Kleisli Categories

Prev: Categories Great and Small Next: Products and Coproducts

Prev: Categories Great and Small Next: Products and Coproducts

Exercises

A function that is not defined for all possible values of its argument is called a partial function. It’s not really a function in the mathematical sense, so it doesn’t fit the standard categorical mold. It can, however, be represented by a function that returns an embellished type optional:

template<typename A>
class optional {
    bool _isValid;
    A _value;
public:
    optional()    : _isValid(false) {}
    optional(A v) : _isValid(true), _value(v) {}
    bool isValid() const { return _isValid; }
    A value() const { return _value; }
};

For example, here’s the implementation of the embellished function safe_root:

optional<double> safe_root(double x) {
    if (x >= 0) return optional<double>{sqrt(x)};
    else return optional<double>{};
}

Here’s the challenge:

1. Construct the Kleisli category for partial functions (define composition and identity).
2. Implement the embellished function safe_reciprocal that returns a valid reciprocal of its argument, if it’s different from zero.
3. Compose the functions safe_root and safe_reciprocal to implement safe_root_reciprocal that calculates sqrt(1/x) whenever possible.