What does it mean for ultrafilter to be $\kappa$-complete?

Question

What does it mean when ultrafilter is said to be $\kappa$-complete? I cannot find suitable Internet resource, so I am asking here.

Answer 1

If $\cal U$ is a filter, we say that it is $\kappa$-complete if whenever $\gamma<\kappa$, and $\{A_\alpha\mid\alpha<\gamma\}\subseteq\cal U$, then $\bigcap_{\alpha<\gamma} A_\alpha\in\cal U$. (In this context, let an intersection over an empty family to be the set $X$ over which $\cal U$ is taken.)

If $\cal U$ is a $\kappa$-complete ultrafilter, then it is a $\kappa$-complete filter, which is also an ultrafilter.

It should be remarked that for $\kappa>\omega$ the existence of a $\kappa$-complete ultrafilter which does not contain a singleton (or a finite set) is not provable from $\sf ZFC$, and it is in fact a large cardinal axiom (measurable cardinals are the weakest which carry such ultrafilters).