Kerodon

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

Corollary 5.6.1.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and let $\operatorname{LCov}(\operatorname{\mathcal{C}})$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ spanned by the left covering maps $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Then Construction 5.6.1.2 supplies an equivalence of categories

\[ \operatorname{LCov}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) \quad \quad (U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}, \]

with homotopy inverse given by the construction

\[ ( \mathscr {F} \in \operatorname{Fun}(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) ) \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F} \in \operatorname{LCov}(\operatorname{\mathcal{C}}). \]