Subsection 7.3. (0NSA) · ScalingStacks

7.3. Co-representability

This subsection is entirely devoted to the proof of theorem 7.13. The proof will follow from propositions 7.17 and 7.19.

0NNJ

Theorem 7.13. Let ${\mathcal{A}}$ be a small stable $\infty$ -category and ${\mathcal{B}}$ be a compact idempotent-complete small stable $\infty$ -category. Then there is a natural equivalence of spectra

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

When ${\mathcal{B}}$ is the small stable $\infty$ -category ${\mathcal{S}}_{\infty}^{\omega}$ of compact spectra, there is a natural equivalence of spectra

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

In particular, we have isomorphisms of abelian groups

\Hom({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega})),\Sigma^{-n}{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}}))\simeq K_{n}(\Idem({\mathcal{A}}))

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

0NNK

Notation 7.14. Given a small stable $\infty$ -category ${\mathcal{A}}$ , we denote by $K^{w}_{{\mathcal{A}}}$ the object

{\mathcal{B}}\mapsto|(S^{\infty}_{\bullet}(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))))_{\mathrm{iso}}|

in $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*}$ and by $K_{{\mathcal{A}}}$ the object

{\mathcal{B}}\mapsto K(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))

in $\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})$ . Note that the value of $K_{{\mathcal{A}}}$ at ${\mathcal{S}}_{\infty}^{\omega}$ is precisely the $K$ -theory spectrum $K({\mathcal{A}})$ of ${\mathcal{A}}$ , similarly and that $K^{w}_{{\mathcal{A}}}$ is the delooping of the $K$ -theory space.

0NNL

Remark 7.15. Recall that corollary 4.27 allow us to model the small $\infty$ -category of exact functors $\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))$ as the pretriangulated spectral category $\mathrm{rep}({\mathcal{B}},{\mathcal{A}})$ of right-compact $\Upsilon({\mathcal{A}})^{\op}\wedge\Upsilon({\mathcal{B}})$ -modules. Combined with proposition 2.10, this implies that the associated mapping space $(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))_{\mathrm{iso}}$ can be calculated as $|w_{\bullet}\mathrm{rep}({\mathcal{B}},\Idem({\mathcal{A}}))|$ . Moreover, $\mathrm{rep}({\mathcal{B}},\Idem({\mathcal{A}}))$ inherits a natural Waldhausen structure as a full subcategory of the cofibrant objects in the model structure on the category of ${\mathcal{B}}\text{-}\Idem({\mathcal{A}})$ -bimodules. As such, we can also consider the algebraic $K$ -theory space $|w_{\bullet}S_{\bullet}\mathrm{rep}({\mathcal{B}},\Idem({\mathcal{A}}))|$ and associated spectrum.

In the following results, we will use the observation that Waldhausen’s $S_{\bullet}$ construction, applied to a spectral category which is a Waldhausen category with the cofibrations inherited from a spectral model structure with all objects fibrant, produces a spectral category (where the mapping spectra are given by an appropriate end) [11, §3]. To ensure we are in this setting, we will tacitly use the equivalent model of spectral categories enriched in EKMM $S$ -modules, as explained in [10, §15]. Alternatively, we could stay with spectral categories in symmetric spectra and use the “Moore” $S_{\bullet}$ construction from [11, §4], which uses an explicit model of the homotopy end. We also need the following lemma which allows us to bring the $S_{\bullet}$ construction inside:

0NNM

Lemma 7.16. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small stable $\infty$ -categories. Then we have an equivalence of simplicial $\infty$ -categories

S^{\infty}_{\bullet}\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}})\simeq\mathrm{Fun}^{\ex}({\mathcal{B}},S^{\infty}_{\bullet}{\mathcal{A}})

and correspondingly an equivalence of spaces

|(S^{\infty}_{\bullet}\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}}))_{\mathrm{iso}}|\simeq|(\mathrm{Fun}^{\ex}({\mathcal{B}},S^{\infty}_{\bullet}{\mathcal{A}}))_{\mathrm{iso}}|.

0NNN

Proof. First, we show that for each $n$ there is an equivalence of $\infty$ -categories

\Gap([n],\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}}))\simeq\mathrm{Fun}^{\ex}({\mathcal{B}},\Gap([n],{\mathcal{A}})).

