Infinite-loop-space structure of approximations

Determine whether \Omega P_mI(\Omega^\infty X) is an infinite loop space for every spectrum X and every approximation degree m.

Background

The paper proves an infinite-loop-space description for the Goodwillie approximations of S1 and derives a related description for the second approximation of an arbitrary infinite loop space. It conjecturally expects analogous behavior throughout the Goodwillie tower but does not establish it.

References

We believe that this behaviour generalizes to the whole Goodwillie tower, but we were unable to prove it, so we leave it as a question. Is $\Omega P_mI(\Omega\infty X)$ always an infinite loop space?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 2, subsection “The model”

Can the hypothesis ``after sufficiently many loops'' be removed?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 2, subsection “The model”