Timeline for Axiomatizability of image of functor
Current License: CC BY-SA 4.0
16 events
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| Sep 1, 2021 at 8:44 | comment | added | Daniel W. | Okay, I see what you mean. But it seems to me that, if $F$ gives a continuous function between spaces of $L$-structures, it "easier" to show that the isomorphism closure of the image of $F$ is Borel. What if I have the feeling that $imF$ is not axiomatizable in a special case? Are there necessary conditions for a set to be Borel that might be easier to disprove in this case? | |
| Aug 30, 2021 at 12:02 | comment | added | Alex Kruckman | One reference is Kechris's book Classical Descriptive Set Theory. "Reasonably definable" is vague on purpose, but it certainly includes situations where $F(M)$ has the same domain as $M$, but where the interpretations of the symbols In the language of $F(M)$ are defined by formulas in terms of the symbols in the language of $M$. | |
| Aug 30, 2021 at 9:44 | comment | added | Daniel W. | Is there also a general reference for the notions you used. For instance, what does it mean exactly that F is "reasonably definable"? | |
| Aug 30, 2021 at 9:16 | comment | added | Daniel W. | What is a reference for the Lopez-Escobar Theorem you are citing? I can't find any. | |
| Aug 26, 2021 at 14:23 | comment | added | Daniel W. | Thank you very much for this nice comments. I will have time to think about them next week and write back then. | |
| Aug 26, 2021 at 14:12 | comment | added | Alex Kruckman | If your functor is reasonably definable, then you'll get a continuous function between spaces of $L$-structures, and the isomorphism-closure of the image will be an analytic set in general. So if you're willing to work with infinitary logic, the axiomatizability question becomes a topological one: when is this analytic set actually Borel? | |
| Aug 26, 2021 at 14:10 | comment | added | Alex Kruckman | Ah! If you're interested in countable structures, then your question fits into a descriptive set theory context: for any language $L$, the set of $L$-structures with domain $\omega$ has a natural structure as a Polish space. Now the Lopez-Escobar Theorem says that a set of structures in this space is an isomorphism-closed Borel set if and only if it is axiomatizable by a sentence of the infinitary logic $L_{\omega_1,\omega}$. | |
| Aug 26, 2021 at 13:51 | comment | added | Daniel W. | Clearly, for all $H$ we have that $\mathcal L(H)$ is countable. Therefore the image of $\mathcal L$ is not an elementary class. But if one restricts $\mathcal D$ to countable structures, the image might be axiomatizable within this new category. | |
| Aug 26, 2021 at 13:48 | comment | added | Daniel W. | For a monoid $H$, the system of sets of lengths $\mathcal L(H)$ of $H$ is the set of all subsets $L$ of $\mathfrak P(\mathbb N_{\geq 2}) \cup \{\{0\},\{1\}\}$ that appear in the following way: For some fixed element $a \in H$, $L \neq \emptyset$ is the set of all non-negative integers that appear as the number of irreducible elements of a factorization of $a$ into irreducible elements. So for instance, if $H$ is a unique factorization monoid, then $\mathcal L(H) = \{\{0\},\{1\},\{2\},\ldots\}$. | |
| Aug 26, 2021 at 13:43 | comment | added | Daniel W. | Yes, I have applications in mind, where all objects in the image of $F$ are countable. For instance, if $\mathcal C$ is the category of commutative cancellative monoids with elementary embeddings and $\mathcal D$ is the category of all structures of the language including a 2-ary function symbol $+$, a 2-ary relation symbol $\subseteq$ and a constant symbol $\{k\}$ for each non-negative integer $k$, then one has a functor $\mathcal L$ as follows: | |
| Aug 26, 2021 at 12:48 | comment | added | Alex Kruckman | I guess I'm asking why you want to restrict to non-elementary classes. Do you have some examples in mind that you want to cover? | |
| Aug 26, 2021 at 12:41 | comment | added | Daniel W. | I also suspect that instead of the categories of ALL models using arbitrary full subcategories makes the problem very hard. A first step would be of course to look at it for the categories of all models. But on the other hand, at some point one wants to restrict to subcategories that are not elementary classes anymore. | |
| Aug 26, 2021 at 11:59 | comment | added | Alex Kruckman | Is there a reason why you restrict attention to full subcategories C and D? The arbitrary nature of C and D makes the situation much more complicated. | |
| Aug 26, 2021 at 9:50 | history | edited | Daniel W. | CC BY-SA 4.0 |
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| Aug 26, 2021 at 9:40 | history | edited | Daniel W. | CC BY-SA 4.0 |
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| Aug 26, 2021 at 9:18 | history | asked | Daniel W. | CC BY-SA 4.0 |