Timeline for Can passing to a larger Grothendieck universe ever lead to category-theoretic complications?
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| when toggle format | what | by | license | comment | |
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| yesterday | history | became hot network question | |||
| yesterday | comment | added | Timothy Chow | @JoelDavidHamkins Not that it's important, but "one-kilogram cylinder" would have been a slightly better metaphor than "one-meter platinum bar" since the latter was abandoned as a standard in 1960. But even the cylinder lost its significance in 2019. | |
| yesterday | answer | added | James E Hanson | timeline score: 15 | |
| yesterday | comment | added | Joel David Hamkins | No, definitely not. Take the first two with a given first order theory. It is a second order property of the second one that there is such a smaller model with the same first order theory, but this fails in the first one. | |
| yesterday | comment | added | Joe Lamond | @JoelDavidHamkins: Thanks, that seems useful. I have a question which maybe is answered in your paper, but I couldn't quite find it. Work in Tarski-Grothendieck set theory. If two Grothendieck universes, have the same first-order theory, does it follow they have the same second-order theory? | |
| yesterday | comment | added | Joel David Hamkins | This paper is all about the situation described in the beginning of the question: [2009.07164] Categorical large cardinals and the tension between categoricity and set-theoretic reflection arxiv.org/abs/2009.07164. | |
| yesterday | comment | added | Simon Henry | Yes but your example does not really make sense: "Having arbitrary coproduct" (or preserving them) is obviously a property that will change if you change universes. When working in a given universe, arbitrary coproduct means U-small coproduct, so if you change U, you are not talking about the same property anymore. In fact a small category can never have all coproducts (unless it is a poset), so moving to a larger universe generally makes sure that you will not have all coproduct anymore if if your category was originally cocomplete. | |
| 2 days ago | comment | added | Joe Lamond | @SimonHenry: In the examples of changing universe I am familiar with, this is in some sense just a technical device employed to make our categories small. But my question is whether passing to a larger universe could cause some other "change" which is undesirable or unwanted. I give an example above in my comment to Zhen Lin. | |
| 2 days ago | comment | added | Simon Henry | I'm not sure I understand the question: of courses passing to a larger universe changes things. If it didn't we would have no reasons to do it in the first place. What you call "problems" are features. The interesting questions are more about how results about the larger universe say something about the original one. | |
| 2 days ago | comment | added | Zhen Lin | Like I said, that is why you should keep track of your universe parameters. As for intuition, in my mind, the basic technique is to use an “explicit” construction (usually something predicative in flavour if not actually predicative) that is manifestly universe-independent and then prove it has a universal property in every universe. In my (admittedly truncated) experience it is rare to find existence theorems that are so inexplicit that universe-independence is not obvious; the Joyal model structure was what bothered me most when I wrote those papers. | |
| 2 days ago | comment | added | Joe Lamond | @ZhenLin: My concern is that if in the middle of establishing some theorem about category theory, one passes to a larger universe, then usually one tacitly assumes that things which "were" true "stay" true. This is perhaps a naive viewpoint, but I could certainly imagine someone with less experience having it. I suppose what I'm really looking for is a deeper explanation/intuition about which statements "should" be universe independent, and which are not. | |
| 2 days ago | comment | added | Zhen Lin | I don’t understand your concern. Or rather, the point of the exercise is precisely to address those concerns where we believe they should not be. To me it is obvious that such a theorem is universe-dependent and I would not even try to establish universe-independence. | |
| 2 days ago | comment | added | Joe Lamond | @ZhenLin: What I am a little worried about is something like this. Presumably the linked theorem is a theorem of Morse-Kelley, and so in particular it is internally true in the power set of any Grothendieck universe. Now, what happens if $\mathscr U$ is a universe containing no measurable cardinals, and $\mathscr U'$ does contain measurable cardinals? | |
| 2 days ago | comment | added | Zhen Lin | They are certainly universe-dependent prima facie. The whole point is to show that they are not universe dependent despite initial appearances. | |
| 2 days ago | comment | added | Joe Lamond | @MartinBrandenburg: It's a very well-written and useful paper, but as far as I can tell, it mainly focuses on questions which are not universe dependent. As mentioned in the introduction to Zhen Lin's paper, there are questions which are universe dependent, and this is what I want to focus on here. | |
| 2 days ago | comment | added | Martin Brandenburg | Isn't Zhen Lin's paper (the one you mention) the precise answer to your question? | |
| 2 days ago | comment | added | Zhen Lin | In principle, there could be a problem. But I think usually there is no problem if you are careful enough to keep track of the universe parameters. | |
| 2 days ago | comment | added | Ivan Di Liberti | Related: mathoverflow.net/questions/365947/… | |
| 2 days ago | history | asked | Joe Lamond | CC BY-SA 4.0 |