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Todd points out elsewhere that there is a problem with the following sentence in the section Smallness in the context of universes:
$C$ is essentially $U$-small if there is a bijection from its set of morphisms to an element of $U$ (the same for the set of objects follows); this condition is non-evil.
(introduced in revision 11).
It looks to me that first of all this is not the right condition – the right condition must mention equivalence of categories to a U-small category.
Yes, I agree. But the condition “there is a bijection from its set of morphisms to an element of U” is also sometimes an interesting condition, although I don’t know a standard name for it.
So getting back to this, “essentially” is not the right adjective; maybe we can substitute another and do a slight rewrite? How does “structurally $U$-small” sound?
How isn’t this completely straightforward: A category is essentially $U$-small if it is equivalent to a $U$-small category.
Oh, you mean you want a word for the wrong condition that is in the entry?
Let’s first just fix the entry. It’s embarrassing for the category theorists around here to have such elementary mistakes in such elementary pages on category theory. I am busy for the next few hours. If nobody fixed it by then, I will do it.
Mike was just saying in #2 that the condition is sometimes interesting, and so I was suggesting that we give it a different name, as well as add the correct definition of essentially $U$-small. (But permission to edit is being withheld for now.)
“Structurally $U$-small” isn’t unreasonable, I guess. I don’t have any better ideas.
Thanks for fixing the mistake!
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