PROOFS OF THE MODEL AXIOMS FOR TOP

Theorem 1 (Quillen model structure). The category of spaces is a model category with the usual weak equivalences, q-fibrations the Serre fibrations, and q-cofibrations the maps that satisfy the RLP with respect to the acyclic q-fibrations.

Lemma 2 (Small object argument). Let ℐ be a set of maps of spaces with compact domain. Then any map f : X ⟶ Y of spaces factors as a composite

\xymatrix{X \ar[r]^{i} & X′ \ar[r]^{p} & Y,}

where p satisfies the RLP with respect to each map in ℐ and i satisfies the LLP with respect to any map that satisfies the RLP with respect to each map in ℐ.

Proof. Let X = X_{0}. We construct a commutative diagram

(1)  \xymatrix{X_{0} \ar[d]_{f=p_{0}} \ar[r]^{i_{0}} & X_{1} \ar[r] \ar[d]^{p_{1}} & ⋯ \ar[r] & X_{n} \ar[r]^{i_{n}} \ar[d]^{p_{n}} & X_{n+1} \ar[r] \ar[d]^{p_{n+1}} & ⋯ \\ Y \ar[r]_{id} & Y \ar[r] & ⋯ \ar[r] & Y \ar[r]_{id} & Y \ar[r] & ⋯}

as follows. Suppose inductively that we have constructed p_{n}. Consider all maps from a map in ℐ to p_{n}. Each such map is a commutative diagram of the form

(2)  \xymatrix{A \ar[d]_{i} \ar[r]^{α} \ar[d] & X_{n} \ar[d]^{p_{n}} \\ B \ar[r]_{β} & Y}

with i ∈ ℐ. Summing over such diagrams, we construct a pushout diagram

\xymatrix{∐ A \ar[r]^{∑ α} \ar[d] & X_{n} \ar[d]^{i_{n}} \\ ∐ B \ar[r] & X_{n+1}.}

The maps β and p_{n} induce a map p_{n+1} : X_{n+1} ⟶ Y such that p_{n+1} ∘ i_{n} = p_{n}. Let X′ = colim X_{n}, let i : X ⟶ X′ be the canonical map, and let p : X′ ⟶ Y be obtained by passage to colimits from the p_{n}. Constructing lifts by passage to coproducts, pushouts, and colimits of sequences, we see that each i_{n} and therefore also i satisfies the LLP with respect to maps that satisfy the RLP with respect to maps in ℐ. Assume given a commutative square

\xymatrix{A \ar[r]^{α′} \ar[d]_{i} & X′ \ar[d]^{p} \\ B \ar[r]_{β} \ar@{-->}[ur]^{g} & Y,}

where i is in ℐ. To verify that p satisfies the RLP with respect to i, we must construct a map g that makes the diagram commute. Clearly X′ is constructed as the colimit of a sequence of inclusions. Since A is compact, the natural map

(3)  colim T op(A, X_{n}) ⟶ T op(A, X′)

is a bijection. Therefore α′ : A ⟶ X′ factors through some X_{n}, giving one of the commutative squares used in the construction of X_{n+1}. By construction, there is a map B ⟶ X_{n+1} whose composite with the natural map to X′ gives a map g. □

We use this to prove a refined version of one of the factorization axioms.

Lemma 3. Any map f : X ⟶ Y factors as p∘i, where i is an acyclic q-cofibration that satisfies the LLP with respect to any q-fibration and p is a q-fibration.

Proof. Let 𝒥 be the set of maps i_{0} : D^{n} ⟶ D^{n} × I, n ≥ 0. A map is a qfibration if and only if it satisfies the RLP with respect to every map in 𝒥 and every map in 𝒥 is an acyclic q-cofibration. Use the lemma to factor f. Then p is a q-fibration and i satisfies the LLP with respect to all q-fibrations. In particular, i is a q-cofibration. Since each i_{n} is the inclusion of a deformation retract, so is i. Therefore i is an acyclic q-cofibration. □

By definition, the q-cofibrations satisfy the LLP with respect to the acyclic qfibrations. The other lifting axiom is now formal.

Lemma 4. The q-fibrations satisfy the RLP with respect to the acyclic q-cofibra-tions.

Proof. Let f : X ⟶ Y be any acyclic q-cofibration. We must show that f satisfies the LLP with respect to q-fibrations. By the previous lemma, we may factor f as f = p ∘ i, where i : X ⟶ X′ is an acyclic q-cofibration that does satisfy the LLP with respect to q-fibrations and p : X′ ⟶ Y is a q-fibration. Since f and i are weak equivalences, so is p. Since f satisfies the LLP with respect to acyclic q-fibrations, there exists g : Y ⟶ Y′ such that g ∘ f = i and p ∘ g = id_{Y}. Clearly p and g, together with the identity map on X, express f as a retract of i. Since i satisfies the LLP with respect to q-fibrations, so does f. □

Finally, here is the proof of the other factorization axiom.

Lemma 5. Any map f : X ⟶ Y factors as p ∘ i, where i is a q-cofibration and p is an acyclic q-fibration.

Proof. This is another application of the first lemma. Let ℐ be the set of inclusions i : S^{n−1} ⟶ D^{n}, n ≥ 0, where S^{−1} is empty. A map is an acyclic q-fibration if and only if it satisfies the RLP with respect to all maps in ℐ. The proof of this statement is an exercise in the meaning of homotopy groups and weak equivalences. Each map in ℐ is thus a q-cofibration. In the factorization f = p ∘ i that we now obtain from the lemma, p is an acyclic q-fibration and i is a q-cofibration. □

We define cell ℐ -complexes and relative cell ℐ -complexes in the evident way. With ℐ and 𝒥 as in the previous two lemmas, we deduce the following corollary. It says that Top is a compactly generated model category with generators ℐ for the q-cofibrations and 𝒥 for the acyclic q-cofibrations.

Lemma 6. The q-cofibrations are the retracts of the relative ℐ -cell complexes. The acyclic q-cofibrations are the retracts of the relative 𝒥 -cell complexes.