{n}} X_{n}.

It is now a straightforward exercise to prove

Proposition 1.4. If X ∈ sC is constant and K ∈ S, then there is a natural isomorphism

sk_{n} X ⊗ sk_{n} K ≅ sk_{n}(X ⊗ K).

To explain how sk_{n} X is built from sk_{n−1} X we define the n^{th} latching object L_{n}X of X by the formula

(1.5)  L_{n}X = (sk_{n−1} X)_{n}

≅ \underrightarrow{lim}_{n↓Δ Δ Δ_{n−1}^{+}} X_{k}.

If Z ∈ C, we may regard Z as a constant object in sC and there is an adjoint isomorphism

hom_{C}(Z, X_{n}) ≅ hom_{sC}(Z ⊗ Δ^{n}, X)

for all n ≥ 0. This immediately supplies maps in sC

L_{n}X ⊗ Δ^{n} → sk_{n−1} X

and

X_{n} ⊗ Δ^{n} → sk_{n} X.

Furthermore, by Proposition 1.4, sk_{n−1}(X_{n} ⊗Δ^{n}) = X_{n} ⊗sk_{n−1} Δ^{n} = X_{n} ⊗∂Δ^{n} and we obtain a diagram

(1.6)

L_{n}X ⊗ ∂Δ^{n}  w u  L_{n}X ⊗ Δ^{n} u

X_{n} ⊗ ∂Δ^{n} wsk_{n−1} X.

Proposition 1.7. For all X ∈ sC there is a natural pushout diagram, n ≥ 0,

X_{n} ⊗ ∂Δ^{n} ∪_{L_{n}X⊗∂Δ^{n}} L_{n}X ⊗ Δ^{n} w u sk_{n−1} X u

X_{n} ⊗ Δ^{n} wsk_{n} X

Proof: In light of Proposition 1.4, if we apply i_{n}^{∗}i_{n∗} = sk_{n}(·) to this diagram we obtain an isomorphic diagram. Since i_{n}^{∗} : s_{n}C → sC is a left adjoint, we need only show this is a pushout diagram in degrees less than or equal to n.

In degrees m < n, the map (L_{n}X⊗∂Δ^{n})_{m} → (L_{n}X⊗Δ^{n})_{m} is an isomorphism, so

(X_{n} ⊗ ∂Δ^{n} ∪_{L_{n}X⊗∂Δ^{n}} L_{n}X ⊗ Δ^{n})_{m} → (X_{n} ⊗ Δ^{m})_{m}

is an isomorphism, and the assertion is that (sk_{n−1} X)_{m} ≅ (sk_{n} X)_{m}, which is true: both are isomorphic to X_{m}. In degree n, the left vertical map is isomorphic to

⨆_{(∂Δ^{n})_{m}} X_{n} ⊔ L_{n}X → ⨆_{(∂Δ^{n})_{m}} X_{n} ⊔ X_{n}

so the assertion is that the right vertical map is isomorphic to the natural map L_{n}X → X_{n}. This is true by the definition of L_{n}X. ■

Morphisms also have skeletal fibrations. If f : A → X is a morphism in sC, define sk_{n}^{A} X by setting sk_{−1}^{A} = A and, for n ≥ 0, defining sk_{n}^{A} by the pushout diagram

sk_{n} A  w u_{sk_{n} f}  Au

sk_{n} X  wsk_{n}^{A} X.

The adjunct to Proposition 1.7 is the next result, which is proved in an identical manner. Let L_{n}(f) = (sk_{n−1}^{A} X)_{n} = A_{n} ∪_{L_{n}A} L_{n}X.

Proposition 1.8. For all morphisms A → X in sC there is a pushout diagram

X_{n} ⊗ ∂Δ^{n} ∪_{L_{n}(f)⊗∂Δ^{n}} L_{n}(f) ⊗ Δ^{n}  w u  sk_{n−1}^{A} X u

X_{n} ⊗ Δ^{n}  wsk_{n}^{A} X.

■

There is a situation under which the pushout diagrams of Proposition 1.7 and 1.8 simplify considerably. This we now explain.

If I is a small category, let I^{δ} be the discrete category with the same objects as I, but no non-identity morphism. The left adjoint r^{∗} to the restriction functor r_{∗} : C^{I} → C^{I^{δ}} has a very simple form

(r^{∗}Z)_{j} = ⨆_{i→j} Z_{i}.

We call such a diagram I-free. More generally, a morphism f : A → X in C^{I} is I-free if there is an I-free object X′ ∈ C^{I} and an isomorphism under A of f with the inclusion of the summand A → A ⊔ X′. This implies that there is an object {Z_{i}} ∈ C^{I^{δ}} so that

X_{j} ≅ A_{j} ⊔ ( ⨆_{i→j} Z_{i}).

Definition 1.9. An object X ∈ sC is degeneracy free if the underlying degeneracy diagram is free. That is, if Δ Δ Δ_{+} ⊆ Δ Δ Δ is the subcategory with same objects but only surjective morphisms, then X regarded as an object in CΔ Δ Δ{op \atop +} is Δ Δ Δ_{+}^{op}-free. A morphism A → X is degeneracy free if, when regarded as a morphism in CΔ Δ Δ{op \atop +} it is Δ Δ Δ_{+}^{op} free.

If X is degeneracy free, then there is a sequence {Z_{n}}_{n≥0} of objects in C so that

X_{n} ≅ ⨆_{ϕ:n→m} Z_{m}

where ϕ runs over the epimorphisms in Δ Δ Δ. A degeneracy free map A → X yields a similar decomposition:

(1.10)  X_{n} ≅ A_{n} ⊔ ⨆_{ϕ:n→m} Z_{m}.

We say A → X is degeneracy free on {Z_{m}}. Although we have not made this explicit before, notice an object X is I-free if and only if the morphism ϕ → X from the initial object in I-free.

The following result says that degeneracy free maps are closed under a variety of operations.

Lemma 1.11.

1) Let f_{α} : A_{α} → X_{α} be a set of maps so that f_{α} is degeneracy free on {Z_{n}}. Then ⨆_{α} f_{α} is free on {⨆_{α} Z_{n}^{α}}.

2) Suppose f : A → X is degeneracy free on {Z_{n}} and

\begin{matrix}A  w u_{f} \\ Bu_{g} \end{matrix} ; {X  w \atop Y}

is a pushout diagram. Then g is degeneracy free on {Z_{n}}.

3) Let A_{0} → A_{1} → A_{2} → ⋯ be a sequence of morphisms so that A_{j−1} → A_{j} is degeneracy free on \{Z_{n}^{j}\}. Then A_{0} → \underrightarrow{lim} A_{j} is degeneracy free on \{⨆_{j} Z_{n}^{j}\}.

4) Let F : C → D be a functor that preserves coproducts. If f : A → X in sC is degeneracy free on {Z_{n}}, then F f : F A → F X is degeneracy free on {F Z_{n}}.

Lemma 1.12. A morphism f : A → X is degeneracy free if and only if there are objects Z_{n} and maps Z_{n} → X_{n} so that the induced map

(A_{n} ∪_{L_{n}A} L_{n}X) ⊔ Z_{n} → X_{n}

is an isomorphism.

Proof: If f is degeneracy free on {Z_{n}}, the decomposition follows from the formulas (1.5) and (1.10). To prove the converse, fix the given isomorphisms. Then Proposition 1.8 implies there is a pushout diagram in CΔ Δ Δ{op \atop +}

Z_{n} ⊗ ∂Δ^{n}  w u  sk_{n−1}^{A} X u

Z_{n} ⊗ Δ^{n}  wsk_{n}^{A} X.

However, the morphism of simplicial ∂Δ^{n} → Δ^{n} is degeneracy free on the canonical n-simplex in (Δ^{n})_{n}; hence, sk_{n−1}^{A} X → sk_{n}^{A} X is degeneracy free on Z_{n} in degree n. The result now follows from Lemma RefVII.1.11.3; indeed A → X is degeneracy free on {Z_{n}}. ■

Proposition 1.13. Suppose A → X is degeneracy-free on {Z_{n}} in sC. Then for all n ≥ 0 there is a pushout diagram,

Z_{n} ⊗ ∂Δ^{n}  w u  sk_{n−1}^{A} X u

Z_{n} ⊗ Δ^{n}  wsk_{n}^{A} X.

Proof: Combine Proposition 1.8 and Lemma 1.12. ■

Notice that one can interpret this result as saying sk_{n}^{A} X is obtained from sk_{n−1}^{A} X by attaching n-cells.

Example 1.14. Lemmas 1.11 and 1.12 provide any number of examples of degeneracy free morphisms. For example, a cofibration in simplicial sets is degeneracy free, by Lemma 1.12. Also, consider a category C equipped with a functor G : C → Sets with a left adjoint F and satisfying the hypotheses of Theorem II.5.1. Define a morphism f : A → X in sC to be free (this terminology is from Quillen) if there is a sequence of sets {Z_{n}} so that f is degeneracy free on {F Z_{n}}. Then a morphism in sC is a cofibration if and only if it is a retract of a free map. To see this, note that the small object argument of Lemma II.5.2, coupled with Lemma 1.11 factors any morphism A → B as

A ⟶^{j} X ⟶^{q} Y

where j is a free map and a cofibration and q is a trivial fibration. Thus any cofibration is a retract of a free map. Conversely, Proposition 1.13 and Proposition II.3.4 imply any free map is a cofibration. Similar remarks apply to the model categories supplied in Theorems II.6.8 and II.7.8. In the latter case one must generalize Lemma 1.11.3 to longer colimits.

Dual to the notion of skeleta is the notion of coskeleta. The theory is analogous, and we give only an outline. The restriction functor i_{n∗} : sC → s_{n}C has right adjoint i_{n}^{!} with

(i_{n}^{!}X)_{m} = \underleftarrow{lim}_{k→m} X_{k},

with the limit over all morphism k → m in Δ Δ Δ with k ≤ n. Equally, one can take the limit over morphisms k → m which are injections. The composite gives the n^{th} coskeleton functor:

(1.15)  cosk_{n} X = i_{n}^{!}i_{n∗}X.

More generally, if f : X → B is a morphism in sC, let cosk_{−1}^{B} X = B and let cosk_{n}^{B} X be defined by the pullback

(1.16)

cosk_{n}^{B} X w u

cosk_{n} X u

B wcosk_{n} B.

Then there are maps cosk_{n}^{B} X → cosk_{n−1}^{B} X and

X ≅ \underleftarrow{lim}_{n} cosk_{n}^{B} X.

Note that if B = ∗, the terminal object, then cosk_{n}^{B} X = cosk_{n} X.

For X ∈ sC, define the n^{th} matching object H_{n}X by the formula

(1.17)  M_{n}X = (cosk_{n−1} X)_{n} = \underleftarrow{lim}_{k→n} X_{k}

where k → n runs over all morphisms (or all monomorphisms) in Δ Δ Δ with k < n.

To fit the map cosk_{n}^{B} X → cosk_{n−1}^{B} X into a pullback diagram, we make some definitions. Let ρ_{n} : sC → C be the functor ρ_{n}X = X_{n}. This is a restriction functor between diagram categories and has a right adjoint ρ_{n}^{!}. The functor L_{n} : sC → C assigning each simplicial object the latching object L_{n}X also has a right adjoint, which we will call μ_{n}^{!}. To see this, see Lemma 1.23 below, or note that we may write

L_{n}X = (sk_{n−1} X)_{n} = ρ_{n}i_{n−1}^{∗}i_{(n−1)∗}X

where i_{(n−1)∗} : sC → s_{n−1}C is the restriction functor. Hence

μ_{n}^{!}Z = i_{n−1}^{!}i_{(n−1)∗}ρ_{n}^{!}Z = cosk_{n−1}(ρ_{n}^{!}Z).

The natural map s : L_{n}X → X gives a natural transformation ρ_{n}^{!} → μ_{n}^{!}. The reader is invited to prove the following result.

Proposition 1.18. Let f : X → B be a morphism in sC. Then for all n ≥ 0 there is a pullback diagram

cosk_{n}^{B} X  w u  ρ_{n}^{!}X_{n} u

cosk_{n−1}^{B} X wρ_{n}^{!}M_{n}(f) ×_{μ_{n}^{!}M_{n}(f)} μ_{n}^{!}X_{n}

where M_{n}(f) = B_{n} ×_{M_{n}B} M_{n}X.

We have never encountered the analog of degeneracy-free morphisms (“face free”?) and don’t include an exposition here.

The latching and matching object functors L_{n} and M_{n} are examples of a much more general sort of functor, which we now introduce and analyze. We begin with generalized matching objects.

Proposition 1.19. Let K ∈ S be a simplicial set. Then the functor

− ⊗ K : C → sC

has right adjoint M_{K}. For fixed X ∈ sC, the assignment K ↦ M_{K}X induces a functor S^{op} → C which has a left adjoint.

Proof: If Z ∈ sC is constant, hom_{sC}(Z, Y) = hom_{C}(Z, Y_{0}). Hence we may set M_{K}X = (X^{K})_{0}, since

hom_{sC}(Z ⊗ K, X) ≅ hom_{sC}(Z, X^{K}).

Furthermore

hom_{C}(Z, M_{K}X) ≅ hom_{sC}(Z ⊗ K, X) ≅ hom_{S}(K, Hom_{sC}(Z, X)).  ■

Since hom_{sC}(Z ⊗ Δ^{n}, X) ≅ hom_{C}(Z, X_{n}), M_{Δ^{n}}X ≅ X_{n}. More generally, we can give a decomposition of M_{K}X as follows. Let Δ ↓ K be the simplex category of K. Thus an object of Δ ↓ K is a morphism σ : Δ^{n} → K in S and a morphism in Δ ↓ K is a triangle in S

{Δ^{n} w^{σ} u_{ϕ} \atop K} ; Δ^{m}.^{[ [[]}^{τ}

If we define a functor F_{K} : Δ ↓ K → S by

F_{K}(σ : Δ^{n} → K) = Δ^{n}

then

\underrightarrow{lim}_{Δ↓K} F_{K} ≅ K.

Since M_{K}X sends colimits in the K variable to limits we have

(1.20)  M_{K}X ≅ \underleftarrow{lim}_{Δ↓K^{op}} M_{F_{K}} X ≅ \underleftarrow{lim}_{Δ↓K^{op}} X

where we write X : Δ ↓ K^{op} → C to be

X(σ : Δ^{n} → K) = X_{n}.

Example 1.21. Let ϕ : Δ^{k} → Δ^{n} be any morphism in S. Then ϕ factors uniquely as a composition

Δ^{k} ⟶^{ϕ′} Δ^{m} ⟶^{ψ} Δ^{n}

where ϕ′ is a surjection and ψ is an injection. Thus if K ⊆ Δ^{n} is any subcomplex, the full subcategory Δ ↓ K_{0} ⊆ Δ ↓ K with objects

σ : Δ^{m} → K

with σ an injection is cofinal. Equivalently Δ ↓ K_{0}^{op} ⊆ Δ ↓ K^{op} is final; hence

M_{K}X ≅ \underleftarrow{lim}_{Δ↓K_{0}^{op}} X.

In particular M_{∂Δ^{n}}X ≅ \underleftarrow{lim}_{ϕ:k→n} X_{k} where ϕ ∈ Δ Δ Δ is an injection and k < n. Thus M_{∂Δ^{n}}X ≅ M_{n}X. Note that the inclusion map ∂Δ^{n} → Δ^{n} induces the projection

X_{n} → M_{n}X.

To generalize the latching objects we appropriate the formulation of matching objects presented in (1.20). Let J be a small category and F : J → Δ Δ Δ^{op}. For X ∈ sC = CΔ Δ Δ^{op}, define the generalized latching object to be

(1.22)  L_{J}X = \underrightarrow{lim}_{J} X ∘ F

= \underrightarrow{lim}_{j} X_{F (j)}.

We are primarily interested in sub-categories J of Δ Δ Δ^{op}. We write X|_{J} for X ∘ F.

Lemma 1.23. For fixed F : J → Δ Δ Δ^{op}, the functor L_{J} : sC → C has a right adjoint and, hence, preserves colimits.

Proof: For hom_{C}(L_{J}X, Z) ≅ hom_{C^{J}} (X ∘ F, Z) where Z is regarded as a constant diagram. But

hom_{C^{J}} (X|_{J}, Z) = hom_{sC}(X, F^{!}Z)

where F^{!} is the right Kan extension functor. ■ This restriction to sub-categories of Δ Δ Δ^{op} is practical, but perhaps gratuitous. An example of a generalized latching object is the following. Let O_{n} be the

category with objects the morphisms ϕ : n → m with ϕ surjective and m < n. The morphisms in O_{n} are commutative triangles in Δ Δ Δ under n. Define F : O_{n}^{op} → Δ Δ Δ^{op} by F (ϕ : n → m) = m. Then there is a natural isomorphism

L_{n}X ≅ L_{O_{n}^{op}}X.

In the next section we will need a decomposition of L_{n}X. To accomplish this define sub-categories M_{n,k} ⊆ O_{n}, 0 ≤ k ≤ n, to be the full sub-category of surjections ϕ : n → m, m < n, with ϕ(k) < k. Define, for X ∈ sC

L_{n,k}X = L_{M_{n,k}^{op}}X.

Then L_{n,0}X = ϕ is the initial object in C (since M_{n,0} is empty) and L_{n,n}X = L_{n}X, since M_{n,n} = O_{n}.

Also define M(k) ⊆ O_{n} to be the full subcategory of surjections ϕ : n → m, m < n, with ϕ(k) = ϕ(k + 1). Notice that s^{k} : n → n − 1 is the initial object of M(k). Hence for X ∈ sC

L_{M(k)^{op}}X ≅ X_{n−1}.

The reader can verify the following statements about these subcategories.

Lemma 1.24.

1) M_{n,k} and M(k) are subcategories of M_{n,k_{1}}, and every object in M_{n,k+1} is in M_{n,k} or M(k) (or both).

2) There is an isomorphism of categories

− ∘ s^{k} : M_{n−1,k} → M(k) ∩ M_{n,k}

sending ϕ to ϕ ∘ s^{k}.

3) If ϕ is an object of M_{n,k} (or M(k)) and ϕ → ψ is a morphism in M_{n,k+1}, then ϕ → ψ is a morphism in M_{n,k} (or M(k)).

The following example, illustrating the case n = 3, k = 2, might be helpful. In the following diagram the symbol 012 near a dot (•) indicates the object

s^{0}s^{1}s^{2} in the appropriate category. The unlabeled arrows indicated composition with s^{i} for some i; for example 0• → •02 means s^{0} ↦ s^{1}s^{0} = s^{0}s^{2}.

\begin{matrix}0 & 1 \\ •u & • ȷ ȷ ȷ fl \\ • \\ 01 \\ M_{2,2} = O_{2} \end{matrix}  0  1 ; \begin{matrix}•_{’ ’ ’ ’)} u \\ • [ [ [ [ [ ^’ ’ ’ ’) \end{matrix} ; 01 • _{’ ’ ’ ’)} • 02 u  • 12 ; • ; 012 ; M_{3,2}

\begin{matrix}2 \\ •u h h h k \\ 02 •u & • 12 ` ` ` \\ ˜ • \\ 012 \\ M(2) \end{matrix} 0  1  2 •u_{4 4 44 6} • {h \atop h} h h k _{’ ’ ’ ’)} •u [ [ [ h h k

01 •_{A A AA C} • 02 u  • 12 ` ` ` `˜

• 012

M_{3,3} = O_{3}

Proposition 1.25. Let X ∈ sC. Then there is a pushout diagram in C

\begin{matrix}L_{n−1,k}X  w u \\ X_{n−1} u \end{matrix} ; \begin{matrix}L_{n,k}X  w \\ L_{n,k+1}X. \end{matrix}

Proof: Lemma 1.24 implies that

\begin{matrix}M_{n−1,k}X  w u \\ M(k) u \end{matrix} ; \begin{matrix}M_{n,k}X  w \\ M_{n,k+1} \end{matrix}

is a pushout in the category of small categories. The same is then true of the opposite categories. The result follows. ■

2. Reedy model category structures.

Let C be a closed model category. Using the considerations of the previous section, this structure can be promoted to a closed model category structure on the category sC of simplicial objects in C that is particularly useful for dealing with geometric realization. The results of this section are a recapitulation of the highly influential, but unpublished paper of C.L. Reedy [80].

