$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.6.4.3. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $f$ is a weak homotopy equivalence, in the sense of Definition 3.1.6.12.
- $(2)$
The induced map of topological spaces $|X| \rightarrow |Y|$ is a weak homotopy equivalence, in the sense of Definition 3.6.3.1.
- $(3)$
The induced map of topological spaces $|X| \rightarrow |Y|$ is a homotopy equivalence.
Proof.
We have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{u_{X}} & Y \ar [d]^{u_{Y}} \\ \operatorname{Sing}_{\bullet }( |X| ) \ar [r]^-{ \operatorname{Sing}_{\bullet }(|f| ) } & \operatorname{Sing}_{\bullet }( |Y| ), } \]
where the vertical maps are weak homotopy equivalences by virtue of Theorem 3.6.4.1. The equivalence $(1) \Leftrightarrow (2)$ now follows from Remark 3.1.6.16. The implication $(3) \Rightarrow (2)$ follows from Example 3.6.3.3, and the reverse implication is a special case of Proposition 3.6.3.8 (since the topological spaces $|X|$ and $|Y|$ are CW complexes; see Remark 1.2.3.12).
$\square$