Subsection 4.1. (0NRV) · ScalingStacks

4.1. Stable envelopes of spectral categories

Given any spectral category ${\mathcal{C}}$ , we can produce an $\infty$ -category by passing to the associated simplicial category, fibrantly replacing, and applying the simplicial nerve to obtain $\mathrm{N}(\Omega^{\infty}({\mathcal{C}})^{\textrm{fib}})$ . This process yields a functor

\Cat_{\mathcal{S}}\longrightarrow\Set_{\Delta},

from the category $\Cat_{\mathcal{S}}$ of small spectral categories to the category of simplicial sets. Precomposing with the functors $\widehat{(-)}_{\perf}$ and $\widehat{(-)}_{\tri}$ , we obtain functors

\psi_{\tri},\psi_{\perf}\colon\Cat_{\mathcal{S}}\longrightarrow\Set_{\Delta}

and a natural transformation $\psi_{\tri}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\psi_{\perf}$ . First, we observe that these functors are compatible with the weak equivalences of Theorem 2.2.

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Lemma 4.1. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small spectral categories, and let $f\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ be a DK-equivalence. Then the induced maps $\psi_{\tri}(f)$ and $\psi_{\perf}(f)$ are categorical equivalences of simplicial sets.

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Proof. If $f\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a DK-equivalence, then one can check that $(f_{!},f^{*})$ gives a Quillen equivalence between the spectral model categories $\widehat{{\mathcal{A}}}$ of ${\mathcal{A}}$ -modules and the spectral model category $\widehat{{\mathcal{B}}}$ of ${\mathcal{B}}$ -modules. Passing to underlying simplicial categories of cofibrant and fibrant objects, we see that $\Omega^{\infty}(\widehat{{\mathcal{A}}})^{\cf}=\Mod({\mathcal{A}})^{\cf}$ and $\Omega^{\infty}(\widehat{{\mathcal{B}}})=\Mod({\mathcal{B}})^{\cf}$ are DK-equivalent simplicial categories. Finally, applying the simplicial nerve yields categorically equivalent simplicial sets. Restricting to various full subcategories yields the result for $\psi_{\tri}(f)$ and $\psi_{\perf}(f)$ . ∎

Therefore, we have induced functors $\Psi_{\tri}$ and $\Psi_{\perf}$ connecting $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ and $\mathrm{N}((\Set_{\Delta})^{\mathrm{c}})[W^{-1}]$ , equipped with a natural transformation connecting them:

\Psi_{\tri}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}\colon\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\longrightarrow\mathrm{N}((\Set_{\Delta})^{\mathrm{c}})[W^{-1}]\simeq\Cat_{\infty}.

In fact, by construction these functors preserve triangulated and Morita equivalences respectively. Furthermore, $\Psi_{\tri}$ lands in small stable $\infty$ -categories and $\Psi_{\perf}$ lands in idempotent-complete stable $\infty$ -categories.

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Lemma 4.2. $\Psi_{\tri}{\mathcal{C}}$ factors through the subcategory $\Cat_{\infty}^{\ex}\subset\Cat_{\infty}$ , and $\Psi_{\perf}{\mathcal{C}}$ factors through the subcategory $\Cat_{\infty}^{\perf}\subset\Cat_{\infty}$ .

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Proof. As noted in remark 2.13, since $\psi_{\tri}$ is the underlying simplicial category associated to a pretriangulated spectral category, $\Psi_{\perf}$ is characterized as the idempotent-completion of $\psi_{\tri}$ given by proposition 3.2 coupled with corollary 3.3. Finally, note that maps of spectral categories induce, by left Kan extension, finite colimit-preserving on the level of stable $\infty$ -categories. ∎

Consequently, we may regard $\Psi_{\tri}$ as a functor $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}$ and $\Psi_{\perf}$ as a functor $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}$ .

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Remark 4.3. Using the machinery of combinatorial simplicial model categories, we can also localize the combinatorial model structure of corollary 2.4 on $\Cat_{\mathcal{S}}$ at the triangulated or Morita equivalences directly to obtain “triangulated” or “Morita” simplicial model categories on small spectral categories and then pass to simplicial nerves; this is equivalent to localizing the $\infty$ -category $\mathrm{N}((\Cat_{\mathcal{S}})^{\cf})$ .

The content of theorem 1.10 is that these functors are equivalences. We prove this theorem by producing an “inverse” to $\Psi_{\tri}$ and $\Psi_{\perf}$ such that the composite is a localization functor on $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ . We begin with the following definition:

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Definition 4.4. A simplicial category ${\mathcal{A}}$ is stable if the simplicial nerve of a fibrant replacement of ${\mathcal{A}}$ is a stable $\infty$ -category. A spectral category ${\mathcal{A}}$ is stable if its underlying simplicial category $\Omega^{\infty}{\mathcal{A}}$ is stable.

We have the following characterization of equivalences between stable spectral categories (see also [11, 5.7]).

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Proposition 4.5. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be stable spectral categories. Then a spectral functor $f\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a DK-equivalence if and only if

\Ho(\Omega^{\infty}f)\colon\Ho(\Omega^{\infty}{\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho(\Omega^{\infty}{\mathcal{B}})

is an equivalence.

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Proof. Certainly essential surjectivity is determined on the level of the homotopy category, so it suffices to show that, for all pairs of objects $a$ and $b$ of ${\mathcal{A}}$ , $\pi_{n}\mathrm{Map}(a,b)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{n}\mathrm{Map}(fa,fb)$ for all integers $n$ whenever this is the case for $n=0$ . Since ${\mathcal{A}}$ and ${\mathcal{B}}$ are stable,

\pi_{0}\mathrm{Map}(\Sigma^{n}a,b)\cong\pi_{n}\mathrm{Map}(a,b)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{n}\mathrm{Map}(fa,fb)\cong\pi_{0}\mathrm{Map}(\Sigma^{n}fa,fb),

so this is immediate. ∎

We write $\Cat_{\mathcal{T}}^{\ex}$ for the simplicial category of small stable simplicial categories. We can model this as the subcategory of the simplicial category $L^{H}(\Cat_{\mathcal{T}})$ of small simplicial categories where the objects are the stable simplicial categories and the mapping spaces are computed by restriction of vertices to those simplicial functors which represent exact functors upon passage to the simplicial nerve.

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Proposition 4.6. The $\infty$ -category obtained by applying the simplicial nerve to (a fibrant replacement of) $\Cat_{\mathcal{T}}^{\ex}$ is equivalent to the $\infty$ -category $\Cat_{\infty}^{\ex}$ . That is, the equivalence (induced by the simplicial nerve [52, 2.2.0.1])

\mathrm{N}((L^{H}(\Cat_{\mathcal{T}})^{\mathrm{fib}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}

restricts to an equivalence

\mathrm{N}((\Cat_{\mathcal{T}}^{\ex})^{\mathrm{fib}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}.

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Proof. It suffices to show that the mapping spaces in $\Cat_{\mathcal{T}}^{\ex}$ have the correct homotopy type, and this follows from the comparison between the mapping spaces of $\Cat_{\mathcal{T}}$ and $\Cat_{\infty}$ [52, 2.2.0.1] and the fact that on both sides we define the mapping spaces by the same restriction of vertices. ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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