This communication deals with positive model theory, a non first order model theoretic setting which preserves compactness at the cost of giving up negation. Positive model theory deals transparently with hyperimaginaries, and accommodates various analytic structures which defy direct first order treatment. We describe the development of simplicity theory in this setting, and an application to the lovely pairs of models of simple theories without the weak non finite cover property.

We prove that for every simple theory $T$ (or even simple thick compact abstract theory) there is a (unique) compact abstract theory $T^fP$ whose saturated models are the lovely pairs of $T$. Independence-theoretic results that were proved in [Ben Yaacov, Pillay, Vassiliev - Lovely pairs of models] when $T^fP$ is a first order theory are proved for the general case: in particular $T^fP$ is simple and we characterise independence.

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.