Modelling polymorphism with categories

In this work we describe a category of domains, whose objects are in general categories instead of posets, such that J.-Y. Girard's category of qualitative domains and stable functions is contained in it as a full subcategory. We describe two ways of interpreting Martin-Lof type theory in this category, the first one allowing $ Sigma$ and $ Pi$, the second one only $ Pi$. Finally we show how to extend the second interpretation to a model of the theory of constructions