The Higher Cicho\'n Diagram

Abstract

For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799) - the ideal of nowhere stationary subsets of a (naturally defined) stationary set $S_{\rm pr}^\kappa \subseteq \kappa$. In particular, we analyse the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable. While some results from the classical case ($\kappa=\omega$) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null)$\le$non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant. We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with $\mathord<\kappa$-support (and a strong "Knaster" version of $\kappa^+$-cc) and one iteration with ${\le}\kappa$-support (and a version of $\kappa$-properness).