Non-Commutative Geometry, Categories and Quantum Physics

Non-Comm utativ e Geometry , Categorie s and Quan tum Ph ysics Paolo Bertozzini a ∗ , Ro be r to Conti b ∗ † , Wicharn Lewkeeratiyutkul b ∗ a Dep artment of Mathematics and Statistics, F aculty of Scienc e and T e chnolo gy Thammasat University, Bangkok 12121, Thailand e-mail: paolo. th@gm ail.com b Dep artment of Mathematics and Computer S cienc e F aculty of S cienc e, Chulalongkorn University, Bangkok 10330 , Thailand e-mail: conti@ sci.u nich.it e-mail: Wichar n.L@c hula.ac.th 08 Nov ember 2 0 09, revised: 26 December 2011 Abstract After an introduction to some basic issues in non-commutativ e geometry (Gel’fand duality , s p ectral triples), we presen t a “panora mic v iew” of the sta tus of our current researc h progra m on the use of categorica l metho ds in the setting of A. Connes’ non- comm utative geometry: morphisms/categories of sp ectral trip les, categoriﬁcation of Gel’fand d ualit y . W e conclude with a summary of the exp ected applications of “cat- egorical non-commutative geometry” to structural questions in relativistic q uantum physics: (hyp er)cov ariance, quantum space-time, (algebraic) q uantum gravit y . Keywords: Non - comm utative Geometry , Sp ectral T rip le, Category , Morphism, Quantum Physics, Space-Time. MSC-2000: 46L87, 46M15, 16D90, 18F99, 81R60, 81T05, 83C65. Con ten ts 1 In tro duction. 2 2 Categories. 3 2.1 Ob jects a nd Mo rphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 F unctors, Natural T ransfor mations, Dualities. . . . . . . . . . . . . . . . . . 4 3 Non-commutativ e Geometry (Ob jects). 5 3.1 Non-commutativ e T op olo gy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1 Gel’fand Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2 Serre-Swan and T ak ahashi Theorems. . . . . . . . . . . . . . . . . . 7 ∗ Pa rtially supported by the Thai Researc h F und: grant n. RSA4780022. † Current address: Dipartimento di Scienze, Univ ersit` a di Chieti-Pe scara “G. D’Annunzio”, Viale Pin- daro 42, I-65127 Pescara, Italy . 1 3.2 Non-commutativ e (Spin) Diﬀeren tial Geometry . . . . . . . . . . . . . . . . . 9 3.2.1 Connes Sp e ctral T riples . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Other Sp ectral Geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Categories in Non-Commutativ e Geometry . 15 4.1 Morphisms of Sp ectral T riples. . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1.1 T otally Geo desic Spin-Morphisms. . . . . . . . . . . . . . . . . . . . 15 4.1.2 Metric Morphis ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.1.3 Riemannian Mor phis ms . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.1.4 Morita Morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Categoriﬁca tion (T op ologica l Level). . . . . . . . . . . . . . . . . . . . . . . 20 4.2.1 Horizontal Categor iﬁcation o f Gel’fand Dualit y . . . . . . . . . . . . . 21 4.2.2 Higher C*-categ ories. . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Categoric a l NCG and T op oi . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5 Applications to Ph ysics. 30 5.1 Categorie s in Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Categoric a l Cov aria nce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3 Non-commutativ e Space- Time. . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.4 Spec tr al Space-Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.5 Quantum Gravit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.5.1 A. Connes’ Non-commutativ e Geometry and Gravit y . . . . . . . . . 39 5.5.2 A Prop os al for (Mo dular ) Alge br aic Quantum Gravit y . . . . . . . . . 41 1 In tro duction. The pu rp ose of this r eview pap er is to presen t the status of our resear ch w ork on ca tegorica l non-commutativ e ge o metry and to contextualize it providing a ppropriate references. The pap er is o rganized as follows. In section 2 w e introduce the basic elemen tary deﬁnitions ab out catego ries, functors, natura l tr ansformations and dualities just to ﬁx o ur notation. In section 3, w e ﬁrs t provide a review o f the basic dua lities (Gelf ’a nd, Serre-Swan a nd T ak ahashi) that co ns titute the ma in categorical motiv ation for non-commutativ e geometry and then we pass to intro duce the deﬁnition of A. Connes sp ectra l tr iple. In the ﬁrst part o f sec tio n 4, we give an ov erview of our prop osed deﬁnitions o f mor phisms betw een sp ectr al triples and ca tegories o f sp ectral tr iple s . In the second part of section 4 we sho w how to generalize Gel’fand duality to the setting o f commutativ e full C*-catego ries and we sug gest how to apply this insigh t to the purpo se of deﬁning “ biv a riant” sp ectral triples as a cor rect no tion of metric morphism. The last se c tio n 5, is mainly intended for an audience of mathematicians and tr ie s to ex- plain how catego rical and non- c ommut ative notions enter the co nt ext of quantum mathe- matical physics and how we hop e to see such notions emerge in a non-p erturbative treat- men t of quantum gravit y . The last part (section 5.5 .2) is m ore speculative and c o nt ains a short ov erview of our present res earch pr ogra m in quantum gr avit y based on T omita-T akesaki mo dular theo r y and categorica l non-co mm utative geometry . W e hav e tried to provide an extensive biliogra ph y (upda ted till Octob er 2009 and s upple- men ted by a few additional references in appendix) in or der to help to place our research in a broader lands c ap e a nd to sugg est as m uch a s p ossible future link s with interesting ideas 2 already dev elop ed. Of course missing references are s ole respo ns ability o f the igno rance of the authors, tha t ar e still try ing to lea rn their w ay through the ma ter ial. W e will b e grateful for any sugg estion to improv e the on- line version of the do cument . Notes and ac kno wledgm en ts The pa rtial r e search supp ort provided b y the Thai Re- search F und (grant n. RSA4780022 ) is kindly ackno wledged. The pap e r o riginates fro m notes prepa r ed in o ccasion of a ta lk at the “ In ternatio nal Confer e nc e on Analysis and its Applications” in Ch ulalongkorn University in May 200 6. Most of the results hav e been announced in the form of research se mina rs in Norway (Universit y o f Oslo ), in Australia (ANU in Can b erra , Macqua rie Universit y in Sydney , University of Queensland in Bris- bane, La T rob e University in Melbourne , Universit y of Newcastle) a nd in Italy (SISSA T rieste, Universit` a di Roma I I, Universit` a di Bologna and Politecnico di Mila no). One of the a uthors (P .B.) thanks Chulalongkorn Universit y fo r the weekly hospita lit y dur ing the last three years o f r e search work. Notes and ac kno wledgments for the revis ed version A preliminar version of the pa- per app eared in the pr o ceedings of the “International Conference on Mathematics a nd Its Applications” (ICMA-MU 2007) in Ma hidol Universit y in May 200 7 and w as subseq ue ntly published in a very shor tened form in the specia l volume 2007 of East W est Journal of Mathematics. The pr esent pa p er is the s econd (and ﬁnal) on- line v ersio n for the arXiv, upda ting and replacing the original s ubmission in January 2008 . It contains, apar t fro m correctio ns of several t yp os, sig niﬁcant impro vemen ts in several sections: the bibliography has b een up dated to Octob er 2009; section 5 on a pplica tions to physics ha s b een consid- erably expanded; r eferences to some imp or tant developmen ts (i.e. those by A. C o nnes on the re construction theorem and by B. Mes la nd on “ K K -morphisms” of sp ectra l triples) hav e b een a dded; a n a pp endix at the end o f the manuscript contains s e lected additional references app eared after Octob er 2 0 09. W e thank Pr of. S. J. Summers and Prof. W. Lawton for reading the original man uscript and suggesting v arious improv ement s. 2 Categories. Just for the pur p o s e to ﬁx our notation, w e recall some g eneral deﬁnitions on catego ry theory , for a full in tro duction to the sub ject the reader ca n consult S. MacLane [Mc] or M. Barr -C. W ells [BW]. 2.1 Ob jects and Morphism s. A category C consists of a) a clas s 1 of ob jects Ob C , b) for any tw o ob ject A, B ∈ Ob C a set of morphi s ms Hom C ( A, B ), c) for any three o b jects A, B , C ∈ Ob C a comp o sition map ◦ : Hom C ( B , C ) × Hom C ( A, B ) → Hom C ( A, C ) that satisﬁes the following prop erties for all mo rphisms f , g , h that ca n b e comp os ed: ( f ◦ g ) ◦ h = f ◦ ( g ◦ h ) , ∀ A ∈ Ob C , ∃ ι A ∈ Hom C ( A, A ) : ι A ◦ f = f , g ◦ ι A = g . 1 The famil y of ob jetcs can be a proper class. T he category is called smal l if the class of ob jects is actually a set. 3 A morphism f ∈ Hom C ( A, B ) is called an isomorphis m if there exists another morphism g ∈ Ho m C ( B , A ) such that f ◦ g = ι B and g ◦ f = ι A . 2.2 F unctors, Natural T ransformations, Du alities. Given tw o ca tegories C , D , a cov arian t functor F : C → D is a pa ir of maps F : Ob C → Ob D , F : A 7→ F A , ∀ A ∈ Ob C , F : Hom C → Hom D , F : x 7→ F ( x ) , ∀ x ∈ Hom C , such that x ∈ Hom C ( A, B ) implies F ( x ) ∈ Hom D ( F A , F B ) and suc h that, f or any tw o comp osable morphisms f , g and any ob ject A , F ( g ◦ f ) = F ( g ) ◦ F ( h ) , F ( ι A ) = ι F A . F or the deﬁnition of a con tra v arian t functor we re q uire F ( x ) ∈ Hom D ( F B , F A ), when- ever x ∈ Hom C ( A, B ). A natura l transformation η : F → G b etw een tw o functors F , G : C → D , is a map η : O b C → Hom D , η : A 7→ η A ∈ Hom D ( F A , G A ), such that the following diagram F A η A / / F ( x )   G A G ( x )   F B η B / / G B . is comm utative for all x ∈ Hom C ( A, B ), A, B ∈ Ob C . A natural transfo r mation η : F → G is a natural isomorphi s m (or natural equiv a lence) if η A is an isomorphism for all ob jects A ; in this case we s ay that the functors F and G are natur ally equiv alent. The functor F : C → D is • faithful if, for all A, B ∈ Ob C , its restr iction to the set Hom C ( A, B ) is injectiv e; • full if its restric tio n to Hom C ( A, B ) is surjective; • representa tive if for all X ∈ Ob D there exists A ∈ Ob C such th at F A is isomorphic to X in D . A duali t y (a contra v ariant e q uiv a lence) of tw o ca tegories C and D is a pair of cont rav a ri- ant functors Γ : C → D and Σ : D → C suc h that Γ ◦ Σ and Σ ◦ Γ a re naturally equiv a le nt to the resp ective identit y functors I D and I C . A dua lity is actually sp eciﬁed by tw o func- tors, but given any one of the tw o functors in the dual pair, the o ther one is unique up to natural iso morphism. A functor Γ is in a duality pair if and only if it is full, faithful a nd representative (see for example M. Barr-C. W ells [BW, Deﬁnition 3.4.2 ]). Catego ries that are in duality ar e cons ider ed “essentially” the sa me (mo dulo the r eversing of arrows). Some impo rtant examples of “geometrica l catego ries” i.e. categories whose ob jects are sets equipp e d with a suitable structure , whose mor phisms are “structure preserving maps” and with comp osition alwa ys given by the usual comp os ition of functions are : • sets and functions; • top ologica l space s and co n tinuous maps; 4 • diﬀerentiable manifolds and diﬀer ent iable maps; • Riemannian manifolds (or also metr ic spa ces) with global metric isometr ie s ; • Riemannian manifolds with Riemannian (totally geo desic) immersions/submersio ns; • orientable (Riemannian) n -dimensional manifolds with orien tation pr eserving maps . 2 # Proble m: we are not a ware of any deﬁnition in the liter ature of “s pin-preserving map” betw een spin-manifolds of diﬀerent dimension. In the case of manifolds with the same dimensio n, it is of co urse possible to say that a map preserves the spin- structure if ther e is a n isomorphism (usually no n-unique), b etw een the pull-back of the spin-bundle of the tar g et manifold and the spin-bundle on the s ource manifold, that “in tertwines” the c harge conjugation operator s. An ywa y , ev en in this case, since spin-bundles are not “natural bundles” on a manifold, there is no in trinsic notion of “pull-back” for spinor ﬁelds (unless we co nsider s o me spec ial clas ses o f manifolds such as K¨ ahler spin-manifolds of a g iven dimension 3 ). The correct solution of this problem (as in the case of “or ientation preserving” maps) consists o f equipping the morphisms (consider ed a s “rela tion submanifolds” of the Cartesian pro duct of the so urce and targ et (oriented) spin-manifolds) with their own additional “spin-str uctur e” (orientation). W ork o n this issue is in pro g ress 4 . Other examples of immediate interest for us include v ector bundles and bundl e maps , with comp os itio n of bundle maps and Hermitian v ector bundles and (co)isome tric bundle maps . F or example, note that K -theo r y is the study of some specia l functors from the categor y of vector bundles to the categor y of (Ab elian) groups. 