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We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. We extend some notions from classification theory to positive logic and position the category of existentially closed exponential fields in the stability hierarchy as NSOP 1 but TP 2 .Kind 3, where Kind 1 is the usual setting of a complete first-order theory, and Kind 2, also known as Robinson theories, is like Kind 3 but where there is amalgamation over every subset of every model. (There is also Kind 4, which is now known as homogeneous abstract elementary classes.) More recently there has been interest in positive model theory, which is only slightly more general than Kind 3. This is the approach taken in this paper, and it means that we look at embeddings and existentially definable sets instead of elementary embeddings and all first-order definable sets. Following Pillay ( 2000) we call it the Category of existentially closed models rather than Kind 3. Model-theoretic backgroundWe give some background on the model theory of the category of existentially closed models of a (usually incomplete) inductive first-order theory. More details can be found in Hodges (1993), Pillay (2000 and Ben Yaacov and Poizat (2007). The only novelty in this section is the notion of a JEP-refinement of an inductive theory T (Definition 2.11), which is a useful syntactic counterpart to the choice of a monster model.

We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full firstorder model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

Quasiminimal pregeometry classes were introduces by Zilber [2005a] to isolate the model theoretical core of several interesting examples. He proves that a quasiminimal pregeometry class satisfying an additional axiom, called excellence, is categorical in all uncountable cardinalities. Recently Bays et al. [2014] showed that excellence follows from the rest of axioms. In this paper we present a direct proof of the categoricity result without using excellence.

The correspondence between definable connected groupoids in a theory T and internal generalised imaginary sorts of T , established by Hrushovski in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaboration on Hrushovki's original constructions are also included.