ScalingStacks

Theorem 9.8. Let ${\mathcal{A}}$ be a small stable $\infty$-category. Then there is a natural equivalence of spectra

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

In particular, for each integer $n$, we have isomorphisms of abelian groups

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

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