arXiv subjects

Paolo Bertozzini

Publications and source records attributed to Paolo Bertozzini.

At least 19 recordsLinked to original sources

arXiv2025

On Strict Higher C*-categories

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.

math.OA↗

arXiv2025

Spectral Theory for Non-full Commutative C*-categories

We extend the spectral theory of commutative C*-categories to the non full-case, introducing a suitable notion of spectral spaceoid provinding a duality between a category of "non-trivial" *-functors of non-full commutative C*-categories and a category of Takahashi morphisms of "non-full spaceoids" (here defined). As a byproduct we obtain a spectral theorem for a non-full generalization of imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras via continuous sections vanishing at infinity of a Hilbert C*-line-bundle over the graph of a homeomorphism between open subsets of the corresponding Gel'fand spectra of the C*-algebras.

math.OA↗

arXiv2025

Involutive Weak Cubical $ω$-categories

We investigate the notion of involutive weak cubical $ω$-categories via Penon's approach: as algebras for the monad induced by the free involutive strict $ω$-category functor on cubical $ω$-sets. A few examples of involutive weak cubical $ω$-categories are provided.

math.CT↗

arXiv2024

Isometric Spectral Subtriples

We investigate the notion of subsystem in the framework of spectral triple as a generalized notion of noncommutative submanifold. In the case of manifolds, we consider several conditions on Dirac operators which turn embedded submanifolds into isometric submanifolds. We then suggest a definition of spectral subtriple based on the notion of submanifold algebra and the already existing notions of Riemannian, isometric, and totally geodesic morphisms. We have shown that our definitions work at least in some relevant almost commutative examples.

math-ph↗

arXiv2023

Involutive Weak Globular $ω$-categories

We investigate the notion of involutive weak globular $ω$-categories via T.Leinster's approach: as algebras for the initial contracted globular operad in the bicategory of globular collections induced by the Cartesian monad of the free involutive strict $ω$-category functor on globular $ω$-sets. An apparently more restrictive notion of involutive weak globular $ω$-categories as algebras for the initial operadic-contraction in the bicategory of globular contracted-collections induced by the previous Cartesian monad (where here the operadic multiplications and units satisfy further compatibility axioms with the contractions) is also considered.

math.CT↗

arXiv2021

Duality for Multimodules

Suitable duals of multimodules are introduced and used to provide transposition contravariant right semi-adjunctions (and dualitites under reflexivity). Several additional notions on multimodules are discussed: generalized morphisms and involutions; tensor products and contractions; inner products; first-order differential operators. A construction of an (involutive) colored properad of multimodules is suggested.

math.CT↗

arXiv2017

Involutive Weak Globular Higher Categories

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.

math.CT↗

arXiv2014

Categorical Operator Algebraic Foundations of Relational Quantum Theory

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. Parts of this work are joint collaborations with: Dr.Roberto Conti (Sapienza Universita' di Roma), Assoc.Prof.Wicharn Lewkeeratiyutkul (Chulalongkorn University, Bangkok), Dr.Rachel Dawe Martins (Istituto Superior Tecnico, Lisboa), Dr.Matti Raasakka (Paris 13 University), Dr.Noppakhun Suthichitranont.

math-ph↗

arXiv2014

Spectral Theory on Commutative Krein C*-algebras

A Banach involutive algebra is called a Krein C*-algebra if there is a fundamental symmetry (an involutive automorphism of period 2) such that the C*-property is satisfied when the original involution is replaced with the new one obtained by composing the automorphism with the old involution. For a given fundamental symmetry, a Krein C*-algebra decomposes as a direct sum of an even part (a C*-algebra) and an odd part (a Hilbert C*-bimodule on the even part). Our goal here is to develop a spectral theory for commutative unital Krein C*-algebras when the odd part is a symmetric imprimitivity C*-bimodule over the even part and there exists an additional suitable "exchange symmetry" between the odd and even parts.

math.OA↗

arXiv2014

Remarks on Morphisms of Spectral Geometries

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.

math.OA↗

arXiv2014

Krein C*-modules

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.

math.OA↗

arXiv2014

Categorical Non-commutative Geometry

The purpose of this short note was to outline the current status, then in 2011, of some research programs aiming at a categorification of parts of A.Connes' non-commutative geometry and to provide an outlook on some possible subsequent developments in categorical non-commutative geometry.

math.OA↗

arXiv2011

Non-Commutative Geometry, Categories and Quantum Physics

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.

math.OA↗

arXiv2011

A Remark on Gelfand Duality for Spectral Triples

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.

math.OA↗

arXiv2011

Krein C*-categories

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.

math.OA↗

arXiv2011

Enriched Fell Bundles and Spaceoids

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.

math.OA↗

arXiv2010

Modular Theory, Non-Commutative Geometry and Quantum Gravity

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.

gr-qc↗

arXiv2010

A Horizontal Categorification of Gelfand Duality

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.

math.OA↗