  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.

Algebraic Topology