Kerodon

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

Construction 5.6.1.2 (The Covariant Transport Representation). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets (Definition 4.2.3.8). Then $U$ is a left fibration (Remark 4.2.3.11), and therefore a cocartesian fibration (Proposition 5.1.4.14). For each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a discrete simplicial set (Remark 4.2.3.17). It follows that the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ of Construction 5.2.3.2 can be regarded as a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets (regarded as a full subcategory of the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). We will denote this functor by $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ and refer to it as the covariant transport representation of $U$.