Suspension spectra and higher stabilization

Abstract: 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.