The n-birational motivic homotopy category is made into a presheaf on a correspondence category, and the birational version of a scheme with finitely many generic points splits as the product of the categories of its generic points.
Noncommutative motives
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Schematic Functorialities of Birational Motivic Homotopy Categories
The n-birational motivic homotopy category is made into a presheaf on a correspondence category, and the birational version of a scheme with finitely many generic points splits as the product of the categories of its generic points.