This chapter unravels the mysteries of categories---what category they are, how they are related to domains and packages, category:constructor how they are defined in Axiom, and how you can extend the constructor:category system to include new categories of your own.
We assume that you have read the introductory material on domains and categories in ugTypesBasicDomainCons . There you learned that the notion of packages covered in the previous chapter are special cases of domains. While this is in fact the case, it is useful here to regard domains as distinct from packages.
Think of a domain as a datatype, a collection of objects (the objects of the domain). From your ``sneak preview'' in the previous chapter, you might conclude that categories are simply named clusters of operations exported by domains. As it turns out, categories have a much deeper meaning. Categories are fundamental to the design of Axiom. They control the interactions between domains and algorithmic packages, and, in fact, between all the components of Axiom.
Categories form hierarchies as shown on the inside cover pages of this book. The inside front-cover pages illustrate the basic algebraic hierarchy of the Axiom programming language. The inside back-cover pages show the hierarchy for data structures.
Think of the category structures of Axiom as a foundation for a city on which superstructures (domains) are built. The algebraic hierarchy, for example, serves as a foundation for constructive mathematical algorithms embedded in the domains of Axiom. Once in place, domains can be constructed, either independently or from one another.
Superstructures are built for quality---domains are compiled into machine code for run-time efficiency. You can extend the foundation in directions beyond the space directly beneath the superstructures, then extend selected superstructures to cover the space. Because of the compilation strategy, changing components of the foundation generally means that the existing superstructures (domains) built on the changed parts of the foundation (categories) have to be rebuilt---that is, recompiled.
Before delving into some of the interesting facts about categories, let's see how you define them in Axiom.