Since $\mathrm{Fun}(-,-)$ is defined simply as the mapping simplicial set [52, 1.2.7.2], we have the equivalence

\mathrm{Fun}(\mathrm{N}(\Ar[n]),\mathrm{Fun}({\mathcal{B}},{\mathcal{A}}))\simeq\mathrm{Fun}({\mathcal{B}},\mathrm{Fun}(\mathrm{N}(\Ar[n]),{\mathcal{A}})).

Since colimits in functor $\infty$ -categories are computed pointwise [52, §5.1.2.3] and the $\infty$ -category $\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}})$ is the full subcategory of $\mathrm{Fun}({\mathcal{B}},{\mathcal{A}})$ spanned by the exact functors, we have a map

\Gap([n],\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{B}},\Gap([n],{\mathcal{A}})),

and lemma 7.3 implies that it is an equivalence. It is now straightforward to check that these comparison maps assemble into the desired simplicial equivalence. ∎

We can now relate ${\mathcal{M}}_{\mathrm{add}}$ to the algebraic $K$ -theory presheaf.

0NNP

Proposition 7.17. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then, we have a natural equivalence $\Sigma({\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}}))\simeq K^{w}_{\mathcal{A}}$ in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ (see notation 6.6) and a natural equivalence $\Sigma{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}})\simeq\Sigma K_{{\mathcal{A}}}$ in ${\mathcal{M}}_{\mathrm{add}}$ .

0NNQ

Proof. We begin by handling the unstable case. Theorem 4.23 implies that we can model ${\mathcal{A}}$ by a small spectral category (which we still denote by ${\mathcal{A}}$ ). Following [56, 3.3], we consider the following sequence of simplicial spectral categories

{\mathcal{A}}_{\bullet}\stackrel{{\scriptstyle I}}{{\longrightarrow}}PS_{\bullet}{\mathcal{A}}\stackrel{{\scriptstyle Q}}{{\longrightarrow}}S_{\bullet}{\mathcal{A}}\,,

where ${\mathcal{A}}_{\bullet}$ is a constant simplicial object and $PS_{\bullet}{\mathcal{A}}$ is the simplicial path object of $S_{\bullet}{\mathcal{A}}$ . By applying the functor ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ to this sequence, we obtain an induced morphism

\Theta\colon{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(PS^{\infty}_{\bullet}{\mathcal{A}})/{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}}_{\bullet})\longrightarrow{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{\bullet}{\mathcal{A}})

of simplicial objects in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ . We now show that each component $\Theta_{n}$ of $\Theta$ is an equivalence. For each $n\geq 0$ , we have a split-exact sequence

in which

	$\displaystyle I_{n}$	$\displaystyle(A)=(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A\stackrel{{\scriptstyle\mathrm{Id}}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}A\stackrel{{\scriptstyle\mathrm{Id}}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}\cdots\stackrel{{\scriptstyle\mathrm{Id}}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}A),$
	$\displaystyle Q_{n}$	$\displaystyle(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n})=(A_{1}/A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n}/A_{0}),$
	$\displaystyle S_{n}$	$\displaystyle(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n-1})=(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n-1}),$
	$\displaystyle R_{n}$	$\displaystyle(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\longrightarrow A_{n-1})=A_{0}.$

By the construction of ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ (and of ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ ), we conclude that the induced morphisms

\Theta_{n}\colon{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(PS^{\infty}_{n}{\mathcal{A}})/{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}})\longrightarrow{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{n}{\mathcal{A}})\qquad n\geq 0\,,

are equivalences in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ . This allow us to obtain the following cocartesian square

and so a natural equivalence

\Sigma({\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}}))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}|{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{\bullet}{\mathcal{A}})|

in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ . By combining this equivalence with the equivalences

(7.18)	$\displaystyle{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{\bullet}{\mathcal{A}})$	$\displaystyle=$	$\displaystyle\|(\mathrm{Fun}^{\ex}(-,\Idem(S^{\infty}_{\bullet}{\mathcal{A}})))_{iso}\|$
		$\displaystyle\simeq$	$\displaystyle\|(S^{\infty}_{\bullet}\mathrm{Fun}^{\ex}(-,\Idem({\mathcal{A}})))_{iso}\|$
		$\displaystyle=$	$\displaystyle K^{w}({\mathcal{A}})\,,$

where (7.18) follows from lemma 7.16, we conclude that $\Sigma({\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}}))\simeq K^{w}_{\mathcal{A}}$ in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ . The identification in the stable setting follows from the unstable considerations and the usual passage from results on the $K$ -theory space to the $K$ -theory spectrum. ∎

