Categories of Spectral Geometries. · 2008-10-08 · Introduction Categories. Non-commutative...

IntroductionCategories.

Non-commutative Geometry (Objects).Categories in Non-Commutative Geometry.

Applications to Physics.

Categories of Spectral Geometries.

Paolo Bertozzini

Department of Mathematics and Statistics - Thammasat University - Bangkok.

Second Workshop on Categories Logic and PhysicsImperial College - London

14 May 2008.

Paolo Bertozzini Categories of Spectral Geometries.




Introduction.Outline.

Introduction 1.In A. Connes’ non-commutative geometry, “spaces” are described“dually” as spectral triples. We provide an overview of some of thenotions that we deem necessary for the development of acategorical framework in the context of spectral geometry, namely:

I several notions of morphism of spectral geometries,

I a spectral theory for commutative full C*-categories,

I a definition of strict-n-C*-categories,

I spectral geometries over C*-categories.

If time will allow, we will speculate on possible applications tofoundational issues in quantum physics:

I categorical covariance,

I spectral quantum space-time,

I modular quantum gravity.






Introduction 2

This is an ongoing joint research with

I Dr. Roberto Conti(University of Newcastle - Australia) and

I Dr. Wicharn Lewkeeratiyutkul(Chulalongkorn University - Bangkok - Thailand).

MSC-2000: 46L87, 46M15, 16D90, 18F99, 81R60, 81T05, 83C65.

Keywords: Non-commutative Geometry, Spectral Triple, Category,Morphism, C*-category, Quantum Physics.






Outline 1.

I Introduction and Outline.I Categories.

I Object/Morphisms.I Functors, Natural Transformations, Dualities.

I Non-commutative Geometry.I Non-commutative Topology

(C*-algebras, Gel’fand Theorem, Hilbert C*-modules,Serre-Swan Theorem, Takahashi Theorem).

I Non-commutative Spin Geometry(Non-commutative Manifolds, Connes Spectral Triples, ConnesTheorem, Connes, Rennie-Varilly Theorem).

I Other Spectral Geometries.(Riemannian Spectral Triples, Lord-Rennie Theorem)






Outline 2.I Categories in Non-commutative Geometry.

** Morphisms(Totally Geodesic Spin, Metric, Riemannian, Lord-RennieDuality, Spectral Congruences, Spectral Spans, General,Morita, Other).

* Categorification (Vertical/Horizontal, C*-categories, FellBundles, Spectral Theorem for Commutative FullC*-categories, Spectral Theorem for Imprimitivity Bimodules).

** Strict Higher C*-categories.** Spectral Geometries over C*-categories.

* Categorical NCG.I Applications to Physics.

I Categories in Physics.* Categorical Covariance.I Spectral/Quantum Space-Time.* Algebraic Quantum Gravity.





Objects and Morphisms.Functors, Natural Transformations, Dualities.

Objects and Morphisms.A category C consists of

a) a class of objects ObC ,

b) for any two object A,B ∈ ObC a set of morphismsHomC (A,B),

c) for any three objects A,B,C ∈ ObC a composition map

: HomC (B,C )× HomC (A,B)→ HomC (A,C )

that satisfies the following properties for all morphisms f , g , hthat can be composed:

(f g) h = f (g h),

∀A ∈ ObC ∃ιA ∈ HomC (A,A) : ιA f = f , g ιA = g .






Functors.

Given two categories C ,D , a covariant functor F : C → D is apair of maps

F : ObC → ObD , F : A 7→ FA, ∀A ∈ ObC ,

F : HomC → HomD , F : x 7→ F(x), ∀x ∈ HomC ,

such that x ∈ HomC (A,B)⇒ F(x) ∈ HomD(FA,FB), and that,for any two composable morphisms f , g and any object A, satisfies

F(g f ) = F(g) F(h),

F(ιA) = ιFA.

For a contravariant functor we require F(x) ∈ HomD(FB ,FA).






Natural Transformations.A morphism f ∈ HomC (A,B) is called an isomorphism if thereexists another morphism g ∈ HomC (B,A) such that

f g = ιB and g f = ιA.

A natural transformation η : F→ G between two functorsF,G : C → D , is a map

η : ObC → HomD , η : A 7→ ηA ∈ HomD(FA,GA),

such that the diagram FAηA //

F(x)

GA

G(x)

FB ηB

// GB .

is commutative for all x ∈ HomC (A,B), A,B ∈ ObC .Paolo Bertozzini Categories of Spectral Geometries.





Dualities.

The functor F : C → D is

I faithful if, for all A,B ∈ ObC , its restriction to the setsHomC (A,B) is injective;

I full if its restriction to HomC (A,B) is surjective;

I representative if for all X ∈ ObD there exists A ∈ ObC suchthat FA is isomorphic to X in D .

A duality (a contravariant equivalence) of two categories C and Dis a pair of contravariant functors Γ : C → D and Σ : D → C suchthat Γ Σ and Σ Γ are naturally equivalent to the respectiveidentity functors ID and IC . A duality is actually specified by twofunctors, but given any one of the two functors in the dual pair,the other one is unique up two natural isomorphism. A functor Γ isin a duality pair if and only if it is full, faithful and representative.





Non-commutative Topology.Non-commutative (Spin) Differential Geometry.Other Spectral Geometries.

Non-commutative Geometry 0 - Introduction.

Non-commutative Geometry

||[Operator Algebra] ∩ [Differential Geometry]

Operator Algebra⋂Functional Analysis

||[Linear Algebra] ∩ [General Topology]






Non-commutative Geometry 1.

Non-commutative geometry, created by A. Connes, is the name ofa very young and fast developing mathematical theory that ismaking use of operator algebras (itself a branch of functionalanalysis created by J. von Neumann in 1929) to find algebraicgeneralizations of most of the structures currently available inmathematics: measurable, topological, differential, metric etc.

From the algebraic point of view, mathematicians have beendealing with “non-commutative” algebraic structures since arelatively long time ago (for example under the form of groups,matrices in linear algebra and Hamilton’s quaternions).







The first seeds of non-commutative geometry (i.e. the idea tosubstitute a commutative algebra with a non-commutative one) isstrictly linked with physics’ developments and can be traced backto the beginning of quantum mechanics in the form ofHeisenberg’s matrix mechanics in 1925, where non-commutingmatrices (operators) take the place of classical commutingobservables of a physical system (hence the fashionable name of“quantum mathematics” often cited in the recent literature).

Anyway, it is only starting in 1980, with the extraordinary work ofA. Connes, the real founder of non-commutative geometry, that asystematic theory capable of describing differential and metricstructures becomes available.







The fundamental idea, implicitly used in A. Connes’non-commutative geometry is a powerful extension of R. Decartes’analytic geometry:

I to “trade” “geometrical spaces” X of points with theirAbelian algebras of (say complex valued) functions f : X → C;

I to “translate” the geometrical properties of spaces intoalgebraic properties of the associated algebras (a line ofthought already present in J.L. Koszul algebraization ofdifferential geometry);







I to “reconstruct” the original geometric space X as a derivedentity (the spectrum of the algebra), a technique thatappeared for the first time in the work of I. Gel’fand onAbelian C*-algebras in 1939 (although similar ideas, previouslydeveloped by D. Hilbert, are well known and used also inalgebraic geometry in P. Cartier-A. Grothendieck’s definitionof schemes).







In order to develop non-commutative geometries, we usuallyproceed as follows:

1) First we find a suitable way to “codify” or translate thegeometric properties of a space X (topology, measure,differential structure, metric, . . . ) in algebraic terms, using acommutative algebra of functions over X .

2) Then we try to see if this codification “survives” generalizingto the case of non-commutative algebras.







3) Finally the generalized properties are taken as axioms definingwhat a “dual” of a “non-commutative” (topological,measurable, differential, metric, . . . ) space is, withoutreferring to any underlying point space.

Of course the process of generalization of the properties from thecommutative to the non-commutative algebra case is highly nontrivial and, as a result, several alternative possible axiomatizationsarise in the non-commutative case, corresponding to a unique“commutative limit”.







Here is a short “dictionary” with some of the “translations” from“commutative” to “quantum” mathematics:

Commutative Non-commutative

Set Algebra

Point Pure State

Topology (locally compact, T2) C*-algebra

Measure (Radon) State on a C*-algebra

Vector Bundle Module over a C*-algebra

Riemannian Manifold(compact, spin)

Connes’ SpectralTriple (A,H,D)

Gel’fand Serre-Swan Connes, Rennie-Varilly







The existence of dualities between categories of “geometricalspaces” and categories “constructed from Abelian algebras” is thestarting point of any generalization of geometry to thenon-commutative situation.Here we will present Gel’fand duality and Takahashi duality ageneralization of Gel’fand that also subsumes Serre-Swanequivalence. Duality Gel’fand Serre-Swan Takahashi Connes, Rennie-Varilly






Non-commutative Geometry 9 - References.

For general introductions to the subject see:

I A. Connes,Noncommutative Geometry, Academic Press (1994).

I G. Landi,An Introduction to Noncommutative Spaces and TheirGeometries, Springer (1997).

I H. Figueroa-J. Gracia-Bondia-J. Varilly,Elements of Noncommutative Geometry, Birkhauser (2000).






C*-algebras.

A complex unital algebra A is a vector space over C with anassociative unital bilinear multiplication.A is Abelian (commutative) if ab = ba, for all a, b ∈ A.An involution on A is a conjugate linear map ∗ : A→ A such that(a∗)∗ = a and (ab)∗ = b∗a∗, for all a, b ∈ A. An involutivecomplex unital algebra is A called a C*-algebra if A is a Banachspace with a norm a 7→ ‖a‖ such that ‖ab‖ ≤ ‖a‖ · ‖b‖ and‖a∗a‖ = ‖a‖2, for all a, b ∈ A.Notable examples are the algebras of continuous complex valuedfunctions C (X ; C) on a compact topological space with the “supnorm” and the algebras of linear bounded operators B(H) on agiven Hilbert space H.






Operator Algebras - References.For all the details on operator algebras, the reader may refer to

I R. Kadison-J. Ringrose,Fundamentals of the Theory of Operator Algebras, vol. 1-2,AMS (1998).

I M. Takesaki,The Theory of Operator Algebras I-II-III, Springer(2001-2002).

I B. Blackadar,Operator Algebras, Springer (2006).

For an elementary introduction to functional analysis for operatoralgebras:

I G. Pedersen,Analysis Now, Springer (1998).






Gel’fand Theorem 1

There exists a duality (Γ,Σ) between the category T (1), ofcontinuous maps between compact Hausdorff topological spaces,and the category A (1), of unital homomorphisms of commutativeunital C*-algebras, where

I Γ is the functor that to every compact Hausdorff topologicalspace X ∈ ObT (1) associates the unital commutativeC*-algebra C(X ; C) of complex valued continuous functionson X (with pointwise multiplication and conjugation andsupremum norm) and that to every continuous mapf : X → Y associates the unital ∗-homomorphismf • : C (Y ; C)→ C (X ; C) given by the pull-back of continuousfunctions by f ;







I Σ is the functor that to every unital commutative C*-algebraA associates its spectrum

Sp(A) := ω | ω : A→ C, is a unital ∗-homomorphism

(as a topological space with the weak topology induced by theevaluation maps ω 7→ ω(x), for all x ∈ A) and that to everyunital ∗-homomorphism φ : A→ B of algebras associates thecontinuous map φ• : Sp(B)→ Sp(A) given by the pull-backunder φ.

Serre-Swan Theorem Takahashi Theorem Connes Rennie-Varilly Theorem







I The natural isomorphism G : IA (1) → Γ Σ is given by theGel’fand transforms GA : A→ C (Sp(A)) defined byGA : a 7→ a where a : Sp(A)→ C is the Gelf’and transform ofa i.e. a : ω 7→ ω(a).

