[Haskell-cafe] Are all arrows functors?

[Neil Brown]

Hi,

I was thinking about some of my code today, and I realised that where I 
have an arrow in my code, A b c, the type (A b) is also a functor.  The 
definition is (see 
http://www.haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html):

fmap = (^<<)
-- Or, in long form:
fmap f x = arr f <<< x

Out of curiosity, and since this is a typical haskell-cafe question, 
does this definition of fmap hold for all arrows?

And is there a wiki page somewhere that has a table of all of these 
Haskell type-classes (Functor, Monad, Category, Arrow, Applicative and 
so on), and says that if you are an instance of class A you must have 
some corresponding instance of B? (e.g. all Monads are Functors and 
Applicatives)  I'm fairly certain my arrow isn't a Monad or Applicative, 
although of course it must be a Category, given the type-class 
dependency, but it would be nice when using one of these things to see 
what other instances you should automatically supply.

Thanks,

Neil.