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Non-commutative geometry, categories and quantum physics. (English) Zbl 1182.46058

Summary: 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 “categorical non-commutative geometry” to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.

MSC:

46L87 Noncommutative differential geometry
46M15 Categories, functors in functional analysis
16D90 Module categories in associative algebras
18F99 Categories in geometry and topology
81R60 Noncommutative geometry in quantum theory
81T05 Axiomatic quantum field theory; operator algebras
83C65 Methods of noncommutative geometry in general relativity
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