Proposition 9.5 (0NP7) · ScalingStacks

0NP7

Proposition 9.5. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ be an exact sequence of small stable $\infty$ -categories. Then the induced sequences

are exact.

0NP8

Proof. It suffices to show the result for ${\mathcal{F}}_{\kappa}$ , as the statement for $\Sigma_{\kappa}$ follows because colimits commute. Thus, we need to verify that

(\Ind_{\omega}({\mathcal{A}}))^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind_{\omega}({\mathcal{B}}))^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind_{\omega}({\mathcal{C}}))^{\kappa}

is exact. The sequence

\Ind_{\omega}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{C}}

is exact by Definition 5.12 and Proposition 5.15. Now the result follows from Proposition 5.17. ∎