ScalingStacks

Proof. By construction, the object $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact in $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$. Let $S$ denote the set of maps in (8.4), $\overline{S}$ the strongly saturated collection of arrows generated by $S$ [52, 5.5.4.5], and let $X$ be an $S$-local object such that the map $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is an $S$-local equivalence (i.e., $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is in $\overline{S}$). Then by definition,

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

so it suffices to show that the functor

(9.11)

$$R:=\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),-):\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}\longrightarrow{\mathcal{S}}_{\infty}$$

sends the maps in $\overline{S}$ to equivalences of spectra. Since $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ is a stable $\infty$-category and $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact, $R$ preserves small colimits, so the two-out-of-three property allows us to reduce to checking that $R$ sends the elements of $S$ to equivalences.

Consider the following diagram

(9.12)

By applying the functor (9.11) to the above diagram (9.12) we obtain by theorem 9.9 a diagram in $\mathcal{S}$

(9.13)

where the upper row is a homotopy cofiber sequence. Now, an argument analogous to the one used in the proof of proposition 7.19 (where we make use of Waldhausen’s fibration theorem) allow us to conclude that the lower row in the above diagram (9.13) is also a homotopy cofiber sequence. This completes the argument. ∎