Arrows that are also Functors

[Maciej Piechotka]

On Tue, 2011-04-26 at 15:13 -0400, Tyson Whitehead wrote:
> On April 26, 2011 13:50:10 Maciej Marcin Piechotka wrote:
> > On Tue, 2011-04-19 at 23:48 -0400, Tyson Whitehead wrote:
> > > On April 19, 2011 23:22:12 Tyson Whitehead wrote:
> > > > ArrowLoop from MonadFix
> > > > 
> > > >   loop' f = fst' .' loop'' (f .' arr' (second snd))
> > > >   
> > > >     where loop'' f = mfix (\y -> f .' arr' (,y))
> > > 
> > > BTW haskellers, I've been wondering if mfix would better be defined as
> > > 
> > >   mfix' :: (m a -> m a) -> m a
> > > 
> > > where "mfix' f = mfix (f . pure)" for the computational monads.  The
> > > advantage being you can give a useful definition for structural monads
> > > as well.
> > 
> > What would be the difference with fix?
> 
> For the IO monad, for example, normal fix would give you an IO value which 
> would repeat the underlying IO action.  The mfix' above give you the result of 
> the first run of the IO action lazily bound in an IO context.  Consider
> 
> Prelude> fix $ fmap (1:)
> ... never returns (overflows the stack and dies) ...
> 
> Prelude> mfix' $ fmap (1:)
> ... returns "return $ repeat 1" ...
> 
> The difference between mfix and mfix' being you can't (or at least I couldn't) 
> implement it for non-singleton monads such as list.

But... list is monad fix. From base:

> instance MonadFix [] where
>     mfix f = case fix (f . head) of
>                []    -> []
>                (x:_) -> x : mfix (tail . f)

Prelude> mfix (fmap (1:) . return)
[1,1,1,1,...

> The problem was mfix had 
> to do a map like operation across the underlying structure to invoke the 
> function on the lazily bound singletons before any of the structure existed.
> 
> Cheers!  -Tyson

I still cannot find any use which would not be covered by either mfix or fix.

Regards
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