coboundary

A coboundary is a cocycle that is trivial for a boring reason: it is manufactured automatically from data one degree lower, with no real content of its own. If a cocycle records a measurement, a coboundary records a measurement that is entirely an artifact of how you set up your coordinates — change the bookkeeping and it disappears. Cohomology is precisely the operation of throwing away all such artifacts, keeping only the cocycles that no change of coordinates can erase.

Formally, for the coboundary operator d on cochains G^n -> A, an n-coboundary is a cochain of the form d(c) where c is an (n-1)-cochain. The set of n-coboundaries is B^n(G, A) = im(d : C^{n-1} -> C^n). Since d^2 = 0, every coboundary is a cocycle, so B^n is a subgroup of Z^n, and the n-th cohomology group is the quotient H^n(G, A) = Z^n / B^n. A cocycle is a coboundary exactly when its class in H^n is zero.

In degree 1 the principal crossed homomorphisms are the coboundaries: f(g) = g·a - a for some fixed a in A. In degree 2 the coboundaries are exactly the factor sets coming from split extensions, those equivalent to a semidirect product. Thus declaring two cocycles equivalent when they differ by a coboundary is not an arbitrary convention — it identifies precisely the cocycles that describe the same underlying object up to a harmless reparametrization.