Corollary 3.6.5.4. Let $f: Y \rightarrow Z$ be a continuous function between topological spaces. The following conditions are equivalent:
- $(1)$
The function $f$ is a weak homotopy equivalence (Definition 3.6.3.1).
- $(2)$
For every simplicial set $S$, the induced map $\operatorname{Fun}( S, \operatorname{Sing}_{\bullet }(Y) ) \rightarrow \operatorname{Fun}(S, \operatorname{Sing}_{\bullet }(Z) )$ is a homotopy equivalence of Kan complexes.
- $(3)$
For every simplicial set $S$, the induced map $\operatorname{Hom}_{\operatorname{Top}}( |S|, Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Top}}( |S|, Z)_{\bullet }$ is a homotopy equivalence of Kan complexes.
- $(4)$
For every topological space $X$ which has the homotopy type of a CW complex, the induced map $\operatorname{Hom}_{\operatorname{Top}}( X, Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Top}}( X, Z)_{\bullet }$ is a homotopy equivalence of Kan complexes.