ScalingStacks

Theorem 1.3. (see theorems 7.13 and 9.8) Let ${\mathcal{A}}$ be an idempotent-complete small stable $\infty$-category. Then, there are natural equivalences of spectra

(1.4)		$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}}))$	$\displaystyle\simeq$	$\displaystyle K({\mathcal{A}})$
(1.5)		$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))$	$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}})\,,$

where ${\mathcal{S}}_{\infty}^{\omega}$ is the small stable $\infty$-category of compact spectra. In particular, for all $n\in\mathbb{Z}$, we have isomorphisms of abelian groups

(1.6)		$\displaystyle\Hom({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega})),\Sigma^{-n}{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}}))$	$\displaystyle\simeq K_{n}({\mathcal{A}})$
(1.7)		$\displaystyle\Hom({\mathcal{U}}_{\mathrm{loc}}\,({\mathcal{S}}_{\infty}^{\omega}))\,,\Sigma^{-n}{\mathcal{U}}_{\mathrm{loc}}\,({\mathcal{A}}))$	$\displaystyle\simeq I\mspace{-6.mu}K_{n}({\mathcal{A}})$

in the triangulated categories $\Ho({\mathcal{M}}_{\mathrm{add}})$ and $\Ho({\mathcal{M}}_{\mathrm{loc}})$.