Theorem 10.3 (0NQY) · ScalingStacks

0NQY

Theorem 10.3. Given an additive invariant $E\colon\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{S}}_{\infty}$ with values in the stable $\infty$ -category of spectra, we have a natural equivalence

\Nat(K,E)\simeq E({\mathcal{S}}_{\infty}^{\omega}),

where $\Nat(K,E)$ denotes the spectrum of natural transformations from $K$ to $E$ as additive invariants from small stable $\infty$ -categories to spectra.

0NQZ

Proof. By theorem 6.10, we can describe the additive invariants $K$ and $E$ as elements of $\mathrm{Fun}^{\mathrm{L}}({\mathcal{M}}_{\mathrm{add}},{\mathcal{S}}_{\infty})$ . The equivalence

\Nat(\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega}),-),E)\simeq E({\mathcal{S}}_{\infty}^{\omega})

follows from 7.13 and the spectral Yoneda lemma. ∎