I Similarly the natural isomorphism E : IT (1) → Σ Γ is givenby the evaluation homeomorphisms EX : X → Sp(C (X ))defined by EX : p 7→ evp, where evp : C (X )→ C is thep-evaluation i.e. evp : f 7→ f (p).






Hilbert C*-modules 1.A left pre-Hilbert C*-module AM over A,1 is a unital leftmodule M over the unital ring A that is equipped with anA-valued inner product M ×M → A denoted by (x , y) 7→ A〈x | y〉such that:2

〈x + y | z〉 = 〈x | z〉+ 〈y | z〉, ∀x , y , z ∈ M,

〈a · x | z〉 = a〈x | z〉, ∀x , y ∈ M, ∀a ∈ A,

〈y | x〉 = 〈x | y〉∗, ∀x , y ∈ M,

〈x | x〉 ∈ A+, ∀x ∈ M,

〈x | x〉 = 0A,⇒ x = 0M .

AM is a left Hilbert C*-module if it is complete in the normdefined by x 7→

√‖A〈x | x〉‖.

1A unital C*-algebra whose positive part is denoted by A+.2The definition for right module requires linearity in the second variable.






Hilbert C*-modules 2.AM is full if span〈x | y〉 | x , y ∈ M = A, where the closure is inthe norm topology of the C*-algebra A. A pre-HilbertC*-bimodule AMB over the unital C*-algebras A,B, is a leftpre-Hilbert module over A and a right pre-Hilbert C*-module overB such that: (a · x) · b = a · (x · b), ∀a ∈ A, x ∈ M, b ∈ B.A full pre-Hilbert C*-bimodule is said to be an imprimitivitybimodule or a Morita equivalence bimodule if:

A〈x | y〉 · z = x · 〈y | z〉B, ∀x , y , z ∈ M.

A bimodule AMA (over an Abelian A) is called symmetric ifax = xa for all x ∈ M and a ∈ A.A module AM is free if it is isomorphic to a module of the form⊕JA for some index set J. A module AM is projective if thereexists another module AN such that M ⊕ N is a free module.






Serre-Swan Theorem 1.

An “equivalence result” strictly related to Gel’fand theorem, is thefollowing “Hermitian” version of Serre-Swan theorem (see forexample Theorem 7.1 in M. Frank3 or Theorem 9.1.6 inN. Weaver4) that provides a “spectral interpretation” of symmetricprojective finite bimodules over a commutative unital C*-algebraas finite rank Hermitian vector bundles over the spectrum of thealgebra.5

3M. Frank Geometrical Aspects of Hilbert C*-modules, Positivity, 3, n. 3,215-243 (1999).

4N. Weaver, Mathematical Quantization, Chapmann and Hall, 2001.5The result is actually true also without the finiteness condition (with

Hilbert bundles in place of Hermitian bundles).Paolo Bertozzini Categories of Spectral Geometries.





Serre-Swan Theorem 2.

Let X be a compact Hausdorff topological space. Let MC(X )

denote the weak monoidal ∗-category6 of symmetric projectivefinite Hilbert C*-bimodules over the commutative C*-algebraC (X ; C) with C (X )-bimodule morphisms. Let EX be the weakmonoidal ∗-category7 of finite rank Hermitian vector bundles overX with bundle morphisms8.The functor Γ : EX →MC(X ), that to every Hermitian vectorbundle associates its symmetric C (X )-bimodule of sections, is anequivalence of weak monoidal ∗-categories.

Takahashi’s Theorem Connes Rennie-Varilly Theorem

6Were the monoidal structure is the usual tensor product of bimodules andthe ∗ is the usual Rieffel dual of bimodules.

7Were the monoidal structure comes from the fiberwise tensor product ofHermitian bundles and the ∗ is the dualization of Hermitian bundles.

8Continuous, fiberwise linear maps, preserving the base points.Paolo Bertozzini Categories of Spectral Geometries.





Problems.

I Serre-Swan theorem deals only with categories of bundles overa fixed topological space (categories of modules over a fixedalgebra, respectively).

I Serre-Swan theorem, in its actual form, gives an equivalenceof categories (and not a duality), this will create problems of“covariance” for any generalization of the well-knowncovariant functors between categories of manifolds andcategories of their associated vector (tensor, Clifford) bundles.

Work on these issues (considering “conguences” of bimodules andreformulating Serre-Swan theorem in terms of “relators”) is inprogress. A first immediate solution is provided by Takahashiduality theorem here below.






Takahashi Theorem 1

Serre-Swan theorem is actually a particular case of the followinggeneral (and surprisingly almost unnoticed) Gel’fand duality resultobtained in 1971 by A. Takahashi in his Ph.D. thesis9 under thesupervision of K. Hofmann.

Note that our Gel’fand duality result for commutative fullC*-categories (that we will present later) can be seen as“strict”-∗-monoidal version of Takahashi duality.

9A. Takahashi, Hilbert Modules and their Representation,Rev. Colombiana Mat., 13, 1-38 (1979).A. Takahashi, A Duality between Hilbert Modules and Fields of Hilbert Spaces,Rev. Colombiana Mat., 13, 93-120 (1979).






Takahashi Theorem 2

There is a (weak ∗-monoidal) category •M of left HilbertC*-modules AM,BN over unital commutative C*-algebras, whosemorphisms are given by pairs (φ,Φ) where φ : A→ B is a unital∗-homomorphism of C*-algebras and Φ : M → N is a continuousmap such that Φ(ax) = φ(a)Φ(x), for all a ∈ A and x ∈ M.There is a (weak ∗-monoidal) category E of Hilbert bundles(E, π,X), (F, ρ,Y) over compact Hausdorff topological spaces withmorphisms given by pairs (f ,F) with f : X→ Y a continuous mapand F : f •(F)→ E satisfies π F = ρf , where (f •(F), ρf ,X)denotes the pull-back of the bundle (F, ρ,Y) under f .There is a duality of (weak ∗-monoidal) categories given by thefunctor Γ that associates to every Hilbert bundle (E, π,X) the setof sections Γ(X; E) and that to every section σ ∈ Γ(Y; F)associates the section F f •(σ) ∈ Γ(X; E). Connes Rennie-Varilly Theorem






Non-commutative Manifolds.

What are non-commutative manifolds?In order to define “non-commutative manifolds”, we have to find acategorical duality between a category of manifolds and a suitablecategory constructed out of Abelian C*-algebras of functions overthe manifolds. The complete answer to the question is not known,but (at least in the case of compact finite dimensional orientableRiemannian spin manifolds), the notion of Connes’ spectral triplesand Connes-Rennie-Varilly reconstruction theorem provide andadeguate starting point, specifying the objects of ournon-commutative category.






Connes Spectral Triples 1.A (compact) spectral triple (A,H,D) is given by:

I a unital pre-C*-algebra A closed under holomorphic functionalcalculus;

I a representation π : A→ B(H) of A on the Hilbert space H;I a (non-necessarily bounded) self-adjoint operator D on H,

called the Dirac operator, such that:

a) the resolvent (D − λ)−1 is a compact operator, ∀λ ∈ C \ R,b) [D, π(a)]− ∈ B(H), for every a ∈ A,

where [x , y ]− := xy − yx denotes the commutator ofx , y ∈ B(H).

Several additional technical conditions (grading, real structure,orientability, regularity, summability, finiteness, Poincare duality)should be imposed on a spectral triple in order to formulate thefollowing results.






Connes Spectral Triples 2.

Given an orientable compact Riemannian spin m-dimensionaldifferentiable manifold M, with a given complex spinor bundleS(M), a given spinorial charge conjugation CM and a given volumeform µM , define by

I AM := C∞(M; C) the algebra of complex valued regularfunctions on the differentiable manifold M,

I HM :=L2(M; S(M)) the Hilbert space of “square integrable”sections of the given spinor bundle S(M) of the manifold Mi.e. the completion of the space Γ∞(M; S(M)) of smoothsections of the spinor bundle S(M) equipped with the innerproduct 〈σ | τ〉 :=

∫M〈σ(p) | τ(p)〉p dµM , where 〈 | 〉p, with

p ∈ M, is the unique inner product on Sp(M) compatible withthe Clifford action and the Clifford product.






Connes Spectral Triples 3.

I DM the Atiyah-Singer Dirac operator i.e. the closure of theoperator that on Γ∞(M; S(M)) is obtained by “contracting”with the Clifford multiplication, the unique spinorial covariantderivative ∇S(M) (induced on Γ∞(M; S(M)) by theLevi-Civita covariant derivative of M);

I JM the unique antilinear unitary extension JM : HM → HM ofthe operator determined by the spinorial charge conjugationCM by (JMσ)(p) := CM(σ(p)) for σ ∈ Γ∞(M; S(M)), p ∈ M;

I ΓM the unique unitary extension on HM of the operator givenby fiberwise grading on Sp(M), with p ∈ M.10

10The grading is actually the identity in odd dimension.Paolo Bertozzini Categories of Spectral Geometries.





Connes, Rennie-Varilly Theorems 1.

Theorem (Connes)

The data (AM ,HM ,DM) define an Abelian regular finitem-dimensional spectral triple that is real, with real structure JM ,orientable, with grading ΓM , and that satisfies Poincare duality.






Connes, Rennie-Varilly Theorems 2.

Theorem (Connes, Rennie-Varilly)

Let (A,H,D) be an irreducible Abelian real (with real structure Jand grading Γ) regular m-dimensional orientable finite spectraltriple satisfying Poincare duality (and, in the Rennie-Varillyformulation, some additional “metric/regularity conditions”).The spectrum of (the norm closure of) A can be endowed, withthe structure of an m-dimensional connected compact spinRiemannian manifold M with an irreducible complex spinor bundleS(M), a charge conjugation JM and a grading ΓM such that:A ' C∞(M; C), H ' L2(M, S(M)), D ' DM , J ' JM , Γ ' ΓM .






Connes-Rennie Theorems 3.

A. Connes proved the previous theorem under the additionalcondition that A is already given as the algebra A = C∞(M; C) ofsmooth functions over a differentiable manifold M andconjectured11 the result for general commutative pre-C*-algebras.

A proof of this fact, under a slightly different set of assumptions,has been presented by A. Rennie-J. Varilly.12

11A. Connes, Brisure de Symetrie Spontanee et Geometrie du Pont de VueSpectral, J. Geom. Phys., 23, 206-234 (1997).

12A. Rennie, Commutative Geometries are Spin Manifolds, Rev. Math. Phys.,13, 409 (2001), math-ph/9903021.Some gaps that were pointed out in the original argument and a differentrevised proof has appeared: A. Rennie, J. Varilly, Reconstruction of Manifoldsin Noncommutative Geometry, math.OA/0610418.






Connes-Rennie-Varilly Theorems 4.

It has been announced that A. Connes is providing a completeproof of the theorem under the original assumptions.13

As a consequence, a one-to-one correspondence should existbetween unitary equivalence classes of these Abelian spectraltriples and connected compact oriented Riemannian spin manifolds,up to spin-preserving isometric diffeomorphisms. morphisms

13A. Connes: On the Spectral Characterization of Manifolds, Lectures, SixthAnnual Spring Institute on NCG and OA, Vanderbilt University, 7-13 May 2008.






Other Spectral Geometries 1.Although NCG, following A. Connes, has been mainly developed inthe axiomatic framework of spectral triples, that essentiallygeneralize the structures available for the Atiyah-Singer theory offirst order differential elliptic operators of the Dirac type, it is verylikely that suitable “spectral geometries” might be developed usingoperators of higher order (the Laplacian type being the firstnotable example). Since “topological obstructions” (such usnon-orientability, non-spinoriality) are expected to surviveessentially unaltered in the transition from the commutative to thenon-commutative world, these “higher-order non-commutativegeometries” will deal with more general situations compared tousual spectral triples. In this direction we are working 14 in thehope to obtain Connes-Rennie type theorems also in these cases.

14P.B., R. Conti, W. Lewkeeratiyutkul, Second OrderNon-commutative Geometry, work in progress.






