Timeline for $\mathscr{U}$-categories and $\mathsf{Hom}$-functors
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20 events
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| Nov 24, 2018 at 13:50 | comment | added | Peter LeFanu Lumsdaine | @FredRohrer: I agree, OP is (very reasonably) motivated by foundations for ordinary mathematics more than by set theory per se, as are many other people. My point is that for that purpose, the differences between ZFC and Bourbaki are essentially negligble. Certainly the differences between (ZFC+global choice) and Bourbaki are entirely negligible, and the differences between ZFC and (ZFC+global choice) are generally inconsequential. The differences only show up when one is really looking at strictly set-theoretic issues. | |
| Nov 24, 2018 at 13:15 | comment | added | Fred Rohrer | Moreover, I think this is not about choosing foundations in order to solve a certain problem, but about choosing foundations such that whatever problem we will study we will not have to care (much) about them. (For example, the axiom UA provides probably far more universes than are necessary in any reasonable concrete problem.) | |
| Nov 24, 2018 at 13:11 | comment | added | Fred Rohrer | @Peter: Sure, ZFC is certainly far more studied by set theorists than Bourbaki's system. But I think the OP is not interested in set theory per se, but in finding rigorous foundations for mathematics (probably mostly outside of set theory). Of course one can say "Take ZFC plus global choice", but if one wishes to build this up from scratch, including a formal language, then I guess one will end up more or less with Bourbaki's system (up to the axiom of foundation) - because, as you say, there are not that many differences. | |
| Nov 24, 2018 at 12:21 | comment | added | Peter LeFanu Lumsdaine | On the other hand, the OP is I think greatly overstating the differences between Bourbaki and ZFC: there’s no issue whatsoever in “adapting” SGA to ZFC. Are worst, one can assume global choice in addition to ZFC, and then use that to uniformly interpret the τ/ε operator. But hardly anything really requires that extra assumption of global choice — the τ/ε operator is used mostly just as a notational convenience, and uses of it can be replaced straightforwardly by appeals to AC on a case-by-case basis (as my answer spells out for this example of hom-functors). | |
| Nov 24, 2018 at 12:21 | comment | added | Peter LeFanu Lumsdaine | @FredRohrer: Most maths that purports to be done over Bourbaki’s set theory is done at the sort of high level that doesn’t look at details of foundations carefully, and doesn’t really depend on the precise differences between them. For work that actually deals precisely with the foundations, there is (at least as what I’m familiar with) far more work in ZFC than in Bourbaki set theory; ZFC is the “industry” standard among modern set theorists. [cont’d] | |
| Nov 24, 2018 at 10:59 | comment | added | Jxt921 | @FredRohrer Truth be told, almost any approach outside of naive set theory and $\mathsf{ZFC}$ is unconventional as of now. Still, it can't be true that the theory of universes is exclusive to Bourbaki set theory, can it? | |
| Nov 24, 2018 at 10:50 | comment | added | Fred Rohrer | Is Bourbaki's Theory of Sets really "unconventional"? It is true that here on MO, some people have complained about it, but often their comments show that they have not really studied it. If you consider the vast amount of mathematics that was built on this foundations I think it is completely okay to use it. | |
| Nov 24, 2018 at 10:31 | comment | added | Jxt921 | Bourbaki set theory. | |
| Nov 24, 2018 at 10:31 | comment | added | Jxt921 | @FredRohrer Dear Fred, I'm trying to understand how to correctly use universes without using Bourbaki set theory as it is quite nonconventional. There are approaches to handle $\mathscr{U}$-smallness: to say that a $\mathscr{U}$-small set an element of $\mathscr{U}$ of to say it is merely equinumerous to an element of $\mathscr{U}$. Grothendieck used the second approach, and it apparently heavily relies on Bourbaki machinery such as tau operator. I'm trying to understand if we can use this approach in conventional foundations such a $\mathsf{ZFC}$ without specifically resorting to | |
| Nov 24, 2018 at 8:04 | history | edited | Fred Rohrer | CC BY-SA 4.0 |
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| Nov 23, 2018 at 23:23 | comment | added | Qfwfq | "...let $\mathscr{U}\text{-}\mathsf{Set}$ be a set of all sets which belong to $\mathscr{U}$". In ZFC, the set of all sets that belong to $\mathscr{U}$ is $\mathscr{U}$ itself, isn't it? Or you mean the category of all $\mathscr{U}$-small sets? In the latter case $\mathscr{U}\text{-}\mathsf{Set}$, though a $\mathscr{U}$-small category according to your definition, will not be a set in ZFC. | |
| Nov 23, 2018 at 23:19 | history | edited | Qfwfq | CC BY-SA 4.0 |
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| Nov 23, 2018 at 23:01 | answer | added | Peter LeFanu Lumsdaine | timeline score: 7 | |
| Nov 23, 2018 at 22:31 | comment | added | Fred Rohrer | @Harry: No, see SGA4.I.1.1.2. | |
| Nov 23, 2018 at 22:25 | comment | added | Harry Gindi | @FredRohrer Can't you get away with demanding this, though? | |
| Nov 23, 2018 at 22:19 | comment | added | Fred Rohrer | @Harry: That is not the definition used in SGA4. | |
| Nov 23, 2018 at 22:19 | comment | added | Fred Rohrer | Dear Vasilii, it seems from your recent bunch of questions that you are trying to "translate" the whole Grothendieck universe stuff from SGA to ZFC. May I ask you to explain why? | |
| Nov 23, 2018 at 22:11 | answer | added | Fred Rohrer | timeline score: 5 | |
| Nov 23, 2018 at 19:08 | answer | added | Rick Sternbach | timeline score: 4 | |
| Nov 23, 2018 at 18:27 | history | asked | Jxt921 | CC BY-SA 4.0 |