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.