$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.3.6.16. Let $\operatorname{\mathcal{C}}$ be a simplicial set. The following conditions are equivalent:
- $(1)$
The simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category.
- $(2)$
For every vertex $X$ of $\operatorname{\mathcal{C}}$, the projection map $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets.
- $(3)$
For every vertex $Y$ of $\operatorname{\mathcal{C}}$, the projection map $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration of simplicial sets.
Proof.
The implications $(1) \Rightarrow (2)$ and $(1) \Rightarrow (3)$ are special cases of Proposition 4.3.6.1. We will complete the proof by showing that $(3)$ implies $(1)$; the proof that $(2)$ implies $(1)$ is similar. Assume that $(3)$ is satisfied, and suppose that we are given a map $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$, where $0 < i < n$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$. Setting $Y = \sigma _0(n)$ and using the isomorphism $\Lambda ^{n}_{i} \simeq \Delta ^{n-1} \coprod _{ \Lambda ^{n-1}_{i} } (\Lambda ^{n-1}_{i})^{\triangleright }$ supplied by Lemma 4.3.6.15, we are reduced to solving a lifting problem of the form
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n-1}_{i} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{/Y} \ar [d] \\ \Delta ^{n-1} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}. } \]
Since $0 < i \leq n-1$, the desired solution exists by virtue of our assumption that the projection map $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.
$\square$