Fibration theorems for TQ-completion of structured ring spectra

N. Schonsheck

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 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 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-π* isomorphism. More generally, we also show that similar results remain true if we replace homotopy fibration sequence with homotopy pullback square. 

Tbilisi Mathematical Journal, Special Issue (HomotopyTheorySpectra - 2020), pp. 1-15