Acióly-Scott Interval Categories

In this work, from the category sight, we provide a generalization for the real interval theory. This generalization allows us to study generic properties of data which are “intervals” of another data, providing a categorical foundation of intervals as a parametric data type. In doing so we obtain some properties which holds for real intervals, complex intervals, interval vectors, interval matrices, and so on. For this purpose we introduce a categorical interval constructor on POSET based on the information order introduced by Dana Scott and used by Benedito Acióly to provide a computational foundation of interval mathematics. We study the categorical properties which this constructor satisfies in order to define the notion of Acióly-Scott interval category. We prove also that several subcategories of POSET are Acióly-Scott interval categories and we show also that the quasi-metric spaces category, which is important from a computational point of view and is not a subcategory of POSET, is an Acióly-Scott interval category.