 In relation with the third definition of the pure Hodge structure, one can say that a mixed Hodge structure cannot be described using the action of the group formula_39 An important insight of Deligne is that in the mixed case there is a more complicated noncommutative proalgebraic group that can be used to the same effect using Tannakian formalism.Moreover, the category of (mixed) Hodge structures admits a good notion of tensor product, corresponding to the product of varieties, as well as related concepts of "inner Hom" and "dual object", making it into a Tannakian category.By Tannaka–Krein philosophy, this category is equivalent to the category of finite-dimensional representations of a certain group, which Deligne, Milne and et el.