ScalingStacks

Theorem 1.12. (see theorem 10.3) Given an additive invariant

$$E:\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{S}}_{\infty}$$

with values in the stable $\infty$-category of spectra, we have a natural equivalence of spectra

$$\mathrm{Map}(K,E)\simeq E({\mathcal{S}}_{\infty}^{\omega}),$$

where $\mathrm{Map}(K,E)$ denotes the spectrum of natural transformations of additive invariants. The analogous result for localizing invariants holds. In the particular case where $E$ is topological Hochschild homology, we obtain an isomorphism

$$\pi_{0}\mathrm{Map}(K,THH)\simeq\pi_{0}THH({\mathcal{S}}_{\infty}^{\omega})\simeq\pi_{0}THH(\mathbb{S})\simeq{\mathbb{Z}}.$$