# 2020 seminar talk: $\Pi_1^1$-subcompactness and type omission

Talk held by Yair Hayut (KGRC) at the KGRC seminar on 2020-03-05.

### Abstract

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 Magidor.

Slides for this talk are available.