[Haskell-cafe] MonadTrans lift implementation

[Mauro J. Jaskelioff]

Ross Paterson wrote:
> On Mon, Jan 19, 2009 at 01:13:37PM -0800, Jonathan Cast wrote:
>   
>> (On the other hand, your hunch that lift = return is correct --- so you
>> get a cookie for that; it's just that return here is neither the return
>> of the monad for m nor the return of the monad for ReaderT m.  It is,
>> instead, the return of the *applicative functor* --- on the category of
>> monads and monad homomorphisms --- associated to the monad transformer
>> ReaderT.)
>>     
>
> It's also a monad in the category of monads, as are ErrorT, StateT and
> WriteT (see Moggi, An Abstract View of Programming Languages, 1989, s4).
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>   
Monads in the (2-)category of monads correspond to distributive laws
(see Formal theory of Monads by Street, 1972).
StateT does not arise from a distributive law.
In particular, you would need a map

tjoin :: Monad m => StateT (StateT m) a -> StateT m a
        :: Monad m = (s -> s -> m ((a,s),s)) ->  s ->  m (a,s)

subject to the monad laws (where
 lift :: Monad m => m a -> StateT m a
is the unit, i.e. return)

All the best

- Mauro

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