In the following definition, let L_{0}X and M_{0}X denote the initial and final object of D respectively.

Definition 2.1. A morphism f : X → Y in sC is a

1) Reedy weak equivalence if f : X_{n} → Y_{n} is a weak equivalence for all n ≥ 0;

2) a Reedy fibration if

X_{n} → Y_{n} ×_{M_{n}Y} M_{n}X

is a fibration for all n ≥ 0;

3) a Reedy cofibration if

X_{n} ∪_{L_{n}X} L_{n}Y → Y_{n}

is a cofibration for all n ≥ 0.

The main result is that this defines a model category structure on sC. Before proving this we give the following lemma.

2. Reedy model category structures 377

Lemma 2.2. A morphism f : X → Y in sC is a

1) Reedy trivial fibration if and only if

X_{n} → Y_{n} ×_{M_{n}Y} M_{n}X

is a trivial fibration for n ≥ 0;

2) a Reedy trivial cofibration if and only if

X_{n} ∪_{L_{n}X} L_{n}Y → Y_{n}

is a trivial cofibration for all n ≥ 0. Proof: Let Δ^{n,k} = d^{0}Δ^{n−1} ∪ ⋯ ∪ d^{k}Δ^{n−1} ⊆ ∂Δ^{n}, −1 ≤ k ≤ n. Then Δ^{n,−1} = ϕ and Δ^{n,n} = ∂Δ^{n}. There are pushout diagrams −1 ≤ k ≤ n − 1

Δ^{n−1,k} w u Δ^{n−1} u

Δ^{n,k} wΔ^{n,k+1}.

Taking matching objects yields a natural pullback square

\begin{matrix}M_{n,k+1}X  w u \\ X_{n−1} u \end{matrix} ; \begin{matrix}M_{n,k}X  w \\ M_{n−1,k}X \end{matrix}

where we have written M_{n,k}X for M_{Δ^{n,k}}X. This yields a pullback square

(2.3)  \begin{matrix}Y_{n} ×_{M_{n,k+1}Y} M_{n,k+1}X  w u \\ X_{n−1} u \end{matrix} ; \begin{matrix}Y_{n} ×_{M_{n,k}Y} M_{n,k}X  w \\ Y_{n−1} ×_{M_{n−1,k}Y} M_{n−1,k}X. \end{matrix}

Assume f : X → Y is a Reedy fibration. Then X_{0} → Y_{0} is a fibration and this begins an induction where the induction hypothesis is that

(2.4)  X_{n−1} → Y_{n−1} ×_{M_{n−1,k}Y} M_{n−1,k}X

is a fibration. To complete the inductive step, one uses (2.3) to show

(2.5)  Y_{n} ×_{M_{n,k+1}Y} M_{n,k+1}X → Y_{n} ×_{M_{n,k}Y} M_{n,k}X

is a fibration for all k, −1 ≤ k ≤ n − 1. Since composites of fibrations are fibrations and X_{n} → Y_{n} ×_{M_{n}Y} M_{n}X is a fibration, we close the loop.

Now suppose f : X → Y is a Reedy trivial fibration. Then, inductively, each of the maps of (2.4) is a trivial fibration. For the inductive step, use (2.3) to show each of the maps (2.5) is a trivial fibration. Then the axiom CM2 of closed model categories and the fact that X_{n} → Y_{n} a trivial fibration finishes the argument. In particular X_{n} → Y_{n} ×_{M_{n}Y} M_{n}X is a trivial fibration for all n.

Conversely, suppose X_{n} → X_{n} ×_{M_{n}Y} M_{n}X is a trivial fibration. Then one again runs the same sort of argument to conclude X_{n} → Y_{n} is a trivial fibration.

The argument for part 2) is similar, using the pushout diagram

\begin{matrix}L_{n−1,k}X  w u \\ X_{n−1} u \end{matrix} ; \begin{matrix}L_{n,k}X  w \\ L_{n,k+1}X \end{matrix}

of Proposition 1.25. ■

We break the proof of the verification of the Reedy model category structure into several steps.

Lemma 2.6. The Reedy structure on sC satisfies the lifting axiom CM4.

Proof: Suppose we are given a lifting problem in sC

\begin{matrix}A  w u_{j} \\ Xu_{q} \end{matrix} ; {B  w \atop Y}

where j is cofibration, q is a fibration and either j or q is a weak equivalence. We inductively solve the lifting problems

sk_{n−1}^{A} B  w u  Xu

sk_{n}^{A} B wY

for n ≥ 0. Because of the pushout diagram of Proposition 1.8 it is sufficient to solve the lifting problems, with L_{n}(j) = A_{n} ∪_{L_{n}A} L_{n}B

L_{n}(j) ⊗ Δ^{n} ∪_{L_{n}(j)⊗∂Δ^{n}} B_{n} ⊗ ∂Δ^{n}  w u  Xu

B_{n} ⊗ Δ^{n} wY.

By an adjunction argument this is equivalent to the lifting problem

\begin{matrix}A_{n} ∪_{L_{n}A} L_{n}B  w u \\ X_{n} u \end{matrix} ; \begin{matrix}B_{n}  w \\ Y_{n} ×_{M_{n}Y} M_{n}X. \end{matrix}

This is solvable by Lemma 2.2. ■

For the proof of the factorization axiom we need to know how much data we need to build a simplicial object.

Lemma 2.7. Let X ∈ sC. Then sk_{n} X is determined up to natural isomorphism by the following natural data: sk_{n−1} X, X_{n}, and maps L_{n}X → X_{n} → M_{n}X so that the composite L_{n}X → M_{n}X is the canonical map.

Proof: The given map X_{n} → M_{n}X is adjoint to a map X_{n} ⊗ ∂Δ^{n} → X which factors uniquely through sk_{n−1} X. The data listed thus yields the pushout square

L_{n}X ⊗ Δ^{n} ∪_{L_{n}X⊗∂Δ^{n}} X_{n} ⊗ ∂Δ^{n} w u sk_{n−1} X u

X_{n} ⊗ Δ^{n} wsk_{n} X of Proposition 1.7. ■

This can be restricted: let s_{n}C be the functor category CΔ Δ Δ{op \atop n} and X ∈ s_{n}C. Let r_{∗}X ∈ s_{n−1}C be the restriction and r^{∗} : s_{n−1}C → s_{n}C the left Kan extension functor. Hence (r^{∗}r_{∗}X)_{n} = L_{n}X.

Lemma 2.7 immediately implies

Lemma 2.8. Let X ∈ s_{n}C. Then X is determined up to natural isomorphism by the following natural data: r_{∗}X, X_{n}, and maps L_{n}X → X_{n} → M_{n}X so that the composite L_{n}X → M_{n}X is the canonical map.

Proof: Let i_{n∗} : sC → s_{n}C be the restriction and i_{n}^{∗} : s_{n}C → sC the left adjoint. Then i_{n∗}i_{n}^{∗} ≅ 1 and i_{n}^{∗}i_{n∗} = sk_{n}. Thus

i_{n}^{∗}X ≅ sk_{n} i_{n}^{∗}X

is determined up to natural isomorphism by sk_{n−1} i_{n}^{∗}X ≅ i_{n−1}^{∗}r_{∗}X (where i_{n−1}^{∗} : s_{n−1}C → sC is left adjoint to restriction), X_{n} and maps L_{n}i_{n}^{∗}X → X_{n} → M_{n}i_{n}^{∗}X that compose to the canonical map L_{n}i_{n}^{∗}X → M_{n}i_{n}^{∗}X. But L_{n}X ≅ L_{n}i_{n}^{∗}X and M_{n}X ≅ M_{n}i_{n}^{∗}X. ■

Lemma 2.9. The Reedy structure on sC satisfies the factorization axiom CM5.

Proof: Let us do the trivial cofibration-fibration factorization (compare the proof of Lemma IV.3.5).

Let X → Y be a morphism in sC and let i_{n∗}X → i_{n∗}Y be the induced morphism in s_{n}C. For each n ≥ 0 we construct a factorization

i_{n∗}X → Z(n) → i_{n∗}Y

in s_{n}C with the property that restricted to s_{n−1}C we get the factorization i_{(n−1)∗}X → Z(n − 1) → i_{(n−1)∗}Y.

For n = 0, simply factor X_{0} → Y_{0} and X_{0} ⟶^{j} Z(0) ⟶^{q} Y_{0} where j is a trivial cofibration in C and q is a fibration in C. Suppose the factorization in s_{n−1}C has been constructed. Let r_{∗} : s_{n}C → s_{n−1}C be the restriction functor and r^{∗}, r^{!} : s_{n−1}C → s_{n}C be respectively, the left and right Kan extension functors. Thus

r^{∗}i_{(n−1)∗}X = i_{(n−1)∗} sk_{n−1} X and r^{!}i_{(n−1)∗}X = i_{n∗} cosk_{n−1} X.

Furthermore, the morphism r_{∗}r^{∗}Z(n−1) → Z(n−1) yields a map r^{∗}Z(n−1) → r^{!}Z(n − 1) so that the following diagram commutes

\begin{matrix}r^{∗}i_{(n−1)∗}X  w u \\ r^{∗}Z(n − 1)  w \\ r^{!}Z(n − 1)  w \\ r^{!}i_{(n−1)∗}Y \end{matrix} ; i_{n∗}X  wi_{n∗}Y. u

At level n this yields a map (write L_{n}Z = L_{n}Z(n − 1), etc.)

X_{n} ∪_{L_{n}X} L_{n}Z → Y_{n} ×_{M_{n}Y} M_{n}Z.

Factor this map as

(2.10)  X_{n} ∪_{L_{n}X} L_{n}Z ⟶^{j} Z_{n} ⟶^{q} Y_{n} ×_{M_{n}Y} M_{n}Z

where j is a trivial cofibration and q is a fibration. The morphisms j and q yield diagrams

L_{n}X  w u  X_{n} u  wM_{n}X u

L_{n}Z  w u  Z_{n} u  wM_{n}Z u

L_{n}Y  wY_{n}  wM_{n}Y

such that all the horizontal maps are induced from the canonical natural transformation L_{n} → M_{n}. Hence Lemma 2.8 produces

i_{n∗}X → Z(n) → i_{n∗}Y.

Finally, define Z ∈ sC by Z_{k} = Z(n)_{k}, k ≤ n. There is a factoring

X ⟶^{j} Z ⟶^{q} Y

and using (2.10) j is a trivial cofibration by Lemma 2.2 and q is a fibration by definition.

The other factorization is similar. ■ We now state Theorem 2.11. With the definitions of Reedy weak equivalence, cofibration, and fibration given in 2.1, sC is a closed model category.

Proof: The axioms CM1–CM3 are easy and Lemmas 2.6 and 2.9 prove CM4 and CM5 respectively. ■

Next suppose C is in fact a simplicial model category. For K ∈ S and Y, Z ∈ C write Z□K, Z^{K}, and Hom_{C}(Y, Z) for the tensor object, the exponential object, and the mapping space. We use this notation to distinguish it from Z ⊗ K defined by the simplicial structure.

The category sC inherits a simplicial structure. If X ∈ sC and K ∈ S, then X□K and X^{K} are defined level-wise

(X□K)_{n} = X_{n}□K and (X^{K})_{n} = X_{n}^{K}.

The mapping space is defined by the usual formula

Hom_{sC}(X, Y)_{n} ≅ hom_{sC}(X□Δ^{n}, Y).

We call this the internal structure on sC.

Corollary 2.12. With this internal simplicial structure on sC, the Reedy model category structure is a simplicial model category.

Proof: We claim M_{n}(Y^{K}) ≅ M_{n}(Y)^{K}. For there is a sequence of natural isomorphisms, Z ∈ C,

hom_{C}(Z, M_{n}(Y^{K})) ≅ hom_{sC}(Z ⊗ ∂Δ^{n}, Y^{K})
≅ hom_{sC}((Z ⊗ ∂Δ^{n})□K, Y)
≅ hom_{sC}((Z□K) ⊗ ∂Δ^{n}, Y)
≅ hom_{sC}(X□K, M_{n}Y) ≅ hom_{sC}(Z(M_{n}Y)^{K}).

The isomorphism (Z ⊗ ∂Δ^{n})□K ≅ (Z□K) ⊗ ∂Δ^{n} follows by a level-wise calculation. The result follows from the claim using Proposition II.3.13. ■

One can ask if sC in the Reedy model category is a simplicial model category in the standard simplicial structure obtained by Quillen’s method (as in the previous section). The answer is no; for if Z ∈ C is cofibrant in C, then

1 ⊗ d^{0} : Z ⊗ Δ^{0} → Z ⊗ Δ^{1}

is a Reedy cofibration, but not, in general, a Reedy weak equivalence. However we do have Proposition 2.13.

1) Let f : X → Y be a Reedy cofibration in sC and j : K → L a cofibration in S. Then

X ⊗ L ∪_{X⊗K} Y ⊗ K → Y ⊗ L

is a Reedy cofibration which is a Reedy weak equivalence if f is a Reedy weak equivalence.

2) Suppose f : X → Y is a Reedy fibration in sC and j : K → L is a cofibration in S. Then

X^{L} → X^{K} ×_{Y^{K}} Y^{L}

is a Reedy fibration which is a Reedy weak equivalence if f is a Reedy weak equivalence.

Proof: By an adjunction argument these two statements are equivalent. Let K ∈ S. Define L_{n}K ⊆ K_{n} to be the degenerate n-simplices; that is

L_{n}K = \underrightarrow{lim}_{ϕ:n→k} K_{k}

where ϕ runs over the surjective morphisms. Then if X ∈ sC

L_{n}(X ⊗ K) ≅ ⨆_{x∈L_{n}K} L_{n}X.

We leave the rest of the proof as an exercise. ■

3. Geometric realization.

Suppose C is a simplicial category and X ∈ sC. Then the geometric realization |X| ∈ C is defined by the coequalizer diagram

(3.1)  ⨆_{ϕ:n→m} X_{m}□Δ^{n} ⇉_{d_{1}}^{d_{0}} ⨆_{n≥0} X_{n}□Δ^{n} → |X|

where ϕ runs over the morphisms of Δ Δ Δ and d_{0} and d_{1} or the factor associated to ϕ : n → m are respectively

X_{m}□Δ^{n}  w^{ϕ^{∗}□1} X_{n}□Δ^{n}  w⨆_{n≥0} X_{n}□Δ^{n}

X_{m}□Δ^{n}  w^{1□ϕ} X_{m}□Δ^{m}  w⨆_{n≥0} X_{n}□Δ^{n}.

This is the obvious generalization of the geometric realization of Chapters I and III. Note that |X| is a coend:

|X| = ∫ ^{Δ Δ Δ} X□Δ,

where Δ denotes the covariant functor n ↦ Δ^{n} on Δ Δ Δ. We discuss the homotopical properties of |X|.

First note that | · | : sC → C has a right adjoint

(3.2)  Y ↦ Y^{Δ} = {Y^{Δ^{n}}}.

If we give sC the internal (or level-wise) simplicial structure induced from C, it follows immediately that if X ∈ sC and K ∈ S, then

(3.3)  |X□K| ≅ |X|□K.

Indeed, hom_{sC}(X□K, Y^{Δ}) ≅ hom_{sC}(X, (Y^{K})^{Δ}).

Now assume C is a simplicial model category. Endow sC with the Reedy model category structure. By Corollary 2.12, this is a simplicial model category in the internal simplicial structure.

Lemma 3.4. The functor (·)^{Δ} : C → sC preserves fibrations and trivial fibrations.

Proof: The first point to be proved is this: if K ∈ S and Y ∈ C, then

(3.5)  M_{K}(Y^{Δ}) ≅ Y^{K}.

Let Δ ↓ K be the simplex category of K (cf. Example 1.21). Then

M_{K}(Y^{Δ}) ≅ \underleftarrow{lim}_{Δ↓K^{op}}(Y^{Δ})_{n}

≅ \underleftarrow{lim}_{Δ↓K^{op}} Y^{Δ^{n}} ≅ Y^{\underrightarrow{lim}_{Δ↓K} Δ^{n}} ≅ Y^{K}.

Now let X → Y be a fibration in C. Then

(X^{Δ})_{n} → (Y^{Δ})_{n} ×_{M_{n}(Y^{Δ})} M_{n}(X^{Δ})

is isomorphic to

X^{Δ^{n}} → Y^{Δ^{n}} ×_{Y^{∂Δ^{n}}} X^{∂Δ^{n}}

and the result follows from Lemma 2.2 and SM7 for C. ■

Proposition 3.6. The geometric realization functor | · | : sC → C preserves cofibrations, trivial cofibrations and weak equivalences between Reedy cofibrant objects.

Proof: Use Lemma 3.4 and Lemma II.8.9. ■

Implicit in the proof of Lemma 3.4, is the assertion that if Z ∈ sC is constant and K ∈ S then there is a natural isomorphism

(3.7)  |Z ⊗ K| ≅ Z□K.

Indeed, using the isomorphism (3.5), we have

hom_{C}(|Z ⊗ K|, Y) ≅ hom_{C}(Z, Y^{K}),

and the assertion follows. Therefore, for X ∈ sC Reedy cofibrant, the realization comes with a natural skeletal filtration. Define

sk_{n} |X| = | sk_{n} X|.

Then Proposition 1.7 and the natural isomorphism of (3.7) yield that there are natural pushout squares

(3.8)

X_{n}□∂Δ^{n} ∪_{L_{n}X□∂Δ^{n}} L_{n}□Δ^{n} w u sk_{n−1} |X| u

X_{n}□Δ^{n} wsk_{n} |X|.

This is because | · |, as a left adjoint commutes with all colimits. If X is cofibrant, Proposition 3.6 implies that each of the maps sk_{n−1} |X| → sk_{n} |X| is a cofibration. Furthermore, again since | · | commutes with colimits

(3.9)  \underrightarrow{lim}_{n} sk_{n} |X| ≅ |X|.

Finally if X happened to s-free on same set {Z_{n}} of objects in C, then (3.8) can be refined (as in Proposition 1.13) to a pushout diagram

(3.10)

Z_{n}□∂Δ^{n}  w u  sk_{n−1} |X| u

Z_{n}□Δ^{n} wsk_{n} |X|.

The object sk_{n} |X| can also be described as a coend. Let sk_{n} Δ be the functor from Δ Δ Δ to S with

m ↦ sk_{n} Δ^{m}.

Proposition 3.11. Let X ∈ sC. Then there is a natural isomorphism

sk_{n} |X| ≅ ∫ ^{Δ Δ Δ} X□ sk_{n} Δ

and this isomorphism is compatible with the skeletal filtrations of source and target.

