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If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that the theory of a countable structure $\mathcal{M}$ is $\aleph_0$ categorical if and only if the action of $\text{Aut}\mathcal{M}$ on $M$ is oligomorphic.

So if we start with an $\aleph_0$ categorical theory, the automorphism group of the countable model gives an oligomorphic action. Conversely, if a group $G$ acts oligomorphically on a set $X$, then the theory of $X$ with an $n$-ary predicate for each orbit of $X^n$ is $\aleph_0$-categorical.

Is there any structure of a category on groups with an action and on $\aleph_0$-categorical theories that can make this correspondence functorial? (For example, two bi-interpretable theories should give rise to isomorphic group actions, and the reduct of a theory should give a richer group)

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  • $\begingroup$ I have never thought of interpretations as morphisms. Interesting idea. But are you looking for functoriality, or some sort of category equivalence? At a glance, functoriality seems rather obvious between the category of omega categorical theories and the category of oligomorphic, faithful actions on a fixed countably infinite set, and it looks like it's quite likely to be a category equivalence, though I haven't the time to check it now. $\endgroup$ – tomasz Jan 16 '14 at 12:34
  • $\begingroup$ I am interested in categorical equivalence and more generally in reference for this perspective (looking at the category of theories with interpretations) $\endgroup$ – user115940 Jan 17 '14 at 22:20
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Yes. See Section 1 of the paper Quasi finitely axiomatizable totally categorical theories by Ahlbrandt and Ziegler.

Here $\text{Aut}$ is shown to be a functor from the category of $\aleph_0$-categorical theories in countable languages, with interpretations as arrows, to the category of topological groups, with continuous homomorphisms as arrows. Instead taking the category of topological groups as the target, one could work with a category of oligomorphic group actions - but then you have to be careful about the definition of morphism: A morphism from $G\curvearrowright X$ to $H \curvearrowright Y$ would have to consist of a set $U\subseteq X^n$ which is closed under the induced action of $G$ on $X^n$, a surjective map $f\colon U\twoheadrightarrow Y$, and a group homomorphism $\varphi\colon G\to H$, such that for all $g\in G$ and all $\overline{x}\in U$, $f(g(\overline{x})) = \varphi(g)(f(\overline{x}))$.

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