Safe Haskell | Safe |
---|---|

Language | Haskell98 |

A monad morphism is a natural transformation:

morph :: forall a . m a -> n a

... that obeys the following two laws:

morph $ do x <- m = do x <- morph m f x morph (f x) morph (return x) = return x

... which are equivalent to the following two functor laws:

morph . (f >=> g) = morph . f >=> morph . g morph . return = return

Examples of monad morphisms include:

`lift`

(from`MonadTrans`

)`squash`

(See below)

(See below), if`hoist`

f`f`

is a monad morphism`(f . g)`

, if`f`

and`g`

are both monad morphisms`id`

Monad morphisms commonly arise when manipulating existing monad transformer
code for compatibility purposes. The `MFunctor`

, `MonadTrans`

, and
`MMonad`

classes define standard ways to change monad transformer stacks:

- class MFunctor t where
- generalize :: Monad m => Identity a -> m a
- class (MFunctor t, MonadTrans t) => MMonad t where
- class MonadTrans t where
- squash :: (Monad m, MMonad t) => t (t m) a -> t m a
- (>|>) :: (Monad m3, MMonad t) => (forall a. m1 a -> t m2 a) -> (forall b. m2 b -> t m3 b) -> m1 c -> t m3 c
- (<|<) :: (Monad m3, MMonad t) => (forall b. m2 b -> t m3 b) -> (forall a. m1 a -> t m2 a) -> m1 c -> t m3 c
- (=<|) :: (Monad n, MMonad t) => (forall a. m a -> t n a) -> t m b -> t n b
- (|>=) :: (Monad n, MMonad t) => t m b -> (forall a. m a -> t n a) -> t n b

# Functors over Monads

A functor in the category of monads, using `hoist`

as the analog of `fmap`

:

hoist (f . g) = hoist f . hoist g hoist id = id

hoist :: Monad m => (forall a. m a -> n a) -> t m b -> t n b #

Lift a monad morphism from `m`

to `n`

into a monad morphism from
`(t m)`

to `(t n)`

MFunctor ListT # | |

MFunctor Lift # | |

MFunctor MaybeT # | |

MFunctor (ErrorT e) # | |

MFunctor (ExceptT e) # | |

MFunctor (StateT s) # | |

MFunctor (StateT s) # | |

MFunctor (WriterT w) # | |

MFunctor (WriterT w) # | |

MFunctor (Backwards *) # | |

MFunctor (IdentityT *) # | |

MFunctor (Product * f) # | |

MFunctor (ReaderT * r) # | |

Functor f => MFunctor (Compose * * f) # | |

MFunctor (RWST r w s) # | |

MFunctor (RWST r w s) # | |

generalize :: Monad m => Identity a -> m a #

A function that `generalize`

s the `Identity`

base monad to be any monad.

# Monads over Monads

class (MFunctor t, MonadTrans t) => MMonad t where #

A monad in the category of monads, using `lift`

from `MonadTrans`

as the
analog of `return`

and `embed`

as the analog of (`=<<`

):

embed lift = id embed f (lift m) = f m embed g (embed f t) = embed (\m -> embed g (f m)) t

class MonadTrans t where #

The class of monad transformers. Instances should satisfy the
following laws, which state that `lift`

is a monad transformation:

lift :: Monad m => m a -> t m a #

Lift a computation from the argument monad to the constructed monad.

MonadTrans ListT | |

MonadTrans MaybeT | |

MonadTrans (ErrorT e) | |

MonadTrans (ExceptT e) | |

MonadTrans (StateT s) | |

MonadTrans (StateT s) | |

Monoid w => MonadTrans (WriterT w) | |

Monoid w => MonadTrans (WriterT w) | |

MonadTrans (IdentityT *) | |

MonadTrans (ContT * r) | |

MonadTrans (ReaderT * r) | |

(MFunctor f, MonadTrans f, MonadTrans g) => MonadTrans (ComposeT f g) # | |

Monoid w => MonadTrans (RWST r w s) | |

Monoid w => MonadTrans (RWST r w s) | |

(>|>) :: (Monad m3, MMonad t) => (forall a. m1 a -> t m2 a) -> (forall b. m2 b -> t m3 b) -> m1 c -> t m3 c infixr 2 #

(<|<) :: (Monad m3, MMonad t) => (forall b. m2 b -> t m3 b) -> (forall a. m1 a -> t m2 a) -> m1 c -> t m3 c infixl 2 #

# Tutorial

Monad morphisms solve the common problem of fixing monadic code after the fact without modifying the original source code or type signatures. The following sections illustrate various examples of transparently modifying existing functions.

