[Haskell-beginners] Category question

[Brent Yorgey]

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').

-Brent