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Riley Shahar (Nov 12 2024 at 21:09):The nlab page on localizations ([[category of fractions]]) says that "there is a general construction of , if it exists." Is there any reason other than size that it could fail to exist? I think I can show that the zig-zag construction given there is general, and e.g. Borceux (Proposition I.5.2.2 of Handbook of Categorical Algebra) asserts that it always exists via the same construction, although Borceux is proving a slightly different universal property (Borceux's is essentially 1-categorical, whereas the nlab's is 2-categorical). I just want to make sure I'm not missing something strange that can break with the version of the universal property on that page.
(Thanks in advance!)
Mike Shulman (Nov 12 2024 at 22:27):I can't think of a reason in ordinary mathematics that it would fail to exist except size. I suppose in some exotic kinds of foundations where you don't have quotient sets you might have problems, but so many things are different in that kind of world that it's not really worth mentioning unless you're an afficionado already.
Riley Shahar (Nov 12 2024 at 22:32):Mike Shulman said:
I can't think of a reason in ordinary mathematics that it would fail to exist except size. I suppose in some exotic kinds of foundations where you don't have quotient sets you might have problems, but so many things are different in that kind of world that it's not really worth mentioning unless you're an afficionado already.
Great, thank you!
David Michael Roberts (Nov 13 2024 at 01:08):It might not exist by virtue of assumptions about what a category is, as Mike says: if one is working in the 2-category of locally small categories then the result of the zig-zag construction might not be locally small and so not be an object of the 2-category.
Matteo Capucci (he/him) (Nov 15 2024 at 10:22):Thanks for asking this @Riley Shahar, I had this doubt too for a while...