The Category Dichotomy for Ideals

Abstract

We provide a counterexample to the Category Dichotomy in the framework of $\textsf{ZFC}$. That is, we prove the existence of an ideal on $\omega$ that is not Kat\v{e}tov below $\mathsf{nwd}$ and does not have restrictions above $\mathcal{ED}$. We also prove that in the Laver model every tall $\mathsf{P}$-ideal is Kat\v{e}tov-Blass above $\mathcal{ED}_{\mathsf{fin}}$ and that it is consistent that every $\mathsf{Q}^{+}$ ideal is meager.