Other Spectral Geometries 2.

In the last few years several other variants for the axioms ofspectral triples have been considered or proposed:

I non-compact spectral triples,15

I spectral triples for quantum groups,16

I Lorentzian spectral triples,17

15V. Gayral, J. M. Gracia-Bondia, B. Iochum, T. Schuker, J. C. Varilly,Moyal Planes are Spectral Triples, Commun. Math. Phys. 246, no. 3 (2004),569-623, hep-th/0307241.

16L. Dabrowski, G. Landi, A, Sitarz, W. van Suijlekom, J. Varilly, The DiracOperator on SUq(2), Commun. Math. Phys., 259, 729-759 (2005),math.QA/0411609.

17A. Strohmaier, On Noncommutative and semi-Riemannian Geometry,math-ph/0110001.






Other Spectral Geometries 3.

I Riemannian non-spin geometries,18,19

I “von Neumann” spectral triples.20,21,22

18J. Frohlich, O. Grandjean, A. Recknagel, Supersymmetric Quantum Theoryand Differential Geometry, Commun. Math. Phys., 193, 527-594 (1998),hep-th/9612205; J. Frohlich, O. Grandjean, A. Recknagel, SupersymmetricQuantum Theory and Non-commutative Geometry, Commun. Math. Phys.,203, 119-184 (1999), mat-ph/9807006.

19S. Lord, Riemannian Geometries, math-ph/0010037.20M-T. Benameur, T. Fack, On von Neumann Spectral Triples,

math.KT/0012233.21M-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev,

K. Wojciechowski, An Analytic Approach to Spectral Flow in von NeumannAlgebras, math.OA/0512454.

22A. Carey, J. Phillips, A. Rennie, Semifinite Spectral Triples Associated withGraph C*-algebras, arXiv:0707.3853.






Riemannian Spectral Triples 1.Let (M, g) be a compact orientable m-dimensional Riemannianmanifold (not necessarily spinorial) and define by

I AM :=C∞(M; C) the algebra of smooth complex-valuedfunctions on the differentiable manifold M.

I HΛM the Hilbert space obtained completing the pre-Hilbert

spaceΓ∞(M; Λ(M)) = ⊕m

q=0Γ∞(M; Λq(M))

of smooth sections of the Grassmann bundle Λ(M) of M(i.e. the smooth differential forms) with respect to the innerproduct defined (on each Γ∞(M; Λq(M))) by

〈σ | τ〉 :=

∫Mσ(p) ∧ ∗τ(p) dµM , ∀σ, τ ∈ Λq(M).






Riemannian Spectral Triples 2.

I πM : AM → B(HΛM) the representation of the algebra AM

onto the Hilbert space HM obtained by unique linear extensionfrom the action by multiplication of AM on Γ∞(M; Λ(M)).

I DΛM the Dirac operator on forms i.e. the densely defined linear

map given by DΛM := d + ∗d∗ where

d : Γ∞(M; Λ(M))→ Γ∞(M; Λ(M))

is the exterior differential and∗ : Γ∞(M; Λ(M))→ Γ∞(M; Λ(M)) is the Hodge-dual ofdifferential forms.






Lord-Rennie Theorem.

In this context of Riemannian (non spinorial manifolds) we have:

TheoremFor any compact orientable m-dimensional Riemannian manifold(M, g), the data (AM ,HM ,DM) give an Abelian spectral triple.

S. Lord23 has proposed axioms for ”Riemannian spectral triples”and a reconstruction theorem based on Rennie’s original proof ofConnes’ reconstruction theorem. A detailed proof of this result isunder investigation24

23S. Lord, Riemannian Geometries, math-ph/0010037.24A. Rennie, personal communication, La Trobe, September 2007.





Morphisms in NCG.Categorification (Topological Level).Strict Higher C*-categories.Spectral Geometries over C*-categories.Categorical NCG - Non-commutative Topoi.

Categories in Non-commutative Geometry - Introduction.

[Non-commutative Geometry] ∩ [Category Theory]

||Categorical Non-commutative Geometry






Morphisms.

Having described A. Connes spectral triples and somehow justifiedthe fact that spectral triples are a possible definition for“non-commutative” compact finite dimensional orientableRiemannian spin manifolds, our next goal here is to discussdefinitions of “morphisms” between spectral triples and toconstruct categories of spectral triples that might support dualitieswith categories of manifolds.There are actually several possible notions of morphism, accordingto the amount of “background structure” of the manifold that wewould like to see preserved (and also depending on the kind oftopological properties that we would like to “attach” to ourmorphisms: orientation, spinoriality, . . . ).






Totally-Geodesic-Spin Morphisms 1.

In our first paper25, we proposed this notion of morphism: giventwo spectral triples (Aj ,Hj ,Dj), with j = 1, 2, a morphism ofspectral triples is a pair (φ,Φ), where φ : A1 → A2 is a∗-morphism between the pre-C*-algebras A1,A2 and Φ : H1 → H2

is a bounded linear map in B(H1,H2) that “intertwines” therepresentations π1, π2 φ and the Dirac operators D1,D2:

π2(φ(x)) Φ = Φ π1(x), ∀x ∈ A1,

D2 Φ(ξ) = Φ D1(ξ), ∀ξ ∈ Dom D1.

With this definition of morphism, isomorphisms are (unitary)equivalences of spectral triples.

25P.B., R. Conti, W. Lewkeeratiyutkul, A Category of Spectral Triples andDiscrete Groups with Length Function, Osaka J. Math., 43, n. 2, (2006).






Totally-Geodesic-Spin Morphisms 2.

This definition of morphism implies quite a strong relationshipbetween the spectra of the Dirac operators of the two spectraltriples. Loosely speaking, for φ epi and Φ coisometric (respectivelymono and isometric), in the commutative case, one should expectsuch definition to become relevant only for maps that “preservethe geodesic structures” (totally geodesic immersions andrespectively totally geodesic submersions)26.Furthermore these morphisms depend, at least in some sense, onthe spin structures: this “spinorial rigidity” (at least in the case ofmorphisms of real even spectral triples) require that suchmorphisms between spectral triples of different dimensions mightbe possible only when the difference in dimension is a multiple of 8.

26See P.B., R. Conti, W. Lewkeeratiyutkul, Non-commutative TotallyGeodesic Submanifolds and Quotient Manifolds, in preparation.






Metric Morphisms.Another notion of morphism that is essentially blind to the spinstructures has been proposed.27

Given two spectral triples (Aj ,Hj ,Dj), with j = 1, 2, denote by

dDj(ω1, ω2) := sup|ω1(x)− ω2(x)| | x ∈ A, ‖[Dj , π(x)]‖ ≤ 1

the quasi-distance induced on the sets P(Aj) of pure states of Aj .A metric morphism of spectral triples is a unital epimorphism28

φ : A1 → A2 of pre-C*-algebras whose pull-backφ• : P(A2)→ P(A1), φ•(ω) := ω φ is an isometry, i.e.

dD1(φ•(ω1), φ•(ω2)) = dD2(ω1, ω2), ∀ω1, ω2 ∈ P(A2).

27P.B., R. Conti, W. Lewkeeratiyutkul, A Remark on Gel’fand Duality forSpectral Triples, preprint (2005).

28Note that if φ is an epimorphism, its pull-back φ• maps pure states intopure states.






Riemannian Morphisms.Work is in progress29 on a weaker notion of morphisms30: giventwo spectral triples (Aj ,Hj ,Dj), with j = 1, 2, a Riemannianmorphism of spectral triples is a pair (φ,Φ) where φ : A1 → A2

is a ∗-morphism between the pre-C*-algebras A1,A2 andΦ : H1 → H2 is a bounded linear map in B(H1,H2) that“intertwines” the representations π1, π2 φ and the commutatorsof the Dirac operators D1,D2:

π2(φ(x)) Φ = Φ π1(x), ∀x ∈ A1,

[D2, φ(x)] Φ = Φ [D1, x ], ∀x ∈ A1.

29P.B., R. Conti, W. Lewkeeratiyutkul, Categories of Spectral Triples andMorita Equivalence, in progress.

30In this case, isomorphisms reduce to the unitary maps considered inM. Paschke-R. Verch, Local Covariant Quantum Field Theory over SpectralGeometries, gr-qc/0405057.






Lord-Rennie Duality.

Modulo Lord-Rennie reconstruction theorem for Riemannianspectral triples, we already have duality results31:

I the pull-back of functions φ := f • : C∞(N)→ C∞(M) andforms Φ = f • : L2(Λ(N))→ L2(Λ(M)) give a duality from thecategory of Riemannian immersions f : M → N to thecategory of spectral triples with morphisms (φ,Φ) such thatΦ[DN , x ]−Φ∗ = [DM , φ(x)]−, with x ∈ C∞(N);

I similarly, the pull back of functions and forms give a dualityfrom the category of Riemannian submersions f : N → M tothe category of spectral triples with morphisms (φ,Φ) suchthat Φ∗[DN , φ(y)]−Φ = [DM , y ]−, with y ∈ C∞(M).

31P.B., R. Conti, W. Lewkeeratiyutkul, Morphisms of Non-commutativeRiemannian Manifolds, in preparation.






Spectral Congruences.32

Let (Aj ,Hj ,Dj), for j = 1, 2, be two spectral triples.A spectral conguence is a triple (φ,Φ,DΦ) where:

I φ ⊂ A1 ×A2 is a conguence of involutive unital C*-algebrasi.e. a unital C*-subalgebra of A1 ⊕A2,

I Φ ⊂ H1 × H2 is a closed congruence of Hilbert spaces i.e. aclosed subpace of H1 ⊕ H2,

I DΦ is subspace of D1 ⊕ D2 ⊂ (H1 ⊕ H1)⊕ (H2 ⊕ H2),

such that Φ is stable under componentwise action of φ and DΦ

and (φ,Φ,DΦ) satisfies the additional axioms imposed on spectraltriples, for example, for every (x , y) ∈ φ, [DΦ, (x , y)]− extends to abounded operator on the Hilbert space Φ ⊂ H1 ⊕ H2.

32P.B., R. Conti, W. Lewkeeratiyutkul, Spectral Geometries overC*-categories and Morphisms of Spectral Geometries, in preparation.






Spectral Spans.33

Given two spectral triples (Aj ,Hj ,Dj), for j = 1, 2, a spectralspan is a spectral triple (A,H,D) equipped with a pair ofmorphisms of spectral triples (φ1,Φ1), (φ2,Φ2):

(A,H,D)(φ1,Φ1)

wwooooooooooo(φ2,Φ2)

''OOOOOOOOOOO

(A1,H1,D1) (A2,H2,D2)

Every spectral span is associated to a spectral congruence andsuch map is surjective (every spectral correspondence comes froma spectral span). Two spectral spans are equivalent if they inducethe same spectral congruence.







Categories of Spectral Congruences and Spans.34

Spectral congruences form a category under composition:

(ψ,Ψ,DΨ) (φ,Φ,DΦ) := (ψ φ,Ψ Φ,DΨ DΦ)

where all the compositions are compositions of relations and wherethe identity of the triple (A,H,D) is given by (ιA, ιH ,D ⊕ D).

This category can also be described as the category of equivalenceclasses of spectral spans.







General Morphisms.

The several notions of morphism of spectral triples described aboveare not as general as possible. In a wider perspective understudy35, a morphism of spectral triples (Aj ,Hj ,Dj), for j = 1, 2,might be formalized as a “suitable” functor F : A2

M → A1M ,

between the categories AjM of Aj -modules, having “appropriate

intertwining” properties with the Dirac operators Dj .Under some “mild” hypothesis, by Eilenberg-Gabriel-Watttheorem, any such functor is given by “tensorization” by abimodule. These bimodules, suitably equipped with spectral data(as in the case of spectral triples), provide, in our opinion, thenatural setting for a general theory of morphisms ofnon-commutative spaces. categorification







Morita Morphisms 1.

