Cofibration and fibration structures on topological spaces

Example 2.2.6 (Structures on \(\Top \)). As will be discussed below, the category \(\Top \) can be equipped with such structures relevant to \(\An \simeq \Top [W^{-1}]\) where \(W\) is the class of weak homotopy equivalences:

• \((\Top , W, I=\text {relative cell complexes})\) is an \(\infty \)-category with weak equivalences and cofibrations (see Theorem 2.3.8). The cofibrant objects are the cell complexes.
• \((\Top , W, F=\text {Serre fibrations})\) is an \(\infty \)-category with weak equivalences and fibrations (see Theorem 2.3.18). All objects in \(\Top \) are fibrant in this structure.