Beautiful pairs

Abstract: We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called \textit{beautiful pairs}, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with properties analogous to properties in Fra\"{i}ss\'e classes. After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically, $p$-adically and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When $T$ is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces and our main result specializes to their strict pro-definability. Most notably, we exhibit the strict pro-definability of a natural space of types associated to Huber's analytification. In this way, we also recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields, which corresponds to Berkovich's analytification.