ScalingStacks

Remark 1.8. In fact, stronger results are true. In equivalence (1.4), ${\mathcal{S}}_{\infty}^{\omega}$ can be replaced by any compact idempotent-complete small stable $\infty$-category ${\mathcal{B}}$ and the right-hand side by the $K$-theory spectrum of $\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}})$; see theorem 7.13. In equivalence (1.5), ${\mathcal{S}}_{\infty}^{\omega}$ can be replaced by any smooth and proper (i.e., dualizable) small stable $\infty$-category ${\mathcal{B}}$ and the right-hand side by $I\mspace{-6.mu}K({\mathcal{B}}^{\op}\widehat{\otimes}{\mathcal{A}})$; see theorem 9.36.