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.
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? Note that there is no particular difference between calling these equations "axioms" or a "definition". That is, "there is an 'identity morphism' satisfying the following axioms..." and "there is an 'identity morphism' defined by..." are just two different ways of saying the exact same thing. -Brent