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## 12.1 Definitions

A category is defined by a function with exactly the same format as category:definition any other function in Axiom.

The definition of a category has the syntax:

CategoryForm : Category == Extensions [ with Exports ]

The brackets [ ] here indicate optionality.

The first example of a category definition is SetCategory, the most basic of the algebraic categories in Axiom. SetCategory

SetCategory(): Category ==
Join(Type,CoercibleTo OutputForm) with
"=" : ($, $) -> Boolean

The definition starts off with the name of the category (SetCategory); this is always in column one in the source file. All parts of a category definition are then indented with respect to this indentation first line.

In Chapter ugTypes , we talked about Ring as denoting the class of all domains that are rings, in short, the class of all rings. While this is the usual naming convention in Axiom, it is also common to use the word ``Category'' at the end of a category name for clarity. The interpretation of the name SetCategory is, then, ``the category of all domains that are (mathematical) sets.''

The name SetCategory is followed in the definition by its formal parameters enclosed in parentheses (). Here there are no parameters. As required, the type of the result of this category function is the distinguished name Category.

Then comes the ==. As usual, what appears to the right of the == is a definition, here, a category definition. A category definition always has two parts separated by the reserved word with with. Debugging hint: it is very easy to forget the with!

The first part tells what categories the category extends. Here, the category extends two categories: Type, the category of all domains, and CoercibleTo(OutputForm). CoercibleTo(OutputForm) can also be written (and is written in the definition above) without parentheses. The operation Join is a system-defined operation that Join forms a single category from two or more other categories.

Every category other than Type is an extension of some other category. If, for example, SetCategory extended only the category Type, the definition here would read ``Type with ...''. In fact, the Type is optional in this line; ``with ...'' suffices.

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