# Monad transformer

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In [functional programming](/source/functional_programming), a '''monad transformer''' is a [type constructor](/source/type_constructor) which takes a [monad](/source/monads_in_functional_programming) 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](/source/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 <code>t</code> of [kind](/source/kind_(type_theory)) <code>(* -> *) -> * -> *</code>
# Monad operations <code>return</code> and <code>bind</code> (or an equivalent formulation) for all <code>t m</code> where <code>m</code> is a monad, satisfying the [monad laws](/source/Monad_(functional_programming))
# An additional operation, <code>lift :: m a -> t m a</code>, satisfying the following laws:<ref name="modular-interpreters">
{{cite conference
 | first = Sheng
 | last = Liang |author2=Hudak, Paul |author3=Jones, Mark
 | title = Monad transformers and modular interpreters
 | book-title = Proceedings of the 22nd ACM SIGPLAN-SIGACT symposium on Principles of programming languages
 | pages = 333–343
 | publisher = ACM
 | year = 1995
 | location = New York, NY
 | url = http://portal.acm.org/citation.cfm?id=199528
 | format = PDF
 | doi = 10.1145/199448.199528
| doi-access = free
 }}
</ref> (the notation <code>`bind`</code> below indicates infix application):
## <code>lift . return = return</code>
## <code>lift (m `bind` k) = (lift m) `bind` (lift . k)</code>

==Examples==
===The option monad transformer===
Given any monad <math>\mathrm{M} \, A</math>, the option monad transformer <math>\mathrm{M} \left( A^{?} \right)</math> (where <math>A^{?}</math> denotes the [option type](/source/option_type)) is defined by:
:<math>\begin{array}{ll}
\mathrm{return}: & A \rarr \mathrm{M} \left( A^{?} \right)\\
& a \mapsto \mathrm{return} (\mathrm{Just}\,a) \\
\mathrm{bind}: & \mathrm{M} \left( A^{?} \right) \rarr \left( A \rarr \mathrm{M} \left( B^{?} \right) \right) \rarr \mathrm{M} \left( B^{?} \right)\\
& m \mapsto f \mapsto \mathrm{bind} \, m \, \left(a \mapsto \begin{cases} \mbox{return Nothing} & \mbox{if } a = \mathrm{Nothing}\\ f \, a' & \mbox{if } a = \mathrm{Just} \, a' \end{cases} \right) \\
\mathrm{lift}: & \mathrm{M} (A) \rarr \mathrm{M} \left( A^{?} \right)\\
& m \mapsto \mathrm{bind} \, m \, (a \mapsto \mathrm{return} (\mathrm{Just} \, a)) \end{array}</math>
===The exception monad transformer===
Given any monad <math>\mathrm{M} \, A</math>, the exception monad transformer <math>\mathrm{M} (A + E)</math> (where {{mvar|E}} is the type of exceptions) is defined by:
:<math>\begin{array}{ll}
\mathrm{return}: & A \rarr \mathrm{M} (A + E)\\
& a \mapsto \mathrm{return} (\mathrm{value}\,a) \\
\mathrm{bind}: & \mathrm{M} (A + E) \rarr (A \rarr \mathrm{M} (B + E)) \rarr \mathrm{M} (B + E)\\
& m \mapsto f \mapsto \mathrm{bind} \, m \,\left( a \mapsto \begin{cases} \mbox{return err } e & \mbox{if } a = \mathrm{err} \, e\\ f \, a' & \mbox{if } a = \mathrm{value} \, a' \end{cases} \right) \\
\mathrm{lift}: & \mathrm{M} \, A \rarr \mathrm{M} (A + E)\\
& m \mapsto \mathrm{bind} \, m \, (a \mapsto \mathrm{return} (\mathrm{value} \, a)) \\
\end{array}</math>
===The reader monad transformer===
Given any monad <math>\mathrm{M} \, A</math>, the reader monad transformer <math>E \rarr \mathrm{M}\,A</math> (where {{mvar|E}} is the environment type) is defined by:
:<math>\begin{array}{ll}
\mathrm{return}: & A \rarr E \rarr \mathrm{M} \, A\\
& a \mapsto e \mapsto \mathrm{return} \, a \\
\mathrm{bind}: & (E \rarr \mathrm{M} \, A) \rarr (A \rarr E \rarr \mathrm{M}\,B) \rarr E \rarr \mathrm{M}\,B\\
& m \mapsto k \mapsto e \mapsto \mathrm{bind} \, (m \, e) \,( a \mapsto k \, a \, e) \\
\mathrm{lift}: & \mathrm{M} \, A \rarr E \rarr \mathrm{M} \, A\\
& a \mapsto e \mapsto a \\
\end{array}</math>
===The state monad transformer===
Given any monad <math>\mathrm{M} \, A</math>, the state monad transformer <math>S \rarr \mathrm{M}(A \times S)</math> (where {{mvar|S}} is the state type) is defined by:
:<math>\begin{array}{ll}
\mathrm{return}: & A \rarr S \rarr \mathrm{M} (A \times S)\\
& a \mapsto s \mapsto \mathrm{return} \, (a, s) \\
\mathrm{bind}: & (S \rarr \mathrm{M}(A \times S)) \rarr (A \rarr S \rarr \mathrm{M}(B \times S)) \rarr S \rarr \mathrm{M}(B \times S)\\
& m \mapsto k \mapsto s \mapsto \mathrm{bind} \, (m \, s) \,((a, s') \mapsto k \, a \, s') \\
\mathrm{lift}: & \mathrm{M} \, A \rarr S \rarr \mathrm{M}(A \times S)\\
& m \mapsto s \mapsto \mathrm{bind} \, m \, (a \mapsto \mathrm{return} \, (a, s)) \end{array}</math>
===The writer monad transformer===
Given any monad <math>\mathrm{M} \, A</math>, the writer monad transformer <math>\mathrm{M}(W \times A)</math> (where {{mvar|W}} is endowed with a [monoid](/source/monoid) operation {{math|&lowast;}} with identity element <math>\varepsilon</math>) is defined by:
:<math>\begin{array}{ll}
\mathrm{return}: & A \rarr \mathrm{M} (W \times A)\\
& a \mapsto \mathrm{return} \, (\varepsilon, a) \\
\mathrm{bind}: & \mathrm{M}(W \times A) \rarr (A \rarr \mathrm{M}(W \times B)) \rarr \mathrm{M}(W \times B)\\
& m \mapsto f \mapsto \mathrm{bind} \, m \,((w, a) \mapsto \mathrm{bind} \, (f \, a) \, ((w', b) \mapsto \mathrm{return} \, (w * w', b))) \\
\mathrm{lift}: & \mathrm{M} \, A \rarr \mathrm{M}(W \times A)\\
& m \mapsto \mathrm{bind} \, m \, (a \mapsto \mathrm{return} \, (\varepsilon, a)) \\
\end{array}</math>

