[Haskell-cafe] Very freaky

[Andrew Coppin]

Alexis Hazell wrote:
> On Thursday 12 July 2007 04:40, Andrew Coppin wrote:
>
>   
>> I once sat down and tried to read about Category Theory. I got almost
>> nowhere though; I cannot for the life of my figure out how the
>> definition of "category" is actually different from the definition of
>> "set". Or how a "functor" is any different than a "function". Or...
>> actually, none of it made sense.
>>     
>
> Iiuc,
>
> "Set" is just one type of category; and the morphisms of the category "Set" 
> are indeed functions. But morphisms in other categories need not be 
> functions; in the category "Rel", for example, the morphisms are not 
> functions but binary relations.
>
> A "functor" is something that maps functions in one category to functions in 
> another category. In other words, functors point from one or more functions 
> in one category to the equivalent functions in another category. Perhaps they 
> could be regarded as 'meta-functions'.
>
> Hope that helps,
>   

It helps a little...

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!)