Kerodon

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

Corollary 4.6.6.12. Let $X$ be a Kan complex and let $x \in X$ be a vertex. The following conditions are equivalent:

$(1)$

The vertex $x$ is initial when viewed as an object of the $\infty $-category $X$.

$(2)$

The vertex $x$ is final when viewed as an object of the $\infty $-category $X$.

$(3)$

The Kan complex $X$ is contractible.

In particular, these conditions are independent of the choice of vertex $x \in X$.

Proof. If the Kan complex $X$ is contractible, then the projection map $X_{x/} \rightarrow X$ is a trivial Kan fibration (Corollary 4.3.7.19), so the object $x \in X$ is initial by virtue of Proposition 4.6.6.11. Conversely, if the projection map $X_{x/} \rightarrow X$ is a trivial Kan fibration, then it is a homotopy equivalence (Proposition 3.1.6.10). Since the Kan complex $X_{x/}$ is contractible (Corollary 4.3.7.14), it follows that $X$ is contractible. This proves the equivalence of $(1)$ and $(3)$; the equivalence of $(2)$ and $(3)$ follows by a similar argument. $\square$