January 16, 2012 4 Comments
We’ve mentioned the category (the category of sets with functions as morphisms). Consider another category which has two objects and and a morphism between them (along with the required identity morphisms)
First of all, the small-caps notation and naming convention is not standard, there is no standard, but I think it is as good as any other, so I’ll use it.
Secondably [sic], there’s a fundamental difference between these two categories. The difference is in the objects. Not the objects themselves, but and . The first one is not a set. After all, we can’t talk about the set of all sets. In some sense, the collection of all sets is just to big to be a set. We can however talk about the set . That is is a set. (One caveat here is that and must be sets themselves, but we can choose them to be, and it’s not the point of this post, nor will it be relevant later).
This is why, in the definition of categories, I specifically mentioned that we had a collection of objects, not a set of objects. But if it’s not a set, what is it? It’s called a class, and it pushes us closer to axiomatic set theory than I want to go. It does however give rise to the following definitions that the pedantic reader will care to think about:
A category is small if is an honest-to-goodness set. Otherwise, we say that is large (in the situation that is a class and not a set).
I must say that really this stuff is important despite how I’ve presented it. But if you trust me not to lie to you (probably a bad move), you can just read and trust that I’m not breaking any mathematical laws.