Abstract
Working in the context of symmetric spectra, we consider algebraic structures that can be described as algebras over an operad O. Topological Quillen homology, or TQ-homology, is the spectral algebra analog of Quillen homology and stabilization. Working in a framework for homotopical descent, the associated TQ-homology completion map appears as the unit map of an adjunction comparing the homotopy categories of O-algebra spectra with coalgebra spectra over the associated comonad via TQ-homology. We prove that this adjunction between homotopy categories can be turned into an equivalence by replacing O-algebras with the full subcategory of 0-connected O-algebras. We also construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups TQ⁎(X) of an O-algebra X, and prove that it converges strongly to π⁎(X) when X is 0-connected.
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