Subsection 5.2. (0NS0) · ScalingStacks

5.2. The Thomason-Neeman localization theorem

In fact, we can further reduce to a criterion on the level of homotopy categories. For this, we need the following proposition. In the proof, we take advantage of the detailed study of localization in the context of (well-generated) triangulated categories by Neeman [61, 62] and Krause [51] and the fact that the homotopy category of a presentable stable $\infty$ -category is a well-generated triangulated category (see [53, 1.4.5.2] and [50]).

0NM0

Proposition 5.14. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ be a fully faithful and $\kappa$ -small colimit preserving functor of $\kappa$ -cocomplete small stable $\infty$ -categories. Then the natural map

\Ho({\mathcal{B}})/\Ho({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{B}}/{\mathcal{A}})

is an equivalence. In other words, the functor $\Ho(-)$ preserves quotients of fully faithful functors.

0NM1

Proof. We have equivalences

\Ho(\Ind_{\kappa}({\mathcal{B}}))/\Ho(\Ind_{\kappa}({\mathcal{A}}))\simeq\Ho(\Ind_{\kappa}({\mathcal{B}})/\Ind_{\kappa}({\mathcal{A}}))\simeq\Ho(\Ind_{\kappa}({\mathcal{B}}/{\mathcal{A}})),

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

\Ho\Ind_{\kappa}({\mathcal{B}})^{\kappa}/\Ho\Ind_{\kappa}({\mathcal{A}})^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho\Ind_{\kappa}({\mathcal{B}})/\Ho\Ind_{\kappa}({\mathcal{A}})

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. ∎

Now we can obtain the following correspondence between exact sequences of small stable $\infty$ -categories and exact sequences of triangulated categories.

0NM2

Proposition 5.15. A sequence of $\kappa$ -cocomplete small stable $\infty$ -categories and $\kappa$ -small colimit preserving functors ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ is exact if and only if the associated sequence $\Ho({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{B}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{C}})$ of triangulated categories is exact, in the sense that the composite is trivial, $\Ho({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{B}})$ is fully faithful, and the map $\Ho({\mathcal{B}})/\Ho({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{C}})$ is an equivalence after idempotent completion.

0NM3

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

\Ho({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{B}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{C}})

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. ∎

Finally, we record a technical proposition that is used in the context of our construction of non-connective $K$ -theory. First, we need a technical lemma about the behavior of the $\Ind$ functor.

0NM4

Lemma 5.16. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ be an exact functor of small stable $\infty$ -categories. Then the induced map $\Ind({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind({\mathcal{B}})$ of presentable stable $\infty$ -categories preserves $\kappa$ -compact objects for all infinite regular cardinals $\kappa$ .

0NM5

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. ∎

Note that in the statement of the following proposition, we implicitly use the facts that stable $\infty$ -categories have all finite colimits and exact functors preserve finite colimits.

0NM6

Proposition 5.17. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ be an exact sequence of small stable $\infty$ -categories. Then for any infinite regular cardinal $\kappa$ ,

\Ind({\mathcal{A}})^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind({\mathcal{B}})^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind({\mathcal{C}})^{\kappa}

is an exact sequence of idempotent-complete small stable $\infty$ -categories.

0NM7

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

\Ho(\Ind({\mathcal{A}})^{\kappa})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho(\Ind({\mathcal{B}})^{\kappa})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho(\Ind({\mathcal{C}})^{\kappa})

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

\Ho(\Ind({\mathcal{B}}))^{\kappa}/\Ho(\Ind({\mathcal{A}}))^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho(\Ind({\mathcal{B}}/{\mathcal{A}}))^{\kappa}

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. ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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