Generic representations of orthogonal groups: The functor category Fquad

In this paper, we define the functor category Fquad associated to F2F2-vector spaces equipped with a quadratic form. We show the existence of a fully faithful, exact functor ι:F→Fquad, which preserves simple objects, where FF is the category of functors from the category of finite-dimensional F2F2-vector spaces to the category of all F2F2-vector spaces. We define the subcategory Fiso of Fquad, which is equivalent to the product of the categories of modules over the orthogonal groups; the inclusion is a fully faithful functor κ:Fiso→Fquad which preserves simple objects.