ScalingStacks

Proof. The result follows from the analogue of the argument for lemma 6.4 in the context of $\kappa$-compact objects and $\Ind_{\kappa}$, and from the universal property of Bousfield localization (see section 2.5 the functor $\psi$ preserves $\kappa$-filtered colimits and proposition 5.27 shows that any split-exact sequence can be approximated by a $\kappa$-filtered colimit of split-exact sequences in $\underline{{\mathcal{E}}_{\mathrm{A}}^{\kappa}}$). ∎

Proof. The result follows from propositions 8.3 and 5.30, and from the universal property of localization (see section 2.5). ∎

Proof. The fact that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ preserves $\kappa$-filtered colimits is clear. Since a functor ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a Morita equivalence if and only if $\Idem({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{B}})$ is an equivalence, proposition 5.31 allow us to conclude that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ inverts Morita equivalences. We now show that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ sends exact sequences to cofiber sequences. Let

$${\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{C}}$$

be an exact sequence. Since we have an induced Morita equivalence ${\mathcal{B}}/{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$, it suffices to show that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ sends exact sequences of shape

$${\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{B}}/{\mathcal{A}}$$

to cofiber sequences in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$. Since $\Idem:\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}$ is a localization, it commutes with colimits and therefore the right-hand vertical map in the diagram

is a Morita equivalence. The bottom line is a strict-exact sequence, and so we conclude that ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ sends exact sequences to cofiber sequences. Finally, the universality of ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}$ follows from propositions 8.3 and 8.5, and from the universal property of localization (see section 2.5). ∎

Proof. Let us denote by ${\mathcal{M}}_{\mathrm{add}}^{\kappa}$ the small stable $\infty$-category constructed as in section 6 but where we use $(\Cat_{\infty}^{\perf})^{\kappa}$ instead of $(\Cat_{\infty}^{\perf})^{\omega}$. Similarly to ${\mathcal{U}}_{\mathrm{add}}$ we have a well-defined functor ${\mathcal{U}}_{\mathrm{add}}^{\kappa}\colon\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{M}}_{\mathrm{add}}^{\kappa}$ and so by performing a localization analogous to the one of subsection 8.3 (with ${\mathcal{E}}_{A}$ instead of ${\mathcal{E}}_{\mathrm{L}}$) we obtain a small stable $\infty$-category which we denote by ${\mathcal{M}}_{\mathrm{add}}^{\omega}$ and a composed functor

$${\mathcal{U}}_{\mathrm{add}}^{\omega}\colon\Cat_{\infty}^{\ex}\stackrel{{\scriptstyle{\mathcal{U}}_{\mathrm{add}}^{\kappa}}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\kappa}\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\omega}\,.$$

Let us start by showing that ${\mathcal{M}}_{\mathrm{add}}^{\omega}$ agrees with ${\mathcal{M}}_{\mathrm{add}}$ and that ${\mathcal{U}}_{\mathrm{add}}^{\omega}$ agrees with ${\mathcal{U}}_{\mathrm{add}}$. For this (and because of the universal property of ${\mathcal{U}}_{\mathrm{add}}$ and ${\mathcal{U}}_{\mathrm{add}}^{\omega}$) it suffices to show that ${\mathcal{U}}_{\mathrm{add}}^{\omega}$ preserves filtered colimits. Consider the composite

$$(\Cat_{\infty}^{\perf})^{\omega}\hookrightarrow(\Cat_{\infty}^{\perf})^{\kappa}\subset\Cat_{\infty}^{\ex}\stackrel{{\scriptstyle{\mathcal{U}}_{\mathrm{add}}^{\kappa}}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\kappa}\,.$$

