Algebraic Domains of Natural Transformations

Abstract Motivated by the semantics of polymorphic programming languages and typed λ-calculi, by formal methods in functor category semantics, and by well-known categorical and domain-theoretical constructs, we study domains of natural transformations F → G of functors F, G:Ω→C with a small category Ω as source and a cartesian closed category of Scott-domains C as target. We put constraints on the image arrows of the functors to obtain that F → G is an object in C. Inf-faithful domains F → G allow that infima in F → G can be computed in each component [FA→GA] separately. If F, G:Ω→SCOTT are two functors such that for all f in mor(Ω) the maps F(f) preserve finite elements and G(f) preserve all nonempty infima, then F → G is inf-faithful, and all inf-faithful domains are Scott-domains. Familiar notions like “inverse limits”, “small products”, and “strict function spaces” are special instances of functors that meet the conditions above. We extend these results to retracts of Scott-domains.