3 Non-comm utativ e Geometry (Ob jects). F or an introduction to the sub ject w e refer the rea ders to the b o oks by A. Connes [C3], G. Landi [La n1], H. Fig ueroa-J . Gracia -Bondia-J . V arilly [FGV] (see also [V ar ]) and M. K halkhali [Kha]; for spe c tr al triples a nd their r elation to index theory we also sugges t A. Rennie’s lectures notes [Re4 ]. Non-commutativ e geometry , cre a ted b y A. Connes, is a powerful extens io n of the ideas of R. Decartes’ analytic geometry: to substitute “geometr ic al ob jects” with their Abelia n algebras o f functions; to “ tr anslate” the geometrical prope r ties o f spaces into algebraic prop erties of the asso ciated algebras 5 and to “ reconstruct” the original g eometric spaces as derived entities (the spectra of the algebras ), a technique that app e ared for the ﬁrst time in the work of I. Gel’fand o n Ab elian C*-a lgebras in 19 39. 6 2 Note that, in general, it has no i ntrinsic meaning to s a y that a map b et wee n manifolds of diﬀerent dimension preserv e (or r ev erse) the orienta tion: a map betw een oriented m anifolds, determines only a unique orientation for the normal bundle of the manifol d. 3 P . Bertozzini, R. Con ti, W. Lewkeerat iyutkul, Non-commu tative (T ota lly Ge o desic) Submanifolds and Quotien t Manifolds, in preparation. 4 P . Bertozzini, R. Con ti, W. Lewk eeratiyutkul, Categories of Spectral T riples and Morita Equiv alence, wo rk in progress. 5 A line of though t alr eady present in J. L. Koszul al gebraization of diﬀer ential geometry . 6 Although similar ideas, previously deve lop ed by D. Hilb ert, are we ll known and used als o i n P . Cartier- A. Grothendiec k’s deﬁnition of schemes in algebraic geometry . 5 Whenever such “co diﬁcations” of ge o metry in algebr a ic terms s till ma ke sense if the Abelia n condition is dropp ed, 7 we can simply work with non-comm utative algebras con- sidered as “duals” of “no n-commutativ e space s ”. The existence of dualities b etw een categ ories of “geometrica l spaces ” and categories “con- structed fro m Abelian alg ebras” is the s tarting p oint of any genera lization of geometry to the non-commutativ e s itua tion. Here are some exa mples . 3.1 Non-comm utativ e T op ology . 3.1.1 Gel’ fand Theorem. F or the details on oper ator alg ebras, the rea der may refer to R. Kadis on-J. Ringro se [KR], M. T akesaki [T] and B. Black adar [Bl]. A complex unital algebra A is a vector space ov er C with an a sso ciative unital bilinear multiplication. A is Ab elian (co mm utative) if ab = ba , for all a, b ∈ A . An in v olution on A is a conjugate linear map ∗ : A → A such that ( a ∗ ) ∗ = a and ( ab ) ∗ = b ∗ a ∗ , f or all a, b ∈ A . An involutiv e c omplex unital algebra is A ca lled a C*-algebra if A is a Ba na ch s pace with a norm a 7→ k a k such that k ab k ≤ k a k · k b k and k a ∗ a k = k a k 2 , for a ll a, b ∈ A . Notable examples are the a lgebras of contin uous complex v a lued functions C ( X ; C ) on a co mpact top olo g ical space with the “sup-norm” and the algebr as o f linea r bo unded op erator s B ( H ) on the Hilb e rt space H . Theorem 3 .1 (Gel’fand) . 8 Ther e exists a duality (Γ (1) , Σ (1) ) b etwe en t he c ate gory T (1) , of c ontinuous maps b etwe en c omp act Hausdorﬀ top olo gic al sp ac es, and the c ate gory A (1) , of unital homomorphisms of c ommutative u nital C*-algebr as. Γ (1) is the functor that asso cia tes to compact Hausdorﬀ top ologica l spa ces X ∈ Ob T (1) the unital commut ative C*-alg ebras C( X ; C ) of complex v alued contin uous functions on X (with p o int wise m ultiplication a nd conjugatio n and supremum-norm) and that to con- tin uous maps f : X → Y asso c ia tes the unital ∗ -homomo rphisms f • : C ( Y ; C ) → C ( X ; C ) given by the pull-back of co ntin uous C -v alued functions by f . Σ (1) is the functor that asso cia tes to every unital commut ative C* - algebra A its sp ectrum Sp( A ) := { ω | ω : A → C is a unital ∗ -homomorphism } (as a topolog ical space with the weak topolo gy induced by the ev a luation maps ω 7→ ω ( x ), fo r all x ∈ A ) and that to every unital ∗ -homomorphis m φ : A → B of algebras asso ciates the contin uous map φ • : Sp( B ) → Sp( A ) given by the pull-back under φ . The natural isomor phism G : I A (1) → Γ (1) ◦ Σ (1) is given by the Gel ’ fand transforms G A : A → C (Sp ( A )) deﬁned by G A : a 7→ ˆ a , where ˆ a : Sp ( A ) → C is the Gel’fand transform of a i.e. ˆ a : ω 7→ ω ( a ). The natur al is omorphism E : I T (1) → Σ (1) ◦ Γ (1) is g iven b y the e v aluation homeomor - phisms E X : X → Sp( C ( X )) deﬁned by E X : p 7→ ev p , where ev p : C ( X ) → C is the p -ev aluation i.e. ev p : f 7→ f ( p ). In view of this result, compact Hausdorﬀ spaces and Ab elian unital C*-algebra s are es- sentially the same thing and w e can free ly translate pro per ties of the geo metrical space in algebraic prop erties of its Ab elian algebra of functions. 9 In the spirit o f no n-commutativ e ge o metry , we can simply consider non-Ab elian unital C*-algebr as as “duals ” of “non- commutativ e compact Hausdor ﬀ top o lo gical spaces”. 7 Usually in the non-commutativ e case, there are sev eral inequiv alen t generalizations of the same con- dition f or Ab elian algebras. 8 See for example [ Bl, Theorems II.2.2.4, II.2.2.6] or [ La3 , Section 6] 9 F or p ossible extensions of Gel’f and theorem to Tyc honoﬀ spaces and lo cally con vex ∗ -algebras see M. Car r i´ on- ´ Alv arez [CA]. A Gel’fand duality theory f or ordered topological spaces has b een elab orated b y F. Besnard [Be2]. 6 3.1.2 Serre-Swan and T ak ahashi Theorems. A left pre-Hilb ert-C*-mo dule A M ov er the unita l C*-algebr a A (who se p ositive part is denoted by A + := { x ∗ x | x ∈ A } ) is a unital left mo dule M ov er the unital ring A that is equipp ed with an A -v a lued inner pr o duct M × M → A denoted b y ( x, y ) 7→ A h x | y i such that, for all x, y , z ∈ M and a ∈ A , h x + y | z i = h x | z i + h y | z i , h a · x | z i = a h x | z i , h y | x i = h x | y i ∗ , h x | x i ∈ A + , h x | x i = 0 A ⇒ x = 0 M . A similar deﬁnition of a rig h t pre-Hilb ert-C*-mo dule is given with multip lication by elemen ts of the algebra on t he right. A left Hilbert C* -mo dule A M is a left pre-Hilber t C*-mo dule tha t is complete in the norm deﬁned b y x 7→ p k A h x | x ik . 10 W e sa y that a left pr e - Hilber t C*-mo dule A M is full if span {h x | y i | x, y ∈ M } = A , where the closure is in the norm topo lo gy of the C*-algebr a A . A pre-Hi lb ert-C*-bi mo dul e A M B ov er the unital C*- algebras A , B , is a left pre-Hilb ert mo dule over A and a right pre- Hilber t C* -mo dule ov er B such that: ( a · x ) · b = a · ( x · b ) , ∀ a ∈ A , ∀ x ∈ M , ∀ b ∈ B . A full Hilber t C* -bimo dule is said to b e an im primitivity bimo dule or an equiv alence bimo dul e if: A h x | y i · z = x · h y | z i B , ∀ x, y , z ∈ M . A bimo dule A M A is called s ymmetric if ax = xa for a ll x ∈ M and a ∈ A . 11 A mo dule A M is free if it is iso mo rphic to a mo dule of the form ⊕ J A for some index set J . A mo dule A M is pro jectiv e if there exists a no ther mo dule A N such that M ⊕ N is a free mo dule. An “equiv alence result” s tr ictly related to Gel’fand theorem, is the following “Hermitian” version of Serre-Swan theorem (se e for example M. K aroubi [Ka, Theorem 6 .18] for the usual Serre-Swan equiv alence and, for its Her mitia n version, M. F rank [F r, Theorem 7.1], N. W eav er [W e2, Theo rem 9.1.6] and also H. Figuero a-J. Gracia -Bondia-J . V a rilly [FGV, Theorem 2.10 and page 68]) that pr ovides a “ sp ectral interpretation” of symmetric ﬁnite pro jective Hilb ert C*-bimo dules over a comm utative unital C*-algebr a a s Hermitian vector bundles ov er the sp ectrum of the algebr a. 12 Theorem 3.2 (Serr e-Swan) . L et X b e a c omp act H au s dorﬀ top olo gic al sp ac e. L et M C ( X ) b e the c ate gory of symmetric pr oje ctive ﬁn ite Hilb ert C*-bimo dules over the c ommu tative C*-algebr a C ( X ; C ) with C ( X ; C ) -bimo dule morphisms. L et E X b e t he c ate gory of Hermi- tian ve ct or bund les over X with bu nd le morphisms 13 . The fun ct or Γ : E X → M C ( X ) , that to every Hermitian ve ctor bund le asso ciates its sym- metric C ( X ) -bimo dule of s e ctions, is an e quivalenc e of c ate gories. In practice, to every Hermitian vector bundle π : E → X ov er the compact Hausdorﬀ spa c e X , w e asso ciate the symmetric Hilbert C* -bimo dule Γ( X ; E ), the contin uous sections of E , ov er the C* -algebra C ( X ; C ). 10 A simi lar deﬁnition applies for r ight modules. 11 Of course this deﬁnition mak e sense only for bimo dules o ver a comm utativ e algebra A . 12 The result, as it is stated in the previously giv en references [F r, W e2] and [FG V, page 68], is actually formulated without the ﬁnitness and pro j ectivit y condition s on th e modules and with Hilb ert bundles (see J. F ell -R. Dor an [FD, Section 13] or [FGV, Deﬁnition 2.9] for a detailed deﬁnition) in place of Hermitian bundles. Note that Hilb ert bundles are not necessarily lo cally tri vi al, but they b ecome so if they hav e ﬁnite constant rank (see for example J. F ell-R. Doran [FD, Remark 13.9]) and hence the mor e general equiv alenc e b et ween the cat egory of Hi lb er t bundles and the category of Hilb ert C*- modules actually en tails the Hermitian ve rsion of Serr e- Sw an theorem present ed here. 13 Con tinuou s, ﬁb erwise linear maps, pr eserving the base p oints. 7 Since, in the lig ht of Gel’fand theorem, non-Ab elian unital C*- algebras a re to be inter- preted as “non-commutative co mpact Hausdorﬀ top ologica l spaces ”, Serr e-Swan theo rem suggests that ﬁnite pro jective Hilbert C*-bimo dules o ver unital C*-algebras should be considered as “Her mitia n bundles over non-commutativ e Hausdorﬀ co mpact spaces” . # Proble m: Serre- Swan theorem dea ls only wit h categories of bundles ov er a ﬁxed top ological space (catego ries of mo dules over a ﬁxed alg ebra, resp ectively). In or der to extend the theo rem to categorie s o f bundles ov er diﬀerent spaces, it is necessar y to deﬁne genera lized notions o f mor phis m b etw een mo dules ov er diﬀerent alg e bras. The easiest solution is to deﬁne a morphism fro m the A -module A M to the B - mo dule B N as a pair ( φ, Φ), where φ : A → B is a homomorphism of algebras and Φ : M → N is a C -linear map o f the bimo dules such tha t Φ( am ) = φ ( a )Φ( m ), for all a ∈ A and m ∈ M . This is the notion that we have used in [BCL1] and that app eared also in [T a1, T a2, F GV, Ho]. A mo re appro priate solution w ould b e to consider “ congruences ” of bimo dules a nd refo rmulate Serre- Sw an theo rem in terms of relators (as deﬁned in [BCL1]). W ork on this topic is in progre ss 14 . # Proble m: note that Serre-Swan theorem gives an equiv alence of ca tegories (and not a duality), this will create pr oblems of “cov a riance” for any gene r alization of the well- known cov ariant functors b etw een categ ories of m anifolds and categories of their asso ciated v ector (tensor, Cliﬀo r d) bundles, to the case of no n-commutativ e spaces and their “bundles”. Again a more a ppropriate approa ch using rela tors should deal with this issue. A ﬁrst immediate so lution to b oth the ab ove problems is provided by T ak ahashi duality theorem b elow. Serre-Swan equiv alence is actually a particular case o f the following general (and surpris ing ly almo st unnoticed) Gel’fand duality result that was obtained in 197 1 by A. T ak ahashi [T a1, T a2]. 15 In this formulation, one a ctually consider mu ch more gener al C*-mo dules and Hilber t bundles at the price of lo sing contact with K -theory; a n ywa y (a s describ ed in the fo otonote 12 at page 7) the Hermitian version of Serre - Swan theorem c an be r ecov ered c o nsidering bundles with constant ﬁnite rank (ov er a ﬁx ed compact Hausdorﬀ top ological space). Theorem 3. 