On Mon, 28 May 2012 10:57:11 -0400
Brent Yorgey
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