Functors

Functor is a mapping between two categories. It maps both objects and morphisms from one category to the other.

Here’s an example of morphism F that is a mapping between categories C and D:

F maps:

• a to F a
• b to F b
• f to F f

A category would translate to a generic type like List<T> in C#. A Select method on it allows us to go from List<T> to List<U>. List<T> is a functor.

In general, generics are often candidates for functors, they just need to adhere to functor laws. It doesn’t mean though that only generic types fall under the functors umbrella.

Functor Laws

1. Mapping an indentity function returns the functor unchanged.

   var input = new[] {1, 2, 3};
   var identity = (int n) => n;
   var result = input.Select(identity);
   
   // input and result are the same

2. if you have two functions, f and g, then mapping over one after the other should be the same as mapping over the composition of f and g.

   var input = new[] {1, 2, 3};
   var f = (int n) => n + 1;
   var g = (int n) => n.ToString();
   
   var result1 = input.Select(f).Select(g);
   var result2 = input.Select(n => g(f(n)));
   
   // result1 and result2 are the same

Examples

Examples of functors include:

• List
• Maybe
• Tree
• Either