ScalingStacks

We begin by observing that $THH$ provides a localizing invariant of small stable $\infty$-categories. Although it is possible to do this directly in the setting of $\infty$-categories (see for instance the more general discussion of topological chiral homology in [53, §5.3] or the constructions outlined in [8, 5.1.1]), we use existing constructions in the setting of spectral categories in order to ease technical difficulties that arise in the subsequent construction of $TR$ and $TC$. Our basic sources for this material are [10] and [11].

Recall that for a small spectral category ${\mathcal{C}}$ we can define $THH({\mathcal{C}})$ in terms of the Hochschild-Mitchell cyclic nerve for spectral categories [10, §3]. The cyclic nerve is defined as the simplicial object

$$N^{\cyc}_{q}{\mathcal{C}}=\bigvee{\mathcal{C}}(c_{q-1},c_{q})\wedge\dotsb\wedge{\mathcal{C}}(c_{0},c_{1})\wedge{\mathcal{C}}(c_{q},c_{0}),$$

where the sum is over the $(q+1)$-tuples $(c_{0},\dotsc,c_{q})$ of objects of ${\mathcal{C}}$. This becomes a simplicial object using the usual cyclic bar construction face and degeneracy maps: The unit maps of ${\mathcal{C}}$ induce the degeneracy maps, and the composition maps in ${\mathcal{C}}$ (along with the twist map at the end) induce the face maps. We denote the geometric realization as $N^{\cyc}{\mathcal{C}}$.

The spectrum $N^{\cyc}{\mathcal{C}}$ has the correct homotopy type only when ${\mathcal{C}}$ has cofibrant mapping spectra [10, 3.1]. Since the cofibrant objects in the Morita model structure on small spectral categories reviewed in theorem 2.2 have cofibrant mapping spectra [74, 4.18], we can define the functor

$$THH:=N^{\cyc}\circ Q\colon\Cat_{\mathcal{S}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{S}$$

where $Q$ denotes the cofibrant replacement functor in the Morita model structure on $\Cat_{\mathcal{S}}$. Since this construction preserves Morita equivalences [10, 5.12] (and in fact DK-equivalences [10, 5.9]) the functor descends to the level of $\infty$-categories.

This definition of $THH$ as a functor of $\infty$-categories lets us deduce the following proposition from known properties of $THH$ in the setting of spectral categories.

The force of the co-representability result for algebraic $K$-theory (theorem 7.13) is that it implies, via the spectral Yoneda lemma, the following identification of the spectrum of natural transformations of additive functors $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E$.

In particular, applying theorem 10.3 to $THH$ yields the following corollary.

Next, we want to characterize the topological Dennis trace $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH$ in terms of the classification of corollary 10.4. To do this, we briefly recall the construction of the topological Dennis trace map for spectral categories and verify that it descends to provide a natural transformation of additive invariants from $K$-theory to $THH$ on $\infty$-categories. We rely on the work of [10, §5].

Recall that any small spectral category ${\mathcal{A}}$ is equivalent in the Morita model structure to the small spectral category $\widehat{{\mathcal{A}}}_{\perf}$ of perfect modules. The category $\widehat{{\mathcal{A}}}_{\perf}$ admits the structure of a Waldhausen category by restriction from the spectral model structure on $\widehat{{\mathcal{A}}}$. As explained in [10, §15], without loss of generality we can work with categories enriched in EKMM $S$-modules. In this case, because all objects are fibrant the weak equivalences of the Waldhausen structure on $\widehat{{\mathcal{A}}}_{\perf}$ are compatible with the spectral enrichment in the sense of [11, §1].

Now, following [11, §5], we construct a trace using the perspective of [57] by “mixing” a cyclic bar construction and Waldhausen’s $S_{\bullet}$ construction; this is definition [11, 5.12]. Upon passage to underlying $\infty$-categories, we end up with a natural transformation of localizing invariants.

As a corollary, we know that there exists an element $x\in{\mathbb{Z}}$ such that $x$ corresponds to the homotopy class of the topological Dennis trace under the identification of corollary 10.4. The following theorem identifies this element as the unit.

