  For a variety with a Whitney stratification by affine spaces, we study
categories of motivic sheaves which are constant mixed Tate along the strata.
We are particularly interested in those cases where the category of mixed Tate
motives over a point is equivalent to the category of finite-dimensional
bigraded vector spaces. Examples of such situations include rational motives on
varieties over finite fields and modules over the spectrum representing the
semisimplification of de Rham cohomology for varieties over the complex
numbers. We show that our categories of stratified mixed Tate motives have a
natural weight structure. Under an additional assumption of pointwise purity
for objects of the heart, tilting gives an equivalence between stratified mixed
Tate sheaves and the bounded homotopy category of the heart of the weight
structure. Specializing to the case of flag varieties, we find natural
geometric interpretations of graded category $\mathcal O$ and Koszul duality.
