We’ve mentioned the category Image may be NSFW.
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(the category of sets with functions as morphisms). Consider another category Image may be NSFW.
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which has two objects Image may be NSFW.
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and Image may be NSFW.
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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 Image may be NSFW.
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and Image may be NSFW.
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. 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 Image may be NSFW.
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. That is Image may be NSFW.
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is a set. (One caveat here is that Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
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is small if Image may be NSFW.
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is an honest-to-goodness set. Otherwise, we say that Image may be NSFW.
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Clik here to view. 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.
Image may be NSFW.
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