This is the title of a talk given at the 2015 North American meeting of the ASL (University of Illinois at Urbana-Champaign) on March 25, 2015. The talk presents the corresponding paper. Here are the slides.
Independence (or forking) is one of the central notion of modern classification theory. In first-order model theory, it was introduced by Shelah and is one of the main device of his book. One can ask whether there is such a notion outside of elementary classes. We will focus on abstract elementary classes (AECs), a very general axiomatic framework introduced by Shelah in 1985. In Shelah's book on AECs, the central concept is that of a good frame (a local independence notion for types of singletons) and conditions are given for their existence. However, the question of when there is a global independence notion (for types of all lengths) is still left open.
We show how to construct such a notion assuming categoricity (in a high-enough cardinal), amalgamation, tameness (a locality condition for types introduced by Grossberg and VanDieren), type-shortness (a relative of tameness introduced by Boney), and an extra condition called dense type locality. We also introduce a definition of superstability that can replace categoricity in our construction.
Sebastien Vasey, Independence in abstract elementary classes, In preparation. Preprint: pdf arXiv.
Sebastien Vasey, Forking and superstability in tame AECs, Submitted. Preprint: pdf arXiv.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vol. 18, College Publications, 2009.