[Haskell-cafe] Very freaky

[Dominic Steinitz]

> 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.

Another example: a partially ordered set is a category. The objects are the 
elements and there is an arrow between two objects a & b if a <= b. An 
element isn't (necessarily) a set. Nothing magical here.

A functor is then an order preserving function (homomorphism).

This question has come up more than once so it may be worth a wiki page if 
anyone has time.

>
> I'm also a little puzzled that a binary relation isn't considered to be
> a function...

That's the definition of a function: a restricted relation in which there is 
at most one range element for a given domain element - see any book on set 
theory e.g. Halmos.

>
>  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!)

Sometimes but clearly the forgetful functor doesn't.