Notas Documentation

4. Kleisli Categories

4. Kleisli Categories


A Kleisli category is a category based on a monad. It has, as its objects, types. The morphism between the type A to type B, for example, is a function that goes from type A to an embellished type derived from B. Morphism composition as well as identity morphisms are defined in a particular way for each Kleisli category.


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

As an 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>{};


Construct the Kleisli category for partial functoins (define composition and identity).


template<class A, class B, class C>
function<optional<C>(A)> compose(function<optional<B>(A)> m1,
                                 function<optional<C>(B)> m2)
   return [m1, m2](A x) {
      auto p1 = m1(x);
      if (p1.isValid()) {
         return m2(p1.value());
      } else {
         return <optional><C>{};


template<class A> optional<A> identity(A x) {
   return optional<a>{x};


Implement the embellished function safe_reciprocal that returns a valid reciprocal of its argument, if it’s different from zero.

optional<double> safe_reciprocal(double x) {
   return (x == 0) ? optional<double>{} : optional<double>{1/x};


Compose safe_root and safe_reciprocal to implement safe_root_reciprocal that calculates sqrt(1/x) whenever possible.

optional<double> safe_root_reciprocal(double x) {
   return compose<double, double, double>(safe_reciprocal, safe_root)(x);
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