Proposition 4.20 (0NL3) · ScalingStacks

0NL3

Proposition 4.20. Let ${\mathcal{C}}\simeq\colim_{i}{\mathcal{C}}_{i}$ be a filtered colimit of stable $\infty$ -categories. Then there is an equivalence of spectral categories

\Upsilon({\mathcal{C}})\simeq\colim_{i}\Upsilon({\mathcal{C}}_{i}).

0NL4

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. ∎