Forking in the category of existentially closed structures

If T is a (first order) universal theory then we can consider the category of existentially closed models of T with embeddings as morphisms. This class of structures is in general not the class of models of a first order theory T ′. (It will be, just if T has a model companion T ′.) The “model theory” of such a “non-elementary class” was a substantial line of research in the 1970’s, mainly among the group of model-theorists around Abraham Robinson. In [10], Shelah developed some aspects of stability theory for such a category, concentrating on counting (existential) types. He asked implicitly whether the theory of forking can be developed in such a context. We give a positive solution in this paper, developing the theory of forking for the category of existentially closed models of an arbitrary universal theory T , assuming a suitable notion of “simplicity”. Under the additional assumption that T has the amalgamation property (AP) and joint embedding property (JEP), Shelah did develop forking in some form. Hrushovski ([4]) rediscovered this latter class of theories (universal T with AP and JEP) calling them Robinson theories, and pointing out that all model-theoretic methods should apply to the category of e.c. models of such T . For Robinson theories, quantifier-free types are the main object of study, and as quantifier-free formulas are closed ∗Supported by NSF grant DMS-96-96268