ScalingStacks

Proof. If $S$ is of small generation then $S^{-1}{\mathcal{C}}$ is presentable and corepresents the functor $\mathrm{Fun}^{\mathrm{L}}_{S}({\mathcal{C}},-)$ by [52, 5.5.4.14, 5.5.4.20]. Conversely, if this functor is corepresentable by ${\mathcal{C}}^{\prime}$ then the identity ${\mathcal{C}}^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}^{\prime}$ determines a colimit-preserving functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}^{\prime}$. Let $T$ be the class of arrows in ${\mathcal{C}}$ which become invertible in ${\mathcal{C}}^{\prime}$, and note that $S\subseteq T$, $T$ is strongly saturated [52, 5.5.4.10], and $T$ is of small generation [52, 5.5.4.16] (the last claim uses the fact that the equivalences in ${\mathcal{C}}^{\prime}$ is the strongly saturated class generated by the identity of the initial object of ${\mathcal{C}}^{\prime}$, which follows from [52, 5.5.4.5, 5.5.4.6]). Thus $T^{-1}{\mathcal{C}}\simeq{\mathcal{C}}^{\prime}$, so ${\mathcal{C}}^{\prime}$ also corepresents the functor $\mathrm{Fun}^{\mathrm{L}}_{T}({\mathcal{C}},-)$, showing that a colimit-preserving functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ inverts the arrows of $S$ if and only if it inverts the arrows of $T$. Since $S$ is strongly saturated, we conclude that $S=T$. ∎

Proof. Let ${\mathcal{C}}$ be a presentable stable $\infty$-category, and note that a colimit-preserving functor ${\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ sends the arrows in $S$ to equivalences in ${\mathcal{C}}$ if and only if its restriction to ${\mathcal{A}}$ is trivial. We therefore may identify

with the full subcategory spanned by those colimit-preserving functors ${\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ which send the arrows in $S$ to equivalences in ${\mathcal{C}}$. It follows from lemma 5.5 that ${\mathcal{B}}/{\mathcal{A}}\simeq T^{-1}{\mathcal{B}}$, where $T$ is the strongly saturated class of arrows of ${\mathcal{B}}$ which become equivalences in ${\mathcal{B}}/{\mathcal{A}}$.

We now show that $S$ is strongly saturated, so that $S=T$. First, suppose given a cofiber sequence $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z$ in ${\mathcal{B}}$ such that $Z$ lies in the essential image of ${\mathcal{A}}$, and let $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X^{\prime}$ be any map. Then the cofiber of $X^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X^{\prime}\coprod_{X}Y$ is equivalence to $Z$, so is also in the essential image of ${\mathcal{A}}$. Second, given a diagram $f_{\alpha}\colon X_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y_{\alpha}$ in $\mathrm{Fun}(\Delta^{1},{\mathcal{B}})$ with colimit $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$, and suppose that the cofibers $Z_{\alpha}$ of each $f_{\alpha}$ lies in the essential image of ${\mathcal{A}}$. Commuting colimits implies that the cofiber $Z$ of $f$ is computed as the colimit of the $Z_{\alpha}$, and this lies in the essential image of ${\mathcal{A}}$ since ${\mathcal{A}}$ is closed under colimits and the functor ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ preserves colimits. Lastly, suppose $h=g\circ f$ is a composite of $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$ followed by $g\colon Y\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z$, and write $Y/X$, $Z/Y$, and $Z/X$ for the cofibers of $f$, $g$, and $h$, respectively. Then we have a cofiber sequence $Y/X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z/X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z/Y$, so if any two lie in the essential image of ${\mathcal{A}}$ then so does the third. ∎

Proof. Without loss of generality we may identify ${\mathcal{A}}$ with its essential image in ${\mathcal{B}}$, so that an arrow $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$ is in $T$ if and only if any cofiber $Z$ of $f$ lies in ${\mathcal{A}}$. By [52, 5.5.4.15] it suffices to show that $T\subseteq\overline{S}$, the strongly saturated class of arrows of ${\mathcal{B}}$ generated by $S$ (see [52, 5.5.4.5]). To see this, let $X\overset{f}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}Y\overset{g}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}Z$ be a cofiber sequence in ${\mathcal{B}}$ such that $Z$ is in ${\mathcal{A}}$. Then $Z=\colim_{\alpha}Z_{\alpha}$ is a $\kappa$-filtered colimit of objects $Z_{\alpha}\in{\mathcal{A}}^{\kappa}\subset{\mathcal{B}}^{\kappa}$ and $Y=\colim_{\alpha}Y_{\alpha}$ is a $\kappa$-filtered colimit of objects $Y_{\alpha}=Y\times_{Z}Z_{\alpha}$. Now $Y_{\alpha}$ may not be $\kappa$-compact, so write $Y_{\alpha}=\colim_{\beta}Y_{\alpha\beta}$ for some $Y_{\alpha\beta}\in{\mathcal{B}}^{\kappa}$ and consider the resulting diagram of cofiber sequences

