Timeline for Is any choice axiom other than WISC inherited by Grothendieck topoi?
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10 events
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| Sep 20, 2022 at 18:25 | comment | added | Gro-Tsen | Somewhat related question about statements true in every spatial (also: localic, Grothendieck) topos. | |
| Sep 20, 2022 at 17:53 | history | edited | saolof | CC BY-SA 4.0 |
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| Sep 20, 2022 at 17:51 | comment | added | saolof | @JamesHanson I believe that is for Grothendieck topoi over set (rather than over an arbitrary topos with the same starting property). I clarified my question to describe what I mean by "inherited by". | |
| Sep 20, 2022 at 17:46 | history | edited | saolof | CC BY-SA 4.0 |
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| Sep 18, 2022 at 23:18 | comment | added | Simon Henry | @JamesHanson No it doesn't holds in any topos - in most sheaves topos (ex Sh([0,1]) ) the axiom of choice is nowhere true, that is $\neg AC$ holds. There are also boolean toposes that don't satisfies AC. Note that there are some technical difficulties here as AC involves quantification on objects, so it is not technically a proposition in the internal logic and so it doesn't make sense to take its negation - however one can make sense of this for Grothendieck toposes using the stack semantics and the fact that Grothendieck toposes are "autological" (arxiv.org/abs/1004.3802) | |
| Sep 18, 2022 at 16:55 | comment | added | James E Hanson | Does $\neg \neg \mathsf{AC}$ hold in every Grothendieck topos? | |
| Sep 18, 2022 at 16:55 | comment | added | James E Hanson | Don't you need to be a little careful how your phrase this question? Specifically, different formulations of classically equivalent choice principles may fail to be intuitionistically equivalent. The nLab article on Grothendieck topoi says that some constructive version of the axiom of multiple choice holds, for instance. | |
| Sep 18, 2022 at 16:21 | comment | added | Simon Henry | The special adjoint functor theorem - and its consequence like the existence of free model for infinitary algebraic theories - hold in all Grothendieck topos (assuming choice in the base - though assuming the special adjoint functor theorem in the base is probably enough). This result fails in ZF, but I don't know if it is a consequence of WISC or some other choice principle that holds in Grothendieck topos. | |
| Sep 18, 2022 at 7:32 | history | edited | saolof | CC BY-SA 4.0 |
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| Sep 8, 2022 at 15:56 | history | asked | saolof | CC BY-SA 4.0 |