Subsection 9.2. (0NSH) · ScalingStacks

9.2. Co-representability

This subsection is entirely devoted to the proof of the following co-representability result.

Theorem 9.8. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then there is a natural equivalence of spectra

\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),\,{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))\simeq I\mspace{-6.mu}K({\mathcal{A}})\,.

In particular, for each integer $n$ , we have isomorphisms of abelian groups

\Hom({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),\,\Sigma^{-n}{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))\simeq I\mspace{-6.mu}K_{n}({\mathcal{A}})

in the triangulated category $\Ho({\mathcal{M}}_{\mathrm{loc}})$ .

The proof of theorem 9.8 will follow from theorems 9.9 and 9.10, and from propositions 9.17 through 9.26.

0NPD

Theorem 9.9. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small stable $\infty$ -categories such that ${\mathcal{B}}$ is $\kappa$ -compact. Then there is a natural equivalence of spectra

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}}),\,\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}}))\simeq K(\mathrm{Fun}^{ex}({\mathcal{B}},{\mathcal{A}}))\,.

If ${\mathcal{B}}={\mathcal{S}}_{\infty}^{\omega}$ is the $\infty$ -category of compact spectra, this reduces to an equivalence

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}}))\simeq K({\mathcal{A}})\,.

0NPE

Proof. The proof is analogous to the argument for theorem 7.13; instead of the idempotent-complete stable $\infty$ -category $\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))$ we consider the small stable $\infty$ -category $\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}})$ . Note that since $\kappa>\omega$ , ${\mathcal{S}}_{\infty}^{\omega}$ belongs to $(\Cat_{\infty}^{\ex})^{\kappa}$ . ∎

0NPF

Theorem 9.10. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then there is a natural equivalence of spectra

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}}))\simeq K({\mathcal{A}})\,.

0NPG

Proof. By construction, the object $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact in $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ . Let $S$ denote the set of maps in (8.4), $\overline{S}$ the strongly saturated collection of arrows generated by $S$ [52, 5.5.4.5], and let $X$ be an $S$ -local object such that the map $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is an $S$ -local equivalence (i.e., $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is in $\overline{S}$ ). Then by definition,

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}}))\simeq\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),X),

so it suffices to show that the functor

(9.11)

R:=\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),-):\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}\longrightarrow{\mathcal{S}}_{\infty}

sends the maps in $\overline{S}$ to equivalences of spectra. Since $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ is a stable $\infty$ -category and $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact, $R$ preserves small colimits, so the two-out-of-three property allows us to reduce to checking that $R$ sends the elements of $S$ to equivalences.

Consider the following diagram

(9.12)

By applying the functor (9.11) to the above diagram (9.12) we obtain by theorem 9.9 a diagram in $\mathcal{S}$

(9.13)

where the upper row is a homotopy cofiber sequence. Now, an argument analogous to the one used in the proof of proposition 7.19 (where we make use of Waldhausen’s fibration theorem) allow us to conclude that the lower row in the above diagram (9.13) is also a homotopy cofiber sequence. This completes the argument. ∎

Let ${\bf V}$ be the partially ordered set $\{(i,j):|i-j|\leq 1,\,i,j\geq 0\}\subset\mathbb{N}\times\mathbb{N}$ . Given a small stable $\infty$ -category ${\mathcal{A}}$ , we denote by $\mathrm{Dia}({\mathcal{A}})$ the $\mathrm{N}({\bf V})$ -diagram

(9.14)

where ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ are as in Definition 9.1.

0NPH

Lemma 9.15. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then $\Sigma_{\kappa}^{(n)}({\mathcal{A}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ and ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ become trivial after application of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ .

