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Let $\sigma$ and $\tau$ be first-order languages and let $S$ resp. $T$ be theories (sets of sentences) for $\sigma$ resp. for $\tau$ ($S$ and $T$ possibly empty). Let $\mathcal C$ resp. $\mathcal D$ be full subcategories of the categories of models for $S$ resp. $T$, that is, objects are models and maps are elementary embeddings. Let $ F: \mathcal C \to \mathcal D$ be a functor.

Question: What are necessary and sufficient conditions for the image $F(Ob(\mathcal C))$ to be axiomatizable within $\mathcal D$ with respect to $\tau$? In other words, when is there a theory $\Sigma$ for $\tau$ such that $F(Ob(\mathcal C)) = \{O \in Ob(\mathcal D) \mid O \models \Sigma \}$.

For instance, the class of additive groups of rings is not axiomatizable, see this post. Equivalently, the image of the forgetful functor from rings to abelian groups cannot be axiomatized within the language of groups. The same is true for multiplicative monoids of rings.

It seems that if the functor does not preserve enough information of the symbols in $\sigma$, it is not possible to axiomatize its image. Can this be made more concrete?

Thank you in advance for any help!

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    $\begingroup$ 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. $\endgroup$
    – Alex Kruckman
    Commented Aug 26, 2021 at 11:59
  • $\begingroup$ 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. $\endgroup$
    – Daniel W.
    Commented Aug 26, 2021 at 12:41
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    $\begingroup$ 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 $L_{\omega_1,\omega}$. $\endgroup$
    – Alex Kruckman
    Commented Aug 26, 2021 at 14:10
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    $\begingroup$ 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? $\endgroup$
    – Alex Kruckman
    Commented Aug 26, 2021 at 14:12
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    $\begingroup$ 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 $M$. $\endgroup$
    – Alex Kruckman
    Commented Aug 30, 2021 at 12:02

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