Hypercompact

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Hypercompactness is a large cardinal property that is a strengthening of supercompactness. A cardinal $\kappa$ is $\alpha$-hypercompact if and only if for every ordinal $\beta < \alpha$ and for every cardinal $\lambda\geq\kappa$, there exists a cardinal $\lambda'\geq\lambda$ and an elementary embedding $j:V\to M$ generated by a normal fine ultrafilter on $P_\kappa\lambda$ such that $\kappa$ is $\beta$-hypercompact in $M$. $\kappa$ is hypercompact if and only if it is $\beta$-hypercompact for every ordinal $\beta$.

Every cardinal is 0-hypercompact, and 1-hypercompactness is equivalent to supercompactness.

Excessive hypercompactness

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