The higher Cichoń diagram

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