ScalingStacks

Proposition 5.22. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ be a (split-) exact sequence of small stable $\infty$-categories. Then there exists a (split-) exact sequence of small stable spectral categories

$$\widetilde{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{B}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{C}}}$$

such that $\Psi_{\perf}(\widetilde{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{B}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathcal{C}}})$ is naturally equivalent to ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$.