Timeline for Examples of statements that are valid in every spatial topos
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 27, 2022 at 19:45 | comment | added | მამუკა ჯიბლაძე | Note also that $G$-sets have IC, and, accordingly, I believe, for a $G$-poset $P$ satisfying the internal Zorn condition (that is, the projection from the object of pairs $(\text{chain of $P$}, \text{its upper bound})$ to the object of chains of $P$ is epi), $\max(P)$ is as inhabited as $P$ (but may fail to have any global elements, that is, $G$-fixed points). | |
| Nov 27, 2022 at 19:41 | comment | added | მამუკა ჯიბლაძე | @TimCampion The topos of $G$-sets for a group $G$ has LEM but not AC, hence also no Zorn's lemma. | |
| Nov 27, 2022 at 15:51 | comment | added | Tim Campion | What's an example of a Grothendieck topos where Zorn's lemma doesn't hold? | |
| Nov 26, 2022 at 8:32 | comment | added | Gro-Tsen | Thanks! For completeness of MO, Zorn's lemma is stated as follows for a poset $P$: “If every part of $P$ that is linearly ordered has an upper bound (in $P$) then $P$ has a maximal element”, where a poset is defined by a lax order $≤$ that is reflexive, antisymmetric and transitive, “linearly ordered” means $∀x.∀y.(x≤y∨y≤x)$, and a “maximal” element is an $m$ such that $∀z.(m≤z⇒m=z)$. | |
| Nov 26, 2022 at 3:41 | history | answered | Ingo Blechschmidt | CC BY-SA 4.0 |