*Hypercompactness* is a large cardinal property that is a strengthening of [supercompactness](Supercompact.md "Supercompact"). 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 [](Filter.md) 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 This article is a stub. Please help us to improve Cantor's Attic by adding information.