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The fine structure of operator mice

Farmer Schlutzenberg, Nam Trang

TL;DR

The paper addresses the challenge of developing fine-structure theory for operator-premice built from an abstract operator $F$, extending Jensen-structure methods to a broader operator framework and enabling core model induction. It builds a comprehensive fine-structural framework—covering hierarchical models, potential opms, Q-opms, and full operator-premice—and introduces coarse and fine condensation notions, along with mouse operators to ground concrete constructions. A central result shows that if $F$ satisfies fine condensation and appropriate iterability, then $F$-iterable $F$-premice inherit fundamental fine-structural properties, including solidity of the standard parameter and universality, with a robust copying/realization toolkit and a weak Dodd– Jensen apparatus to manage iteration strategies. This framework broadens the operator-mice theory, enabling more flexible inner model constructions and strengthening the toolkit for core model induction in the presence of general operators and superstrong extenders.

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.

The fine structure of operator mice

TL;DR

The paper addresses the challenge of developing fine-structure theory for operator-premice built from an abstract operator , extending Jensen-structure methods to a broader operator framework and enabling core model induction. It builds a comprehensive fine-structural framework—covering hierarchical models, potential opms, Q-opms, and full operator-premice—and introduces coarse and fine condensation notions, along with mouse operators to ground concrete constructions. A central result shows that if satisfies fine condensation and appropriate iterability, then -iterable -premice inherit fundamental fine-structural properties, including solidity of the standard parameter and universality, with a robust copying/realization toolkit and a weak Dodd– Jensen apparatus to manage iteration strategies. This framework broadens the operator-mice theory, enabling more flexible inner model constructions and strengthening the toolkit for core model induction in the presence of general operators and superstrong extenders.

Abstract

We develop the fine structure theory of operator-premice. These are a generalization of standard premice, in which an abstract operator is used to form the successor steps in the internal hierarchy of the premouse, instead of Jensen's -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 and show that fine condensation and iterability together ensure that -mice have the fundamental fine structural properties including universality and solidity of the standard parameter.

Paper Structure

This paper contains 18 sections, 31 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Conventions and Notation
  3. The fine structural framework
  4. Hierarchical models
  5. Potential operator-premice
  6. Fine structure
  7. Q-operator-premice
  8. Operator-premice
  9. Fine structure and iterations
  10. ${\mathcal{F}}$-mice for operators ${\mathcal{F}}$
  11. Abstract operators $\mathcal{F}$ and $\mathcal{F}$-premice
  12. Coarse condensation of operators
  13. Operators which (don't) condense well
  14. Mouse operators
  15. Fine condensation
  16. ...and 3 more sections

Key Result

Lemma 2.5

Let ${\mathcal{M}}$ be a hierarchical model and ${\mathcal{N}}\trianglelefteq{\mathcal{M}}$. Then ${\mathcal{N}}$ is a hierarchical model.

Theorems & Definitions (131)

  • Definition 2.1
  • Definition 2.2: Hulls
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof : Proof Sketch
  • ...and 121 more