As categories have parameters, the actual operations exported by a conditional category can depend on these parameters. As an example, the operation determinantdeterminantMatrixCategory from category MatrixCategory is only exported when the underlying domain R has commutative multiplication:
if R has commutative("*") then determinant: $ -> R
Conditionals can also define conditional extensions of a category. Here is a portion of the definition of QuotientFieldCategory: QuotientFieldCategory
QuotientFieldCategory(R) : Category == ... with ... if R has OrderedSet then OrderedSet if R has IntegerNumberSystem then ceiling: $ -> R ...
Think of category QuotientFieldCategory(R) as denoting the domain Fraction(R), the class of all fractions of the form a/b for elements of R. The first conditional means in English: If the elements of R are totally ordered ( R is an OrderedSet), then so are the fractions a/b. Fraction
The second conditional is used to conditionally export an operation ceiling which returns the smallest integer greater than or equal to its argument. Clearly, ceiling makes sense for integers but not for polynomials and other algebraic structures. Because of this conditional, the domain Fraction(Integer) exports an operation ceiling: Fraction Integer->Integer, but Fraction Polynomial Integer does not.
Conditionals can also appear in the default definitions for the operations of a category. For example, a default definition for ceilingceilingField within the part following the add reads:
if R has IntegerNumberSystem then ceiling x == ...
Here the predicate used is identical to the predicate in the Exports part. This need not be the case. See ugPackagesConds for a more complicated example.