0NPI

Proof. The object $\Sigma_{\kappa}^{(n)}({\mathcal{A}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ is already trivial in $\Cat_{\infty}^{\ex}$ . Since Proposition 2.18 implies that ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ admits all $\kappa$ -small colimits, for any ${\mathcal{B}}$ in $(\Cat_{\infty}^{\ex})^{\kappa}$ the small stable $\infty$ -category $\mathrm{Fun}({\mathcal{B}},{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$ also admits all $\kappa$ -small colimits. Thus, the connective $K$ -theory spectrum $K(\mathrm{Fun}({\mathcal{B}},{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$ is trivial. Finally, theorem 9.9 and the fact that the objects $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}})$ , with ${\mathcal{B}}$ in $(\Cat_{\infty}^{\ex})^{\kappa}$ generate the category $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ [52, 5.5.7.3] allow us to conclude that ${\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})$ becomes trivial after application of $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}$ , and thus after application of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ . ∎

Let ${\mathcal{A}}$ be a small stable $\infty$ -category. We denote by $V({\mathcal{A}})$ the object

V({\mathcal{A}})=\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))

in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ whose indexing maps are induced from the above diagram (9.14). Note that $V({\mathcal{A}})$ is functorial in ${\mathcal{A}}$ and that we have a natural map $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}V({\mathcal{A}})$ . We obtain then a well-defined functor $V$ along with a natural transformation:

(9.16)

\displaystyle V(-):\Cat_{\infty}^{\ex}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

\displaystyle\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}\Rightarrow V(-)\,.

0NPJ

Proposition 9.17. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then, there is a natural equivalence of spectra

\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,V({\mathcal{A}}))\simeq I\mspace{-6.mu}K({\mathcal{A}})\,.

0NPK

Proof. This follows from the following equivalences

(9.18)	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}})$	$\displaystyle=$	$\displaystyle\underset{n\geq 0}{\mathrm{hocolim}}\,\Omega^{n}K(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$
(9.18)		$\displaystyle\simeq$	$\displaystyle\underset{n\geq 0}{\mathrm{hocolim}}\,\Omega^{n}\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}})))$
(9.19)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty})^{\kappa}),\,\underset{n\geq 0}{\mathrm{colim}}\,\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}})))$
(9.19)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\kappa}),\,V({\mathcal{A}}))\,.$

Equivalence (9.18) comes from theorem 9.10 and equivalence (9.19) comes from the compactness of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ . ∎

0NPL

Proposition 9.20. The functor $V$ (9.16) inverts Morita equivalences.

0NPM

Proof. It suffices to show that $V(-)$ sends maps of shape ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{A}})$ to isomorphisms. Consider the following diagram

Proposition 2.18 implies that ${\mathcal{F}}_{\kappa}(P)$ is an equivalence. Therefore, since both rows are strict-exact sequences and $\Ho({\mathcal{A}})$ and $\Ho(\Idem({\mathcal{A}}))$ differ by direct summands, we conclude that $\Sigma_{\kappa}(P)$ is an equivalence. The definition of the functor $V(-)$ allow us to conclude the proof. ∎

0NPN

Proposition 9.21. The functor

V(-)\colon\Cat_{\infty}^{\ex}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

inverts Morita equivalences, preserves $\kappa$ -filtered colimits, and sends exact sequences to cofiber sequences.

0NPP

Proof. Proposition 9.20 implies that $V(-)$ inverts Morita equivalences. Furthermore, by Lemma 9.7, ${\mathcal{F}}_{\kappa}$ and $\Sigma_{\kappa}$ preserve $\kappa$ -filtered colimits for $\kappa>\omega$ , and so $V(-)$ does as well. Now, let

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

be an exact sequence. Proposition 9.20 implies that we can assume that $\Ho({\mathcal{A}})$ is a thick triangulated subcategory of $\Ho({\mathcal{B}})$ . Consider the following diagram

(9.22)

where $\mathrm{Dia}({\mathcal{A}},{\mathcal{B}})$ is obtained by passage to the cofiber objectwise. Note that since in the above diagram (9.22) the upper row is objectwise a strict-exact sequence, we obtain a cofiber sequence

V({\mathcal{A}})\longrightarrow V({\mathcal{B}})\longrightarrow V({\mathcal{B}},{\mathcal{A}})\longrightarrow\Sigma V({\mathcal{A}})

in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ , where

V({\mathcal{B}},{\mathcal{A}}):=\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{B}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\,.

