hello, i think what is importnat is that one can tell when two arrows are the same (how to specify that of course is tricky in general). equality on objects kind of follows from there. in fact one can describe a category without referring to objects at all: 1. pick a collection of arrows (call it A) 2. pick a sub-collection if A that you will call "identity" arrows (call that I) 3. give the following mappings (these are not arrows!) 3.1 source: A -> I 3.2 target: A -> I 3.3 compose: (A,A) -> A (this one is partial, and has to be defined, when the source and target match as usual). the trick is that essentially objects are represented by the identity arrows on them (as there is a 1-1 correspondance between the two). my favourite way to think of a category is as follows: 1. i think of objects as types 2. i think of an arrow: A -> B as value of type B, that has a free variable of type A 3. i think of composition as substitution hope this helps iavor Graham Klyne wrote:
(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@NineByNine.org
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-- ================================================== | Iavor S. Diatchki, Ph.D. student | | Department of Computer Science and Engineering | | School of OGI at OHSU | | http://www.cse.ogi.edu/~diatchki | ==================================================