0NNR

Proposition 7.19. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then, the presheaves $K^{w}_{{\mathcal{A}}}$ and $K_{{\mathcal{A}}}$ (see notation 7.14) are local, i.e., given any split-exact sequnce ${\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ in ${\mathcal{E}}$ , the induced maps of spectra (see (6.5) and (6.9))

\map(\phi({\mathcal{D}}),K^{w}_{{\mathcal{A}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Map}(\phi({\mathcal{C}})/\phi({\mathcal{A}}),K^{w}_{{\mathcal{A}}})

\map(\psi({\mathcal{D}}),K_{{\mathcal{A}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Map}(\psi({\mathcal{C}})/\psi({\mathcal{A}}),K_{{\mathcal{A}}})

are equivalences.

0NNS

Proof. The argument is exactly the same in both cases. Therefore, we discuss only the stable $K_{{\mathcal{A}}}$ . Since ${\mathcal{B}}$ , ${\mathcal{C}}$ and ${\mathcal{D}}$ belong to $(\Cat_{\infty}^{\perf})^{\omega}$ , the spectral Yoneda lemma shows us that we need to prove that the induced sequence of spectra

is a cofiber sequence. Using corollary 4.27 it suffices to consider the split-exact sequence of small spectral categories

Note that, again by corollary 4.27, all of these spectral categories carry a natural Waldhausen structure inherited from the usual model structure on spectral modules. We will apply Waldhausen’s fibration theorem [84, 1.6.4]. We have the Waldhausen category $v\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ , whose weak equivalences are the morphisms $f$ such that $\Cone(f)$ is contractible, as well as the Waldhausen category $w\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ , with the same cofibrations as $v\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ but whose weak equivalences are those $f$ such that $\Cone(f)$ belongs to $\mathrm{rep}({\mathcal{D}},{\mathcal{A}})$ . Moreover, we have a natural inclusion $v\mathrm{rep}({\mathcal{C}},{\mathcal{A}})\subset w\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ and an equivalence $\mathrm{rep}({\mathcal{C}},{\mathcal{A}})^{w}\simeq\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ ; see [84, § 1.6]. The conditions of [84, 1.6.4] are satisfied, so we obtain a cofiber sequence of spectra

K(\mathrm{rep}({\mathcal{D}},{\mathcal{A}}))\longrightarrow K(\mathrm{rep}({\mathcal{C}},{\mathcal{A}}))\longrightarrow K(\mathrm{rep}({\mathcal{B}},{\mathcal{A}})).

∎

Propositions 7.17 and 7.19 allow us to prove theorem 7.13 as follows : let ${\mathcal{A}}$ be stable $\infty$ -category and ${\mathcal{B}}$ a compact small idempotent-complete stable $\infty$ -category. By proposition 7.17 we have an equivalence ${\mathcal{U}}_{\mathrm{add}}({\mathcal{A}})\simeq K_{{\mathcal{A}}}$ and by proposition 7.19 $K_{{\mathcal{A}}}$ is local. Therefore, we have the following natural equivalence

\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{B}}),{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}}))\simeq\mathrm{Map}(\psi({\mathcal{B}}),K_{{\mathcal{A}}})\,,

where the right-hand side is calculated in $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega};{\mathcal{S}}_{\infty})$ . Since ${\mathcal{B}}$ belongs to $(\Cat_{\infty}^{\perf})^{\omega}$ , the presheaf $\psi({\mathcal{B}})$ is representable and so by the spectral Yoneda lemma we have $\mathrm{Map}(\psi({\mathcal{B}}),K_{{\mathcal{A}}})\simeq K_{{\mathcal{A}}}({\mathcal{B}})$ . Finally, since by definition of $K_{{\mathcal{A}}}$ we have $K_{{\mathcal{A}}}({\mathcal{B}})=K(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))$ the proof is finished.

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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