ScalingStacks

Proof. By proposition 4.11, we have natural equivalences

These allows us to identify the smallest stable subcategory $\Psi_{\tri}{\mathcal{A}}\subseteq\Psi_{\perf}{\mathcal{A}}$ spanned by the representable functors $\widehat{a}\colon{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ with the stably representable functors $\Sigma^{\infty}\widehat{a}\colon\Psi_{\tri}{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$. ∎

Proof. By Yoneda’s lemma, mapping spectra in $M{\mathcal{A}}$ between stably representable objects are given by mapping spectra between the representing spectrum objects, giving an equivalence

Since $\Psi_{\tri}{\mathcal{A}}$ is a stable $\infty$-category of spectral functors, Yoneda’s lemma also gives an equivalence

Hence $\mathrm{Map}_{{\mathcal{A}}}(a,b)\simeq\mathrm{Map}_{M{\mathcal{A}}}(\eta_{{\mathcal{A}}}(a),\eta_{{\mathcal{A}}}(b))$. ∎

Proof. Indeed, ${{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{{\mathcal{A}}}$ is essentially surjective if and only if $\Omega^{\infty}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{\infty}M{\mathcal{A}}$ is essentially surjective, which is the case if and only if ${\mathcal{A}}$ is already stable. ∎

Proof. First note that $M\eta_{{\mathcal{A}}}\colon M{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M^{2}{\mathcal{A}}$ sends $x\colon\widehat{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ to the functor $\widehat{\eta_{{\mathcal{A}}}}_{!}x\colon\widehat{M{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ induced by homotopy left Kan extension along $\widehat{\eta_{{\mathcal{A}}}}:\widehat{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{M{\mathcal{A}}}$. If $x=\mathrm{Map}_{A}(-,a)$ is represented by the object $a$ of ${\mathcal{A}}$, then the universal properties of representable functors and homotopy left Kan extensions force an equivalence $\widehat{\eta_{{\mathcal{A}}}}_{!}x\cong\mathrm{Map}(-,\eta_{{\mathcal{A}}}(a))$, so that $\widehat{\eta_{{\mathcal{A}}}}_{!}x$ is represented by $\eta_{{\mathcal{A}}}(a)$. It follows that the restrictions of $\eta_{M{{\mathcal{A}}}}$ and $M\eta_{{\mathcal{A}}}$ to ${\mathcal{A}}$ are equivalent. ∎

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. ∎

Proof. This follows from the previous proposition and corollary by [52, 5.2.7.4]. ∎

Proof. We must show that the natural map

is a DK-equivalence of spectral categories. Since ${\mathcal{C}}$ and the ${\mathcal{C}}_{i}$ are all stable spectral categories and $\Omega^{\infty}$ and $\mathrm{N}$ commute with filtered colimits, this follows from propositions 4.5 and 4.8. ∎

Proof. The follows from the factorization of $M$ given in equation 4.12 and proposition 4.20. ∎

Proof. The $\infty$-category of idempotent-complete stable $\infty$-categories is a localizing subcategory of the $\infty$-category of stable $\infty$-categories, and idempotent-completion is an accessible functor as the inclusion $\Psi_{\tri}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}$ preserves filtered colimits. ∎

Proof. This follows from theorem 4.23 and corollary 3.3. ∎

Proof. This is a consequence of [52, 4.2.4.4]. Given a diagram of small stable $\infty$-categories, the equivalence in theorem 4.22 gives rise to a diagram in the localization of $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$. Including the localization into $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$, we now obtain a diagram in $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\simeq\mathrm{N}((\Cat_{\mathcal{S}})^{\cf})$ and we can use [52, 4.2.4.4] to lift this to a rigid diagram in $\Cat_{\mathcal{S}}$. ∎