Model approximations for relative homological algebra

Recently, Chach\'olski, Neeman, Pitsch, and Scherer studied, in a series of three papers, model approximations for the unbounded category of cochain complexes over a commutative ring. These approximations allow to construct relative injective resolutions with respect to particular choices of injectives. In this paper we define similar model approximations for cochain complexes on general Grothendieck categories generalizing the previous constructions and reaching a better understanding of the whole picture.