Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions

We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal $\kappa $ has its strong compactness indestructible under $\kappa $-directed closed forcing. This is in analogy to and generalizes the author’s result in Arch. Math. Logic 46 (2007), but without the restriction that no cardinal is supercompact up to an inaccessible cardinal.