  Given a rigid tensor-triangulated category and a vector space valued
homological functor for which the K\"{u}nneth isomorphism holds, we construct a
universal graded-Tannakian category through which the given homological functor
factors. We use this to (unconditionally) construct graded-Tannakian categories
of pure motives associated to a fixed Weil cohomology theory, with a fiber
functor realizing the given cohomology theory. For $\ell$-adic cohomology and a
ground field which is algebraic over a finite field, this category is
Tannakian. In this case, we obtain in particular motivic Galois groups which
act naturally on $\ell$-adic cohomology without assuming any of the standard
conjectures. We show that these graded-Tannakian categories are equivalent to
Grothendieck's category of pure motives if the standard conjecture D holds.

Algebraic Geometry;Algebraic Topology;Category Theory