(Off-topic) Question about categories

[Graham Klyne]

(I'm asking here because I believe there is some overlap between category 
theory and functional programming cognoscenti...)

It has been suggested to me that categories may be a useful framework (more 
useful than set theory) for conceptualizing things for which there is no 
well-defined equality relationship.  (The discussion is about resources in 
WWW.)  I've done a little digging about category theory, and find some 
relatively approachable material at:
   http://www.wikipedia.org/wiki/Category_theory
   http://www.wikipedia.org/wiki/Class_(set_theory)
and nearby.

But I'm hitting a mental block with this (and other places I've look don't 
add anything I can grok).  The definition of a category depends on the 
definition of a morphism, and in particular the existence of an identity 
morphism for every object in a category.  The definition of an morphism is 
in terms of equality of compositions of morphisms:
     for f : A -> B we have Id[B]. f = f = f . Id[A]

My problem is this:  how does it make sense to define an equality of 
morphisms without some well-defined concept of equality on the underlying 
objects to which they apply?  That is, given object X and an object Y, it 
is possible to examine them and determine whether or not they are the same 
object.  And if the underlying objects have such a concept of equality, 
what prevents them from being sets or members of sets?  But categories are 
presented as being more general than sets.

Does anyone see the cause of my non-comprehension here?

#g


------------
Graham Klyne
GK at NineByNine.org