Theorem 9.9 (0NPD) · ScalingStacks

0NPD

Theorem 9.9. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small stable $\infty$ -categories such that ${\mathcal{B}}$ is $\kappa$ -compact. Then there is a natural equivalence of spectra

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

If ${\mathcal{B}}={\mathcal{S}}_{\infty}^{\omega}$ is the $\infty$ -category of compact spectra, this reduces to an equivalence

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

0NPE

Proof. The proof is analogous to the argument for theorem 7.13; instead of the idempotent-complete stable $\infty$ -category $\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))$ we consider the small stable $\infty$ -category $\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}})$ . Note that since $\kappa>\omega$ , ${\mathcal{S}}_{\infty}^{\omega}$ belongs to $(\Cat_{\infty}^{\ex})^{\kappa}$ . ∎