Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

The aim of this paper is to show that homotopy pro-nilpotent structured ring spectra are 𝖳𝖰 -local, where structured ring spectra are described as algebras over a spectral operad 𝒪 . Here, 𝖳𝖰 is short for topological Quillen homology, which is weakly equivalent to 𝒪 -algebra stabilization. An 𝒪 -algebra is called homotopy pro-nilpotent if it is equivalent to a limit of nilpotent 𝒪 -algebras. Our result provides new positive evidence to a conjecture by Francis–Gaisgory on Koszul duality for general operads. As an application, we simultaneously extend the previously known 0-connected and nilpotent 𝖳𝖰 -Whitehead theorems to a homotopy pro-nilpotent 𝖳𝖰 -Whitehead theorem.