Proof: There is a sequence of natural isomorphisms, where

i_{n∗} : sC → s_{n}C

is the restriction functor

hom_{C}(sk_{n} |X|, Y) ≅ hom_{sC}(sk_{n} X, Y^{Δ})
≅ hom_{sC}(i_{n}^{∗}i_{n∗}X, Y^{Δ})
≅ hom_{sC}(X, i_{n}^{!}i_{n∗}Y^{Δ})

Now i_{n}^{!}i_{n∗}Y^{Δ} ≅ cosk_{n}(Y^{Δ}) ≅ Y^{sk_{n} Δ}. Hence

hom_{C}(sk_{n} |X|, Y) ≅ hom_{sC}(X, Y^{sk_{n} Δ})
≅ hom_{C}(∫ ^{Δ} X□ sk_{n} Δ, Y).  ■

4. Cosimplicial spaces.

The language and technology of the previous three sections can be used to give a discussion of the homotopy theory of cosimplicial spaces; that is, of the category cS = SΔ Δ Δ of functors from the ordinal number category to simplicial sets. We go through some of the details and give some examples. It turns out that cofibrations in cS have a very simple characterization; we close the section with a proof of this fact.

We begin with two important examples.

Example 4.1. Let R = 𝔽_{p}, the prime field with p > 0, or let R be a subring of the rationals. The forgetful functor from R-modules to sets has left adjoint X ↦ RX, where RX is the free R-module on X. These functors prolong to an adjoint pair between simplicial R-modules and simplicial sets. By abuse of notation we write

R : S → S

for the composite of these two functors. Then R is the functor underlying a cotriple on S and, if X ∈ S,

π_{∗}RX ≅ H_{∗}(X; R).

Let T : S → S be any cotriple on S with natural structure maps η : X → T X and ϵ : T^{2}X → T X. If X ∈ S is any object, there is a natural augmented cosimplicial space

X → T^{•}X

with (T^{•}X)^{n} = T^{n+1}X and

\begin{aligned}d^{i} & = T^{i}ηT : (T^{•}X)^{n} → (T^{•}X)^{n+1} \\ s^{i} & = T^{i}ϵT^{n−i} : (T^{•}X)^{n+1} → (T^{•}X)^{n}. \end{aligned}

The augmentation is given by η : X → T X = (T^{•}X)^{0}; note that

d^{0}η = d^{1}η : X → (T^{•}X)^{1}.

In particular, if we let T = R : S → S we get an augmented cosimplicial space

X → R^{•}X

with the property that d^{i}, i ≥ 1, and s^{i}, i ≥ 0, are all morphisms of simplicial Rmodules. Furthermore, if we apply R one more time, the augmented cosimplicial R-module

RX → R(R^{•}X)

has a cosimplicial contraction; hence

π^{p}H_{∗}(R^{•}X; R) ≅ \{ \begin{matrix}H_{∗}(X; R), & p = 0 \\ 0, & p > 0. \end{matrix}

The object X → R^{•}X is a variation on the Bousfield-Kan R-resolution of X.

Example 4.2. Let J be a small category and S^{J} the category of J-diagrams in S. Let J^{δ} be the category with some objects as J but no non-identity morphisms— J^{δ} is J made discrete. There is an inclusion functor J^{δ} → J, hence a restriction functor

r_{∗} : S^{J} → S^{J^{δ}}.

The functor r_{∗} has a right adjoint r^{!} given by right Kan extension; in formulas

(r^{!}X)_{j} = ∏_{j→i} X_{i}

where the product is over morphisms in J with source j. Let

T = r^{!}r_{∗} : S^{J} → S^{J}.

Then T is the functor of a cotriple on S^{J}, and if Y ∈ S^{J}, there is a natural cosimplicial object in S^{J}

(4.3)  Y → T^{•}Y.

This cosimplicial object has the property that the underlying J^{δ} diagram has a cosimplicial contraction. Put another way, for each j ∈ J, the augmented cosimplicial space

Y_{j} → (T^{•}Y)_{j}

has a cosimplicial contraction. We can apply the functor \underleftarrow{lim}_{J}(·) to (4.3) to obtain an augmented cosimplicial space

(4.4)  \underleftarrow{lim}_{J} Y → \underleftarrow{lim}_{J} T^{•}Y.

Note that \underleftarrow{lim}_{J}(T^{•}Y)^{n} can be easily computed because

\underleftarrow{lim}_{J} r^{!}X ≅ ∏_{j} X_{j}

where j runs over the objects on J. This last follows from the isomorphisms

hom_{S}(Z, \underleftarrow{lim}_{J} r_{!}X) ≅ hom_{S}_{J^{δ}} (Z, r_{!}X)

≅ hom_{S}_{J^{δ}} (Z, X) ≅ ∏_{j} hom_{S}(Z, X_{j})

where Z ∈ S is regarded as a constant diagram in S^{J} or S^{J^{s}}.

The cosimplicial object (4.4) is used in analyzing the homotopy inverse limit of Y ∈ S^{J} — see Section IX.5.

With these examples in hand, we begin to analyze the homotopy theory of cS. Since S is a simplicial model category, so is the opposite category S^{op}. Thus the category (s(S^{op}))^{op} = cS acquires a Reedy model category structure as in section 2 of this chapter. We take some care with the definitions as the opposite category device can be confusing.

To begin with, cS is a simplicial category: if K ∈ S and X ∈ cS define X□K and X^{K} in cS by

(4.5)  (X□K)^{n} = X^{n} × K

and

(4.6)  (X^{K})^{n} = (X^{n})^{K}.

One often writes X ⊗ K for X□K; however, we are—in this chapter—reserving the tensor product notation for the external operation on sC. The mapping spaces functor is then

(4.7)  Hom_{cS}(X, Y)_{n} ≅ hom_{cS}(X□Δ^{n}, Y).

There are also latching and matching objects. At this point the literature goes to pieces. The matching objects in cS defined in [BK] are the latching objects in s(S^{op}) as defined in Section 1. Since we would hope the reader will turn to the work of Bousfield and Kan as needed, we will be explicit.

Let X ∈ cS. The n^{th} matching space M^{n}X ∈ S is the (n + 1)^{st} latching object L_{n+1}X of X ∈ s(S^{op}). Specifically,

(4.8)  M^{n}X ≅ \underleftarrow{lim}_{ϕ:n+1→k} X^{k}

where ϕ : n → k runs over the surjections in Δ Δ Δ with k ≤ n. Thus n ≥ −1. There are natural maps s : X^{n+1} → M^{n}X and f : X → Y in cS is a fibration if and only if

(4.9)  X^{n+1} → Y^{n+1} ×_{M^{n}Y} M^{n}X

is a fibration for n ≥ −1.

The reader should be aware of 1) the shift in indices M^{n}X = L_{n+1}X and 2) the superscript versus the subscript: M^{n}X ≠ M_{n}X.

Similarly, there are latching objects. If X ∈ cS, L^{n}X = M_{n+1}X, the matching spaces of X ∈ s(S^{op}); thus

L_{n}X = \underrightarrow{lim}_{ϕ:k→n+1} X^{k}

where ϕ runs over the injections in Δ Δ Δ. A morphism X → Y in cS is a cofibration if and only if

(4.10)  X^{n+1} ∪_{L_{n}X} L_{n}Y → Y^{n+1}

is a cofibration (that is, inclusion) of simplicial sets.

Finally, we define a morphism X → Y in cS to be a weak equivalence if X^{n} → Y^{n} is a weak equivalence for all n ≥ 1. Theorem 2.11 and Corollary 2.12 now imply the following result.

Proposition 4.11. With the definitions above, the category cS is a simplicial model category.

We can give a simple characterization of cofibrations in cS, and, along the way, show that cS is cofibrantly generated. First, let us define a set of specific cofibrations.

The functors from cS → S

ρ_{n} : X ↦ X^{n}, n ≥ 0

and

μ_{n} : X ↦ M^{n}X, n ≥ −1

all have left adjoints, given by variations on left Kan extension. Indeed, the adjoint to ρ_{n} is given by the formula

(ρ_{n}^{∗}Y)^{k} = ⨆_{ϕ:n→k} Y

where ϕ runs over all morphisms in Δ Δ Δ with source n. This is a left Kan extension. The adjoint to μ_{n} is slightly more complicated: if J is the category with objects surjections n + 1 → k in Δ Δ Δ, and r : J → Δ Δ Δ the functor sending n + 1 → k to k, then the left adjoint μ_{n}^{∗} to μ_{n} is characterized by

hom_{cS}(μ_{n}^{∗}Z, X) ≅ hom_{S^{J}} (Z, r_{∗}X)
≅ hom_{S}(Z, M_{n}X)

where Z ∈ S^{J} is the constant diagram. Thus μ_{n}^{∗}Z is a left Kan extension of a constant diagram. Note that the natural transformation

X_{n} → M_{n−1}X

gives a natural transformation

μ_{n−1}^{∗}Z → ρ_{n}^{∗}Z.

Define morphisms in cS as follows:

(4.12)  ρ_{n}^{∗}∂Δ^{m} ∪_{μ_{n−1}^{∗}∂Δ^{m}} μ_{n−1}^{∗}Δ^{m} = ∂Δ \left[{m \atop n}\right] ⟶^{i_{n}^{m}} Δ \left[{m \atop n}\right] = ρ_{n}^{∗}Δ^{m},

for n ≥ 0, m ≥ 0, and

(4.13)  Δ [ \begin{matrix}m k \\ n \end{matrix} ] = ρ_{n}^{∗}Δ_{k}^{m} ∪_{μ_{n−1}^{∗}Δ_{k}^{m}} μ_{n−1}^{∗}Δ^{m} ⟶^{j_{n}^{m,k}} Δ [ \begin{matrix}m \\ n \end{matrix} ] = ρ_{n}^{∗}Δ^{m},

for n ≥ 0, 0 ≤ k ≤ m, m ≥ 1.

Lemma 4.14. A morphism f : X → Y in cS is a fibration if and only if it has the right lifting property with respect to the morphisms j_{n}^{m,k} of (4.13). A morphism in cS is a trivial fibration if and only if it has the right lifting property with respect to the morphisms i_{n}^{m} of (4.12).

Proof: We prove the trivial fibration case; the other is similar. A lifting problem

\begin{matrix}∂Δ [ \begin{matrix}m \\ n \end{matrix} ]  w u \\ Xu \end{matrix} ; {Δ [ \begin{matrix}m \\ n \end{matrix} ]  w \atop Y}

is equivalent, by adjointness to a lifting problem

\begin{matrix}∂Δ^{m}  w u \\ X_{n} u_{s} \end{matrix} ; \begin{matrix}Δ^{m}  w \\ Y_{n} ×_{M^{n−1}Y} M^{n−1}X. \end{matrix}

Lemma 2.2.2 (remember M^{n−1}X = L_{n}X, where X ∈ s(S^{op})) implies that f : X → Y is a trivial fibration if and only if s : X_{n} → Y_{n} ×_{M^{n−1}Y} M^{n−1}X is a trivial fibration for all n. The result follows. ■

Proposition 4.15. The simplicial model category structure on cS is cofibrantly generated: the morphisms i_{n}^{m} of (4.12) generate the cofibrations and the morphisms j_{n}^{m,k} of (4.13) generate the trivial cofibrations.

Proof: In light of Lemma 4.14, the small object argument now applies. ■

We can use this result to characterize cofibrations. If X ∈ cC is a cosimplicial object in any category, define the maximal augmentation aX by the equalizer diagram in C

aX → X^{0} ⇉_{d^{1}}^{d^{0}} X^{1}.

Let d^{0} : aX → X^{0} be the natural map.

Proposition 4.16. A morphism f : X → Y in cS is a cofibration if and only if X^{n} → Y^{n} is a cofibration in S for all n ≥ 0 and the induced map aX → aY of maximal augmentations is an isomorphism.

We give the proof below, after some technical preliminaries.

Let Δ Δ Δ_{−1} be the augmented ordinal number category; this has objects

n = {0, 1, …, n − 1}, n ≥ −1,

(−1 = ϕ) and ordering preserving maps. An augmented cosimplicial object in C is a functor X : Δ Δ Δ_{−1} → C. We write d^{0} : X^{−1} → X^{0} for the unique map. Note that the maximal augmentation extends any cosimplicial object to an augmented cosimplicial object.

Lemma 4.17. Let X be an augmented cosimplicial set, and

Z^{n} = X^{n} − ⋃_{i=0}^{n} d^{i}X^{n−1}

If d^{0} : X^{−1} → X^{0} isomorphic to the inclusion of the maximal augmentation, then

⨆_{ϕ} Z^{k} → X^{n}

with ϕ : k → n running over all injections, −1 ≤ k ≤ n, is an isomorphism.

Proof: Out of any cosimplicial set X we may construct a simplicial set Y “without d^{0}” as follows: Y_{n} = X^{n} and

\begin{aligned}d_{i} & = s^{n−i} : Y_{n} → Y_{n−1}, 1 ≤ i ≤ n \\ s_{i} & = d^{n−i} : Y_{n} → Y_{n+1}, 0 ≤ i ≤ n. \end{aligned}

Notice that this construction does not use d^{n} : X^{n−1} → X^{n}. Let Z_{n} ⊆ Y_{n} be the non-degenerate simplices:

Z_{n} = Y_{n} − ⋃_{i} s_{i}Y_{n−1}.

The standard argument for simplicial sets (see Example 1.14) shows that

Y_{n} ≅ ⨆_{ψ:n→k} Z_{k}

where ψ runs over the surjections in Δ Δ Δ. Unraveling the definitions shows that our claim will follow if we can show that d^{n} : X^{n−1} → X^{n} induces an isomorphism

Z^{n} ⊔ Z^{n−1} ⟶^{≅} Z_{n} or, equivalently, an isomorphism

Z^{n−1} ⟶^{≅} Z_{n} ∩ d^{n}X_{n−1}.

First note that d^{n} does induce an injection

(4.18)  d^{n} : Z^{n−1} → Z_{n} ∩ d^{n}X_{n−1}.

To show this it is sufficient to show that if y ∈ Z^{n−1}, then d^{n}y ∈ Z_{n}; that is, if

y ∉ ⋃_{i=0}^{n−1} d^{i}X_{n−2}, then d^{n}y ∉ ⋃_{i=0}^{n−1} d^{i}X_{n−1}. The contrapositive of this statement

reads: if d^{n}y ∈ ⋃_{i=0}^{n−1} d^{i}X_{n−1}, then y ∈ ⋃_{i=0}^{n−1} d^{i}X_{n−2}. So assume d^{n}y = d^{i}z with

i < n. If i < n − 1, then

z = s^{i}d^{n}y = d^{n−1}s^{i}y

so d^{n}y = d^{i}z = d^{i}d^{n−1}s^{i}y = d^{n}d^{i}s^{i}y and y = d^{i}s^{i}y. If i = n − 1 and n > 1, then

y = s^{n−1}d^{n}y = s^{n−1}d^{n−1}z = z

so d^{n}y = d^{n−1}y; hence

y = s^{n−2}d^{n−1}y = s^{n−2}d^{n}y = d^{n−1}s^{n−2}y.

If n = 1 and i = n − 1, we have d^{1}y = d^{0}z, hence y = z; since d^{0} : X^{−1} → X^{0} is the inclusion of the maximal augmentation, y = d^{0}w for some w This is where the hypothesis is used.

We now must show that d^{n}, as in (4.18) is onto. If n = 0, Z_{0} = X_{0} and the

result is clear. If n ≥ 1, we need to show that if x = d^{n}y and x ∉ ⋃_{i=0}^{n−1} d^{i}X_{n−1},

then y ∉ ⋃_{i=0}^{n−1} d^{i}X^{n−2}. The contrapositive of this statement is: if x = d^{n}y and

y = d^{i}w, i ≤ n − 1, then x = d^{j}z for j ≤ n − 1. But

x = d^{n}y = d^{n}d^{i}w = d^{i}d^{n−1}w.  ■

We can now prove Proposition 4.16: Proof of 4.16: For the purposes of this argument, we say a morphism f : X → Y in cS has Property C if X^{n} → Y^{n} is a cofibration in S for all n ≥ 0 and aX ≅ aY. We leave it as an exercise to show that the class of morphisms satisfying Property C is closed under isomorphisms, coproducts, retracts, cobase change, and colimits over ordinal numbers. Only the statement about cobase change is non-trivial. Furthermore, the generating cofibrations ι_{n}^{m} : ∂Δ \left[{m \atop n}\right] → Δ \left[{m \atop n}\right] have Property C. Hence Proposition 4.15 implies all cofibrations have Property C.

For the converse, suppose f : X → Y has Property C. Referring to Lemma 4.17, write Z^{n}(X_{m}) for Z^{n} obtained from the cosimplicial set of m simplices X_{k}. Then, Lemma 4.17 implies

(L^{0}X)_{m} ≅ Z^{0}(X_{m}) ⊔ Z^{−1}(X_{m}) ⊔ Z^{0}(X_{m}) ⊔ Z^{−1}(X_{m})

and if n > 1,

(L^{n−1}X)_{m} ≅ ⨆_{ϕ:k→n} Z^{k}(X_{m})

with ϕ running over injections with −1 ≤ k < n. Since f : X_{m}^{n} → Y_{m}^{n} is one-to-one, Z^{k}(·) is natural in f. Since Z^{−1}(X_{m}) ≅ Z^{−1}(Y_{m}),

[X^{n} ∪_{L^{n−1}X} L^{n−1}Y]_{m} ≅ Z^{n}(X_{m}) ⊔ ⨆_{ϕ:k→n} Z^{k}(Y_{m})

for all n ≥ 0. Again ϕ runs over injections k, −1 ≤ k < n. The result follows. ■

Proposition 4.16 makes it very easy to decide when an object of cS is cofibrant; for example, a constant object on a non-empty simplicial set is not cofibrant, but the standard simplices Δ^{n} form a cosimplicial space {Δ^{n}}_{n} = Δ which is cofibrant.

5. The total tower of a cosimplicial space.

The results on geometric realization from Section 3, interpreted in the context of cosimplicial spaces, leads to the study of the total space of a cosimplicial space. The purpose of this section is to work through these results.

If X ∈ cS ≅ (s(S^{op}))^{op}, we can define the geometric realization |X| ∈ (S^{op})^{op} = S; we will write Tot(X) for this object. Using the coequalizer (3.1), suitably interpreted, we find that Tot(X) fits into an equalizer diagram

(5.1)  Tot(X) → ∏_{n≥0} X_{n}^{Δ^{n}} ⇉ ∏_{ϕ:n→m} X_{m}^{Δ^{n}}.

If Δ = {Δ^{n}} ∈ cS is the cosimplicial space composed of the standard simplices, (5.1) immediately implies there is a natural isomorphism in S

(5.2)  Tot(X) ≅ Hom_{sC}(Δ, X).

Since Δ ∈ cS is cofibrant, SM7 for cS implies that Tot(·) preserves weak equivalences between fibrant objects; alternatively, this fact is a consequence of Proposition 3.6.

The skeletal filtration on the geometric realization (as in (3.8) and (3.9) writes Tot(X) as the inverse limit of a tower. We define Tot_{n}(X) = sk_{n} |X| ∈ (S^{op})^{op}; arguing as in (5.1) and (5.2) and using Proposition 3.11, we have

(5.3)  Tot_{n}(X) ≅ Hom_{cS}(sk_{n} Δ, X)

and the natural projection Tot_{n}(X) → Tot_{n−1}(X) is induced by the inclusion sk_{n−1} Δ → sk_{n} Δ. Note

(5.4)  Tot(X) ≅ \underleftarrow{lim} Tot_{n}(X).

By Proposition 4.16, sk_{n−1} Δ → sk_{n} Δ is a cofibration in cS, hence, if X ∈ cS is fibrant, SM7 for cS implies

(5.5)  {Tot_{n}(X)}_{n≥0}

is a tower of fibrations. This fact equally follows from Proposition 3.6 This is the total tower of X.

