Interpolative fusions II: Preservation results

Abstract

We study interpolative fusion, a method of combining theories $T_1$ and $T_2$ in distinct languages in a "generic" way over a common reduct $T_\cap$, to obtain a theory $T_\cup^*$. When each $T_i$ is model-complete, $T_\cup^*$ is the model companion of the union $T_1\cup T_2$. Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of $T_1$ and $T_2$ are inherited by $T_\cup^*$. We first prove preservation results for quantifier elimination, model-completeness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of $T_\cap$, the property $\mathrm{NSOP}_1$ is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in $T_1$ and $T_2$. This generalizes many previous results; for example, simplicity of $\mathrm{ACFA}$ and the random $n$-hypergraph are both non-obvious corollaries. We also address preservation of stability, $\mathrm{NIP}$, and $\aleph_0$-categoricity, and we describe examples which witness that these results are sharp.