I Y. Manin36 has been the first to propose such “Moritamorphisms” (tensorizations with Hilbert C*-bimodules) as thenatural notion of morphism of non-commutative spaces.

I A. Connes37 already discussed how to transfer a given Diracoperator using Morita equivalence bimodules and compatibleconnections on them.

36Y. Manin, Real Multiplication and Noncommutative Geometry, in: TheLegacy of Niels Henrik Abel, Springer (2004), 685-727. math.AG/0202109.

37A. Connes, Gravity coupled with Matter and the Foundations ofNoncommutative Geometry, Commun. Math. Phys., 182, 155-176 (1996).






Morita Morphisms 2.

I In a forthcoming paper38, we define a strictly related categoryof Morita-Connes morphism of spectral triples (thatcontains “inner deformations” as isomorphisms) as a pair(X ,∇) where X is a A1-A2-bimodule that is aHilbert-C*-module over A2, ∇ is a Riemannian connection onthe bimodule X and the composition is given byX 3 := X 2 ⊗A1

X 1 with connection:

∇3(ξ1 ⊗ ξ2)(h3) := ξ1 ⊗ (∇2ξ2)(h3) + (∇1ξ1)(ξ2 ⊗ h3).







Morita Morphisms 3.

I In a remarkable paper, A. Connes-C. Consani-M. Marcolli39

have been pushing even further the notion of “Moritamorphism” defining morphisms between two algebras A,B as“homotopy classes” of bimodules in G. Kasparov KK -theoryKK (A,B). In this way, every morphism is an equivalence classdetermined by a bimodule that is further equipped withadditional structure (Fredholm module).

I In the same paper, A. Connes and collaborators provideground for considering “cyclic cohomology” as an “absolutecohomology of non-commutative motives” and the category ofmodules over the “cyclic category” (already defined byA. Connes-H. Moskovici) as a “NC motivic cohomology”.

39A. Connes, C. Consani, M. Marcolli, Noncommutative Geometry andMotives: the Thermodynamics of Endomotives, math.QA/0512138.






Morita Morphisms 4.

I A. Connes-M. Marcolli (in chapter 8.4 of their book40) andM. Marcolli-A. Zainy41 give a definition of “spectralcorrespondences” as Hilbert C*-bimodules providing a“bivariant version” of spectral triple.

40A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields andMotives, AMS, 2008.

41M. Marcolli, A. Zainy, Covering, Correspondences and NoncommutativeGeometry, preprint, 2007.






Morita Morphisms 5.

I S. Mahanta42 is trying to relate “spectral correspondences”with the “geometric morphisms” of derived categories of thedifferential graded categories already used in thenon-commutative algebraic geometry approach tonon-commutative spaces43.

42S. Mahanta, Noncommutative Correspondence Categories, work inprogress.

43See S. Mahanta, On Some Approaches to Non-commutative AlgebraicGeometry, arXiv:0501166 for a survey.






Other Approaches 1.

Work is in progress on several other variants of morphisms ofspectral triples, their mutual relations and their duality withgeometrical categories:

I Modifications of the notion of spin morphisms that satisfysome “graded intertwining relations”44 with the relevantoperators, according to sign rules (depending on thedimension of the triple modulo 8) as proposed by A. Sitarz45.

44P.B., R. Conti, W. Lewkeeratiyutkul, Morphism of Spectral Triples andSpin Manifolds, in progress.

45A. Sitarz, Habilitation Thesis, Jagellonian University (2002).Paolo Bertozzini Categories of Spectral Geometries.





Other Approaches 2.

I Morphisms of AF-algebras and categories for spectral triplesarising from AF-algebras46 with C. Antonescu-E. Christensenconstruction47.

46P.B., R. Conti, W. Lewkeeratiyutkul, Morphisms of Spectral Triples andAF-Algebras, in progress.

47C. Antonescu, E. Christensen, Spectral Triples for AF C* Algebras andMetrics on the Cantor Set, math.OA/0309044.






Categorification 1.

Categorification (L. Crane-D. Yetter48, J. Baez-J. Dolan49) denotesthe generic process in which ordinary algebraic set theoreticstructures are replaced with categorical counterparts.Vertical categorification is performed by promoting sets tocategories, functions to functors, . . . , hence replacing a categorywith a bi-category and so on, increasing the “depth” of morphisms.A kind of vertical categorification is a compulsory step in NCG:spaces are defined “dually” by “spectra” i.e. categories ofrepresentations of their algebras of functions and morphisms ofnon-commutative spaces are particular functors between “spectra”.

48L. Crane, D. Yetter, Examples of Categorification, Cahiers de Topologie etGeometrie Differentielle Categoriques, 39 n. 1, 3-25 (1998).

49J. Baez, J. Dolan, Categorification, in: Higher Category Theory,eds. E. Getzler, M. Kapranov, Contemp. Math., 230, 1-36 (1998),math/9802029.






Categorification 2.

In horizontal categorification50, ordinary algebraic associativestructures are interpreted as categories with only one object andsuitable analog categories with more than one object are defined.In this case the passage is from endomorphisms of a single objectto morphisms between different objects:

Monoids Small Categories (Monoidoids)

Groups Groupoids

Associative Unital Rings Ringoid

Associative Unital Algebras Algebroids

Unital C*-algebras C*-categories (C*-algebroids)

50We use here the term “horizontal” in order to stress the difference from theproper “vertical categorification” process in which n-categories are substitutedwith n + 1-categories. J. Baez prefers to use the term oidization for this case.






Horizontal Categorification of Gel’fand Duality 1.It is our purpose here to find:

I suitable horizontal categorifications T of T (1) and A ofA (1);

I to extend the categorical duality (Γ(1),Σ(1)) between T (1)

and A (1) of Gel’fand Theorem, to a natural categoricalequivalence between T and A :

T (1) _

Γ(1)/ A (1)

Σ(1)o

_

T

Γ / A .Σ

o

Since A (1) is a full subcategory of the category of C*-algebras, weexpect to identify the horizontal categorification of A (1) as asubcategory of a category of small C*-categories.






Horizontal Categorification of Gel’fand Duality 2.In our forthcoming paper51, in the setting of C*-categories, weprovide a definition of “spectrum” of a commutative fullC*-category as a one dimensional saturated unital Fell-bundle overa suitable groupoid (equivalence relation) and we prove acategorical Gel’fand duality theorem generalizing the usual Gel’fandduality between the categories of Abelian C*-algebras and compactHausdorff spaces. categorical gelfand theorem

As a byproduct, we also obtain a spectral theorem for imprimitivitybimodules over Abelian C*-algebras: every such bimodule isobtained by “twisting” (by the 2 projection homeomorphisms) thesymmetric bimodule of sections of a unique Hermitian line bundleover the graph of a unique homeomorphism between the spectra ofthe two C*-algebras. spectral theorem for bimodules generalizations gravity

51P.B., R. Conti, W. Lewkeeratiyutkul, Horizontal Categorification ofGel’fand’s Theory, preprint (2007).






C*-categories 1.

A C*-category52,53 is a category C such that:

I the sets CAB := HomC(B,A) are complex Banach spaces,

I the compositions are bilinear maps,

I the norm satisfies:

‖xy‖ ≤ ‖x‖ · ‖y‖, ∀x ∈ CAB , ∀y ∈ CBC ,

‖ιA‖ = 1, ∀A ∈ ObC,

52P. Ghez, R. Lima, J. Roberts, W∗-categories, Pacific J. Math., 120, 79-109(1985).

53P. Mitchener, C*-categories, Proceedings of the London MathematicalSociety, 84, 375-404 (2002).






C*-categories 2.

I there is an involution ∗ : HomC → HomC such that:

x∗ ∈ HomC(B,A), ∀x ∈ HomC(A,B),

(x + y)∗ = x∗ + y∗, ∀x , y ∈ CAB ,

(α · x)∗ = α · x∗, ∀α ∈ C, ∀x ∈ CAB ,

(xy)∗ = y∗x∗, ∀y ∈ CBC , ∀x ∈ CAB ,

(x∗)∗ = x , ∀x ∈ CAB ,

‖x∗x‖ = ‖x‖2, ∀x ∈ CBA,

x∗x ∈ CAA+, ∀x ∈ CBA.






C*-categories 3.

I In a C*-category C, the sets CAA := HomC(A,A) are unitalC*-algebras for all A ∈ ObC.

I The sets CAB := HomC(B,A) have a natural structure ofunital Hilbert C*-bimodule on the C*-algebras CAA on theright and CBB on the left.

A C*-category is commutative if the C*-algebras CAA are Abelianfor all A ∈ ObC.The C*-category C is full if all the bimodules CAB are full54.A standard example is the C*-category of linear bounded mapsbetween Hilbert spaces.

54In this case CAB are imprimitivity bimodules.Paolo Bertozzini Categories of Spectral Geometries.





Fell Bundles 1.

A Banach bundle is a C-vector bundle (E , p,X ), such that eachfiber Ex := p−1(x) is a Banach space for all x ∈ X in such a waythat the maps x 7→ ‖σ(x)‖ are continuous for every sectionσ ∈ Γ(X ; E ).If the topological space X is equipped with the algebraic structureof category (let X o be the set of its units, by r , s : X → X o itsrange and source maps and byX n := (x1, . . . , xn) ∈ ×n

j=1X | s(xj) = r(xj+1) its set ofn-composable morphisms), we further require that the composition : X 2 → X is a continuous map.If X is an involutive category i.e. there is a map ∗ : X → X withthe properties (x∗)∗ = x and (x y)∗ = y∗ x∗, for all (x , y) ∈ X 2,we also require ∗ to be continuos.






Fell Bundles 2.A Fell bundle55,56,57 over the (involutive) category X is a Banachbundle (E , p,X ) whose total space E is equipped with

i) a multiplication defined on the setE 2 := (e, f ) | (p(e), p(f )) ∈ X 2, denoted by (e, f ) 7→ ef ,that satisfies the following properties:

e(fg) = (ef )g , ∀(p(e), p(f ), p(g)) ∈ X 3,

p(ef ) = p(e) p(f ), ∀e, f ∈ E 2,

∀x , y ∈ X 2, the restriction of (e, f ) 7→ ef to Ex × Ey is bilinear,

‖ef ‖ ≤ ‖e‖ · ‖f ‖, ∀e, f ∈ E 2,

55J. Fell, R. Doran, Representations of C*-algebras, Locally Compact Groupsand Banach ∗-algebraic Bundles, Vol. 1, 2, Academic Press (1998).

56A. Kumjian, Fell Bundles over Groupoids, math.OA/9607230.57R. Martins, Double Fell Bundles and Spectral Triples, arXiv:0709.2972v2.






Fell Bundles 3.

ii) an involution ∗ : E → E that satisfies:

(e∗)∗ = e, ∀e ∈ E ,

p(e∗) = p(e)∗, ∀e ∈ E ,

∀x ∈ X , the restriction of e 7→ e∗ to Ex is conjugate linear,

iii) and moreover such that:

(ef )∗ = f ∗e∗, ∀e, f ∈ E 2,

‖e∗e‖ = ‖e‖2,∀e ∈ E ,

e∗e ≥ 0, ∀e ∈ E ,

where, in the last line we mean that e∗e is a positive elementin the C*-algebra Ex with x = p(e∗e)






Fell Bundles 4.

It is in fact easy to see that for every x ∈ X o , Ex is a C*-algebra.A Fell bundle (E , p,X ) is said to be unital if the C*-algebras Ex ,for x ∈ X o , are unital.Note that the fiber Ex has a natural structure of HilbertC*-bimodule over the C*-algebras Er(x) on the left and Es(x) onthe right.A Fell bundle is said to be saturated if the above HilbertC*-bimodules Ex are full.Note also that in a saturated Fell bundle, the Hilbert C*-bimodulesEx are imprimitivity bimodules.






