Kerodon

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

Construction 5.5.2.17. Let $\operatorname{\mathcal{C}}$ be a simplicial category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $V: (\operatorname{\mathcal{C}}_{/X} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the right cone contraction functor of Construction 5.5.2.15. Passing to homotopy coherent nerves (and invoking Remark 5.5.2.14, we obtain a map

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( ( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \]

carrying the cone point to the vertex $X$, which we can further identify with a morphism of simplicial sets $c: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X}$. We will refer to $c$ as the slice comparison morphism. Similarly, the left cone contraction functor $V': : (\operatorname{\mathcal{C}}_{X/} )^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ induces a morphism of simplicial sets $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/}$, which we will refer to as the coslice comparison morphism.