Integral chains and Bousfield-Kan completion

TL;DR

This paper establishes a new integral chains characterization of the homotopy type of simply connected spaces using derived adjunctions and equivalences in the Arone-Ching framework for homotopical descent.

Contribution

It proves that the derived adjunction comparing spaces with coalgebra complexes over integral homology can be upgraded to a derived equivalence for simply connected spaces.

Findings

Derived adjunction with integral chains forms a derived equivalence for simply connected spaces.

Provides an integral chains characterization of homotopy types.

Connects Bousfield-Kan completion with homotopical descent framework.

Abstract

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.