Spaceoids

Let O be a set and X a compact Hausdorff topological space.We denote by

RO := (A,B) | A,B ∈ O

the “total” equivalence relation in O and by

∆X := (p, p) | p ∈ X

the “diagonal” equivalence relation in X .

DefinitionA topological spaceoid (E, π,X) is a saturated unital rank-oneFell bundle over the product involutive topological categoryX := ∆X × RO.

categorical gelfand theorem






Morphisms of Spaceoids.

Let (Ej , πj ,Xj), for j = 1, 2, be two spaceoids58

DefinitionA morphism of spaceoids (E1, π1,X1)

(f ,F)−−−→ (E2, π2,X2) is a pair(f ,F) where

I f := (f∆, fR) with f∆ : ∆1 → ∆2 a continuous map oftopological spaces and fR : R1 → R2 an isomorphism ofequivalence relations;

I F : f •(E2)→ E1 is a fiberwise linear ∗-functor such thatπ1 F = (π2)f , where (f •(E2), πf

2 ,X1) denotes an f -pull-backof (E2, π2,X2).

58Where Xj = ∆Xj × ROj , with Oj sets and Xj compact Hausdorfftopological spaces for j = 1, 2.






Category of Spaceoids 1.

Topological spaceoids constitute a category if composition isdefined by

(g ,G) (f ,F) := (g f ,F f •(G))

with identities given by

ι(E, π,X) := (ιX, ιE).

Note that f •(g•(E3)) is naturally a (g f )-pull-back of(E3, π3,X3) and that (E, π,X) is a natural ιX-pull-back of itself.






Category of Spaceoids 2.

The category T (1) of continuous maps between compactHausdorff spaces can be naturally identified with the fullsubcategory of the category T of spaceoids with index set O

containing a single element.To every object X ∈ ObT (1) we associate the trivial C-line bundleXX × C over the involutive category XX := ∆X × ROX

withOX := X the one point set.To every continuous map f : X → Y in T (1) we associate themorphism (g ,G) with g∆(p, p) := (f (p), f (p)),gR : (X ,X ) 7→ (Y ,Y ) and G := ιXX×C.Note that the trivial bundle over XX is naturally a f -bull-back ofthe trivial bundle over XY hence G can be taken as the identitymap.






The Category of Small C*-categories.

Let C and D be two full commutative small C*-categories (withthe same cardinality of the set of objects). Denote by Co and Do

their sets of identities.A morphism Φ : C→ D is an object bijective ∗-functor i.e. a mapsuch that

Φ(x + y) = Φ(x) + Φ(y), ∀x , y ∈ CAB ,

Φ(a · x) = a · Φ(x), ∀x ∈ C, ∀a ∈ C,Φ(x y) = Φ(x) Φ(y), ∀x ∈ CCB , y ∈ CBA

Φ(x∗) = Φ(x)∗, ∀x ∈ CAB ,

Φ(ι) ∈ Do , ∀ι ∈ Co ,

Φo := Φ|Co : Co → Do is bijective.






The Section Functor Γ on Objects.To every spaceoid (E, π,X), with X := ∆X ×RO, we can associatea full commutative C*-category Γ(E) as follows:

I ObΓ(E) := O;I ∀A,B ∈ ObΓ(E), HomΓ(E)(B,A) := Γ(∆X × (A,B); E),

where Γ(∆X × (A,B); E) denotes the set of continuoussections σ : ∆X × (A,B) → E, σ : pAB 7→ σAB

p ∈ EpABof

the restriction of E to the base space ∆X × (A,B) ⊂ X.I for all σ ∈ HomΓ(E)(A,B) and ρ ∈ HomΓ(E)(B,C ):

ρ σ : pAC 7→ (ρ σ)ACp := ρAB

p σBCp ,

σ∗ : pBA 7→ (σ∗)BAp := (σAB

p )∗,

‖σ‖ := supp∈∆X

‖σABp ‖E,

with operations taken in the total space E of the Fell bundle.






The Section Functor Γ on Morphisms.We extend now the definition of Γ to the morphism of T in orderto obtain a contravariant functor.Let (f ,F) be a morphism in T from (E1, π1,X1) to (E2, π2,X2).Given σ ∈ Γ(E2), by we consider the unique sectionf •(σ) : X1 → f •(E2) such that f π2 f •(σ) = σ f and thecomposition F f •(σ). In this way we get a map

Γ(f ,F) : Γ(E2)→ Γ(E1), Γ(f ,F) : σ 7→ F f •(σ), ∀σ ∈ Γ(E2).

Proposition

For any morphism (E1, π1,X1)(f ,F)−−−→ (E2, π2,X2) in T , the map

Γ(f ,F) : Γ(E2)→ Γ(E1) is a morphism in A .The pair of maps Γ : (E, π,X) 7→ Γ(E) and Γ : (f ,F) 7→ Γ(f ,F)

gives a contravariant functor from the category T of spaceoids tothe category A of small full commutative C*-categories.






The Spectrum functor Σ on Objects 1.We proceed to associate to every commutative full C*-category C

its spectral spaceoid Σ(C) := (EC, πC,XC).

I The set [C; C] of C-valued ∗-functors ω : C→ C, with theweakest topology making all evaluations continuous, is acompact Hausdorff topological space.

I By definition two ∗-functors ω1, ω2 ∈ [C; C] are unitarilyequivalent if there exists a “unitary” natural trasformationA 7→ νA ∈ T between them. This is true iff ω1|CAA

= ω2|CAA

for all A ∈ ObC.

I Let Spb(C) := [ω] | ω ∈ [C; C] denote the base spectrumof C, defined as the set of unitary equivalence classes of∗-functors in [C; C]. It is a compact Hausdorff space with thequotient topology induced by the map ω 7→ [ω].






The Spectrum functor Σ on Objects 2.I Let XC := ∆C × RC be the direct product of the compact

Hausdorff ∗-category ∆C := ∆Spb(C) and the topologically

discrete ∗-category RC := C/C ' RObC.

I For ω ∈ [C; C], the set Iω := x ∈ C | ω(x) = 0 is an ideal inC and Iω1 = Iω2 if [ω1] = [ω2].

I Denoting [ω]AB the point ([ω], (A,B)) ∈ XC, we define:

I[ω]AB:= Iω ∩ CAB , EC

[ω]AB:=

CAB

I[ω]AB

, EC :=⊎

[ω]AB∈XC

EC[ω]AB

.

Proposition

The map πC : EC → XC, that sends an element e ∈ EC[ω]AB

to the

point [ω]AB ∈ XC has a natural structure of unital saturated rankone Fell bundle over the topological involutive category XC.






The Spectrum functor Σ on Morphisms 1.

Let Φ : C→ D be an object-bijective ∗-functor between two smallcommutative full C*-categories with spaceoids Σ(C),Σ(D) ∈ T .

We define a morphism ΣΦ : Σ(D)(λΦ,ΛΦ)−−−−−→ Σ(C) in the category

T :

I λΦ : XD(λΦ

∆,λΦR)

−−−−−→ XC whereλΦ

R(A,B) := (Φ−1o (A),Φ−1

o (B)), for all (A,B) ∈ RObD;

λΦ∆([ω]) := [ω Φ] ∈ ∆Spb(C), for all [ω] ∈ ∆Spb(D).

I The bundle⊎

[ω]AB∈XD

CλΦ

R(AB)

IλΦ([ω]AB )

with the maps

πΦ : ([ω]AB , x + IλΦ([ω]AB)) 7→ [ω]AB ∈ XD, x ∈ CλΦR(AB),

Φπ : ([ω]AB , x +IλΦ([ω]AB)) 7→ (λΦ([ω]AB), x +IλΦ([ω]AB)) ∈ EC

is a λΦ-pull-back (λΦ)•(EC) of the Fell bundle (EC, πC,XC).






The Spectrum functor Σ on Morphisms 2.

I Since Φ(IλΦ([ω]AB)) ⊂ I[ω]ABfor [ω]AB ∈ XD, we define

ΛΦ : (λΦ)•(EC)→ ED by([ω]AB , x + IλΦ([ω]AB)

)7→(

[ω]AB , Φ(x) + I[ω]AB

).

Proposition

For any morphism CΦ−→ D in A , the map Σ(D)

ΣΦ

−−→ Σ(C) is amorphism of spectral spaceoids. The pair of maps Σ : C 7→ Σ(C)and Σ : Φ 7→ ΣΦ give a contravariant functor Σ : A → T , fromthe category A of object-bijective ∗-functors between smallcommutative full C*-categories to the category T of spaceoids.






Gel’fand Duality Theorem for C*-categories.There exists a duality (Γ,Σ) between the category T ofobject-bijective morphisms between spaceoids and the category Aof object-bijective ∗-functors between small commutative fullC*-categories, where

I Γ is the functor that to every spaceoid (E, π,X) ∈ ObT

associates the small commutative full C*-category Γ(E) andthat to every morphism between spaceoids(f ,F) : (E1, π1,X1)→ (E2, π2,X2) associates the ∗-functorΓ(f ,F);

I Σ is the functor that to every small commutative fullC*-category C associates its spectral spaceoid Σ(C) and thatto every object-bijective ∗-functor Φ : C→ D of C*-categoriesin A associates the morphism ΣΦ : Σ(D)→ Σ(C) betweenspaceoids. C*-categories spectral theorem for bimodules categorification






The Gel’fand Natural Transform G.

I The natural isomorphism G : IA → Γ Σ is provided by thehorizontally categorified Gel’fand transformsGC : C→ Γ(Σ(C)) defined by

GC : C→ Γ(EC), GC : x 7→ x where

xAB[ω] := x + I[ω]AB

, ∀x ∈ CAB .

In particular:

Proposition

The functor Γ : T → A is representative i.e. given a commutativefull C*-category C, the Gel’fand transform GC : C→ Γ(Σ(C)) is afull isometric (hence faithful) ∗-functor.






The Evaluation Natural Transform E.

I The natural isomorphism E : IT → Σ Γ is provided by thehorizontally categorified “evaluation” transforms

EE : (E, π,X)(ηE,ΩE)−−−−−→ Σ(Γ(E)), defined as follows:

I ηER(A,B) := (A,B), ∀(A,B) ∈ RO.

I ηE∆ : p 7→ [γ evp] ∀p ∈ ∆X , where the evaluation map

evp : Γ(E)→ ](AB)∈RO EpABgiven by evp : σ 7→ σAB

p is a∗-functor with values in a one dimensional C*-category thatdetermines59 a unique point [γ evp] ∈ ∆Spb(Γ(E)).

I⊎

pAB∈X Γ(E)ηER(AB)/IηE(pAB ) equipped with the natural

projection (pAB , σ + IηE(pAB )) 7→ pAB , and with the

EΓ(E)-valued map (pAB , σ + IηE(pAB )) 7→ σ + IηE(pAB ), is a

ηE-pull-back (ηE)•(EΓ(E)) of Σ(Γ(E)).59There is always a C valued ∗-functor γ : ](AB)∈RO EpAB → C and any two

compositions of evp with such ∗-functors are unitarily equivalent because theycoincide on the diagonal C*-algebras EpAA .






The Evaluation Natural Transform E.

I ΩE : (ηE)•(EΓ(E))→ E is defined byΩE : (pAB , σ + IηE(pAB )) 7→ σAB

p , ∀σ ∈ Γ(E)AB , pAB ∈ X.

In particular, with such definitions we can prove:

Proposition

The functor Σ : A → T is representative i.e. given a spaceoid(E, π,X), the evaluation transform EE : (E, π,X)→ Σ(Γ(E)) is anisomorphism in the category of spaceoids.






Spectral Theorem for Imprimitivity Abelian C*-bimodules.