Much of the interest in the topological Dennis trace comes from the fact that the $THH$ spectrum comes with an additional equivariant structure (generalizing the classical connection between the cyclic bar construction of a space and the free loop space) which allows a refinement into a theory called $TC$, the topological cyclic homology. The topological Dennis trace lifts to a map

$$K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC$$

called the cyclotomic trace map [17].

We will once again apply co-representability and point-set models to characterize the cyclotomic trace. However, when attempting to apply our co-representability results to $TC$, we run into certain obstacles. Although $TC$ is Morita invariant and satisfies localization [10], it does not preserve filtered colimits and is therefore not a localizing (or additive) invariant. Nonetheless, we can adapt our results to characterize the cyclotomic trace in this setting.

We begin by recalling the definition of $TC$ in the context of spectral categories. Our review is brief; once again, we refer the interested reader to [11, §5] and [10, §4] for authoritative treatment. Fix a prime $p$. For a spectral category ${\mathcal{C}}$, we can realize $THH({\mathcal{C}})$ as a cyclotomic $S^{1}$-spectrum. It is convenient to use the Bokstedt model of $THH$, which is revised in detail in [10, §4]. Since the Bokstedt model is naturally weakly equivalent to $N^{\cyc}Q{\mathcal{C}}$ for any spectral category ${\mathcal{C}}$ [10, 3.1], we can leverage the work above. A cyclotomic structure is additional structure on an equivariant spectrum arising from the self-equivalence $S^{1}/H\cong S^{1}$ (for finite $H\subset S^{1}$) that models the structure of the free loop space.

Roughly speaking, what we have is a set of compatible maps

$$\phi^{H}T\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}T$$

for finite $H\subset S^{1}$. The equivariant structure allows us to consider the associated non-equivariant spectra

$$TR^{n}({\mathcal{C}})=THH({\mathcal{C}})^{C_{p^{n-1}}},$$

the fixed points with respect to the induced $C_{p^{n-1}}$ action. The inclusion of fixed points and the cyclotomic structure give rise to maps $F$ and $R$ respectively

$$F,R\colon TR^{n}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TR^{n-1}.$$

We define $TC^{n}({\mathcal{C}})$ to be the homotopy equalizer

$$\holim_{F,R}TR^{n}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TR^{n-1}({\mathcal{C}}).$$

We then have that

$$TC({\mathcal{C}})=\holim_{n}TC^{n}({\mathcal{C}}),$$

where we form the homotopy limit over the maps induced by the restriction $R$; this definition is equivalent to the one originally given in [17].

The work of [10, §5] produces a description of $THH({\mathcal{C}})$ for a spectral category ${\mathcal{C}}$ as a cyclotomic spectrum, and hence constructions of $TR^{n}$, $TC^{n}$, and $TC$. Moreover, a cyclotomic trace map $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC$ is constructed which arises from compatible maps $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC^{n}$. As above, we import these constructions into the setting of $\infty$-categories. First, we have the following lemma:

Next, we observe that each of the objects $TC^{n}$ provides a localizing invariant.

The topological Dennis trace lifts through the constructions of $TC^{n}$, essentially by construction. Roughly speaking (see [11, 5.12] for a detailed construction), the trace is induced by an “inclusion of objects” map $S_{\bullet}{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH(S_{\bullet}{\mathcal{C}})$, given by taking an object to its identity map in the 0-skeleton. Since the trace lands in the fixed set with respect to the spacewise $S^{1}$-action on the cyclotomic $THH$ spectrum, it is compatible with the maps $R$ and $F$ (see e.g., [40, 1.2] for a more detailed discussion of this). Moreover, we can check that the trace descends to a natural transformation of localizing invariant using lemma 10.5.

Furthermore, the cyclotomic trace $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC$ provides a natural transformation in this setting which is assembled from natural transformations of localizing invariants.

Although $TC=\holim TC^{n}$ is not itself a localizing invariant (it does not preserve filtered colimits in general), any natural transformation of functors $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC$ is equivalent to the data of compatible maps to each $TC^{n}$. Therefore, if we consider the spectrum of natural transformations of functors to spectra from $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC$ which restrict to localizing invariants on each component, the spectrum is given by the limit (in the $\infty$-category of spectra)

$$\lim_{n}\Nat(K(-),TC^{n}(-)).$$

Finally, this yields the following characterization of the cyclotomic trace.