On the strong equality between supercompactness and strong compactness

Abstract:

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if $V \models$ ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension $V[{G}] \models$ ZFC + GCH in which, (a) (preservation) for $\kappa \le \lambda$ regular, if $V \models {}$ “$\kappa$ is $\lambda$ supercompact”, then $V[G] \models {}$ “$\kappa$ is $\lambda$ supercompact” and so that, (b) (equivalence) for $\kappa \le \lambda$ regular, $V[{G}] \models {}$ “$\kappa$ is $\lambda$ strongly compact” iff $V[{G}] \models {}$ “$\kappa$ is $\lambda$ supercompact”, except possibly if $\kappa$ is a measurable limit of cardinals which are $\lambda$ supercompact.