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Wikipedia / Monad transformer

In functional programming, a monad transformer is a type constructor which takes a monad as an argument and returns a monad as a result.

Monad transformers can be used to compose features encapsulated by monads – such as state, exception handling, and I/O – in a modular way. Typically, a monad transformer is created by generalising an existing monad; applying the resulting monad transformer to the identity monad yields a monad which is equivalent to the original monad (ignoring any necessary boxing and unboxing).

Definition

A monad transformer consists of:

  1. A type constructor t of kind (* -> *) -> * -> *

  2. Monad operations return and bind (or an equivalent formulation) for all t m where m is a monad, satisfying the monad laws

An additional operation, lift :: m a -> t m a, satisfying the following laws: (the notation `bind` below indicates infix application):

  1. lift . return = return

  2. lift (m `bind` k) = (lift m) `bind` (lift . k)

Examples

The option monad transformer

denotes the option type) is defined by:

The exception monad transformer

(where is the type of exceptions) is defined by:

The reader monad transformer

(where is the environment type) is defined by:

The state monad transformer

(where is the state type) is defined by:

The writer monad transformer

(where is endowed with a monoid operation ) is defined by:

The continuation monad transformer

, the continuation monad transformer maps an arbitrary type , where is the result type of the continuation. It is defined by:

Note that monad transformations are usually not commutative (a computation which yields a final state and an optional return value).

See also

References