nLab strongly compact cardinal

Definition

A cardinal $\kappa$ is strongly compact if any $\kappa$-complete filter can be extended to a $\kappa$-complete ultrafilter.

Here a filter is $\kappa$-complete if it is closed under intersections of families with fewer than $\kappa$ elements.

Properties

Strongly compact cardinals are measurable cardinals.

The existence of a proper class of strongly compact cardinals implies that images of accessible functors are accessible as long as they are complete or cocomplete.

Related notions

• accessible functor

References

Strongly compact cardinals were introduced by Keisler and Tarski in 1963.

For a basic theory, see

• Thomas Jech, Set theory.