Kerodon

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

Proposition 3.1.3.13. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r]^{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^{g} \ar@ {-->}[ur]^{ \overline{f} } & S, } \]

where $i$ is a monomorphism and $q$ is a Kan fibration. Then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ is a Kan complex. If $i$ is anodyne, then the Kan complex $\operatorname{Fun}_{A/ \, /S}(B,X)$ is contractible.

Proof. By virtue of Remark 3.1.3.11, the simplicial set $\operatorname{Fun}_{A/ \, /S}( B, X)$ can be identified with a fiber of the restriction map

\[ \theta : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,X). \]

Theorem 3.1.3.1 guarantees that $\theta $ is a Kan fibration, so its fibers are Kan complexes by virtue of Remark 3.1.1.9. If $i$ is anodyne, then $\theta $ is a trivial Kan fibration (Theorem 3.1.3.5), so its fibers are contractible Kan complexes (Remark 1.4.5.9). $\square$