The Category Dichotomy for Ideals

We provide a counterexample to the Category Dichotomy in the framework of $\mathsf{\text{ZFC}}$. That is, we prove the existence of an ideal on $\omega$ that is not Katětov below $\mathsf{n}\mathsf{w}\mathsf{d}$ 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 ${\mathcal{E}\mathcal{D}}_{\mathsf{f}\mathsf{i}\mathsf{n}}$ and that it is consistent that every ${\mathsf{Q}}^{+}$ ideal is meager.