  We prove that the stabilization of spaces functor---the classical
construction of associating a spectrum to a pointed space by tensoring with the
sphere spectrum---satisfies homotopical descent on objects and morphisms. This
is the stabilization analog of the Quillen-Sullivan theory main result that the
rational chains (resp. cochains) functor participates in a derived equivalence
with certain coalgebra (resp. algebra) complexes, after restriction to
1-connected spaces up to rational equivalence. In more detail, we prove that
the stabilization of spaces functor participates in a derived equivalence with
certain coalgebra spectra (where the stabilization construction naturally
lands), after restriction to 1-connected spaces up to weak equivalence. This
resolves in the affirmative the infinite case, involving stabilization and
suspension spectra, of a question/conjecture posed by Tyler Lawson on
(iterated) suspension spaces almost ten years ago. A key ingredient of our
proof, of independent interest, is a higher stabilization theorem for spaces
that provides strong estimates for the uniform cartesian-ness of certain
cubical diagrams associated to n-fold iterations of the spaces-level
stabilization map; this is the stabilization analog of Dundas' higher Hurewicz
theorem.
