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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 17th 2020
• (edited Jan 17th 2020)

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.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJan 18th 2020

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.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJan 18th 2020

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?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJan 19th 2020
• (edited Jan 19th 2020)

How isn’t this completely straightforward: A category is essentially $U$-small if it is equivalent to a $U$-small category.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 19th 2020

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.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeJan 19th 2020

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.)

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeJan 19th 2020

Fixed the mistaken use of “essentially”.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJan 21st 2020

“Structurally $U$-small” isn’t unreasonable, I guess. I don’t have any better ideas.

Thanks for fixing the mistake!