Grothendieck categories of enriched functors

It is shown that the category of enriched functors [C,V] is Grothendieck whenever V is a closed symmetric monoidal Grothendieck category and C is a category enriched over V. Localizations in [C,V] associated to collections of objects of C are studied. Also, the category of chain complexes of generalized modules Ch(CR) is shown to be identified with the Grothendieck category of enriched functors [modR,Ch(ModR)] over a commutative ring R, where the category of finitely presented R-modules mod R is enriched over the closed symmetric monoidal Grothendieck category Ch(ModR) as complexes concentrated in zeroth degree. As an application, it is proved that Ch(CR) is a closed symmetric monoidal Grothendieck model category with explicit formulas for tensor product and internal Hom-objects. Furthermore, the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of Hovey-Palmieri-Strickland [14] is introduced. It is shown that the derived category of generalized modules D(CR) over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category.