I-Categories as a framework for solving domain equations

Abstract

An abstract notion of category of information systems or I-category is introduced as a generalisation of Scott's well-known category of information systems. As in the theory of partial orders, I-categories can be complete or ω-algebraic, and it is shown that ω-algebraic I-categories can be obtained from a certain completion of countable I-categories. The proposed axioms for a complete I-category introduce a global partial order on the morphisms of the category, making them a cpo. An initial algebra theorem for a class of functors continuous on the cpo of morphisms is proved, thus giving canonical solution of domain equations; an effective version of these results for ω-algebraic I-categories is also provided. Some basic examples of I-categories representing the categories of sets, Boolean algebras, Scott domains and continuous Scott domains are constructed.