Kerodon

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

Corollary 4.6.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the initial objects of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{\mathrm{fin}} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the final objects of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ contains an initial object, then $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex. If $\operatorname{\mathcal{C}}$ contains a final object, then $\operatorname{\mathcal{C}}^{\mathrm{fin}}$ is a contractible Kan complex.

Proof. Assume that $\operatorname{\mathcal{C}}$ contains an initial object, we will show that $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex (the analogous assertion for final objects follows by a similar argument). Suppose we are given a morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}^{\mathrm{init} }$; we wish to show that $\sigma $ can be extended to a morphism $\overline{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}^{\mathrm{init}}$. If $n=0$, this follows from our assumption that $\operatorname{\mathcal{C}}$ contains an initial object. If $n > 0$, then we can regard $\sigma $ as a morphism from $\operatorname{\partial \Delta }^ n$ to $\operatorname{\mathcal{C}}$ with the property that $\sigma (i) \in \operatorname{\mathcal{C}}$ is initial for $0 \leq i \leq n$. Setting $i=0$, we conclude that $\sigma $ can be extended to a morphism $\overline{\sigma }: \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$, which automatically factors through the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$. $\square$