in which the lower right and upper left squares are cartesian, which implies that these two squares are also cocartesian and that the maps $X_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ and $Z_{\alpha\beta}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Z_{\alpha}$ are equivalences. Hence $Z_{\alpha\beta}\in{\mathcal{A}}^{\kappa}\subseteq{\mathcal{B}}^{\kappa}$ and we conclude that $g_{\alpha\beta}$ and therefore $f_{\alpha\beta}$ as well are maps in ${\mathcal{B}}^{\kappa}$; in particular, $f_{\alpha\beta}$ is an arrow in $S$. It follows from [52, 5.5.4.5] that the pushout $f_{\alpha}$ of $f_{\alpha\beta}$ along $X_{\alpha\beta}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X_{\alpha}\simeq X$ is in $\overline{S}$, and we see from [52, 5.5.4.12] that $f\simeq\colim_{\alpha}f_{\alpha}:X\simeq\colim_{\alpha}X_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\colim Y_{\alpha}\simeq Y$ is then also in $\overline{S}$. ∎

Proof. By construction, ${\mathcal{B}}/{\mathcal{A}}\subseteq{\mathcal{B}}$ is the full subcategory on those objects $b$ such that $\map(a,b)\simeq*$ for all objects $a$ in the image of ${\mathcal{A}}$. This shows that, as full subcategories of $\Ho({\mathcal{B}})$, $\Ho({\mathcal{B}}/{\mathcal{A}})\subseteq\Ho({\mathcal{B}})/\Ho({\mathcal{A}})$. Conversely, if $b$ is in $\Ho({\mathcal{B}})/\Ho({\mathcal{A}})$, then $\pi_{0}\map(a,b)\simeq\ast$ for each object $a$ in the image of ${\mathcal{A}}$, and we claim that in fact $\map(a,b)\simeq*$. Indeed, ${\mathcal{A}}$ is a stable subcategory of ${\mathcal{B}}$, so that $\pi_{n}\map(a,b)\simeq\pi_{0}\map(\Sigma^{n}a,b)\simeq\ast$. Hence $\Ho({\mathcal{B}})/\Ho({\mathcal{A}})\subseteq\Ho({\mathcal{B}}/{\mathcal{A}})$ as well. ∎

Proof. The fully faithful inclusions ${\mathcal{A}}\subset\Ind_{\kappa}({\mathcal{A}})$ and ${\mathcal{B}}\subset\Ind_{\kappa}({\mathcal{B}})$ show that ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is fully faithful if and only if $\Ind_{\kappa}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\kappa}({\mathcal{B}})$ is fully faithful (for the reverse direction, this follows from the definition of the mapping spaces in $\Ind_{\kappa}(-)$). Thus it remains to check that ${\mathcal{B}}/{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ is an equivalence upon idempotent completion if and only if $\Ind_{\kappa}({\mathcal{B}})/\Ind_{\kappa}({\mathcal{A}})\simeq\Ind_{\kappa}({\mathcal{B}}/{\mathcal{A}})$. Since $\Ind_{\kappa}$ preserves cofibers, it is enough to check that the equivalence $\Ind_{\kappa}({\mathcal{B}})/\Ind_{\kappa}({\mathcal{A}})\simeq\Ind_{\kappa}({\mathcal{B}}/{\mathcal{A}})$ implies the equivalence ${\mathcal{B}}/{\mathcal{A}}\simeq{\mathcal{C}}$ whenever the latter are idempotent complete. Thus, given a $\kappa$-cocomplete small stable $\infty$-category ${\mathcal{D}}$ (which we assume is idempotent complete if $\kappa=\omega$), we must show that

is a fiber sequence of $\infty$-categories. Since ${\mathcal{D}}\simeq(\Ind_{\kappa}({\mathcal{D}}))^{\kappa}$, by adjunction this is equivalent to the sequence

which is a fiber sequence by assumption. ∎

Proof. We have equivalences

where the first equivalence follows from proposition 5.9 and the last equivalence follows from the fact that $\Ind_{\kappa}$ preserves cofibers. We therefore obtain a commutative (up to natural isomorphism) square

where the right vertical map is fully faithful and the bottom map is an equivalence.

To see that the top vertical map is fully faithful, we use Neeman’s generalization of Thomason’s localization theorem (see [61, 4.4.9] or  [62]) to show that the left vertical map is fully faithful. First, since $\Ind_{\kappa}({\mathcal{A}})$ and $\Ind_{\kappa}({\mathcal{B}})$ are presentable, the criterion of [50] (characterizing well-generated triangulated categories) and [53, 1.4.5.2] imply that $\Ho(\Ind_{\kappa}({\mathcal{B}}))$ is well-generated and (since the map $\Ind_{\kappa}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\kappa}({\mathcal{B}})$ is fully-faithful) the image of $\Ho(\Ind_{\kappa}({\mathcal{A}}))$ is a localizing subcategory generated by a small set of objects. Applying the form of Neeman’s theorem proved by Krause in [51, 7.2.1] now implies that

is a fully faithful map. Since [53, 1.4.5.1] implies that there is an equivalence $\Ho(\Ind_{\kappa}({\mathcal{A}})^{\kappa})\simeq\Ho(\Ind_{\kappa}({\mathcal{A}}))^{\kappa}$ and $\Ind_{\kappa}({\mathcal{A}})^{\kappa}\simeq{\mathcal{A}}$ up to idempotent completion (and similarly for ${\mathcal{B}}$), we conclude that the left vertical map is fully faithful.

