Kerodon

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

Lemma 1.4.7.5. Let $f: X_{\bullet } \hookrightarrow Y_{\bullet }$ and $f': X'_{\bullet } \hookrightarrow Y'_{\bullet }$ be monomorphisms of simplicial sets. If $f$ is inner anodyne, then the induced map

\[ u_{f,f'}: (Y_{\bullet } \times X'_{\bullet } ) \coprod _{ (X_{\bullet } \times X'_{\bullet })} (X_{\bullet } \times Y'_{\bullet } ) \hookrightarrow Y_{\bullet } \times Y'_{\bullet } \]

is inner anodyne.

Proof. Let us regard the morphism $f': X'_{\bullet } \hookrightarrow Y'_{\bullet }$ as fixed. Let $T$ be the collection of all morphisms $f: X_{\bullet } \rightarrow Y_{\bullet }$ for which the map $u_{f,f'}$ is inner anodyne. Then $T$ is weakly saturated. To prove Lemma 1.4.7.5, we must show that $T$ contains all inner anodyne morphisms of simplicial sets. By virtue of Lemma 1.4.6.8, it will suffice to show that $T$ contains every morphism of the form

\[ u_{i,j}: (B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq B_{\bullet } \times \Delta ^2, \]

where $i: A_{\bullet } \hookrightarrow B_{\bullet }$ is a monomorphism of simplicial sets and $j: \Lambda ^2_1 \hookrightarrow \Delta ^2$ is the inclusion. Setting

\[ A'_{\bullet } = (B_{\bullet } \times X'_{\bullet }) \coprod _{ (A_{\bullet } \times X'_{\bullet } )} ( A_{\bullet } \times Y'_{\bullet }) \quad \quad B'_{\bullet } = B_{\bullet } \times Y'_{\bullet }, \]

we are reduced to the problem of showing that the map

\[ u_{i',j}: (B'_{\bullet } \times \Lambda ^2_1) \coprod _{A'_{\bullet } \times \Lambda ^2_1 } (A'_{\bullet } \times \Delta ^2) \subseteq B'_{\bullet } \times \Delta ^2, \]

is inner anodyne, which follows from Lemma 1.4.6.8. $\square$