Realization of Levine's motives and p-adic Hodge cohomology

We construct a realization functor from Levine's triangulated category of motives, for any cohomology theory that takes values in a tensor category that is not necessarily a category of vector spaces, on smooth quasi-projective schemes over a reduced separated noetherian scheme. As a sample application, we construct a realization functor associated with the triple that consists of the rigid cohomology, the de Rham cohomology and the specialization map between them, whose domain is Levine's motivic category for the case in which the base scheme is the spectrum of a ring of p-adic integers.