Structure theorems above a strongly compact cardinal

TL;DR

This paper explores how certain large cardinal axioms can imply that the universe of sets has a structure similar to canonical inner models, highlighting rare instances of this phenomenon.

Contribution

It presents two new examples where large cardinals imply a universe structure akin to canonical inner models, expanding understanding of their interplay.

Findings

Large cardinals can imply inner model-like universe structures

Two new instances demonstrating this phenomenon

Enhances comprehension of large cardinal and inner model relationships

Abstract

While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of the universe of sets is analogous to the canonical inner models. This note provides two new instances of this phenomenon.