The diagrams (3.8) and (3.10) tell us how Tot_{n}(X) is built from Tot_{n−1}(X). Recalling that for X ∈ cS, M^{n−1}X ≅ L_{n}X, X ∈ s(S^{op}), (3.8) implies that for all X ∈ cS there is a pullback diagram

(5.6)

Tot_{n}(X)  w u  X_{n}^{Δ^{n}} u

Tot_{n−1}(X)  wX_{n}^{∂Δ^{n}} ×_{M_{n−1}^{∂Δ^{n}} X} (M_{n−1}X)^{Δ^{n}}.

Note Tot_{0}(X) ≅ X^{0}. If X is fibrant, the vertical maps in (5.6) are fibrations. Finally, if the cosimplicial object X ∈ cS is codegeneracy-free in the sense that the underlying codegeneracy diagram is the right Kan extension of a discrete diagram {Z^{n}}, then (3.10) implies that there is a pullback diagram in S.

(5.7)

Tot_{n} X  w u  (Z^{n})^{Δ^{n}} u

Tot_{n−1} X  w(Z^{n})^{∂Δ^{n}}.

The rest of the section is devoted to showing that our two examples 4.1 and 4.2 are codegeneracy free in this sense.

Example 5.8. The category of cosimplicial abelian groups is equivalent to the category (s(Ab^{op}))^{op}. Since Ab is abelian, so is Ab^{op}; hence the normalization functor N : s(Ab^{op}) → C_{∗}Ab^{op} defines an equivalence of categories. See Section III.2. In particular every object A ∈ s(Ab^{op}) is naturally degeneracy free on {NA_{s}}. An examination of the proof shows that d_{0} plays no role in the decomposition of A ∈ s(Ab^{op}); specifically if nAb^{op} denotes the category with objects {X_{n}}_{n≥0}, X_{n} ∈ Ab^{op}, then normalization and its adjoint define an equivalence of categories

N : s_{0}(Ab)^{op} → nAb^{op}.

Here s_{0}C denotes the category of functors Δ Δ Δ_{0}^{op} → C where Δ Δ Δ_{0} ⊆ Δ Δ Δ is the subcategory with some objects and morphisms restricted ϕ(0) = 0. These are simplicial objects without d_{0}.

If c^{+}Ab denotes the cosimplicial abelian groups with without d^{0}, these remarks supply an equivalence categories

N^{∗} : c^{+}Ab → nAb

6. The homotopy spectral sequence of a tower of fibrations397

sending A to N^{∗}A with

(N^{∗}A)^{n} = ⋂_{s=0}^{n−1} ker{s^{i} : A^{n} → A^{n−1}}.

In particular any object A ∈ c^{+}Ab is naturally codegeneracy free on N^{∗}A. Now let X ∈ S and R^{•}X the Bousfield-Kan R-resolution of Example 4.1. Then R^{•}X ∈ cS is not a cosimplicial simplicial abelian group; however, when restricted an object R^{•}X ∈ c^{+}S is has a natural structure as an object in c^{+}(sAb). Since products in sAb are products in S, we have that R^{•}X is naturally codegeneracy free on N^{∗}R^{•}X. Then (5.7) gives the existence of a pullback diagram

Tot_{n} R^{•}X w u

Hom(Δ^{n}, N^{∗}R^{n}X) u

Tot_{n−1} R^{•}X w Hom(∂Δ^{n}, N^{∗}R^{n}X).

Example 5.9. Consider the cosimplicial object T^{•}X ∈ cS^{J} constructed in Example 4.2. Examining the definitions we have

(T^{•}X)^{n} = T^{n+1}(X)_{j} = ∏_{j→i_{0}→···→i_{n}} X_{i_{n}}.

It follows that T^{•}X is naturally codegeneracy free as an object in cS^{J} on Z^{n}(X) where

Z^{n}(X) = ∏_{j→i_{0}→i_{1}→···→i_{n}} X_{i_{n}}

where i_{t} → i_{t+1} is not the identity for any t, 0 ≤ t ≤ n − 1. It follows that \underleftarrow{lim}_{J} T^{•}X at cS is also codegeneracy free, since \underleftarrow{lim} preserves products.

6. The homotopy spectral sequence of a tower of fibrations. Given a tower of pointed fibrations, the exact sequences in homotopy give

rise to a spectral sequence. There are a number of problems with this spectral sequence. First, there is the problem that π_{0} and π_{1} are not generally abelian groups; second, the spectral sequence need not converge to anything homotopical; third, the spectral sequence is “fringed” in the sense that E_{∞} can contain extra elements; and, fourth, the E_{2} term may not have a sensible description in terms of homological algebra. We analyze these difficulties in turn. The first two difficulties are addressed in this section, and the last in the next. The fringing effect will be dealt with in Section 8.

So let

⋯  wX_{n}  w^{p_{n}} X_{n−1}  w⋯  wX_{1}  w^{p_{1}} X_{0}  w∗

F_{n} u^{i_{n}}  F_{n−1} u^{i_{n−1}}  F_{1} u^{i_{1}}  F_{0} u^{=}

be a tower of pointed fibrations, with F_{n} the fiber of p_{n}. Applying homotopy to each of the fiber sequences, yields

⋯  w π_{∗}X_{n} w π_{∗}X_{n−1} w i i i k i i i k ⋯  w π_{∗}X_{1} w π_{∗}X_{0} i i i k

π_{∗}F_{n} u  π_{∗}F_{n−1} u  π_{∗}F_{1} u  π_{∗}F_{0}

where each of the dotted maps is of degree −1. This gives an exact couple and hence a spectral sequence. We will be explicit because π_{1} is not abelian and because

π_{1}X_{n−1} → π_{0}F_{n} → π_{0}X_{n} → π_{0}X_{n−1}

is exact only in the sense of pointed sets, with the additional proviso that the action of π_{1}X_{n−1} on π_{0}F_{n} extends to an injection of sets

∗ → π_{0}F_{n}/π_{1}X_{n−1} → π_{0}X_{n}.

Note that π_{0}X_{n} → π_{0}X_{n−1} need not be onto, although the pre-image of the basepoint is π_{0}F_{n}/π_{1}X_{n−1}. (Here and throughout, if a group G acts on a set Z, Z/G denotes the orbits.)

The spectral sequence is now defined as follows. Let

(6.1)  π_{i}X_{n}^{(r)} = Im(π_{i}X_{n+r} → π_{i}X_{n}) ⊆ π_{i}X_{n},

and for t − s ≥ 0, let

(6.2)  Z_{r}^{s,t} = Ker(π_{t−s}F_{s} → π_{t−s}X_{s}/π_{t−s}X_{s}^{(r−1)}).

Then let

(6.3)  B_{r}^{s,t} = Ker(π_{t−s+1}X_{s−1} → π_{t−s+1}X_{s−r−1}).

Then we set

(6.4)  E_{r}^{s,t} = Z_{r}^{s,t}/B_{r}^{s,t}

where this formula must be interpreted: if t − s ≥ 1 and j_{∗} : π_{i+1}X_{n−1} → π_{i}F_{n} is the connecting homomorphism, j_{∗}B_{r}^{s,,t} ⊆ Z_{r}^{s,t} is a normal subgroup and E_{r}^{s,t} is the quotient group, and if t − s = 0, B_{r}^{s,t} acts on Z_{r}^{s,t} and E_{r}^{s,t} is the set of orbits. Define differentials for t − s ≥ 1

(6.5)  d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t+r−1}

by the composite map

(6.6)  E_{r}^{s,t} → π_{t−s}X_{s}^{(r−1)} → E_{r}^{s+r,t+r−1}

where the first map is induced by noticing that if x ∈ Z_{r}^{s,t} ⊆ π_{t−s}F_{s}, then (i_{s})_{∗}x ∈ π_{t−s}X_{s}^{(r−1)} and the second by noticing that if y is in π_{t−s}X_{s}^{(r−1)} we may choose z ∈ π_{s−1}X_{s+r−1} mapping to y and

j_{∗}z ∈ E_{r}^{s+r,t+r−1}

is independent of that choice. Notice that no differential is defined on E_{r}^{s,s} and, indeed, that E_{r}^{s,t} is not defined for t − s < 0.

The following result is left as an exercise:

Lemma 6.7.

1) E_{r}^{s,t} is a pointed set if t − s ≥ 0, a group if t − s ≥ 1, and an abelian group if t − s ≥ 2;

2) The function d_{r} : E_{r}^{s,t} → E_{s}^{s+r,t+r−1} is a homomorphism if t − s ≥ 2 and a map of pointed sets if t − s ≥ 1.

3) The image of d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t+r−1} is in the center of E_{r}^{s+r,t+r−1} if t−s ≥ 2 and if t − s ≥ 1, E_{r+1}^{s,t} = Ker(d_{r})/Im(d_{r}).

4) The map d_{r} : E_{r}^{s,s+1} → E_{r}^{s+r,s+r} extends to an action of the source on the target and E_{r+1}^{s,s} ⊆ E_{r}^{s,s}/E_{r}^{s−r,s−r+1}.

Proof: All these facts follow from the properties of the long exact sequence in homotopy of a fibration. ■

Remark 6.8. The fact that E_{r+1}^{s,s} ⊆ E_{r}^{s,s}/E_{r}^{s−r,s−r+1} and is not necessarily equal to what is meant by saying the spectral sequence is fringed. The failure of equality arises from the fact that π_{0}X_{n} → π_{0}X_{n−1} need not be onto.

The collection {E_{r}^{s,t}; d_{r}}_{r≥1} is the homotopy spectral sequence of the tower of fibrations. It is natural in the tower and we may write {E_{r}^{s,t}X}. We now begin to address what it might converge to.

Notice that if r > s, E_{r+1}^{s,t} ⊂ E_{r}^{s,t}. Define

(6.9)  E_{∞}^{s,t} = \underleftarrow{lim}_{r>s} E_{r}^{s,t} = ⋂_{r>s} E_{r}^{s,t}.

One hopes that E_{∞}^{s,t} may have something to do with the homotopy groups of \underleftarrow{lim} X_{n}. For this, we must first decide how to compute π_{∗}(\underleftarrow{lim} X_{n}) in terms of

the tower of groups \{π_{∗}X_{n}\}. This is done by a \underleftarrow{lim}^{1} exact sequence, once we

have defined \underleftarrow{lim}^{1} for non-abelian groups.

First recall that \underleftarrow{lim}^{1} for a tower of abelian groups A = \{A_{n}\}_{n≥0} is defined by the exact sequence

0 → \underleftarrow{lim} A_{n} → ∏_{n} A_{n} ⟶^{∂} ∏_{n} A_{n} → \underleftarrow{lim}^{1} A_{n} → 0

where ∂(a_{n}) = (a_{n} − f(a_{n+1})), with f : A_{n+1} → A_{n} the maps in the tower. The following is an easy exercise in homological algebra.

Lemma 6.10. The functor \underleftarrow{lim}^{1} : tow(Ab) → Ab is characterized up to natural isomorphism by:

1) If 0 → A → B → C → 0 is a short exact of towers of abelian groups, then there is a six-term exact sequence

0 → \underleftarrow{lim} A → \underleftarrow{lim} B → \underleftarrow{lim} C → \underleftarrow{lim}^{1} A → \underleftarrow{lim}^{1} B → \underleftarrow{lim}^{1} C → 0.

2) If A is any tower so that A_{n+1} → A_{n} is surjective for all n, then \underleftarrow{lim}^{1} A = 0.

We give a similar statement for non-abelian groups. Let G = {G_{n}} be a tower of not necessarily abelian groups. Then the group ∏_{n} G_{n} acts on the set Z = ∏_{n} G_{n} by (g_{n}) ∘ (x_{n}) = (g_{n}x_{n}f(g_{n+1})^{−1}). Then the stabilizer subgroup of e ∈ Z is \underleftarrow{lim} G_{n} and we define \underleftarrow{lim}^{1} G_{n} to be the pointed set of orbits. If G is a tower of abelian groups these definitions agree with those above.

Lemma 6.11. The functor \underleftarrow{lim}^{1} : tow(Gr) → pointed sets is characterized up to natural isomorphism by

1) If ∗ → G_{1} → G_{2} → G_{2} → ∗ is short exact sequence of towers of groups then there is a six term sequence

∗ → \underleftarrow{lim} G_{1} → \underleftarrow{lim} G_{2} → \underleftarrow{lim} G_{3} → \underleftarrow{lim}^{1} G_{1} → \underleftarrow{lim}^{1} G_{2} → \underleftarrow{lim}^{1} G_{3} → ∗

which is exact as groups at the first three terms, as pointed sets at the last three terms, and \underleftarrow{lim} G_{3} acts on \underleftarrow{lim}^{1} G_{1} and there is an induced injection

\underleftarrow{lim}^{1} G_{1}/ \underleftarrow{lim} G_{3} → \underleftarrow{lim}^{1} G_{2}.

2) If G is any tower so that G_{n+1} → G_{n} is surjective for all n, then \underleftarrow{lim}^{1} G_{n} = ∗.

Proof: The proof is the same as in the abelian case. ■

Remark 6.12. It is good to have examples where \underleftarrow{lim}^{1} is non-zero. Let B be any group and let B also be the constant tower on that group. let A_{n} ⊆ B be a descending (A_{n+1} ⊂ A_{n}) sequence of normal subgroups so that ⋂_{n} A_{n} = e. Let A = {A_{n}} be the resulting tower. Then one gets a short exact sequence

(6.13)  ∗ → B → \underleftarrow{lim}_{n}(B/A_{n}) → \underleftarrow{lim}^{1} A → ∗

so that \underleftarrow{lim}^{1} A = ∗ if and only if B is complete in the topology defined by the A_{n}. For example, if B = ℤ and A_{n} = p^{n}ℤ for some prime p, (6.13) becomes

0 → ℤ → ℤ_{p} → ℤ_{p}/ℤ → 0

where ℤ_{p} is the group of p-adic integers, which is uncountable. One could take B = ⨁_{s≥1} ℤ and A_{n} = ⨁_{s≥n} ℤ ⊆ B. Then (6.13) becomes

0 → ⨁_{s≥1} ℤ → ∏_{s≥1} ℤ → ∏_{s≥1} ℤ/ ⨁_{s≥1} ℤ → 0.

Now let X = {X_{n}} be a tower of pointed fibrations.

Proposition 6.14. For every t ≥ 0 there is a sequence

∗ → \underleftarrow{lim}^{1} π_{t+1}X_{n} → π_{t}(\underleftarrow{lim} X_{n}) → \underleftarrow{lim} π_{t}X_{n} → ∗

which is exact as groups if t ≥ 1 and as pointed sets if t = 0.

Proof: Consider the description of the homotopy inverse limit given in Section IX.5. Then one has a homotopy fibration sequence

∏_{n} ΩX_{n} → \underleftarrow{holim} X_{n} → ∏ X_{n}.

A simple calculation with the homotopy exact sequence of this fibration gives the result except possibly for the claim that π_{0} \underleftarrow{holim} X_{n} → \underleftarrow{lim} π_{0}X_{n} is onto. But it is an easy exercise to show that for a tower of fibrations π_{0} \underleftarrow{lim} X_{n} → \underleftarrow{lim} π_{0}X_{n} is onto. ■

We now return to computing π_{∗} \underleftarrow{lim} X_{n} by the homotopy spectral sequence. Because of E_{r}^{s,t} given in (6.1)–(6.4) and E_{∞}^{s,t} in (6.9) we can think of E_{∞}^{s,t} as an equivalence class of elements x ∈ π_{t−s}F_{s} so that for each r ≥ 0 there is an element y_{r} ∈ π_{t−s}X_{s+r} mapping to i_{∗}x ∈ π_{t−s}X_{s}. (Notice we do not say y_{r} maps to y_{r−1}; see Example 6.17.2 below.) If z ∈ π_{t−s} \underleftarrow{lim} X_{n} maps to zero in π_{t−s}X_{s−1}, then one gets such an equivalence class. To make this precise, let

G_{s}π_{i}X = Im(π_{i}X → π_{i}X_{s})

and

(6.15)  e_{∞}^{s,t} = ker{G_{s}π_{t−s}X_{s} → G_{s}π_{t−s}X_{s−1}}.

If y ∈ e_{∞}^{s,t}, let x ∈ π_{t−s}F_{s} be any element so that i_{∗}x = y.

Lemma 6.16.

1) The assignment y ↦ x induces a well-defined monomorphism e_{∞}^{s,t} → E_{∞}^{s,st}.

2) The inclusion maps G_{s}π_{∗}X → π_{∗}X_{s} induce an isomorphism

\underleftarrow{lim}_{s} G_{s}π_{∗}X ⟶^{≅} \underleftarrow{lim}_{s} π_{∗}X_{s}.

Proof: This is an exercise in unraveling the definitions. ■

Notice that any element of \underleftarrow{lim}^{1} π_{∗}X maps to zero in all G_{s}π_{∗}X, so will not be detected in any e_{∞}^{s,t}. It also turns out that e_{∞}^{s,t} → E_{∞}^{s,t} need not be an isomorphism.

Here are two examples of the potential difficulties.

Example 6.17.

1) Consider the tower of abelian groups

⋯ ⟶^{×p} ℤ ⟶^{×p} ℤ ⟶^{×p} ℤ ⟶^{×p} ℤ

where p ∈ ℤ is a prime number. The resulting tower of Eilenberg-MacLane spaces

→ K(ℤ, n) ⟶^{×p} K(ℤ, n) ⟶^{×p} ⋯ ⟶^{×p} K(ℤ, n)

can be transformed into a tower of fibrations. One easily calculates that E_{1}^{0,n} = ℤ and E_{1}^{s,s+n−1} = ℤ/pℤ and E_{1}^{s,t} = 0 otherwise. Next E_{r}^{0,n} = p^{r−1}ℤ ⊆ ℤ = E_{1}^{0,n} and d_{r} : E_{r}^{0,n} → E_{r}^{r,r+n−1} = ℤ/pℤ is onto.. Thus E_{∞}^{s,t} = 0 for all t − s ≥ 0. However

π_{n−1} \underleftarrow{lim} X_{s} = ℤ_{p}/ℤ ≅ \underleftarrow{lim}_{r}^{1}(p^{r}ℤ) ≠ 0.

Note that in this case \underleftarrow{lim}^{1} E_{r}^{0,n} ≠ 0.

2) Consider the tower of abelian groups, where a is the addition map,

→ ⨁_{s≥3} ℤ → ⨁_{s≥2} ℤ → ⨁_{s≥1} ℤ ⟶^{a} ℤ

and let {X_{n}} be the resulting tower of fibrations obtained by applying K(·, n) and the converting tower of fibrations. Now

π_{n−1} \underleftarrow{lim} X_{m} ≅ lim^{1}π_{n}X_{m} ≅ ∏_{s≥1} ℤ/ ⨁_{s≥1} ℤ

and π_{k} \underleftarrow{lim} X_{m} = 0 otherwise. Then

E_{1}^{s,t} = \{ \begin{matrix}ℤ & s = 0, t = n \\ ⨁_{i≥1} ℤ & s = 1, t = n + 1 \\ ℤ & s ≥ 2, t = s + n − 1 \end{matrix}

and d_{r} : E_{r}^{1,n+1} → E_{r}^{r+1,n+r} is onto. Thus E_{∞}^{0,n} = ℤ and E_{∞}^{s,t} = 0 otherwise. Thus e_{∞}^{0,n} ≠ E_{∞}^{0,n}. Note also \underleftarrow{lim}^{1} E_{r}^{1,n+1} ≠ 0. ■

The following definition now seems appropriate.

