The initial segment condition for $\kappa^+$-supercompactness

TL;DR

This paper develops a fine structural hierarchy for mice with long extenders at $oldsymbol{ ext{kappa}^+}$-supercompact cardinals, introducing stronger initial segment conditions and new condensation properties.

Contribution

It introduces a new hierarchy for mice with long extenders that aligns more with short extender frameworks and establishes novel condensation and Dodd-structure analyses.

Findings

Established a hierarchy with stronger initial segment conditions.

Proved a form of fine structural condensation involving non-identity embeddings.

Adapted Dodd-structure analysis to $oldsymbol{ ext{kappa}^+}$-supercompact mice.

Abstract

We give a development of the fine structure of mice with long extenders, to the level of $\kappa^+$-supercompact cardinals $\kappa$. We do this using a hierarchy with features more analogous to those familiar in the short extender context than the hierarchies introduced by Woodin and by Neeman-Steel. In particular, the mice we consider satisfy stronger versions of the initial segment condition. We establish a form of fine structural condensation involving embeddings $\pi:H\to M$ which need not be the identity below the projectum of $H$ (under special assumptions). We also adapt the analysis of the Dodd structure of short extenders on the sequence to mice at this level.