===The continuation monad transformer===
Given any monad <math>\mathrm{M} \, A</math>, the continuation monad transformer maps an arbitrary type {{mvar|R}} into functions of type <math>(A \rarr \mathrm{M} \, R) \rarr \mathrm{M} \, R</math>, where {{mvar|R}} is the [result type](/source/result_type) of the continuation.  It is defined by:
:<math>\begin{array}{ll}
\mathrm{return} \colon & A \rarr \left( A \rarr \mathrm{M} \, R \right) \rarr \mathrm{M} \, R\\
& a \mapsto k \mapsto k \, a \\
\mathrm{bind} \colon & \left( \left( A \rarr \mathrm{M} \, R \right) \rarr \mathrm{M} \, R \right) \rarr \left( A \rarr \left( B \rarr \mathrm{M} \, R \right) \rarr \mathrm{M} \, R \right) \rarr \left( B \rarr \mathrm{M} \, R \right) \rarr \mathrm{M} \, R\\
& c \mapsto f \mapsto k \mapsto c \, \left( a \mapsto f \, a \, k \right) \\
\mathrm{lift} \colon & \mathrm{M} \, A \rarr (A \rarr \mathrm{M} \, R) \rarr \mathrm{M} \, R\\
& \mathrm{bind} 
\end{array}</math>
Note that monad transformations are usually not [commutative](/source/commutative): for instance, applying the state transformer to the option monad yields a type <math>S \rarr \left(A \times S \right)^{?}</math> (a computation which may fail and yield no final state), whereas the converse transformation has type <math>S \rarr \left(A^{?} \times S \right)</math> (a computation which yields a final state and an optional return value).

==See also==
*[Monads in functional programming](/source/Monads_in_functional_programming)

==References==
{{Reflist}}

==External links==
{{Wikibooks|Haskell|Monad transformers}}
*[http://conway.rutgers.edu/~ccshan/wiki/blog/posts/Monad_transformers/ A blog post briefly reviewing some of the literature on monad transformers and related concepts, with a focus on categorical-theoretic treatment]
{{Expand section|date=May 2008}}

Category:Functional programming

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Adapted from the Wikipedia article [Monad transformer](https://en.wikipedia.org/wiki/Monad_transformer) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Monad_transformer?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.