Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.1.2.5. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Then:

$(1)$

The morphism $f$ is a left fibration 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 $i$ is left anodyne, there exists a dotted arrow rendering the diagram commutative.

$(2)$

The morphism $f$ is a right fibration 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 $i$ is right anodyne, there exists a dotted arrow rendering the diagram commutative.

Proof. The “only if” directions are immediate from the definitions, and the “if” directions follow from Proposition 1.4.4.16. $\square$