Computation in Hodge cohomology using the plectic conjecture, for abelian characters of CM fields and Hilbert modular forms. Yields a definition of a category of mixed Hodge structures: Expected to exist for arithmetic cohomology of a Shimura variety, using toroidal compactification See Tannakian categories.

See tannakian categories and fiber functor.

See also fiber functor

tannaka duality and tannaka reconstruction :