ScalingStacks

Proof. Proposition 9.20 implies that $V(-)$ inverts Morita equivalences. Furthermore, by Lemma 9.7, ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve $\kappa$-filtered colimits for $\kappa>\omega$, and so $V(-)$ does as well. Now, let

$${\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{C}}$$

be an exact sequence. Proposition 9.20 implies that we can assume that $\Ho({\mathcal{A}})$ is a thick triangulated subcategory of $\Ho({\mathcal{B}})$. Consider the following diagram

(9.22)

where $\mathrm{Dia}({\mathcal{A}},{\mathcal{B}})$ is obtained by passage to the cofiber objectwise. Note that since in the above diagram (9.22) the upper row is objectwise a strict-exact sequence, we obtain a cofiber sequence

$$V({\mathcal{A}})\longrightarrow V({\mathcal{B}})\longrightarrow V({\mathcal{B}},{\mathcal{A}})\longrightarrow\Sigma V({\mathcal{A}})$$

in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$, where

$$V({\mathcal{B}},{\mathcal{A}}):=\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{B}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\,.$$

We now show that the induced map

(9.23)

$$V({\mathcal{B}},{\mathcal{A}})\longrightarrow V({\mathcal{C}})$$

is an equivalence. For this, consider the following commutative diagram

Since the induced triangulated functor

$$\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}}))$$

preserves $\kappa$-small colimits, [70, §3.1] implies that the triangulated category

$$\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}})/{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$$

is idempotent complete. Therefore, $\theta_{n}$ is an equivalence, and we obtain maps

$$\psi_{n}:\Sigma_{\kappa}^{(n)}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}})/{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$$

which induce maps

$$\Psi_{n}:\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{C}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Sigma^{-n-1}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n+1)}({\mathcal{B}})/\Sigma^{(n+1)}_{\kappa}({\mathcal{A}})).$$

It follows that the natural map

$$\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{B}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{C}}))$$

is an equivalence, which implies that the map (9.23) is an equivalence. ∎