The Category Dichotomy for Ideals

Abstract

We provide a counterexample to the Category Dichotomy in the framework of $\text{ZFC}$. That is, we prove the existence of an ideal on $\ddot{I}‰$ that is not Katětov below $\mathsf{nwd}$ and does not have restrictions above $\mathcal{E}\mathcal{D}$. We also prove that in the Laver model every tall $\mathsf{P}$-ideal is Katětov-Blass above ${\mathcal{E}\mathcal{D}}_{\mathsf{fin}}$ and that it is consistent that every ${\mathsf{Q}}^{+}$ ideal is meager.