  Working in the context of symmetric spectra, we consider any higher algebraic
structures that can be described as algebras over an operad O. We prove that
the fundamental adjunction comparing O-algebra spectra with coalgebra spectra
over the associated comonad K, via topological Quillen homology (or
TQ-homology), can be turned into an equivalence of homotopy theories by
replacing O-algebras with the full subcategory of 0-connected O-algebras. This
resolves in the affirmative the 0-connected case of a conjecture of
Francis-Gaitsgory.
  This derived Koszul duality result can be thought of as the spectral algebra
analog of the fundamental work of Quillen and Sullivan on the rational homotopy
theory of spaces, and the subsequent p-adic and integral work of Goerss and
Mandell on cochains and homotopy type---the following are corollaries of our
main result: (i) 0-connected O-algebra spectra are weakly equivalent if and
only if their TQ-homology spectra are weakly equivalent as derived
K-coalgebras, and (ii) if a K-coalgebra spectrum is 0-connected and cofibrant,
then it comes from the TQ-homology spectrum of an O-algebra. We 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 pi_*(X) when X is 0-connected.
