Kerodon

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

Example 5.7.5.6 (Weighted Nerves). Let $\operatorname{\mathcal{C}}$ be an ordinary category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor, and let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the weighted nerve of Definition 5.3.3.1. Then the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration (Proposition 5.3.3.15). Moreover, the equivalence

\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]

of Proposition 5.7.4.8 exhibits $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ as a covariant transport representation for $U$.