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

TL;DR

This paper demonstrates that under certain large cardinal hypotheses, the least weakly compact cardinal can possess properties like unfoldability, weak measurability, and near supercompactness, and shows how entire classes of such cardinals can be aligned.

Contribution

It establishes the possibility of the least weakly compact cardinal having multiple strong properties simultaneously and aligns entire classes of large cardinals under these properties.

Findings

Least weakly compact can be unfoldable, weakly measurable, nearly supercompact.

Entire class of weakly compact cardinals can coincide with other large cardinal classes.

Answers open questions and extends the identity-crises phenomenon to these cardinals.

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.