Cocyclelike

Cocyclelike is an adjective used in several branches of mathematics to describe objects that resemble a cocycle but do not strictly satisfy the defining cocycle condition. In its most common setting, a cocycle is a function that encodes a compatibility relation for a group action or a dynamical system. A cocyclelike object, by contrast, typically obeys a related identity only approximately or under weaker assumptions.

In group cohomology, a 1-cocycle with coefficients in a G-module M satisfies the exact identity c(gh) = c(g) + g·c(h). A cocyclelike object may satisfy a relaxed or approximate version of this equation. Common variations include defects that are bounded, defects that vanish only on a subgroup or subcategory, or defects that are well-controlled up to a coboundary. Such objects are often called almost cocycles, quasi-cocycles, near-cocycles, or local/partial cocycles, depending on the precise relaxation employed.

Cocyclelike notions appear in several contexts. In dynamical systems and ergodic theory, cocycles describe skew-product extensions; cocyclelike objects arise when the skew-product identity holds only approximately, which is central to perturbation arguments and stability questions. In geometric group theory and representation theory, quasi-cocycles or almost cocycles can be used to study bounded cohomology, rigidity phenomena, and the stability of actions under perturbations.

Because the concept is not rigidly standardized, the exact meaning of cocyclelike depends on the field and the chosen relaxation of the cocycle condition. In general, however, it signals a structure that mirrors a cocycle’s compatibility condition while allowing controlled deviations. See also cocycle, almost cocycle, quasi-cocycle.