Kerodon

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

Example 5.4.2.18. Let $\operatorname{\mathcal{C}}$ be an ordinary category, which we identify with the associated constant simplicial category $\underline{\operatorname{\mathcal{C}}}$ of Example 2.4.2.3. For every object $X \in \operatorname{\mathcal{C}}$, the slice and coslice comparison morphisms

\[ c: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X} \quad \quad c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} \]

of Construction 5.4.2.17 can be identified with the isomorphisms $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{/X}) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/X}$ and $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{X'}) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/}$ described in Example 4.3.5.8.