Homology, Homotopy and Applications

Volume 21 (2019)

Number 2

Integral chains and Bousfield–Kan completion

Pages: 29 – 58

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n2.a4


Jacobson R. Blomquist (Department of Mathematics, Ohio State University, Columbus, Oh., U.S.A.)

John E. Harper (Department of Mathematics, Ohio State University, Newark, Oh., U.S.A.)


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.


completion, homotopical descent, coalgebra, integral chains

2010 Mathematics Subject Classification

55N10, 55P60, 55P99

Received 6 September 2017

Received revised 6 September 2018

Published 19 December 2018