# Hypercompact

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Hypercompactness is a large cardinal property that is a strengthening of supercompactness. It was defined by Apter in , but with a mistake. Perlmutter called that form excessive hypercompactness.

Definition:

• 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.

If the cardinal $δ$ is Woodin for supercompactness (i.e. Vopěnka), then in the model $V_δ$, there is a proper class of hypercompact cardinals.

The relationship between extendible, hypercompact and enhanced supercompact cardinals is not known.

## Excessive hypercompactness

Definition:

• A cardinal $κ$ is excessively $0$-hypercompact iff $κ$ is supercompact.
• For $α > 0$, a cardinal $κ$ is excessively $α$-hypercompact iff for any cardinal $δ ≥ κ$, there is an elementary embedding $j : V → M$ witnessing the $δ$-supercompactness of $κ$ (i.e. $crit(j) = κ$, $j(κ) > δ$ and $M^δ ⊆ M$) generated by a supercompact ultrafilter over $P_κ (δ)$ such that $M \models$ “$κ$ is excessively $β$-hypercompact for every $β < α$”.
• A cardinal κ is excessively hypercompact iff $κ$ is excessively $α$-hypercompact for every ordinal $α$.

The key difference between the definitions of hypercompact and excessively hypercompact is that in the definition of hypercompact, the embedding $j$ need not be witnessed by a normal fine measure on $P_κλ$, but can be witnessed instead by a larger supercompactness measure. (Besides, Perlmutter changed the handling of limit stages.)

There are no excessively hypercompact cardinals. In particular, there is no cardinal $κ$ such that $κ$ is excessively $(2^κ)^+$-hypercompact.

On the other hand, the $β$-hypercompactness and excessive $β$-hypercompactness hierarchies align below $κ^+$ (this alignment is off by one because of the handling of limit stages):

For a cardinal $κ$ and an ordinal $β ≤ κ^+$, if $κ$ is $β$-hypercompact, then for every ordinal $α < β$ and for every cardinal $λ ≥ κ$, there is an elementary embedding $j : V → M$ generated by a normal fine measure on $P_κ λ$ such that $κ$ is $α$-hypercompact in $M$.

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