Iterated suspension spaces and higher Freudenthal suspension

We establish a higher Freudenthal suspension theorem and prove that the derived fundamental adjunction comparing spaces with coalgebra spaces over the homotopical iterated suspension-loop comonad, via iterated suspension, 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 iterated suspension spaces; that homotopical descent for iterated suspension is satisfied on objects and morphisms---the corresponding iterated desuspension space can be built as the homotopy limit of a cosimplicial cobar construction encoding the homotopical coalgebraic structure. It also provides a homotopical recognition principle for iterated suspension spaces. In a nutshell, we show that the iterated loop-suspension completion map studied by Bousfield participates in a derived equivalence between spaces and coalgebra spaces over the associated homotopical comonad, after restricting to 1-connected spaces.