Kerodon

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

Proposition 4.1.1.10. Let $\operatorname{\mathcal{C}}$ be a category, and let $q: X \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. Then $q$ is an inner fibration if and only if $X$ is an $\infty $-category.

Proof. If $q$ is an inner fibration, then Remark 4.1.1.9 guarantees that $X$ is an $\infty $-category. Conversely, suppose that $X$ is an $\infty $-category and that we are given a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & X_{} \ar [d]^{q} \\ \Delta ^{n} \ar@ {-->}[ur]^-{\sigma } \ar [r]^-{ \overline{\sigma } } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \]

for integers $0 < i < n$. Our assumption that $X$ is an $\infty $-category guarantees that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$. The equality $q \circ \sigma = \overline{\sigma }$ is automatic by virtue of Proposition 1.2.3.1. $\square$