Kerodon

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

Proposition 4.3.7.9. Let $X$ and $Y$ be simplicial sets. If $X$ is weakly contractible and $Y'$ is a simplicial subset of $Y$, then the inclusion $X \star Y' \hookrightarrow X \star Y$ is left anodyne. If $Y$ is weakly contractible and $X'$ is a simplicial subset of $X$, then the inclusion $X' \star Y \hookrightarrow X \star Y$ is right anodyne.

Proof. We will prove the first assertion; the second follows by a similar argument. Fix a vertex $x \in X$, so that the inclusion morphism $\iota : X \star Y' \hookrightarrow X \star Y$ factors as a composition

\[ X \star Y' \xrightarrow {\iota '} (X \star Y') \coprod _{(\{ x\} \star Y')} (\{ x\} \star Y) \xrightarrow {\iota ''} X \star Y. \]

The morphism $\iota '$ is a pushout of the inclusion $Y'^{\triangleleft } \hookrightarrow Y^{\triangleleft }$, and is left anodyne by virtue of Lemma 4.3.7.8. It will therefore suffice to show that $\iota ''$ is left anodyne. This is a special case of Proposition 4.3.7.1, since the inclusion map $\{ x\} \hookrightarrow X$ is a weak homotopy equivalence (by virtue of our assumption that $X$ is weakly contractible) and therefore anodyne (by virtue of Corollary 3.3.7.5). $\square$