Kerodon

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

Example 11.3.0.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories which is a fibration in sets (Definition 4.2.3.1), so that the induced map $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a right fibration, and in particular a cartesian fibration. Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{N}_{\bullet }(q)}$ of Construction 5.2.5.7 is given by the composition

\[ \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { \chi _{q} } \operatorname{Set}\hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty } }, \]

where $\chi _{q}$ is the transport representation of Construction 11.10.5.7 and $\operatorname{Set}\hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ is the fully faithful embedding which associates to each set $X$ the associated discrete simplicial set, regarded as an $\infty $-category.