A cochain complex in an Ab-enriched category is nothing but a chain complex in , but with the sign of the -grading reversed.
This means that for a cochain complex the differential (or coboundary) increases the degree
An archetypical example is the deRham cochain complex of differential forms on a manifold , or more generally of differential forms in synthetic differential geometry.
The dual (in some suitable sense) of a chain complex is canonically a cochain complex with and .
A monoid in chain complexes as well as cochain complexes is a differential graded algebra.
By the dual Dold-Kan correspondence cochain complexes in non-negative degree are equivalent to cosimplicial abelian groups.
Last revised on October 25, 2012 at 11:44:16. See the history of this page for a list of all contributions to it.