Kerodon

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

Example 5.5.3.17. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. For each object $C \in \operatorname{\mathcal{C}}$, the comparison map

\[ F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \]

of Construction 5.5.3.16 induces an isomorphism of fibers

\[ F_{C}: \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \simeq \mathscr {F}(C) \]

(see Example 5.5.3.7).