[Haskell-beginners] Category question

[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.

> 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