$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Theorem 4.4.2.6 (Joyal). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $u$ is an isomorphism.
- $(2)$
Let $n \geq 2$ and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the initial edge
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1\} ) \hookrightarrow \Lambda ^ n_0 \xrightarrow {\sigma _0} \operatorname{\mathcal{C}} \]
is equal to $u$. Then $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$.
- $(3)$
Let $n \geq 2$ and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the final edge
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n-1 < n\} ) \hookrightarrow \Lambda ^ n_ n \xrightarrow {\sigma _0} \operatorname{\mathcal{C}} \]
is equal to $u$. Then $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$.
Proof of Theorem 4.4.2.6.
The implication $(1) \Rightarrow (3)$ is a special case of Proposition 4.4.2.13. We will complete the proof by showing that $(3) \Rightarrow (1)$ (a similar argument shows that $(1)$ and $(2)$ are equivalent). Let $u: X \rightarrow Y$ be a morphism in an $\infty $-category $\operatorname{\mathcal{C}}$, and consider the map $\sigma _0: \Lambda ^{2}_{2} \rightarrow \operatorname{\mathcal{C}}$ depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{u} & \\ Y \ar@ {-->}[ur]^{v} \ar [rr]^{ \operatorname{id}_ Y} & & Y. } \]
If $u$ satisfies condition $(3)$, then we can complete $\sigma _0$ to a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$, which witnesses the morphism $v = d^{2}_2(\sigma )$ as a right homotopy inverse of $u$. The tuple $(\sigma , s^{1}_0(u), s^{1}_1(u), \bullet )$ then determines a morphism of simplicial sets $\tau _0: \Lambda ^3_3 \rightarrow \operatorname{\mathcal{C}}$ (see Proposition 1.2.4.7). Invoking assumption $(3)$ again, we can extend $\tau _0$ to a $3$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. The face $d^{3}_3(\tau )$ then witnesses that $v$ is also a left homotopy inverse to $u$, so that $u$ is an isomorphism as desired.
$\square$