Section 4. (0NRU) · ScalingStacks

4. Morita theory

There is a close connection between stable $\infty$ -categories and spectral categories. On the one hand, for every pair of objects in a stable $\infty$ -category we can extract a mapping spectrum, as we discussed in definition 2.15. On the other hand, given a category ${\mathcal{A}}$ enriched in spectra, the category of (right) ${\mathcal{A}}$ -modules has a standard projective model structure and the associated $\infty$ -category is stable.

The purpose of this section is to provide a precise account of the relationship between small spectral categories and small stable $\infty$ -categories. The moral of the story is that the homotopy theory of small spectral categories, localized at the Morita equivalences, is the same as the homotopy theory of small stable idempotent-complete $\infty$ -categories. Specifically, we prove theorem 1.10 from the introduction, which can be thought of as a generalization of the Morita theory of [69]; that is, small stable idempotent-complete $\infty$ -categories are $\infty$ -categories of modules, and the $\infty$ -category of exact functors between two such is a stable subcategory of the $\infty$ -category of bimodules.

Establishing this correspondence serves several purposes for us. For one thing, having models of $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$ as accessible localizations of an $\infty$ -category which arises as the nerve of a model category provides technical control on $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$ ; we use this to show that $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$ are compactly generated in corollary 4.25. For another, it permits us to rectify diagrams of small stable $\infty$ -categories to strict diagrams in $\Cat_{\mathcal{S}}$ . We exploit this to pass to rigid models for the purposes of using Waldhausen’s $K$ -theory machinery in Section 7.

As described in Section 2.2, we have several equivalent options for producing a model of the $\infty$ -category of spectral categories (with respect to the DK-equivalences): we can use the combinatorial simplicial model structure of Corollary 2.4 and take $\mathrm{N}((\Cat_{\mathcal{S}})^{\cf})$ , we can use the Dwyer-Kan simplicial localization followed by fibrant replacement to obtain $\mathrm{N}((L^{H}\Cat_{\mathcal{S}})^{\mathrm{fib}})$ , or we can invert the weak equivalences and form $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ . We will refer interchangably to the underlying $\infty$ -category as “the” $\infty$ -category of small spectral categories.

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.

0NK6

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.

0NK7

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.

0NK8

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 $\Cat_{\infty}^{\perf}\subset\Cat_{\infty}$ .

0NK9

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 $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}$ .

0NKA

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]).

0NKC

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.

0NKD

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.

0NKE

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}.

0NKF

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. ∎

4.2. Spectral enrichment of stable $\infty$ -categories

The mapping spaces of a stable $\infty$ -category ${\mathcal{C}}$ are naturally the underlying spaces of mapping spectra, as discussed in section 2.3. We now use this to construct a cofibrant and fibrant spectral category $\Upsilon(\mathcal{C})$ whose underlying $\infty$ -category $\mathrm{N}(\Omega^{\infty}\Upsilon(\mathcal{C}))$ is equivalent to $\mathcal{C}$ . Indeed, the simplicial category of presheaves of spectra

\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})

on the associated (cofibrant) simplicial category $\mathfrak{C}[\mathcal{C}]$ is simultaneously a simplicial model category as well as a spectral category, where the spectral enrichment is inherited from the spectral structure on ${\mathcal{S}}$ itself. Moreover, we have an equivalence

\mathrm{N}(\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})^{\cf})[W^{-1}]\simeq\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty}),

so it makes sense to ask whether or not a given presheaf of spectra is stably representable (in the underlying $\infty$ -category $\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$ ).

0NKG

Definition 4.7. Let

\Upsilon(\mathcal{C})\subset\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})

denote the full spectral subcategory spanned by those (projectively) cofibrant and fibrant functors which are stably representable.

0NKH

Proposition 4.8. For a small stable $\infty$ -category $\mathcal{C}$ , there is a natural equivalence of $\infty$ -categories $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}(\Omega^{\infty}\Upsilon(\mathcal{C}))$ .

0NKI

Proof. The spectral Yoneda embedding

\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})\simeq\mathrm{N}(\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})^{\cf})

is adjoint to a simplicial functor

\mathfrak{C}[\mathcal{C}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})^{\cf})

which evidently factors through the full simplicial subcategory $\Omega^{\infty}\Upsilon(\mathcal{C})$ spanned by the stably representable functors. The map $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}(\Omega^{\infty}\Upsilon(\mathcal{C}))$ is the adjoint of the resulting map $\mathfrak{C}[\mathcal{C}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{\infty}\Upsilon(\mathcal{C})$ .

