[Haskell-cafe] Very freaky

[Andrew Coppin]

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