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: A type constructor t of kind (* -> *) -> * -> * 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): lift . return = return lift (m `bind` k) = (lift m) `bind` (lift . k)
Examples The option monad transformer Given any monad , the option monad transformer (where denotes the option type) is defined by:
The exception monad transformer Given any monad , the exception monad transformer (where is the type of exceptions) is defined by:
The reader monad transformer Given any monad , the reader monad transformer (where is the environment type) is defined by:
The state monad transformer Given any monad , the state monad transformer (where is the state type) is defined by:
The writer monad transformer Given any monad , the writer monad transformer (where is endowed with a monoid operation with identity element ) is defined by:
The continuation monad transformer Given any monad , the continuation monad transformer maps an arbitrary type into functions of type , where is the result type of the continuation. It is defined by:
Note that monad transformations are usually not commutative: for instance, applying the state transformer to the option monad yields a type (a computation which may fail and yield no final state), whereas the converse transformation has type (a computation which yields a final state and an optional return value).
See also Monads in functional programming
References
External links
A blog post briefly reviewing some of the literature on monad transformers and related concepts, with a focus on categorical-theoretic treatment