Proposition 9.2.5.15 (04N5)

Proposition 9.2.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pushout diagram

9.15

\begin{equation} \begin{gathered}\label{equation:pushout-of-weakly-left-orthogonal} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{f} & A' \ar [d]^{f'} \\ B \ar [r] & B'. } \end{gathered} \end{equation}

If $f$ is weakly left orthogonal to a morphism $g: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, then $f'$ is also weakly left orthogonal to $g$.

Proof. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Remark 9.2.5.12, it will suffice to show that for any morphism $\widetilde{f}'$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f}' ) = f'$, the object $\widetilde{X}$ is weakly $\widetilde{f}'$-local. Since $\pi $ is a right fibration, we can lift (9.15) to a diagram

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{A} \ar [r] \ar [d]^{ \widetilde{f} } & \widetilde{A}' \ar [d]^{ \widetilde{f}'} \\ \widetilde{B} \ar [r] & \widetilde{B}' } \]

in the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, which is also a pushout square (Proposition 7.1.4.20). By virtue of Proposition 9.2.3.14, it will suffice to show that $\widetilde{X}$ is weakly $\widetilde{f}$-local, which follows from our assumption that $f$ is weakly left orthogonal to $g$ (Remark 9.2.5.12). $\square$