ncocycles
An n-cocycle is a type of cochain that satisfies a specific compatibility condition in a cohomology theory. In a typical setting, one fixes a space or a topological object X and an abelian group A (or a suitable coefficient system). The n-cochains C^n(X; A) are functions that assign coefficients in A to oriented n-simplices (or to local data in other models). The coboundary operator δ maps C^n to C^{n+1} and satisfies δ∘δ = 0, so the image of δ is contained in the kernel of δ.
An n-cocycle, then, is an element z in C^n(X; A) with δz = 0. The set of all
Cocycles appear in several flavors. In singular or simplicial cohomology, they are explicit cochains on chains;