Kerodon

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

Example 5.5.6.3 (The Comparison Map on Edges). 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 $(C,X)$ and $(D,Y)$ be vertices of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$. Using Remark 5.5.3.12, we can identify edges of $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ having source $(C,X)$ and target $(D,Y)$ with pairs $(f,u)$, where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: \mathscr {F}(f)(X) \rightarrow Y$ is an edge of the simplicial set $\mathscr {F}(D)$. 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 edges by the construction $(f,u) \mapsto (f,u)$, where we identify $(f,u)$ with an edge of the simplicial set $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ using Example 5.5.4.13. In particular, the morphism $\theta $ is bijective at the level of edges.