The fine structure of operator mice

TL;DR

This paper develops the detailed fine structure theory for operator-premice, a generalization of premice using an abstract operator, to support core model induction and related set-theoretic applications.

Contribution

It introduces the concept of fine condensation for operators and proves that combined with iterability, it guarantees key fine structural properties of F-mice.

Findings

Established fine condensation for operators F.

Proved that F-mice are universal and solid under certain conditions.

Enhanced the understanding of operator-premice in inner model theory.

Abstract

We develop the fine structure theory of operator-premice. These are a generalization of standard premice, in which an abstract operator $F$ is used to form the successor steps in the internal hierarchy of the premouse, instead of Jensen's $J$-operator (which computes rudimentary closure). Such notions have seen applications in core model induction arguments, but their theory has not previously been developed in detail. We define fine condensation for operators $F$ and show that fine condensation and iterability together ensure that $F$-mice have the fundamental fine structural properties including universality and solidity of the standard parameter.