Workshop on Accessible Categories and their Connections – set theory – model theory – homotopy theory, University of Leeds, July 2018.
We provide a blueprint for the study of interpretations of Abstract Elementary Classes: we first revisit interpretability and internality in a category-theoretical language (for first order theories). This reframes work of Hrushovski and Kamensky in a formalism derived from Makkai. We then describe the issue of recovering the biintepretability class of a theory in terms of the automorphism group of a saturated model, and the role of the “Small Index Property” (SIP). An SIP theorem for AECs with strong amalgamation properties we published in 2017 is now placed in the context of reconstruction: we propose a notions of interpretation between some specific kinds of AECs.
This is joint work with Zaniar Ghadernezhad.