| _version_ | 1866901220709367808 |
|---|---|
| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the foundational role of category theory in developing and understanding non-commutative geometry (NCG). NCG extends classical geometry by considering spaces whose coordinate algebras are non-commutative, a concept crucial for quantum mechanics and quantum field theory. We argue that category theory provides a robust, flexible, and unifying language for this extension, offering powerful tools to construct, analyze, and relate various non-commutative geometric structures. The paper reviews the historical interplay between geometry, algebra, and category theory, highlighting how Gelfand duality serves as a prototype for non-commutative generalizations. We delve into specific categorical frameworks, such as Grothendieck topoi, monoidal categories, and module categories, demonstrating their utility in defining non-commutative spaces, spectral triples, quantum groups, and deformation quantizations. The discussion emphasizes how category theory offers new avenues for axiomatizing and constructing quantum spaces, revealing deep structural insights that transcend particular algebraic models. We conclude by outlining future directions for a fully categorical approach to NCG, suggesting its potential to bridge diverse areas of mathematics and theoretical physics. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17687771 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Categorical Foundations for Non-commutative Geometry SÉRGIO DE ANDRADE, PAULO This paper explores the foundational role of category theory in developing and understanding non-commutative geometry (NCG). NCG extends classical geometry by considering spaces whose coordinate algebras are non-commutative, a concept crucial for quantum mechanics and quantum field theory. We argue that category theory provides a robust, flexible, and unifying language for this extension, offering powerful tools to construct, analyze, and relate various non-commutative geometric structures. The paper reviews the historical interplay between geometry, algebra, and category theory, highlighting how Gelfand duality serves as a prototype for non-commutative generalizations. We delve into specific categorical frameworks, such as Grothendieck topoi, monoidal categories, and module categories, demonstrating their utility in defining non-commutative spaces, spectral triples, quantum groups, and deformation quantizations. The discussion emphasizes how category theory offers new avenues for axiomatizing and constructing quantum spaces, revealing deep structural insights that transcend particular algebraic models. We conclude by outlining future directions for a fully categorical approach to NCG, suggesting its potential to bridge diverse areas of mathematics and theoretical physics. |
| title | Categorical Foundations for Non-commutative Geometry |
| url | https://doi.org/10.5281/zenodo.17687771 |