Notation 8.5.1.24. Let $\operatorname{Ret}$ denote the category introduced in Construction 8.5.0.2. By construction, the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Ret})$ contains a retraction diagram $\sigma : \Delta ^2 \rightarrow \operatorname{N}_{\bullet }( \operatorname{Ret})$, which we depict as
\[ \xymatrix@R =50pt@C=50pt{ & \widetilde{X} \ar [dr]^{ \widetilde{r} } & \\ \widetilde{Y} \ar [ur]^{\widetilde{i}} \ar [rr]^{ \operatorname{id}} & & \widetilde{Y}. } \]
We let $\mathcal{R}$ denote the image of $\sigma $, which we regard as a simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{Ret})$.