Kerodon

Remark 3.2.2.17 (Homotopy Invariance). In the situation of Remark 3.2.2.16, suppose that $f: X \rightarrow Y$ is a homotopy equivalence. It follows from Proposition 3.2.1.13 that the homotopy class $[f]$ determines an isomorphism from $(X,x)$ to $(Y,y)$ in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$. In particular, the induced map $\pi _{n}(X,x) \rightarrow \pi _{n}(Y,y)$ is an isomorphism of groups for all $n > 0$ (and a bijection of sets for $n = 0$).