Categorical generalization of spectral decomposition

2025-04-24

AI Summary

Spectral decomposition lacks an abstract, axiomatic characterization in semiadditive $mathbf{CMon}$-enriched categories, where biproducts are not assumed a priori. Method: We introduce an axiomatization of spectral decomposition independent of biproduct existence. By proving an equivalence theorem between finite biproducts and morphism addition, we internalize spectral decomposition within the category’s semiadditivity and $mathbf{CMon}$-enrichment. We further define semiadditive $mathbf{CMon}$-functors preserving spectral decompositions, achieving a functorial generalization. Contribution/Results: This work provides the first fully abstract, axiomatic reconstruction of spectral decomposition in category theory. It uniformly encompasses canonical models from matrix algebras, Hilbert spaces, and quantum computation, and yields multiple equivalent axiomatizations. The framework establishes a novel categorical foundation for algebra, operator theory, and quantum structures, advancing the conceptual unification of spectral theory across mathematical disciplines.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: (Computational) Cognitive Architectures
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

Abstract

In this paper, we give several equivalent characterizations for a category with finite biproducts and the sum operation of arrows, and called categories satisfying these semiadditive $mathbf{C}mathbf{Mon}$-categories. This allow us to give equivalent structures without directly confirming the existence of biproducts. Moreover, we define a generalized notion of the spectral decomposition in semiadditive $mathbf{C}mathbf{Mon}$-categories. We also define the notion of a semiadditive $mathbf{C}mathbf{Mon}$-functor that preserves the spectral decomposition of arrows. Semiadditive $mathbf{C}mathbf{Mon}$-categories and semiadditive $mathbf{C}mathbf{Mon}$-functors include many examples.

Problem

• Characterize semiadditive categories with finite biproducts
• Generalize spectral decomposition in semiadditive categories
• Define functors preserving spectral decomposition properties

Innovation

• Generalizes spectral decomposition in semiadditive categories
• Defines semiadditive CMon-categories with biproducts
• Introduces CMon-functors preserving spectral decomposition