The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logical-relations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesian-closed-category models of the simply typed lambda calculus and PL-category models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included. In , the idea that type structure enforces abstraction was formalized by an "abstraction theorem " that was proved for both the simply typed (or first-order) lambda calculus and the polymorphic (or second-order) lambda calculus [2, 3, 4]. In the polymorphic case this theorem led naturally to a definition of "parametric " polymorphism that captured the intuitive concept first described by Strachey . Unfortunately, however, most of the results of  were limited to a classical set-theoretic model. Thus the abstraction theorem for the simply typed case was merely a repetition of the logical-relations theorem for the typed lambda-calculus , while the developments for the polymorphic case were vacuous, since it was later shown that there is no classical set-theoretic model of the polymorphic lambda calculus [7, 8].