ScalingStacks

Proof. If $f\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a DK-equivalence, then one can check that $(f_{!},f^{*})$ gives a Quillen equivalence between the spectral model categories $\widehat{{\mathcal{A}}}$ of ${\mathcal{A}}$-modules and the spectral model category $\widehat{{\mathcal{B}}}$ of ${\mathcal{B}}$-modules. Passing to underlying simplicial categories of cofibrant and fibrant objects, we see that $\Omega^{\infty}(\widehat{{\mathcal{A}}})^{\cf}=\Mod({\mathcal{A}})^{\cf}$ and $\Omega^{\infty}(\widehat{{\mathcal{B}}})=\Mod({\mathcal{B}})^{\cf}$ are DK-equivalent simplicial categories. Finally, applying the simplicial nerve yields categorically equivalent simplicial sets. Restricting to various full subcategories yields the result for $\psi_{\tri}(f)$ and $\psi_{\perf}(f)$. ∎