Theorem (P.B.-R. Conti-W. Lewkeeratiyutkul)

Given an imprimitivity Hilbert C*-bimodule AMB over the Abelianunital C*-algebras A,B, there exists a canonical homeomorphism60

RBA : Sp(A)→ Sp(B) and a Hermitian line bundle E over RBA

such that AMB is isomorphic to the (left/right) “twisting”61 of thesymmetric bimodule Γ(RBA; E )C(RBA;C) of sections of the bundle Eby the two “pull-back” isomorphisms π•A : A→ C (RBA; C),π•B : B→ C (RBA; C). categorification generalizations

60RBA is a compact Hausdorff subspace of Sp(A)× Sp(B) homeomorphic toSp(A) via the projection πA : RBA → Sp(A) and to Sp(B) via the projectionπB : RBA → Sp(B).

61If M is a left module over C and φ : A→ C is an isomorphism, the lefttwisting of M by φ is the module over A defined by a · x := φ(a)x for a ∈ A

and x ∈ M.Paolo Bertozzini Categories of Spectral Geometries.





Generalizations and Applications of Gel’fand Duality 1.We are now working on a number of generalizations of ourhorizontal categorified Gel’fand duality:

I Gel’fand duality for general ∗-functors and ∗-relators.

I Gel’fand duality for non-full C*-categories.

I Categorification of Dauns-Hofmann spectral theorem anddualities for non-commutative C*-categories or more generallyhigher rank Fell bundles.

I Gel’fand dualities for commutative higher C*-categories and“higher-spaceoids”.62

I Spectral triples over C*-categories and horizontalcategorification of spectral triples and other spectralgeometries.

62Very interesting is the possible relation between such “higher” spectra andthe notions of stacks and gerbes already used in higher gauge theory.






Generalizations and Applications of Gel’fand Duality 2.

Extremely intriguing for its possible physical implications is theappearence of a natural local gauge structure on the spectra: thespectrum is no more just a (topological) space, but a special fiberbundle.Every isomorphism class of a full commutative C*-category can beidentified with an equivalence relation in the Picard-Morita1-category of Abelian unital C*-algebras. In practice a C*-categoryis just a “strict implementation” of an equivalence relationsubcategory of Picard-Morita. Since morphism of spectral triplesare essentially “special cases” of Morita morphisms, we are nowtrying to develop a notion of horizontal categorification of spectraltriples (and of other spectral geometries) in order to identify acorrect definition of morphism of spectral triples that supports aduality with a suitable spectrum (in the commutative case).






Strict Higher C*-categories 1.63

Given n ∈ N, a globular n-set

C0 ⇔ C1 ⇔ · · ·Cm−1 ⇔ Cm ⇔ · · ·⇔ Cn,

is given by:

I for all m = 0, . . . , n, a collections of classes Cm whoseelements are called m-arrows,

I for all m = 1, . . . , n, a pair of source, target mapssm, tm : Cm → Cm−1 such that for all m = 1, . . . , n − 1:

sm sm+1 = sm tm+1

tm sm+1 = tm tm+1.

63P.B., Roberto Conti, Wicharn Lewkeeratiyutkul, NoppakhunSuthichitranont, Strict Higher C*-categories, in preparation.






Strict Higher C*-categories 2.

A (globular) strict n-category is a globular n-set such that

I for all 0 ≤ p < m ≤ n, there is a partial p-composition map

mp : Cm ×Cp Cm → Cm, (x , y) 7→ x mp y ,

defined on the set Cm ×Cp Cm of p-composable m-arrows(x , y) ∈ Cm ×Cp Cm ⇔ tp+1 · · · tm(y) = sp+1 · · · sm(x),

I for all m = 0, . . . , n − 1 there is an identity map

ιm : Cm → Cm+1,

in such a way that the following axioms are satisfied:







I for all m = 0, . . . , n, for all p = 0, . . . ,m − 1,for all (x , y) ∈ Cm ×Cp Cm,

sm(x mp y) = sm(y), tm(x mp y) = tm(x), if p = m − 1;

sm(x mp y) = sm(x) m−1p sm(y), if p = 0, . . . ,m − 2,

tm(x mp y) = tm(x) m−1p tm(y), if p = 0, . . . ,m − 2;

I for all x ∈ Cm,

sm+1(ιm(x)) = x , tm+1(ιm(x)) = x ;

I for all m = 1, . . . , n and p = 0, . . . ,m − 1 and for allx , y , z ∈ Cm with (x , y), (y , z) ∈ Cm ×Cp Cm,

(x mp y) mp z = x mp (y mp z);






Strict Higher C*-categories 4.I for all m = 1, . . . , n, for all p = 0, . . . ,m − 1,

for all x ∈ Cm,(ιm−1 · · · ιp

(tp+1 · · · tm(x)

))mp x = x ,

x = x mp(ιm−1 · · · ιp

(sp+1 · · · sm(x)

));

I for all m = 2, . . . , n, for all p, q = 0, . . . ,m − 1, with q < p,for all w , x , y , z ∈ Cm such that (w , x), (y , z) ∈ Cm ×Cp Cm

and (w , y), (x , z) ∈ Cm ×Cq Cm,

(w mp x) mq (y mp z) = (w mq y) mp (x mq z);

I for all m = 1, . . . , n − 1, for all p = 0, . . . ,m − 1,for all (x , y) ∈ Cm ×Cp Cm,

ιm(x mp y) = ιm(x) m+1p ιm(y).







A (globular) strict involutive n-category is a strict n-categorythat is equipped with a family of “involutions” ∗m : Cm → Cm, for0 < m ≤ n, that satisfy the following properties:

I for all x ∈ Cm, sm(x∗m

) = tm(x), tm(x∗m

) = sm(x),

I for all x , y ∈ Cm ×Cp Cm with m = 1, . . . , n,

(x mp y)∗m

= y∗m mp x∗

m, for 0 ≤ p = m − 1,

(x mp y)∗m

= x∗m mp y∗

m, for 0 ≤ p < m − 1,

I for all x ∈ Cm, (x∗m

)∗m

= x .

For m = 0 we do not require an involution, alternatively we canassume the property of “hermitianity of zero arrows” i.e.

I for all x ∈ C0, x∗0

= x .







It is also possible to require involutions ∗mq : Cm → Cm of depth qfor 0 ≤ q < m ≤ n with the properties:

I for all x ∈ Cm, sm(x∗mm−1) = tm(x), tm(x∗

mm−1) = sm(x),

and, for q < m − 1,

sm(x∗mq ) = (sm(x))∗

m−1q , tm(x∗

mq ) = (tm(x))∗

m−1q ;

I for all x , y ∈ Cm ×Cp Cm with m = 1, . . . , n andp = 0, . . . ,m − 1,

(x mp y)∗mq = y∗

mq mp x∗

mq , if q = m − 1,

(x mp y)∗mq = x∗

mq mp y∗

mq , if q 6= m − 1;

I for all x ∈ Cm, (x∗mq )∗

mq = x .







As “very tentative” proposal, we define a strict-n-C*-category tobe a strict involutive n-category such that

I for all m = 1, . . . , n, for all x , y ∈ Cm−1, the setsCm(x , y) := z ∈ Cm | sm(z) = y , tm(z) = xare Banach spaces with norm denoted by x 7→ ‖x‖m.

I for 0 ≤ p < m, for all w , x , y , z ∈ Cm−1 such thatCm(w , x)× Cm(y , z) ⊂ Cm ×Cp Cm, the mapsmp : Cm(w , x)× Cm(y , z)→ Cm are “bilinear”,

I for all m = 1, . . . , n, for all x , y ∈ Cm−1, the maps∗m : Cm(x , y)→ Cm are conjugate linear;







I for all m = 1, . . . , n, for all p = 0, . . . ,m − 1, for all pairs(x , y) ∈ Cm ×Cp Cm,

‖x mp y‖m ≤ ‖x‖m · ‖y‖m;

I for all m = 1, . . . , n and 0 ≤ p < m, for all(x∗

m, x) ∈ Cm ×Cp Cm,

‖x∗m mp x‖m = ‖x‖2m, (4.1)







Note that the above properties already imply that, for allm = 1, . . . , n and for all x ∈ Cm−1, the set Cm(x , x) is aC*-algebra with multiplication mm−1 and involution ∗m and hencethe following final condition is meaningful:

I for all m = 1, . . . , n, for all x ∈ Cm(u, v),

x∗m mm−1 x ∈ Cm(u, u)+

i.e. x∗m mp x is a positive element in the C*-algebra Cm(u, u).







A left module CM over the n-category C is given by

C0 C1stoo · · ·

stoo Cn−1

stoo Cn

stoo

M1

τ

aaBBBBBBBB

M2

τ

aaCCCCCCCC· · · Mn

τ

bbEEEEEEEE

where for all m = 0, . . . , n, τ : Mm → Cm−1 is a fibered categoryover the (m − 1)-category Cm−1 and, for all 0 ≤ p < m ≤ n, thereis a left action µm

p : Cm ×Mm →Mm of the bi-fibered(m− 1)-category Cm ⇒ Cm−1 × Cm−1 over Mm → Cm−1 such thatµm

p (Cm(x , y)×Mm(z) ⊂Mm(x)) whenever(y , z) ∈ Cm−1 ×Cp Cm−1.64

64For p = m − 1 we assume Cm−1 ×Cp Cm−1 = ∆Cm−1 .Paolo Bertozzini Categories of Spectral Geometries.






Similar definitions can be given for right modules MC andbimodules CMC over the n-C*-category C.

The notion of left Hilbert C*-module CM over a strictn-C*-category C can be given imposing that for all m = 1, . . . n,τ : Mm → Cm−1 is a “Fell bundle”(for all the compositions andinvolutions in Cm−1) equipped with an inner product〈· | ·〉m : Mm ×Mm → Cm such that〈Mm(x) |Mm(y)〉m ⊂ Cm(y , x).

Similar notions can be given for right Hilbert C*-modules andright/left bimodules over a strict n-C*-category.65.

65It is necessary to distiguish right and left structures also for bimodules.Paolo Bertozzini Categories of Spectral Geometries.





Strict Higher C*-categories 12.Examples of rank-one strict-n-C*-categoriesi.e. strict-n-C*-categories such that for every m = 1, . . . , n, theBanach space Cm(x , y) is one-dimensional, can be manuallyconstructed by recursion.In the theory of higher C*-categories they play the role of C.Rank-one Hilbert C*-modules play the role of n-Hilbert spaces.Examples of non-commutative strict-n-C*-categories can beconstructed via the following:

TheoremEvery left Hilbert module CM over the strict-n-C*-category C

determines a strict-n-C*-category B(CM) of fiberwise adjointablemaps. The strict n-C*-category C is represented (i.e. there is strict∗-n-functor) into B(CM) via the left covariant actionsµ : C×M→M.






Spectral Geometries over C*-categories 1.

In a forthcoming paper66 we give the following extremely tentativedefinitions and we will also examine their relation with the notionsof morphisms of spectral geometries already presented.

As always, in “horizontal categorification”, modules over acategory are indexed by the objects of the category, whetherbimodules over a category are indexed by pairs of objects of thecategory and hence we have to distinguish carefully between“monovariant” and several “bivariant” versions of the axioms.







Spectral Geometries over C*-categories 2.A categorical spectral geometry (C,H,D) is given by:

I a pre-C*-category C;

I a module H over C that is also a Hilbert C*-module over C;in other terms a family of Hilbert spaces H equipped with anobject bijective ∗-functor π : C→ B(H) with values in theC*-category of bounded linear maps between the Hilbertspaces in the family H;

I the generator D of a unitary one-parameter group on H

(i.e. the generator of a one-parameter group whose adjointaction in the eveloping C*-algebra of B(H) leaves B(H)invariant) such that, for all x ∈ C,

[D, π(x)]− is extendable to an operator in B(H).