It inverts Morita equivalences and sends split-exact sequences to split cofiber sequences. By the construction of ${\mathcal{M}}_{\mathrm{add}}$ we obtain then an additive invariant $\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{M}}_{\mathrm{add}}^{\kappa}$ and hence by the universal property of ${\mathcal{U}}_{\mathrm{add}}$ a colimit preserving functor $\Phi\colon{\mathcal{M}}_{\mathrm{add}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{M}}_{\mathrm{add}}^{\kappa}$ and a natural transformation $\eta\colon\Phi\circ{\mathcal{U}}_{\mathrm{add}}\Rightarrow{\mathcal{U}}_{\mathrm{add}}^{\kappa}$. We now observe that the two functors

$\displaystyle\Cat_{\infty}^{\ex}\stackrel{{\scriptstyle{\mathcal{U}}_{\mathrm{add}}^{\kappa}}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\kappa}\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\omega}$

$\displaystyle\Cat_{\infty}^{\ex}\stackrel{{\scriptstyle{\mathcal{U}}_{\mathrm{add}}}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\kappa}\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\omega}$

agree. Since they preserve $\kappa$-filtered colimits and every object in $\Cat_{\infty}^{\ex}$ can be expressed as a $\kappa$-filtered colimit of $\kappa$-small objects, it suffices to show that they agree for every $\kappa$-compact small stable $\infty$-category ${\mathcal{A}}$. The stable $\infty$-category ${\mathcal{A}}$ can be expressed as a filtered colimit $\underset{\alpha}{\mathrm{colim}}({\mathcal{A}}_{\alpha})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}$, with ${\mathcal{A}}_{\alpha}\in(\Cat_{\infty}^{\ex})^{\omega}$, and the evaluation of the natural transformation $\eta$ at ${\mathcal{A}}$ identifies with

$$\underset{\alpha}{\mathrm{colim}}\,\,{\mathcal{U}}_{\mathrm{add}}^{\kappa}({\mathcal{A}}_{\alpha})\longrightarrow{\mathcal{U}}_{\mathrm{add}}^{\kappa}({\mathcal{A}})\,.$$

Since these map belongs to ${\mathcal{E}}_{A}$, they become invertible in ${\mathcal{M}}_{\mathrm{add}}^{\omega}$, and so we conclude that the above two functors agree. Since the one on the right-hand side preserves filtered colimits we conclude that ${\mathcal{U}}_{\mathrm{add}}^{\omega}$ also preserves filtered colimits. This shows that ${\mathcal{U}}_{\mathrm{add}}^{\omega}$ agrees with ${\mathcal{U}}_{\mathrm{add}}$ (and hence that ${\mathcal{M}}_{\mathrm{add}}^{\omega}$ agrees with ${\mathcal{M}}_{\mathrm{add}}$). Now, let ${\mathcal{M}}_{\mathrm{loc}}^{\omega}$ be the category defined as ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ but with $\kappa$ replaced by $\omega$. Clearly, the associated functor ${\mathcal{U}}_{\mathrm{loc}}^{\omega}$ is the universal localizing invariant and so in order to conclude the proof of Theorem 8.7 it suffices to show that ${\mathcal{M}}_{\mathrm{loc}}$ agrees with ${\mathcal{M}}_{\mathrm{loc}}^{\omega}$ and that ${\mathcal{U}}_{\mathrm{loc}}$ agrees with ${\mathcal{U}}_{\mathrm{loc}}^{\omega}$. Starting with ${\mathcal{M}}_{\mathrm{add}}^{\kappa}$ we can perform the following two localizations

$\displaystyle{\mathcal{M}}_{\mathrm{add}}^{\kappa}\longrightarrow{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow{\mathcal{M}}_{\mathrm{loc}}^{\omega}$

$\displaystyle{\mathcal{M}}_{\mathrm{add}}^{\kappa}\longrightarrow{\mathcal{M}}_{\mathrm{add}}^{\omega}\simeq{\mathcal{M}}_{\mathrm{add}}\longrightarrow{\mathcal{M}}_{\mathrm{loc}}\,.$

Since these localizations are independent of the order in which they are performed, our claim follows and so the proof is finished. ∎