ScalingStacks

Proof. We must show that the natural map

$$\colim_{i}\Upsilon({\mathcal{C}}_{i})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Upsilon({\mathcal{C}})$$

is a DK-equivalence of spectral categories. Since ${\mathcal{C}}$ and the ${\mathcal{C}}_{i}$ are all stable spectral categories and $\Omega^{\infty}$ and $\mathrm{N}$ commute with filtered colimits, this follows from propositions 4.5 and 4.8. ∎