Diamond, square, and level by level equivalence

Abstract

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, ♦δ holds for every regular uncountable cardinal δ, and below the least supercompact cardinal κ, □δ holds on a stationary subset of κ. There are no restrictions in our model on the structure of the class of supercompact cardinals.