Remark 8.5.1.23 (04KB): Lifting Retraction Diagrams

Remark 8.5.1.23 (Lifting Retraction Diagrams). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of $\infty $-categories. Suppose we are given a retraction diagram

8.66

\begin{equation} \begin{gathered}\label{equation:lifting-retraction-diagrams} \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y } \end{gathered} \end{equation}

in $\operatorname{\mathcal{C}}$ and an object $\widetilde{Y} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{Y} ) = Y$. Our assumption that $U$ is a cartesian fibration guarantees the existence of a $U$-cartesian morphism $\widetilde{r}: \widetilde{X} \rightarrow \widetilde{Y}$ in $\operatorname{\mathcal{E}}$ satisfying $U( \widetilde{r} ) = r$. Since $\widetilde{r}$ is $U$-cartesian, we can lift (8.66) to a retraction diagram

\[ \xymatrix@R =50pt@C=50pt{ & \widetilde{X} \ar [dr]^{ \widetilde{r} } & \\ \widetilde{Y} \ar [ur]^{ \widetilde{i} } \ar [rr]^{ \operatorname{id}_{\widetilde{Y}} } & & \widetilde{Y} } \]

In particular, the object $\widetilde{Y}$ can be realized as a retract of an object $\widetilde{X}$ satisfying $U( \widetilde{X} ) = X$.