# Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

## TL;DR

This paper proves that homotopy pro-nilpotent structured ring spectra are TQ-local, providing evidence for a conjecture on Koszul duality and extending Whitehead theorems in the context of spectral operads.

## Contribution

It establishes that homotopy pro-nilpotent structured ring spectra are TQ-local, advancing understanding of operad-based algebraic structures in homotopy theory.

## Key findings

- Homotopy pro-nilpotent structured ring spectra are TQ-local.
- Extension of TQ-Whitehead theorems to homotopy pro-nilpotent cases.
- Supports conjecture on Koszul duality for general operads.

## Abstract

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.