Drafts by Paolo Bertozzini
We provide definitions for strict involutive higher categories (a vertical categorification of da... more We provide definitions for strict involutive higher categories (a vertical categorification of dagger categories), strict higher C*-categories and higher Fell bundles (over arbitrary involutive higher topological categories). We put forward a proposal for a relaxed form of the exchange property for higher (C*)-categories that avoids the Eckmann-Hilton collapse and hence allows the construction of explicit non-trivial " non-commutative " examples arising from the study of hypermatrices and hyper-C*-algebras, here defined. Alternatives to the usual globular and cubical settings for strict higher categories are also explored. Applications of these non-commutative higher C*-categories are envisaged in the study of morphisms in non-commutative geometry and in the algebraic formulation of relational quantum theory.
Papers by Paolo Bertozzini
We investigate the notion of involutive weak globular ω-categories via Jacque Penon's approach. I... more We investigate the notion of involutive weak globular ω-categories via Jacque Penon's approach. In particular, we give the constructions of a free self-dual globular ω-magma, of a free strict involutive globular ω-category, over an ω-globular set, and a contraction between them. The monadic definition of involutive weak globular ω-categories is given as usual via algebras for the monad induced by a certain adjunction. In our case, the adjunction is obtained from the " free functor " that associates to every ω-globular set the above contraction. Some examples of involutive weak globular ω-categories are also provided.
Extended version (censored/rejected by arXiv) of a paper published in the News Bulletin of the In... more Extended version (censored/rejected by arXiv) of a paper published in the News Bulletin of the International Association of Mathematical Physics (IAMP) 31 January (2017):17-24
We provide an algebraic formulation of C.Rovelli's relational quantum theory that is based on sui... more We provide an algebraic formulation of C.Rovelli's relational quantum theory that is based on suitable notions of "non-commutative" higher operator categories, originally developed in the study of categorical non-commutative geometry. As a way to implement C.Rovelli's original intuition on the relational origin of space-time, in the context of our proposed algebraic approach to quantum gravity via Tomita-Takesaki modular theory, we tentatively suggest to use this categorical formalism in order to spectrally reconstruct non-commutative relational space-time geometries from categories of correlation bimodules between operator algebras of observables.
Proceedings of AMM2014, "19th Annual Meeting in Mathematics (2014)" 138-147 , 2014
The main purpose of this paper is to identify suitable axiom defining strict “hybrid” 2-categorie... more The main purpose of this paper is to identify suitable axiom defining strict “hybrid” 2-categories and to apply such new structure as an instrument for the unified treatment of important examples of strict categories consisting of maps that are either linear or conjugate-linear between complex vector spaces (and more generally between bimodules over involutive complex algebras). We provide definitions of strict hybrid categories both via “2-quivers” (globular sets) and via “partial algebraic monoids” (arrows only).
We introduce a notion of Krein C*-module over a C*-algebra and more generally over a Krein C*-alg... more We introduce a notion of Krein C*-module over a C*-algebra and more generally over a Krein C*-algebra. Some properties of Krein C*-modules and their categories are investigated.
Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain... more Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain Connes' spectral triples, in this very preliminary review, we first present a description of the relevant categories of geometrical spaces, namely compact Hausdorff smooth finite-dimensional orientable Riemannian manifolds (or more generally Hermitian bundles of Clifford modules over them); we give some tentative definitions of the relevant categories of algebraic structures, namely "propagators" and "spectral correspondences" of commutative Riemannian spectral triples; and we provide a construction of functors that associate a naive morphism of spectral triples to every smooth (totally geodesic) map. The full construction of spectrum functors (reconstruction theorem for morphisms) and a proof of duality between the previous "geometrical' and "algebraic" categories are postponed to subsequent works, but we provide here some hints in this direction. We also show how the previous categories of "propagators" of commutative C*-algebras embed in the mildly non-commutative environments of categories of suitable Hilbert C*-bimodules, factorizable over commutative C*-algebras, with composition given by internal tensor product.
Noncommutative Geometry and Physics 3
We propose a definition of involutive categorical bundle (Fell bundle) enriched in an involutive ... more We propose a definition of involutive categorical bundle (Fell bundle) enriched in an involutive monoidal category and we argue that such a structure is a possible suitable environment for the formalization of different equivalent versions of spectral data for commutative C*-categories.
We present a duality between the category of compact Riemannian spin manifolds (equipped with a g... more We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable "metric" category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a "quotient" of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.
This paper contains the first written exposition of some ideas (announced in a previous survey) o... more This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A.Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
In the setting of C*-categories, we provide a definition of "spectrum" of a commutative full C*-c... more In the setting of C*-categories, we provide a definition of "spectrum" of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand duality theorem generalizing the usual Gelfand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces.
Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.
After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and ... more After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and imprimitivity, we discuss a spectral reconstruction theorem for imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras and consider some of its applications in the theory of commutative full C*-categories.
C*-categories are essentially norm-closed *-categories of bounded linear operators between Hilber... more C*-categories are essentially norm-closed *-categories of bounded linear operators between Hilbert spaces.
The purpose of this work is to identify suitable axioms defining Krein C*-categories, i.e. those categories that play the role of C*-categories whenever Hilbert spaces are replaced by more general indefinite inner product Krein spaces, and provide some basic examples. Finally we provide a Gel'fand-Naimark representation theorem for Krein C*-algebras and Krein C*-categories.
