On the Intermediate Models of Strongly Compact Prikry Forcing

Abstract

We analyze the intermediate models of the strongly compact Prikry forcing. We exhibit a simple combinatorial property which, for a given supercompact cardinal $\hat{I}º$, characterize the projections of all projections of the strongly compact Prikry forcing using $\hat{I}º$-complete fine measures. Considering level-by-level results, if $\hat{I}º$ is $2^{\hat{I}}»$-strongly compact, we characterize the forcings of size $\le \hat{I}»$ which are projections of that $\hat{I}»$-strongly compact Prikry forcing. Our characterization generalizes several known results, including those of Benhamou-Hayut-Gitik and folklore results regarding the class of $\hat{I}º$-distributive forcing notions which are embedded into the supercompact Prikry forcing. Fixing a $\hat{I}º$-complete fine measure $\mathcal{U}$ on $P_{\hat{I}}º(\hat{I}»)$, we also provide Rudin-Keisler like critiria for the existence projections from the strongly compact Prikry forcing with $\mathcal{U}$. Finally, we prove that among all projections of the $\hat{I}»$-strongly compact Prikry forcing, the class of forcings of cardinality $\hat{I}»$ are exactly those for which there is a projection map which depends only on the stem of the Prikry condition. We also give partial results regarding projections of arbitrary cardinality.