In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category of a graded category. We show that those constructions agree with the usual ones for algebras. In case of the smash product for an infinitegroup our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units in. We recover in a categorical generalized setting the duality theorems of M. Cohen and S. Montgomery and we provide a unification with the results on coverings of quivers and relations by E. Green. We obtain a confirmation in a quiver and relations free categorical setting that twoconstructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representations theories of a category and of a Galois cover of it.