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Tuesday, February 27, 2018

**Abstract:** Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is "definable", i.e. really just the points in all models of some imaginary sort of our theory?

In the '80s, Michael Makkai provided the following answer to this question: a functor $\mathrm{Mod}(T) \to \mathrm{Set}$ is definable if and only if it preserves all ultraproducts and all "formal comparison maps" between them, called ultramorphisms (generalizing e.g. the diagonal embedding into an ultrapower). This is known as strong conceptual completeness.

Any general framework which reconstructs theories from their categories of models should be considerably simplified for $\aleph_0$-categorical theories. Indeed, we show:

If $T$ is $\aleph_0$-categorical, then $X : \mathrm{Mod}(T) \to \mathrm{Set}$ is definable, i.e. isomorphic to ($M \mapsto \psi(M)$) for some formula $\psi \in T$, if and only if $X$ preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when $T$ is $\aleph_0$-categorical. We show this definability criterion fails if we remove the $\aleph_0$-categoricity assumption, by constructing examples of theories and non-definable functors $\mathrm{Mod}(T) \to \mathrm{Set}$ which exhibit this.