ScalingStacks

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.