[Haskell-beginners] two typos in my last. Was: Re: Making sense of currying in the context of the Hask Category

[Jay Sulzberger]

On Fri, 28 Sep 2012, Jay Sulzberger wrote:

>
> On Thu, 27 Sep 2012, Lino Rosa <lino.aguiar.rosa at gmail.com> wrote:
>
>> Hi,
>> 
>> I'm a Haskell newbie and I couldn't quite make sense of how currying
>> maps to the the Hask Category.
>> How would I map, for instance (+) to a Hask 'arrow'?
>
> There are several categories that might be called "the Hask
> category".  At
>
>  http://www.haskell.org/haskellwiki/Hask
>
> there are pointers to papers on what a useful Hask might be.
>
> Let us consider the category SET whose objects are sets and whose
> morphisms are everywhere defined single valued maps.  So a
> morphism f: X -> B, where X and B are sets, would just be a
> function, usual modern sense, from X to B.  Now the set of all
> morphisms from X to B, let us call it SET(X;B), is itself a set,
> so SET(X;B) is an object of SET.
>
> Now suppose B is the set SET(Y;Z) of morphisms from the set X to
> the set Y.  In this case given an element x of X, we have that

Oi, typo:

Above lines should be:

Now suppose B is the set SET(Y;Z) of morphisms from the set Y to
the set Z.  In this case given an element x of X, we have that

> f(x) is a morphism of SET, and f(x) has source Y and target Z, that
> is, f(x): Y -> Z.  So given x, and now given y in Y, we have a new two
> place function, call it g: X x Y -> Z, defined by:
>
>  for all x in X and y in Y, g(x, y) = [f(x)](y)
>
> where the square brackets are just for grouping.
>
> Note that we have the "operation" of Cartesian product on objects
> of SET, which operation is shown above as the "x" in the
> expression "X x Y".  Note also that SET is an odd sort of
> category in that SET(X;B), for any two objects X, B, is itself an
> object of SET.
>
> Now in SET we may also, for any g: X x Y -> Z get an
> f: X -> SET(Y;Z), such that our condition holds.
>
> Category theorists apparatus to make explicit what we have just

and another, above line should be

Category theorists have apparatus to make explicit what we have just

oo--JS.


> claimed in vague and not quite precise terms.  This apparatus is
> the theory of Cartesian Closed Categories:
>
>  http://en.wikipedia.org/wiki/Cartesian_closed_category
>  [page was last modified on 28 September 2012 at 19:08]
>
> Let me give an example of our imprecision, which formal category
> theory clarifies:
>
>  We said that g is a two place function.  We also wrote
>  g: X x Y -> Z.  What does this mean?  In any category every
>  morphism goes from exactly one object to exactly one object.
>  So how can there be such a thing as a "function with two
>  inputs"?
>
> oo--JS.
>
>
>> 
>> If objects are types on Hask, then would a -> a -> a be the first
>> object on this chain?
>> 
>> In that case, for the first arrow, would I have as many arrows as
>> there are possible partial applications on this type? In other words,
>> would I have (+) 1, (+) 2, (+) 3 ... all those transitioning to the
>> second object ( a -> a )
>> Or, do I have ONE arrow only, like (+) a ?
>> 
>> In either case, what happens after 'm left with the object a -> a?
>> What function (arrow) mutates it to the final value 'a'? That's the
>> function resulting from the previous partial application of (+), but
>> that fuction only exists at run time, after you apply the first one. I
>> guess the question is, if you'd have to write a diagram for this, what
>> would you write beside each object and beside each arrow?
>> 
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>> 
>
>