The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-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 compact cardinals, 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.

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