On the cofinality of the least $\lambda$-strongly compact cardinal

Abstract

In this paper, we characterize the possible cofinalities of the least $\lambda$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $\delta$, that carries a $\lambda$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $\delta$, that the least $\lambda$-strongly compact cardinal has cofinality $\delta$. On the other hand, provably the cofinality of the least $\lambda$-strongly compact cardinal always carries a $\lambda$-complete uniform ultrafilter.