ScalingStacks

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

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

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

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

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

We then have that

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)

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