Definition 6.18. The spectral sequence of the pointed tower of fibrations {X_{n}} converges completely if

1) \underleftarrow{lim}^{1} π_{i}X_{n} = ∗ for i ≥ 1, and

2) e_{∞}^{s,t} ≅ E_{∞}^{s,t} for all (s, t) with t − s ≥ 1.

When 6.18.2 holds, the spectral sequence effectively computes \underleftarrow{lim} π_{∗}X_{n}, subject to the fringing effect when t − s = 0, and when 6.18.1 holds, we have π_{∗}(\underleftarrow{lim} X_{n}) ≅ \underleftarrow{lim} π_{∗}X_{n}. One would like to be able to decide when one has complete convergence from knowledge of the spectral sequence, rather than using knowledge of π_{∗} \underleftarrow{lim} X_{n}, which Definition 6.18 demands — after all one is trying to compute π_{∗} \underleftarrow{lim} X_{n}. The next lemma is the best possible result. Compare the examples of 6.17.

Lemma 6.19 (Complete Convergence Lemma). The spectral sequence of the pointed tower of fibrations {X_{n}} converges completely if and only if

\underleftarrow{lim}_{r}^{1}E_{r}^{s,t} = ∗, t − s ≥ 1.

This will be proved below, after some examples and preliminaries.

Corollary 6.20. Suppose that for each integer n ≥ 0 there are only finitely many s, so that E_{2}^{s,s+n} ≠ ∗. Then the spectral sequence converges completely.

More generally we have

Corollary 6.21. Suppose that for each integer n ≥ 0 and each integer s, there is an integer N so that

E_{∞}^{s,s+n} ≅ E_{N}^{s,s+n}.

Then the spectral sequence converges completely.

Proof: For \underleftarrow{lim}^{1} E_{r}^{s,t} = ∗. See 6.22.1 below. ■

The phenomenon encountered in Corollary 6.21 is called Mittag-Leffler convergence. The wider implications are explored in [14, p.264].

As a preliminary to proving the complete convergence lemma, we discuss derived towers of group. Suppose {G_{n}} is a tower of groups. Define G_{n}^{(r)} = Im{G_{n+r} → G_{n}}. We already encountered π_{∗}X_{s}^{(r)} = (π_{∗}X_{s})^{(r)} in (6.1)–(6.4). There are maps

G_{n}^{(r)} → G_{n−1}^{(r+1)} → G_{n−1}^{(r)}

and one has two new towers of groups

{G_{n}^{(r)}}_{n≥0} {G_{n}^{(r)}}_{r≥0}.

Lemma 6.22.

1) The inclusions G_{n}^{(r)} ⊆ G_{n} induce isomorphisms

\underleftarrow{lim}_{n} G_{n}^{(r)} ≅ \underleftarrow{lim}_{n} G_{n} and \underleftarrow{lim}^{1}_{n} G_{n}^{(r)} ≅ \underleftarrow{lim}^{1}_{n} G_{n}.

2) There is a natural isomorphism

\underleftarrow{lim}_{n} \underleftarrow{lim}_{r} G_{n}^{(r)} ≅ \underleftarrow{lim}_{n} G_{n}

and a natural short exact sequence

∗ → \underleftarrow{lim}^{1}_{n} \underleftarrow{lim}_{r} G_{n}^{(r)} → \underleftarrow{lim}^{1} G_{n} → \underleftarrow{lim}_{n} \underleftarrow{lim}^{1}_{r} G_{n}^{(r)} → ∗.

Proof: For 1) note that the functor {G_{n}}_{n≥0} ↦ {G_{n}^{(r)}}_{n≥0} is exact and takes tower of surjections to surjections. Since \underleftarrow{lim}_{n} G_{n}^{(n)} ≅ \underleftarrow{lim}_{n} G_{n} by inspection, the

result follows from the characterization of \underleftarrow{lim}^{1} given in Lemma 6.11: the functor

\{G_{n}\} → \underleftarrow{lim}^{1} G_{n}^{(r)}

satisfies the requirements there.

For 2), note that if the groups were abelian, what we would have is a degenerate form of a composite functor spectral sequence. In the non-abelian case, it is better to proceed topologically. For fixed r define a tower Y_{r} = {Y_{r,n}} with Y_{r,n} = BG_{n}^{(r)}. There are maps of towers Y_{r+1} → Y_{r}; convert the resulting tower of towers into a tower of fibrations in tow(S_{∗}):

⋯ → Z_{2} → Z_{1} → Z_{0}.

Thus Y_{0} → Z_{0} is a weak equivalence in tow(S_{∗}) with Z_{0} fibrant (i.e., a tower of fibrations) and Y_{r} → Z_{r} → Z_{r−1} factors Y_{r} → Y_{r−1} → Z_{r−1} as a weak equivalence followed by a fibrations. Then

⋯ → \underleftarrow{lim}_{n} Z_{2} → \underleftarrow{lim}_{n} Z_{1} → \underleftarrow{lim}_{n} Z_{0}

is a tower of fibrations and for all r, π_{1} \underleftarrow{lim}_{n} Z_{r} ≅ \underleftarrow{lim}_{n} G_{n} and π_{0} \underleftarrow{lim}_{n} Z_{r} ≅ \underleftarrow{lim}_{n} ^{1}G_{r}, by part 1). Hence

π_{1}(\underleftarrow{lim}_{r} \underleftarrow{lim}_{n} Z_{r,n}) ≅ \underleftarrow{lim}_{n} G_{n}
π_{0}(\underleftarrow{lim}_{r} \underleftarrow{lim}_{n} Z_{r,n}) ≅ \underleftarrow{lim}^{1}_{n} G_{n}.

Now one takes limits in the other direction: for fixed n, {Z_{r,n}}_{r≥0} is a tower fibration, so

π_{1} \underleftarrow{lim}_{r} Z_{r,n} ≅ \underleftarrow{lim}_{r} G_{n}^{(r)}
π_{0} \underleftarrow{lim}_{r} Z_{r,n} ≅ \underleftarrow{lim}^{1}_{r} G_{n}^{(r)}

Now we calculate π_{∗} \underleftarrow{lim}_{n} \underleftarrow{lim}_{r} Z_{r,n} and use that limits commute to finish the result. ■ Proof of the Complete Convergence Lemma 6.19: We prove only that if \underleftarrow{lim}^{1}_{r} E_{r}^{s,t} = ∗, then we have complete convergence. The reverse implication is

left as an exercise—it is a matter of reversing the arguments.

For each pair of positive integers r ≥ n, let us define K_{n}^{r} by the exact sequence of groups

(6.23)  ∗ → K_{n}^{r} → π_{t−s}X_{s+n}^{(r−n)} → π_{t−s}X_{s}^{(r)} → ∗.

If n = 1 and r > s, K_{n}^{r} = E_{r}^{s+1,t+1}; therefore, if r > s+n, there is a composition series for K_{n}^{r} with all of the horizontal maps surjections

\begin{matrix}K_{n}^{r}  w & K_{n−1}^{r+1}  w & ⋯  w & K_{1}^{r+n−1} \\ E_{r}^{s+n,t+n} u & E_{r+1}^{s+n−1,t+n−1} u & E_{r+n−1}^{s+1,t+1} \end{matrix}

An induction argument shows \underleftarrow{lim}_{r}^{1} K_{n}^{r} = ∗ and that each of the maps \underleftarrow{lim}_{r} K_{n}^{r} →

\underleftarrow{lim}_{r} K_{n−1}^{r+1} is onto. Taking the limit over r in (6.23) yields a short exact sequence

∗ → \underleftarrow{lim}_{r} K_{n}^{r} → \underleftarrow{lim}_{r} π_{t−s}X_{s+n}^{(r−n)} → \underleftarrow{lim}_{r} π_{t−s}X_{s}^{(r)} → ∗.

Now we take the limit over n. By the previous lemma

\underleftarrow{lim}_{n} \underleftarrow{lim}_{r} π_{t−s}X_{s+n}^{(r−n)} ≅ \underleftarrow{lim}_{n} π_{t−s}X_{n},

and since \underleftarrow{lim}^{1} vanishes on towers of surjections, we get a short exact sequence

(6.24)  ∗ → \underleftarrow{lim}_{n} \underleftarrow{lim}_{r} K_{n}^{r} → \underleftarrow{lim}_{n} π_{t−s}X_{n} → \underleftarrow{lim}_{r} π_{t−s}X_{s}^{(r)} → ∗

and an injection

(6.25)  \underleftarrow{lim}^{1}_{n} \underleftarrow{lim}_{r} π_{t−s}X_{s+n}^{(r−n)} → \underleftarrow{lim}^{1}_{n} π_{t−s}X_{s}^{(r)}.

In particular, (6.24) yields that the natural map \underleftarrow{lim}_{n} π_{t−s}X_{n} → \underleftarrow{lim}_{r} π_{t−s}X_{s}^{(r)} is a surjection. Thus G_{s}π_{t−s} \underleftarrow{lim} X_{n} ≅ \underleftarrow{lim}_{r} π_{t−s}X_{s}^{(r)} and, by induction, e_{∞}^{s,t} = E_{∞}^{s,t}.

Also, since n does not appear in \underleftarrow{lim}_{n}^{1} π_{t−s}X_{s}^{(r)}, this term is trivial, so we have that the term \underleftarrow{lim}_{n}^{1} \underleftarrow{lim}_{r} π_{t−s}X_{s+n}^{(r−n)} = ∗. If \underleftarrow{lim}_{n} \underleftarrow{lim}_{r}^{1} π_{t−s}X_{s+n}^{(r−n)} = ∗, the previous lemma implies \underleftarrow{lim}^{1} π_{t−s}X_{n} = ∗ and we’ll be done. So we’ll close by showing \underleftarrow{lim}_{r}^{1} π_{t−s}X_{s+n}^{(r−n)} = ∗.

As with K_{n}^{r}, there is a composition series for π_{t−s}X_{s}^{(r)}: if r > s

\begin{matrix}π_{t−s}X_{s}^{(r)}  w & π_{t−s}X_{s−1}^{(r+1)}  w & ⋯  w & π_{t−s}X_{0}^{(r+s)} \\ E_{r+1}^{s,t} u & E_{r+2}^{s−1,t−1} u & E_{r+s+1}^{0,t−s} u \end{matrix}

A similar inductive argument as with K_{n}^{r}, shows \underleftarrow{lim}_{r}^{1} π_{t−s}X_{s}^{(r)} = ∗. ■

It is worth pointing out that even without convergence one has a comparison result. Proposition 6.26. Suppose f : X = {X_{n}} → Y = {Y_{n}} is a map of towers of pointed fibrations and suppose there is a (finite) N ≥ 1 so that

f_{∗} : E_{N}^{s,t}X → E_{N}^{s,t}Y

is an isomorphism for all (s, t). If E_{r}^{s,s}X = ∗ for all s, then \underleftarrow{lim} X_{n} → \underleftarrow{lim} Y_{n} is a weak equivalence of connected spaces.

Proof: Note that f_{∗} : E_{r}^{s,t}X → E_{r}^{s,t}Y is an isomorphism for all r ≥ N. Then induction on n and the exact sequence

E_{r}^{s+t,t+1} → π_{t−s}X_{s+1}^{(r−1)} → π_{t−s}X_{s}^{(r)}

(see (6.23)) implies π_{k}X_{n}^{(n)} → π_{k}Y_{n}^{(r)} is an isomorphism for all r ≥ N−1. Finally, the hypotheses on E_{r}^{s,s} imply π_{0}X_{n}^{(r)} = ∗ = π_{0}Y_{n}^{(r)} and π_{1}X_{n}^{(r)} → π_{1}X_{n−1}^{(r)} is onto. Thus the result follows from Lemmas 6.22.1, 6.11, and Proposition 6.14.■

7. The homotopy spectral sequence of a cosimplicial space. If X is a pointed cosimplicial space, then the tower {Tot_{n} X}_{n≥0} has a homo-

topy spectral sequence. In this section, we work out the E_{1} and E_{2} terms of the spectral sequence, and give an extended example—the original Bousfield-Kan spectral sequence, which is a kind of unstable Adams spectral sequence.

Fix a fibrant pointed cosimplicial space X, and form the tower {Tot_{n} X}_{n≥0}, where Tot_{n} X = Hom_{cS}(sk_{n} Δ, X). The constant map is the basepoint of Tot_{n} X and by the diagram (5.6), the fiber of Tot_{n} X → Tot_{n−1} X is Ω^{n}N^{n}X, where N^{n}X is the fiber of s : X^{n} → M^{n−1}X. Thus the E_{1} term of the associate spectral sequence is, for t − s ≥ 0,

E_{1}^{s,t} = π_{t−s}Ω^{s}N^{s}X ≅ π_{t}N^{s}X.

If G is a cosimplicial (not necessarily abelian) group, define

N^{s}G = ⋂_{i=0}^{j−1} ker{s^{i} : G^{s} → G^{s−1}}.

If G is a cosimplicial abelian group, then NG = {N^{s}G, Σ(−1)^{i}d^{i}} becomes a cochain complex. Define Nπ_{0}X = π_{0}X^{0}.

Lemma 7.1. For a fibrant pointed cosimplicial space X, there is a natural isomorphism

E_{1}^{s,t} ≅ π_{t}N^{s}X ≅ N^{s}π_{t}X, t − s ≥ 0

Proof: First assume t ≥ 2. The cases t = 0, 1 will be handled at the end. We use the pullback square

(7.2)  \begin{matrix}M^{n−1,k+1}X  w u \\ X^{n−1} u \end{matrix} ; \begin{matrix}M^{n−1,k}X  w \\ M^{n−2,k}X \end{matrix}

7. The homotopy spectral sequence of a cosimplicial space 409

of Proposition 1.25, and a double induction on the following two statements. Notice that M^{n,k} is a functor defined as cC where C is any category with finite limits.

H(n, k): The natural map π_{t}M^{n−1,k}X → M_{n−1,k}π_{t}X is an isomorphism. I(n, k): Let N^{n,k}X be the fiber of s : X^{n} → M_{n−1,k}X. Then

0 → π_{t}N^{n,k}X → π_{t}X^{n} → π_{t}M_{n−1,k}X → 0

is short exact.

The result follows from setting n = k = s in I(n, k) and H(n, k).

First notice that I(0, 0) is trivial since M^{−1,0}X = ∗. Also H(n, 0) and I(n, 0) are trivial since M^{n−1,0}X = ∗ and M^{n−1,0}π_{t}X = 0.

Next, H(n, k), H(n−1, k), and I(n−1, k) yield H(n, k +1), since π_{t}(·) applied to (7.2) yields a pullback diagram

\begin{matrix}π_{t}M^{n−1,k+1}X  w u \\ π_{t}X_{n−1} u \end{matrix} ; \begin{matrix}M^{n−1,k}π_{t}X  w \\ M_{n−2,k}π_{t}X. \end{matrix}

Finally, H(n, k) ⇒ I(n, k) because π_{t}X^{n} → M^{n−1,k}π_{t}X is surjective. Indeed, as in Example 5.8,

M^{n−1,k}π_{t}X ≅ ⨁_{ϕ:n→m} N^{m}π_{t}X

where ϕ runs over the objects of M_{n,k}.

For the cases t = 0 and 1 of Lemma 7.1, note that if t = 0, s = 0, so one need only check π_{0}N^{0}X = π_{0}X^{0} = Nπ^{0}X which is true since N^{0}X = X^{0}. If t = 1, s = 0 or 1. If s = 0, one has π_{1}N^{0}X = π_{1}X^{0} as required. If s = 1,

∗ → π_{1}N^{1}X → π_{1}X^{1} ⟶^{s^{0}} π_{1}X^{0} → ∗ is exact, since s^{0}d^{0} = 1 and X^{0} = M^{0}X. ■

We use this calculation as the input for the calculation of the E_{2} term. In fact, we wish to claim that E_{2}^{s,t} = π^{s}π_{t}X, but we need to take some care with the definitions, especially for t = 1.

If A is a cosimplicial abelian group, we can make A into a cochain complex

⋯ → A_{2} ⟶^{∂} A_{1} ⟶^{∂} A_{0}

with ∂ = Σ(−1)^{i}d^{i}. Then π^{s}A = H^{s}(A). This is equivalent to computing H^{s}N^{∗}A, where N^{∗}A is the normalized cochain complex with

N^{∗}A = {N^{s}A, Σ(−1)^{i}d^{i}}.

One has

(7.3)  π^{s}A = H^{s}A ≅ H^{s}N^{∗}A.

If G is a cosimplicial (not necessarily abelian) group, one still has π^{0}G and π^{1}G. The group π^{0}G is the equalizer of d^{0}, d^{1}:

(7.4)  π^{0}G → G^{0} ⇉_{d^{1}}^{d^{0}} G^{1}.

The pointed set π^{1}G is defined by the formula

(7.5)  π^{1}G = Z^{1}G/G^{0}

where Z^{1}G is the pointed set of cocycles

Z^{1}G = {x ∈ G^{1}|(d^{2}x)(d^{1}x)^{−1}(d^{0}x) = e}

and G^{0} acts on Z^{1}G by

g ∘ x = d^{0}g · x(d^{1}g)^{−1}.

Notice that as G is a cosimplicial abelian group, the two definitions of π^{1}G agree. One can also normalize in the non-abelian case; indeed, the situation is already normalized in the sense that if x ∈ Z^{1}G, then x ∈ N^{1}G because

e = s^{0}[(d^{2}x)(d^{1}x)^{−1}(d^{0}x)] = d^{1}s^{0}x

so

e = s^{0}d^{1}s^{0}x = s^{0}x.

Thus N^{1}G ∩ Z^{1}G = Z^{1}G and

(7.6)  π^{1}G = (N^{1}G ∩ Z^{1}G)/G^{0}.

If X is a cosimplicial set we define π^{0}X to be the equalizer of d^{0}, d^{1} : X^{0} → X^{1}.

Proposition 7.7. Let X be a pointed fibrant cosimplicial space. Then the E_{2} term of the spectral sequence of the tower of fibrations {Tot_{n} X} can be calculated by natural isomorphism

E_{2}^{s,t} ≅ π^{s}π_{t}X, t − s ≥ 0.

Because of this result, one often writes

π^{s}π_{t}X ⇒ π_{t−s} Tot X

for the homotopy spectral sequence arising from a pointed cosimplicial space. The double arrow is not meant to prejudice anyone for or against convergence of the spectral sequence; for this, one must appeal to the results of Section 6.

Before proving Proposition 7.7, we line up notation and ideas. We are interested in the tower {Tot_{n} X}, where the tower maps

Tot_{s} X = Hom_{cS}(sk_{s} Δ, X) → Hom_{cS}(sk_{s−1} Δ, X) = Tot_{s−1} X

are induced by the inclusion sk_{s−1} Δ → sk_{s} Δ. Thus the fiber at the constant map is the space of pointed maps

Hom_{cS_{∗}}(sk_{s} Δ/ sk_{s−1} Δ, X).

If f : sk_{s} Δ/ sk_{s−1} Δ → X is a pointed map one gets a diagram

\begin{matrix}Δ^{s}/∂Δ^{s}  w^{f} u \\ X^{s} u \end{matrix} ; \begin{matrix}∗  w \\ M_{s}X \end{matrix}

and hence a map of spaces

Hom_{cS_{∗}}(sk_{s} Δ/ sk_{s−1} Δ, X) → Hom_{∗}(Δ^{s}/∂Δ^{s}, N_{s}X).

