Relative cell complexes and Serre fibrations

Proof. Choose homotopy equivalences \(u\colon X' \to X\) and \(v\colon Y' \to Y\) with \(X'\) and \(Y'\) cell complexes, and let \(w\colon Y \to Y'\) be a homotopy inverse to \(v\). The composite \(wfu\colon X' \to Y'\) is a weak homotopy equivalence, hence a homotopy equivalence by the usual Whitehead theorem. Since \(u\) and \(w\) are homotopy equivalences, so is \(f\). □

Proof. In light of the homotopy hypothesis, Axiom M, it suffices to show that the functor \[ \Cell [\{\text {homotopy equivalences}\}^{-1}] \to \Top [\{\text {weak homotopy equivalences}\}^{-1}] \] induced by the inclusion is an equivalence of \(\infty \)-categories. By Theorem 2.3.4 there exists a functor \(Z\colon \Top \to \Cell \) equipped with a natural transformation \(q\colon Z \Rightarrow \id _{\Top }\) such that the map \(q_X\colon Z(X) \to X\) is a weak homotopy equivalence for all \(X\). Given a weak homotopy equivalence \(f\colon X \to Y\), the 2-out-of-3 property applied to the diagram

shows that the map \(Z(f)\) is again a weak homotopy equivalence, and by Whitehead’s theorem it is in fact a homotopy equivalence. It follows that \(Z\) induces a functor \(\Top [\{\text {weak homotopy equivalences}\}^{-1}] \to \Cell [\{\text {homotopy equivalences}\}^{-1}]\), and the natural transformation \(q\colon Z \to \id \) exhibits this as the requisite inverse functor. □

Proof. It suffices to observe that a functor \(F\colon \Cell \to C\) into a 1-category inverts homotopy equivalences if and only if it identifies homotopic maps. One implication follows by applying \(F\) to homotopy inverses. Conversely, if \(H\colon f \sim g\) is a homotopy, then \[ F(f)=F(H)F(i_0)=F(H)F(\pr )^{-1}=F(H)F(i_1)=F(g), \] where \(i_0,i_1\colon X \to X\times [0,1]\) are the endpoint inclusions and \(\pr \colon X\times [0,1]\to X\) is the projection. Thus \(\h \Cell \) has the same universal property as the homotopy category of the localization in Proposition 2.3.6. □

Proof sketch. We need to verify the axioms from Definition 2.2.4 (dualized for cofibrations):

(1)

(2-out-of-3): It is clear from the definition that the class \(W_{\text {whe}}\) satisfies the 2-out-of-3 property.

(2)

(Class of cofibrations): The class \(I_{\text {cell}}\) contains all isomorphisms (by attaching no cells) and is closed under composition (attaching cells iteratively). It is also closed under pushouts: if \(i\colon A \to X\) is a relative cell complex and \(f\colon A \to A'\) is any map, consider the pushout square

Then \(i'\) is also a relative cell complex, where the filtration is inductively defined by \(X'_{\lambda +1} := X_{\lambda +1} \sqcup _{X_{\lambda }} X'_{\lambda }\).

(3)

(Pushout axiom): We need to show that if \(i\colon A \to X\) is a trivial cofibration (i.e., a relative cell complex and a weak homotopy equivalence) between cofibrant objects (cell complexes), then the pushout \(i'\colon A' \to X'\) is also a weak homotopy equivalence. This is a non-trivial fact in model category theory. One first shows that every trivial cofibration has the left lifting property with respect to the Serre fibrations (see Definition 2.3.12 below), then observes that this property is preserved by pushouts, and finally deduces that every map with this lifting property is a weak homotopy equivalence. See Hovey (1999), Section 2.4.

(4)

(Factorization axiom): The small object argument functorially factors every map \(f\colon X \to Y\) as \(X \xrightarrow {i} Z \xrightarrow {p} Y\), where \(i\) is a relative cell complex and \(p\) is a trivial Serre fibration, hence in particular a weak homotopy equivalence. See Hovey (1999), Sections 2.1 and 2.4.

Since all axioms are satisfied, \((\Top , W_{\text {whe}}, I_{\text {cell}})\) is an \(\infty \)-category with weak equivalences and cofibrations (with functorial factorizations). It is straightforward to see that a small coproduct of (trivial) cofibrations is again a (trivial) cofibration, so that \(\Top \) is in fact homotopy cocomplete. □

Proof. This follows directly from Theorem 2.3.8 and the right exactness of localization functors stated in Corollary 2.2.9. □

Proof. Consider the mapping cylinder of the map \(f\): \[ M(f) \quad := \quad (X \sqcup (Z \times [0,1])) / (z,0) \sim f(z). \] The map \(f\) factors as \(Z \xrightarrow {i} M(f) \xrightarrow {p} X\), where \(i(z) := [(z,1)]\), where \(p\vert _X = \id _X\) and where \(p\vert _{Z \times [0,1]}(z,t) := f(z)\). Observe that the map \(p\colon M(f) \to X\) is a homotopy equivalence, with homotopy inverse sending \(x\) to \([x]\). In particular, it induces an equivalence \(\Pi _{\infty }(M(f)) \iso \Pi _{\infty }(X)\) of animae. Further observe that the homotopy pushout \(X \sqcup _Z^h Y\) can be identified with the strict pushout \(M(f) \sqcup _Z Y\) via the maps \(g\) and the projection \(M(f) \to X\):

We will show that the functor \(\Pi _{\infty }\colon \Top \to \An \) preserves this pushout square.