3 (T ak ahashi) . Ther e is a (we ak ∗ -monoidal) c ate gory • M of left Hilb ert C*-mo dules A M , B N over unital c ommutative C*-algebr as, whose morphisms ar e given by p ai rs ( φ, Φ) wher e φ : A → B is a unital ∗ -homomorphism of C*-algebr as and Φ : M → N is a c ontinuous additive map such that Φ( ax ) = φ ( a )Φ( x ) , for al l a ∈ A and x ∈ M . Ther e is a (we ak ∗ -monoidal) c ate gory E of Hilb ert bund les ( E , π , X ) , ( F , ρ , Y ) over c om- p act Hausdorﬀ top olo gic al sp ac es with morphisms given by p airs ( f , F ) with f : X → Y c ontinuous and F : f • ( F ) → E a c ontinuous ﬁb erwise line ar map that satisﬁes π ◦ F = ρ f , wher e ( f • ( F ) , ρ f , X ) denotes the pul l-b ack of the bund le ( F , ρ, Y ) under f . Ther e is a duality (of we ak ∗ -monoidal) c ate gories give n by the functor Γ that asso ciates to every Hilb ert bun d le ( E , π , X ) the set of se ctions Γ( X ; E ) and that to every morphism of bund les ( f , F ) : ( E , π , X ) → ( F , ρ, Y ) asso ciates the morphism of mo dules ( f • , Φ) , wher e Φ is the map that to evey se ction σ ∈ Γ( Y ; F ) asso ciates the se ction F ◦ f • ( σ ) ∈ Γ( X ; E ) . Of co urse, m uch more deserves to b e said abo ut the v ast landscap e of resea rch currently developing in non- commutativ e to po logy , but it is not o ur purpo se to provide here an 14 P . Bertozzini, R. Con ti, W. Lewk eeratiyutkul, Categories of Spectral T riples and Morita Equiv alence, wo rk in progress. 15 Note that our Gel’f and dualit y result for c ommutat ive full C*-cate gories (that we wi l l presen t later in section 4.2. 1) can b e seen as “strict”- ∗ -monoidal version of T ak ahashi dualit y . 8 ov erview of this huge sub ject. F airly detailed treatmen ts of some of the usual techniques in algebraic top olog y are already av ailable in their non-commutativ e coun terpar t (see [FGV] or the exp osito ry article by J . Cuntz [Cu] for more details): non-c o mmu tative K -theor y ( K -theor y of C*-algebr as), K - homology (G. Ka s parov’s K K -theory ) and (co)homolog y (Hochschild and A. Connes, B. Tsygan cyclic co ho mologies). Among the mos t recent achiev ements, we limit ours e lves to mention the extremely int eres ting deﬁnitions of quan- tum principa l a nd asso cia ted bundles by P . Baum-P . Ha jac-R. Ma tthes-W. Szy man- ski [BHMS] and of non-co mmutative CW-complexes b y D. N. Diep [Di ]. A t the (diﬀeren tial) t op olo g ical level, we ment ion that impor tant connections b etw een non-commutativ e geo metry a nd signal proce s sing are emerging in the w orks b y O. Bratteli- P . Jorgensen [BJ] (wa velets and Cuntz algebra s ) and by F. Luef [Lu, Lu2 ] (Gab or analysis and Hilber t C*-mo dules for non-commutativ e tori). 3.2 Non-comm utativ e ( Spin) Diﬀeren tial Geometry . What are “non- commutativ e manifolds”? In order to deﬁne “no n-commutativ e manifolds”, we hav e to ﬁnd a categor ical dua lit y betw een a c ategory of manifolds and a s uitable catego ry cons tructed out of Ab elian C * - algebras o f functions ov er the manifolds. The complete a nswer to the question is not yet known, but (at least in the case of compac t ﬁnite-dimensio nal orientable Riemannian spin- manifolds) the notion of Connes sp ectral triples and Co nnes-Rennie-V arilly [C5, C11, R V1] reconstructio n theor em provide an appropr iate star ting point, sp ecifying the ob jects of our non-commutativ e ca teg ory . 16 3.2.1 Connes Sp ectral T riples. A. Connes (see [C3, FGV]) has prop os e d a set of axioms for “non-commutativ e manifolds” (at least in the ca se of a co mpact ﬁnite-dimensional or ie ntable Riemannian spin-manifolds), called a (compact) sp ectral triple or an (un b ounded) K -c ycle. • A (compact) sp ectral triple ( A , H , D ) is g iven by: – a unital pre-C* -algebra A ; 17 – a (faithful) r epresentation π : A → B ( H ) o f A on the Hilb ert space H ; – a (genera lly unbounded) self-adjoin t op erator D on H , called the Dirac op er ator, such that: a) the reso lvent ( D − λ ) − 1 is a compact op erator , ∀ λ ∈ C \ R , 18 b) [ D , π ( a )] − ∈ B ( H ), for every a ∈ A , where [ x, y ] − := xy − y x deno tes the c ommut ator o f x, y ∈ B ( H ). 19 • A sp ectra l triple is called even if there exists a grading oper ator, i.e. a b ounded self-adjoint op erato r Γ ∈ B ( H ) such that: Γ 2 = Id H ; [Γ , π ( a )] − = 0 , ∀ a ∈ A ; [Γ , D ] + = 0 , 16 W e will of course deal later with the morphisms in section 4.1. 17 Sometimes A is r equir ed to be closed under holomorphic functional calculus. 18 As alr eady noticed by Connes, t his condition has to b e weak ened in the case of non-compact m anifolds, cf. [GLMV , GGISV, Re2 , Re3]. 19 Since the Dirac operator D can be unbounded, the condition [ D , π ( a )] − ∈ B ( H ) actually means that the domain of D is inv ar i an t under all the elements a ∈ π ( A ) and that the op erators [ D , π ( a )] − = D ◦ π ( a ) − π ( a ) ◦ D , deﬁned on Dom( D ) ⊂ H , can b e extended to bounded linear op erators on H . 9 where [ x, y ] + := xy + y x is the anticomm utator o f x, y . A sp ectral triple that is no t even is called o dd . • A sp ectral triple is reg u l ar if the function Ξ x : t 7→ exp( it | D | ) x exp( − i t | D | ) is regular, i.e. Ξ x ∈ C ∞ ( R , B ( H )) , 20 for every x ∈ Ω D ( A ), where 21 Ω D ( A ) := span { π ( a 0 )[ D , π ( a 1 )] − · · · [ D , π ( a n )] − | n ∈ N , a 0 , . . . , a n ∈ A } . • A spectra l triple is n -di mensional iﬀ there exists an in teger n suc h that th e Dixmier trace o f | D | − n is ﬁnite nonzero. • A sp ectral triple is θ -s ummable if exp( − tD 2 ) is a trace-c la ss o p er ator for all t > 0 . • A spectr a l triple is real if there exists an a nt iunitary op erator J : H → H such that: [ π ( a ) , J π ( b ∗ ) J − 1 ] − = 0 , ∀ a, b ∈ A ; [ [ D , π ( a )] − , J π ( b ∗ ) J − 1 ] − = 0 , ∀ a, b ∈ A , ﬁrst order condition ; J 2 = ± Id H ; [ J, D ] ± = 0; and, only in the even c a se, [ J , Γ] ± = 0 , where the choice of ± in the las t three formulas dep ends on the “dimension” n of the sp ectral triple mo dulo 8 in a ccordance to the following table: n 0 1 2 3 4 5 6 7 J 2 = ± Id H + + − − − − + + [ J, D ] ± = 0 − + − − − + − − [ J, Γ] ± = 0 − + − + • A sp ectral triple is ﬁnite if H ∞ := ∩ ∞ k =1 Dom D k is a ﬁnite pr o jective A -bimo dule and absolutely contin uous if, there exists an Hermitian form ( ξ , η ) 7→ ( ξ | η ) o n H ∞ such that, for all a ∈ A , h ξ | π ( a ) η i is the Dixmier trac e o f π ( a )( ξ | η ) | D | − n . • An n -dimensional sp ectra l triple is said to b e orientable if there is a Hochschild cycle c = P m j =1 a ( j ) 0 ⊗ a ( j ) 1 ⊗ · · · ⊗ a ( j ) n such that its “repr e s ent ation” on the Hilb ert space H , π ( c ) = P m j =1 π ( a ( j ) 0 )[ D , π ( a ( j ) 1 )] − · · · [ D , π ( a ( j ) n )] − is the gra ding o p er ator in the ev en ca s e or the identit y op er ator in the o dd ca se 22 . • A real sp e ctral triple is said to satisfy P oincar´ e dualit y if its fundament al cla ss in the K R -homo logy of A ⊗ A op induces (via Kasparov in tersection pr o duct) an isomorphism be t ween the K -theory K • ( A ) and the K -homolog y K • ( A ) of A . 23 20 This condition i s equiv alen t to π ( a ) , [ D , π ( a )] − ∈ ∩ ∞ m =1 Dom δ m , for all a ∈ A , where δ is the deriv ation given by δ ( x ) := [ | D | , x ] − . 21 W e assume that for n = 0 ∈ N the term in the formula simply r educes to π ( a 0 ). 22 In the f ollowing, i n order to simpl ify the discussion, w e will alwa ys refer to a “grading op erator” Γ that actually coincides with the grading operator in the even case and that is by deﬁnition the ident ity operator in the odd case. 23 In [ R V1] s ome of the axioms are reformulated in a diﬀerent for m , in particular this condition is r eplaced b y the requiremen t that the C*-m odule completion of H ∞ is a Morita equiv alence bimodule betw een (the norm completions of ) A and Ω D ( A ). 10 • A sp ectral triple will b e ca lled Ab elian or commutativ e whenever A is Ab elian. • A sp ectral triple is i rreducible if there is no non-trivia l closed s ubspace in H that is in v ariant for π ( A ) , D , J, Γ. T o every sp ectra l triple ( A , H , D ) there is a natura lly asso ciated quasi-metric 24 on the set of pure states P ( A ), called Connes ’ distance and given for all pure states ω 1 , ω 2 by: d D ( ω 1 , ω 2 ) := s up {| ω 1 ( x ) − ω 2 ( x ) | | x ∈ A , k [ D , π ( x )] k ≤ 1 } . Theorem 3. 4 (Connes; see e.g. [C3, F GV]) . Given an orientable c omp act Riemannian spin m -dimensional diﬀer ent iable manifold M , with a given c ompl ex spinor bun d le S ( M ) , a given spino rial char ge c onjugation C M and a given volume form µ M , 25 deﬁne: A M := C ∞ ( M ; C ) the algebr a of c omplex value d r e gular functions on the diﬀer en- tiable manifold M , H M := L 2 ( M ; S ( M )) the Hilb ert sp ac e of “squar e inte gr able” se ctions of the given spinor bund le S ( M ) of t he manifold M i.e. the c ompletion of the sp ac e Γ ∞ ( M ; S ( M )) of smo oth se ctions of t he spinor bund le S ( M ) e quipp e d with the inner pr o duct given by h σ | τ i := R M h σ ( p ) | τ ( p ) i p d µ M , wher e h | i p , with p ∈ M , is the unique inner pr o duct on S p ( M ) c omp atible with the Cliﬀor d action and the Cliﬀor d pr o duct. D M the Atiya h-Singer Dir ac op er ator i.e. the closur e of the op er ator that is ob- taine d by “c ontr acting” the unique spinorial c ovari ant derivative ∇ S ( M ) (induc e d on Γ ∞ ( M ; S ( M )) by t he L evi-Civita c ovariant d erivative of M , se e [F GV, The o- r em 9.8]) with the Cliﬀor d mult iplic ation; J M the unique antiline ar unitary ext ension J M : H M → H M of the op er ator de- termine d by the spinorial char ge c onjugation C M as ( J M σ )( p ) := C M ( σ ( p )) for σ ∈ Γ ∞ ( M ; S ( M )) and p ∈ M ; Γ M the unique unitary extension on H M of t he op er ator given by ﬁb erwise gr ading on S p ( M ) , with p ∈ M . 26 The data ( A M , H M , D M ) deﬁn e a sp e ctr al triple that is Ab elia n r e gu lar ﬁnite absolutely c ontinuous m -dimensional r e al , with r e al structure J M , orientable, with gr ading Γ M , and that satisﬁes Poinc ar´ e duality. Theorem 3.5 (Connes [C5, C11]) . L et ( A , H , D ) b e an irr e ducible c ommutative r e al (with r e al stru ctur e J and gr ading Γ ) str ongly r e gular 27 m -dimensional ﬁnite absolutely c ontinuous orientable sp e ctr al triple satisfying Poinc ar ´ e duality. The sp e ct ru m of (the norm closur e of ) A c an b e endowe d, essen tial ly in a unique way, w ith t he structur e of an m -dimensional c onne cte d c omp act spin Ri emannian manifold M with an irr e ducible c omplex spinor bund le S ( M ) , a char ge c onjugation J M and a gr ading Γ M such t hat: A ≃ C ∞ ( M ; C ) , H ≃ L 2 ( M , S ( M )) , D ≃ D M , J ≃ J M , Γ ≃ Γ M . 24 In general d D can take the v alue + ∞ unless the sp ectral triple is irreducible. 25 Remem b er that an orientab le manifolds admits tw o di ﬀer ent orientation s and that, on a Riemannian manifold, the c hoice of an orientat ion canonically dete rmines a volume form µ M . Recall also [S] that a spin-manifold M admits sev eral inequiv alen t s pinor bundles and for every cho ice of a complex spinor bundle S ( M ) (whose isomorphism class deﬁne the spin c -structure of M ) there are inequiv alen t choices of spinorial charge conjugations C M that deﬁne, up to bundle isomorphisms, the spin-s tructure of M . 26 The grading is actually the identit y in o dd dim ension. 27 In the sense of [C11, Deﬁnition 6.1]. 11 # A. Connes ﬁrst prov ed the previous theorem under the additional condition that A is already given as the algebr a of smooth complex-v alued functions o ver a diﬀerentiable manifold M , namely A = C ∞ ( M ; C ), and conjecture d [C6, Theorem 6, Remar k (a)] [C5] the result for general comm utative pre- C* -algebr as A . A tentativ e pro o f of this last fact has b een published by A. Rennie [Re1]; some gaps were p ointed out in the original a rgument, a diﬀerent revis e d, but still incorrect, pro of app ears in [R V1] (se e also [R V2]) under some additiona l technical conditions. Recen tly A. Connes [C11] (see also [C12]) ﬁnally provided the missing steps in the pro of of the result. As a conseque nc e , there exists a o ne-to-one corr esp ondence b etw een unitar y equiv alence classes of sp ectr al triples and co nnected compact orie nted Riema nnian spin-manifolds up to spin-preserving isometric diﬀeomo r phisms. Similar results are als o av a ila ble for spin c -manifolds [C6, Theorem 6, Remark (e)]. 