Kerodon

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

Proposition 1.4.6.7. Let $S_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S_{\bullet }$ is an $\infty $-category.

$(2)$

For every inner anodyne map of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$ and every map $f_0: A_{\bullet } \rightarrow S_{\bullet }$, there exists a map $f: B_{\bullet } \rightarrow S_{\bullet }$ such that $f_0 = f \circ i$.

Proof. The implication $(2) \Rightarrow (1)$ is immediate (since every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is inner anodyne). Conversely, if $(1)$ is satisfied, then every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ has the left lifting property with respect to the projection map $p: S_{\bullet } \rightarrow \Delta ^0$. It then follows from Remark 1.4.4.17 that every inner anodyne map has the left lifting property with respect to $p$. $\square$