Kerodon

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

Example 9.5.0.2. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote its Duskin nerve (Construction 2.3.1.1), and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$ (which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$). Specializing Remark 8.1.5.5 to the case where $\operatorname{\mathcal{A}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{$n$-simplices of $\operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$} \} \ar [d]^{\sim } \\ \{ \textnormal{$n$-simplices of $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)$} \} . } \]

In other words, we can identify $n$-simplices of $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)$ with commutative diagrams

\[ \xymatrix@C =20pt{ f_{0,0} \ar@ {=>}[dr] & & f_{1,1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{n-1,n-1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{n,n} \ar@ {=>}[dl] \\ & \cdots \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dr] \ar@ {=>}[dl] & & \cdots \ar@ {=>}[dl] \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dl] & \\ & & f_{0,n-2} \ar@ {=>}[dr] & & f_{1,n-1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{2,n} \ar@ {=>}[dl] & & \\ & & & f_{0,n-1} \ar@ {=>}[dr] & & f_{1,n} \ar@ {=>}[dl] & & & \\ & & & & f_{0,n} & & & & } \]

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.