Kerodon

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Proposition 4.3.7.12. Let $q: X \rightarrow S$ and $f: K \rightarrow X$ be morphisms of simplicial sets. Then:

• If $q$ is a right fibration and $K$ is weakly contractible, then the induced map $X_{/f} \rightarrow S_{/(q \circ f)}$ is a trivial Kan fibration.

• If $q$ is a left fibration and $K$ is weakly contractible, then the induced map $X_{f/} \rightarrow S_{(q \circ f)/}$ is a trivial Kan fibration.

Proof. We will prove the first assertion; the second follows by a similar argument. To show that the morphism $X_{/f} \rightarrow S_{/(q \circ f)}$ is a trivial Kan fibration, we must prove that every lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & X_{/f} \ar [d] \\ \Delta ^{n} \ar [r] \ar@ {-->}[ur] & S_{/q \circ f} }$

admits a solution. Unwinding the definitions, we can rephrase this as a lifting problem

$\xymatrix@R =50pt@C=50pt{ ( \operatorname{\partial \Delta }^{n} ) \star K \ar [r] \ar [d] & X \ar [d]^{q} \\ (\Delta ^{n} ) \star K \ar@ {-->}[ur] \ar [r] & S. }$

This lifting problem admits a solution, since $q$ is assumed to be a right fibration and the left vertical map is right anodyne (Proposition 4.3.7.9). $\square$