+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Foundations +-- {: .hide} [[!include foundations - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea Supercompact cardinals are among the [[large cardinal]]s. ## Definition For $S$ a [[set]] and $\kappa$ a [[cardinal]], let $P_\kappa(S)$ be the set of [[subset]]s of $S$ of cardinality less than $\kappa$. For $\lambda$ an [[ordinal]] a cardinal $\kappa$ is called **$\lambda$-supercompact** if $P_\kappa(\lambda)$ admits a [[normal measure]]. It is **supercompact** if it is $\lambda$-supercompact for every $\lambda$. $\kappa$ being $\lambda$-supercompact is equivalent to there being an [[elementary embedding]] $j : V \to M$ such that $j(\alpha) = \alpha$ for all $\alpha \lt \kappa$ and $j(\kappa) \gt \lambda$, where $M$ is an [[inner model]] such that $\{f | f : \lambda \to M\} \subset M$, i.e. every $\lambda$-sequence of elements of $M$ is an element of $M$. ## Properties By invoking [[Vopěnka's principle]] one can make strong statements about the existence of [[reflective subcategories]]. The assumption of supercompact cardinals is much weaker, and accordingly they similarly imply existence of reflective subcategories only under some more additional assumptions. The following theorems are all from ([BCMR](#BagariaCasacubertaMathiasRosicky)). +-- {: .num_theorem} ###### Theorem Suppose there are arbitrarily large supercompact cardinals. Then if $L$ is a reflection on an accessible category $C$ and the class of $L$-equivalences is $\Sigma_2$-[[Levy hierarchy|definable]], then the $L$-local objects are a small-orthogonality class (so that $L$ is a localization with respect to some [[set]] of morphisms). =-- +-- {: .num_theorem} ###### Theorem Suppose there are arbitrarily large supercompact cardinals. Then any full subcategory of a locally presentable category which is closed under limits and $\Sigma_2$-[[Levy hierarchy|definable]] is reflective. =-- There is also a generalization to $\Sigma_n$-definability involving [[C(n)-extendible cardinals]]; see [[Vopenka's principle]]. ## References Supercompact cardinals are discussed for instance in * T. Jech _Set Theory_ The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, Heidelberg (2003) * A. Kanamori, _The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings_ Perspectives in Mathematical Logic. Springer-Verlag, Berlin, Heidelberg (1994) The relation to refelctive subcategories is discusssed in * Joan Bagaria, [[Carles Casacuberta]], Adrian Mathias, _Epireflections and supercompact cardinals_ Journal of Pure and Applied Algebra 213 (2009), 1208-1215 ([pdf](http://atlas.mat.ub.es/personals/casac/articles/bcm.pdf)) {#BagariaCasacubertaMathias} * Joan Bagaria, [[Carles Casacuberta]], Adrian Mathias, [[Jiri Rosicky]] _Definable orthogonality classes in accessible categories are small_, [arXiv](http://arxiv.org/abs/1101.2792) {#BagariaCasacubertaMathiasRosicky} [[!redirects supercompact cardinal]] [[!redirects supercompact cardinals]] [[!redirects supercompact cardinal number]] [[!redirects supercompact cardinal numbers]]