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
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