3.3 Examples. Of course, the mo s t ins piring ex amples of sp ectral triples (sta rting fr o m t hose arising from Riemannian spin-manifolds ) a re contained in A. Connes ’ b o ok [C3] and an up dated account of most of the a v ailable co nstructions is contained in A. Connes-M. Marcolli’s lecture notes [CM1]. Her e b elow we provide a short guide to some of the relev ant liter a ture: • Ab elian spectral triples arising fr om the A tiyah-Singer Dirac o per ator on Rieman- nian spin-manifolds, A. Connes [C3], and classica l compact homog eneous spaces, M. Rieﬀel [Ri3]. • Sp ectral triples for the non-c ommut ative tor i, A. Connes [C3]. • Discrete sp ectral triples, T. K ra jewski [K r], M. Paschk e-A. Sitarz [PS1]. • Sp ectral triples from Moy al planes (these ar e e x amples of “non-compact” triples), V. Gayral-J.M. Gracia - Bondia-B. Io chu m-T. Sch¨ uker-J. V a r illy [GGISV]. • Examples of Non-co mm utative Lor ent zian Sp ectral T r iples (following the deﬁnition given b y A. Stro hmaier [Str]), W. D. v an Suijlek om [Sui]. • Sp ectral T riple s related to the Kronecker foliation (follo wing the gene r al co nstruction by A. Connes-H. Moscovici [CMo1] of spec tral triples a sso ciated to c r ossed pro duct algebras related to foliations ), R. Matthes-O. Rich ter-G. Rudolph [MRR ]. • Dirac op erator s as multip lication by length functions on ﬁnitely generated discrete (amenable) groups, A. Connes [C1], M. Rieﬀel [Ri1]. • K -c y cles and (t wisted) sp ectra l triples a r ising fro m sup ers ymmetric quantum ﬁeld theories, A. Jaﬀe-A. Lesniewsk i-K. O sterwalder [JLO1, JLO2], D. K astler [K1], A. Connes [C3], D. Goswami [Go2]; cyclic co cycles from supe r KMS-s tates in alge- braic quantum ﬁeld theory , D. Buchholz-H. Grundling [BGr1] and s p ectr al triples on sup e r-Viraso ro algebra s in co nfo r mal ﬁeld theory , S. Car pi-R. Hillier-Y. Kaw ahiga shi- R. Lo ngo [CHKL]. • Sp ectral triples a s so ciated to quantum groups (in some ca se it is necessa ry to mo dify the ﬁrs t order condition inv olving the Dirac opera tor, r equiring it to hold only up to co mpact ope rators), P . Chakrab orty-A. Pal [ChP1, ChP2, ChP3, ChP4, ChP5, 12 ChP6, ChP7, ChP8, ChP9], D. Goswami [Go1], A. Connes [C8], L. Dabr owski- G. Landi-A. Sitarz-W. v an Suijlek om-J. V a rilly [DLSSV1 , DLSSV2 ], J. Kustermans- G. Murph y-L. T uset [KMT], S. Nesh vey ev-L. T uset [NT]; a nd also sp ectral triples as- so ciated to ho mogeneus space s of quan tum groups: L. Dabrowski [Da], L. Dabrowski- G. Landi-M. Paschk e-A. Sita rz [DLPS], F. D’Andrea-L. Dabrowski [DD1, DD2], F. D’Andrea- G. Landi [DAL], F. D’Andrea-L. Dabrowski-G. La ndi [DDL1, DDL2], [D] (the latter is “twisted” according to A. Connes-H. Moscovici [CMo3, Mos]). • Non-commutativ e manifolds and instantons, A. Connes-G. Landi [CL], L. Dabrowski G. La ndi-T. Ma suda [DLM], L. Dabr owski-G. L a ndi [DL], G. Landi [Lan3, Lan4], G. La ndi-W. v an Suijlekom [LS1, LS2]. • Non-commutativ e spherical manifolds A. Connes-M. D ub ois-Violette [CDV1, CD V2, CD V3]. • Sp ectral triples for some clas ses of fractal spaces, A. Connes [C3], D. Guido-T. Isola [GI1, GI2, GI3], C. An tonescu-E. Christensen [AC], E. Chris tensen C. Iv an-M. La pidus [CI L ]. • Sp ectral T riples for AF C*- a lgebras, C . An tonescu-E. Christensen [A C]. • Sp ectral tr iple s in n um b er theory: A . Connes [C3], A. Co nnes-M. Mar colli [CM1], R. Meyer [Me2]; sp ectr a l triples f rom Arak elov Geometry , from Mumfor d curves and hyperb olic Riemann s urfaces, C. Consani-M. Marcolli [CoM1, CoM2, CoM3, CoM4], G. Cornelissen-M. Marco lli-K. Reihani-A. Vdovina [CMR V], G. Cornelis s en- M. Marco lli [CMa]; sp ectra l triples for cer tain clas ses of ﬁnite c onnected unoriented graphs, J. W. de Jong [DJ]. • Sp ectral triples of the standar d mo del in par ticle ph ysics, A. Connes-J. Lott [CLo], J. Gracia-Bondia -J.V arilly [GV], D. Kastler [K3, K5, KaS], A. Connes [C4, C5, C1 0], J. Bar r ett [Bar], A. Chams eddine-A. Connes [CC1, CC2, CC3, CC4], A. Chamsed- dine [Ch], A. Connes-M. Mar colli [CM1, CM2], A. Chamseddine- A. Connes-M. Mar- colli [CCMa]. 3.4 Other Sp ect r al Geometries. In the la st few years several others v a riants and ex tensions of “ sp ectral geometries ” have bee n considered or prop os e d: • Lorentzian sp ectra l geometries: A. Strohmaier [Str], M. P aschke-R. V erch [P V2], M. Paschk e-A. Sita r z [PS2] and also M. Borr is -R. V erch [BV], • Riemannian non-spin: S. Lor d [Lo], • Laplacia n, K ¨ ahler: J. F r¨ o hlich-O. Gr a ndjean-A. Recknagel [FGR1, FGR2, FGR 3, F GR4] (for a study of no n- commutativ e La pla ce op era tors a nd elliptic partial dif- ferential equations in non-co mm utative geo metr y see J. Rosenber g [Ros]), • F ollowing works by M. Breuer [Br1, Br 2] o n F redho lm mo dules o n von Neumann alge- bras, M-T. Bena meur-T. F ack [BF, BF2] and mo re re cent ly in a remar k able series of pap ers [CP , CPS1, CPS2, CP RS1 , CPRS2, CPRS3, CPRS4, CRSS, BCPRSW, PaR, CPR1, CPR2, CPR3, CPR4, CR T], M-T. Benameur- A. Car ey-D. Pask-J. Phillips- A. Rennie-F. Suk o chev-K. T ong-K. W o jciecho wski (see als o J. Kaad-R. Nest-A. Ren- nie [KNR] and A. Carey -S. N eshv eyev-R. Nest-A. Rennie [CNNR]), hav e been trying 13 to genera lize the formalism o f Connes sp ectra l triples when the algebr a o f b ounded op erators on the Hilb ert space of the triple is replaced by a mo r e gener a l v on Neu- mann algebra that is e ither semiﬁnite or that ca rries a perio dic action of the mo dular group of a KMS-state. Among e x amples of semiﬁnite sp ectral triples a sp ecial mention des erve those con- structed on algebr as of holonomy lo ops in canonica l quantum gravity b y J . Aas trup- J. Grimstrup-R. Nest [AGN1, AGN2, AGN3, A GN4] (see also s ection 5.5.1). # Although non-co mmutative diﬀerential geometr y , following A. Connes, has been mainly develop ed in the axiomatic framework of sp ectr a l triples, that essentially generalize the structures av aila ble for the Atiy ah-Singer theory of ﬁrst order diﬀer- ent ial elliptic o pe r ators of the Dirac type, it is very lik ely that suita ble “spec tral geometries” might be developed using op er ators of higher o r der (the Laplacia n type being the ﬁrst notable example). Since “top ologica l obstructions” (such us non- orientabilit y , non- spinoriality) are exp ected to survive essentially unaltered in the transition from the commutativ e to the non commu tative world, these “higher- order non-commutativ e geometr ies” will deal w ith mo re g eneral situatio ns compa red to usual sp ectral triples. In this direction we ar e developing 28 deﬁnitions in the hope to obtain Connes Rennie-V arilly reco nstruction theor ems also in these cases . # Apart from the “s pec tr al approaches” to non-comm utative geometry , more or less directly inspired by A. Connes sp ectra l triples , ther e a re other lines of development that ar e worth inv estigating and whose “relation” with sp ectra l triples is not yet clear: – J.-L. Sauv ageot [Sa] and F. Cipr iani [CS] are developing a version of non- commutativ e geometry descr ibed by Hilb ert C* - bimo dules associa ted to a semi- group of co mpletely p ositive contractions, an approach that is directly r elated to the analysis of the prop erties of the heat-kernel o f the Laplacian on Riemannian manifolds (see N. Berline - E. Getzler -M. V ergne [BGV]); – M. Rieﬀel [Ri2 ], and a long s imilar lines N. W eav er [W e1, W e 2 ], hav e developed a theory of non-commutativ e compact metric spaces ba sed on Lipschitz a lgebras. – F ollowing a n idea of G. Parﬁonov-R. Zapatrin [PZ], V. Moretti [Mo] has gen- eralized Connes’ distance formula (using the D’Alem b ert o p er ator) to the case of Lorentzian globally h yp erb olic manifolds and has develop ed a n approa ch to Lorentzian non-co mmutative geometry based on C*-a lg ebras who se r elations with Strohmaier’s sp ectr al triples is in triguing. – In algebraic quantum ﬁeld theory (see section 5 .3), S. Doplicher-K. F reden- hagen J. Ro ber ts [DFR1, DFR2] (a nd a lso S. Doplicher [Do 2, Do 3, Do4]) hav e developed a mo del of Poincar´ e c ov ariant q uantum spacetime. – O. Bratteli and collab orator s [B, BR] and more recently M. Mado re [Mad] hav e been approaching the deﬁnition of non-commutativ e diﬀerential g eometries through mo dules of deriv ations ov er the algebra of “smo oth functions”. – Strictly related to the previous appro ach there is a for midable literature (see for example S. Ma jid [Ma j1, Ma j2]) o n non-commutativ e geometry based on “quantum gro ups ” structur es (Hopf algebras ). 28 P . Bertozzini, R. C onti, W. Lewkeerat iyutkul, Second Order Non-commuta tive Geometry , w ork in progress. 14 – Most of the physics literatur e use the term non- commut ative ge ometry to indi- cate no n-commutativ e spaces obta ined by a quantum “deforma tio n” of a clas- sical commutativ e space. 4 Categories in Non-Comm utati v e Geometry . After the discussion of “ob jects” in non-commutativ e geometry , we no w shift our attention to some very tentativ e deﬁnitions of morphism o f non-commutativ e spaces and of ca tegories of non-commutativ e spaces. In the ﬁrst subse c tio n we presen t morphisms of “sp ectra l geo metries”. W e limit o ur discussion essentially to the case of mor phisms of A. Connes spec tral triple s , although we exp ect that similar notio ns might be dev elop ed also for o ther s pec tr al geometries. In the second subsection we describ e some other extremely imp ortant categ ories of “non- commutativ e spaces” that arise, a t the “top olog ical level”, from “v ariations on the theme” of Morita equiv alence. Finally we indicate some direc tion o f future resea rch. 4.1 Morphisms of Sp ec t ral T riples. Having descr ib ed A. Connes spec tral triples and someho w justiﬁed the fact that s pec- tral tr iples ar e a possible deﬁnit ion for “no n-commutativ e” co mpact ﬁnite-dimensio nal orientable Riemannian s pin- manifolds, our next goal her e is to discuss deﬁnitions of “mor - phisms” b etw een sp ectra l triples and to c o nstruct ca teg ories of sp ectra l triples (for further details and an up dated ov erview of this line of r esearch see also the slides [B2]). Even for sp ectral triples , ther e are actually se veral p oss ible notions of morphism, acco rd- ing to the amount of “background structure ” of the manifold that we would like to see preserved: 29 • the metric, globa lly (isometrie s ), • the metric, lo cally (totally geo desic ma ps, in the diﬀerentiable ca se), • the Riemannian structure, • the diﬀerentiable str uctur e, 4.1.1 T otally Ge o desic Spin-Morphi s ms. This is the notion of mor phism of sp ectral triples that we pro p o sed in [B CL1]. Given t wo spec tral triples ( A j , H j , D j ) , with j = 1 , 2 , a morphism of sp ectral triples is a pair ( A 1 , H 1 , D 1 ) ( φ, Φ) − − − → ( A 2 , H 2 , D 2 ) , where φ : A 1 → A 2 is a ∗ -mor phism betw een the pr e -C*-alge bras A 1 , A 2 and Φ : H 1 → H 2 is a b ounded 30 linear map in B ( H 1 ; H 2 ) that “intert wines” the r epresentations π 1 , π 2 ◦ φ and the Dirac op er ators D 1 , D 2 : π 2 ( φ ( x )) ◦ Φ = Φ ◦ π 1 ( x ) , ∀ x ∈ A 1 , D 2 ◦ Φ = Φ ◦ D 1 , (4.1) 29 And also dep ending on the kind of topological prop erties that we w ould li ke to “attac h” to our morphisms: orientation, spinoriality , . . . 30 It might b e necessary to r elax this condition and to consider also cases in which Φ is unbounded. 