Higher-dimensional amalgamation properties played a key role in settling several questions in classification theory. It turns out that these properties, suitably formulated, are non-trivial even for totally categorical first order theories. The main goal of this project was to understand and characterize the failure of higher-dimensional amalgamation properties in stable theories. We show that the failure of n-dimensional amalgamation is detected by a suitable homology group; this group must be abelian profinite and is isomorphic to a certain automorphism group. Along the way, we establish that the failure of n dimensional amalgamation is witnessed by certain canonical objects, with the higher category-theoretic flavor, that are definable in the models of the theory.

Joint work with John Goodrick and Byunghan Kim. Speaker(s): Alexei Kolesnikov (Towson University)