Tameness is a locality property of Galois types in AECs. Since its isolation by Grossberg and VanDieren 10 years ago, it has been used to prove new results (upward categoricity transfer, stability transfer) and replace set-theoretic hypotheses (existence of independence notions). In this talk, we will outline the basic definitions, summarize some key results, and discuss some open questions related to tameness.

Shelah’s Categoricity Conjecture is a central test question in the study of Abstract Elementary Classes (AECs) in model theory. Recently Boney has shown that under the assumption that sufficiently large strongly compact cardinals exist, the Shelah Categoricity Conjecture holds at successor cardinals. Lieberman and Rosicky have subsequently shown that AECs can be characterised in a very natural way in a category-theoretic setting, and with this perspective Boney’s result can actually be seen as a corollary of an old category-theoretic result of Makkai and Pare. Rosicky and I have now been able to improve upon this result of Makkai and Pare (and consequently Boney’s Theorem), obtaining it from α-strongly compact cardinals.