Spectral Geometries over C*-categories 3.A bivariant spectral geometry over two pre-C*-categories (withthe same objects) A and B is given by a quintuple(A,B,H,DA,DB), where

I H is a bimodule over A-B that is also a Hilbert C*-bimoduleover C and hence it is equipped with two ∗-representationsρ : A→ Bρ(H) and λ : B→ Bλ(H) into the right, andrespectively left, C*-category of the bimodule;

I DA (acting on the left) and DB (acting on the right) are two(generally unbounded) self-adjoint operators on H thatgenerate on the enveloping algebras of Bρ(H), andrespectively of Bλ(H), one-parameter groups leaving Bρ(H),and respectively Bλ(H), invariant and such that [DA, ρ(x)]−and [DB, λ(y)]− are extendable to bounded operators inBρ(H), Bλ(H), for all x ∈ A and y ∈ B.






Spectral Congruences and Bivariant Spectral Geometries.

As we already conjectured, the notion of morphism of spectralgeometries is strictly related to that of bivariant spectralgeometry.67

Every spectral congruence (φ,Φ,DΦ) between two spectralgeometries (Aj ,Hj ,Dj), j =, 2, is naturally associated to abivariant spectral geometry given by:

(A1,A2,Φ⊗ Φ,D1 ⊗ I , I ⊗ D2).







Categorical Non-commutative Geometry and Topoi 1.

One of the main goals of our investigation is to discuss theinterplay between ideas of categorification and NCG. Here we canpresent only a few suggestions. Work is in progress.

I Following higher category and multicategory theory68,69, wewould like to define “vertical categorifications” for spectraltriples or multicategories of non-commutative spaces70.

68J. Baez, An Introduction to n-Categories, in: 7th Conference on CategoryTheory and Computer Science, eds. E. Moggi, G. Rosolini, Lecture Notes inComputer Science, 1290, 1-33, Springer (1997), q-alg/9705009,

69T. Leinster, Higher Operads, Higher Categories, Cambridge (2004).70P.B. C* Polycategories, in progress.






Categorical Non-commutative Geometry and Topoi 2.

I Extremely intriguing is the possible relation between thenotions of (category of) spectral triples and A. Grothendiektopoi. Speculations in this direction have been given byP. Cartier71 and are also discussed by A. Connes72.

A full (categorical) notion of non-commutative space (NC-Kleinprogram / NC-Grothendiek topos) is still waiting to be defined73.

quantum gravity

71P. Cartier, A Mad Day’s Work: from Grothendiek to Connes andKontsevich, The Evolution of Concepts of Space and Symmetry,Bull. Amer. Math. Soc., 38, n. 4, 389-408 (2001).

72A. Connes, A View of Mathematics, on-line on: www.alainconnes.org73P.B., R. Conti, W. Lewkeeratiyutkul, Non-commutative Klein-Cartan

Program, in progress.Paolo Bertozzini Categories of Spectral Geometries.




Categories in Physics.Categorical Covariance.Spectral Space-Time.Quantum Gravity via NCG.

A Panorama of Mathematical-Physics 1.

Finite Degrees of Freedom

Covariance Classical Physics Quantum Physics ?

Aristotle Aristotle Mechanics - ?

Galilei Classical Mechanics Quantum Mechanics ?

PoincareSpecialRelativisticMechanics

Special RelativisticQuantumMechanics*

?

EinsteinGeneralRelativisticMechanics

Generally CovariantQuantum Mechanics ?

?

? ? ? ?







Infinite Degrees of Freedom

Covariance Classical Physics Quantum Physics ?

Aristotle - - ?

GalileiClassical ContinuumStatistical Mechanics

Quantum StatisticalMechanics*

?

PoincareClassicalField Theory

Relativistic QuantumField Theory*

?

EinsteinGeneral RelativisticField Theory

Quantum Gravity ? ?

? ? ? ?







Some explanations of terms and notations in the tables are inorder:

Classical = Commutative,

Quantum = Non-commutative,

Aristotle (= rotation group), Galilei, Poincare, Einstein (=diffeomorphisms group) refer to the covariance groups of thetheory,

∗ means that, at present, we do NOT have a soundmathematical theory,

? means that we do not have any serious theory yet,

- means that the theory is missing for historical reasons.






Categories in Physics 1.

I S. Doplicher-J. Roberts’ theory of superselection sectors inalgebraic quantum field theory provides a generalTannaka-Kreın duality for compact groups, where the dual is aparticular monoidal W∗-category.74,75

I G. Segal76 and M. Atiyah77 in conformal/topological QFT.

74S. Doplicher, J. Roberts, A New Duality Theory for Compact Groups,Inventiones Mathematicae, 98 (1), 157-218 (1989).

75S. Doplicher, J. Roberts, Why there Is a Field Algebra with CompactGauge Group Describing the Superselection Structure in Particle Physics,Commun. Math. Phys, 131, 51-107 (1990).

76G. Segal, The Definition of Conformal Field Theory, in: Topology,Geometry and Quantum Field Theory, Cambridge University Press, 421-577,(2004).

77M. Atiyah, Topological Quantum Field Theories,Inst. Hautes Etudes Sci. Publ. Math., 68, 175-186 (1988).







I C. Isham-J. Butterfield and A. Doring-C. Isham78,79,80,81

suggest topoi as basic structures for the construction ofphysical theories in which ordinary set theoretic concepts(including real/complex numbers and classical two valuedlogic) are replaced by more general topos theoretic notions.

78See for example: C. Isham, J. Butterfield, Some Possible Roles for ToposTheory in Quantum Theory and Quantum Gravity, Found. Phys., 30,1707-1735 (2000), gr-qc/9910005.

79A. Doring-C. Isham, A Topos Foundation for Theories of Physics I-II-III-IV,quant-ph/0703060-62-64-66.

80A. Doring, C. Isham, ‘What is a Thing?’: Topos Theory in the Foundationsof Physics, arXiv:0803.0417v1.

81A. Doring, Topos Theory and ‘Neo-realist’ Quantum Theory,arXiv:0712.4003v1.







I Along similar lines, C. Heunen-N. Landsman-B. Spitters82

have recently introduced a topos theoretic basis for quantumtheory in the C*-algebraic approach.

I S. Abramsky-B. Coecke83,84 are developing a categoricalaxiomatic for quantum mechanics, via monoidal categories,with intriguing links to knot theory and computer science.85

82C. Heunen-N. Landsman-B. Spitters, A Topos for Algebraic QuantumTheory, arXiv:0709.4364v2.

83S. Abramsky-B. Coecke, A Categorical Semantic of Quantum Protocols,quant-ph/0402130.

84B. Coecke, Kindergarten Quantum Mechanics, quant-ph/0510072.85S. Abramsky, Temperly-Lieb Algebra: from Knot Theory to Logic and

Computation via Quantum Mechanics, 2007.Paolo Bertozzini Categories of Spectral Geometries.






I J. Baez86,87 advocates the usage of categorical methods(higher category theory, categorification) in quantum physicsand in quantum gravity as well as in connection with logic andcomputation. 88 A new field of “categorical quantum gravity”is emerging (see L. Crane89,90 for an intriguing overview).

86J. Baez, Higher-Dimensional Algebra and Planck-Scale Physics, in: PhysicsMeets Philosophy at the Planck Length, eds. C. Callender, N. Huggett,Cambridge University Press, 177-195 (2001). gr-qc/9902017.

87J. Baez, Quantum Quandaries: A Category Theoretic Perspective,quant-ph/0404040.

88J. Baez, M. Stay, Physics, Topology, Logic and Computation: a RosettaStone, http://math.ucr.edu/baez/rosetta.pdf.

89L. Crane, Categorical Geometry and the Mathematical Foundations ofQuantum Gravity, gr-qc/0602120.

90L. Crane, What is the Mathematical Structure of Quantum Spacetime,arXiv:0706.4452.






Covariance.

Covariance of physical theories has been always discussed in thelimited domain of groups acting on spaces:

I Aristoteles covariance: the group of rotations in R3.

I Galilei covariance: the ten parameters symmetry group of theNewton space-time generated by 3 space translations, 1 timetranslation, 3 rotations and 3 boosts.

I Poincare covariance: the symmetry group of the fourdimensional Minkowski space i.e. the semidirect product ofLorentz group with the group of translations in R4.

I Einstein covariance: the group of diffeomorphisms of fourdimensional Lorentzian manifolds.

Different observers are “related” through transformations in thegiven covariance group.






Categorical Covariance 1.

There is no deep physical or operational reason to think that onlygroups (or quantum groups) might be the right mathematicalstructure to capture the “translation” between different observersand actually, in our opinion, categories provide a much moresuitable environment in which also the concept of “partialtranslations” between observers can be described. Work is inprogress on these issues91.

91P.B., Hypercovariant Theories and Spectral Space-time.Paolo Bertozzini Categories of Spectral Geometries.






The substitution of groups with categories as the basic covariancestructure of theories should be a key ingredient in thereconstruction of physics from operationally founded principles ofinformation theory (see for example C. Rovelli92,A. Greenbaum93,94) and, in the context of quantum gravity, also inthe formalism of quantum casual histories95.

92C. Rovelli, Relational Quantum Mechanics, Int. J. Theor. Phys., 35, 1637(1996). quant-ph/9609002.

93A. Grinbaum, Elements of Information-Theoretic Derivation of theFormalism of Quantum Theory, International Journal of Quantum Information,1(3), 289-300 (2003), quant-ph/0306079.

94A. Grinbaum, The Significance of Information in Quantum Theory,Ph.D. Thesis, Ecole Polytechnique, Paris (2004), quant-ph/0410071.

95F. Markopoulou, New Directions in Background Independent QuantumGravity, gr-qc/0703097.







In the direction of the idea of categorical covariance, we mentionseveral new works by R. Brunetti-K. Fredenhagen-R. Verch96, andR. Brunetti-M. Porrmann-G. Ruzzi97 that following and idea ofJ. Dimock98 aim at a generalization ofH. Araki-R. Haag-D. Kastler algebraic quantum field theoryaxiomatic, that is suitable for an Einstein covariant background.

96R. Brunetti, K. Fredenhagen, R. Verch, The Generally Covariant LocalityPrinciple - A New Paradigm for Local Quantum Physics,Commun. Math. Phys., 237, 31-68 (2003), math-ph/0112041.

97R. Brunetti, M. Porrmann, G. Ruzzi, General Covariance in AlgebraicQuantum Field Theory, math-ph/0512059.

98J. Dimock, Algebras of Local Observables on a Manifold,Commun. Math. Phys., 77, 219 (1980); Dirac Quantum Fields on a Manifold,Trans. Amer. Math. Soc., 269, 133 (1982).







Similar ideas are also used in the non-commutative versions of theaxioms recently proposed by M. Paschke and R. Verch99.

In the framework of topos theoretic foundations for physics,C. Heunen-N. Landsman-B. Spitters have proposed a principle ofgeneral tovariance, 100 in which covariance is implemented via thecategory of geometric morphisms between topoi.

99M. Paschke, R. Verch, Local Covariant Quantum Field Theory overSpectral Geometries, gr-qc/0405057.

100C. Heunen, N. Landsman, B. Spitters, The Principle of General Tovariance,http://philsci-archive.pitt.edu/archive/00003931.






Non-commutative Space-Time 1.

There are 3 main reasons for the introduction of non-commutativespace-time in physics:

1) Quantum effects (Heisenberg uncertainty principle), coupledto the general relativistic effect of the stress-energy tensor onthe curvature of space-time (Einstein equation), entail that atvery small scales the space-time manifold structure might be“unphysical”. (B. Riemann, A. Einstein,S. Doplicher-K. Fredenhagen-J. Roberts101).

101S. Doplicher, K. Fredenhagen, J. Roberts, The Structure of Spacetime atthe Planck Scale and Quantum Fields, Commun. Math. Phys., 172, 187(1995), hep-th/0303037.






Non-commutative Space-Time 2.

2) Modification to the short scale structure of space-time mighthelp to resolve the problems of “ultraviolet divergences” inQFT (W. Heisenberg, H. Snyder102 and many others) and of“singularities” in General Relativity.

