12.10 Parameters

Like domain constructors, category constructors can also have parameters. For example, category MatrixCategory is a parameterized category for defining matrices over a ring R so that the matrix domains can have different representations and indexing schemes. Its definition has the form:

MatrixCategory(R,Row,Col): Category ==
    TwoDimensionalArrayCategory(R,Row,Col) with ...

The category extends TwoDimensionalArrayCategory with the same arguments. You cannot find TwoDimensionalArrayCategory in the algebraic hierarchy listing. Rather, it is a member of the data structure hierarchy, given inside the back cover of this book. In particular, TwoDimensionalArrayCategory is an extension of HomogeneousAggregate since its elements are all one type.

The domain Matrix(R), the class of matrices with coefficients from domain R, asserts that it is a member of category MatrixCategory(R, Vector(R), Vector(R)). The parameters of a category must also have types. The first parameter to MatrixCategory R is required to be a ring. The second and third are required to be domains of category FiniteLinearAggregate(R). This is another extension of HomogeneousAggregate that you can see in the data structure hierarchy. In practice, examples of categories having parameters other than domains are rare.

Adding the declarations for parameters to the definition for MatrixCategory, we have:

R: Ring
(Row, Col): FiniteLinearAggregate(R)
MatrixCategory(R, Row, Col): Category ==
    TwoDimensionalArrayCategory(R, Row, Col) with ...