ScalingStacks

Theorem 7.13. Let ${\mathcal{A}}$ be a small stable $\infty$-category and ${\mathcal{B}}$ be a compact idempotent-complete small stable $\infty$-category. Then there is a natural equivalence of spectra

$$\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{B}}),{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}}))\simeq K(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))\,.$$

When ${\mathcal{B}}$ is the small stable $\infty$-category ${\mathcal{S}}_{\infty}^{\omega}$ of compact spectra, there is a natural equivalence of spectra

$$\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}}))\simeq K(\Idem({\mathcal{A}}))\,.$$

In particular, we have isomorphisms of abelian groups

$$\Hom({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega})),\Sigma^{-n}{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}}))\simeq K_{n}(\Idem({\mathcal{A}}))$$

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