Tue, 15 March 2022, 1 pm MST
We first revisit notions of interpretability and internality in a category-theoretical language (for first order theories), reframing work of Hrushovski and Kamensky in a formalism derived from Makkai's early work. We then describe the issue of recovering the bi-intepretability class of a theory in terms of the automorphism group of a saturated model, and the role of the "Small Index Property" (SIP) - a way of recovering the topology of a group action from purely algebraic information.
We then turn to abstract elementary classes, and discuss the same notions, in the opposite order: first, two situations where a Small Index Property holds (joint work with Ghadernezhad), and then some applications to the problem of interpretation and reconstruction, adapted to abstract elementary classes.