[Haskell-cafe] Very freaky
Andrew Coppin
andrewcoppin at btinternet.com
Thu Jul 12 14:44:50 EDT 2007
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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