ScalingStacks

Let $\Cat_{\infty}^{\ex}$ be the $\infty$-category of small stable $\infty$-categories and $\Cat_{\infty}^{\perf}$ the full subcategory of idempotent-complete small stable $\infty$-categories; see §2.2. An exact functor $F:{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ in $\Cat_{\infty}^{\ex}$ is called a Morita equivalence when its idempotent completion $\Idem(F):\Idem({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{B}})$ is an equivalence of $\infty$-categories; see definition 4.21. This is equivalent to the condition that the induced map $F_{!}:\Mod({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Mod({\mathcal{B}})$ is an equivalence of $\infty$-categories.

A sequence ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ in $\Cat_{\infty}^{\perf}$ is called exact if the composite is zero, ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is fully faithful, and the map ${\mathcal{B}}/{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$, from the cofiber of the inclusion of ${\mathcal{A}}$ into ${\mathcal{B}}$ to ${\mathcal{C}}$, is an equivalence; see proposition 5.15. The sequence is called split-exact if there exist adjoint splitting maps such that the relevant composites are the respective identities; see definition 5.18. More generally, a sequence ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ in $\Cat_{\infty}^{\ex}$ is (split-) exact if $\Idem({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{B}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{C}})$ is (split-) exact. Now, let

be a functor with values in a stable presentable $\infty$-category ${\mathcal{D}}$. We say that $E$ is an additive invariant if it inverts Morita equivalences, preserves filtered colimits, and sends split-exact sequences to (split) cofiber sequences; see definition 6.1. This last condition corresponds to the stable $\infty$-categorical analogue of Waldhausen’s additivity theorem. If $E$ in fact sends all exact sequences to cofiber sequences we say that it is a localizing invariant; this property corresponds to Neeman’s generalization of the Thomason-Trobaugh localization theorem [62, 79].

Every localizing invariant is additive, but not every additive invariant is necessarily localizing. Examples of localizing invariants include the non-connective version of algebraic $K$-theory and topological Hochschild homology. Connective algebraic $K$-theory is an example of an additive invariant which is not localizing. Note that, as we discuss in sections 7 and 9, the $\infty$-categorical versions of these invariants agree with the “classical” definitions. For instance, we give a precise comparison in section 7 between Waldhausen’s algebraic $K$-theory of a Waldhausen category ${\mathcal{C}}$ and the $\infty$-categorical algebraic $K$-theory of the $\infty$-category underlying ${\mathcal{C}}$.

Our first main result is the following.

From a motivic perspective, the $\infty$-categories ${\mathcal{M}}_{\mathrm{add}}$ and ${\mathcal{M}}_{\mathrm{loc}}$ should be considered as candidate categories of non-commutative motives. In fact, theorem 1.1 shows us that every additive (respectively localizing) invariant factors uniquely through ${\mathcal{M}}_{\mathrm{add}}$ (resp., through ${\mathcal{M}}_{\mathrm{loc}}$). That is, all the information concerning additive (resp., localizing) invariants is encoded in ${\mathcal{M}}_{\mathrm{add}}$ (resp., in ${\mathcal{M}}_{\mathrm{loc}}$).

Our second main result is the following characterization of the higher algebraic $K$-theory of a stable $\infty$-category. Any stable $\infty$-category (and in particular ${\mathcal{M}}_{\mathrm{add}}$ and ${\mathcal{M}}_{\mathrm{loc}}$) admits natural mapping spectra; that is, a stable $\infty$-category is naturally enriched over the $\infty$-category ${\mathcal{S}}_{\infty}$ of spectra (see sections 2.3 and  4).

In particular, when ${\mathcal{A}}$ is the $\infty$-category of perfect complexes over a suitable scheme (or stack), we recover the $K$-theory spectra of the scheme (stack). Taking ${\mathcal{A}}$ to be the $\infty$-category of compact modules over a ring spectrum $R$, we recover the $K$-theory of $R$. When $R$ is a connective ring spectrum, we show in theorem 9.53 that the negative homotopy groups of the non-connective $K$-theory of $R$ are the same as those of $\pi_{0}R$. However, we expect the non-connective $K$-theory of a non-connective ring spectrum to be an interesting new invariant.

Note that the left-hand sides of the natural equivalences and isomorphisms of theorem 1.3 are defined solely in terms of universal constructions on presheaf categories; algebraic $K$-theory is not used in their construction. Rather, a variant of Waldhausen’s path-fibration argument (see proposition 7.17) shows that Waldhausen’s $S_{\bullet}$ construction acts as the suspension functor in ${\mathcal{M}}_{\mathrm{add}}$ and also ${\mathcal{M}}_{\mathrm{loc}}$.

Therefore, theorem 1.3 (combined with theorem 1.1) provides an intrinsic characterization of algebraic $K$-theory as a functor of stable $\infty$-categories. Furthermore, theorem 1.12 below (see also theorem 10.3 in the text) shows that the co-representability result coupled with the Yoneda lemma provides a complete classification of all natural transformations from algebraic $K$-theory to an arbitrary additive (or localizing) functor from small stable categories to spectra.