Corollary 3.3.7.7. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $f$ is anodyne.
- $(2)$
The morphism $f$ is both a monomorphism and a weak homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 3.1.6.14 and Remark 3.1.2.3. To prove the converse, assume that $f$ is a weak homotopy equivalence and apply Proposition 3.1.7.1 to write $f$ as a composition $X \xrightarrow {f'} Q \xrightarrow {f''} Y$, where $f'$ is anodyne and $f''$ is a Kan fibration. Then $f'$ is a weak homotopy equivalence (Proposition 3.1.6.14), so $f''$ is a weak homotopy equivalence (Remark 3.1.6.16). Invoking Proposition 3.3.7.6, we conclude that $f''$ is a trivial Kan fibration. If $f$ is a monomorphism, then the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r]^-{ f' } & Q \ar [d]^{f''} \\ Y \ar [r]^-{\operatorname{id}_ Y} \ar@ {-->}[ur] & Y } \]
admits a solution. It follows that $f$ is a retract of $f'$ (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). Since the collection of anodyne morphisms is closed under retracts, we conclude that $f$ is anodyne. $\square$