ScalingStacks

Proof. Since ${\mathcal{A}}$ generates $M{\mathcal{A}}$ under finite homotopy colimits and desuspensions, it suffices to show that $M\eta_{{\mathcal{A}}}$ preserves finite homotopy colimits and desuspensions. The fact that $M\eta_{{\mathcal{A}}}$ preserves finite homotopy colimits follows from the fact that $M\eta_{{\mathcal{A}}}$ is a homotopy left Kan extension along $\widehat{\eta_{{\mathcal{A}}}}$. But suspension is an example of a finite homotopy colimit, so we have that $M\eta_{{\mathcal{A}}}(\Sigma x)\simeq\Sigma M\eta_{{\mathcal{A}}}(x)$. Hence $M\eta_{{\mathcal{A}}}(x)\simeq\Sigma M\eta_{{\mathcal{A}}}(\Sigma^{-1}x)$, and as $M^{2}{\mathcal{A}}$ is stable we see that $M\eta_{{\mathcal{A}}}(\Sigma^{-1}x)\simeq\Sigma^{-1}M\eta_{{\mathcal{A}}}(x)$. The final statement is a consequence of corollary 4.16 and the fact that $M{\mathcal{A}}$ is a stable spectral category. ∎