#
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

- Thomas Jech?,
*Set theory*.

Created on June 11, 2020 at 02:01:54.
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