Kerodon

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

Example 5.5.6.2 (The Comparison Map on Vertices). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. Let us identify vertices of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ with pairs $(C,X)$, where where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is a vertex of the simplicial set $\mathscr {F}(C)$ (Remark 5.5.3.11). Under this identification, the comparison map

\[ \theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]

of Construction 5.5.6.1 is given on vertices by the construction $(C,X) \mapsto (C,X)$, where we identify $(C,X)$ with a vertex of $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ using Example 5.5.4.12. In particular, the morphism $\theta $ is bijective at the level of vertices.