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).

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
3. An additional operation, lift :: m a -> t m a, satisfying the following laws:^[1] (the notation `bind` below indicates infix application):
   1. lift . return = return
   2. lift (m `bind` k) = (lift m) `bind` (lift . k)

Given any monad ${\displaystyle \mathrm {M} \,A}$, the option monad transformer ${\displaystyle \mathrm {M} \left(A^{?}\right)}$ (where ${\displaystyle A^{?}}$ denotes the option type) is defined by:

${\displaystyle {\begin{array}{ll}\mathrm {return} :&A\rightarrow \mathrm {M} \left(A^{?}\right)\\&a\mapsto \mathrm {return} (\mathrm {Just} \,a)\\\mathrm {bind} :&\mathrm {M} \left(A^{?}\right)\rightarrow \left(A\rightarrow \mathrm {M} \left(B^{?}\right)\right)\rightarrow \mathrm {M} \left(B^{?}\right)\\&m\mapsto f\mapsto \mathrm {bind} \,m\,\left(a\mapsto {\begin{cases}{\mbox{return Nothing}}&{\mbox{if }}a=\mathrm {Nothing} \\f\,a'&{\mbox{if }}a=\mathrm {Just} \,a'\end{cases}}\right)\\\mathrm {lift} :&\mathrm {M} (A)\rightarrow \mathrm {M} \left(A^{?}\right)\\&m\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} (\mathrm {Just} \,a))\end{array}}}$

Given any monad ${\displaystyle \mathrm {M} \,A}$, the exception monad transformer ${\displaystyle \mathrm {M} (A+E)}$ (where E is the type of exceptions) is defined by:

${\displaystyle {\begin{array}{ll}\mathrm {return} :&A\rightarrow \mathrm {M} (A+E)\\&a\mapsto \mathrm {return} (\mathrm {value} \,a)\\\mathrm {bind} :&\mathrm {M} (A+E)\rightarrow (A\rightarrow \mathrm {M} (B+E))\rightarrow \mathrm {M} (B+E)\\&m\mapsto f\mapsto \mathrm {bind} \,m\,\left(a\mapsto {\begin{cases}{\mbox{return err }}e&{\mbox{if }}a=\mathrm {err} \,e\\f\,a'&{\mbox{if }}a=\mathrm {value} \,a'\end{cases}}\right)\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow \mathrm {M} (A+E)\\&m\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} (\mathrm {value} \,a))\\\end{array}}}$

Given any monad ${\displaystyle \mathrm {M} \,A}$, the reader monad transformer ${\displaystyle E\rightarrow \mathrm {M} \,A}$ (where E is the environment type) is defined by:

${\displaystyle {\begin{array}{ll}\mathrm {return} :&A\rightarrow E\rightarrow \mathrm {M} \,A\\&a\mapsto e\mapsto \mathrm {return} \,a\\\mathrm {bind} :&(E\rightarrow \mathrm {M} \,A)\rightarrow (A\rightarrow E\rightarrow \mathrm {M} \,B)\rightarrow E\rightarrow \mathrm {M} \,B\\&m\mapsto k\mapsto e\mapsto \mathrm {bind} \,(m\,e)\,(a\mapsto k\,a\,e)\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow E\rightarrow \mathrm {M} \,A\\&a\mapsto e\mapsto a\\\end{array}}}$

Given any monad ${\displaystyle \mathrm {M} \,A}$, the state monad transformer ${\displaystyle S\rightarrow \mathrm {M} (A\times S)}$ (where S is the state type) is defined by:

${\displaystyle {\begin{array}{ll}\mathrm {return} :&A\rightarrow S\rightarrow \mathrm {M} (A\times S)\\&a\mapsto s\mapsto \mathrm {return} \,(a,s)\\\mathrm {bind} :&(S\rightarrow \mathrm {M} (A\times S))\rightarrow (A\rightarrow S\rightarrow \mathrm {M} (B\times S))\rightarrow S\rightarrow \mathrm {M} (B\times S)\\&m\mapsto k\mapsto s\mapsto \mathrm {bind} \,(m\,s)\,((a,s')\mapsto k\,a\,s')\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow S\rightarrow \mathrm {M} (A\times S)\\&m\mapsto s\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} \,(a,s))\end{array}}}$

Given any monad ${\displaystyle \mathrm {M} \,A}$, the writer monad transformer ${\displaystyle \mathrm {M} (W\times A)}$ (where W is endowed with a monoid operation ∗ with identity element ${\displaystyle \varepsilon }$) is defined by:

${\displaystyle {\begin{array}{ll}\mathrm {return} :&A\rightarrow \mathrm {M} (W\times A)\\&a\mapsto \mathrm {return} \,(\varepsilon ,a)\\\mathrm {bind} :&\mathrm {M} (W\times A)\rightarrow (A\rightarrow \mathrm {M} (W\times B))\rightarrow \mathrm {M} (W\times B)\\&m\mapsto f\mapsto \mathrm {bind} \,m\,((w,a)\mapsto \mathrm {bind} \,(f\,a)\,((w',b)\mapsto \mathrm {return} \,(w*w',b)))\\\mathrm {lift} :&\mathrm {M} \,A\rightarrow \mathrm {M} (W\times A)\\&m\mapsto \mathrm {bind} \,m\,(a\mapsto \mathrm {return} \,(\varepsilon ,a))\\\end{array}}}$

Given any monad ${\displaystyle \mathrm {M} \,A}$, the continuation monad transformer maps an arbitrary type R into functions of type ${\displaystyle (A\rightarrow \mathrm {M} \,R)\rightarrow \mathrm {M} \,R}$, where R is the result type of the continuation. It is defined by:

${\displaystyle {\begin{array}{ll}\mathrm {return} \colon &A\rightarrow \left(A\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R\\&a\mapsto k\mapsto k\,a\\\mathrm {bind} \colon &\left(\left(A\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R\right)\rightarrow \left(A\rightarrow \left(B\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R\right)\rightarrow \left(B\rightarrow \mathrm {M} \,R\right)\rightarrow \mathrm {M} \,R\\&c\mapsto f\mapsto k\mapsto c\,\left(a\mapsto f\,a\,k\right)\\\mathrm {lift} \colon &\mathrm {M} \,A\rightarrow (A\rightarrow \mathrm {M} \,R)\rightarrow \mathrm {M} \,R\\&\mathrm {bind} \end{array}}}$

Note that monad transformations are usually not commutative: for instance, applying the state transformer to the option monad yields a type ${\displaystyle S\rightarrow \left(A\times S\right)^{?}}$ (a computation which may fail and yield no final state), whereas the converse transformation has type ${\displaystyle S\rightarrow \left(A^{?}\times S\right)}$ (a computation which yields a final state and an optional return value).

• Monads in functional programming

The Wikibook Haskell has a page on the topic of: Monad transformers

• A highly technical blog post briefly reviewing some of the literature on monad transformers and related concepts, with a focus on categorical-theoretic treatment