In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil. Weil cohomology theories play an important role in the theory of motives, insofar as the category of Chow motives is a universal Weil cohomology theory in the sense that any Weil cohomology function factors through Chow motives. Note that, however, the category of Chow motives does not give a Weil cohomology theory since it is not abelian.
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