# Arrows that are also Functors

Edward Kmett ekmett at gmail.com
Tue Apr 26 21:19:53 CEST 2011

```On Tue, Apr 19, 2011 at 11:48 PM, Tyson Whitehead <twhitehead at gmail.com>wrote:

> On April 19, 2011 23:22:12 Tyson Whitehead wrote:
> >
> >   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
> being you can give a useful definition for structural monads as well.

Note: This does not generalize the signature of mfix, it only overlaps
slightly, as not every monad m permits the extraction of the value a
injected (consider Cont r), so you necessarily change the meaning or
obliterate a number of instances.

Recall the main motivation for mfix was to support Erkoek and Launchbury's
recursive do:

This necessitates 4 laws for mfix, which don't translate nicely.

- mfix (return . h) = return (fix h)
- mfix (\x -> a >>= \y -> f x y) = a >>= \y -> mfix (\x -> f x y)
- if
h
is strict,
mfix (liftM h . f) = liftM h (mfix (f . h))
- mfix (\x -> mfix (\y -> f x y)) = mfix (\x -> f x x)

The other commonly proposed mfix replacement is to define it once, as guided
by the types, but while this works for fix and the the comonadic equivalent,
it doesn't generate a useful mfix for recursive do either.

-Edward
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