Suspension spectra and higher stabilization

We establish a higher stabilization theorem and prove that the fundamental adjunction comparing spaces with coalgebra spectra over the associated stabilization comonad, via tensoring with the sphere spectrum, can be turned into an equivalence of homotopy theories by replacing spaces with the full subcategory of 1-connected spaces. This resolves in the affirmative a conjecture of Lawson on stabilization; it also provides a homotopical recognition principle for suspension spectra---the corresponding destabilization space can be built as the homotopy limit of a cosimplicial cobar construction encoding the appropriate coalgebraic structure.