We now show that the induced map

(9.23)

V({\mathcal{B}},{\mathcal{A}})\longrightarrow V({\mathcal{C}})

is an equivalence. For this, consider the following commutative diagram

Since the induced triangulated functor

\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}}))

preserves $\kappa$ -small colimits, [70, §3.1] implies that the triangulated category

\Ho({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}})/{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))

is idempotent complete. Therefore, $\theta_{n}$ is an equivalence, and we obtain maps

\psi_{n}:\Sigma_{\kappa}^{(n)}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{B}})/{\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}})

which induce maps

\Psi_{n}:\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{C}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Sigma^{-n-1}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n+1)}({\mathcal{B}})/\Sigma^{(n+1)}_{\kappa}({\mathcal{A}})).

It follows that the natural map

\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{B}})/\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{C}}))

is an equivalence, which implies that the map (9.23) is an equivalence. ∎

0NPQ

Corollary 9.24 (of proposition 9.21). There is a functor

\mathrm{Loc}:{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

such that $\mathrm{Loc}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}}))\simeq V({\mathcal{A}})$ , for every small stable $\infty$ -category ${\mathcal{A}}$ .

0NPR

Proof. This follows from propositions 9.21 and 8.6. ∎

0NPS

Proposition 9.25. The two functors

\mathrm{Loc},\gamma^{\ast}\colon{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

are canonically equivalent, where $\gamma^{\ast}$ is the right adjoint of the localization functor.

0NPT

Proof. Let us denote by ${\bf L}$ the endofunctor $\mathrm{Loc}\circ\gamma$ of $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ . Note that we have a natural transformation $\Id\Rightarrow{\bf L}$ . Making use of the definition of $V(-)$ and of the fact that colimits in $\infty$ -categories commute, we observe that ${\bf L}$ is a localization functor on $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ [52, 5.2.7.4]. Therefore, it suffices to show that a map in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ becomes an equivalence in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ if and only if it becomes an equivalence after application of ${\bf L}$ . This follows from the fact that for every small stable $\infty$ -category ${\mathcal{A}}$ , we have an equivalence $\gamma(V({\mathcal{A}}))\simeq{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})$ : note that we have cofiber sequences in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$

{\mathcal{U}}_{\mathrm{loc}}^{\kappa}(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow{\mathcal{U}}_{\mathrm{loc}}^{\kappa}(\Sigma_{\kappa}^{(n+1)}({\mathcal{A}}))\,.

∎

0NPU

Proposition 9.26. Let ${\mathcal{A}}$ be small stable $\infty$ -category. We have a natural isomorphism in the stable homotopy category of spectra

\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),\,{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}}))\simeq I\mspace{-6.mu}K({\mathcal{A}})\,.

0NPV

Proof. This follows from the following equivalences

(9.27)	$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}}))$	$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\gamma^{\ast}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})))$
(9.27)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\mathrm{Loc}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})))$
(9.28)		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),V({\mathcal{A}}))$
(9.29)		$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}})\,.$

Equivalence (9.27) comes from proposition 9.25, equivalence (9.28) comes from corollary 9.24, and equivalence (9.29) is proposition 9.17. ∎

0NPW

Proof of theorem 9.8. Recall from subsection 8.3 that ${\mathcal{M}}_{\mathrm{loc}}$ is obtained by localizing ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ with respect to the set ${\mathcal{E}}_{\mathrm{L}}$ . Since ${\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega})$ is compact in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ , it is sufficient by proposition 9.26 and the universal property of localization (see section 2.5) to show that the functor

\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),-)\colon{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow{\mathcal{S}}_{\infty}

sends the elements of ${\mathcal{E}}_{\mathrm{L}}$ to equivalences. This follows from the fact that the non-connective $K$ -theory construction preserves filtered colimits (see [70, §7, Lemma 6]), and so the proof is finished. ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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