 Supercompact cardinal

A cardinal $\kappa$ is supercompact if for every cardinal $\lambda \ge \kappa$ there is an elementary embedding $j:V\to M$ into a transitive class with critical point of an elementary embedding $\kappa$, $j(\kappa )>\lambda$, and $M^{\lambda }\subseteq M$. This closure supplies every sufficiently small rank segment needed to transfer existential witnesses into $M$.