Kerodon

Example 5.3.3.21. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category, so that the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration (Corollary 5.3.3.16). Define $\mathscr {F}^{\simeq }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {F}^{\simeq }( C) = \mathscr {F}(C)^{\simeq }$. Then $\operatorname{N}_{\bullet }^{\mathscr {F}^{\simeq }}(\operatorname{\mathcal{C}})$ can be identified with with simplicial subset of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ spanned by those $n$-simplices which carry each edge of $\Delta ^ n$ to a $U$-cocartesian edge of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. That is, the projection map $U^{\simeq }: \operatorname{N}_{\bullet }^{\mathscr {F}^{\simeq }}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the underlying left fibration of the cocartesian fibration $U$ (see Corollary 5.1.4.16).