ScalingStacks

Proof. Given a point $\phi$ in $\Nat(K(-),THH(-))$ (a specific natural transformation, that is), we can describe the corresponding element in $\pi_{0}(\mathbb{S})$ as the homotopy class represented by the composite

where the first map picks out the identity map in $\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega}))$. There is also a classical map $i\colon\mathbb{S}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K(\mathbb{S})$ constructed (for instance) as the canonical inclusion of the finite sets into finite spaces. Waldhausen’s calculations [86, §5] imply that the homotopy class of $i$ is represented by $1\in\pi_{0}(K(\mathbb{S}))$. On the other hand, since the identity map is the unit for the multiplication on $\pi_{0}(\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega})))$ induced by the composition, it must also be represented by $1\in\pi_{0}(K(\mathbb{S}))$. Finally, specializing to the case when $\phi$ is the topological Dennis trace, Waldhausen [86, 5.2] proves that the composite

is homotopic to the identity. ∎