Pedantry

We’ve mentioned the category $\textsc{Set}$ (the category of sets with functions as morphisms). Consider another category $\textsc{Two}$ which has two objects $A$ and $B$ 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 $Ob(\textsc{Set})$ and $Ob(\textsc{Two})$. 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 $\{A, B\}$. That is $Ob(\textsc{Fin})$ is a set. (One caveat here is that $A$ and $B$ 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:

Definition:

A category $\mathcal C$ is small if $Ob(\mathcal C)$ is an honest-to-goodness set. Otherwise, we say that $\mathcal C$ is large (in the situation that $Ob(\mathcal C)$ 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.

4 Responses to Pedantry

1. soffer801 says:

It was pointed out to me that I lied. The category $\textsc{Fin}$ of all finite sets is indeed large. We can fix this by talking about the category of equivalence classes of finite sets. Then there is only one for each cardinality (of which there are only countably many). Some set theory hocus pocus will make this work.

I’ve edited the post to reflect something more correct.

Thanks all, for stopping me when I’m wrong. It is much needed.

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