ScalingStacks

Proof. The object $\Sigma_{\kappa}^{(n)}({\mathcal{A}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ is already trivial in $\Cat_{\infty}^{\ex}$. Since Proposition 2.18 implies that ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ admits all $\kappa$-small colimits, for any ${\mathcal{B}}$ in $(\Cat_{\infty}^{\ex})^{\kappa}$ the small stable $\infty$-category $\mathrm{Fun}({\mathcal{B}},{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$ also admits all $\kappa$-small colimits. Thus, the connective $K$-theory spectrum $K(\mathrm{Fun}({\mathcal{B}},{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$ is trivial. Finally, theorem 9.9 and the fact that the objects $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}})$, with ${\mathcal{B}}$ in $(\Cat_{\infty}^{\ex})^{\kappa}$ generate the category $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ [52, 5.5.7.3] allow us to conclude that ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ becomes trivial after application of $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}$, and thus after application of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$. ∎