Model theory working seminar Ical Atom

We shall rewrite in the simplicial language the standard definitions of a complete first order theory, a model of it, and various characterisations of stability of a complete first order theory. In our reformulations the simplicial language replaces the standard definitions based on syntax, making them formally unnecessary. However, in the lectures that I shall explain these definitions both in standard language, and in the simplicial, diagram-chasing language. We shall assume only basic familiarity with category theory and model theory.

In this approach we view a complete first-order theory as a symmetric simplicial object in the category of profinite sets and open continuous maps, defined by the functor sending a finite set of variables into the Stone space of complete types in those variables. A model of a complete first-order theory is then a morphism from a representable simplicial set satisfying certain lifting properties reminiscent of, but weaker then, those in the definition of a fibration. The class of simplicial profinite sets corresponding to complete first order theories is characterised by the same lifting properties required of the map from the simplicial covering space (decalage) forgetting the extra degeneracy.

In a concise manner our simplicial reformulations are presented in the notes

The conceptual completeness theorem answers a fundamental question: given the category of models of a theory, how much of the theory’s syntax can be recovered? After a gentle introduction to categorical logic and classifying categories, I will explain Makkai’s remarkable result showing that the semantic category, together with ultraproduct-preserving functors, determines the theory up to definitional equivalence.

I will give an overview of some recent and some older results on rational dynamics and on rational difference equations. The main model theoretic tool here is the theory ACFA, the model companion of the theory of difference fields.

The talk focuses on the general behavior of discrete weakly o-minimal structures, and their definable sets and functions. Comparisons are drawn with the dense weakly o-minimal case and with the discrete o-minimal case. We develop tools to describe definable maps in these structures, and ultimately state a prove a monotonicity theorem. We also state, using the results given so far, how we hope to prove that no definable groups exist in the discrete, weakly o-minimal setting.