T-motives

Considering a (co)homology theory T on a base category C as a fragment of a first-order logical theory we here construct an abelian category A[T] which is universal with respect to models of T in abelian categories. Under mild conditions on the base category C, e.g. for the category of algebraic schemes, we get a functor from C to Ch(Ind(A[T])) the category of chain complexes of ind-objects of A[T]. This functor lifts Nori's motivic functor for algebraic schemes defined over a subfield of the complex numbers. Furthermore, we construct a triangulated functor from D(Ind(A[T])) to Voevodsky's motivic complexes.