  We consider categories of equivariant mixed Tate motives, where equivariant
is understood in the sense of Borel. We give the two usual definitions of
equivariant motives, via the simplicial Borel construction and via algebraic
approximations of it. The definitions turn out to be equivalent and give rise
to a full six-functor formalism. For rational \'etale motives over a finite
field or the homotopical stable algebraic derivator arising from the
semisimplified Hodge realization, the equivariant mixed Tate motives provide a
graded version of the equivariant derived category. We show that, in
sufficiently nice and clean cases, these categories admit weight structures;
moreover, a tilting result holds which identifies the category of equivariant
mixed Tate motives with the bounded homotopy category of the heart of its
weight structure. This can be seen as a formality result for equivariant
derived categories. We also discuss convolution functors on equivariant mixed
Tate motives, and consequences for the categorification of the Hecke algebra
and some of its modules.
