# Kerodon

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Proposition 4.2.4.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then:

$(1)$

The morphism $q$ is a left fibration if and only if, for every square diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S }$

where $i$ is left anodyne, there exists a dotted arrow rendering the diagram commutative.

$(2)$

The morphism $q$ is a right fibration if and only if, for every square diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S }$

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. $\square$