To see that this map is an equivalence, we observe first that it is essentially surjective: indeed, a stably representable cofibrant and fibrant functor $X\colon\mathfrak{C}[\mathcal{C}]^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ is necessarily of the form $X\simeq\mathrm{Map}(-,A)$ for some spectrum object $A=\{a_{i}\}$ of $\mathcal{C}\simeq\mathrm{N}(\mathfrak{C}[\mathcal{C}]^{\mathrm{fib}})$ . Since $\mathcal{C}$ is stable, $a_{i}\simeq\Sigma^{i}a$ for some object $a$ of $\mathcal{C}$ , so $X$ is in the image of $\mathcal{C}$ (which sends $a$ to the presheaf represented by $\Sigma^{\infty}a$ ). This map is also fully faithful, because if $a$ and $b$ are any pair of objects of $\mathcal{C}$ , then

\map(\Sigma^{\infty}b,\Sigma^{\infty}a)\simeq\map(b,\Omega^{\infty}\Sigma^{\infty}a)\simeq\map(b,a)

since $a\simeq\Omega^{\infty}\Sigma^{\infty}a$ . ∎

We have the following description of $\Upsilon$ in terms of the stable Yoneda embedding.

0NKJ

Proposition 4.9. Let $\mathcal{C}$ be a small stable $\infty$ -category. The fully-faithful inclusion

\Upsilon(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})

factors, on the level of underlying $\infty$ -categories, as the composite

	$\displaystyle\mathrm{N}(\Omega^{\infty}\Upsilon(\mathcal{C}))\simeq\mathcal{C}\rightarrow\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$	^ω
	$\displaystyle\subseteq\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$	$\displaystyle\simeq\mathrm{N}\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})^{\cf}[W^{-1}].$

0NKK

Proof. Any stably representable functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is exact, giving the factorization

\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})\subseteq\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty}).

By proposition 3.2, we may rewrite this as $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind(\mathcal{C})\simeq\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$ to see that, as an exact functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ , any stably representable functor is also compact. ∎

Our model of $\Cat^{\ex}_{\mathcal{T}}$ allows us to check that the construction of $\Upsilon$ induces a simplicial functor:

0NKL

Proposition 4.10. The assignment which associates to the stable simplicial category $\mathcal{C}$ the spectral category $\Upsilon(\mathcal{C})$ defines a simplicial functor

\Upsilon\colon\Cat^{\ex}_{\mathcal{T}}\longrightarrow L^{H}(\Cat_{\mathcal{S}})

and hence a functor of $\infty$ -categories

\mathrm{N}(\Upsilon)\colon\Cat_{\infty}^{\ex}\longrightarrow\mathrm{N}((L^{H}(\Cat_{\mathcal{S}}))^{\mathrm{fib}}).

0NKM

Proof. We first check that the construction of $\Upsilon$ induces a functor $\Cat^{\ex}_{\mathcal{T}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\mathcal{S}}$ . Let $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ be a map of stable simplicial categories and write

f_{!}^{\mathrm{cf}}\colon\mathrm{Fun}_{\Delta}(\mathcal{C}^{\op},{\mathcal{S}})^{\mathrm{cf}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}_{\Delta}({\mathcal{D}}^{\op},{\mathcal{S}})^{\mathrm{cf}}

for the induced spectral functor. Suppose that $X\colon\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ is projectively cofibrant and fibrant and that $\mathrm{N}(X)\colon\mathrm{N}(\mathcal{C})^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is stably representable via the spectrum object $A=\{a_{i}\}$ in $\mathrm{N}\mathcal{C}$ . Since the diagram

commutes (where the vertical maps are the stable Yoneda embeddings), we see that $f_{!}$ restricts to a spectral functor $\Upsilon({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Upsilon({\mathcal{D}})$ .

To verify that $\Upsilon$ induces a simplicial functor, we must check that it preserves equivalences of stable simplicial categories. So suppose that $f\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ is an equivalence of stable simplicial categories. Then it follows that $f_{!}^{\mathrm{cf}}$ is a DK-equivalence of spectral categories, as is its restriction to the stably representable objects. ∎

0NKN

Proposition 4.11. Let ${\mathcal{A}}$ be a spectral category. Then there are natural equivalences of compactly-generated stable $\infty$ -categories

\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\simeq\Ind(\Psi_{\tri}{\mathcal{A}})\simeq\mathrm{Fun}^{\ex}(\Psi_{\tri}{\mathcal{A}}^{\op},{\mathcal{S}}_{\infty}).

