The Quillen cofibration structure on topological spaces

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. □