[Haskell-beginners] Category question

[Manfred Lotz]

On Tue, 29 May 2012 09:31:01 -0400
Brent Yorgey <byorgey at seas.upenn.edu> wrote:

> On Mon, May 28, 2012 at 06:50:34PM +0200, Manfred Lotz wrote:
> > On Mon, 28 May 2012 10:57:11 -0400
> > Brent Yorgey <byorgey at seas.upenn.edu> wrote:
> > 
> > > On Mon, May 28, 2012 at 04:14:40PM +0200, Manfred Lotz wrote:
> > > > 
> > > > For me id: A -> A could be defined by: A morphism id: A -> A is
> > > > called identity morphism iff for all x of A we have  id(x) = x.
> > > 
> > > This is not actually a valid definition; the notation id(x) = x
> > > does not make sense.  It seems you are assuming that morphisms
> > > represent some sort of function, but that is only true in certain
> > > special categories.
> > > 
> > 
> > Ok, it is a valid definition only in a certain context. In the
> > far wider context of category theory this indeed makes no sense.
> 
> Right.
> 
>  
> > In 'Conceptual Mathematics' by F. William Lawvere, Stephen H.
> > Schanuel they define an identity map with fa = a for each a in A.
> > Then on page 17 they define category and say 
> > 
> > ...
> > Identity Maps: (one per object) 1A: A -> A
> > ...
> > Rules for a category
> > 1. The identity laws:
> > where they say g . 1A = g and 1B . f = f
> > 2. associatlve laws
> > ...
> > 
> > It seems that this definition of a category is not as general as it
> > could be. Here 1. is something which follows easily from the
> > definition of an identity map.
> 
> I am guessing (though I have not looked at 'Conceptual Mathematics' in
> detail) that they use 'an identity map with fa = a for each a in A'
> simply as an *example* to help build intuition; then on page 17 they
> generalize this example to the fully abstract definition of a
> category. It does seem unfortunate that they continue to use the name
> 'identity map', because morphisms/arrows are more general than 'maps'
> (to me, 'map' is synonymous with 'function').
> 

Yes, even in the general definition they use identity map. IMHO, they
should have made it clearer that there is a broader context. 

But anyway, the discussion here was fruitful and thanks to you and the
others it is now clear to me. 



-- 
Manfred