ScalingStacks

Proof. Using theorem 10.3 as in the proof of corollary 10.4, we see that the spectrum of natural transformations $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC$ which restrict to localizing invariants on each component can be computed as the limits (in the $\infty$-category of spectra)

$$\lim_{n}\Nat(K(-),TC^{n}(-))\simeq\lim_{n}TC^{n}(\mathbb{S})=TC(\mathbb{S}).$$

Completing at the prime $p$, recall that $TC(\mathbb{S})\simeq\mathbb{S}\vee\Sigma CP^{\infty}_{-1}$ [68, §1]. Since $\pi_{0}(\Sigma CP^{\infty}_{-1})=0$, we deduce that the set of homotopy classes of compatible invariants is $\mathbb{Z}_{p}$. Furthermore, using the argument for theorem 10.6 and passing to the limit, we can identify the class of the cyclotomic trace by understanding the homotopy class of the composite

$$\mathbb{S}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K(\mathbb{S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TC(\mathbb{S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH(\mathbb{S})\simeq\mathbb{S}$$

(after $p$-completion). An elaboration of Waldhausen’s results [86, §5] (see [17, §5] or [68, §1]) implies that this homotopy class is the identity (i.e., the unit splits the trace $TC(\mathbb{S})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH(\mathbb{S})\simeq\mathbb{S}$, which gives the identification of $TC(\mathbb{S})$ above), and so using the work of theorem 10.6 we again deduce that the cyclotomic trace is represented by the unit. ∎