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arXiv · 2002.00038

Fibration theorems for TQ-completion of structured ring spectra

Nikolas Schonsheck
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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-$\pi_*$ isomorphism. More generally, we also show that similar results remain true if we replace "homotopy fibration sequence" with "homotopy pullback square."

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Nikolas Schonsheck. 2020-01-31. Fibration theorems for TQ-completion of structured ring spectra. https://arxiv.org/abs/2002.00038

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