ScalingStacks

Definition 9.6. Let ${\mathcal{A}}$ be a small stable $\infty$-category. Its non-connective $K$-theory spectrum $I\mspace{-6.mu}K({\mathcal{A}})$ is given by

$$I\mspace{-6.mu}K({\mathcal{A}}):=\colim_{n}\Omega^{n}K(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\,.$$

Here, $K$ stands for the $K$-theory spectrum of §7.1, and the structure maps are induced from the exact sequences

$$\Sigma_{\kappa}^{(n)}({\mathcal{A}})\longrightarrow{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})\longrightarrow\Sigma_{\kappa}^{(n+1)}({\mathcal{A}})\qquad n\geq 0\,.$$