[Haskell-cafe] A question about "monad laws"

[ajb at spamcop.net]

G'day all.

Quoting askyle <valgarv at gmx.net>:

> Yup: bind f = f <=< id         -- whence you can say (>>=) = flip bind

Ah, yes.

> My point is that (as far as I can see) you cannot prove the properties of
> bind by only assuming identity and associativity for (<=<).

One thing that may help is that if you can prove that fmap is sane:

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

then the naturality of return is precisely its free theorem, and ditto
for bind.

So perhaps this law:

     (f <=< g) . h === f <=< (g . h)

is actually the fmap law in disguise?

Cheers,
Andrew Bromage