Monad (functional Programming) - Formal Definition

Formal Definition

A monad is a construction that, given an underlying type system, embeds a corresponding type system (called the monadic type system) into it (that is, each monadic type acts as the underlying type). This monadic type system preserves all significant aspects of the underlying type system, while adding features particular to the monad.

The usual formulation of a monad for programming is known as a Kleisli triple, and has the following components:

  1. A type constructor that defines, for every underlying type, how to obtain a corresponding monadic type. In Haskell's notation, the name of the monad represents the type constructor. If M is the name of the monad and t is a data type, then M t is the corresponding type in the monad.
  2. A unit function that maps a value in an underlying type to a value in the corresponding monadic type. The unit function has the polymorphic type t→M t. The result is normally the "simplest" value in the corresponding type that completely preserves the original value (simplicity being understood appropriately to the monad). In Haskell, this function is called return due to the way it is used in the do-notation described later.
  3. A binding operation of polymorphic type (M t)→(t→M u)→(M u), which Haskell represents by the infix operator >>=. Its first argument is a value in a monadic type, its second argument is a function that maps from the underlying type of the first argument to another monadic type, and its result is in that other monadic type. Typically, the binding operation can be understood as having four stages:
    1. The monad-related structure on the first argument is "pierced" to expose any number of values in the underlying type t.
    2. The given function is applied to all of those values to obtain values of type (M u).
    3. The monad-related structure on those values is also pierced, exposing values of type u.
    4. Finally, the monad-related structure is reassembled over all of the results, giving a single value of type (M u).

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