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

TL;DR

This paper develops a categorical framework for algebraic phases, establishing rigidity, invariance, and equivalence results that deepen the understanding of phase interactions and their structural properties.

Contribution

It introduces a 2-categorical structure for algebraic phases, proving rigidity theorems, invariance of boundaries, and the universal nature of phase completion.

Findings

Phase morphisms are uniquely determined by rigid cores.

Under bounded defect, various equivalences coincide.

Structural boundaries are invariant under Morita-type equivalence.

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.