Cohomology Theory

A contravariant assignment (usually a functor) from a category of spaces (topological, differentiable, or algebraic) to graded abelian groups, modules, or rings that captures global obstruction classes and supports natural product structures and long exact sequences.

A contravariant algebraic invariant that assigns graded groups or rings to spaces (or spectra) and satisfies axioms such as homotopy invariance and exactness in classical forms; cohomology theories record global obstructions, classify structures, and carry multiplicative or cohomological operations.