Establish the unresolved general comparison with Nardin’s genuine spectra

Determine whether the canonical colimit-preserving normed functor from Nardin’s category of genuine spectra to the constructed stable parametrised category of spectra is an equivalence for general atomic orbital categories, and determine whether Nardin’s category admits norms in that generality.

Background

For a general atomic orbital category, the paper constructs a stable parametrised category of spectra by formally inverting norms of the circle in pointed anima. Under additional hypotheses, a canonical colimit-preserving normed functor exists from Nardin’s category of genuine spectra to this construction. The finite-group case is known to give an equivalence, but the general case remains unresolved, including the more basic question of whether Nardin’s category itself supports the required norms.

References

This is an equivalence in the finite-group equivariant situation, but we do not know if this is an equivalence in general or whether Nardin's category admits norms, although our expectation is that this is not so.

— Normed idempotence of spectra and distributivity in equivariant higher algebra  (2610.22074 - Hilman et al., 15 Oct 2026) in Section 1, subsection “Relation to other work”