Kerodon

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

Corollary 5.5.5.4. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets which factors through the full subcategory $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) ) \subset \operatorname{\mathcal{QC}}$ of Remark 5.4.4.9. Then the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is an inner covering map of simplicial sets.

Proof. By virtue of Corollary 4.1.5.11, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In this case, we wish to show that the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is isomorphic to the nerve of an ordinary category (see Proposition 4.1.5.10), which is a special case of Proposition 5.5.5.3. $\square$