0NKP

Proof. The first equivalence follows from the definition of $\Psi_{\tri}{\mathcal{A}}$ as the smallest stable subcategory of the stable $\infty$ -category $\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ containing the representables, together with the observations that $\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ is compactly generated with compact objects

\Psi_{\perf}{\mathcal{A}}\simeq\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]^{\omega}

and $\Ind(\Psi_{\tri}{\mathcal{A}})\simeq\Ind(\Psi_{\perf}{\mathcal{A}})$ (as $\Ind$ -categories are automatically idempotent-complete). The second equivalence follows immediately from proposition 3.2. ∎

4.3. The triangulated and Morita localizations

Let

M\colon\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]

denote the composite functor

(4.12)

As the previous proposition suggests, $M{\mathcal{A}}$ is essentially the same as the pretriangulated spectral closure $\widehat{{\mathcal{A}}}_{\tri}$ of ${\mathcal{A}}$ .

0NKQ

Proposition 4.13. There is an equivalence

\widehat{{\mathcal{A}}}_{\tri}\simeq M{\mathcal{A}}

in $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ , natural in spectral categories ${\mathcal{A}}$ .

0NKR

Proof. By proposition 4.11, we have natural equivalences

\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]^{\omega}\simeq\Psi_{\perf}{\mathcal{A}}\simeq\mathrm{Fun}^{\ex}(\Psi_{\tri}{\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})^{\omega}.

These allows us to identify the smallest stable subcategory $\Psi_{\tri}{\mathcal{A}}\subseteq\Psi_{\perf}{\mathcal{A}}$ spanned by the representable functors $\widehat{a}\colon{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ with the stably representable functors $\Sigma^{\infty}\widehat{a}\colon\Psi_{\tri}{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ . ∎

There is a natural transformation $\eta\colon\id\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M$ which can be described as follows: On the level of spectral categories, the Yoneda embedding

{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{A}}}

factors through the inclusion of the essentially small full spectral subcategory

\widehat{{\mathcal{A}}}_{\tri}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{A}}}.

The result is a natural transformation $\id\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{(-)}_{\tri}$ of endofunctors of $\Cat_{\mathcal{S}}$ . By proposition 4.13, there is a natural equivalence

\widehat{{\mathcal{A}}}_{\tri}\overset{\sim}{\longrightarrow}\Upsilon(\Psi_{\tri}{\mathcal{A}})=M{\mathcal{A}}

in $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ ; composing with this natural equivalence gives the desired natural transformation $\eta\colon\id\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M$ .

0NKS

Proposition 4.14. For any spectral category ${\mathcal{A}}$ , $\eta_{{\mathcal{A}}}\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{\mathcal{A}}$ is fully faithful.

0NKT

Proof. By Yoneda’s lemma, mapping spectra in $M{\mathcal{A}}$ between stably representable objects are given by mapping spectra between the representing spectrum objects, giving an equivalence

\mathrm{Map}_{M{\mathcal{A}}}(\eta_{{\mathcal{A}}}(a),\eta_{{\mathcal{A}}}(b))_{n}\simeq\map_{\Psi_{\tri}{\mathcal{A}}}(\widehat{a},{\Sigma^{n}}{\widehat{b}}).

Since $\Psi_{\tri}{\mathcal{A}}$ is a stable $\infty$ -category of spectral functors, Yoneda’s lemma also gives an equivalence

\mathrm{Map}_{{\mathcal{A}}}(a,b)_{n}\simeq\map_{\Psi_{\tri}{\mathcal{A}}}(\widehat{a},{\Sigma^{n}}{\widehat{b}}).

Hence $\mathrm{Map}_{{\mathcal{A}}}(a,b)\simeq\mathrm{Map}_{M{\mathcal{A}}}(\eta_{{\mathcal{A}}}(a),\eta_{{\mathcal{A}}}(b))$ . ∎

0NKU

Proposition 4.15. The functor $\eta_{{\mathcal{A}}}\colon{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{\mathcal{A}}$ is essentially surjective if and only if ${{\mathcal{A}}}$ is stable.

0NKV

Proof. Indeed, ${{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{{\mathcal{A}}}$ is essentially surjective if and only if $\Omega^{\infty}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{\infty}M{\mathcal{A}}$ is essentially surjective, which is the case if and only if ${\mathcal{A}}$ is already stable. ∎

Combining propositions 4.14 and 4.15, we obtain the following corollary.