## Generalizing base monads

Imagine that some library provided the following `State`

code:

import Control.Monad.Trans.State tick :: State Int () tick = modify (+1)

... but we would prefer to reuse `tick`

within a larger
`(`

block in order to mix in `StateT`

Int `IO`

)`IO`

actions.

We could patch the original library to generalize `tick`

's type signature:

tick :: (Monad m) => StateT Int m ()

... but we would prefer not to fork upstream code if possible. How could
we generalize `tick`

's type without modifying the original code?

We can solve this if we realize that `State`

is a type synonym for
`StateT`

with an `Identity`

base monad:

type State s = StateT s Identity

... which means that `tick`

's true type is actually:

tick :: StateT Int Identity ()

Now all we need is a function that `generalize`

s the `Identity`

base monad
to be any monad:

import Data.Functor.Identity generalize :: (Monad m) => Identity a -> m a generalize m = return (runIdentity m)

... which we can `hoist`

to change `tick`

's base monad:

hoist :: (Monad m, MFunctor t) => (forall a . m a -> n a) -> t m b -> t n b hoist generalize :: (Monad m, MFunctor t) => t Identity b -> t m b hoist generalize tick :: (Monad m) => StateT Int m ()

This lets us mix `tick`

alongside `IO`

using `lift`

:

import Control.Monad.Morph import Control.Monad.Trans.Class tock :: StateT Int IO () tock = do hoist generalize tick :: (Monad m) => StateT Int m () lift $ putStrLn "Tock!" :: (MonadTrans t) => t IO ()

`>>>`

Tock! ((), 1)`runStateT tock 0`

## Monad morphisms

Notice that `generalize`

is a monad morphism, and the following two proofs
show how `generalize`

satisfies the monad morphism laws. You can refer to
these proofs as an example for how to prove a function obeys the monad
morphism laws:

generalize (return x) -- Definition of 'return' for the Identity monad = generalize (Identity x) -- Definition of 'generalize' = return (runIdentity (Identity x)) -- runIdentity (Identity x) = x = return x

generalize $ do x <- m f x -- Definition of (>>=) for the Identity monad = generalize (f (runIdentity m)) -- Definition of 'generalize' = return (runIdentity (f (runIdentity m))) -- Monad law: Left identity = do x <- return (runIdentity m) return (runIdentity (f x)) -- Definition of 'generalize' in reverse = do x <- generalize m generalize (f x)

## Mixing diverse transformers

You can combine `hoist`

and `lift`

to insert arbitrary layers anywhere
within a monad transformer stack. This comes in handy when interleaving two
diverse stacks.

For example, we might want to combine the following `save`

function:

import Control.Monad.Trans.Writer -- i.e. :: StateT Int (WriterT [Int] Identity) () save :: StateT Int (Writer [Int]) () save = do n <- get lift $ tell [n]

... with our previous `tock`

function:

tock :: StateT Int IO ()

However, `save`

and `tock`

differ in two ways:

We can mix the two by inserting a `WriterT`

layer for `tock`

and
generalizing `save`

's base monad:

import Control.Monad program :: StateT Int (WriterT [Int] IO) () program = replicateM_ 4 $ do hoist lift tock :: (MonadTrans t) => StateT Int (t IO) () hoist (hoist generalize) save :: (Monad m) => StateT Int (WriterT [Int] m ) ()

`>>>`

Tock! Tock! Tock! Tock! [1,2,3,4]`execWriterT (runStateT program 0)`

## Embedding transformers

Suppose we decided to `check`

all `IOException`

s using a combination of
`try`

and `ErrorT`

:

import Control.Exception import Control.Monad.Trans.Class import Control.Monad.Trans.Error check :: IO a -> ErrorT IOException IO a check io = ErrorT (try io)

... but then we forget to use `check`

in one spot, mistakenly using `lift`

instead:

program :: ErrorT IOException IO () program = do str <- lift $ readFile "test.txt" check $ putStr str

`>>>`

*** Exception: test.txt: openFile: does not exist (No such file or directory)`runErrorT program`

How could we go back and fix `program`

without modifying its source code?

Well, `check`

is a monad morphism, but we can't `hoist`

it to modify the
base monad because then we get two `ErrorT`

layers instead of one:

hoist check :: (MFunctor t) => t IO a -> t (ErrorT IOException IO) a hoist check program :: ErrorT IOException (ErrorT IOException IO) ()

We'd prefer to `embed`

all newly generated exceptions in the existing
`ErrorT`

layer:

embed check :: ErrorT IOException IO a -> ErrorT IOException IO a embed check program :: ErrorT IOException IO ()

This correctly checks the exceptions that slipped through the cracks:

`>>>`

`import Control.Monad.Morph`

`>>>`

Left test.txt: openFile: does not exist (No such file or directory)`runErrorT (embed check program)`