Relative stability, relative categoricity and internal covers

Abstract

Let $T$ be a countable complete theory with a distinguished unary predicate $P$, and let $T^{P}$ be the theory of the $P$-parts of models of $T$ with the induced structure. $T$ is said to be relatively categorical or categorical over $P$ if any isomorphism between the $P$-parts of two models of $T$ lifts to an isomorphism of the models in question. We study the special case of relative categoricity where $T$ is internal to $T^{P}$ (that is, every model $M$ of $T$ is in the definable closure of $P(M)$ together with additional parameters from $M$). We first give a structure theory for such $T$: after passing to $T^{eq}$ and naming a parameter, $T$ is the same thing as a "pure torsor cover" of $T^{P}$, namely simply adjoining to $T^{P}$ a new sort for a torsor $S$ for a $\emptyset$-definable group $G$ in $T^{P}$, with no additional structure. We discuss relative stability, or stability over $P$, and give a characterization of relative stability, superstability, and $\omega$-stability of $T$ in terms of $H$ having the stable chain condition, superstable chain condition, and $\omega$-stable chain condition, respectively.