Kerodon

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

Theorem 3.1.3.1. Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, and let $i: A_{} \hookrightarrow B_{}$ be any monomorphism of simplicial sets. Then the induced map

\[ \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]

is a Kan fibration.

Proof. By virtue of Remark 3.1.2.6, it will suffice to show that if $i': A' \hookrightarrow B'$ is an anodyne morphism of simplicial sets, then every lifting problem of the form

\[ \xymatrix@C =100pt{ A'_{} \ar [d]^{i'} \ar [r] & \operatorname{Fun}( B_{}, X_{} ) \ar [d] \\ B'_{} \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) } \]

admits a solution. Equivalently, we must show that every lifting problem

\[ \xymatrix@C =100pt{ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ B_{} \times B'_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]

admits a solution. This follows from Remark 3.1.2.6, since the left vertical map is anodyne (Proposition 3.1.2.7) and the right vertical map is a Kan fibration. $\square$