15 i.e. such that the following dia grams commute for every x ∈ A 1 : H 1 D 1   Φ / /  H 2 D 2   H 1 Φ / / H 2 H 1 π 1 ( x )   Φ / /  H 2 π 2 ◦ φ ( x )   H 1 Φ / / H 2 Here the in tertwining relation betw een the Dirac op erator s ho lds on the domain of D 1 , since we supp ose that Φ(Dom( D 1 )) ⊂ Dom( D 2 ). It is p os sible (in the ca s e of even and/or real sp ectral triples) to require also comm utations betw een Φ and the grading op erators and/or the real structure s. More sp e ciﬁcally: a m orphism of real sp ectral triple s ( A j , H j , D j , J j ), is a morphism of sp ectral triples, as ab ove, such that Φ also “intert wines” the real structure op e rators J 1 , J 2 : J 2 ◦ Φ = Φ ◦ J 1 ; a morphism of ev en sp ectral triples ( A j , H j , D j , Γ j ), with j = 1 , 2, is a mor- phism of spec tr al triples , as a b ove, suc h that Φ als o “intert wines” the g rading op er - ators Γ 1 , Γ 2 : Γ 2 ◦ Φ = Φ ◦ Γ 1 . Clearly this deﬁnition of morphism con tains as a specia l case the notion of (un itary ) equiv alence of sp ectr al triples [FGV , pp. 485 -486] and implies quite a strong r elationship betw een the spectra o f the Dirac op e rators of the tw o s pec tr al triples. Lo osely sp eaking , for φ epi and Φ coisometric (resp ectively mono and isometric), in the commutativ e case 31 , one e xpe cts such deﬁnition to become r elev an t only for maps that “preserve the geo desic structur es” (totally g eo desic immersio ns and r esp ectively tota lly geo desic submer sions). Note that (alrea dy in the co mm utative case) these maps might not necessar ily b e metr ic isometries: totally geo des ic maps are lo cal isometries but not alwa ys g lobal iso metries (but we do not hav e a counterexample yet). F urthermore these morphisms dep end, at least in some sense, on the spin structures: 32 this “spinoria l rigidity” (at least in the case of morphisms of re a l ev en sp ectral triples ) requires tha t such morphisms b etw een sp ectral triples of diﬀeren t dimensio ns might b e po ssible only when the diﬀerence in dimension is a multiple of 8 . It might b e interesting to examine alternative sets o f conditions on the pairs ( φ, Φ) that allow for example to for malize the no tion of “ immersion” o f a non- commut ative manifold int o another with arbitrary higher dimension, av oiding the requiremen ts coming from the spinor ial structures. Some prelimina ry consider ations along similar lines hav e b een independently prop os ed by A. Sitarz [Si] in his habilitation thesis. There it was suggested that t he appropria te morphisms satisfy some “gra ded intert wining r elations” with the relev an t op erato r s, indicating the p ossibility to formalize s uitable sig n rules dep ending on the inv olved dimensions (modulo 8). W e plan to elab or ate on this topic elsewhere 33 . 4.1.2 Metri c Morphisms. In [BCL2] w e intro duce the follo wing notio n of metric morphism. Given t wo sp ectral triples ( A j , H j , D j ), with j = 1 , 2, deno te by P ( A j ) the sets o f pure sta tes ov er (the norm 31 The details ar e dev elop ed in: P . Bertozzini, R. Conti, W. Lewke eratiyutkul, Non-commutativ e (T otally Geodesic) Submanifolds and Quotien t Manifolds, i n preparation. 32 In the case of morphisms of eve n real sp ectral tri ples, the map should preserve in the strongest p ossible sense the spin and orien tation structures of the manifolds (whatev er this might mean). 33 P . Be rtozzini, R. Conti, W. Le wkeeratiyut kul, Morphism of Spectral T riples and Spin Manifolds, w ork in progress. 16 completion of ) A j . A metric mo rphism o f sp ectral triples ( A 1 , H 1 , D 1 ) φ − → ( A 2 , H 2 , D 2 ) is by deﬁnition a unital epimorphism 34 φ : A 1 → A 2 of pre-C* -algebra s whose pull-back φ • : P ( A 2 ) → P ( A 1 ) is an isometry , i.e. d D 1 ( φ • ( ω 1 ) , φ • ( ω 2 )) = d D 2 ( ω 1 , ω 2 ) , ∀ ω 1 , ω 2 ∈ P ( A 2 ) . This no tion o f metric morphism is “ess ent ially blind” to the s pin struc tur es o f the non- commutativ e ma nifo lds (that in this case app ears only as a nec essary co mplication 35 ). 4.1.3 Ri emannian Morphisms . A less rigid notion of morphis m of spectr al triples 36 (a deﬁnition that, for unitary ma ps, was in tro duced by R. V erch and M. Pasc hke [PV1]) consis ts of relaxing the “intert wining” condition (4.1) be tween Φ and the Dirac o per ators, imp osing only “intert wining r elations” with the comm utators o f Dirac o pe rators with elemen ts o f the a lgebras. In mo re detail: given tw o spectral triples ( A j , H j , D j ) , with j = 1 , 2 , a Ri emannian morphism of sp ectral triples is a pa ir ( A 1 , H 1 , D 1 ) ( φ, Φ) − − − → ( A 2 , H 2 , D 2 ) , where φ : A 1 → A 2 is a ∗ -mor phism betw een the pr e -C*-alge bras A 1 , A 2 and Φ : H 1 → H 2 is a bounded linear ma p in B ( H 1 ; H 2 ) that “int ertwines” the r epresentations π 1 , π 2 ◦ φ and the commutators of the Dir a c op erators D 1 , D 2 with the elements x ∈ A 1 , φ ( x ) ∈ A 2 : π 2 ( φ ( x )) ◦ Φ = Φ ◦ π 1 ( x ) , ∀ x ∈ A 1 , [ D 2 , φ ( x )] ◦ Φ = Φ ◦ [ D 1 , x ] , ∀ x ∈ A 1 , i.e. such that the following dia grams commute for every x ∈ A 1 : H 1 [ D 1 ,x ]   Φ / /  H 2 [ D 2 ,φ ( x )]   H 1 Φ / / H 2 H 1 π 1 ( x )   Φ / /  H 2 π 2 ◦ φ ( x )   H 1 Φ / / H 2 In the comm utative case, when φ is epi and Φ is coisometric (r esp ectively mo no and isomet- ric), this deﬁnition is expec ted to co r resp ond to the Riemannian iso metr ies (r esp ectively coisometries ) of compact ﬁnite-dimensiona l orientable Riemannian spin-manifolds . # These no tions of morphism o f s pectr al triples are o nly tentativ e and mor e examples need to b e tested. As p o inted out by A. Rennie, it is likely that the “ correct” deﬁni- tion of mor phism will evolve, but it will sure ly reﬂect the bas ic structure s uggested here. At the “top ologica l lev el” pair of ma ps ( φ, Φ) that intert wine the a c tions of the algebras on the resp ective Hilber t spaces (but not the Dirac op erators or their com- m utators ), ha ve r ecently bee n used by P . Iv anko v-N. Iv a nko v [I I] for the deﬁnition of ﬁnite cov ering (a nd fundamental group) of a sp ectr a l triple. 34 Note that if φ is an epimorphism, its pull-back φ • maps pure states into pure states. 35 Since it is possible to deﬁne functional distanc es using also Laplacian operators, we exp ect this notion to contin ue to mak e sense once a suitable notion of “Laplacian non-comm utativ e manifol d” is deve lop ed. 36 P . Bertozzini, R. Conti, W. Lewkee ratiyutkul, Morphisms of Non-commut ative Riemannian Manifolds, in preparation. See also the slides [B2]. 17 # The sev eral notions of morphism of spectra l triples describ ed a b ove are no t a s general as possible. In a wider per spe ctive, 37 a morphism of sp ectr a l tr iples ( A j , H j , D j ), where j = 1 , 2, might b e formalized as a “ suitable” functor F : A 2 M → A 1 M , b e- t ween t he catego ries A j M of A j -mo dules, having “appropriate in tert wining” prop er- ties with the Dira c oper ators D j . Now, under some “mild” hyp o thesis, by Eilenberg - Gabriel-W atts theorem (see for example [Me1]), any such functor is given by “ten- sorizatio n” by a bimodule. Thes e bimo dules, suitably equipp ed with s pec tral data (as in the cas e of sp ectral triples), provide the natural setting for a general theory of morphisms of non-co mm utative spaces (see [B2] for some concr ete prop osal). In this direction we ment ion the notion o f “sp ectral corre sp o ndences” develope d by A. Connes-M. Marcolli [CM2] and further utilized in M. Marcolli- A. Zainy [MZ]. 4.1.4 Mori ta Morphisms . In the previous subsections w e describ ed in some detail some prop osed notio ns of morphism of “non-commutativ e spaces” (describ ed as spectr a l triple s ) at the “ metric” level. A few other discussions of non-co mm utative geometry in a suitable catego rical framework, hav e already app ear ed in the litera tur e in a mo re or less e x plicit form. Most of them deal essentially with morphisms at the “ top o logical level” and a r e making us e o f the notion of Morita equiv alence that we are going to introduce. Deﬁnition 4.1. Two unital C*-algebr as A , B ar e said to b e str ongly Morita e quivalent if ther e exists an imprimitivity bimo dule A X B . It is a sta nda rd pro cedure in algebr aic geometry , to deﬁne “s paces” dually b y their “sp ec- tra” i.e. by the categories of (equiv a lence classes of ) representations of their alg ebras. Hence, for a given unital C*-algebra A , w e consider its category A M o f (isomorphism classes o f ) left C*-Hilb ert A -mo dules with morphisms given by (equiv alence cla sses of ) A -linear mo dule maps. Morphisms b etw een these “non-commutativ e sp ectra” are given b y cov ariant functors b e- t ween the ca teg ories of mo dules . 38 The Eilenberg- Gabriel-W atts theorem (see e.g. [Me1]) assures that under suitable c o n- ditions every functor F : A M → B M coincides “up to a natur al e quiv alence” with the functor given by left tensorization with a C* -Hilber t A - B -bimo dule B X A (with X unique up to isomorphism of bimo dules) i.e.: F ( A E ) ≃ B X A ⊗ A E . Y. Manin [M] has been a dvocating the use of suc h “Morita morphisms” (tensorizations with Hilbert C*-bimo dules) as the natural no tion of m or phis m of non-comm utative spac e s. In [C4, C5, C7] A. Connes alrea dy discussed how to tra nsfer a given Dira c op erato r using Morita equiv a lence bimodules and compatible connections on them, th us leading to the concept o f “inner deformations” of a sp ectral g e ometry underlying the “transfo rmation rule” e D = D + A + J AJ − 1 (where A denotes the “ connection”). In our w ork 39 , we try to deﬁne a str ic tly related catego ry of sp ectral triples, based on the notions of connection o n a Morita morphism, that co ntains “inner defo rmations” as isomor phis ms. 37 P . Bertozzini, R. Con ti, W. Lewk eeratiyutkul, Categories of Spectral T riples and Morita Equiv alence, wo rk in progress. 38 This kind of “ideology” ab out categories of “non-comm utativ e sp ectra” is very fashionable in “non- comm utativ e algebraic geomet ry” (see for example M. Kontsevic h and A. Rosen berg [ KR1, KR2, R]). 39 P . Bertozzini, R. Con ti, W. Lewk eeratiyutkul, Categories of Spectral T riples and Morita Equiv alence, wo rk in progress. 18 More sp eciﬁcally , giv en tw o s pec tral triples ( A j , H j , D j ) , with j = 1 , 2 , b y a Morita- Connes morphism of spec tral triples, we mean a pa ir ( X, ∇ ) where X is Morita morphism from A 1 to A 2 i.e. an A 2 - A 1 -bimo dule that is a Hilb ert C*-module ov er A 2 and ∇ is a Riemannian connection on the bimodule X (the Dirac op erator s are r elated to the connection ∇ b y the “inner deformation” formula). The comp osition ( X 3 , ∇ 3 ) of t w o Morita-Co nnes mor phisms ( X 1 , ∇ 1 ) and ( X 2 , ∇ 2 ) is deﬁned by taking the tensor pr o duct X 3 := X 1 ⊗ A 2 X 2 of the bimo dules a nd tak ing the connection ∇ 3 on X given by: ∇ 3 ( ξ 1 ⊗ ξ 2 )( h 1 ) := ξ 1 ⊗ ( ∇ 2 ξ 2 )( h 1 ) + ( ∇ 1 ξ 1 )( ξ 2 ⊗ h 1 ) , h 1 ∈ H 1 , ξ j ∈ X j . In a remark able recent pap er, A. Connes-C. Consani-M. Marcolli [CCM] ha ve b e en pushing even further the notio n of “Morita mor phism” deﬁning morphis ms b etw een tw o algebra s A , B a s “homoto p y classe s ” of bimo dules in G. Kaspa r ov K K -theory K K ( A , B ). In this wa y , every mo rphism is determined by a bimodule that is further equippe d with additional structure (F redholm mo dule). 40 In the same pap er [CCM], A. C o nnes and collab ora tors provide g round for consider ing “cyclic cohomo lo gy” a s an “abs olute cohomolog y o f no n- commutativ e motiv es” and the category of mo dules ov er the “cyclic categ ory” (already deﬁned by A. Connes -H. Moscovici [CMo2]) as a “non-commutativ e motivic co homology” . # All the notions of catego r ies of non-commutativ e spaces developed from the notion of Morita morphism, seem to b e co nﬁned to the top ological setting. Morita eq uiv a - lence in itself is a non-commutativ e “top ologica l” notion. It is widely b elieved that Morita equiv alent algebr a s sho uld b e co nsidered as descr ibing the “sa me” space. This comes from the fact that mos t of the “ geometric functor s” for commutativ e spac es when suitably extended to the non-comm utative ca s e are in v ariant under Morita equiv alences (becaus e Morita equiv ale nce reduces to isomorphism for co mm utative algebras ). Anywa y , most of the success o f A. Connes’ non-commut ative geo metry actually comes from the fact tha t some co mm utative alg ebras ar e replaced with some other Morita equiv alen t non-commutativ e alg ebras that are able to describ e in a m uch b etter way the g eometry of the “singula r spa ce”. In a more direct w ay , it seems that the corr ect way to a sso ciate a C*- algebra to a spa ce, requires the direct input of the natural symmetries of the space (hence Morita eq uiv alence is broken). Along these lines we hav e some w ork in prog ress on non-commutativ e Kle in pro gram 41 . Although the for malization of the notion of mo rphism a s a bimo dule is pr obably here to sta y , additional structures on the bimo dule will b e required to account for diﬀerent lev el of “r igidity” (metr ic , Riemannian, diﬀerential, . . . ) and some o f these, are proba bly go ing to br eak Mo rita eq uiv ariance a s long as non-top ologica l prop erties are concerned. A. Connes-M. Mar colli [CM2 , Chapter 8.4] and M. Marcolli-A. Zain y [MZ] give a deﬁnition of “sp ectra l co r resp ondences” as Hilber t C*- bimo dules pr oviding a “biv ariant version” of a sp ectral triple. The problem o f deﬁning a “metric” categ ory of sp ectral triples via morphisms in Ka sparov K K -theory suitably equipp ed with s mo oth and metr ic structures, has b een recen tly ad- dressed in a remark able pap er by B. Mesland [Me s]: a morphism fro m the sp ectral triple ( B , H ′ , D ′ ) to the sp ectral triple ( A , H , D ) is g iven by a unitary isomorphism c la ss of an un b ounded “s mo oth” A - B - bimo dule ( E , S, ∇ ) with co nnection ∇ such that: 40 Other im portant results in this direction are obtained by S. Mahan ta [ Mah4]. 