[Haskell-beginners] Category question

[Manfred Lotz]

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.



> > My point is that in the books about category theory those two
> > statements are stated as axioms, and id is (in many books) just
> > self understood or defined as I have defined it above.
> > 
> > If in a book about category the author would say that for each
> > object A there must exist a morphism id: A -> A (called identity
> > morphism) which is defined by idB . f = f and f . idA = f then this
> > would be clearer (and better, IMHO).
> 
> This is exactly what category theory books do (or should) say.  Do you
> have a particular example of a book which does not state things in
> this way?
> 

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 guess that this made me think of idA as idA(x) = x for each x of A.
Later when I saw other (more general) definitions I did not read
carefully to realize the difference.


Thanks a lot for making this clear to me.


-- 
Manfred