Forking in the category of existentially closed structures (Q2758394)

In the last years, the theory of forking was developed in the category of models of a complete first-order theory with elementary embeddings as morphisms, under the assumption of simplicity of the theory. The author develops the theory of forking in the category of existentially closed models of a universal theory with embeddings as morphisms, assuming suitably defined ``simplicity''. Earlier, S. Shelah and E. Hrushovski did it in some form under the additional assumption that the theory has the amalgamation property and joint embedding property. In general, the class of existentially closed models of a universal theory is not elementary; so in the class one cannot freely play with compactness as in the classical theory. For the class, the existential types of tuples realized in existentially closed models play the role of complete types. Certain ``big'' existentially closed models (roughly speaking, the models saturated and homogeneous for existential types) play the role of saturated models. Such existentially closed models are called e-universal domains. In the paper under review the theory of simplicity is developed in the context of a single e-universal domain (or, equivalently, for a subcategory of existentially closed models with the same universal theory). Here one of the main technical observations is that the Erdős-Rado theorem enables us to obtain existentially indiscernible sequences. As an example, it is shown that if \(T\) is a complete stable theory with quantifier elimination, and \(T_\sigma\) is the theory of expansions of models of \(T\) by automorphisms then the category of existentially closed models of \(T_\sigma\) (equivalently, of the universal part of \(T_\sigma\)) is simple.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00010].