Timeline for Sheaf-theoretic approach to forcing
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Mar 5, 2021 at 13:26 | comment | added | Peter Scholze | Fun fact: On Hamkins' webpage the blackboard on the title photo is about boolean ultrapowers. | |
| Mar 5, 2021 at 12:27 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Mar 5, 2021 at 11:58 | comment | added | მამუკა ჯიბლაძე | @AsafKaragila take your time... :P | |
| Mar 5, 2021 at 11:57 | comment | added | Asaf Karagila♦ | @Jacob: ... yet! I'm writing one, it's just taking a while! :-) | |
| Mar 5, 2021 at 11:56 | comment | added | Jacob Lurie | The functor definitely doesn't preserve infinite coproducts. But the axioms of ZFC do not involve infinite disjunctions... | |
| Mar 5, 2021 at 11:54 | comment | added | მამუკა ჯიბლაძე | @JacobLurie Oh I see, thank you! So only infinite coproducts remain. Well, I think I can be persuaded that they do not create any problems... | |
| Mar 5, 2021 at 11:45 | comment | added | Jacob Lurie | It is an isomorphism. Abstract proof: since $i_* i^*$ is a left exact functor, the map $i_* i^*( \mathscr{F} ) \coprod i_* i^*(\mathscr{G} ) \rightarrow i_* i^* (\mathscr{F} \coprod \mathscr{G} )$ is automatically a monomorphism. The epimorphism statement is the one you have to worry about (and is the content of that Lemma in Moerdijk-MacLane). Concretely: the operation $V \mapsto \overline{V}$ and $\overline{V} \mapsto U \cap \overline{V}$ define mutually inverse bijections from clopen subsets of $U$ to clopen subsets of $\overline{U}$, because extremally disconnected spaces are bizarre. | |
| Mar 5, 2021 at 11:40 | comment | added | მამუკა ჯიბლაძე | @JacobLurie But $1\sqcup1\to i_*i^*(1\sqcup1)$ is not an isomorphism, right? So, some of its stalks must also fail to be isomorphisms, as $\operatorname{Sh}(S)$ has enough points, being just the topos of sheaves on a space. | |
| Mar 5, 2021 at 11:37 | comment | added | Jacob Lurie | Open and closed subsets of $U$ are the same as open and closed subsets of the closure of $U$, since $S$ is extremally disconnected. Alternatively, you can compute the stalk at $s$ using only clopen neighborhoods. The functor preserves only finite coproducts, not arbitrary ones. | |
| Mar 5, 2021 at 11:32 | comment | added | მამუკა ჯიბლაძე | Accordingly, I don't think the endofunctor on Sets mentioned by @JacobLurie in that comment preserves coproducts. So then, might not there be problems with axioms involving disjunction? Also, what about infinite epimorphic families? These might become involved if there are some axiom schemata, for example. | |
| Mar 5, 2021 at 11:31 | comment | added | მამუკა ჯიბლაძე | @PeterScholze Thank you for the explanation! And, sorry - another doubt. That lemma says that the pushforward preserves finite epimorphic families. But I don't think it preserves coproducts, even finite ones. For example, if I am not mistaken, pushforward of $1\sqcup1$ is not $1\sqcup1$: the value of the latter on $U$ is the set of open and closed subsets of $U$, while for the former it is the set of open and closed subsets of the closure of $U$, which is the same only if $U$ is itself open and closed. | |
| Mar 5, 2021 at 11:31 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Mar 5, 2021 at 11:13 | comment | added | Asaf Karagila♦ | @ZhenLin: That's fine, I suppose. But it's not that important for you to spend more time on this than you have to. Thanks! | |
| Mar 5, 2021 at 11:00 | comment | added | Zhen Lin | I'm not comfortable with the forcing language / names approach, but I don't think so? Also there is the subtlety with boolean valued models I mentioned in the comments to my answer that I haven't figured out, so I'm not really sure how well this translates to material set theory. I can probably write a precise question if someone wants to try to answer... | |
| Mar 5, 2021 at 10:43 | comment | added | Asaf Karagila♦ | @ZhenLin: And so in that case, asking if the functor is itself an inverse image functor would be assigning a canonical name for every set in the generic extension, in a way that "makes sense" somehow. (Which I suspect is something that you can't definably do in ZFC, even with global choice, because it would somehow let you decode the generic filter in the ground model or something, and that should be a known thing.) | |
| Mar 5, 2021 at 10:35 | comment | added | Asaf Karagila♦ | @ZhenLin: I suppose that you're describing the maps $i_p$, for a condition $p$, which are interpreted as $p\Vdash\dot x\in\dot y$ and $p\Vdash\dot x=\dot y$, right? (So $i_p(\dot x)=\{\dot y\mid p\Vdash\dot y\in\dot x\}$ or something like that.) | |
| Mar 5, 2021 at 10:32 | comment | added | Zhen Lin | @AsafKaragila Not really... the functor corresponds to the inclusion of the ultrapower into the extension (of the ultrapower), which in material set theory obviously has pleasant properties such as preserving finite disjoint unions and quotients and finite products and so on. If the inclusion were actually an inverse image functor then it should be the case that every set in the extension is functorially isomorphic to a set in the ultrapower, which should imply that the extension is trivial. But I haven't thought about this properly. | |
| Mar 5, 2021 at 10:03 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Mar 5, 2021 at 9:59 | comment | added | Asaf Karagila♦ | @ZhenLin: Can you cast that statement about filtered colimit of inverse image functors into the world of material set theory? | |
| Mar 5, 2021 at 9:52 | comment | added | Peter Scholze | @მამუკაჯიბლაძე I think (as mentioned by Jacob Lurie in the comments below Mike Shulman's answer) Lemma 4 on page 517 of MacLane-Moerdijk is relevant to this question. In the notation of his comment, this ensures that this endofunctor of $\mathrm{Set}$ preserves surjections, which should lead to preservation of existential axioms. | |
| Mar 5, 2021 at 9:47 | comment | added | Zhen Lin | The relationship between $\textbf{Set}_s$ and $\textbf{Set}_s [G]$ is something I never thought about before, very interesting. The functor $\textbf{Set}_s \to \textbf{Set}_s [G]$ is a filtered colimit of inverse image functors so it should have pleasant properties, but I suppose it is not actually an inverse image functor in general. (I think it would have to be an equivalence if it were.) | |
| Mar 5, 2021 at 9:38 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Mar 5, 2021 at 9:27 | comment | added | მამუკა ჯიბლაძე | Could you tell more about the place where you push forward a model from double negation sheaves to ordinary sheaves on $S$? The only pushforward that seems to be available is the right adjoint to the restriction: this restriction hardly ever in this case has a left adjoint. Now the right adjoint preserves anything equational, but is not obliged to preserve any existential statements that might happen to be among axioms of the theory in question, so it might destroy models of some important theories, no? | |
| Mar 5, 2021 at 8:57 | history | answered | Peter Scholze | CC BY-SA 4.0 |