3) A. Connes’ view of the standard model in particle physics as a“classical” non-commutative geometry of space-time (withspectral triples).103

102H. Snyder, Quantized Spacetime, Phys. Rev., 71, 38-41 (1947).103A. Connes, Essay on Physics and Noncommutative Geometry, in: The

Interface of Mathematics and Particle Physics, ed. D. Quillen, Clarendon Press,(1990).






Spectral Space-Time 1.

By “spectral space-time” we mean the idea that space-time(commutative or not) has to be “reconstructed a posteriori”, in aspectral way, from other operationally defined degrees of freedom(geometrical or not). The origin of this “pregeometricalphilosophy” is not clear:

I Space-time as a “relational” a posteriori entity originate fromG.W. Leibnitz, G. Berkeley, E. Mach.







I Pregeometrical speculations date as back as Pythagoras, butin their modern form, they start with J.A. Wheeler’s“pregeometry”104,105 and “it from bit”106 proposals.

I R. Geroch 107 has been the first to suggest a “shift” fromspace-time to algebras of functions over it, in order to addressthe problems of singularities in general relativity.

104J.A. Wheeler, Pregeometry: Motivations and Prospects, in: QuantumTheory and Gravitation, ed. A. Marlov, Academic Press (1980).

105D. Meschini, M. Lehto, J. Piilonen, Geometry, Pregeometry and Beyond,Stud. Hist. Philos. Mod. Phys., 36, 435-464 (2005), gr-qc/0411053.

106J.A. Wheeler, It from Bit, Sakharov Memorial Lectures on Physics, vol. 2,Nova Science (1992).

107R. Geroch, Einstein Algebras, Commun. Math. Phys., 26, 271-275 (1972).Paolo Bertozzini Categories of Spectral Geometries.





Spectral Space-Time 3.I R. Feynman-F. Dyson proof of Maxwell equations, from

non-relativistic QM of a free particle, indicates that essentialinformation about the underlying space-time is alreadycontained in the algebra of observables of the system108.

I The “reconstruction” of (classical Minkowski) space-timefrom suitable states over the observable algebra in algebraicquantum field theory has been considered by S. Doplicher109,A. Ocneanu110, U. Bannier111.

108An argument recently revised and extended to non-commutativeconfiguration spaces byT. Kopf-M. Paschke arXiv:math-ph/0301040,0708.0388

109S. Doplicher, private conversation, Rome, April 1995.110As reported in: A. Jadczyk, Algebras Symmetries, Spaces, in: Quantum

Groups, H. D. Doebner, J. D. Hennig ed., Springer (1990).111U. Bannier, Intrinsic Algebraic Characterization of Space-Time Structure,

Int. J. Theor. Phys., 33, 1797-1809 (1994).Paolo Bertozzini Categories of Spectral Geometries.






I Extremely important rigorous results on the “reconstruction ofclassical Minkowski space-time” from the vacuum state inalgebraic quantum field theory, via Tomita-Takesaki modulartheory, have been obtained in the “geometric modular action”program by D. Buchholz-S. Summers112,113,114.

112D. Buchholz, O. Dreyer, M. Florig, S. Summers, Geometric ModularAction and Spacetime Symmetry Groups, Rev. Math. Phys., 12, 475-560(2000), math-ph/9805026.

113S. Summers, Yet More Ado About Nothing: The Remarkable RelativisticVacuum State, arXiv:0802.1854v1

114S. J. Summers, R. K. White, On Deriving Space-Time from QuantumObservables and States, hep-th/0304179.







I Tomita-Takesaki modular theory is also used in the “modularlocalization program” by R. Brunetti-D. Guido-R. Longo115.In this context a reconstruction of space-time has beenconjectured by N. Pinamonti116.

115R. Brunetti, D. Guido, R. Longo, Modular Localization and WignerParticles, arXiv:math-ph/0203021.

116N. Pinamonti, On Localization of Position Operators in Mobius covariantTheories, math-ph/0610070.







I That non-commutative geometry provides a suitableenvironment for the implementation of spectral reconstructionof space-time from states and observables in quantum physicshas been my research motivation since 1990 and it is still anopen work in progress117.

I The idea that space-time might be spectrally reconstructed,via non-commutative geometry, from Tomita-Takesakimodular theory applied to the algebra of physical observableswas elaborated in 1995 by myself and independently byR. Longo. Since then, this conjecture is the main subject andgoal of our investigation118.

117P.B., Hypercovariant Theories and Spectral Space-time (2001).118P.B., Modular Spectral Triples in Non-commutative Geometry and Physics,

Research Report, Thai Research Fund, (2005).Paolo Bertozzini Categories of Spectral Geometries.





Quantum Gravity via NCG 1.It is often claimed that NCG provides the right mathematics (akind of quantum version of Riemannian geometry) for amathematically sound theory of quantum gravity, 119,120.Among the available approaches to quantum gravity via NCG:

I J. Madore’s “derivation based approach”121;

I S. Majid’s “quantum group approach”122.119L. Smolin, Three Roads to Quantum Gravity, Weidenfeld & Nicolson

(2000).120P. Martinetti, What Kind of Noncommutative Geometry for Quantum

Gravity?, Mod. Phys. Lett., A20, 1315 (2005), gr-qc/0501022.121J. Madore, An Introduction to Non-commutative Geometry and its

Physical Applications, Cambridge University Press (1999).122S. Majid, Hopf Algebras for Physics at the Planck Scale, J. Classical and

Quantum Gravity, 5, 1587-1606 (1988). S. Majid, Algebraic Approach toQuantum Gravity I,II,III, arXiv:hep-th/0604130, arXiv:hep-th/0604182,http://philsci-archive.pitt.edu/archive/00003345.






Quantum Gravity via NCG 2.

Current applications of NCG to quantum gravity have been limitedto some example or to attempts to make use of its mathematicalframework “inside” some already established theories such as“strings” or “loops”. Among these, we mention:

I the interesting examples studied by C. Rovelli123 andF. Besard124;

I the applications to string theory in the work byA. Connes-M. Douglas-A. Schwarz125,

123C. Rovelli, Spectral Noncommutative Geometry and Quantization: aSimple Example, Phys. Rev. Lett., 83, 1079-1083 (1999), gr-qc/9904029.

124F. Besnard, Canonical Quantization and Spectral Action, a Nice Example,gr-qc/0702049.

125A. Connes, M. Douglas, A. Schwarz, Noncommutative Geometry andMatrix Theory: Compactification on Tori, hep-th/9711162.







I the links between loop quantum gravity (spin networks),quantum information and NCG described byF. Girelli-E. Livine126.

I the intriguing interrelations with loop quantum gravity in therecent works by J. Aastrup-J. Grimstrum-R. Nest127,128,129.

126F. Girelli, E. Livine, Reconstructing Quantum Geometry from QuantumInformation: Spin Networks as Harmonic Oscillators, Class. Quant. Grav., 22,3295-3314 (2005), gr-qc/0501075.

127J. Aastrup, J. Grimstrup, Spectral Triples of Holonomy Loops,arXiv:hep-th/0503246.

128J. Aastrup, J. Grimstrup, Intersecting Connes Noncommutative Geometrywith Quantum Gravity, hep-th/0601127.

129J. Aastrup, J. Grimstrup, R. Nest, On Spectral Triples in Quantum GravityI-II, arXiv:0802.1783v1, arXiv:0802.1784v1.







Unfortunately, with the only notable exception of two programspartially outlined in

I M. Paschke, An Essay on the Spectral Action and its Relationto Quantum Gravity, in: Quantum Gravity, MathematicalModels and Experimental Bounds, Birkauser (2007),

I A. Connes, M. Marcolli, Noncommutative Geometry QuantumFields and Motives, July 2007,

a foundational approach to quantum physics based on A. Connes’NCG has never been proposed.The obstacles are both technical and conceptual.






Modular Algebraic Quantum Gravity 1.

Our ongoing research project130 is developing a new approach tothe foundations of quantum physics technically based on algebraicquantum theory (operator algebras - AQFT) and A. Connes’ NCGwhose main objective is a “spectral” reconstruction ofnon-commutative space-time from Tomita-Takesaki modulartheory:

130P.B., R. Conti, W. Lewkeeratiyutkul, Algebraic Quantum Gravity, work inprogress.







In the specific:

I Building on our previous research on “modularspectral-triples”131, we make use of Tomita-Takesaki modulartheory of operator algebras to associate non-commutativegeometrical objects (only formally similar to A. Connes’spectral-triples) to suitable states over involutive normedalgebras (in the same direction we stress the important recentwork on semifinite spectral triples by A. Carey, J. Phillips,A. Rennie, F. Sukochev as reported in arXiv:0707.3853).

131P.B., Modular Spectral Triples in Non-commutative Geometry and Physics,Research Report, Thai Research Fund, (2005); P.B., R. Conti,W. Lewkeeratiyutkul, Modular Spectral Triples, in preparation.







I We are developing132 an “event interpretation” of theformalism of states and observables in algebraic quantumphysics that is in line with C. Isham’s “history projectionoperator theory” 133 and C. Rovelli’s “relational quantummechanics”134.

132P.B., Algebraic Formalism for Rovelli Quantum Theory, in preparation.133See for example C. Isham, Quantum Logic and the Histories Approach to

Quantum Theory, J. Math. Phys., 35, 2157-2185 (1994), gr-qc/9308006.134C. Rovelli, Relational Quantum Mechanics, arXiv:/quant-ph/9609002.







I Making contact with our current research project on“categorical non-commutative geometry” and with otherprojects in categorical quantum gravity135,136, we willgeneralize the diffeomorphism covariance group of generalrelativity in a categorical context and use it to “indentify” thedegrees of freedom related to the spatio-temporal structure ofthe physical system.

135J. Baez, Higher-Dimensional Algebra and Planck-Scale Physics,arXiv:gr-qc/9902017; J. Baez, Quantum Quandaries: a Category TheoreticPerspective, arXiv:quant-ph/0404040.

136L. Crane, Categorical Geometry and the Mathematical Foundations ofQuantum Gravity, gr-qc/0602120; L. Crane, What is the MathematicalStructure of Quantum Spacetime, arXiv:0706.4452.







I Using techniques from “decoherence/einselection”137,138,“emergence/noiseless subsystems”139,140, or the “cooling”procedure developed by A. Connes-M. Marcolli141, we will tryto extract from our spectrally defined “modular”non-commutative geometries, a macroscopic space-time forthe pair state/system and its classical “residue”.

137H.D. Zeh, Roots and Fruits of Decoherence, quant-ph/0512078.138W. Zurek, Decoherence and the Transition from Quantum to Classical, Los

Alamos Science 27 (2002).139T. Konopka, F. Markopoulou, Constrained Mechanics and Noiseless

Subsystems, gr-qc/0601028; F. Markopoulou, Towards Gravity form theQuantum, hep-th/0604120.

140O. Dreyer, Emergent Probabilities in Quantum Mechanics,quant-ph/0603202.; O. Dreyer, Emergent General Relativity, gr-qc/0604075.

141A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields andMotives, July 2007






Algebraic Quantum Gravity 9.

I Possible reproduction of quantum geometries already definedin the context of loop quantum gravity142 orS. Doplicher-J. Roberts-K. Fredenhagen models143 will beinvestigated.

I Important connections of these ideas to “quantuminformation theory” and “quantum computation” arecurrently under consideration144.

142C. Rovelli, Quantum Gravity, Cambridge University Press (2004).143S. Doplicher, K. Fredenhagen, J. Roberts, The Structure of Spacetime at

the Planck Scale and Quantum Fields, Commun. Math. Phys., 172, 187(1995), hep-th/0303037.

144P.B., Hypercovariant Theories and Spectral Space-time, unpublished(2001).






Reference

P.B., R. Conti, W. Lewkeeratiyutkul,Non-commutative Geometry, Categories and Quantum Physics,preprint (submitted to East-West Journal of Mathematics,prooceedings of the “International Conference in Mathematics andApplications”, Mahidol University, 15-17 August 2007),arXiv:0801.2826v1.






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