Kerodon

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

Proposition 5.4.6.2. The simplicial set $\operatorname{\mathcal{QC}}_{\ast }$ is an $\infty $-category, and the projection map $\operatorname{\mathcal{QC}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}$ is a left fibration of $\infty $-categories.

Proof. By virtue of Proposition 5.4.4.3, the simplicial set $\operatorname{\mathcal{QC}}$ is an $\infty $-category. It follows that for every object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}$, the projection map $\operatorname{\mathcal{QC}}_{\operatorname{\mathcal{C}}/} \rightarrow \operatorname{\mathcal{QC}}$ is a left fibration (Corollary 4.3.6.11). Taking $\operatorname{\mathcal{C}}= \Delta ^{0}$, we conclude that the projection map $\operatorname{\mathcal{QC}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}$ is a left fibration, so that $\operatorname{\mathcal{QC}}_{\ast }$ is an $\infty $-category (Remark 4.2.1.4). $\square$