Subsection 9.1. (0NSG) · ScalingStacks

9.1. Non-connective $K$ -theory of $\infty$ -categories

In order to construct the non-connective $K$ -theory spectrum associated to a small stable $\infty$ -category, we use a generalization of the axiomatic framework due to Schlichting [70]. For an uncountable regular cardinal $\kappa$ , we will produce functors ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ from $\Cat_{\infty}^{\ex}$ to $\Cat_{\infty}^{\ex}$ such that for any small stable $\infty$ category ${\mathcal{A}}$ :

(i)

$K({\mathcal{F}}_{\kappa}{\mathcal{A}})$ is contractible,
(ii)

there are natural transformations

$\Id\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Sigma_{\kappa}$

such that ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Sigma_{\kappa}{\mathcal{A}}$ is exact,
(iii)

the functors ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve exact sequences,
(iv)

and ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve $\kappa$ -filtered colimits in $\Cat_{\infty}^{\ex}$ .

The idea is that ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ is a “ $K$ -theoretic cone” and so $\Sigma_{\kappa}{\mathcal{A}}$ is a “suspension” of ${\mathcal{A}}$ . Fix an uncountable regular cardinal $\kappa$ , and for a stable $\infty$ -category ${\mathcal{C}}$ recall from Section 2.4 that ${\mathcal{C}}^{\kappa}$ denotes the $\kappa$ -compact objects in ${\mathcal{C}}$ .

0NP3

Definition 9.1. Using proposition 2.18, we define ${\mathcal{F}}_{\kappa}{\mathcal{A}}=(\Ind_{\omega}({\mathcal{A}}))^{\kappa}$ and $\Sigma_{\kappa}{\mathcal{A}}$ to be the cofiber $(\Ind_{\omega}({\mathcal{A}}))^{\kappa}/{\mathcal{A}}$ .

0NP4

Remark 9.2. One might wish to simply use $\Ind_{\omega}{\mathcal{A}}$ as the cone construction; however, this will rarely turn out to be a small $\infty$ -category, whereas passing to the $\kappa$ -compact objects yields an (essentially) small $\infty$ -category by construction.

Observe that ${\mathcal{F}}_{\kappa}$ is a composite functor

(9.3)

\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}_{\omega}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex(\!\kappa)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}.

By construction and Propositions 5.6 and 5.9, we have an exact sequence

{\mathcal{A}}\longrightarrow{\mathcal{F}}_{\kappa}{\mathcal{A}}\longrightarrow\Sigma_{\kappa}{\mathcal{A}},

which is natural in small stable $\infty$ -categories ${\mathcal{A}}$ . Next, we check that ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ satisfies property (i) above.

0NP5

Lemma 9.4. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then $K({\mathcal{F}}_{\kappa}{\mathcal{A}})$ is trivial.

0NP6

Proof. Since $\kappa$ is uncountable, ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ has countable coproducts, and so the usual Eilenberg swindle argument implies that the identity map is null-homotopic on $K$ -theory and so its $K$ -theory vanishes. Specifically, the functor $F\colon{\mathcal{F}}_{\kappa}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}{\mathcal{A}}$ defined by $X\mapsto\coprod_{{\mathbb{N}}}X$ is exact. Moreover, there is a natural equivalence of exact functors $\id\coprod F\simeq F$ induced by the equivalence $X\coprod(\coprod_{{\mathbb{N}}}X)\simeq\coprod_{{\mathbb{N}}}X$ . Applying $K$ -theory, we can split off the $F$ component of the resulting equivalence of spectra and deduce that the identity of ${\mathcal{F}}_{\kappa}{\mathcal{A}}$ is null-homotopic. ∎

We must check that ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve exact sequences of small stable $\infty$ -categories.

0NP7

Proposition 9.5. 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 the induced sequences

are exact.

0NP8

Proof. It suffices to show the result for ${\mathcal{F}}_{\kappa}$ , as the statement for $\Sigma_{\kappa}$ follows because colimits commute. Thus, we need to verify that

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

is exact. The sequence

\Ind_{\omega}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{C}}

is exact by Definition 5.12 and Proposition 5.15. Now the result follows from Proposition 5.17. ∎

Passing to the triangulated homotopy category by composing with the functor $\Ho$ , we get a series of functors which satisfies Schlichting’s setup of [70, §2.2] and so produces negative $K$ -groups. Furthermore, we can define the non-connective $K$ -theory spectrum as follows, following [70, §12].

0NP9

Definition 9.6. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Its non-connective $K$ -theory spectrum $I\mspace{-6.mu}K({\mathcal{A}})$ is given by

I\mspace{-6.mu}K({\mathcal{A}}):=\colim_{n}\Omega^{n}K(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\,.

Here, $K$ stands for the $K$ -theory spectrum of §7.1, and the structure maps are induced from the exact sequences

\Sigma_{\kappa}^{(n)}({\mathcal{A}})\longrightarrow{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})\longrightarrow\Sigma_{\kappa}^{(n+1)}({\mathcal{A}})\qquad n\geq 0\,.

Schlichting’s axiomatic framework implies that this construction agrees with his when both are defined, and therefore we deduce from his comparison results [70, §8] that the non-connective $K$ -theory spectrum of Definition 9.6 agrees with the various classical constructions of non-connective $K$ -theory spectra.

Finally, we establish the final technical condition; this will be needed in the following sections.

0NPA

Lemma 9.7. The functors ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve $\kappa$ -filtered colimits.

0NPB

Proof. Recall that ${\mathcal{F}}_{\kappa}$ is the composite (9.3). Hence, the claim follows from the fact that the passage to $\Ind_{\kappa}$ and to $\kappa$ -compact objects preserves $\kappa$ -filtered colimits [52, 5.5.7.8, 5.5.7.10, 5.5.7.11]. Since $\Sigma_{\kappa}$ is the cofiber of the inclusion ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}$ and colimits commute, we deduce that $\Sigma_{\kappa}$ preserves $\kappa$ -filtered colimits if ${\mathcal{F}}_{\kappa}$ does. ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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9.1. Non-connective KK-theory of ∞\infty-categories

9.1. Non-connective $K$ -theory of $\infty$ -categories