# 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.

## References

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

For a basic theory, see

Created on June 11, 2020 at 02:01:54. See the history of this page for a list of all contributions to it.