$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.7.3.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then:
- $(1)$
The object $X$ is final if and only if the projection map $F: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ admits a section $G$ satisfying $G(X) = \operatorname{id}_{X}$.
- $(2)$
The object $X$ is initial if and only if the projection map $F': \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ admits a section $G'$ satisfying $G'(X) = \operatorname{id}_{X}$.
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. If $X$ is a final object, then the projection map $F: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration (Corollary 4.7.3.12), so the construction $X \mapsto \operatorname{id}_ X$ can be extended to a section of $F$. Conversely, suppose that $F$ admits a section $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{/X}$ satisfying $G(X) = \operatorname{id}_ X$. Let $C$ be an object of $\operatorname{\mathcal{C}}$: we wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is contractible. The functors $G$ and $F$ induce morphisms of Kan complexes
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( G(C), \operatorname{id}_ X ) \xrightarrow {F} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,X), \]
whose composition is the identity. In particular, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is a retract of $ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( G(C), \operatorname{id}_ X )$. It will therefore suffice to show that the Kan complex $ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( G(C), \operatorname{id}_ X )$ is contractible. This is clear, since $\operatorname{id}_{X}$ is a final object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ (Proposition 4.7.3.20).
$\square$