Timeline for What about the enough points requirement in Bekes "Theories of presheaf type"?
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13 events
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| Jun 20, 2021 at 14:25 | comment | added | Matthias Hutzler | About Example 3.3: (Sorry for coming back to it again, but I am glad we are discussing it!) The propositional theory of a surjection N --> I is still of the form you suggest, except for the sequents (f(n) = y and f(n) = y') |-- bottom, which can't destroy presheaf type. I think the issue is that Beke argues that a structure (for a "relational" language) containing finitely many elements besides the constants is finitely presentable, which doesn't need to be true if there are infinitely many relation symbols or infinitely many constants. | |
| Jun 20, 2021 at 13:50 | vote | accept | Matthias Hutzler | ||
| Jun 19, 2021 at 14:09 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 19, 2021 at 14:06 | comment | added | Simon Henry | @IvanDiLiberti : I didn't know, thanks ! Though I do not how if it can be used to strengthen my answer: I don't know if there is an easily definable class of theories (other than coherent theories) that has lfp classyfing topos. | |
| Jun 19, 2021 at 14:02 | comment | added | Simon Henry | I agree that there is a problem with 3.5 if R is uncountable. I guess there is a way to make it work using that each ring each a directed colimit of countable rings, but I don't think there is a general argument applying. At this point, maybe you need to ask Beke directly. Regarding 3.3, I shouldn't have insisted on 'quantifier-free'. The point is that a general propositional theory has axiom of the form $\phi \Rightarrow \psi$ with $\phi$ and $\psi$ geometric, and here it is justs $\phi$. | |
| Jun 19, 2021 at 11:33 | comment | added | Matthias Hutzler | Actually, what about Example 3.5 (Flat modules)? If we choose an uncountable ring R (e.g. R = C[X]), the signature of the theory is uncountable, and the flatness axioms (formulated as in (ii) in the example) use countable disjunctions, so the quotient theory is also not coherent. Is there still a quick way to see that it has enough Set-models? | |
| Jun 19, 2021 at 10:52 | comment | added | Matthias Hutzler | (Note about "negated positive sentences": Theorem 8.2.8 in Caramello's Book "Theories, Sites, Toposes" is more general and doesn't need an enough models assumption.) | |
| Jun 19, 2021 at 10:29 | comment | added | Matthias Hutzler | I guess then, all the fully concrete examples in the paper are covered by these two theorems; and if you want to read the general examples (such as adding "negated positive sentences") as actual corollaries of Theorem 1.1 (as I first did), you need to keep the enough models condition in mind. | |
| Jun 19, 2021 at 9:55 | comment | added | Matthias Hutzler | Nice! About Example 3.3: In my understanding, propositional theories can't use quantifiers at all, since a quantifier needs to quantify over one of the sorts of the theory. Am I mixing up terminology? (For example, the language of the theory of a surjection N --> I consists only of proposition symbols "f(n) = y" for all n and y, and the axioms use infinite disjunctions but no quantifiers, right?) | |
| Jun 18, 2021 at 23:08 | comment | added | Ivan Di Liberti | Notice that the recent result of Rogers generalizes Deligne's completeness to lfp topoi too. | |
| Jun 18, 2021 at 15:51 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 18, 2021 at 15:43 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Jun 18, 2021 at 15:35 | history | answered | Simon Henry | CC BY-SA 4.0 |