Fibration theorems for TQ-completion of structured ring spectra

Abstract

The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the context of algebraic structures that can be described as algebras over an operad $\mathcal{O}$ in symmetric spectra. Our main result is that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total $\mathcal{O}$-algebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-completion tower is a pro-${\pi }_{\ast }$ isomorphism. More generally, we also show that similar results remain true if we replace "homotopy fibration sequence" with "homotopy pullback square."

14 pages; updated to final version; accepted for publication

References in corpus (1)

• Iterated delooping and desuspension of structured ring spectra

Cited by in corpus (2)

• On the Goodwillie derivatives of the identity in structured ring spectra
• On the chromatic localization of the homotopy completion tower for $\mathcal{O}$-algebras