ScalingStacks

Proof of theorem 9.8. Recall from subsection 8.3 that ${\mathcal{M}}_{\mathrm{loc}}$ is obtained by localizing ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ with respect to the set ${\mathcal{E}}_{\mathrm{L}}$. Since ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega})$ is compact in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$, it is sufficient by proposition 9.26 and the universal property of localization (see section 2.5) to show that the functor

$$\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),-)\colon{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow{\mathcal{S}}_{\infty}$$

sends the elements of ${\mathcal{E}}_{\mathrm{L}}$ to equivalences. This follows from the fact that the non-connective $K$-theory construction preserves filtered colimits (see [70, §7, Lemma 6]), and so the proof is finished. ∎