Corollary 3.1.7.5 (00X8)

Corollary 3.1.7.5. 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 weakly left orthogonal to all Kan fibrations. That is, if $g: Z \rightarrow S$ is a Kan fibration of simplicial sets, then every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r] & Z \ar [d]^{g} \\ Y \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 3.1.2.7. To deduce the converse, we first 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. If $f$ satisfies condition $(2)$, then the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r]^-{f'} & Q \ar [d]^{f''} \\ Y \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur] & Y } \]

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, it follows that $f$ is anodyne. $\square$