Remarks on relative categoricity
Anand Pillay
TL;DR
This paper investigates relative categoricity for a complete theory $T$ in a relational language with a distinguished unary predicate $P$, focusing on when isomorphisms between $P$-substructures $M^{P}$ lift to full isomorphisms between $M$ and $N$ and on the Gaifman property that every $T^{P}$-model is of the form $M^{P}$ for some $M\models T$. It recalls and develops the framework of relative $(\omega,\omega)$-categoricity, $1$-cardinality, and co-analyzability (and almost internality to $P$), establishing two main results: (i) if $T$ is relatively $(\omega,\omega)$-categorical then any $T^{P}$-model of size at most $\aleph_{1}$ is $M^{P}$ for some $M\models T$; (ii) if, in addition, every $M\models T$ lies in $acl(P(M)\cup F)$ for a finite $F$, then $T$ is relatively categorical and has the Gaifman property. The work further analyzes how stability/embeddedness notions and co-analyzability influence these properties, connects to stability/simplicity results, and discusses recent related advances by Usvyatsov and Shelah. Overall, it clarifies when relative categoricity implies the Gaifman property and highlights the influence of internality conditions and cardinality constraints on this transfer.
Abstract
We make some elementary observations about relative categoricity and the Gaifman property. T will be a complete theory in a countable language L with a distinguished unary predicate P. We will assume L is relational and T has quantifier elimination. For M a model of of T, M^P is the substructure of M with universe P(M), and T^P is the common L-theory of these M^P. T is said to be relatively categorical if for any models M_1, M_2 of T any isomorphism between M_1^P and M_2^P lifts to an isomorphism between M_1 and M_2. T has the Gaifman property (or P-existence) if every model of T^P is of the form M^P for a model M of T. It was conjectured that if T is relatively categorical then T has the Gaifman property. T is said to be relatively (omega, omega) categorical if relative categoricity holds when restricted to countable models of T. We observe that (i) if T is relatively (omega, omega) categorical then any model of T^P of cardinality at most aleph_1 is of the form M^P for M a model of T, and (ii) if in addition every model M of T is in the algebraic closure of P(M) together with a (finite) subset of M, then T is relatively categorical and has the Gaifman property.