41 P . Bertozzini, R. Conti, W. Lewk eeratiyutkul, Non-commutativ e Klein-Cartan Program, work in progress. 19 • [ ∇ , S ] is a completely b ounded op er a tor, • H is iso morphic to E ⊗ B H ′ , • D = S ⊗ Id + Id ⊗ ∇ D ′ with Id ⊗ ∇ D ′ ( e ⊗ f ) := ( − 1) ∂ e ( e ⊗ D ′ f + ∇ D ′ ( e ) f ). S. Mahanta [Mah4] is trying to relate “s pec tr al corr esp ondences” with the “geometric morphisms” of derived categor ies of the diﬀerential graded categ ories alr eady used in the non-commutativ e algebr a ic geometry appro ach to non-commutativ e spaces [Ma h1, Mah2, Mah3]. # Finally we note that we have not b een discussing here the role o f quantum gro ups as possible s ymmetries of spectra l triples (see for example the recen t pape r s b y D. Goswami [Go3, Go4, Go5, Go 6] and J . Bhowmic k-D. Go swami-A. Sk alski [BG1, BG2, BG3, BG4, BG5, BGS] discussing qua nt um isometr ies of s pec tr al triples). 4.2 Categoriﬁcation (T o p ological Lev el). Categoriﬁca tion is the term, intro duced by L. Crane-D. Y etter [CY], to denote the g eneric pro cess to substitute o rdinary a lgebraic s tructures with categor ic al counterparts. The term is now mostly used to denote a wide a rea of r esearch (see J . Baez-J. Dolan [BD2]) whose purpos e is to use hig her order catego ries to deﬁne categoria l analog s of algebraic structures. This v ertical categoriﬁcation 42 is usually done by promoting sets to cate- gories, functions to functors, . . . hence r eplacing a category with a 2-categ ory a nd so on. In non-co mm utative ge ometry , where usua lly spaces a re deﬁned “dually ” b y “ sp ectra” i.e. categories of repr e s ent ations of their algebr as of functions, this is a kind of compul- sory step: mo rphisms of non-commutativ e spac e s a re a ctually particular functors be tw een “sp ectra”. In this sense, non- commut ative geo metry (and also ordina r y commutativ e al- gebraic geometry of schemes) is already a kind of vertical categ oriﬁcation. There are also more “triv ial” forms of horizon tal c ategoriﬁcation in which ordinary algebraic unital asso cia tive structures a re interpreted as categories with only one ob ject and suitable analog categor ies with more than one ob ject ar e deﬁned. In this case the passage is f rom endomorphisms of a single ob ject to morphisms b etw een diﬀerent ob jects 43 : Monoids Small Catego ries (Mono idoids) Groups Group oids Asso ciative Unital Rings Ringoids Asso ciative Unital Algebras Algebro ids Unital C*-alg ebras C*-catego ries (C*-a lgebroids) It is a n ex tremely in teresting future topic o f inv estiga tion to discuss the in terplay b etw een ideas of categoriﬁc a tion and no n- commutativ e geometr y . . . here w e are really only at the beg inning of a long journey and w e can present only a few ideas . 44 42 In general a n -category get replaced with a n + 1 -category , i ncreasing the “depth ” of the a v ailable morphisms, hence the termi nology “vertical” adopted here. 43 Hence the name “horizontal”, adopted here, that implies that no jump in the “depth” of morphisms is requir ed. J. Baez [B] prefers to use the term oidization for this case. 44 Other approache s to the abstract concept of “categoriﬁcation” ha v e turned out to b e useful in the theory of knots and links, see [Kh1, Kh2] . 20 4.2.1 Hori zontal Categoriﬁcation of Gel’fand Duality . As a ﬁrst step in the developmen t of a “ categorica l non-commutativ e geometry”, we hav e bee n lo o king at a p ossible “horizo n tal categ oriﬁcation” of Gel’fand duality (theorem 3 .1). In practice, the purp os e is : • to ﬁnd “suitable embedding functors” F : T (1) → T and G : A (1) → A of the categorie s T (1) (of compact Hausdorﬀ top olo gical spac es) and A (1) (of unital co m- m utative C*- algebras ) in to tw o c ategories T a nd A ; • to e xtend the categor ical duality (Γ (1) , Σ (1) ) b etw een T (1) and A (1) provided b y Gel’fand theorem, to a categorica l duality b etw een T and A in such a wa y that the following diagrams are commutativ e up to natural iso mo rphisms η , ξ : T (1) F   Γ (1) / A (1) Σ (1) o G   F ◦ Σ (1) η / / Σ ◦ G, T Γ / A , Σ o G ◦ Γ (1) ξ / / Γ ◦ F. Since A (1) is a full sub categor y of the ca tegory o f C*-alge bras, we iden tify the horizontal categoriﬁc a tion of A (1) as a sub categ o ry of the ca tegory of small C*-categ o ries. In [BC L 4], in the setting of C* -categor ies, we provide a deﬁnitio n of “sp ectrum” of a com- m utative full C*-catego ry as a one-dimensional unital F ell bundle over a suitable group oid (equiv alence relation) and w e prove a ca tegorica l Gel’fand dualit y theorem genera lizing the usual Gel’fa nd duality b etw een the ca tegories of Ab elian C* -algebra s and compac t Hausdorﬀ spaces. As a bypro duct, in [BCL 3] we also obtain the following spectr a l theore m for imprimitivity bimo dules ov er Ab elian unital C*-alg ebras: e very such bimo dule is obtained b y “t wisting” (b y the t wo pro jection homeomo rphisms) the symmetric bimo dule of sections of a unique Hermitian line bundle over the graph of a unique homeomo rphism b etw een the sp ectra of the tw o C* -algebra s. Theorem 4. 2. (P. Bertozzini-R. Conti-W. L ewke er atiyutkul [BCL3, The or em 3.1]) Given an imprimitivity Hilb ert C*-bimo dule A M B over the Ab elian unital C*-algebr as A , B , ther e exists a c anonic al home omorphism 45 R B A : Sp( A ) → Sp( B ) and a Herm itian line bu nd le E over R B A such that A M B is isomorphic to t he (left/right) “twisting” 46 of the sym- metric bimo dule Γ( R B A ; E ) C ( R BA ; C ) of se ctions of the bund le E by t he two “pul l-b ack ” isomorphi sms π • A : A → C ( R B A ; C ) , π • B : B → C ( R B A ; C ) . # This reconstruction theorem for imprimitivity bimo dules is actually only the starting po int for the dev elopment of a complete “biv ariant” version of Serr e-Swan equiv a- lence and T ak ahashi dua lit y . In this case we will gener alize the pr evious sp ectra l theorem to (classes o f ) bimo dules o ver commutativ e unital C*-algebra s that are more genera l than imprimitivit y bimodules; furthermo re the appropriate notion of morphism will b e intro duced in order to get a categorica l dualit y . W e plan to return to this sub ject elsewhere 47 . 45 R BA is a compact Hausdorﬀ subspace of Sp( A ) × Sp( B ) homeomorphic to Sp( A ) (resp. Sp( B )) via the pr o jections π A : R BA → Sp( A ) (resp. π B : R BA → Sp( B )). 46 If M is a left module o ve r C and φ : A → C i s an isomorphism, the left twisting of M by φ is the module ov er A deﬁned by a · x := φ ( a ) x for a ∈ A and x ∈ M . 47 P . Bertozzini, R. Con ti, W. Lewke eratiyutkul, Biv ariant Serre-Sw an Equiv alenc e, in pr eparation. 21 A C*-category [GLR, Mit] is a category C suc h that the sets C AB := Ho m C ( B , A ) are complex Banach spaces and the comp os itio ns are bilinear maps, there is an inv olutive antilinear contrav aria n t functor ∗ : Hom C → Hom C acting identically on the ob jects s uc h that x ∗ x is a p os itive ele men t in the ∗ -alg ebra C AA for every x ∈ C B A (that is, x ∗ x = y ∗ y for some y ∈ C AA ), k xy k ≤ k x k · k y k , ∀ x ∈ C AB , y ∈ C B C , k x ∗ x k = k x k 2 , ∀ x ∈ C B A . In a C*-categor y C , the sets C AA := Hom C ( A, A ) a re unital C* -algebra s for a ll A ∈ Ob C . The sets C AB := Hom C ( B , A ) hav e a na tur al structure o f unital Hilb ert C*-bimo dule on the C*-alg ebras C AA on the rig h t and C B B on the left. A C*-catego ry is comm utativ e if the C*- algebras C AA are Abe lia n for all A ∈ Ob C . The C*-categor y C is full if all the bimo dules C AB are full 48 . A basic example is the C*-catego ry of linear b ounded maps b etw een Hilb ert spac e s . A Banac h bundle [FD, Section I.13] ( E , p, X ) is given by a cont inuous op en surjection p : E → X of H ausdo rﬀ t op olog ical s paces, who se total spac e E is equipp ed with a contin uous partial op eration of addition + : { ( e 1 , e 2 ) | p ( e 1 ) = p ( e 2 ) } → E , a contin uous op eration of multiplication b y sca la rs · : C × E → E and a contin uous no rm k · k : E → R , making all the ﬁb ers E x := p − 1 ( x ) Banach spa ces a nd such that, for all x ∈ X , the sets of the for m B U,ǫ := { e ∈ E | p ( e ) ∈ U, k e k < ǫ } , where ǫ > 0 and U is a neighbour ho o d of x ∈ X , cons titute a ba se of neighbourho o ds of 0 x ∈ E x in the top olog y of E . If the to p o lo gical base spa ce X is equipped with the algebra ic structure of categ ory (let X o be the set of its units, let r , s : X → X o be its range a nd source maps and let X n := { ( x 1 , . . . , x n ) ∈ × n j =1 X | s ( x j ) = r ( x j +1 ) } b e its set of n -comp osable morphisms), we further req uir e that the compo sition ◦ : X 2 → X is a co nt inuous map. If X is an inv ol utiv e category (a lso known as a ∗ - categor y [GLR, Mit] or a “ da gger category ” [Sel, AbC2]) i.e. there is a map ∗ : X → X with the pro p er ties ( x ∗ ) ∗ = x, ∀ x ∈ X and ( x ◦ y ) ∗ = y ∗ ◦ x ∗ , for all ( x, y ) ∈ X 2 , we also requir e ∗ to be contin uous. An inv olutive category X is called an in v olutiv e in v erse cat egory if x ◦ x ∗ ◦ x = x for all x ∈ X . A F ell bundle 49 o v er the inv olutiv e i n v erse category X (see also [BCL4]) is a Ba nach bundle ( E , p, X ) whose total space E is equipp ed with a mult iplication deﬁned on the set E 2 := { ( e , f ) | ( p ( e ) , p ( f )) ∈ X 2 } , denoted by ( e, f ) 7→ ef , and an in volution ∗ : E → E such that e ( f g ) = ( 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 restrictio n of ( e, f ) 7→ ef to E x × E y is bilinear , k ef k ≤ k e k · k f k , ∀ e, f ∈ E 2 , ( e ∗ ) ∗ = e, ∀ e ∈ E , p ( e ∗ ) = p ( e ) ∗ , ∀ e ∈ E , ∀ x ∈ X , the restriction of e 7→ e ∗ to E x is conjugate linear , ( ef ) ∗ = f ∗ e ∗ , ∀ e, f ∈ E 2 , k e ∗ e k = k e k 2 , ∀ e ∈ E , e ∗ e ≥ 0 , ∀ e ∈ E , where, in the last line we mea n that e ∗ e is a p ositive elemen t in the C*-a lgebra E p ( e ∗ e ) . 48 In this case C AB are i mprimitivity bimodules. 49 F ell bundles ov er topological groups w ere ﬁrs t int ro duced b y J. F ell [FD, Section II.16] and later generalized to the case o f groupoids by S. Y amagami (see A. Kumjian [Ku ] o r P . Muhl y- D. Williams [MW] and references therein) and to the case of inv erse semigroups b y N. Sieb en (see R. Exel [Ex, Section 2]). 