ScalingStacks

Definition 9.30. Let ${\mathcal{C}}$ be a DHKS-saturated Waldhausen category with factorization. Then the non-connective $K$-theory $I\mspace{-6.mu}K({\mathcal{C}})$ of ${\mathcal{C}}$ is defined as the non-connective $K$-theory $I\mspace{-6.mu}K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])$ of the $\infty$-category

$$\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}]\simeq\colim\{\mathrm{N}({\mathcal{C}})[W^{-1}]\overset{\Sigma}{\longrightarrow}\mathrm{N}({\mathcal{C}})[W^{-1}]\overset{\Sigma}{\longrightarrow}\cdots\}$$

obtained by inverting the suspension on the underlying $\infty$-category $\mathrm{N}({\mathcal{C}})[W^{-1}]$ in the $\infty$-category $\Cat_{\infty}^{\mathrm{Rex}}$ of $\infty$-categories with finite colimits and right-exact functors.