Kerodon

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

Example 8.1.3.6 (Tautological Cospans). Let $\operatorname{\mathcal{D}}$ be a simplicial set and let $e: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{D}}$, which we also view as a vertex of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{D}})$. Then the morphism $\operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$ carries $e$ to an edge of the simplicial set $\operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$, which we can identify with a pair of edges

\[ \operatorname{id}_{X} \xrightarrow { e_{L} } e \xleftarrow {e_{R}} \operatorname{id}_{Y} \]

in the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{D}})$. Here $e_{L}$ and $e_{R}$ can be identified with degenerate $3$-simplices of $\operatorname{\mathcal{D}}$, which we depict informally in the diagrams

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{\operatorname{id}_ X} & X \ar [l]^{\operatorname{id}_{X}} & Y \ar [d]^{\operatorname{id}_ Y} & X \ar [l]^{f} \\ X \ar [r]^-{f} & Y & Y \ar [r]^-{\operatorname{id}_{Y}} & Y. } \]