The semantic study of system F stumbles on the problem of variable types for which there was no convincing interpretation; we develop here a semantics based on the category-theoretic idea of direct limit, so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones, thus getting rid of the circularity of variable types. To do so, one has first to simplify somehow the extant semantic ideas, replacing Scott domains by the simpler and more finitary qualitative domains. The interpretation obtained is extremely compact, as shown on simple examples. The paper also contains the definitions of a very small ‘universal model’ of lambda-calculus, and investigates the concept totality.