Iterated suspension spaces and higher Freudenthal suspension

TL;DR

This paper proves a higher Freudenthal suspension theorem and establishes an equivalence between spaces and coalgebra spaces over an iterated suspension-loop comonad, confirming Lawson's conjecture and providing a homotopical recognition principle.

Contribution

It introduces a derived equivalence for 1-connected spaces relating homotopy theory of spaces and coalgebra structures over an iterated suspension-loop comonad.

Findings

Homotopical descent holds for iterated suspension spaces.

The derived fundamental adjunction becomes an equivalence for 1-connected spaces.

Provides a homotopical recognition principle for iterated suspension spaces.

Abstract

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.