Kerodon

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Proposition 4.5.5.20. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is a trivial Kan fibration if and only if it is both an isofibration and a categorical equivalence.

Proof. If $q$ is a trivial Kan fibration, then it is an isofibration by virtue of Example 4.5.5.8 and a categorical equivalence by virtue of Proposition 4.5.3.11. Conversely, suppose that $q$ is both an isofibration and a categorical equivalence. Using Exercise 3.1.7.11, we can write $q$ as a composition $X \xrightarrow {q'} Y \xrightarrow {q''} S$, where $q'$ is a monomorphism and $q''$ is a trivial Kan fibration. Then $q''$ is a categorical equivalence (Proposition 4.5.3.11), so that $q'$ is also a categorical equivalence (Remark 4.5.3.5). Invoking our assumption that $q$ is an isofibration, we conclude that the lifting problem

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

admits a solution. It follows that $q$ is a retract of the morphism $q''$, and is therefore also a trivial Kan fibration. $\square$