Strongly compact cardinals can be characterized in various ways: compactness of \(L_{\kappa,\kappa}\), filter extensions, the existence of fine measures, the strong tree property (Inaccessibility) and many other ways. Localizations of those definitions produce a rich hierarchy. Supercompact cardinals have much fewer parallel characterizations, obtained typically by adding a normality assumption.
In this talk I will present a characterization of supercompact cardinals in terms of compactness of \(L_{\kappa,\kappa}\) with type omission. Using it, I will present a variant of the strong tree property which is (locally) weaker than the ineffable tree property and together with inaccessibility characterize supercompactness. Those characterizations localize to a characterization of \(\Pi^1_1\)-subcomapctness.
This is a joint work with Menachem Magid