Kerodon

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

Corollary 5.6.0.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the construction

\[ ( \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}) \mapsto (\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}) \]

induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})^{\simeq } )$ to the set of equivalence classes of left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having the following property: for every object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small.

Proof. Combine Theorem 5.6.3.6 (applied to the full subcategory $\operatorname{\mathcal{S}}\subset \operatorname{\mathcal{QC}}$) with with Proposition 5.1.4.14. $\square$