The Kleisli Arrow

A Peculiar Kind of Function

Regular functions map values to values:

However, there exists a special class of functions that have an intriguing characteristic - they produce their output wrapped in a container type:

These functions, known as Kleisli arrows (or monadic functions), possess a remarkable capability: they can determine the structure of their output container internally. This distinguishes them from regular functions, as they don’t just compute values - they also define how those values should be organized within a container context.

While the universal function type 'a -> 'b can theoretically express any function (including these container-producing ones), there are compelling reasons to treat Kleisli arrows as a distinct category. Their unique power to shape container structures from within the function itself provides a key insight into understanding Monads, which we’ll explore next.