A Banach involutive algebra (A, ∗) is called a Krein C*-algebra if there is a fundamental symmetr... more A Banach involutive algebra (A, ∗) is called a Krein C*-algebra if there is a fundamental symmetry of (A, ∗), i.e., α ∈ Aut (A, ∗) such that α^2 = Id_A and ||α(x∗ )x|| = ||x||^2 for all x ∈ A. Using α, we can decompose A = A_+ ⊕ A_−, where A+ = {x ∈ A | α(x) = x} and A− = {x ∈ A | α(x) = −x}. The even part A_+ is a C*-algebra and the odd part A_− is a Hilbert C*-bimodule over the even part A+ . The ultimate goal is to develop a spectral theory for commutative Krein C*-algebras when the odd part is an imprimitivity bimodule over the commutative even part.
After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectra... more 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.
In the context of A. Connes' spectral triples, a suitable notion of morphism is introduced. Discr... more In the context of A. Connes' spectral triples, a suitable notion of morphism is introduced. Discrete groups with length function provide a natural example for our definitions. A. Connes' construction of spectral triples for group algebras is a covariant functor from the category of discrete groups with length functions to that of spectral triples. Several interesting lines for future study of the categorical properties of spectral triples and their variants are suggested.
We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal ne... more We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal net of von Neumann algebras on the circle. If the sector has finite index, it has automatically positive energy. In the infinite index case, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on the real line, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Moebius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.
Slides by Paolo Bertozzini
We provide an algebraic formulation of C.Rovelli’s relational
quantum theory that is based on sui... more We provide an algebraic formulation of C.Rovelli’s relational
quantum theory that is based on suitable notions of
“non-commutative” higher operator categories, originally developed in the study of categorical non-commutative geometry. As a way to implement C.Rovelli’s original intuition on the relational origin of space-time, in the context of our proposed algebraic approach to quantum gravity via Tomita-Takesaki modular theory, we tentatively suggest to use this categorical formalism in order to spectrally reconstruct non-commutative relational space-time geometries from categories of correlation bimodules between operator algebras of observables.
A satisfactory marriage between higher categories and operator algebras has never been achieved: ... more A satisfactory marriage between higher categories and operator algebras has never been achieved: although (monoidal) C*-categories have been systematically used since the development of the theory of superselection sectors, higher category theory has more recently evolved along lines closer to classical higher homotopy.
We present axioms for strict involutive n-categories (a vertical
categorification of dagger categories) and a definition for strict
higher C*-categories and Fell bundles (possibly equipped with
involutions of arbitrary depth), that were developed in
collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.
In this talk, motivated by the need to address foundational problems in relativistic quantum phys... more In this talk, motivated by the need to address foundational problems in relativistic quantum physics, we first historically introduce some basic elements of non-commutative geometry (Gel’fand-Naimark duality and Connes’ spectral triples); then (following arXiv:1007.4094) we speculatively suggest that, when coupled with Tomita-Takesaki modular theory and suitable notions of categorical covariance (higher C*-categories), non-commutative geometry provides a possible way for a formulation of a theory of quantum relativity, where non-commutative relational space-time might be spectrally reconstructed from an operational formalism of categories of observable algebras.
Part of this work is a joint collaboration with:
Dr.Roberto Conti (Sapienza Universita' di Roma),
Dr.Matti Raasakka (Paris 13 University).
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Drafts by Paolo Bertozzini
Papers by Paolo Bertozzini
Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.
The purpose of this work is to identify suitable axioms defining Krein C*-categories, i.e. those categories that play the role of C*-categories whenever Hilbert spaces are replaced by more general indefinite inner product Krein spaces, and provide some basic examples. Finally we provide a Gel'fand-Naimark representation theorem for Krein C*-algebras and Krein C*-categories.
Slides by Paolo Bertozzini
quantum theory that is based on suitable notions of
“non-commutative” higher operator categories, originally developed in the study of categorical non-commutative geometry. As a way to implement C.Rovelli’s original intuition on the relational origin of space-time, in the context of our proposed algebraic approach to quantum gravity via Tomita-Takesaki modular theory, we tentatively suggest to use this categorical formalism in order to spectrally reconstruct non-commutative relational space-time geometries from categories of correlation bimodules between operator algebras of observables.
We present axioms for strict involutive n-categories (a vertical
categorification of dagger categories) and a definition for strict
higher C*-categories and Fell bundles (possibly equipped with
involutions of arbitrary depth), that were developed in
collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.
Part of this work is a joint collaboration with:
Dr.Roberto Conti (Sapienza Universita' di Roma),
Dr.Matti Raasakka (Paris 13 University).
Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.
The purpose of this work is to identify suitable axioms defining Krein C*-categories, i.e. those categories that play the role of C*-categories whenever Hilbert spaces are replaced by more general indefinite inner product Krein spaces, and provide some basic examples. Finally we provide a Gel'fand-Naimark representation theorem for Krein C*-algebras and Krein C*-categories.
quantum theory that is based on suitable notions of
“non-commutative” higher operator categories, originally developed in the study of categorical non-commutative geometry. As a way to implement C.Rovelli’s original intuition on the relational origin of space-time, in the context of our proposed algebraic approach to quantum gravity via Tomita-Takesaki modular theory, we tentatively suggest to use this categorical formalism in order to spectrally reconstruct non-commutative relational space-time geometries from categories of correlation bimodules between operator algebras of observables.
We present axioms for strict involutive n-categories (a vertical
categorification of dagger categories) and a definition for strict
higher C*-categories and Fell bundles (possibly equipped with
involutions of arbitrary depth), that were developed in
collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.
Part of this work is a joint collaboration with:
Dr.Roberto Conti (Sapienza Universita' di Roma),
Dr.Matti Raasakka (Paris 13 University).
This is an ongoing joint research with Dr. Roberto Conti
(Universit`a di Chieti-Pescara “G.D’Annunzio” - Italy)
Dr. Wicharn Lewkeeratiyutkul
(Chulalongkorn University - Bangkok - Thailand).