  Working in the Arone-Ching framework for homotopical descent, it follows that
the Bousfield-Kan completion map with respect to integral homology is the unit
of a derived adjunction. We prove that this derived adjunction, comparing
spaces with coalgebra complexes over the associated integral homology comonad,
via integral chains, can be turned into a derived equivalence by replacing
spaces with the full subcategory of simply connected spaces. In particular,
this provides an integral chains characterization of the homotopy type of
simply connected spaces.
