Timeline for On the large cardinals foundations of categories
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2013 at 6:49 | comment | added | Asaf Karagila♦ | Could you post a notice of this answer/a copy on the MSE post as well? | |
| Jun 30, 2013 at 6:48 | vote | accept | Asaf Karagila♦ | ||
| May 15, 2013 at 10:53 | comment | added | Zhen Lin | Yes, that much is easy. Actually I prefer not to use any universes at all and just work in NBG for such simple scenarios: because, say, locally small categories are actual objects in the universe of discourse, I can then form the metacategory of locally small categories at the syntactic level and do what I need there. My belief is that, so long as nothing sophisticated is being done, iterating the ZFC-to-NBG construction $\omega$ times should be enough to guarantee the existence of any definable categories one might like to use. The difficulty is when one wants to do universal constructions... | |
| May 15, 2013 at 9:23 | comment | added | Asaf Karagila♦ | ... I think. No? | |
| May 15, 2013 at 9:23 | comment | added | Asaf Karagila♦ | Zhen Lin, I am certain that I will not be able to understand a lot of the things involved. Exactly like how it can be immensely difficult to understand forcing without knowing quite a bit of set theory at first. And again, it's clear to me there is more than just bookkeeping involved, and that there are things which are probably not feasible without having universes which are very nicely extended from one step to the next. But if you want to talk about the category of large categories, going to $M_3$ and considering the set of those elements of $M_2$ which are large categories in $M_1$ works | |
| May 15, 2013 at 9:17 | comment | added | Zhen Lin | @Asaf: My point is that sometimes one is doing much more than mere bookkeeping. It should go without saying that for the first 2 years that I studied category theory I was in the anti-universe camp because I thought it was all just a matter of being careful with various things, but I have now seen that this just isn't feasible in some areas. I'm afraid there's nothing more I can say if you don't know the category theory involved! | |
| May 15, 2013 at 9:05 | comment | added | Michal R. Przybylek | @Asaf, because you've asked an imprecise question, you get posts answering different issues of the concept of "universe". Perhaps the only answer to the question "is it sufficient..." (and take it really serious) is: "it is sufficient to define a category as number 22; it is also sufficient to define a category as number 2 --- if one does not believe in existence of such large natural numbers as 22". | |
| May 15, 2013 at 8:28 | comment | added | Asaf Karagila♦ | I feel, reading your comment, that I lack the understanding of what people use universes for. I suppose this is a topic for a whole other question though (or rather to find out on my own). Because I still don't see the issue about keeping track in which universe your objects and categories are in, and taking the proper collections at each point. It's just a lot messier. I didn't ask whether it's more convenient to use a chain of models, and I'm sure it's not. Just whether or not from a foundational point of view if that's a solid foundation which can replace large cardinals. | |
| May 14, 2013 at 21:34 | history | answered | Zhen Lin | CC BY-SA 3.0 |