Graham Klyne wrote, on 18 september 2003 14:44 [...]
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.
I am not a specialist, but my understanding is as follows: Category Theory is an axiomatic theory: a language with certain primitive notions and relations between these notions. Equality between objects is not a notion of category theory and you will not find any category-theoretic theorem proving that two objects are equal. (You may say that an object has an identity, as given by its identity morphism. A category actually can be defined using morphisms only.) Equality between morphisms however is a central notion of category theory. If you have a concrete category, you have to denote some things as objects, others as morphisms; before exploiting category theory, you have to prove that the objects and morphisms of your concrete category fulfill the category axioms, and for that you may, of course, use the equality relations in your concrete category. Peter J. Veger, Best Netherlands