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Explicit Morphisms in the Galois-Tukey Category

David Philips

TL;DR

This work reframes cardinal characteristics of the continuum as morphisms in the Galois-Tukey category, assuming the failure of the Continuum Hypothesis to access intermediate cardinals. It defines the GT framework precisely, including objects as relations and morphisms between them, and develops a rich toolkit of constructions (products, coproducts, sigma-relations) to translate inequalities into categorical morphisms. By applying this machinery to Cichoń's diagram and beyond, the paper obtains explicit morphisms witnessing standard inequalities among $ rak b$, $ rak d$, $ rak s$, $ rak r$, and related invariants, and extends the approach to Baire/Lebesgue characteristics, chopped Reals, Ramsey-like invariants, and the lower topology. The results clarify both the power and the limits of the GT framework, offering a unified categorical perspective on the continuum's cardinal characteristics and suggesting where direct proofs or alternative witnesses may be necessary.

Abstract

If the Continuum Hypothesis is false, it implies the existence of cardinalities between the integers and the real numbers. In studying these "cardinal characteristics of the continuum", it was discovered that many of the associated inequalities can be interpreted as morphisms within the "Galois-Tukey" category. This thesis aims to reformulate traditional direct proofs of cardinal characteristic inequalities by making the underlying morphisms explicit. New, purely categorical results are also discussed.

Explicit Morphisms in the Galois-Tukey Category

TL;DR

This work reframes cardinal characteristics of the continuum as morphisms in the Galois-Tukey category, assuming the failure of the Continuum Hypothesis to access intermediate cardinals. It defines the GT framework precisely, including objects as relations and morphisms between them, and develops a rich toolkit of constructions (products, coproducts, sigma-relations) to translate inequalities into categorical morphisms. By applying this machinery to Cichoń's diagram and beyond, the paper obtains explicit morphisms witnessing standard inequalities among , , , , and related invariants, and extends the approach to Baire/Lebesgue characteristics, chopped Reals, Ramsey-like invariants, and the lower topology. The results clarify both the power and the limits of the GT framework, offering a unified categorical perspective on the continuum's cardinal characteristics and suggesting where direct proofs or alternative witnesses may be necessary.

Abstract

If the Continuum Hypothesis is false, it implies the existence of cardinalities between the integers and the real numbers. In studying these "cardinal characteristics of the continuum", it was discovered that many of the associated inequalities can be interpreted as morphisms within the "Galois-Tukey" category. This thesis aims to reformulate traditional direct proofs of cardinal characteristic inequalities by making the underlying morphisms explicit. New, purely categorical results are also discussed.

Paper Structure

This paper contains 13 sections, 33 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Galois-Tukey Category
  3. Morphisms and Relations
  4. The Category $\mathbb{GT}$
  5. Cichoń's Diagram
  6. Bounding and Dominating
  7. Ideals
  8. Baire Characteristics
  9. Lebesgue Characteristics
  10. More Cardinal Characteristics
  11. Splitting and Reaping
  12. Ramsey-like Characteristics
  13. The Lower Topology

Key Result

Theorem 2.9

If there exists a morphism $\varphi:\mathbf{A}\rightarrow \mathbf{B}$ then $\left\lVert\mathbf{A}\right\rVert\geq\left\lVert\mathbf{B}\right\rVert$ and $\left\lVert\mathbf{A^{\perp}}\right\rVert\leq\left\lVert\mathbf{B^{\perp}}\right\rVert$.

Figures (3)

  • Figure 1: An Illustration of a Morphism
  • Figure 2: Cichoń's Diagram
  • Figure 3: A Diagram of the Additional Cardinal Characteristics

Theorems & Definitions (80)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Theorem 2.9
  • proof
  • ...and 70 more