Section 6. (0NS4) · ScalingStacks

6. Additivity

In this section we construct the universal additive invariant of small stable $\infty$ -categories; see theorem 6.10. Its construction is divided in two steps : first, using proposition 5.27, we construct the unstable version; see theorem 6.7. Then, by stabilizing, we obtain the universal additive invariant. Our arguments follow the pattern of the analogous result for dg-categories given in [73].

0NMT

Definition 6.1. Let ${\mathcal{D}}$ be a stable presentable $\infty$ -category. A functor

E:\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{D}}

is called an additive invariant of small stable $\infty$ -categories if it inverts Morita equivalences (see definition 2.14), preserves filtered colimits, and satisfies additivity, i.e., given a split-exact sequence

(6.2)

the functors $i$ and $g$ induce an equivalence in ${\mathcal{D}}$

E({\mathcal{A}})\vee E({\mathcal{B}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}E({\mathcal{C}})\,.

We denote by $\mathrm{Fun}_{\mathrm{add}}(\Cat_{\infty}^{\ex},{\mathcal{D}})$ the $\infty$ -category of additive invariants with values in ${\mathcal{D}}$ .

0NMU

Example 6.3. As we discuss in Sections 7 and 10, appropriate versions of algebraic $K$ -theory and topological Hochschild homology provide additive invariants of small stable $\infty$ -categories.

6.1. Unstable version

Let us denote by $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*}$ the $\infty$ -category

\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},{\mathcal{T}}_{\infty})_{*}

of presheaves of pointed spaces on the essentially small $\infty$ -category $(\Cat_{\infty}^{\perf})^{\omega}$ of compact idempotent-complete small stable $\infty$ -categories.

0NMV

Lemma 6.4. Let ${\mathcal{D}}$ be a pointed presentable $\infty$ -category. Then, we have an equivalence of $\infty$ -categories

\mathrm{Fun}^{\mathrm{L}}(\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*},{\mathcal{D}})\simeq\mathrm{Fun}_{\mathrm{flt}}(\Cat_{\infty}^{\perf},{\mathcal{D}})\,,

where the right-hand side denotes the $\infty$ -category of morphisms of $\infty$ -categories which preserve filtered colimits.

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Proof. The proof is a consequence of the equivalences

	$\displaystyle\mathrm{Fun}^{\mathrm{L}}(\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*},{\mathcal{D}})$	$\displaystyle\simeq\mathrm{Fun}((\Cat_{\infty}^{\perf})^{\omega},{\mathcal{D}})$
		$\displaystyle\simeq\mathrm{Fun}_{\mathrm{flt}}(\Ind((\Cat_{\infty}^{\perf})^{\omega}),{\mathcal{D}})$
		$\displaystyle\simeq\mathrm{Fun}_{\mathrm{flt}}(\Cat_{\infty}^{\perf},{\mathcal{D}})\,,$

where the first follows from [52, 5.1.5.6] and the fact that ${\mathcal{D}}$ is pointed, and the last follows from corollary 4.25. ∎

Let

\phi\colon\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*}

be the functor obtained by first taking the Yoneda embedding and then restricting the presheaves to the category $(\Cat_{\infty}^{\perf})^{\omega}$ . Recall from corollary 5.24 that we can choose a fixed set ${\mathcal{E}}$ of representatives of split-exact sequences in $(\Cat_{\infty}^{\perf})^{\omega}$ . We denote by ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ the localization of $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*}$ [52, 5.5.4.15] with respect to the set of maps

(6.5)

\phi({\mathcal{C}})/\phi({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\phi({\mathcal{B}})\,,

where ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a split-exact sequence in ${\mathcal{E}}$ . Finally, let ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ be the composite

(6.6)

where $\gamma$ is the localization functor.

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Theorem 6.7. The functor ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ inverts Morita equivalences, preserves filtered colimits, and sends split-exact sequences in $\Cat_{\infty}^{\ex}$ to cofiber sequences in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ . Moreover, ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ is universal with respect to these properties, i.e., given any pointed presentable $\infty$ -category ${\mathcal{D}}$ , we have an equivalence of $\infty$ -categories

({\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}})^{\ast}:\mathrm{Fun}^{\mathrm{L}}({\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}},{\mathcal{D}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Fun}_{\mathrm{add}}^{\mathrm{un}}(\Cat_{\infty}^{\ex},{\mathcal{D}})\,,

where the right-hand denotes the full subcategory of $\mathrm{Fun}(\Cat_{\infty}^{\ex},{\mathcal{D}})$ of morphisms of $\infty$ -categories which satisfy the above conditions.

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Proof. The result follows from definition 2.14, lemma 6.4 and from the universal property of Bousfield localization (see section 2.5: The functor $\phi$ preserves filtered colimits and by proposition 5.27 any split-exact sequence can be approximated by a filtered colimit of split-exact sequences in ${\mathcal{E}}$ ). ∎

6.2. Universal additive invariant

Let ${\mathcal{M}}_{\mathrm{add}}$ be the stabilization $\Stab({\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}})$ [53, §1.4] of ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ ; by construction, this is a stable $\infty$ -category. Denote by ${\mathcal{U}}_{\mathrm{add}}$ the following composite

\Cat_{\infty}^{\ex}\stackrel{{\scriptstyle{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}\longrightarrow\Stab({\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}})\,.

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Remark 6.8. Note that we have the equivalences

	$\displaystyle\Stab(\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*})$	$\displaystyle=\Stab(\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},{\mathcal{T}}_{\infty*}))$
		$\displaystyle\simeq\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},\Stab({\mathcal{T}}_{\infty*})))$
		$\displaystyle\simeq\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},{\mathcal{S}}_{\infty}),$

the last of which follows from [53, 1.4.4.11]. Therefore, defining

\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})=\mathrm{Fun}(((\Cat_{\infty}^{\perf})^{\omega})^{\op},{\mathcal{S}}_{\infty})

and writing

\psi\colon\Cat_{\infty}^{\perf}\longrightarrow\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})

for the natural functor, we see that ${\mathcal{M}}_{\mathrm{add}}$ can alternately be described as the localization of $\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})$ with respect to the set of maps

(6.9)

\psi({\mathcal{C}})/\psi({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\psi({\mathcal{B}})\,,

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Theorem 6.10. The functor ${\mathcal{U}}_{\mathrm{add}}$ is the universal additive invariant, i.e., given any stable presentable $\infty$ -category ${\mathcal{D}}$ , we have an equivalence of $\infty$ -categories

({\mathcal{U}}_{\mathrm{add}})^{\ast}\colon\mathrm{Fun}^{\mathrm{L}}({\mathcal{M}}_{\mathrm{add}},{\mathcal{D}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Fun}_{\mathrm{add}}(\Cat_{\infty}^{\ex},{\mathcal{D}})\,.

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Proof. The result follows from theorem 6.7 and from the universal property of stabilization (i.e., [53, 1.4.5.5]). Note that stabilization preserves colimits and ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ sends split-exact sequences to cofiber sequences, so the split-exact sequence (6.2) is sent to a split cofiber sequence ${\mathcal{U}}_{\mathrm{add}}({\mathcal{C}})\simeq{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}})\vee{\mathcal{U}}_{\mathrm{add}}({\mathcal{B}})$ in ${\mathcal{M}}_{\mathrm{add}}$ . ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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