ScalingStacks

Lemma 5.25. Given a filtered diagram of exact sequences ${\mathcal{A}}_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}_{\alpha}$ of compact idempotent-complete small stable $\infty$-categories, the colimit ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ is an exact sequence of idempotent-complete small stable $\infty$-categories; that is, ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is fully faithful with cofiber ${\mathcal{C}}$.