$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.2.4.10. Let $i: A \rightarrow B$ be a morphism of simplicial sets. Then:
- $(1)$
The morphism $i$ is left anodyne if and only if, for every square diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{f} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
where $f$ is left fibration, there exists a dotted arrow rendering the diagram commutative.
- $(2)$
The morphism $i$ is right anodyne if and only if, for every square diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{f} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
where $f$ is right fibration, there exists a dotted arrow rendering the diagram commutative.
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. Using Proposition 4.2.4.8, we can factor $i$ as a composition $A \xrightarrow {i'} Q \xrightarrow {f} B$, where $i'$ is left anodyne and $f$ is a left fibration. If the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r]^-{i'} & Q \ar [d]^{f} \\ B \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur]^{r} & B } \]
admits a solution, then the map $r$ exhibits $i$ as a retract of $i'$ (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). Since the collection of anodyne morphisms is closed under retracts, it follows that $i$ is anodyne. This proves the “if” direction of $(1)$; the reverse implication follows from Proposition 4.2.4.5.
$\square$