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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
(fromMonadTrans
) 
squash
(See below) 
(See below), ifhoist
ff
is a monad morphism 
(f . g)
, iff
andg
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
 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
Monads over Monads
class (MFunctor t, MonadTrans t) => MMonad t whereSource
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 transformer of monads:
MonadTrans MaybeT  
MonadTrans IdentityT  
Monoid w => MonadTrans (WriterT w)  
Monoid w => MonadTrans (WriterT w)  
MonadTrans (StateT s)  
MonadTrans (StateT s)  
MonadTrans (ReaderT r)  
Error e => MonadTrans (ErrorT e)  
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 cSource
(<<) :: (Monad m3, MMonad t) => (forall b. m2 b > t m3 b) > (forall a. m1 a > t m2 a) > m1 c > t m3 cSource
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 ()
>>>
runStateT tock 0
Tock! ((), 1)
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 ) ()
>>>
execWriterT (runStateT program 0)
Tock! Tock! Tock! Tock! [1,2,3,4]
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
>>>
runErrorT program
*** Exception: test.txt: openFile: does not exist (No such file or directory)
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
>>>
runErrorT (embed check program)
Left test.txt: openFile: does not exist (No such file or directory)