nLab cochain complex

Idea

A cochain complex $V^\bullet$ in an Ab-enriched category $C$ may be regarded as a chain complex (see there for more) in $C$, but with the sign of the $\mathbb{Z}$-grading reversed.

This means that for a cochain complex $(V^\bullet,d)$ the differential (or coboundary operator) $d$ increases the degree

$$d \colon V^{k} \to V^{k+1} \,.$$

Examples

• An archetypical example is the deRham cochain complex of differential forms on a manifold $X$, or more generally of differential forms in synthetic differential geometry.

• The dual (in some suitable sense) of a chain complex $(V_\bullet, \partial_k : V_k \to V_{k-1})$ is canonically a cochain complex with $V^k := (V_k)^*$ and $d_k := (\partial_k)^*$.

• A monoid in chain complexes as well as cochain complexes is a differential graded algebra.

  • In their dual formulation L-infinity-algebras and more generall Lie infinity-algebroids are encoded by such monoids in cochain complexes.

• By the dual Dold-Kan correspondence cochain complexes in non-negative degree are equivalent to cosimplicial abelian groups.