Remark Let $f: X \rightarrow Y$ be a continuous function between topological spaces. Then $f$ is a weak homotopy equivalence if and only if it satisfies the following pair of conditions:

  • The induced map of path components $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is a bijection.

  • For every point $x \in X$ and every $n \geq 1$, the map of homotopy groups $\pi _{n}(f): \pi _{n}( X, x) \rightarrow \pi _{n}(Y, f(x) )$ is an isomorphism.

This follows by applying Theorem to the map of Kan complexes $\operatorname{Sing}_{\bullet }(f): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(Y)$ (see Example