0NKW

Corollary 4.16. The functor $\eta_{{\mathcal{A}}}\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{\mathcal{A}}$ is an equivalence of spectral categories if and only if ${\mathcal{A}}$ is a stable spectral category.

Next, we want to verify that $M$ is a localization.

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Proposition 4.17. The pair of natural transformations $\eta_{M{\mathcal{A}}},M\eta_{{\mathcal{A}}}\colon M{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M^{2}{\mathcal{A}}$ induce a homotopy commutative square

0NKY

Proof. First note that $M\eta_{{\mathcal{A}}}\colon M{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M^{2}{\mathcal{A}}$ sends $x\colon\widehat{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ to the functor $\widehat{\eta_{{\mathcal{A}}}}_{!}x\colon\widehat{M{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ induced by homotopy left Kan extension along $\widehat{\eta_{{\mathcal{A}}}}:\widehat{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{M{\mathcal{A}}}$ . If $x=\mathrm{Map}_{A}(-,a)$ is represented by the object $a$ of ${\mathcal{A}}$ , then the universal properties of representable functors and homotopy left Kan extensions force an equivalence $\widehat{\eta_{{\mathcal{A}}}}_{!}x\cong\mathrm{Map}(-,\eta_{{\mathcal{A}}}(a))$ , so that $\widehat{\eta_{{\mathcal{A}}}}_{!}x$ is represented by $\eta_{{\mathcal{A}}}(a)$ . It follows that the restrictions of $\eta_{M{{\mathcal{A}}}}$ and $M\eta_{{\mathcal{A}}}$ to ${\mathcal{A}}$ are equivalent. ∎

0NKZ

Corollary 4.18. The spectral functors $\eta_{M{\mathcal{A}}}$ and $M\eta_{{\mathcal{A}}}$ are equivalent. In particular, both $\eta_{M{\mathcal{A}}}$ and $M\eta_{{\mathcal{A}}}$ are equivalences.

0NL0

Proof. Since ${\mathcal{A}}$ generates $M{\mathcal{A}}$ under finite homotopy colimits and desuspensions, it suffices to show that $M\eta_{{\mathcal{A}}}$ preserves finite homotopy colimits and desuspensions. The fact that $M\eta_{{\mathcal{A}}}$ preserves finite homotopy colimits follows from the fact that $M\eta_{{\mathcal{A}}}$ is a homotopy left Kan extension along $\widehat{\eta_{{\mathcal{A}}}}$ . But suspension is an example of a finite homotopy colimit, so we have that $M\eta_{{\mathcal{A}}}(\Sigma x)\simeq\Sigma M\eta_{{\mathcal{A}}}(x)$ . Hence $M\eta_{{\mathcal{A}}}(x)\simeq\Sigma M\eta_{{\mathcal{A}}}(\Sigma^{-1}x)$ , and as $M^{2}{\mathcal{A}}$ is stable we see that $M\eta_{{\mathcal{A}}}(\Sigma^{-1}x)\simeq\Sigma^{-1}M\eta_{{\mathcal{A}}}(x)$ . The final statement is a consequence of corollary 4.16 and the fact that $M{\mathcal{A}}$ is a stable spectral category. ∎

0NL1

Corollary 4.19. The functor $M$ defines a localization of $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ with essential image the stable spectral categories.

0NL2

Proof. This follows from the previous proposition and corollary by [52, 5.2.7.4]. ∎

To see that we have an accessible localization, we need the following proposition:

0NL3

Proposition 4.20. Let ${\mathcal{C}}\simeq\colim_{i}{\mathcal{C}}_{i}$ be a filtered colimit of stable $\infty$ -categories. Then there is an equivalence of spectral categories

\Upsilon({\mathcal{C}})\simeq\colim_{i}\Upsilon({\mathcal{C}}_{i}).

0NL4

Proof. We must show that the natural map

\colim_{i}\Upsilon({\mathcal{C}}_{i})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Upsilon({\mathcal{C}})

is a DK-equivalence of spectral categories. Since ${\mathcal{C}}$ and the ${\mathcal{C}}_{i}$ are all stable spectral categories and $\Omega^{\infty}$ and $\mathrm{N}$ commute with filtered colimits, this follows from propositions 4.5 and 4.8. ∎

Following Definition 2.7, we make the following definitions.

0NL5

Definition 4.21. A map of small spectral $\infty$ -categories $f:{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is:

•

A triangulated equivalence if $\Psi_{\tri}{f}:\Psi_{\tri}{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\tri}{{\mathcal{B}}}$ is an equivalence of (stable) $\infty$ -categories, and
•

A Morita equivalence if $\Psi_{\perf}f\colon\Psi_{\perf}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}{\mathcal{B}}$ is an equivalence of (idempotent-complete) stable $\infty$ -categories.

