nLab
supercompact cardinal

Contents

Idea

Supercompact cardinals are among the large cardinals.

Definition

For SS a set and κ\kappa a cardinal, let P κ(S)P_\kappa(S) be the set of subsets of SS of cardinality less than κ\kappa.

For λ\lambda an ordinal a cardinal κ\kappa is called λ\lambda-supercompact if P κ(λ)P_\kappa(\lambda) admits a normal measure?. It is supercompact if it is λ\lambda-supercompact for every λ\lambda.

κ\kappa being λ\lambda-supercompact is equivalent to there being an elementary embedding j:VMj : V \to M such that j(α)=αj(\alpha) = \alpha for all α<κ\alpha \lt \kappa and j(κ)>λj(\kappa) \gt \lambda, where MM is an inner model? such that {ff:λM}M\{f | f : \lambda \to M\} \subset M, i.e. every λ\lambda-sequence of elements of MM is an element of MM.

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 LL is a reflection on an accessible category CC and the class of LL-equivalences is Σ 2\Sigma_2-definable, then the LL-local objects are a small-orthogonality class (so that LL 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 Σ 2\Sigma_2-definable is reflective.

There is also a generalization to Σ n\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 Millenium 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)

Revised on September 24, 2012 06:19:41 by Mike Shulman (192.16.204.218)