22 It is in fact easy to see that for every x ∈ X o , and more generally for every Hermitian idempo ten t x = x ◦ x = x ∗ ∈ X , the ﬁber E x is a C*-a lgebra. A F ell bundle ( E , p, X ) is said to b e unital if the C*-alg ebras E x , for x ∈ X o , are unital. Note that the ﬁb er E x has a natural structure of Hilb ert C*-bimo dule ov er the C*-a lgebras E r ( x ) on the left and E s ( x ) on the right. A F ell bundle is sa id to be saturated if the ab ove Hilber t C*-bimo dules E x are full. Note also that in a saturated F ell bundle, the Hilb ert C* -bimo dules E x are imprimitivity bimo dules. Let O be a set and X a compact Hausdor ﬀ top o lo gical space. W e denote b y R O := { ( A, B ) | A, B ∈ O } the “total” equiv alence relation in O a nd b y ∆ X := { ( p, p ) | p ∈ X } the “ diagonal” equiv alence re la tion in X . Deﬁnition 4.3. A top olo gic al sp ac e oid 50 ( E , π , X ) is a satur ate d unital r ank-one F el l bund le over the pr o duct involutive top olo gic al c ate gory X := ∆ X × R O . Let ( E j , π j , X j ), for j = 1 , 2, b e t wo spaceo ids (where X j = ∆ X j × R O j , with O j sets and X j compact Hausdo r ﬀ top olog ical spa ces for j = 1 , 2). Deﬁnition 4.4. A morphism of sp ac e oid s ( E 1 , π 1 , X 1 ) ( f , F ) − − − → ( E 2 , π 2 , X 2 ) is a p air ( f , F ) wher e • f := ( f ∆ , f R ) with f ∆ : ∆ 1 → ∆ 2 a c ontinuous map of t op ol o gic al sp ac es and f R : R 1 → R 2 an isomorph ism of e quivalenc e r elations; • F : f • ( E 2 ) → E 1 is a ﬁb erwise line ar c ontinuous ∗ -functor such t hat π 1 ◦ F = ( π 2 ) f , wher e ( f • ( E 2 ) , π f 2 , X 1 ) denotes a given choic e of an f -pul l-b ack 51 of ( E 2 , π 2 , X 2 ) . T op ological spaceoids constitute a ca teg ory if comp osition is deﬁned by ( g , G ) ◦ ( f , F ) := ( g ◦ f , F ◦ f • ( G ) ◦ Θ) , where Θ is the natural isomorphism from f • ( g • ( E 3 )) to ( g ◦ f ) • ( E 3 ), and (having c hosen ( E , π , X ) to be the ι X -pull-back of itself ) with identit ies g iven by ι ( E , π , X ) := ( ι X , ι E ) . The category T (1) of con tinuous maps b etw een compact Ha us dorﬀ spaces ca n b e natura lly ident iﬁed with the full sub catego ry of the categor y T of spaceo ids with index set O containing a single element. T o ev ery ob ject X ∈ Ob T (1) we asso cia te the trivial C -line bundle X X × C ov er the inv olutiv e categor y X X := ∆ X × R O X with O X := { X } the one p oint set. T o every contin uous map f : X → Y in T (1) we asso ciate the mor phism ( g , G ) with g ∆ ( p, p ) := ( f ( p ) , f ( p )), g R : ( X , X ) 7→ ( Y , Y ) and G := ι X X × C . Note that the trivial bundle ov er X X is naturally a f -bull-back of the trivial bundle ov er X Y and hence G can b e taken as the identit y map. Let C and D be t wo full c ommut ative small C*- categor ie s (with the same cardinality o f the set of ob jects). Denote by C o and D o their sets of identities. 50 Note that , despite the name and the in volv emen t of groupoids, spaceoids are not directly related with the fractaloids introducted by I. Cho -P . Jorgensen [CJ]: our spaceoids are group oids but are equipp ed with a suitable bundle structure and fractaloids (graph group oids with fr actal pr operties) are not a horizontal categoriﬁcat ion of self-simi lar fractal spaces. 51 Here we denote by π f 2 : f • ( E 2 ) → X 1 the pro j ection of the pull-back bundle ( f • ( E 2 ) , π f 2 , X 1 ) and b y f π 2 : f • ( E 2 ) → E 2 the m or phism of bundles such that π 2 ◦ f π 2 = f ◦ π f . 23 A morphism Φ : C → D is an ob ject bijective ∗ -functor, i.e. a map such that Φ( x + y ) = Φ( x ) + Φ( y ) , ∀ x, y ∈ C AB , Φ( a · x ) = a · Φ( x ) , ∀ x ∈ C , ∀ a ∈ C , Φ( x ◦ y ) = Φ( x ) ◦ Φ( y ) , ∀ x ∈ C C B , y ∈ C B A Φ( x ∗ ) = Φ ( x ) ∗ , ∀ x ∈ C AB , Φ( ι ) ∈ D o , ∀ ι ∈ C o , Φ o := Φ | C o : C o → D o is bijectiv e . T o every space oid ( E , π , X ), with X := ∆ X × R O , we can as so ciate a full co mm utative C*-catego ry Γ( E ) as follows: • Ob Γ( E ) := O ; • ∀ A, B ∈ Ob Γ( E ) , Hom Γ( E ) ( B , A ) := Γ(∆ X × { ( A, B ) } ; E ), where Γ(∆ X × { ( A, B ) } ; E ) denotes the set of con tinu ous sections σ : ∆ X ×{ ( A, B ) } → E , σ : p AB 7→ σ AB p ∈ E p AB of the restriction of E to the ba se space ∆ X × { ( A, B ) } ⊂ X ; • for all σ ∈ Hom Γ( E ) ( A, B ) and ρ ∈ Hom Γ( E ) ( B , C ): ρ ◦ σ : p AC 7→ ( ρ ◦ σ ) AC p := ρ AB p ◦ σ B C p , σ ∗ : p B A 7→ ( σ ∗ ) B A p := ( σ AB p ) ∗ , k σ k := sup p ∈ ∆ X k σ AB p k E , with op erations taken in the tota l s pace E of the F ell bundle. W e extend now the deﬁnition o f Γ to the morphism of T in o rder to obtain a con trav ar iant functor. Let ( f , F ) b e a morphism in T fro m ( E 1 , π 1 , X 1 ) to ( E 2 , π 2 , X 2 ). Given a section σ ∈ Γ( E 2 ), we consider the unique section f • ( σ ) : X 1 → f • ( E 2 ) such that f π 2 ◦ f • ( σ ) = σ ◦ f and the compo sition F ◦ f • ( σ ). In this wa y we g et a map Γ ( f , F ) : Γ( E 2 ) → Γ( E 1 ) , Γ ( f , F ) : σ 7→ F ◦ f • ( σ ) , ∀ σ ∈ Γ( E 2 ) . Prop ositio n 4.5. ([BCL4 , Pr op ositio n 4.1]) L et ( E 1 , π 1 , X 1 ) ( f , F ) − − − → ( E 2 , π 2 , X 2 ) b e a mor- phism in T , the map Γ ( f , F ) : Γ( E 2 ) → Γ( E 1 ) is a morphism in A . The p air of maps Γ : ( E , π , X ) 7→ Γ( E ) and Γ : ( f , F ) 7→ Γ ( f , F ) gives a c ontra variant functor fr om the c ate gory T of sp ac e oids to the c ate gory A of smal l ful l c ommutative C*-c ate gories. W e pr o ceed to asso ciate to every commutativ e full C*-category C its sp ectral spaceoid Σ( C ) := ( E C , π C , X C ), see [BCL4, Section 5] for deta ils . • The set [ C ; C ] of C -v alued ∗ -functors ω : C → C , with the weak est top ology making all ev aluations co nt inuous, is a compa ct Haus dorﬀ top o logical space. • By deﬁnition tw o ∗ -functors ω 1 , ω 2 ∈ [ C ; C ] are unitarily equi v alent if there exists a “ unita ry” natural tra sformation A 7→ ν A ∈ T betw een them. This is true iﬀ ω 1 | C AA = ω 2 | C AA for all A ∈ Ob C . 24 • Let Sp b ( C ) := { [ ω ] | ω ∈ [ C ; C ] } deno te the base sp ectrum of C , deﬁned as the set of unita ry equiv alence classes of ∗ -functors in [ C ; C ]. It is a co mpact Hausdo r ﬀ spa ce with the quotient top olog y induced by the map ω 7→ [ ω ]. • Let X C := ∆ C × R C be the dir ect pro duct top ologica l in volutiv e category o f the com- pact Hausdorﬀ ∗ - categor y ∆ C := ∆ Sp b ( C ) and the top olo gically discrete ∗ -categor y R C := C / C ≃ R Ob C . • F or ω ∈ [ C ; C ], the set I ω := { x ∈ C | ω ( x ) = 0 } is an ideal in C and I ω 1 = I ω 2 if [ ω 1 ] = [ ω 2 ]. • Denoting by [ ω ] AB the p oint ([ ω ] , ( A, B )) ∈ X C , we deﬁne: I [ ω ] AB := I ω ∩ C AB , E C [ ω ] AB := C AB I [ ω ] AB , E C := ] [ ω ] AB ∈ X C E C [ ω ] AB . Prop ositio n 4.6 . ([BCL4, Pr op osition 5.7]) The map π C : E C → X C , t hat sends an element e ∈ E C [ ω ] AB to the p oi nt [ ω ] AB ∈ X C has a natur al stru ctur e of unital r ank-one F el l bund le over the top olo gic al involutive c ate gory X C . Let Φ : C → D b e an ob ject-bijective ∗ -functor b etw een t wo small commutativ e full C*-catego ries with spac e oids Σ( C ) , Σ( D ) ∈ T . W e deﬁne a morphism Σ Φ : Σ( D ) ( λ Φ , Λ Φ ) − − − − − → Σ( C ) in the ca tegory T : • λ Φ : X D ( λ Φ ∆ ,λ Φ R ) − − − − − → X C where λ Φ R ( A, B ) := (Φ − 1 o ( A ) , Φ − 1 o ( B )), for all ( A, B ) ∈ R Ob D ; λ Φ ∆ ([ ω ]) := [ ω ◦ Φ ] ∈ ∆ Sp b ( C ) , fo r all [ ω ] ∈ ∆ Sp b ( D ) . • The bundle U [ ω ] AB ∈ X D C λ Φ R ( AB ) I λ Φ ([ ω ] AB ) with the maps π Φ : ([ ω ] AB , x + I λ Φ ([ ω ] AB ) ) 7→ [ ω ] AB ∈ X D , x ∈ C λ Φ R ( AB ) , Φ π : ([ ω ] AB , x + I λ Φ ([ ω ] AB ) ) 7→ ( λ Φ ([ ω ] AB ) , x + I λ Φ ([ ω ] AB ) ) ∈ E C is a λ Φ -pull-back ( λ Φ ) • ( E C ) of the F ell bundle ( E C , π C , X C ). • Since Φ( I λ Φ ([ ω ] AB ) ) ⊂ I [ ω ] AB for [ ω ] AB ∈ X D , we can deﬁne a map Λ Φ : ( λ Φ ) • ( E C ) → E D by  [ ω ] AB , x + I λ Φ ([ ω ] AB )  7→  [ ω ] AB , Φ( x ) + I [ ω ] AB  . Prop ositio n 4.7. ([BCL4, Pr op osition 5.8]) F or any morphism C Φ − → D in A , the map Σ( D ) Σ Φ − − → Σ( C ) is a morphism o f sp e ctr al sp ac e oids. T he p air of maps Σ : C 7→ Σ( C ) and Σ : Φ 7→ Σ Φ give a c ontr avaria nt functor Σ : A → T , fr om the c ate gory A of obje ct- bije ctive ∗ -functors b etwe en smal l c ommutative ful l C*-c ate gories to the c ate gory T of sp ac e oids. W e can now state o ur main dua lity theo rem for commutativ e full C*-ca tegories: Theorem 4.8. (P. Bertozzini-R. Conti-W. L ewke er atiyutkul [BCL4, The or em 6.5]) Ther e exists a duality (Γ , Σ) b etwe en t he c ate gory T of obje ct-bije ctive morphisms b etwe en top o- lo gic al sp ac e oids and the c ate gory A of obje ct- bije ctive ∗ -functors b etwe en smal l c ommu- tative ful l C*-c ate gorie s, wher e 25 • Γ is the fun ct or that to every sp ac e oid ( E , π , X ) ∈ Ob T asso cia tes the smal l c ommu- tative ful l C*-c ate gory Γ( E ) and t hat to every morphism b etwe en top olo gic al sp ac e oids ( f , F ) : ( E 1 , π 1 , X 1 ) → ( E 2 , π 2 , X 2 ) asso ciates t he ∗ -fun ctor Γ ( f , F ) ; • Σ is the functor that to every smal l c ommutative ful l C*-c ate gory C asso cia tes its sp e ctr al sp ac e oi d Σ( C ) and t hat t o every obje ct-bije ctive ∗ -functor Φ : C → D of C*-c ate gories in A asso ciates the morphism Σ Φ : Σ( D ) → Σ( C ) b etwe en sp ac e oids. The natural isomo rphism G : I A → Γ ◦ Σ is pr ovided by the horizon tally categori ﬁe d Gel’fand transforms G C : C → Γ(Σ( C )) deﬁned by G C : C → Γ( E C ) , G C : x 7→ ˆ x where ˆ x AB [ ω ] := x + I [ ω ] AB , ∀ x ∈ C AB . Prop ositio n 4 .9. ([BCL4, The or em 6.3]) The functor Γ : T → A is r epr esent ative i.e. given a c ommutative ful l C*-c ate gory C , the Gel’fand t r ansform G C : C → Γ(Σ( C )) is a fu l l isometric (henc e faithful) ∗ - functor. The natural isomorphism E : I T → Σ ◦ Γ is provided by the ho rizon tally categoriﬁed “ev aluatio n” transforms E E : ( E , π , X ) ( η E , Ω E ) − − − − − → Σ(Γ( E )), deﬁned a s follows: • η E R ( A, B ) := ( A, B ) , ∀ ( A, B ) ∈ R O . • η E ∆ : ∆ X → ∆ Sp b (Γ( E )) , p 7→ [ γ ◦ ev p ], where ev p : Γ( E ) → ⊎ ( AB ) ∈ R O E p AB is the ev aluation ma p given by σ 7→ σ AB p that is a ∗ -functor with v alues in a o ne dimensional C*-catego ry that actually determines 52 a unique p oint [ γ ◦ ev p ] ∈ ∆ Sp b (Γ( E )) . • U p AB ∈ X Γ( E ) η E R ( AB ) / I η E ( p AB ) with the pro jection ( p AB , σ + I η E ( p AB ) ) 7→ p AB , and with the E Γ( E ) -v alued ma p ( p AB , σ + I η E ( p AB ) ) 7→ σ + I η E ( p AB ) , is a η E -pull-back ( η E ) • ( E Γ( E ) ) o f Σ(Γ( E )). • Ω E : ( η E ) • ( E Γ( E ) ) → E is deﬁned b y Ω E : ( p AB , σ + I η E ( p AB ) ) 7→ σ AB p , ∀ σ ∈ Γ( E ) AB , p AB ∈ X . In particular , with such deﬁnitions w e can pr ov e: Prop ositio n 4.1 0. ([BCL4, Th e or em 6.4]) The fun ctor Σ : A → T is r epr esent ative i.e. given a sp ac e oid ( E , π , X ) , the evaluation tr ansform E E : ( E , π , X ) → Σ(Γ( E )) is an isomorphi sm in t he c ate gory of sp ac e oids. W e are no w working on a num ber of generalizatio ns and extensions of our horizontal categoriﬁe d Gel’fand duality: # The ﬁrst immediate possibility is to extend Gel’fand duality to include the ca se of categorie s of g e neral ∗ - functors b etw een full commutativ e C*- categor ie s . This will necessarily r equire the c o nsideration of catego ries o f ∗ - relators (see [BCL1]) b e tween C*-catego ries. # Our dualit y theo r em is for now limited to the case of full commutativ e C*- categor ie s and further work is necessar y in order to e xtend the res ult to a Gel’fand duality for non-full C*-catego ries. 52 There is alwa ys a C v alued ∗ -functor γ : ⊎ ( AB ) ∈ R O E p AB → C and any t wo compositions of ev p with suc h ∗ -functors are unitarily equiv alen t b ecause they coincide on the di agonal C*-algebras E p AA . 