Pointed [Re: 2014 Applicative => Monad proposal]

[Petr Pudlák]

Dne 25.5.2013 17:33, Dan Doel napsal(a):

> On Fri, May 24, 2013 at 2:15 PM, Andreas Abel <andreas.abel at ifi.lmu.de 
> <mailto:andreas.abel at ifi.lmu.de>> wrote:
>
> [...]
> Pointed is pretty bad for this. Or, maybe, it's good: there's nothing 
> to reason about, because there's nothing you can reason about. It just 
> does _something_ unspecified. Hopefully something that is good for 
> combining with some other type class.

-1 on |Pointed|.

>
> I've reluctantly come around to the Apply/Bind classes (despite them 
> having bad names), because they actually follow the way people 
> typically build up structure in algebra. Rarely do you see people 
> defining 'a monoid is a pointed set equipped with an associative 
> binary operation that uses the point as an identity.' It's much more 
> likely that you see, 'a monoid is a semigroup with an identity for the 
> binary operation.' I suspect that's even how many people would like to 
> see the hierarchy at this point: Semigroup m => Monoid m. And not 
> Default m => Monoid m, because Default is just ad-hoc nonsense (even 
> if it's sometimes useful).
>
> To answer your original question, perhaps a somewhat better way of 
> characterizing the problem is probably talking about 'monoids m over 
> type a'. Perhaps something like:
>
>     class Monoid m => MonoidOver a m | m -> a where
>       embed :: a -> m
>
> which expresses that m is expected to be generated (as a monoid) by a 
> in some way. Then maybe you can say something about how it interacts 
> with the Monoid operations, which seems preferable to me to, "the 
> operation of this class does anything, except if the type is also a 
> monoid, or if it's also a monad...."

I think you hit the nail on the head here. The only places when I missed 
something like Pointed were when I wanted to abstract building some kind 
of collection (Set / [] / Seq / ...). They're all monoids, but you need 
some way how to construct singleton collections. Having |MonoidOver| 
would solve precisely this problem. I'd certainly prefer something like 
this over |Pointed|.

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