I'm fixing some type of structure $\Sigma$ (possibly multi-sorted, with functions symbols and relation symbols, though assuming it single sorted with only relation symbols wouldn't change anything). Let $A$ and $B$ be two $\Sigma$-structure. Is there a connection between the following to notion :


1. The locale $[A,B]\_\Sigma$ classyfing morphisms of $\Sigma$-structure form $A$ to $B$.
2. Morphism of $\Sigma$-structure from $A$ to an ultrapower of $B$.


Intuitively both are way to make formal the idea that "their ought to be a morphism from $A$ to $B$, up to potential (infinite) cardinality obstruction"


As a first concrete step, the question would be:


**Is it true that $[A,B]\_\Sigma \neq \varnothing$ if and only if there exists a morphism of $\Sigma$-structure from $A$ to an ultrapower of $B$ ?**


But Ideally (and if the above is indeed true) I would like something more concrete that explain how to go back and forth between a morphism to an ultrapower and some sort of witness that the locale is non-trivial (like maybe points of a compactification or something like this).


If you are a model theorist, I guess you can replace $[A,B]\_\Sigma \neq \varnothing$ by the existence of a forcing extention of the base set theory in which there is a map from $A$ to $B$ and that should give something that can be translated back to my question.


I'm giving an explicit example in order to fix the idea :


Let $\Sigma$ be single sorted with a single binary relation $R$, and take $A$ and $B$ two $\Sigma$ structure where $R$ is interpreted as the relation $\neq$. Then $[A,B]\_{\Sigma}$ is the classifying locale of injection from $A$ to $B$ wich is non trivial as soon as "$B$ is infinite or $|A| \leqslant |B| < \infty$ ". And this is also the condition under which (I'm assuming choice here) you will be able to get an injection from $A$ to some ultrapower of $B$.



The locale of morphisms vs a morphism to an ultrapower?
I believe there are situations where the relevant locale is trivial, but there are many homomorphisms post-ultrapower.


For example, take $\mathcal{B}=(\mathbb{N};<)$ and let $\mathcal{A}$ be a nontrivial ultrapower of $\mathcal{B}$. There are no genuine homomorphisms from $\mathcal{A}$ to $\mathcal{B}$, and moreover this remains true in any forcing extension (since no homomorphism of strict linear orders can "move" infinitely-far-apart points to within finite distance of each other); per the comments, I *think* this means that the relevant locale is trivial too, but I'm not very familiar with locales. On the other hand, we trivially have lots of homomorphisms from $\mathcal{A}$ into ultrapowers of $\mathcal{B}$.




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EDIT: Well after the fact, I've run into a result which while not directly related to this question may still be of similar interest, so I've decided to mention it here. John Bell's paper [*Isomorphism of structures in $S$-toposes*](https://www.jstor.org/stable/2273748). Bell shows that for structures $\mathcal{A},\mathcal{B}$ in the same language, the following are equivalent:


* $\mathcal{A}\cong\mathcal{B}$ in some forcing extension (that is, $\mathcal{A}\cong\_{\infty\omega}\mathcal{B}$ by earlier results).
* $\mathcal{A}$ and $\mathcal{B}$ "become isomorphic" in some topos over $\bf{Set}$ (which Bell calls "$S$").


I think this result adds some nuance to the observations above, although I'm a bit out of my element here.