Diagram (5.6) that this map is an isomorphism. To calculate E_{2}^{s,t} = Z_{2}^{s,t}/B_{2}^{s,t} (see (6.1)–(6.4)) we calculate Z_{2}^{s,t}, B_{2}^{s,t}, and the action of B_{2}^{s,t}. These are given in Lemmas 7.8, 7.14, and 7.15 below. Proposition 7.7 will follow.

Recall that for t ≥ 2, we may define

∂ = ∑_{i=0}^{s+1} d^{i} : π_{t}N^{s}X → π_{t}N^{s+1}X.

This defines the sub-group ker(∂) ⊆ π_{t}N^{s}X, t ≥ 2. If t = 1, define

ker(∂) ⊆ π_{1}N^{1}X

by

ker(∂) = Z^{1}π_{1}N^{∗}X

(see (7.4)–(7.5)). This is a pointed set. If t = 0 or 1 define Ker(∂) ⊆ π_{t}N^{0}X − π_{t}X^{0} to be the equalizer of

d^{0}, d^{1} : π_{t}X^{0} → π_{t}X^{1}.

If t = 1, ker(∂) is a group; if t = 0, ker(∂) is a pointed set.

Lemma 7.8. For all t − s ≥ 0, there is a natural isomorphism

Z_{2}^{s,t} ≅ ker(∂) ⊆ π_{t}N^{s}X.

Proof: First take s ≥ 1. According to (6.2),

f ∈ π_{t−s}Hom_{cS_{∗}}(sk_{s} Δ/ sk_{s−1} Δ, X)

is in Z_{2}^{s,t} if and only if there is a lift

Tot_{s+1} X u

S^{t−s} w^{f}^{´´´´´´´´´´´´´´´´´´´´´}ˆ

Hom_{cS_{∗}}(sk_{s} Δ/ sk_{s−1} Δ, X) w

Tot_{s} X.

Here we have confused f with a representative. We can do this as all objects are fibrant. Thus, by adjointness, f ∈ Z_{2}^{s,t} if and only if there is an extension (with + denoting a disjoint basepoint)

S^{t−s} ∧ (sk_{s} Δ)_{+}  w u  S^{t−s} ∧ (sk_{s} Δ/ sk_{s−1} Δ)  w^{g} X

S^{t−s} ∧ (sk_{s+1} Δ)_{+} ^{´´´´´´´´´´´´´´´´´´}ˆ

in the category of pointed cosimplicial spaces. Here g is adjoint to f. By Diagram (5.6) and adjointnetss, f ∈ Z_{2}^{s,t} if and only if there is a lift

(7.9)

S^{t−s} ∧ (∂Δ^{s+1})_{+}  w u  S^{t−s} ∧ (∂Δ^{s+1}/ sk_{s−1} Δ^{s+1})  w^{g} X^{s+1} u

S^{t−s} ∧ (Δ^{s+1})_{+} w^{´´´´´´´´´´´´´´´´´´´´´}ˆ M_{s}X

in the category of pointed spaces. Let ι : Δ^{s}/∂Δ^{s} → Δ^{s}/∂Δ^{s} be the identity. Then the collapse map ∂Δ^{s+1} → ∂Δ^{s+1}/ sk_{s−1} Δ^{s−1} represents, when we choose 0 ∈ ∂Δ^{s+1} as the basepoint,

∑_{i=0}^{s+1}(−1)^{i}d^{i}ι ∈ π_{s}(∂Δ^{s+1}/ sk_{s−1} Δ^{s+1}) if s + 1 > 2

and

[d^{2}ι][d^{1}ι]^{−1}[d^{0}ι] ∈ π_{1}(∂Δ^{2}/ sk_{0} Δ^{2}) if s + 1 = 2.

Thus if the original map f was adjoint to α ∈ π_{t}N^{s}X, the morphism across the top of (7.9) represents ∂α (if s + 1 > 2) and [d^{2}α][d^{1}α]^{−1}[d^{0}α] (if s + 1 = 2). Thus f ∈ Z_{2}^{s,t} if and only if the morphism across the top is null-homotopic; that is, if and only if α ∈ ker(∂).

If s = 0, one is asking whether there is a lift

Tot_{1} X u

S^{t}^{5 5 5 55 6} w_{f} Tot_{0} X = X^{0}

or, again by (5.6) and adjointness, whether the following diagram can be completed

S^{t} ∧ (∂Δ^{1})_{+}  w u  X_{1} u

S^{t} ∧ (Δ^{1})_{+}^{O O O O P}wX_{0}.

This happens if and only if d^{0}f = d^{1}f ∈ π_{t}X^{1}. ■

In order to calculate the boundaries B_{2}^{s,t} and the quotient E_{2}^{s,t}, we must be explicit about the action of the fundamental group on the components of the fiber. This requires a short digression.

Let C be any simplicial model category. Suppose A is pointed cofibrant and A → B is a pointed cofibration. Then if B/A is define the pushout diagram

\begin{matrix}A  w u \\ Bu \end{matrix} ; \begin{matrix}∗  w \\ B/A, \end{matrix}

the object B/A is pointed and there is a pointed fibration sequence. If A ∈ C is pointed cofibrant and K is a pointed simplicial set

A ∧ K = A ⊗ K/A ⊗ ∗ ∨ ∗ ⊗ K.

If X ∈ C is pointed and fibrant, there is a pointed fibration sequence

Hom_{∗}(B/A, X) → Hom_{∗}(B, X) → Hom_{∗}(A, X)

where we take the constant to be the basepoint. We describe the action of π_{t}Hom_{∗}(A, X) ≅ [A ∧ S^{t}, X]_{∗} on π_{t−s}Hom_{∗}(B/A, X) ≅ [B/A ∧ S^{t−1}, X]_{∗}.

One model for the pointed cone on A is C^{1}A = A∧Δ_{+}^{1}/A∧{1}_{+}. For another model let [0, 2] ∈ S be the simplicial set with two non-degenerate one simplices [0, 1] and [1, 2] and d^{0}[0, 1] = {1} = d^{1}[1, 2]. (We can think of Δ^{1} as [0, 1].) Then C^{2}A = A ∧ [0, 2]_{+}/A ∧ {2}_{+} is another model for the pointed cone on A. Think of A ⊆ C^{1}A or A ⊆ C^{2}A as the subspace A ∧ {0}_{+}. Then there are weak equivalences

B ∪_{A} C^{2}A ≃ B/A ≃ B ∪_{A} C^{1}A

and an isomorphism C^{1}A/A ≅ A ∧ S^{1}. Thus one gets a map

(7.10)  ψ : B ∪_{A} C^{2}A → B ⊔ C^{2}A/A ∧ {1}_{+} = A ∧ S^{1} ∨ B ∪_{A} C^{1}A

If α ∈ π_{t}Hom_{∗}(A, X) and β ∈ π_{t−1}Hom_{∗}(B/A, X) are represented by f : A ∧ S^{t} → X and g : (B∪_{A}C^{1}A)∧S^{t−1} → X respectively α·β ∈ π_{t−1}Hom_{∗}(B/A, X) is represented by

(7.11) (B ∪_{A} C^{2}A) ∧ S^{t−1} ⟶^{ψ∧1} (A ∧ S^{1}) ∧ S^{t−1} ∨ (B ∪_{A} C^{1}A) ∧ S^{t−1} ⟶^{f∨g} X.

If t ≥ 2, π_{t−1}Hom_{∗}(B/A, X) is a group, the action is given by α · β = (j_{∗}α)β where j_{∗} is the connecting homomorphism of the long exact sequence defined by j_{∗}α = α · e.

If A → B is not a pointed cofibration, we still have a fibration sequence

Hom_{∗}(B/A, X) → Hom(B, X) → Hom(A, X).

We can calculate the action π_{t}Hom(A, X) on π_{t−1}Hom_{∗}(B/A, X) via the device of the disjoint basepoint. Define A_{+} = A ⊔ ∗. Then Hom(A, X) ≅ Hom_{∗}(A_{+}, X), so if α ∈ π_{t}Hom(A, X) is represented by f : A_{+} ∧ S^{t} → X and β ∈ π_{t−1}Hom(B/A, X) is represented by g : (B ∪_{A} C^{1}A) ∧ S^{t−1} → X, then α · β is represented by

(7.12) (B ∪_{A} C^{2}A) ∧ S^{t−1} → (A_{+} ∧ S^{1}) ∧ S^{t−1} ∨ (B ∪_{A} C^{1}A) ∧ S^{t−1} ⟶^{f∨g} X.

The disjoint basepoint is crucial in the following.

Example 7.13. Let X ∈ S be pointed and fibrant and consider the cofibration ∂Δ^{1} → Δ^{1}. Then, the associated fiber sequence is

ΩX → Hom(Δ^{1}, X) → X × X

and the action of π_{t}(X × X) ≅ π_{t}X × π_{t}X on π_{t−1}ΩX ≅ π_{t}X is given by

(α_{1}, α_{2}) · β = α_{1}βα_{2}^{−1}.  ■

We are now in a position to calculate with B_{2}^{s,t}. Recall from (6.1)–(6.4), that B_{2}^{s,t} = Ker{π_{t−s+1} Tot_{s−1} X → π_{t−s+1} Tot_{s−2} X}, and, in order to calculate E_{2}^{s,t} we need to know the action of B_{2}^{s,t} on

Z_{2}^{s,t} ⊆ π_{t−s}Ω^{s−1}N^{s−1}X.

We separate out the cases s = 1 and s > 1.

Lemma 7.14. If s = 1, B_{2}^{s,t} = π_{t} Tot_{0} X ≅ π_{t}X and if α ∈ π_{t}X and β ∈ π_{t−1}Ω^{1}N^{1}X ≅ π_{t}N^{1}X, then

α · β = (d^{0}α)β(d^{1}α)^{−1}.

Proof: One is concerned with the fibration sequence in cS

Hom_{∗}(sk_{1} Δ/ sk_{0} Δ, X) → Hom(sk_{1} Δ, X) → Hom(sk_{0} Δ, X).

Thus if α ∈ π_{t}Hom(sk_{0} Δ, X) ≅ π_{t}X and β ∈ π_{t−1}Hom_{∗}(sk_{1} Δ/ sk_{0} Δ, X) ≅ π_{t−1}Ω^{1}N^{1}X, then α · β is represented by (see (7.12))

(sk_{1} Δ ∪_{sk_{0} Δ} C^{2} sk_{0} Δ) ∧ S^{t−1} u

(sk_{0} Δ)_{+} ∧ S^{1} ∧ S^{t−1} ∨ (sk_{1} Δ ∪_{sk_{0} Δ} C^{1} sk_{0} Δ) ∧ S^{t−1} w β ∨ α X.

Specializing to cosimplicial degree 1, we have

Δ^{1}∪_{∂Δ^{1}} C^{2}∂Δ^{1}∧S^{t−1} → (∂Δ^{1})_{+}∧S^{1}∧S^{t−1}∨(Δ^{1}∪_{∂Δ^{1}} C^{2}∂Δ^{1})∧S^{t−1} ⟶^{β^{1}∨α^{1}} X^{1}.

However α^{1} = d^{0}α ∨ d^{1}α : (∂Δ^{1})_{+} ∧ S^{t} → X^{1} and β^{1} is a representative for β ∈ π_{t−1}ΩN^{1}X. Hence Example 7.13 implies the result. ■

Lemma 7.15. If s > 1, then

i_{∗}π_{t−s+1}Ω^{s−1}N^{s−1}X = B_{2}^{s,t} ⊆ π_{t−s+1} Tot_{s−1} X

and j_{∗}B_{2}^{s,t} ≅ Im(∑(−1)^{i}d^{i}) ⊆ π_{t−s}Ω^{s}N^{s}X ≅ π_{t}N^{s}X.

Proof: The equality ι_{∗}π_{∗}Ω^{s−1}N^{s−1}X = B_{2}^{s,∗} follows from the long exact of the fibration

Ω^{s−1}N^{s−1}X → Tot_{s−1} X → Tot_{s−2} X.

For the calculation of j_{∗} : π_{∗} Tot_{s−1} X → π_{∗−1}Ω^{s}N^{s}X we are concerned with the fibration sequence

Hom_{∗}(sk_{s} Δ/ sk_{s−1} Δ, X) → Hom(sk_{s} Δ, X) → Hom(sk_{s−1} Δ, X).

If α ∈ π_{t−s+1}Hom(sk_{s−1} Δ, X), then j_{∗}α is the composite (where f represents α)

(sk_{s} Δ ∪_{sk_{s−1} Δ} C^{2} sk_{s−1} Δ) ∧ S^{t−s} → (sk_{s−1} Δ_{+}) ∧ S^{t−s+1} ⟶^{f} X.

At cosimplicial degree s, this yields

(7.16)  Δ^{s} ∪_{∂Δ^{s}} C^{2}∂Δ^{s} ∧ S^{t−s} → (∂Δ_{+}^{s}) ∧ S^{t−s+1} ⟶^{f^{s}} X^{s}.

If α ∈ B_{2}^{s,t} then f^{s} factors as

(7.17)  \begin{matrix}(∂Δ_{+}^{s}) ∧ S^{t−s+1}  w u \\ X^{s} \end{matrix} ; (∂Δ_{+}^{s}/ sk_{s−2} Δ_{+}^{s}) ∧ S^{t−s+1}^{```}ˆ

Now Δ^{s} ∪_{∂Δ^{s}} C^{2}∂Δ^{s} ≃ Δ^{s}/∂Δ^{s} ≃ S^{s}, ∂Δ^{s} ≃ S^{s−1} and if i : Δ^{s−1} → Δ^{s−1} is the identity then the collapse map

∂Δ^{s} → ∂Δ^{s}/ sk_{s−2} Δ^{s}

represents Σ(−1)^{i}d^{i}i. Combining these facts with (7.16) and (7.17) yields the result. ■

Example 7.18. Let X be a pointed space and let X → \overline{𝔽}_{p}^{•}X be the Bousfield-Kan resolution of X. Then the total space Tot(\overline{𝔽}_{p}^{•}X) = X_{p}^{∧} is called the Bousfield-Kan p-completion of X, and π_{∗}X_{p}^{∧} may be calculated by the spectral sequence

π^{s}π_{t}\overline{𝔽}^{•}X ⇒ π_{t−s}X_{p}^{∧}.

The point of this example is to rewrite the E_{2} term in terms of homological algebra. We assume the reader is familiar with the category CA of unstable coalgebras over the Steenrod algebra. See [71, Sec. 1]. If X is a space, H_{∗}X ⊂ CA. Let CA_{∗} be the category of augmented (or pointed) coalgebras in CA; thus an object in CA_{∗} is an unstable coalgebra C over the Steenrod algebra equipped with a morphism of such coalgebras 𝔽_{p} → C. The homology of a pointed space is in CA_{∗}.

First notice that if V is any simplicial vector space, then, choosing 0 ∈ V as the basepoint,

(7.19)  π_{t}V ≅ hom_{CA_{∗}}(H_{∗}S^{t}, H_{∗}V).

One way to see this is to note that there is a (non-canonical) weak equivalence V ≃ ⨁_{α} K(ℤ/p, n_{α}) for some set of non-negative integers {n_{α}}, and to use the calculations of Serre and Cartan on H_{∗}K(ℤ/p, n_{α}). Now the fact that V is a simplicial abelian group implies (7.19) holds for any choice of basepoint. Since a choice of basepoint for X induces a basepoint for \overline{𝔽}_{p}^{•}X,

(7.20)  π^{s}π_{t}\overline{𝔽}_{p}^{•}X ≅ π^{s} hom_{CA_{∗}}(H_{∗}S^{t}, H_{∗}\overline{𝔽}_{p}^{•}X).

Now, the forgetful functor CA → n𝔽_{p} to the category of graded vector spaces has a right adjoint G. The functor G is characterized by the fact that it commutes with products and filtered colimits, and by the fact that if W ∈ n𝔽_{p} is dimension 1 concentrated in degree n, then

(7.21)  G(W) ≅ H_{∗}K(ℤ/p, n).

(Note that G is the functor of a cotriple on n𝔽_{p} and CA is the category of coalgebras over G.)

Let \overline{G} : CA → CA be the composite of G and the forgetful functor. Then \overline{G} is a triple on CA, so if C ∈ CA, we have an augmented cosimplicial object (see Example 4.1).

(7.22)  C → \overline{G}^{•}C.

This is a resolution in the sense that as an augmented cosimplicial graded vector space it has a retraction, so π^{∗}\overline{G}^{•}C ≅ C via the augmentation. Thus we define, for C ∈ CA_{∗},

(7.23)  Ext_{CA_{∗}}^{s}(H_{∗}S^{t}, C) = π^{s} hom_{CA_{∗}}(H_{∗}S^{t}, \overline{G}^{•}C).

Now if X is a space, H_{∗}\overline{𝔽}X ≅ \overline{G}(H_{∗}X). This follows from (7.19), (7.21) and the properties of \overline{G}. Thus combining (7.20) with (7.23) we have

π^{s}π_{t}\overline{𝔽}_{p}^{•}X ≅ Ext_{CA_{∗}}(H_{∗}S^{t}, H_{∗}X)

and the homotopy spectral sequence of \overline{𝔽}_{p}^{•}X is the Bousfield-Kan spectral sequence

Ext_{CA_{∗}}^{s}(H_{∗}S^{t}, H_{∗}X) ⇒ π_{t−s}X_{p}^{∧}.

It is an unstable Adams spectral sequence.

The reader might object that, for pointed coalgebras, one should define the Ext groups differently. If C ∈ CA_{∗}, define the “coaugmentation ideal” by

JC = coker{𝔽_{p} → C}.

Then J : CA_{∗} → n𝔽_{p} has a right adjoint G_{∗} with G_{∗}V = GV augmented by G(0) ≅ 𝔽_{p} → G(V). Let \overline{G}_{∗} = G_{∗} ∘ J : CA_{∗} → CA_{∗} and one might demand that the Ext object be

π^{s} hom_{CA_{∗}}(H_{∗}S^{t}, \overline{G}_{∗}^{•}C).

However, a bicomplex argument with

hom_{CA_{∗}}(H_{∗}S^{t}, \overline{G}^{•}\overline{G}_{∗}C)

shows that this definition agrees with the previous one (7.23); furthermore, (7.23) offers additional flexibility with basepoints.

We finish this section by constructing a spectral sequence for computing the homotopy groups of a pointed homotopy inverse limit.

We begin with a discussion of the derived functors of limit on I diagrams of abelian groups. Let Ab^{I} be the category of these diagrams. Then the functor G_{i} : Ab^{I} → Ab with G_{i}A = Ai has both a left and a right adjoint; in particular, the right adjoint is given by

(7.24)  (T_{i}B)_{j} = ∏_{j→i} B.

If B ∈ Ab is injective, then T_{i}B is injective in Ab^{I} and this implies Ab^{I} has enough injectives; indeed, if we embed each i in an injective abelian group B_{i}, then A → ∏_{i} T_{i}B_{i} is an embedding of A into an injective in Ab^{I}. We define the functors \underleftarrow{lim}^{s} = \underleftarrow{lim}_{I}^{s} to be the right derived functors of \underleftarrow{lim}_{I}. Since limit is left

exact, \underleftarrow{lim} = \underleftarrow{lim}^{0}. The standard observation is that one need not go all the way to injectives to compute the derived functors. As with sheaf cohomology one can resolve by “flabby” objects rather than injective ones. The crucial result is:

Lemma 7.25. Let B be an abelian group. Then for all i we have \underleftarrow{lim} T_{i}B ≅ B and \underleftarrow{lim}^{s} T_{i}B = 0 if s > 0.

