Andrius wrote:

> In defining Cat, the category of small categories, Wikipedia states that the terminal object (the terminal category) is the trivial category 1 with a single object and morphism.

> Is that trivial category necessarily unique?

No: there are infinitely many different categories with one object and one morphism. These terminal categories are not equal. The reason is that object could be anything you want, e.g. \\(\sqrt{2}\\) or \\(\\{\emptyset\\}\\), and so could the morphism. Since \\(\sqrt{2} \ne \\{\emptyset\\}\\), the terminal category with \\(\sqrt{2}\\) as its own object is not equal to the terminal category with \\(\\{\emptyset\\}\\) as its only object.

However, by virtue of the definition of "terminal", any two terminal categories are isomorphic. Even better, as Matthew pointed out, they are isomorphic in a _unique way_.

This enable category theorists to talk about "the" terminal category. In category theory, unlike set theory, we are happy to talk about "the" thing with a property if any two things with that property are isomorphic in a canonical way. "In a canonical way" requires some explanation, but here we are in a simple case: any two terminal categories are isomorphic in a _unique_ way, and "unique" implies "canonical".

> A category Set and some other category are the same category if they have the same external relationships, which is to say, the same diagram and the same commutative diagram, which can be labelled the same.

Saying "the same" is sloppy, but okay if one is talking to experts who know exactly what that means in this context. It does not mean "equal". It means "isomorphic".

> If you had another category which had the same relationships (though with nominally different objects), then it wouldn't just be isomorphic, it would be EQUAL to the category of sets.

No.

> In defining Cat, the category of small categories, Wikipedia states that the terminal object (the terminal category) is the trivial category 1 with a single object and morphism.

> Is that trivial category necessarily unique?

No: there are infinitely many different categories with one object and one morphism. These terminal categories are not equal. The reason is that object could be anything you want, e.g. \\(\sqrt{2}\\) or \\(\\{\emptyset\\}\\), and so could the morphism. Since \\(\sqrt{2} \ne \\{\emptyset\\}\\), the terminal category with \\(\sqrt{2}\\) as its own object is not equal to the terminal category with \\(\\{\emptyset\\}\\) as its only object.

However, by virtue of the definition of "terminal", any two terminal categories are isomorphic. Even better, as Matthew pointed out, they are isomorphic in a _unique way_.

This enable category theorists to talk about "the" terminal category. In category theory, unlike set theory, we are happy to talk about "the" thing with a property if any two things with that property are isomorphic in a canonical way. "In a canonical way" requires some explanation, but here we are in a simple case: any two terminal categories are isomorphic in a _unique_ way, and "unique" implies "canonical".

> A category Set and some other category are the same category if they have the same external relationships, which is to say, the same diagram and the same commutative diagram, which can be labelled the same.

Saying "the same" is sloppy, but okay if one is talking to experts who know exactly what that means in this context. It does not mean "equal". It means "isomorphic".

> If you had another category which had the same relationships (though with nominally different objects), then it wouldn't just be isomorphic, it would be EQUAL to the category of sets.

No.