Corollary 7.12 (0NNH) · ScalingStacks

Corollary 7.12. In the setting of 7.11, there are equivalences

K({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K({\mathcal{M}}({\mathcal{C}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K(\mathrm{N}(({\mathcal{M}}({\mathcal{C}}))^{\cf}))

which are natural in weakly exact functors.

0NNI

Proof. First, since we have a natural DK-equivalence ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{M}}({\mathcal{C}})$ , there is a natural equivalence $K({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K({\mathcal{M}}({\mathcal{C}}))$ [13, 19, 81]. Next, since the category ${\mathcal{M}}({\mathcal{C}})$ satisfies the hypothesis of Theorem 7.8, the second equivalence holds. ∎