Kerodon

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

Warning 5.4.2.19. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing an object $X$. Then 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 are always bijective at the level of vertices (on the left side, vertices of either of the simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} )$ and $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X}$ can be identified with pairs $(C,f)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $f$ is a morphism from $C$ to $X$). Beware that $c$ and $c'$ are generally not bijective on simplices of dimension $\geq 1$. Unwinding the definitions, we see that edges of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{/X})$ can be identified with diagrams

\[ \xymatrix@R =50pt@C=50pt{ C \ar [dr]_{f} \ar [rr]^{h} & & D \ar [dl]^{g} \\ & X & } \]

in the category $\operatorname{\mathcal{C}}$ which are strictly commutative, while edges of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{/X}$ can be identified with diagrams which commute up to a specified homotopy $\mu : g \circ h \rightarrow f$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet }$.