ScalingStacks

Proof. The result follows from the fact that $\Cat_{\infty}^{\perf}$ is an accessible localization of $\Cat_{\infty}^{\ex}$, and $\Cat_{\infty}^{\ex}$ itself is an accessible localization of the finitely presentable $\infty$-category of small spectral categories via theorem 1.10. ∎

Proof. This follows from the fact that the filtered colimit of fully faithful functors is a fully faithful functor and that the cofiber of a filtered colimit of fully faithful functors is equivalent to the filtered colimit of the cofibers. ∎

Proof. First observe that a split-exact sequence ${\mathcal{A}}\overset{f}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}{\mathcal{B}}\overset{g}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}{\mathcal{C}}$ is completely determined by the projection $g:{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ together with its section $j:{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$. This is because $f:{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is the fiber of $g$, which we may identify with the full subcategory of ${\mathcal{B}}$ spanned by the $b\in{\mathcal{B}}$ such that $g(b)\simeq 0$, and, since $f$ is fully faithful, $i:{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}$ is determined by the composite $f\circ i:{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$, the fiber

of the unit map of the adjunction $(g,j)$. Hence $\mathrm{Split}(\Cat_{\infty}^{\perf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Loc}(\Cat_{\infty}^{\perf})$ has contractible (homotopy) fibers and is therefore and equivalence. ∎

Proof. By proposition 5.26, we may equivalently show that $\mathrm{Loc}(\Cat_{\infty}^{\perf})$ is accessible. Recall that an adjunction of $\infty$-categories can be described as a map ${\mathcal{M}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Delta^{1}$ which is both a cocartesian fibration and a cartesian fibration [52, 5.2.2.1]. This leads us to consider the commutative diagram of pullback squares

in which $\Cat_{\infty/\Delta^{1}}^{\mathrm{cocart}}\subset\Cat_{\infty/\Delta^{1}}$ (respectively, $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart,ff}}\subset\Cat_{\infty/\Delta^{1}}$) denote the subcategories of cocartesian fibrations (respectively, cartesian fibrations whose straightenings are fully faithful) and functors which preserve cocartesian (respectively, cartesian) edges.

Since $\Cat_{\infty}^{\perf}\subseteq\Cat_{\infty}$ is an accessible functor between accessible $\infty$-categories, it suffices, using the [52, 5.4.4.3, 5.4.5.16, 5.4.6.6] and the duality between cartesian and cocartesian fibrations, to show that $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart,ff}}$ is accessible, and that the inclusions $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart,ff}}\subseteq\Cat_{\infty/\Delta^{1}}^{\mathrm{cart}}\subseteq\Cat_{\infty/\Delta^{1}}$ are accessible functors. The straightening functor gives an equivalence $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart}}\simeq\mathrm{Pre}_{\Cat_{\infty}}(\Delta^{1})$ between cartesian fibrations over $\Delta^{1}$ and presheaves of $\infty$-categories on $\Delta^{1}$ [52, 3.2.0.1].

In order to understand the condition of being fully faithful, we write $\Cat_{\infty}$ as an accessible localization $\Cat_{\infty}\subseteq\mathrm{Pre}(\mathrm{N}(\Delta))$ of simplicial spaces [47]. A functor is fully faithful when the corresponding map of (local) simplicial spaces is fully faithful, and recall that a map of simplicial spaces $j\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$ is fully faithful if and only if $\map(\Delta^{1},X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\map(\partial\Delta^{1},X)\times_{\map(\partial\Delta^{1},Y)}\map(\Delta^{1},Y)$ is an equivalence. It follows that $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart,ff}}$ is the accessible localization of $\mathrm{Pre}(\Delta^{1}\times\mathrm{N}(\Delta))$ obtained by also inverting the pushout product of $\Id_{\Delta^{1}}$ and $\partial\Delta^{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Delta^{1}$. Thus $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart,ff}}$ and $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart,ff}}\subseteq\Cat_{\infty/\Delta^{1}}^{\mathrm{cart}}$ are accessible.

Finally, it remains to show that the inclusion $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart}}\subseteq\Cat_{\infty/\Delta^{1}}$ is accessible. First, observe that finite limits commute with filtered colimits in $\Cat_{\infty}$, as $\Cat_{\infty}\simeq\Ind(\Cat_{\infty}^{\omega})$ is compactly generated, the inclusion $\Ind(\Cat_{\infty}^{\omega})\subseteq\mathrm{Pre}(\Cat_{\infty}^{\omega})$ preserves limits and filtered colimits [52, 5.3.5.3], and finite limits commute with filtered colimits in presheaf $\infty$-categories (this uses [52, 5.3.3.3] and the fact that (co)limits in presheaf $\infty$-categories are computed objectwise). It follows that the filtered colimit $\mathcal{C}\simeq\colim_{i}\mathcal{C}_{i}$ of cartesian fibrations $p_{i}\colon\mathcal{C}_{i}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Delta^{1}$, computed in $\Cat_{\infty}$, is itself a cartesian fibration $p\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Delta^{1}$; indeed, the inclusions $\mathcal{C}_{i}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$ preserve cartesian edges over $\Id_{\Delta^{1}}$, and inspection of the fibers

over each vertex $\Delta^{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Delta^{1}$ shows that $p\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Delta^{1}$ is also the colimit in $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart}}$. ∎