Proof: To prove \underleftarrow{lim} T_{i}B ≅ B, let A ∈ Ab^{I} be a constant diagram on an abelian group A. Then

hom_{Ab}(A, \underleftarrow{lim} T_{i}B) ≅ hom_{Ab}^{I} (A, T_{i}B) ≅ hom_{Ab}(A, B).

To obtain the vanishing of the higher derived functors, let B → J^{∗} be an injective resolution of B. Then (see (7.24)), T_{i}B → T_{i}J^{∗} is an injective resolution of T_{i}B. Applying \underleftarrow{lim} to this resolution, one recovers B → J^{∗}, which is acyclic by construction. ■

We use this to build a cosimplicial resolution of A ∈ Ab^{I} suitable for computing \underleftarrow{lim}^{s}. For our category I let dI be the associated discrete category; i.e., dI has the same objects as I but no non-identity morphisms. There is a functor dI → I, inducing a forgetful functor Ab^{I} → Ab^{dI}. The category Ab^{dI} has objects indexed families of abelian groups A = {A_{i}}_{i∈I}. This forgetful functor has a right adjoint given by

T A = ∏_{i} T_{i}A_{i}.

We also write T : Ab^{I} → Ab^{I} be the composite functor. This is the same functor T as in Example 4.2. Then T is the functor of a triple on Ab^{I} and one, therefore, obtains a natural augmented cosimplicial object in Ab^{I}

(7.26)  A → T^{·}A.

If one forgets to Ab^{dI} this cosimplicial object has a retraction and so is acyclic. This implies it is acyclic in Ab^{I} and Lemma 7.25 implies:

Lemma 7.27. For all A ∈ Ab^{I} there is a natural isomorphism

\underleftarrow{lim}^{s} A ≅ π^{s} \underleftarrow{lim} T^{·}A.

By analogy we use this construction to define \underleftarrow{lim}^{1} G where G is an I diagram of non-abelian groups. The reader will have noticed that there is nothing special about abelian group in the construction of the cosimplicial object A → T^{·}A of (7.26). Indeed, if C is any category and C ∈ C^{I} is an I diagram, we obtain a cosimplicial object C → T^{·}C. In particular if G is an I-diagram of (not necessarily abelian) groups, one has a cosimplicial I-diagram of groups

G → T^{·}G

and we define

(7.28)  \underleftarrow{lim}^{1} G = π^{1} \underleftarrow{lim} T^{·}G.

Thus \underleftarrow{lim}^{1} G is a pointed set and is a group of cycles modulo a group of boundaries. See (7.5). At this point we have potentially two different definitions of \underleftarrow{lim}^{1} G_{n} for a tower of groups, one from (7.28) and the other from Lemma 6.11; this difficulty will be resolved by Proposition 7.30 below.

Now let X ∈ S_{∗}^{I} be an I-diagram of pointed spaces. We wish to calculate with \underleftarrow{holim} X. We use the cosimplicial space T^{·}X of Example 4.2. We will see in Section IX.5 that Tot T^{·}X is naturally weakly equivalent \underleftarrow{holim}_{I} X. Note that if t ≥ 0,

π_{t}T^{·}X ≅ T^{·}π_{t}X.

Hence the spectral sequence

π^{s}π_{t}T^{·}X ⟹ π_{t−s} Tot T^{·}X

is naturally isomorphic to a spectral sequence

(7.29)  \underleftarrow{lim}^{s}_{I} π_{t}X ⟹ π_{t−s} \underleftarrow{holim}_{I} X.

This follows from Lemma 7.25 and a simple calculation in the case t = s = 0.

Finally, we show that our two potentially different definitions of \underleftarrow{lim}^{1} G_{n} agree up to natural isomorphism.

For the next few paragraphs, let {G_{n}} be a tower of groups. We define three functors. First, there is \underleftarrow{lim}^{1} G_{n}, as in Lemma 6.7. Then there is \underline{lim}^{1}G_{n}, which

is a temporary name for \underleftarrow{lim}^{1} G_{n} as defined in (7.28). Finally, define a functor by

\{G_{n}\} ↦ π_{0} \underleftarrow{holim} BG_{n}.

Proposition 7.30. There are natural isomorphisms

π_{0} \underleftarrow{holim} BG_{n} ≅ \underline{lim}^{1}G_{n} ≅ \underleftarrow{lim}^{1} G_{n}.

Proof: We will show π_{0} \underleftarrow{holim} BG_{n} ≅ \underline{lim}^{1}G_{n} and π_{0} \underleftarrow{holim} BG_{n} ≅ \underleftarrow{lim}^{1} G_{n}. The second of these isomorphisms was already proved in Proposition 6.14. For the first, we have that

π_{0} \underleftarrow{holim} BG_{n} ≅ π_{0} Tot \underleftarrow{lim} T^{·}\{BG_{n}\}.

By (5.7) and Example 5.9 we have an equality

π_{0} Tot \underleftarrow{lim} T^{·}\{BG_{n}\} = π_{0} Tot_{1} \underleftarrow{lim} T^{·}\{BG_{n}\}.

Since π_{0} Tot_{0} \underleftarrow{lim} T^{·}\{BG_{n}\} = π_{0} ∏_{n} BG_{n} = ∗, we have

π_{0} Tot \underleftarrow{lim} T^{·}\{BG_{n}\} = E_{2}^{1,1} ≅ \underline{lim}^{1}G_{n}

by (6.1)–(6.4). ■

8. Obstruction theory.

The purpose of this section is to develop a small amount of the theory of Bousfield’s paper [10], and to discuss to some extent the meaning of the elements on the fringe of the spectral sequence of a cosimplicial space. We close the section with an extended example intended to make the theory concrete.

In order to make the exposition flow more smoothly we will make the following assumption: we have a fibrant cosimplicial space X with the property that for all n ≥ 0 and every choice of basepoint v ∈ X^{n}, Whitehead products in π_{∗}(X^{n}, v) vanish. Thus π_{1}(X^{n}, v) is abelian and the action of π_{1}(X^{n}, v) on π_{m}(X^{n}, v), m ≥ 1, is trivial. We do not assume π_{0}(X^{n}, v) = ∗, as one of our main examples will be X = Hom(Y, R^{•}Z), where Y ∈ S, and R^{•}Z is the R-resolution of Z ∈ S, as in Example 4.1.

We wish to address the following question: when is an element x ∈ π^{0}π_{0}X in the image of π_{0} Tot X → π_{0} Tot_{0} X = π_{0}X. We will try to lift x in successive stages up the Tot tower. The following three results are our goal.

Proposition 8.1. An element x ∈ π_{0}X^{0} = π_{0} Tot_{0} X lifts to π_{0} Tot_{1} X if and only if x ∈ π^{0}π_{0}X.

An element x ∈ π^{0}π_{0}X determines elements x_{n} ∈ π_{0}X^{n} by

x_{n} = d_{∗}^{i_{n}} ⋯ d_{∗}^{i_{0}}X

where d_{∗}^{i_{t}} : π_{0}X^{t} → π_{0}X^{t+1} is any of the coface operators. We write π_{t}(X, x) for the resulting cosimplicial group. Recall

N^{s}π_{t}(X, x) = ⋂_{i=0}^{s−1} ker{s_{∗}^{i} : π_{t}(X^{s}, x_{s}) → π_{t}(X^{s−1}, x_{s−1})}

and Z^{s}π_{t}(X, s) is the group of cocycles in degree s in N^{∗}π_{t}(X, x).

Proposition 8.2. Let n ≥ 1, y ∈ π_{0} Tot_{n} X, and x ∈ π^{0}π_{0}X the image of y under the projection π_{0} Tot_{n} X → π^{0}π_{0}X ⊆ π_{0} Tot_{0} X. Then there is an element

γ(y) ∈ Z^{n+1}π_{n}(X, x)

so that y lifts to π_{0} Tot_{n+1} X if and only if γ_{n}(x) = 0.

Next recall π^{s}π_{t}(X, x) = H^{s}N^{∗}π_{t}(X, x).

Proposition 8.3. Let n ≥ 1 and suppose z ∈ π_{0} Tot_{n−1} X lifts to π_{0} Tot_{n} X. Let x ∈ π^{0}π_{0}X be the image of z under the projection π_{0} Tot_{n−1} X → π^{0}π_{0}X ⊆ π_{0} Tot_{0} X. Then there is an element

ϵ(z) ∈ π^{n+1}π_{n}(X, x)

so that z lifts to π_{0} Tot_{n+1} X if and only if ϵ(z) = 0.

We will prove these three results in order. Before starting we write down the obvious corollary. Recall our assumptions on X made in the second paragraph of this section.

Proposition 8.4. Let x ∈ π^{0}π_{0}X, and suppose π^{n+1}π_{n}(X, x) = 0 for n ≥ 1. Then x lifts to π_{0} Tot X.

Proof: This follows from Propositions 8.1 and 8.3 since

π_{0} Tot(X) → \underleftarrow{lim} π_{0} Tot_{n} X

is surjective. ■

We begin the proofs of Proposition 8.1 to 8.3. To pass from π_{0} Tot_{n} X to π_{0} Tot_{n−1} X we will have to investigate the pullback square

Tot_{n+1} X  w u  Hom(Δ^{n+1}, X^{n+1}) u

Tot_{n} X  wHom(Δ^{n+1}, M^{n}X) ×_{Hom(∂Δ^{n+1},M^{n}X)} Hom(∂Δ^{n+1}, X^{n+1}).

Let us make an abbreviation

Hom(i_{n+1}, s_{n+1})

for the lower right-hand corner of this square. The following technical lemma is crucial to calculations. Let

Iπ_{0} Tot_{n} X = Im{π_{0} Tot_{n} X → π_{0}Hom(i_{n+1}, s_{n+1})}.

Lemma 8.5. The composition of the map Iπ_{0} Tot_{n} X → π_{0}Hom(i_{n+1}, s_{n+1}) with

π_{0}Hom(i_{n+1}, s_{n+1}) u

π_{0}Hom(Δ^{n+1}, M^{n}X) ×_{π_{0}Hom(∂Δ^{n+1},M^{n}X)} π_{0}Hom(∂Δ^{n+1}, X^{n+1})

is injective.

Proof: Take f ∈ π_{0}Hom(i_{n+1}, s_{n+1}), and let f_{1} ∈ Hom(∂Δ^{n+1}, X^{n+1}), g ∈ Hom(Δ^{n+1}, M^{n}X), and f_{0} ∈ Hom(∂Δ^{n+1}, M^{n}X) be the induced vertices. If we can show

π_{1}(Hom(∂Δ^{n+1}, X^{n+1}), f_{1}) → π_{1}(Hom(∂Δ^{n+1}, M^{n}X); f_{0})

is onto for all f ∈ Iπ_{0} Tot_{n} X, the result will follows from the Barratt-Puppe sequence.

Let v_{0} = d_{n+1} ⋯ d_{1}c_{n} ∈ (Δ^{n+1})_{0} be the basepoint, then let x_{1} = f_{1}(v_{0}) and x_{0} = f_{0}(v_{0}) be the basepoints of X^{n+1} and M^{n}X

Evaluating at the basepoint gives a diagram where the vertical maps are fibrations and the horizontal maps give a fiber sequence

Hom_{∗}(∂Δ^{n+1}, X^{n+1}) w u Hom(∂Δ^{n+1}, X^{n+1}) w u X^{n+1} u

Hom_{∗}(∂Δ^{n+1}, M^{n}X) w Hom(∂Δ^{n+1}, M^{n}X) w_{p} M^{n}X.

The ∗ means pointed maps. This leads to short exact sequences in homotopy, m ≥ 1, (8.6)

0u  0u

π_{m}(Hom_{∗}(∂Δ^{n+1}, X^{n+1}); f_{1}) w^{Hom(∂Δ^{n+1},s)_{∗}} u π_{m}(Hom_{∗}(∂Δ^{n+1}, M^{n}X); f_{0}) u

π_{m}(Hom(∂Δ^{n+1}, X^{n+1}); f_{1}) w u π_{m}(Hom(∂Δ^{n+1}, M^{n}X); f_{0}) u

π_{m}(X^{n+1}  w u  π_{m}(M^{n}X, x_{0}) u

0  0

The right sequence is exact because of the existence of g : Δ^{n+1} → M^{n}X. Thus f_{0} is contractible and the constant map section M^{n}X → Hom(∂Δ^{n+1}, M^{n}X) of p actually shows the right is split exact. The left sequence is short exact because Whitehead products vanish in X^{n+1} : α : Δ^{m}/∂Δ^{m} → X^{n+1} lifts, up to homotopy, to Hom(∂Δ^{n+1}, X^{n+1}) with basepoint f_{1} if and only if the following factors

Δ^{m}/∂Δ^{m} ∨ ∂Δ^{n+1} u w^{α∨f_{1}} X^{n+1}.

Δ^{m}/∂Δ^{m} × ∂Δ^{n+1}^{´´´}ˆ

To finish the proof we use that f (which determines f_{1}, f_{0}, x_{1}, and x_{0}) arises from an element of y ∈ π_{0} Tot_{n} X. By projection, y determines an element x ∈ π^{0}π_{0}X and hence a unique element z_{k} ∈ π_{0}X^{k}, via any composition of coface maps, and, as above, we write π_{m}(X, x) for the resulting cosimplicial abelian group. Now z_{n+1} is the class of x_{1} in π_{0}X^{n+1} and the argument of Lemma 7.1 implies

π_{m}(M^{n}X, x_{0}) ≅ M^{n}π_{m}(X, z).

It follows that, in diagram (8.6), that s_{∗} is surjective. Also, since f_{0} is homotopic to the constant map, if we let \overline{f}_{1} : ∂Δ^{n+1} → X^{n+1} be a lift of f_{0} which is homotopic to the constant map, then

Hom(∂Δ^{n+1}, s)_{∗} : π_{m}Hom_{∗}(∂Δ^{n+1}, X^{n+1}); \overline{f}_{1}) ≅ π_{m+n}(X; x_{1})
→ M^{n}π_{m+n}(X, x) ≅ π_{m}Hom_{∗}(∂Δ^{n+1}, M_{n}X)

is onto. Now \overline{f}_{1} need not be homotopic to the constant map. However, if we let N^{n+1}X be the fiber of s : X^{n+1} → M^{n}X at X_{0}, then Hom_{∗}(∂Δ^{n+1}, N^{n+1}X) acts on Hom_{∗}(∂Δ^{n+1}, X^{n+1}) over Hom_{∗}(∂Δ^{n+1}, M^{n}X). Since sf_{1} = f_{0}, there is a vertex v ∈ Hom_{∗}(∂Δ^{n+1}, N^{n+1}) so that operation by v which is a homotopy equivalence) induces a diagram

Hom_{∗}(∂Δ^{n+1}, X^{n+1}) w^{v}_{N NN P} Hom_{∗}(∂Δ^{n+1}, X^{n+1}) ‚ ‚ ‚ “

Hom_{∗}(∂Δ^{n+1}, M^{n}X)

and so that \overline{f}_{1} = vf_{1}. Thus Hom(∂Δ^{n+1}, s)_{∗}, as in (8.6), is surjective, and the result follows. ■

Proof of Proposition 8.1: There is, by (5.6), a pullback diagram (since Tot_{0} X = X^{0})

Tot_{1} X  w u  Hom(Δ^{1}, X^{1}) u

X^{0} wHom(i_{1}, s_{1}).

Hence

π_{0} Tot_{1} X → π_{0}X^{0} ×_{π_{0}Hom(i_{1},s_{1})} π_{0}Hom(Δ^{1}, X^{1})

is surjective. By Lemma 8.5, the target of this map on π_{0} is isomorphic to the pullback of

π_{0}X^{1} u_{q} ; \begin{matrix}π_{0}X^{0} w^{p} \\ π_{0}X^{0} ×_{π_{0}X^{0}×π_{0}X^{0}} π_{0}X^{1} × π_{0}X^{1} \end{matrix}

where p and q are respectively

\begin{aligned}p(x) & = (x, d^{0}x, d^{1}x) \\ q(y) & = (s^{0}y, y, y). \end{aligned}

But this pullback is exactly π^{0}π_{0}X. ■

Before proving Proposition 8.2, we define the obstruction cocycle γ(y) ∈ Z^{n+1}π_{n}(X, x). We will use heavily that all Whitehead products vanish in X^{s}, ≥ 0, so that we needn’t worry about the difference between unbased and based maps in computing homotopy groups. Components do matter, however.

The element y ∈ π_{0} Tot_{n}(X) is determined by a map

f : sk_{n} Δ → X

and, hence, determines a unique homotopy class of maps

f^{n+1} : ∂Δ^{n+1} = sk_{n} Δ^{n+1} → X^{n+1}.

Then

(8.7)  γ(y) = [f^{n+1}] ∈ π_{n}(X^{n+1}, x).

Since there are commutative diagrams

\begin{matrix}∂Δ^{n+1} u_{s^{i}}  w^{f^{n+1}} \\ X^{n+1} u_{s^{i}} \end{matrix} ; \begin{matrix}Δ^{n}  w^{f^{n}} \\ X^{n} \end{matrix}

we have that γ(y) ∈ N^{n+1}π_{n}(X, x). Since there are commutative diagrams

∂Δ^{n+1} u_{d^{i}} w^{f^{n+1}} X^{n+1} u_{d^{i}}

sk_{n} Δ^{n+2} w^{f^{n+2}} X^{n+2}

and since π_{n}(X^{n+1}, x) is abelian for n ≥ 1, we will have γ(y) ∈ Z^{n+1}π_{n}(X, x) if

0 = ∑_{i=0}^{n+1}(−1)^{i}d_{∗}^{i}a_{n} ∈ H_{n}(sk_{n+2} Δ^{n+2}).

Here a_{n} ∈ H_{n}(∂Δ^{n+1}) is the generator. But this is a standard calculation.

In the next argument we will need the following technical lemma. Recall that X is fibrant in cS and all Whitehead products vanish.

Lemma 8.8. Let x ∈ π^{0}π_{0}X and b ∈ X^{n} any vertex with [b] = x ∈ π_{0}X^{n}. Let s : X^{n} → M^{n−1}X be the canonical map and v = s(b). Then for k ≥ 1,

π_{k}(M^{n−1}X, v) ≅ M^{n−1}π_{k}(X, x).

Furthermore, if N^{n}(X, x) is the fiber of s at v. Then for k ≥ 1,

π_{k}(N^{n}(X, x), b) ≅ N^{n}π_{n}(X, x).

Proof: The argument of Lemma 7.1 goes through verbatim, provided one is explicit about basepoints. Notice the argument extends to π_{1} because of the assumption that π_{1}(X, x) is abelian. ■

Proof of Proposition 8.2: If y lifts to π_{0} Tot_{n+1} X, then there is a factoring

sk_{n} Δ u w^{f} X.

sk_{n+1} Δ^{h hh j}

In particular, f_{n+1} factors as ∂Δ^{n+1} → Δ^{n+1} → X^{n+1} and γ(y) = 0. The converse is a little more difficult. We use the pullback square

Tot_{n+1} X  w u  Hom(Δ^{n+1}, X^{n+1}) u

Tot_{n} X wHom(i_{n+1}, s_{n+1}).

Then

π_{0} Tot_{n+1} X → π_{0} Tot_{n} X ×_{π_{0}Hom(i_{n+1},s_{n+1})} π_{0}Hom(Δ^{n+1}, X^{n+1})

is surjective. By Lemma 8.5, the target of this map is isomorphic to the pullback of

(8.9)

π_{0}Hom(Δ^{n+1}, X) u_{p}

π_{0} Tot_{n} X w^{k} π_{0}Hom(Δ^{n+1},M^{n}X)×_{π0Hom(∂Δn+1,MnX)}π_{0}Hom(∂Δ^{n+1},X^{n+1})

We need to show that k(y) is in the image of p.

