Thickness, and a categoric view of type-space functors

We dene the class of thick cats (compact abstract theories, which con- tains in particular semi-Hausdor, Hausdor and rst order cats), and prove that in this class simplicity behaves as in rst order theories. We consider well-known rst order no- tions, such as interpretability or stable dividing/reduct, and propose analogous notions that can be naturally expressed in terms of maps between type-space functors. We prove several desirable properties of the new notions and show the connection between them and their classical counterparts. We conclude with several scattered results concerning cats and simplicity. Introduction. In (Ben03) we dened cats (or compact abstract theo- ries), which may be viewed as a model-theoretic framework with compact- ness but without negation. It is more general than the rst order framework, and can accommodate, for example, various kinds of analytic structures that do not admit a rst order description. In fact, we showed the equivalence of several quite dieren t approaches to the denition of cats, each having its own merits. The rst and most concrete presentation of a cat is via a particular kind of universal domains which are homogeneous and compact in a language without negation. Types, dividing, etc., are dened more or less as usual inside a universal domain. In (Benb) we showed how simplicity can be de- veloped in this context from the assumption that non-dividing has the local character, even though this does not imply that non-dividing extensions always exist. In the rst section of the present paper we introduce the class of thick cats, i.e., cats where indiscernibility is type-denable. We show that the thickness assumption is very mild, and that with this assumption many