cocycle

A cocycle is a function that obeys a consistency rule strict enough to deserve a name. Imagine assigning a correction to every step you take so that, no matter which path of steps you follow, the accumulated corrections agree. The closure condition that enforces this path-independence is exactly the cocycle equation. A cocycle is a candidate cohomology class: it satisfies the rule, but whether it is genuinely new or secretly trivial is a separate question answered by coboundaries.

In group cohomology a (normalized, inhomogeneous) n-cochain is a function f : G^n -> A into a G-module A. The coboundary operator d sends an n-cochain to an (n+1)-cochain by an alternating-sum formula that mixes the action of G with the slots of f. An n-cocycle is a cochain f with df = 0; the set of n-cocycles is denoted Z^n(G, A) and forms a subgroup of all cochains. Because d^2 = 0, every coboundary is a cocycle, but not conversely.

The cohomology group is the quotient H^n(G, A) = Z^n(G, A) / B^n(G, A), so a cocycle determines a cohomology class and two cocycles give the same class iff they differ by a coboundary. For n = 1 the cocycle condition reads f(gh) = f(g) + g·f(h), the crossed-homomorphism law; for n = 2 it is the associativity-type identity satisfied by the factor set of a group extension. The word “cocycle” is used identically in topology and sheaf theory — same closure rule, different cochains.