To calculate k(y), first note that for any (fibrant) space Y

π_{0}Hom(∂Δ^{n+1}, Y) ≅ ⨆_{x∈π_{0}Y} π_{n}^{fr}(Y, x),

where π_{n}^{fr}(Y, x) denotes (unbased) homotopy classes of maps ∂Δ^{n+1} → Y landing in the component x. If y ∈ Y is a vertex with [y] = x, then there is a quotient

π_{n}(Y, y) → π_{n}^{fr}(Y, x)

which is an isomorphism if Whitehead products vanish in π_{∗}(Y, y). This later will hold if Y = X^{n+1}. If x ∈ π_{0}M_{n}X is a component arising from an element of π^{0}π_{0}X it also holds as (with [y] = x)

π_{n}(M_{n}X, y) = M^{n}π_{n}(X, x), n ≥ 1.

For each x ∈ π_{0}X^{n+1}, let

K_{x} = {α ∈ π_{n}(X, y) | (s_{n+1})_{∗}α = 0 ∈ π_{n}^{fr}(M_{n}X, f(y))}.

Then, since π_{0}Hom(Δ^{n+1}, M^{n}X) ≅ π_{0}M^{n}X

π_{0}Hom(Δ^{n+1}, M^{n}X) ×_{π_{0}Hom(∂Δ^{n+1}M^{n}X)} π_{0}Hom(∂Δ^{n+1}, X^{n+1})

≅ ⨆_{x∈π_{0}X^{n+1}} K_{x}.

Now let us calculate k(y), where k is as in (8.9). Since y is a map

f : sk_{n} Δ → X^{n+1}

we have k(y) is the class of f^{n+1} : sk_{n} Δ^{n+1} → X^{n+1}; that is

k(y) = γ(y) ∈ N^{n+1}π_{n}(X, x) = K_{x} ⊆ ⨆_{x∈π_{0}X^{n+1}} K_{x}.

The image of

p : π_{0}X^{n+1} → ⨆_{x∈π_{0}X^{n+1}} K_{x}

since x ∈ π_{0}X^{n+1} to 0 ∈ K_{x}. Hence k(y) is in the image of p if (and only if) γ(y) = 0. ■

To prove Proposition 8.3, we first define the obstruction class

ϵ(z) ∈ π^{n+1}π_{n}(X, x).

In fact, let y ∈ π_{0} Tot_{n} X be any lift of z and let

(8.10)  ϵ(z) = [γ(y)] ∈ π^{n+1}π_{n}(X, x)

where γ(y) ∈ Z^{n+1}π_{n}(X, x) is the obstruction cocycle of Proposition 8.2, defined in (8.7).

Lemma 8.11. The class ϵ(z) is independent of the choice of lift y ∈ π_{0} Tot_{n} X.

Proof: Let y_{0} and y_{1} be two lifts of z. Then we choose representative

f_{0}, f_{1} : sk_{n} Δ → X

of y_{0}, y_{1} respectively. Since f_{0}|_{sk_{n−1} Δ} and f_{1}|_{sk_{n−1} Δ} both represent z, they are homotopic; since sk_{n−1} Δ → sk_{n} Δ is a cofibration we may, by altering f_{1} by a homotopy, assume

f_{0}|_{sk_{n−1} Δ} = f_{1}|_{sk_{n−1} Δ}.

Thus we obtain a map

f_{0} ⊔ f_{1} : sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ → X.

In cosimplicial degree n

[sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ]^{n} ≅ Δ^{n} ∪_{∂Δ^{n}} Δ^{n}

which is a simplicial model for S^{n}. Let a_{1}, a_{0} be the two obvious n-simplices of Δ^{n} ∪_{∂Δ^{n}} Δ^{n} and let the class [a_{1} −a_{0}] ∈ H_{n}(Δ^{n} ∪_{∂Δ^{n}} Δ^{n}) define the orientation. Let d(f_{0}, f_{1}) ∈ π_{n}(X^{n}, x) be the resulting class.

In degree n − 1,

[sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ]^{n−1} ≅ Δ^{n−1},

so

d(f_{0}, f_{1}) ∈ N^{n}π_{n}(X, x).

In degree n + 1

[sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ]^{n+1} ≅ ∂Δ^{n+1} ∪_{sk_{n−1} Δ^{n+1}} ∂Δ^{n+1}.

Let i_{t} : ∂Δ^{n+1} → [sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ]^{n+1} be the two obvious inclusions and [i_{1}] − [i_{0}] ∈ H^{n}(sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ) the resulting homology class. Then

∂[q_{1} − a_{0}] = [i_{1}] − [i_{0}].

Since (f_{1} ⊔ f_{0}) ∘ i_{t} = γ(y_{t}), it follows from naturality that

∂d(f_{0}, f_{1}) = γ(y_{1}) − γ(y_{0}).  ■

To complete the proof of Proposition 8.3, we must argue that if ϵ(z) = 0 there exists a lift y of z to π_{0} Tot_{n} X so that γ(y) = 0. This follows from the next result.

Lemma 8.12. Let z ∈ π_{0} Tot_{n−1} X and y_{0} ∈ π_{0} Tot_{n} X a left of z. Let ω ∈ N^{n}π_{n}(X, x), where x ∈ π^{0}π_{0}X is the image of z. Then there exists y_{1} ∈ π_{1} Tot_{n} X a left of z so that

γ(y_{1}) = γ(y_{0}) + ∂ω.

Remark 8.13. Note that Proposition 8.3 immediately follows, for if ϵ(z) = 0, γ(y) = ∂ω for some ω, and we may choose y_{1} so that

γ(y_{1}) = γ(y) + ∂(−ω)

and apply Proposition 8.2. ■

To prove 8.12 we use a sequence of lemmas. Let

i_{n}^{n} : ∂Δ \left[{n \atop n}\right] → Δ \left[{n \atop n}\right]

be the cofibration of (4.13). Then Δ \left[{n \atop n}\right] = p_{n}^{∗}Δ^{n}, so

hom_{cS}(Δ \left[{n \atop n}\right] , X) ≅ hom_{S}(Δ^{n}, X^{n}).

Let ϕ_{n} : Δ \left[{n \atop n}\right] → sk_{n} Δ be adjoint to the identity. Also, since

∂Δ \left[{n \atop n}\right] = ρ_{n}^{∗}∂Δ^{n} ∪_{μ_{n−1}^{∗}∂Δ^{n}} μ_{n−1}^{∗}Δ^{n}

we have that hom_{cS}(∂Δ \left[{n \atop n}\right] , X) is isomorphic to the pullback

hom_{S}(Δ^{n}, M^{n−1}X) ×_{hom_{S}(∂Δ^{n},M^{n−1},X)} hom_{S}(∂Δ^{n}, X^{n}).

Let ψ_{n} : ∂Δ \left[{n \atop n}\right] → sk_{n−1} Δ be adjoint to the element (s_{n}, 1_{∂Δ^{n}}) of

hom_{S}(Δ^{n}, M^{n−1} sk_{n−1} Δ) ×_{hom_{S}(∂Δ^{n},M^{n−1} sk_{n−1} Δ)} hom_{S}(∂Δ^{n}, ∂Δ^{n}).

Here we are using that

M^{n−1} sk_{n−1} Δ ≅ M^{n−1} sk_{n} Δ

and s_{n} : Δ^{n} → M^{n−1} sk_{n} Δ is the canonical map. The following diagram commutes

(8.14)

∂Δ \left[{n \atop n}\right]  w^{ψ_{n}} u_{i_{n}^{n}}  sk_{n−1} Δ u

Δ \left[{n \atop n}\right] w_{ϕ_{n}} sk_{n} Δ.

Lemma 8.15. The diagram (8.14) is a pushout diagram.

Proof: It suffices to show that after applying hom_{sC}(·, X), X ∈ sC, one obtains a natural pullback diagram. This follows from (5.6). ■

Now let i_{j} : sk_{n} Δ → sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ be the inclusion of the left functor. Let j_{n} : Δ \left[{n \atop n}\right] → sk_{n} Δ∪_{sk_{n−1} Δ}sk_{n} Δ be adjoint to the inclusion Δ^{n} → Δ^{n}∪_{∂Δ^{n}} Δ^{n} of the right factor.

Lemma 8.16. There is a pushout square

∂Δ \left[{n \atop n}\right]  w^{ϕ_{n}∘i_{n}^{n}} u_{i_{n}^{n}}  sk_{n} Δ u_{i_{0}}

Δ \left[{n \atop n}\right] w^{j_{n}} sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ.

Proof: This follows from Lemma 8.15 and the formula

(B_{2} ∪_{A} B_{1}) ∪_{B_{1}} (B_{1} ∪_{A} B_{2}) = B_{2} ∪_{A} (B_{1} ∪_{A} B_{2}).  ■

This brings us to our final technical argument. Proof of Lemma 8.12: Let f_{0} : sk_{n} Δ → X represent y_{0} and let

ω ∈ N^{n}π_{n}(X, x)_{•} ⊆ π_{n}(X^{n}, x).

Then

f_{0} : (sk_{n} Δ)^{n} = Δ^{n} → X^{n}

can be extended to a map

g : (sk_{n} Δ ∪_{sk_{n−1}Δ} sk_{n} Δ)^{n} = Δ^{n} ∪_{∂Δ^{n}} Δ^{n} → X^{n}

representing ω simply because

π_{0}f_{0} : π_{0}Δ^{n} = {pt} → π_{0}X^{n}

hits x. Let i_{1} : Δ^{n} → Δ^{n} ∪_{∂Δ^{n}} Δ^{n} be the inclusion of the right hand factor and let h : Δ \left[{n \atop n}\right] → X be adjoint to g ∘ i_{1} : Δ^{n} → X^{n}. The claim is that we may choose h so that the following diagram commutes

(8.17)

∂Δ \left[{n \atop n}\right]  w^{ϕ_{n}∘i_{n}^{n}} u_{i_{n}^{n}}  sk_{n} Δ u_{f_{0}}

Δ \left[{n \atop n}\right] w_{h} X.

Assuming this for the moment, we would then have, by Lemma 8.16, a map

g^{∗} : sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ → X

extending g. Then if f_{1} is the composite

sk_{n} Δ ⟶^{i_{1}} sk_{n} Δ ∪_{sk_{n−1} Δ} sk_{n} Δ ⟶^{g^{∗}} X,

the argument for Lemma 8.11 and the fact that g represents ω finishes the argument; in fact, if y_{1} = f_{j} ∈ π_{0} Tot_{n} X, then

d(f_{0}, f_{1}) = ω

and ∂d(f_{0}, f_{1}) = γ(y_{1}) − γ(y_{0}).

To prove the claim about diagram (8.17) we compute in

hom(Δ^{n}, M^{n−1}X) ×_{hom(∂Δ^{n},M^{n−1}X)} hom(∂Δ^{n}, X^{n}).

In this pullback, f_{0} ∘ i_{n}^{n} ∘ ϕ_{n} is the pair (α, f_{0}^{n}) where α is the composite

Δ^{n} ⟶^{s} M^{n−1} sk_{n} Δ ⟶^{M^{n−1}f_{0}} M^{n−1}X

and f_{0}^{n} is short-hand for the composite

∂Δ^{n} ⊆ (sk_{n} Δ)^{n} ⟶^{f_{0}^{n}} X^{n}.

On the other hand h ∘ i_{n}^{n} is the pair (β, g^{n} ∘ i_{1}) where β is the composite

Δ^{n} ⟶^{i_{1}} Δ^{n} ∪_{∂Δ^{n}} Δ^{n} ⟶^{g} X^{n} ⟶^{s} M^{n−1}X

and g ∘ i_{1} denotes the composite

∂Δ^{n} ⟶^{⊆} Δ^{n} ⟶^{i_{1}} Δ^{n} ∪_{∂Δ^{n}} Δ^{n} ⟶^{g} X^{n}

since f_{0} = g ∘ i_{1} when restricted to the right hand factors; the left hand factors are the issue. Thus we need to choose g which provides a lifting in the following diagram

Δ^{n}  w^{f_{0}} u_{i_{0}}  X^{n} u_{g}

Δ^{n} ∪_{sk_{n−1} Δ^{n}} Δ^{n} w^{s∪S}^{‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ›} M^{n−1} sk_{n} Δ w^{M^{n−1}f_{0}} M^{n−1}X.

That is, there is a representative

ω ∈ Nπ_{n}(X, x) ⊆ π_{0}Hom(Δ^{n} ∪_{sk_{n−1} Δ^{n}} Δ^{n}, X^{n})

that maps to (f_{0}, M^{n−1}f_{0} ∘ (s ∪ S)) in

π_{0}(Hom(Δ^{n}, X) ×_{Hom(Δ^{n},M^{n−1}X)} Hom(Δ^{n} ∪_{sk_{n−1} Δ^{n}} Δ^{n}, M^{n−1}X)).

Of course [f_{0}^{n}] = x ∈ π_{0}Hom(Δ^{n}, X^{n}) = π_{0}X^{n}.

Thus we need to find a representative g of

ω ∈ Nπ_{n}(X, x) ⊆ π_{0}Hom(Δ^{n} ∪_{∂Δ^{n}} Δ^{n}, X^{n})

that maps to the class of (f_{0}, M^{n−1}f_{0} ∘ (s ∪ S)) in

π_{0}(Hom(Δ^{n}, X^{n})×_{Hom(Δ^{n},M^{n−1}X)}Hom(Δ^{n} ∪_{∂Δ^{n}} Δ^{n}, M^{n−1}X))

≅ ⨆_{[b]∈π_{0}X^{n}} π_{n}(M^{n−1}X, s(b)).

Now [f_{0}] = x ∈ π_{0}Hom(Δ^{n}, X^{n}) = π_{0}X^{n} and

[M^{n−1}f_{0} ∘ (s ∪ S)]] = 0 ∈ π_{n}(M^{n−1}X, x) ⊆ ⨆_{[b]∈π_{0}X^{n}} π_{n}(M^{n−1}X, s(b)).

By Lemma 8.8,

π_{n}(M^{n−1}X, x) ≅ M^{n−1}π_{n}(X, x).

Since N^{n}π^{n}(X, x) is exactly the kernel of

s_{∗} : π_{n}(X^{n}, x) → M^{n−1}π_{n}(X, x)

such a g can be found. ■

Now we come to the example. Let Y ∈ S be a space and let 𝔽_{p}^{·}Y be the Bousfield-Kan resolution of Y, as in Example 4.1. Given another space X we get a cosimplicial space Hom(X, 𝔽_{p}^{·}Y) with

Tot Hom(X, 𝔽_{p}^{·}Y) ≅ Hom(X, Tot(𝔽_{p}^{·}Y))
≅ Hom(X, Y_{p})

where Y_{p} is the Bousfield-Kan completion of Y. See Example 7.18. We describe the sets π^{s}π_{t}Hom(X, 𝔽_{p}^{·}Y).

Let H^{∗} = H^{∗}(·, 𝔽_{p}) and let K be the category of unstable algebras over the Steenrod algebra. Then for any space Z there is a Hurewicz homomorphism

(8.18)  π_{0}Hom(X, Z) → hom_{K}(H^{∗}Z, H^{∗}Y)

send f to f^{∗}. If f ∈ Hom(X, Z) is a chosen basepoint, and φ : S^{t} → Hom(X, Z) represents a class in π_{t}Hom(X, Z) with this basepoint, then φ is adjoint to a map ψ : S^{t} × X → Z that fits into a diagram

{S^{t} × X w^{ψ} \atop Z} ; ∗ × X. u ^{[ [ []}^{f}

The morphism ψ^{∗} : H^{∗}Z → H^{∗}(S^{t} × X) ≅ H^{∗}S^{t} ⊗ H^{∗}X can be decomposed

ψ^{∗}(x) = 1 ⊗ f^{∗}(x) + x_{t} ⊗ ∂(x)

where x_{t} ∈ H^{t}S^{t} is a chosen generator. Because ψ^{∗} is a morphism in K, the induced map of graded vector spaces

∂ : H^{∗}Z → Σ^{t}H^{∗}X

is a morphism of unstable modules over the Steenrod algebra and a derivation over f^{∗}:

∂(xy) = (−1)^{t|x|}f^{∗}x∂(y) + ∂(x)f^{∗}y.

We write Der_{K}(H^{∗}Z, Σ^{t}H^{∗}X; f^{∗}) or simply Der_{K}(H^{∗}Z, Σ^{t}H^{∗}X) for the vector space of such derivations. Hence we get a Hurewicz map

(8.19)  π_{t}Hom(X, Z) ⟶ Der_{K}(H^{∗}Z, Σ^{t}H^{∗}X; f^{∗}).

If Z is a simplicial 𝔽_{p} vector space and Z and X are of finite type in the sense that H^{n}Z and H^{n}X are finite dimensional for all n, then the Hurewicz maps of (8.18) and (8.19) are isomorphisms. Therefore, if X and Y are of finite type then

(8.20)  π^{0}π_{0}Hom(X, 𝔽_{p}^{·}Y) ≅ hom_{K}(H^{∗}Y, H^{∗}X)

and for any φ ∈ hom_{K}(H^{∗}Y, H^{∗}X)

(8.21)  π^{s}π_{t}Hom(X, 𝔽_{p}^{·}Y) ≅ π^{s} Der_{K}(H^{∗}𝔽_{p}^{·}Y, Σ^{t}X; φ).

In the last equation we have written φ for any composition of face operators

\begin{matrix}H^{∗}(𝔽_{p}^{s}Y) → H^{∗}Y w^{φ} \\ H^{∗}X. \end{matrix}

The category sK of simplicial objects in in K has a simplicial model category structure, as in Example II.6.2. Also the objects H^{∗}K(ℤ/p, n) form a set of projective generators for K. If we regard H^{∗}Y as a constant object in sK, then the augmentation

H^{∗}𝔽_{p}^{·}Y → H^{∗}Y

is a weak equivalence in sK and H^{∗}𝔽_{p}^{·}Y is cofibrant – indeed, almost-free as in Example 1.14. This last assertion follows form Example 5.8. Thus

Der_{K}(H^{∗}𝔽_{p}^{·}Y, Σ^{t}X; φ)

is a model for the total derived functor of Der_{K}(·, Σ^{t}H^{∗}X; φ) applied to H^{∗}Y. Thus we may as well write

(8.22)  π^{s} Der_{K}(H^{∗}𝔽_{p}^{·}Y, Σ^{t}X; φ) = R^{s} Der_{K}(H^{∗}Y, Σ^{t}X; φ).

This is an example of an André-Quillen cohomology and a great deal of work has gone into learning how to compute this object in special cases. For a general viewpoint one might look at [35]. However, the most successful application of this and related techniques is [62].

Combining (8.22) with (8.20) and (8.21) and appealing to Proposition 8.4 we have

Proposition 8.23. Suppose X and Y are spaces of finite type. A morphism φ : H^{∗}Y → H^{∗}X in K can be lifted to a map X → Y_{p} if

R^{s+1} Der_{K}(H^{∗}Y, Σ^{s}X; φ) = 0

for s ≥ 1.

The techniques of Lannes’s paper [62], cited above, show that if X = B(ℤ/p)^{n} is the classifying space of an elementary abelian p-group and A ∈ K is any unstable algebra, then

R^{s+1} Der_{K}(A, Σ^{t}X; φ) = 0

for all φ ∈ hom_{K}(A, H^{∗}B(ℤ/p)^{n}) and all s ≥ 0, t ≥ 0.