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  "body": "\n## Definition\n\nLet $\\kappa$ be a [[cardinal|0EBACQFAZW]], and let $\\lambda \\ge \\kappa$ be an [[ordinal|0YSCSX5YZC]].\nThe cardinal $\\kappa$ is *$\\lambda$-supercompact* if there is an [[elementary embedding|CGH8C8EA9N]] $j \\colon V \\to M$, with $M$ a transitive class, such that:\n\n1. The critical point of $j$ is $\\kappa$, that is, $j(\\alpha) = \\alpha$ for $\\alpha \u003c \\kappa$ and $j(\\kappa) \u003e \\kappa$.\n2. $j(\\kappa) \u003e \\lambda$.\n3. $M$ is closed under $\\lambda$-sequences: every function from $\\lambda$ to $M$ is an element of $M$.\n\nThe cardinal $\\kappa$ is *supercompact* if it is $\\lambda$-supercompact for every ordinal $\\lambda \\ge \\kappa$.\n\n## Remarks\n\nA cardinal is $\\kappa$-supercompact if and only if it is measurable: the ultrapower embedding by a $\\kappa$-complete nonprincipal ultrafilter on $\\kappa$ satisfies the three clauses with $\\lambda = \\kappa$.\nSo the least measurable cardinal is $\\kappa$-supercompact, but it is not supercompact, since a supercompact cardinal has measurable cardinals below it.\nAn equivalent form uses ultrafilters on the set $P_\\kappa(\\lambda)$ of subsets of $\\lambda$ of size less than $\\kappa$.\nThe cardinal $\\kappa$ is $\\lambda$-supercompact if and only if there is a fine normal $\\kappa$-complete ultrafilter on $P_\\kappa(\\lambda)$.\nEvery supercompact cardinal is a Woodin cardinal.\n",
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