Kerodon

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

Example 5.6.2.4 (Left Covering Maps). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ be the functor of Construction 5.6.1.2, which we identify with a morphism of simplicial sets $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set}) \subset \operatorname{\mathcal{QC}}$. Then Proposition 5.6.1.5 supplies an isomorphism of simplicial sets $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, which exhibits $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as a covariant transport representation of $U$ (in the sense of Definition 5.6.2.1).