ScalingStacks

Let

$$\phi\colon\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*}$$

be the functor obtained by first taking the Yoneda embedding and then restricting the presheaves to the category $(\Cat_{\infty}^{\perf})^{\omega}$. Recall from corollary 5.24 that we can choose a fixed set ${\mathcal{E}}$ of representatives of split-exact sequences in $(\Cat_{\infty}^{\perf})^{\omega}$. We denote by ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ the localization of $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*}$ [52, 5.5.4.15] with respect to the set of maps

(6.5)

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

where ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a split-exact sequence in ${\mathcal{E}}$. Finally, let ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ be the composite

(6.6)

where $\gamma$ is the localization functor.