Kerodon

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

Corollary 4.4.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. The following conditions are equivalent:

$(1)$

The objects $X$ and $Y$ are isomorphic.

$(2)$

There exists a connected Kan complex $\operatorname{\mathcal{E}}$, a pair of vertices $\overline{X}, \overline{Y} \in \operatorname{\mathcal{E}}$, and a morphism $f: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $f( \overline{X}) = X$ and $f( \overline{Y} ) = Y$.

$(3)$

There exists a contractible Kan complex $\operatorname{\mathcal{E}}$, a pair of vertices $\overline{X}, \overline{Y} \in \operatorname{\mathcal{E}}$, and a morphism $f: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $f( \overline{X}) = X$ and $f( \overline{Y} ) = Y$.