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Algebraic Phase Theory IV: Morphisms, Equivalences, and Categorical Rigidity

Joe Gildea

TL;DR

This work develops a categorical backbone for Algebraic Phase Theory by introducing phase morphisms, equivalences, and completion within a framework controlled by defect propagation and a canonical filtration. In the strongly admissible regime, phase morphisms are rigidly determined by their action on the rigid core and all reasonable equivalence notions collapse, while completion forms a universal, reflective localization yielding complete phases. The paper also defines a 2-category of phases via filtration-compatible 2-morphisms and proves boundary invariants and Morita-type invariants are intrinsic consequences of phase interaction and finiteness. In the weakly admissible regime, rigidity persists only up to the defect control depth, allowing controlled extension freedom beyond that depth. Together, these results elevate APT from algebraic constructions to a categorical theory with rigidity, equivalence collapse, boundary invariants, and universal completions.

Abstract

We complete the foundational architecture of Algebraic Phase Theory by developing a categorical and $2$-categorical framework for algebraic phases. Building on the structural notions introduced in Papers~I-III, we define phase morphisms, equivalence relations, and intrinsic invariants compatible with the canonical filtration and defect stratification. For finite, strongly admissible phases we establish strong rigidity theorems: phase morphisms are uniquely determined by their action on rigid cores, and under bounded defect, weak, strong, and Morita-type equivalence all coincide. In particular, finite strongly admissible phases admit no distinct models with the same filtered representation theory. We further show that structural boundaries are invariant under Morita-type equivalence and therefore constitute genuine categorical invariants. Algebraic phases, phase morphisms, and filtration-compatible natural transformations form a strict $2$-category in the strongly admissible regime. We also prove that completion defines a reflective localization of this category, with complete phases characterized as universal forced rigidifications. Together, these results elevate Algebraic Phase Theory from a collection of algebraic constructions to a categorical framework in which rigidity, equivalence collapse, boundary invariance, and completion arise as intrinsic consequences of phase interaction, finiteness, and admissibility.

Algebraic Phase Theory IV: Morphisms, Equivalences, and Categorical Rigidity

TL;DR

This work develops a categorical backbone for Algebraic Phase Theory by introducing phase morphisms, equivalences, and completion within a framework controlled by defect propagation and a canonical filtration. In the strongly admissible regime, phase morphisms are rigidly determined by their action on the rigid core and all reasonable equivalence notions collapse, while completion forms a universal, reflective localization yielding complete phases. The paper also defines a 2-category of phases via filtration-compatible 2-morphisms and proves boundary invariants and Morita-type invariants are intrinsic consequences of phase interaction and finiteness. In the weakly admissible regime, rigidity persists only up to the defect control depth, allowing controlled extension freedom beyond that depth. Together, these results elevate APT from algebraic constructions to a categorical theory with rigidity, equivalence collapse, boundary invariants, and universal completions.

Abstract

We complete the foundational architecture of Algebraic Phase Theory by developing a categorical and -categorical framework for algebraic phases. Building on the structural notions introduced in Papers~I-III, we define phase morphisms, equivalence relations, and intrinsic invariants compatible with the canonical filtration and defect stratification. For finite, strongly admissible phases we establish strong rigidity theorems: phase morphisms are uniquely determined by their action on rigid cores, and under bounded defect, weak, strong, and Morita-type equivalence all coincide. In particular, finite strongly admissible phases admit no distinct models with the same filtered representation theory. We further show that structural boundaries are invariant under Morita-type equivalence and therefore constitute genuine categorical invariants. Algebraic phases, phase morphisms, and filtration-compatible natural transformations form a strict -category in the strongly admissible regime. We also prove that completion defines a reflective localization of this category, with complete phases characterized as universal forced rigidifications. Together, these results elevate Algebraic Phase Theory from a collection of algebraic constructions to a categorical framework in which rigidity, equivalence collapse, boundary invariance, and completion arise as intrinsic consequences of phase interaction, finiteness, and admissibility.
Paper Structure (9 sections, 20 theorems, 79 equations)

This paper contains 9 sections, 20 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Phases as Structured Objects
  3. Morphisms of Algebraic Phases
  4. Equivalence and Morita-Type Notions
  5. Intrinsic Invariants of Phases
  6. The Category and 2-Category of Phases
  7. Completion as Reflective Localization
  8. Examples Across Flagship Phases
  9. Conclusion and Outlook

Key Result

Lemma 2.6

Let $\mathcal{P}$ be an algebraic phase with canonical filtration $\mathcal{P}=\mathcal{P}^{(0)}\supseteq \mathcal{P}^{(1)}\supseteq \cdots$. Then for every $i\ge 0$, the stratum $\mathcal{P}^{(i)}$ is a subphase of $\mathcal{P}$. Equivalently, $\mathcal{P}^{(i)}$ is closed under every operation in

Theorems & Definitions (70)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 60 more