ALGEBRAIC DOMAINS OF NATURAL TRANSFORMATIONS

Motivated by the semantics of polymorphic programming languages and typed lambda-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:OMEGA --> C with a small category OMEGA 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:OMEGA --> SCOTT are two functors such that for all f in mor(OMEGA) 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.