The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $${\theta}$$ θ -supercompact

Abstract

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $${\theta}$$?-supercompact, for any desired $${\theta}$$?. In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $${\theta_\kappa}$$??-supercompact cardinals $${\kappa}$$?, for nearly any desired function $${\kappa\mapsto\theta_\kappa}$$????. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.