Finally, this map is essentially surjective because there is a commutative (up to natural isomorphism) triangle

such that both maps from $\Ho({\mathcal{B}})$ are essentially surjective. ∎

Proof. Suppose ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ is exact. Then the composite is trivial, ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is fully faithful, and ${\mathcal{B}}/{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ is an equivalence up to idempotent completion, and so the same must be true on the level of triangulated homotopy categories. Thus it is enough to show that $\Ho({\mathcal{B}})/\Ho({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{C}})$ is an equivalence up to idempotent completion, which follows from proposition 5.14. Conversely, suppose that

is exact. Then ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is fully faithful by proposition 5.10, and the equivalences $\Ho({\mathcal{B}}/{\mathcal{A}})\simeq\Ho({\mathcal{B}})/\Ho({\mathcal{A}})\simeq\Ho({\mathcal{C}})$ (the last up to idempotent completion) implies that ${\mathcal{B}}/{\mathcal{A}}\simeq{\mathcal{C}}$ by corollary 5.11. ∎

Proof. Recall that a right adjoint preserves $\kappa$-filtered colimits if and only if its left adjoint preserves $\kappa$-compact objects [52, 5.5.1.4]. Since functors which preserve filtered colimits also preserve $\kappa$-filtered colimits and $\Ind(-)$ takes exact functors to functors which preserve compact objects, the result follows. ∎

Proof. First, by proposition 5.15, it suffices to check that

is an exact sequence of triangulated categories. Again, we will deduce this from Neeman’s generalization of Thomason’s localization theorem (see [61, 4.4.9] or  [62]), as follows. First, observe that [53, 1.4.5.1] implies that there is an equivalence $\Ho(\Ind({\mathcal{A}})^{\kappa})\simeq\Ho(\Ind({\mathcal{A}}))^{\kappa}$ (and analogous equivalences for the other terms in the sequence). Next, since $\Ind({\mathcal{A}})$ and $\Ind({\mathcal{B}})$ are presentable, the criterion of [50] (characterizing well-generated triangulated categories) and [53, 1.4.5.2] imply that $\Ho(\Ind({\mathcal{B}}))$ is well-generated and (since the map $\Ind({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind({\mathcal{B}})$ is fully-faithful) the image of $\Ho(\Ind({\mathcal{A}}))$ is a localizing subcategory generated by a small set of objects. Once again, the localization theorem [51, 7.2.1] implies that

is an equivalence up to idempotent completion. The hypothesis that ${\mathcal{B}}/{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ is an equivalence up to idempotent completion now implies the result. ∎

Proof. This follows from proposition 4.28. ∎

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 $\Cat_{\infty/\Delta^{1}}^{\mathrm{cart}}$. ∎

Proof. Write ${\mathcal{B}}\simeq\colim_{\alpha}{\mathcal{B}}_{\alpha}$ as a $\kappa$-filtered colimit of $\kappa$-compact stable $\infty$-categories ${\mathcal{B}}_{\alpha}$, and define ${\mathcal{A}}_{\alpha}={\mathcal{A}}\times_{\mathcal{B}}{\mathcal{B}}_{\alpha}$ to be the full subcategory of ${\mathcal{A}}$ consisting of those objects of ${\mathcal{A}}$ which lie in the image of ${\mathcal{B}}_{\alpha}$. Evidently, ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}/{\mathcal{A}}$ is the $\kappa$-filtered colimit of the exact sequences ${\mathcal{A}}_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}_{\alpha}/{\mathcal{A}}_{\alpha}$, and ${\mathcal{A}}_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}_{\alpha}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}_{\alpha}/{\mathcal{A}}_{\alpha}$ is strict-exact because if $Y\in{\mathcal{B}}_{\alpha}$ is a summand of $X\in{\mathcal{A}}_{\alpha}$ then $Y\in{\mathcal{A}}_{\alpha}$ because the image of $Y$ in ${\mathcal{B}}$ lies in ${\mathcal{A}}$. ∎

Proof. Write ${\mathcal{A}}\simeq\colim_{\alpha}{\mathcal{A}}_{\alpha}$ as a $\kappa$-filtered colimit of $\kappa$-compact small stable $\infty$-categories ${\mathcal{A}}_{\alpha}$. Then $\Idem({\mathcal{A}})\simeq\colim_{\alpha}\Idem({\mathcal{A}}_{\alpha})$, since $\Idem$ (viewed as an endofunctor of $\Cat_{\infty}^{\ex}$) commutes with $\kappa$-filtered colimits — this follows from the characterization of $\Idem$ in terms of a subcategory of the $\Ind$ category [52, 5.4.2.4] and the fact that filtered colimits in $\Cat_{\infty}^{\ex}$ can be computed in $\Cat_{\infty}$ [53, 1.1.4.6]. ∎