Cocycles are mathematical objects that arise in the study of cohomology theories, representing elements of a cochain complex that satisfy a certain condition of closure. They are crucial in understanding the relationship between singular cohomology and homology, as well as in group and Lie algebra cohomology. Cocycles can be thought of as the 'good' elements that contribute to the computation of cohomology groups and help in identifying isomorphisms between different algebraic structures.
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