Kerodon

Remark 5.6.0.9. In the statement of Theorem 5.6.0.2, it is not necessary to assume that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category. This additional generality will play an essential role in our proof (which will require us to analyze the restriction of the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ to simplicial subsets of $\operatorname{\mathcal{C}}$). Moreover, it has a number of pleasant consequences: since $\operatorname{\mathcal{QC}}$ is an $\infty $-category, it guarantees that every cocartesian fibration of simplicial sets is equivalent to the pullback of a cocartesian fibration between $\infty $-categories. In §5.6.7, we use this to prove a sharper statement: every cocartesian fibration of simplicial sets is isomorphic to the pullback of a cocartesian fibration between $\infty $-categories (Corollary 5.6.7.3). From this, we deduce that every cocartesian fibration of simplicial sets is an isofibration (Corollary 5.6.7.5), and that the collection of categorical equivalences of simplicial sets is stable under the formation of pullback by cocartesian fibrations (Corollary 5.6.7.6).