Non-Commutative Geometry, Categories and Quantum Physics Paolo Bertozzinia,∗ Roberto Contib∗†, Wicharn Lewkeeratiyutkulb∗ aDepartment of Mathematics and Statistics, Faculty of Science and Technology Thammasat University, Bangkok 12121, Thailand e-mail:
[email protected] bDepartment of Mathematics and Computer Science Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand e-mail:
[email protected] e-mail:
[email protected] 08 November 2009, revised: 26 December 2011 Abstract After an introduction to some basic issues in non-commutative geometry (Gel’fand duality, spectral triples), we present a “panoramic view” of the status of our current research program on the use of categorical methods in the setting of A. Connes’ non- commutative geometry: morphisms/categories of spectral triples, categorification of Gel’fand duality. We conclude with a summary of the expected applications of “cat- egorical non-commutative geometry” to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity. Keywords: Non-commutative Geometry, Spectral Triple, Category, Morphism, Quantum Physics, Space-Time. MSC-2000: 46L87, 46M15, 16D90, 18F99, 81R60, 81T05, 83C65. Contents 1 Introduction. 2 arXiv:0801.2826v2 [math.OA] 27 Dec 2011 2 Categories. 3 2.1 ObjectsandMorphisms.............................. 3 2.2 Functors, Natural Transformations, Dualities. ...... 4 3 Non-commutative Geometry (Objects). 5 3.1 Non-commutativeTopology.. 6 3.1.1 Gel’fandTheorem. ............................ 6 3.1.2 Serre-SwanandTakahashiTheorems. 7 ∗Partially supported by the Thai Research Fund: grant n. RSA4780022. †Current address: Dipartimento di Scienze, Universit`adi Chieti-Pescara “G. D’Annunzio”, Viale Pin- daro 42, I-65127 Pescara, Italy. 1 3.2 Non-commutative (Spin) Differential Geometry.