nLab supercompact cardinal

Idea

Supercompact cardinals are among the large cardinals.

Definition

For $S$ a set and $\kappa$ a cardinal, let $P_\kappa(S)$ be the set of subsets 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).

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 $\Sigma_2$-definable, then the $L$-local objects are a small-orthogonality class (so that $L$ is a localization with respect to some set of morphisms).

Theorem

Suppose there are arbitrarily large supercompact cardinals. Then any full subcategory of a locally presentable category which is closed under limits and $\Sigma_2$-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)

• Joan Bagaria, Carles Casacuberta, Adrian Mathias, Jiri Rosicky Definable orthogonality classes in accessible categories are small, arXiv