Filter extension games with mini supercompactness measures

This is a talk at the Third Berkeley Conference on Inner Model Theory, University of California, Berkeley, June 28, 2025.
Slides

Abstract: Several classical smaller large cardinals $\kappa$, among them weakly compact, ineffable, and Ramsey cardinals, can be characterized by the existence of measure-like filters on $\kappa$-sized families of subsets of $\kappa$. It suffices to consider families that arise as $P(\kappa)^M$ for some $\kappa$-sized $\in$-model of a sufficiently large fragment of set theory. Isolating patterns in the properties of these filters has led to a better understanding of the classical large cardinals and the definition of new ones. Holy and Schlicht introduced the filter extension games of length $1$ up to $\kappa^+$ in which the first player plays an increasing sequence of these $\kappa$-sized models and the second player responds by choosing an increasing sequence of filters for them. They, and later Nielsen and Welch, used the existence of a winning strategy for one of the players in these games to characterize (or imply) classical large cardinals and define new ones. The existence of the strategy for the second player can be used to characterize (or imply) versions of generic measurability, including, as shown by Foreman, Magidor, and Zeman, the existence of precipitous ideals.

We generalize the filter extension games to the two-cardinal setting by considering corresponding filters on $\lambda$-sized families of subsets of $P_\kappa(\lambda)$ (arising as $P(P_\kappa(\lambda))^M$ for $\lambda$-sized $\in$-models). We show that the existence of a winning strategy for the second player in these games characterizes (or implies) several large cardinal notions in the neighborhood of a supercompact, including, nearly $\lambda$-supercompact, completely $\lambda$-ineffable and $\lambda$-$\Pi^1_n$-indescribable cardinals. We also connect the existence of winning strategies for the second player in these games with various notions of generic supercompactness, including the existence of precipitous ideals. This is joint work with Tom Benhamou.