The authors have done an admirable job in integrating the latest results in the field, making the book both a tutorial and a compendium. It would not be surprising if the book turns out to be as influential as Hintikka's classic, Knowledge and Belief . The idea of axiomatic domain theory is to identify properties of categorical structures appropriate for the denotational semantics of programming languages in the traditional style of domain theory. These properties typically guarantee, among other things, the existence of certain fixed-point operators, the existence of a rich collection of type constructors, and the canonical solution of equations based on them. This book presents the development of axiomatic domain theory via categories of partial maps.
Approximation is shown to be a derived concept from the primitive notion of partiality.
Type constructors on the categories of partial maps are investigated through a characterization of colimit diagrams of total maps. Fundamental properties of categories in which recursive types admit canonical interpretations are established based on the notion of algebraically compact categories. The denotational semantics is specified in a categorical setting. Categories of domains which ensure computational adequacy are axiomatized. Chapter 2 is a review of 2-category theory and enriched category theory, the less known part of category theory to be used in the rest of the book.