ScalingStacks

Remark 6.8. Note that we have the equivalences

	$\displaystyle\Stab(\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*})$	$\displaystyle=\Stab(\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},{\mathcal{T}}_{\infty*}))$
		$\displaystyle\simeq\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},\Stab({\mathcal{T}}_{\infty*})))$
		$\displaystyle\simeq\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},{\mathcal{S}}_{\infty}),$

the last of which follows from  [53, 1.4.4.11]. Therefore, defining

$$\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})=\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},{\mathcal{S}}_{\infty})$$

and writing

$$\psi\colon\Cat_{\infty}^{\perf}\longrightarrow\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})$$

for the natural functor, we see that ${\mathcal{M}}_{\mathrm{add}}$ can alternately be described as the localization of $\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})$ with respect to the set of maps

(6.9)

$$\psi({\mathcal{C}})/\psi({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\psi({\mathcal{B}})\,,$$

where ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a split-exact sequence in ${\mathcal{E}}$.