Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of certain strength.

For instance, weakly compact cardinals are precisely the cardinals $\kappa$ where the $\kappa$-compactness theorem holds in $L_{\kappa,\kappa}$ when we bound the number of non-logical symbols by $\kappa$. Strongly compact cardinals are precisely the cardinals where the $\kappa$-compactness theorem holds for $L_{\kappa,\kappa}$. Extendible cardinals are the cardinals where the compactness theorem holds for higher order infinitary logic. Furthermore, there are characterizations of at least weakly compact, measurable, and strongly compact cardinals involving different forms of compactness besides logical compactness.

Is there is a characterization of supercompact cardinals in terms of some sort of compactness theorem? Can any other large cardinal axioms besides the ones already mentioned be characterized in terms of some sort of compactness theorem?