Animae as the localization of cell complexes

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. □