The Monsky-Washnitzer and the overconvergent realizations

We construct the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classic etale site provides a new direct definition of the over-convergent de Rham cohomology and rigid cohomology and shows that their finite dimensionality follows formally from one of Betti cohomology for smooth projective complex varieties.