Proposition 4.13 (0NKQ) · ScalingStacks

Proposition 4.13. There is an equivalence

\widehat{{\mathcal{A}}}_{\tri}\simeq M{\mathcal{A}}

in $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ , natural in spectral categories ${\mathcal{A}}$ .

0NKR

Proof. By proposition 4.11, we have natural equivalences

\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]^{\omega}\simeq\Psi_{\perf}{\mathcal{A}}\simeq\mathrm{Fun}^{\ex}(\Psi_{\tri}{\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})^{\omega}.

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 $\Sigma^{\infty}\widehat{a}\colon\Psi_{\tri}{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ . ∎