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.

The main technical device in our proofs of theorems 1.1 and 1.3 is the Morita theory of stable categories and spectral categories. In particular, we prove theorem 1.3 by using a comparison result between the theory of small spectral categories (see §2.1) and the theory of small stable $\infty$-categories to rigidify questions about the algebraic $K$-theory of $\infty$-categories to corresponding questions in the (classical) Waldhausen $K$-theory of Waldhausen categories.

The category $\Cat_{\mathcal{S}}$ of small spectral categories carries a Quillen model category structure in which the weak equivalences are the DK-equivalences, i.e., the functors that are fully faithful and essentially surjective up to weak homotopy equivalence; see [74] (reprised below in theorem 2.2). As a consequence, we can form the associated $\infty$-category $(\Cat_{\mathcal{S}})_{\infty}$ of small spectral categories.

A spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is called a triangulated equivalence if it induces a weak equivalence on the triangulated closures of ${\mathcal{A}}$ and ${\mathcal{B}}$, and it is called a Morita equivalence if it induces a weak equivalence on the thick closures of ${\mathcal{A}}$ and ${\mathcal{B}}$; see definition 2.7. Our comparison result, which can be regarded as a generalization of the Morita theory of [69], is the following.

We use this comparison result to deduce structural properties of the categories $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$, notably that they are compactly generated, complete, and cocomplete; see corollary 4.25. Furthermore, we prove theorem 1.3 by using theorem 1.10 to lift split-exact sequences of small $\infty$-categories to split-exact sequences of small spectral categories so we can take the $K$-theory in the setting of Waldhausen categories. More generally, this comparison result explains the relationship between the classical versions of algebraic $K$-theory (and topological Hochschild and cyclic homology) and the $\infty$-categorical versions. We believe that theorem 1.10 is of independent interest and expect it will find applications in the future. For instance, this theorem provides clean and concise proofs of the main theorems of Toën’s work [80] on internal $\hom$ objects in the category of dg-categories and its (previously unknown) extension to the context of spectral categories.

The category $\Cat_{\infty}^{\perf}$ is a symmetric monoidal $\infty$-category (in the sense of [53, §2]), in which the tensor product $\widehat{\otimes}$ is characterized by the property that maps out of ${\mathcal{A}}\widehat{\otimes}{\mathcal{B}}$ correspond to maps out of the product ${\mathcal{A}}\times{\mathcal{B}}$ which preserve colimits in each variable; see section 3 for a discussion of this structure, following the work of Lurie [53] and Ben-Zvi, Francis, and Nadler [8]. We will reserve a careful study of the structure of ${\mathcal{M}}_{\mathrm{add}}$ and ${\mathcal{M}}_{\mathrm{loc}}$ as symmetric monoidal $\infty$-categories for the forthcoming paper [9]. However, in order to carry out the extension of the non-connective co-representability theorem described in remark 1.8, we study the theory of dualizable objects in $\Cat_{\infty}^{\perf}$, using the theory of [53, §4.2.5].

In analogy with the situation for dg-categories [21, §4], we obtain a characterization of the dualizable objects in $\Cat_{\infty}^{\perf}$ as the smooth and proper objects. We define these notions as follows. Implicit in the comparison between small stable $\infty$-categories and spectral categories of theorem 1.10 is the fact that for objects $a$ and $b$ in a small stable $\infty$-category ${\mathcal{A}}$ there exists a natural mapping spectrum ${\mathcal{A}}(a,b)$ (see definition 2.15). Using this fact, we say that a small stable $\infty$-category ${\mathcal{A}}$ is proper if, for all pairs of objects $a$ and $b$ of ${\mathcal{A}}$, the mapping spectrum ${\mathcal{A}}(a,b)$ is compact. We say that a small stable $\infty$-category ${\mathcal{A}}$ is smooth if it is perfect as an ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$-module. (Here we use the fact that any small stable $\infty$-category can be regarded as a bimodule over itself.) We then have the following theorem characterizing the dualizable objects in these terms:

One of the major revolutions in the calculational study of algebraic $K$-theory of rings and schemes in the past two decades has been the development of trace methods, following the ideas of Goodwillie and Bokstedt-Hsiang-Madsen [17]. The cyclotomic trace from $K$-theory to topological cyclic homology $TC$ and topological Hochschild homology $THH$ (stable homotopy theory generalizations of negative cyclic homology and Hochschild homology) has allowed major calculational advances. The fiber of this map is well understood by work of Goodwillie, McCarthy, and Dundas [58, 26], and the target is relatively computable using the methods of equivariant stable homotopy theory (e.g., see the extensive body of work by Hesselholt and Madsen on the Quillen-Lichtenbaum conjecture [40]). One application of the co-representability of algebraic $K$-theorem (theorem 1.3) is the complete classification of all natural transformations with source the algebraic $K$-theory functor.

A calculation then provides a canonical construction and conceptual description of the topological Dennis trace map $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH$.

That is, up to scaling, the trace is the only natural transformation of additive invariants between connective algebraic $K$-theory and $THH$. This provides a direct proof that all known constructions of the topological Dennis trace map agree up to homotopy.

Working directly with topological cyclic homology ($TC$) is somewhat more complicated; $TC$ does not preserve filtered colimits in general, and is therefore not an additive or localizing functor. However, we deduce an analogous identification of the cyclotomic trace as determined by the unit map; see section 10.

Finally, we note that in the localizing setting our results provide an extension of the cyclotomic trace from non-connective algebraic $K$-theory to the non-connective versions of $TC$ and $THH$. This generalizes and extends the non-connective traces constructed in [35] and [10] for rings and schemes.

The “motivic” idea of constructing universal invariants is not new and appears in several different subjects: for example, Cortiñas-Thom’s work [22] on bivariant algebraic $K$-theory, Higson’s work [41] on Kasparov’s bivariant $K$-theory, Meyer-Nest’s work [60] on $C^{\ast}$-algebras, Morel-Voevodsky’s work [59] on $\mathbb{A}^{1}$-homotopy theory of schemes, and Voevodsky’s work [82] on (mixed) motives.

In this vein, over the past few years the third author and Cisinski have carried out a program [20, 21, 73] of providing universal characterizations of algebraic $K$-theory in the setting of dg-categories, using the formalism of derivators. One of the main goals of our work in this paper completes this program by extending these results to stable $\infty$-categories and by solving a key open question left open in the previous work of the third author, namely identifying the cyclotomic trace in terms of the co-representability results.

Finally, we would like to mention that Barwick [3] has recent work on a universal characterization of higher algebraic $K$-theory, in the context of a detailed study of the algebraic $K$-theory of $\infty$-categories.