We prove the geometric Satake equivalence for mixed Tate motives over the
integral motivic cohomology spectrum. This refines previous versions of the
geometric Satake equivalence for split groups and power series affine
Grassmannians. Our new geometric results include Whitney-Tate stratifications
of Beilinson-Drinfeld Grassmannians and cellular decompositions of
semi-infinite orbits. With future global applications in mind, we also achieve
an equivalence relative to a power of the affine line. Finally, we use our
equivalence to give Tannakian constructions of the C-group and a modified form
of Vinberg's monoid.Comment: Comments welcome