Timeline for What are projective locales / injective frames?
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20 events
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| Jul 2, 2021 at 22:34 | comment | added | Peter LeFanu Lumsdaine | @მამუკა: If I’m not mistaken, your description is close to right: Simon’s $B_\kappa$ is the algebra of $\lnot\lnot$-closed lower sets of the poset $P(\omega,\kappa)$ finite of partial functions $\omega \rightharpoonup \kappa$ (i.e. lower sets $X$ such that for any $p$, if $X$ is dense below $p$ then $p \in X$). I think his $I_\kappa$ is also the set of ideals in $P(\omega,\kappa)$, but I’d need to check that a bit more carefully to be sure and it’s late here! | |
| Jul 1, 2021 at 15:02 | comment | added | Simon Henry | Maybe this will help: The locale $I_\kappa$ is actually spatial, so it is just the topological space of injection $\omega \to \kappa$ with the topology induced by the product topology on $\kappa^\omega$. I'm not sure how to directly prove that $I_\kappa$ is spatial (except checking it by hand that the two description are equivalent), but if you were just to replace $I_\kappa$ everywhere by this topological space, that wouldn't change anything. | |
| Jul 1, 2021 at 14:52 | comment | added | მამუკა ჯიბლაძე | @SimonHenry Thanks a lot for so much work. Although I still don't see clearly the frame for $I_\kappa$, I agree that it can be constructed in the standard way from the theory that you described. | |
| Jul 1, 2021 at 12:50 | comment | added | Simon Henry | I've edited to try to fill all the gap and give a more self contained answer. But let me know if there are still some unclear things. | |
| Jul 1, 2021 at 10:35 | vote | accept | მამუკა ჯიბლაძე | ||
| Jul 1, 2021 at 9:50 | comment | added | მამუკა ჯიბლაძე | I now tried to look up the literature. Maybe the frame of lower sets of the poset of injective partial functions with finite domain from $\omega$ to $\kappa$ is meant? Not that I see it for that case either, but... | |
| Jul 1, 2021 at 9:39 | comment | added | მამუკა ჯიბლაძე | @PeterLeFanuLumsdaine Unfortunately I know even less about the set-theoretic part. My vague understanding is that with forcing techniques you may e.g. wipe out any intermediate cardinalities. But I have no idea how can one collapse anything from the minimal cumulative hierarchy (i.e. the class generated from $\varnothing$ by powersets and limits). And if I am not mistaken the answer requires to do it for arbitrarily large ones from there, no? | |
| Jul 1, 2021 at 8:54 | comment | added | Peter LeFanu Lumsdaine | @მამუკაჯიბლაძე: Like Simon I don’t know references for these in topos-theoretic language, but there’s lots on these algebras and their properties in the set-theoretic forcing literature — the terminology of “collapsing cardinals” comes from there, and these are the Boolean algebras known as “Cohen forcing for κ”. | |
| Jul 1, 2021 at 0:04 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 30, 2021 at 20:50 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 30, 2021 at 20:17 | comment | added | Simon Henry | $B_\kappa$ is the double negation sublocale of a non-trivial locale (the locale of injection $\omega \to \kappa$ which has plenty of points), so it is non-trivial (because the double negation sublocale is dense). This should be covered in classical reference on the topic. I'm trying to find one... | |
| Jun 30, 2021 at 19:54 | comment | added | მამუკა ჯიბლაძე | Sorry, I still don't see why either $B_\kappa$ or $T_\kappa$ is nontrivial, say, for $\kappa=\omega_1$. | |
| Jun 30, 2021 at 15:15 | comment | added | Simon Henry | Note: By "total number of section" in the previous comment, I mean locale section, so the cardinal of $\coprod_{u \in \mathcal{O}(X)} \mathcal{F}(U)$. | |
| Jun 30, 2021 at 12:48 | comment | added | Simon Henry | $T_\kappa$ exists by the general theory of classyfing toposes, but the construction of $B_\kappa$ sketched above is what shows that $T_\kappa$ is non-trivial. The simplest way to show that a locale $X$ can't collapse arbitrary large cardinal is to give bound for the total number of section of $p^*\kappa$ in terms of $\kappa$ and $X$ and show that if $\kappa$ is large enough (compared to $X$), then $p^* \kappa$ has much more section than $p^* \omega$. | |
| Jun 30, 2021 at 12:43 | comment | added | Simon Henry | I say a locale $X$ "collaps a cardinal $\kappa$ to $\omega$", if $p^* \kappa \simeq p^* \omega$ in the category of sheaves over $X$ where $p:X \to 1$ is the unique map. In particular, if $B_\kappa$ collaps $\kappa$ to $\omega$ and $X \to B_\kappa$ is any map then $X$ also collaps $\kappa$ to $\omega$. If you want, you can replace $B_\kappa$ by $T_\kappa$, the locale that classifies "collapse of $\kappa$ to $\omega$", i.e. a map from $X \to T_\kappa$ is exactly the same as an isomorphism $p^*\kappa \simeq p^* \omega$ in $Sh(X)$. The only difference is that $T_\kappa$ is not boolean. | |
| Jun 30, 2021 at 4:52 | comment | added | მამუკა ჯიბლაძე | I am ready to accept this answer, but could you please say a few words about collapsing cardinals? Seems it has to do with characterizing maps from any $X$ to $B_\kappa$, could you say what they are? | |
| Jun 29, 2021 at 20:52 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 29, 2021 at 20:39 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 29, 2021 at 20:02 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 29, 2021 at 19:56 | history | answered | Simon Henry | CC BY-SA 4.0 |