Categories

• Every domain belongs to the type Type, but it is useful to be
  able to assert more.
• Categories provide information about domain values, indicating what exports they must provide.
• A basic category-valued expression gives a list of exports:

  with { vertex: (%, Integer) -> Complex ;
         new: List Complex -> % }
  

• Categories may be used in declarations.

  Polygon with { vertex: (%, Integer) -> Complex ;
         new: List Complex -> % 
  } == add {
      Rep == List Complex;
      new(l: List Complex): % == per l;
      vertex(p: %, i: Integer): Complex == rep(p).i;
  }
  

• We may create category-valued constants.

  define Monoid: Category == with {
      1: %;
      *: (%, %) -> %
  }
  define Finite: Category == with {
      cardinality: Integer
  }
  

• The Join operator combines catgories, providing multiple inheritance:

  define FiniteMonoid: Category == Join(Monoid, Finite)
  

• One may use functions to compute categories:

  define Module(R: Ring): Category == Ring with {
      *: (R, %) -> %
  }
  
  define ComplexCategory(R: Ring): Category ==  Module(R) with {
      complex: (R, R) -> R;
      real:    % -> R;
      imag:    % -> R;
  }