Algebraic Phase Theory IV: Morphisms, Equivalences, and Categorical Rigidity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917277490741248 |
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| author | Gildea, Joe |
| author_facet | Gildea, Joe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18254 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic Phase Theory IV: Morphisms, Equivalences, and Categorical Rigidity Gildea, Joe Rings and Algebras Category Theory Primary 18B99, Secondary 18D05, 16D90, 81P45 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. |
| title | Algebraic Phase Theory IV: Morphisms, Equivalences, and Categorical Rigidity |
| topic | Rings and Algebras Category Theory Primary 18B99, Secondary 18D05, 16D90, 81P45 |
| url | https://arxiv.org/abs/2601.18254 |