[Haskell-cafe] monad . comonad = id

[Jan Christiansen]

Hi,

On 08.05.2009, at 13:14, Miguel Mitrofanov wrote:

>> I have a question regarding the connection between monads and  
>> comonads. I always thought of the comonad operations as being the  
>> inverse operations of the corresponding monad functions.
>
> That's not true.

I thought there is some kind of duality between monads and comonads in  
category theory. Does this "only" refer to the types of the functions  
or am I totally wrong here?

>> If this is the case I would like to know what the corresponding  
>> monad for the following comond could be. This comonad treats lists  
>> as pointed sets where the first element is focused.
>>  instance Comonad [] where
>>    extract = head
>>    duplicate xs = init (zipWith (++) (tails xs) (inits xs))
>
> First of all, this is NOT a comonad. The term "comonad" doesn't  
> refer just to instances of Comonad class, it refers to instances  
> that satisfy several laws.

I am aware of this. I thought it does satisfy all the laws. Which one  
have I missed?

> Secondly, "[]" is a monad, and there is an instance of Monad class  
> for it. It's working; don't fix it.

I am also aware of this. Perhaps I should have used a data type like

   data Pointed a = Pointed a [a]

but in this case a corresponding duplicate implementation is more  
complicated.

I am very sorry if have asked a stupid question.

Cheers, Jan