I see fmap as a "penetration" of one box.
fmap.fmap as a "penetration" of two boxes, and so on.

So, if you have (UI -> Behavior (Blah, a)), then the value "a" is "buried" inside a (UI->) box, then inside a Behavior box, and then inside a (Blah,) box.
To change the value of "a" you can thus use: (fmap.fmap.fmap) aToB

To access (Blah, a), instead, just use (fmap.fmap) and penetrate just 2 boxes instead of 3.

Additionally, this "trick" also works nicely with liftA2. If you have two values that are buried inside the same 2 boxes (For example, two values of type: UI -> Behavior a -- the boxes here are a (UI->) reader, and a Behavior), then you can apply f to the "a" inside both via:

(liftA2.liftA2) f doublyBoxedA doublyBoxedB

And as usual, (liftA2.liftA2.liftA2) when the values are wrapped in 3 boxes, and so on.

Eyal

2008/11/19 Peter Verswyvelen <bugfact@gmail.com>

Many of the fmap fmaps I've seen come from

(fmap f) . g = (fmap f) `fmap` g = fmap (fmap f) g = (fmap (fmap f)) g = ((fmap.fmap) f) g = (fmap.fmap) f g

But... to me the former is much clearer than the latter... It is less abstract since it uses (.) instead of fmap.

However, I also see (fmap.fmap.fmap), so Conal must have seen a pattern here?

Of course one could also write (fmap `fmap` fmap `fmap` fmap) instead of (fmap.fmap.fmap).

Or as was already mentioned with the generic replacement (.) = fmap, (fmap.fmap.fmap) can be written as ( (.) . (.) . (.) )

But besides this all being very Haskelly Hackery - and maybe ASCII art that needs to be censured - I don't see the light yet :)

Could somebody enlighten me?

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