Timeline for answer to $\mathscr{U}$-categories and $\mathsf{Hom}$-functors by Fred Rohrer
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Nov 24, 2018 at 11:01 | comment | added | Fred Rohrer | In Proposition 1.14 of this paper by Zhen Lin Low something along this line is done, but when trying to provide rigorous foundations it seems questionable to change encodings (here: of maps) whenever you run into problems. | |
| Nov 24, 2018 at 10:57 | comment | added | Jxt921 | @FredRohrer And if I restrict my question to $\mathsf{ZFC}$ alone rather other hypothetical foundations? What about the suggestion of Harry Gindi above? Could it help to adopt this approach to $\mathsf{ZFC}$? | |
| Nov 24, 2018 at 10:45 | comment | added | Fred Rohrer | @Jxt921: I do not enough about foundations different from Bourbaki and ZFC to decide whether they can provide what you're after. Concerning "adopt the approach [...] belong to $U$", you may read SGA4.I.1.1.2 to see that then you run into problems once you consider categories of functors. | |
| Nov 24, 2018 at 10:36 | comment | added | Jxt921 | @FredRohrer Yeah, I have suspected something like that. If I can ask one more question, Fred. Does this mean that if we try to use universes in any other mainstream foundation other than Bourbaki set theory, we have to adopt the approach to universes where $\mathscr{U}$-categories are categories whose $\mathsf{Hom}$-sets are not merely isomorphic to an element of $\mathscr{U}$, but actually belong to $\mathscr{U}$? | |
| Nov 23, 2018 at 22:34 | comment | added | Harry Gindi | Haha, sorry for asking all of these stupid questions! | |
| Nov 23, 2018 at 22:32 | comment | added | Fred Rohrer | @Harry: Essentially yes, see SGA4.I.1.3. | |
| Nov 23, 2018 at 22:31 | comment | added | Harry Gindi | When Grothendieck demands that each Hom set is isomorphic to one in $U$, maybe we should interpret this as meaning that for each pair $X,Y$, there is an actual distinguished isomorphism as part of the data? | |
| Nov 23, 2018 at 22:11 | history | answered | Fred Rohrer | CC BY-SA 4.0 |