Stefan O'Rear wrote:
On Thu, Jul 12, 2007 at 07:19:07PM +0100, Andrew Coppin wrote:
I'm still puzzled as to what makes the other categories so magical that they cannot be considered sets.
I'm also a little puzzled that a binary relation isn't considered to be a function...
From the above, it seems that functors are in fact structure-preserving mappings somewhat like the various morphisms found in group theory. (I remember isomorphism and homomorphism, but there are really far too many morphisms to remember!)
Many categories have more structure than just sets and functions. For instance, in the category of groups, arrows must be homomorphisms.
What the heck is an arrow when it's at home?
Some categories don't naturally correspond to sets - consider eg the category of naturals, where there is a unique arrow from a to b iff a ≤ b.
...um...
Binary relations are more general then functions, since they can be partial and multiple-valued.
What's a partial relation?
indeed, it is possible to form the "category of small categories" with functors for arrows. ("Small" means that there is a set of objects involved; eg Set is not small because there is no set of all sets (see Russel's paradox) but Nat is small.)
OK, see, RIGHT THERE! That's the kind of sentence that I read and three of my cognative processes dump sort with an "unexpected out of neural capacity exception". o_O