Crash Monad Tutorial

Examples of Monads

For any category C an identity monad can be defined. It consists of an identity functor and identity morphisms.
Suppose we have a group G. Let's define a monad M_G in Set. The monad functor will be like this:
X ↦ X × G.
u(X) : X → X × G maps an element x to a pair (x,e), where e is the group's unit.
M_G(M_{_{_G}}(X)) = (id_{_{_X}},m_{_{_G}}), where m_{_{_G}} is the group multiplication.
Lists in Set. For a set X the result of applying the functor, let's call it List, is the set of all lists, (x₁,x₂,x₃...), including the empty one, of elements of X. This functor becomes a popular monad if we add u and m. Let u_{_X} : X → List(X) create a single-element list for each x∊X. And m_{_X}: List(List(X)) → List(X) maps lists of lists to plain lists by flattening them