Case 1: Assume first that the spaces \(X\), \(Y\) and \(Z\) are cell complexes and that the map \(f\) is a relative cell complex. In this case, the map \(i\colon Z \to M(f)\) is also a relative cell complex: it may be written as the composite of \(Z \times \{1\} \hookrightarrow Z \times [0,1]\), which is a relative cell complex by the product cell filtration, and the map \(Z \times [0,1] \to M(f)\), which is a pushout of the map \(Z \to X\) and hence also a relative cell complex. The fact that \(\Pi _{\infty }(-)\) preserves the pushout square in Equation 2.1 is now an instance of Proposition 2.3.9.

Case 2: We will now prove the general case. By Theorem 2.3.4, there exists a cell complex \(Z'\) and a weak homotopy equivalence \(Z' \iso Z\). By factorizing the composite maps \(Z' \to X\) and \(Z' \to Y\) into a relative cell complex followed by a weak equivalence, we obtain a commutative diagram

where \(X'\), \(Y'\) and \(Z'\) are cell complexes, the vertical maps are weak homotopy equivalences, and \(f'\) and \(g'\) are relative cell complexes. Since all six spaces have the homotopy type of a cell complex, it follows from Whitehead’s theorem that the vertical maps are even homotopy equivalences. By homotopy invariance of homotopy pushouts, Exercise 2.1.4, the induced map \(X' \sqcup _{Z'}^h Y' \to X \sqcup _Z^h Y\) is again a homotopy equivalence, and in particular it becomes an equivalence after applying \(\Pi _{\infty }(-)\). This means that we have reduced the problem to Case 1, thus finishing the proof. □

Proof. The NDR condition implies that the canonical map from the homotopy cofiber of \(A\hookrightarrow Y\) to the actual quotient \(Y/A\) is a homotopy equivalence; see May (1999), Chapter 8, Section 4. The claim now follows by applying Proposition 2.3.10 to the diagram \(Y \leftarrow A \to *\). □

Proof. By working separately over each path component of \(B\), we may assume that \(B\) is path-connected. Picking a preferred basepoint \(b_0\) of \(B\), it then suffices to show that \(p\) is a weak homotopy equivalence if and only if the fiber \(F := F_{b_0}\) is weakly contractible. This follows immediately from the long exact sequence from Theorem 2.3.14. □

Proof. Consider a solid commutative diagram of the form

We may define the dashed map \(\overline {H}\) as \(\overline {H}(x,s) := (e(x),K(x,s,-))\), where

• The map \(e\colon D^n \to E\) is the first component of \(g\);

• The map \(K \colon D^n \times [0,1] \times [0,1] \to B\) is a continuous map satisfying the following three constraints:

  • To guarantee that \(\overline {H}(x,s) \in P(f)\) we need \(K(x,s,0) = f(e(x))\);
  • To guarantee commutativity of the top triangle, we need \(K(x,0,t) = \gamma (x)(t)\), where \(\gamma (x)\) is the second component of \(g(x)\);
  • To guarantee commutativity of the bottom triangle, we need \(K(x,s,1) = H(x,s)\).

  Such a continuous map exists, since the inclusion \(\big ([0,1] \times \{0,1\}\big ) \cup \big (\{0\} \times [0,1]\big ) \hookrightarrow [0,1] \times [0,1]\) has a continuous retraction. □

Proof. It is clear that the weak homotopy equivalences satisfy 2-out-of-3, and it is also straightforward to check that Serre fibrations are closed under pullbacks and composition and contain all homeomorphisms. This gives conditions (1) and (2) from Definition 2.2.4.

To see condition (3), consider a pullback square

of topological spaces, where \(p\) is both a Serre fibration and a weak homotopy equivalence. We need to show that also \(p'\) is a weak homotopy equivalence. By Corollary 2.3.15, it suffices to show that for every point \(b' \in B'\) the fiber \(F'_{b'} := {p'}^{-1}(b')\) is weakly contractible. But since \(p'\) is a pullback of \(p\), this fiber is homeomorphic to the fiber \(F_b = p^{-1}(b)\), which in turn is weakly contractible by Corollary 2.3.15.

Finally, for condition (4), note that a continuous map \(f\colon E \to B\) may be factored as \[ E \xrightarrow {i} P(f) \xrightarrow {p} B, \] where \(p\) is the path space fibration from Lemma 2.3.17, and where \(i\) maps \(e\) to \((e,\const _{f(e)})\). The first map is a homotopy equivalence, hence in particular a weak homotopy equivalence, and the second map is a Serre fibration by Lemma 2.3.17. Since the definition of \(P(f)\) is functorial in \(f\), this finishes the proof that \((\Top , W_{\text {whe}}, F_{\text {Serre}})\) is a category with weak equivalences and fibrations, with functorial factorizations. It is also clear that a product of Serre fibrations is again a Serre fibration, finishing the proof. □

Proof. In light of Theorem 2.3.18 this is an instance of Theorem 2.2.8. □

Proof. Consider continuous maps \(f\colon X \to Z\) and \(g\colon Y \to Z\), and consider the pullback square

defining \(X \times _Z^{h} Y\). The path fibration \(p\) is a Serre fibration by Lemma 2.3.17, hence this square is sent to a pullback square in \(\An \) by Proposition 2.3.19. The canonical inclusion \(i\colon X \hookrightarrow P(f)\) is a homotopy equivalence, giving an equivalence \(\Pi _{\infty }(P(f)) \simeq \Pi _{\infty }(X)\). This finishes the proof. □