[Haskell-cafe] Type classes

[Johannes Gerer]

Ok, now I see a difference, why Kleisli can be used to relate
typeclasses (like Monad and ArrowApply) and Cokleisli can not:

"Kleisli m () _"  =  "() -> m _" is isomorphic to "m _"

whereas

"Cokleisli m () _" = "m _ -> ()" is not.

Can somebody point out the relevant category theoretical concepts,
that are at work here?



On Tue, May 28, 2013 at 5:43 PM, Tom Ellis
<tom-lists-haskell-cafe-2013 at jaguarpaw.co.uk> wrote:
> On Tue, May 28, 2013 at 05:21:58PM +0200, Johannes Gerer wrote:
>> That makes sense. But why does
>>
>> instance Monad m => ArrowApply (Kleisli m)
>>
>> show that a Monad can do anything an ArrowApply can (and the two are
>> thus equivalent)?
>
> I've tried to chase around the equivalence between these two before, and
> I didn't find the algebra simple.  I'll give an outline.
>
> In non-Haskell notation
>
> 1) instance Monad m => ArrowApply (Kleisli m)
>
> means that if "m" is a Monad then "_ -> m _" is an ArrowApply.
>
> 2) instance ArrowApply a => Monad (a anyType)
>
> means that if "_ ~> _" is an ArrowApply then "a ~> _" is a Monad.
>
> One direction seems easy: for a Monad m, 1) gives that "_ -> m _" is an
> ArrowApply.  By 2), "() -> m _" is a Monad.  It is equivalent
> to the Monad m we started with.
>
> Given an ArrowApply "_ ~> _", 2) shows that "() ~> _" is a Monad.  Thus by
> 1) "_ -> (() ~> _)" is an ArrowApply.  I believe this should be the same
> type as "_ ~> _" but I don't see how to demonstrate the isomorphsim here.
>
> Tom
>
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