Assembling the work of this section we obtain the following two results:

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Theorem 4.22. The functor

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

admits a fully faithful and accessible right adjoint

\Upsilon\colon\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}].

That is, the $\infty$ -category of stable $\infty$ -categories is an accessible localization of the $\infty$ -category of spectral categories obtained by inverting the triangulated equivalences.

0NL7

Proof. The follows from the factorization of $M$ given in equation 4.12 and proposition 4.20. ∎

Recall that we have a stable idempotent completion functor $\Idem\colon\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}$ . Since $\Idem$ is left adjoint to the (fully faithful) inclusion $\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}$ , $\Cat_{\infty}^{\perf}$ is the localization of $\Cat_{\infty}^{\ex}$ obtain by inverting idempotent completion maps. Further, recall that there is an equivalence $\Idem\circ\Psi_{\tri}\simeq\Psi_{\perf}$ .

0NL8

Theorem 4.23. The functor

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

admits a fully faithful and accessible right adjoint

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

That is, the $\infty$ -category of idempotent-complete stable $\infty$ -categories is an accessible localization of the $\infty$ -category of spectral categories obtained by inverting the Morita equivalences.

0NL9

Proof. The $\infty$ -category of idempotent-complete stable $\infty$ -categories is a localizing subcategory of the $\infty$ -category of stable $\infty$ -categories, and idempotent-completion is an accessible functor as the inclusion $\Psi_{\tri}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}$ preserves filtered colimits. ∎

0NLA

Remark 4.24. Theorems 4.22 and 4.23 imply that computing the localizations of the model category structure on spectral categories from Corollary 2.4 at the triangulated and Morita equivalences (as discussed in Remark 4.3) and passing to the simplicial nerve also yields the $\infty$ -categories $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$ respectively.

We conclude the section with the promised applications of the theory. First, the fact that we have accessible localizations provides the following corollary about the structure of $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$ .

0NLB

Corollary 4.25. The $\infty$ -categories $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$ are compactly generated, complete, and cocomplete.

We will use the comparison above to lift small stable $\infty$ -categories to spectral categories. To this end, we make the following definition.

0NLC

Definition 4.26. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small idempotent-complete stable $\infty$ -categories. We write $\mathrm{rep}({\mathcal{B}},{\mathcal{A}})=\Upsilon(\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}}))$ for the small pretriangulated spectral category associated to the small stable $\infty$ -category of exact functors from ${\mathcal{B}}$ to ${\mathcal{A}}$ .

0NLD

Corollary 4.27. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be idempotent-complete small stable $\infty$ -categories and let $\Upsilon(A)$ and $\Upsilon(B)$ be spectral categories lifting ${\mathcal{A}}$ and ${\mathcal{B}}$ . Then $\mathrm{N}(\mathrm{rep}({\mathcal{A}},{\mathcal{B}}))\simeq\mathrm{Fun}^{\ex}\!({\mathcal{A}},\!{\mathcal{B}})$ is equivalent to the $\infty$ -category of right-compact $\Upsilon(A)^{\op}\wedge\Upsilon(B)$ -modules.

0NLE

Proof. This follows from theorem 4.23 and corollary 3.3. ∎

We also record the following result concerning lifting diagrams of small stable $\infty$ -categories to diagrams of spectral categories.

0NLF

Proposition 4.28. Let $I$ be a small category. Given a diagram ${\mathcal{D}}$ of small stable $\infty$ -categories indexed by $\mathrm{N}(I)$ , there exists an $I$ -diagram of pretriangulated spectral categories $\widetilde{{\mathcal{D}}}$ lifting ${\mathcal{D}}$ .

0NLG

Proof. This is a consequence of [52, 4.2.4.4]. Given a diagram of small stable $\infty$ -categories, the equivalence in theorem 4.22 gives rise to a diagram in the localization of $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ . Including the localization into $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ , we now obtain a diagram in $\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\simeq\mathrm{N}((\Cat_{\mathcal{S}})^{\cf})$ and we can use [52, 4.2.4.4] to lift this to a rigid diagram in $\Cat_{\mathcal{S}}$ . ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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4. Morita theory

4.1. Stable envelopes of spectral categories

4.2. Spectral enrichment of stable ∞\infty-categories

4.3. The triangulated and Morita localizations

4.2. Spectral enrichment of stable $\infty$ -categories