26 # V ery in teresting is the po ssibility to generalize our dua lit y to a full spectr a l theory for non-commutativ e C*-catego r ies and F ell bundles in term o f endofunctors with target in the ca tegory of F e ll line-bundles. This might b e a step in order to make contact with the notion of “F ell bundle geometry ” introduced by R. Ma rtins [Marti1, Marti2, Marti3] for a categor ical reformulation of the spec tral triple in the standa rd mo del. # F urthermore we w ould like to explore if our appro ach will allow to develop ca teg ori- ﬁcations of J. Dauns- K.H. Hofmann theorem [DH], R. Cirelli-A. Mani` a-L. Pizzo c - chero [CMP] sp ectral theorem and G. E lliott-K. Kaw amu ra [E K, Ka w] Serre-Swan equiv alence for g eneral non-commutativ e C*-a lgebras. # Similarly , it might b e impor tant to study the rela tion betw een our sp ectral s pa ceoids and other spec tr al notions such as lo cales a nd top oi that a re alre a dy used in the constructive sp e c tral theorems by B. Banachewki-C. Mulvey [BM] and C. Heunen- K. Landsman-B. Spitters [HLS1, HLS3, HLS4]. In the same order of ideas, mo ti- v ated by a gener a l sp ectra l theor y for C*- categor ie s, it is worth in vestigating in the non-commutativ e case the connection b etw een C* -categor ies, sp ectral spaceoids a nd categoriﬁe d notions of (loca le) quan tale already develop ed for (commutativ e) C*- algebras (see D. Kruml-J. Pelletier-P . Resende-J. Ro sicky [KPRR], D. Kruml-P . Re- sende [KrR] P . Resende [Res], L. Cr a ne [Cr2] and references therein for details). # The existence of a horizo n tal categoriﬁe d Gel’fand transform might b e re lev an t for the study of harmonic ana lysis o n commutativ e group oids. In this direction it is natural to in vestigate the implications for a Pon trjagin dualit y for commutativ e group oids and later, in a fully non-co mm utative co ntext, the r elations with the theory of C*-pseudo-multiplicativ e unitaries that has been recen tly developed by T. T immer mann [Ti1, Ti2, Ti3, Ti4]. # Extremely intriguing for its p ossible physical implica tions in algebraic quantum ﬁeld theory is the app ear ance of a natural “lo cal gauge structure” on the sp ectra : the sp ectrum is no mor e just a (topo logical) space, but a sp ecial ﬁb er bundle. Possible relations with the w or k of E . V as selli [V a1, V a2, V a 3 , V a4] on con tinous ﬁelds of C*-catego ries in the theory of sup erselection secto rs and especia lly with the recent work on net bundles and gaug e theory by J. Rob erts- G. Ruzzi-E. V a s selli [RR V1, RR V2] remain to b e explo r ed. 4.2.2 Hi g her C*-categories. In our last fo r thcoming work 53 , we pr o ceed to further ex tend the ca tegoriﬁcatio n pr o cess of Gel’fand duality theor em to a full “vertical catego riﬁcation” [B a 2]. F or this purpo se we ﬁrst provide, via globular sets (see T. Leins ter ’s b o ok [Le]), a suitable deﬁnition of “str ict” n -C* -catego r y . In pr actice, without entering here in further technical details (see the slides [B2, Pages 93- 104] for a de e per ov erview) a strict higher C*-category C (or more generally a higher F ell bundle over a hig he r inv erse ∗ -categor y X ), is provided b y a strict higher ∗ -c a tegory C ﬁbered over a strict hig her inverse ∗ -categ ory X whose comp ositions and in volutions satisfy , ﬁber wise at all levels, “appropria te versions” of all the proper ties listed in the deﬁnition of a F ell bundle. 53 P . Bertozzini, R. Con ti, W. Lewk eeratiyutkul, N . Suthichitranon t, Stri ct Higher C*-categ ories, in preparation. 27 In the sp ecial case of commutativ e full strict n -C* -categor ies, we dev elop e a sp ectral Gel’fand theorem in term of n -spac e oids i.e. rank-o ne n -C*-F ell bundles o ver a “particular” n - ∗ -categ ory (that is g iven b y the dir ect pro duct o f the diag o nal equiv alence relation o f a compact Hausdorﬀ space and the quotient n - ∗ -ca tegory C / C of an n -C* -categor y C ). # Unfortunately our deﬁnition is for now limited to the ca se of strict hig her C*-c at- egories. O f course, as always the case in higher category theory , a n ev en more int eresting problem will b e the characterizatio n o f suita ble a x ioms for “weak higher C*-catego ries”. 54 This is o ne of the main obstacles in the development of a full categoriﬁc a tion of the notion of sp e ctral triple and of A. Connes non-co mm utative geometry . # Note that several examples a nd deﬁnitions of 2-C*-c a tegories a re already av ailable in the literature (s e e for example R. Longo-J. Rober ts [LR] and P . Zito [Z]). In general such cases will no t e x actly ﬁt with the strict version of our axio ms for n - C*-categories . Actually we exp ect to hav e a co mplete hier arch y o f deﬁnitions of higher C*-categories depe nding on the “depth” at which some ax ioms are required to b e satisﬁed (i.e. some prop erties can b e re q uired to hold o nly for p -a rrows with p higher than a cer tain depth). # In our work, w e deﬁne (Hilbe r t C*-)mo dules ov er strict n -C*-c a tegories and in this way we ca n pr ovide interesting deﬁnitions of n -Hilbe r t s pa ces and start a developmen t of “higher functional analysis ”. Extr emely interesting for us will b e to understand the re la tion with categor iﬁed notions of higher vector and Hilber t spaces develope d by M. Kapranov- V. V o evods ky [KV], J. Baez-A. Cr ans [Ba 1, BC], J. Elgueta [E] and J. Mor ton [Mor 3, Mor4]. 4.3 Categorical Non-comm utativ e Geometry and Non-comm utativ e T op oi. One of the main go als of our inv estigation is to discus s the interplay betw een ideas of categoriﬁc a tion and non-commutativ e geometr y . Here there is still m uch to be done and we can present only a few suggestions. W o rk is in progr ess. # Every isomorphism class of a full co mm utative C*-category can be iden tiﬁed with a n equiv alence relation in the Picard-Mo rita 1-categ ory of Ab elian unital C*-algebr as. In practice a C*-categ ory is just a “strict implement ation” of an equiv alence r elation sub c ategory o f Picard- Morita. Since morphism of sp ectra l triples (mo r e generally morphisms of non-c ommut ative spaces) a re essentially “sp ecia l cases ” of Morita mor phisms, w e s tarted the study of “sp ectral triples ov er C*-categor ies” and we are now trying to develop a notion o f horizontal categ oriﬁcation of sp ectr a l triples (and o f other s pec tr al geometries) in order to identify a correct deﬁnition of morphism of sp ectral triples that supp or ts a duality with a suitable sp ectr um (in the commutativ e case). The g e ne r al picture that is emerging [B2, Pages 105-1 08] 55 is that a corr ect notion of metric morphism betw een sp ectra l triples is g iven b y a kind o f “biv ariant v ersio n” 54 F or the purp ose of the developmen t of a notion of “weak C*-algebra” (where the usual axioms for product, identit y and in v olution are expected to hold on ly up to i somorphism) it is inte resting to consider the recent work by P . Bouwk negt-K. Hannabuss-V. Mathai [BHM], where “C*-algebras” with a s tr i ctly non-associative pro duct are deﬁned as ob jects internal to suitable monoidal dagger categories. 55 P . Bertozzini-R. Cont i-W. Lewk eeratiyutkul, Sp ectral Geometries o ve r C*-categories and Morphisms of Sp ectral Geometries, in preparation; Horizontal Categoriﬁcation of Sp ectral T riples, work i n progress. 28 of spec tr al triple i.e. a bimo dule ov er tw o diﬀerent algebr as that is equipp ed with a left/right action of “Dirac - like” o per ators. # As a very ﬁrst step in the dir ection of a full “ higher no n- commutativ e g eometry” 56 we plan to start the s tudy of a strict version of “highe r sp ectra l triples” i.e. sp ectral triples ov er strict hig he r C*-catego ries. As in the ca se of hor izontal catego riﬁcation, this will provide some hints for a c o rrect deﬁnition of “higher sp ectral triples ”. # Although at the momen t it is only a specula tive idea, it is very interesting to ex - plore the p ossible relation b et ween such “higher s p ectr a” (higher spaceoids) and the notions of stacks a nd ge rb es a lready us e d in higher gaug e theory . The rece nt work by C. Daenzer [Dae] in the co n text of T-dua lit y discuss a P ontry agin duality b e- t ween commutativ e principal bundles and g erb es that mig h t b e connected with our categoriﬁe d Gel’fand transfo rm for commutative C*-catego ries. # Extremely in triguing is the possible connection b etw een the notions of (categ ory of ) sp ectra l triples a nd A. Grothendieck topo i. Sp eculations in this direction have bee n g iven b y P . Cartier [Car] and are a lso dis c us sed b y A. Co nnes [C9]. A full (categorica l) notion of non-commutativ e space (non-co mm utative Klein pr ogra m / non-commutativ e Gro thendieck top os) is still waiting to be deﬁned 57 . Actually some int eresting prop osa l for a deﬁnition of a “quantum top os” is alrea dy av ail- able in the recent work by L. Crane [Cr2 ] based o n the no tion of “quantaloids”, a ca te- goriﬁcatio n of the notion o f q ua nt ale (see P . Res e nde [Res ] and r eferences therein). A t this level o f gener ality , it is imp ortant to emphasize that our discussio n of no n- commutativ e g eometry has b een esse ntially conﬁned to the consideration of A. Connes ’ ap- proach. In the ﬁeld of algebraic geometry (see V. Ginzburg [Gi], M. K ontsevic h-Y. Soib el- man [KS1 , KS2] and S. Mahanta [Mah1, Mah2, Mah3, Ma h4] as recent references), many other p eople ha ve b een trying to prop o s e deﬁnitions o f non-commutativ e schemes and non-commutativ e spaces (see for exa mple A. Ros e n b erg [R] and M. Ko nt sevich-A. Rosen- ber g [KR]) as “sp ectra ” o f Abelian categorie s (or generaliza tion of Ab elian categories such as triangulated, dg, or A ∞ categorie s). Since ev ery Abelia n category is essentially a categ ory of mo dules, it is in fact usually assumed that an Ab elian category should be considered as a top os of sheaves ov er a non-co mm utative space. # It is worth noting that the ca tegories naturally arising in the theory of self-adjoint op erator a lgebras and in A. Connes’ non-commutativ e geometry are ∗ -mono idal cat- egories (see [BCL4] for detailed deﬁnitions). The monoidal pro per ty is p erfectly in line with the recent prop os al b y T. Maszczyk [Mas] to construct a theory of algebraic non-commutativ e g eometry based on Ab elian categ o ries equippe d with a monoidal structure. A t this point it is actua lly tempting (in our opinion) to think that also the in volutiv e structures (and o ther prop erties strictly related to the exis tence of a n in v olution including modular theory 58 ) are going to play so me vital role in the correct deﬁnition of a non-comm utative generaliz a tion of spa c e . But this is still sp eculation in pr ogress ! 56 On this topic the reader is strongly advised to read the in teresting discussions on the “ n -category caf ´ e” http://gol em.ph.utexa s.edu/category/ and in particular: U. Schreiber, Connes Sp ectral Geometry and the Standard M odel II, 06 September 2006. 57 P . Bertozzini, R. Conti, W. Lewk eeratiyutkul, Non-commutativ e Klein-Cartan Program, work in progress. 58 See section 5.5.1 f or some r eferences. 29 # Finally , there ar e strong indications (V. Dolgushev-D. T amark in- B. Tsygan [DTT]) 59 coming again from “algebra ic non-commutativ e geometry” that a pr op er categori- ﬁcation of non-commutativ e g e o metry might actually b e p ossible only consider ing ∞ -categor ies. The implications for a prog r am of ca teg oriﬁcation of A. Connes’ spec- tral triples is not yet clear to us. 5 Applications to Ph ysics. In this ﬁnal section we would lik e to spend some time to int ro duce (in a no n- techn ical wa y) the mathematical readers to the consider ation of some extremely imp ortant topics in quantum physics that are ess e n tially motiv ating the construction of non-commutativ e spaces, the us e o f ca tegorical idea s and the even tual merg ing of these t wo lines of thoug h t. The tw o main sub jects of our discussion, non-commutativ e geo metry and category theory , hav e b een s eparately used and a pplied in theoretical ph ysics (although not as widely as we would have liked to see) and we are go ing to revie w here some of the main historica l steps in these directio ns . An ywa y , o ur feeling is tha t the mo st impor tant input to physics will come from a kind of “combined” approa ch where non-co mm utative and ca tegorical s tructures are applied in a “synerg ic wa y” in an “algebra ic theor y o f qua n tum gr avity” (A QG). A co ncrete pr op osal in this direction is prese nted in section 5.5.2. 5.1 Categories in P h ysics. Category theor y has b een conceived as a too l to formalize ba sic structures (functors, na t- ural transformations ) that ar e omnipresent in algebraic top olo gy . Its lev el of abstr action has b een a n obstac le to its utilization even in the mathematics c o mm unity and so it do es not come as a surpr ise that fruitful applications to physics had to wait. Probably , the ﬁrst p erso n to call for the usage of categor ical methods in physics has b een J. Rob erts in the seven ties. The joint w ork of S. Doplicher and J. Robe rts [DR1, DR2] on the theory o f sup erselectio n sectors in alg ebraic quantum ﬁeld theory 60 is one o f the mos t elo quent exa mples o f the p ow er of catego ry theo r y when applied to fundamental physics: giving a full explana tio n of the origin of compact gauge g r oups of the ﬁrst kind and ﬁeld algebras in quan tum ﬁeld theory and pr oviding at the s ame time a gene r al T anna k a-Kre ˘ ın duality theor y for compact gr oups, wher e t he dual of a compact group is given by a particular monoidal W*-ca tegory . Since then, mono idal ∗ -ca tegories are a commo n topic of inv estigation in algebraic quantum ﬁeld theo ry , 61 where several p eople a re still working on po ssible v ariants and extensions of sup erse le ction theory . 62 The r ole o f categ ories in physics, mo re r ecently , has been stre ssed als o from diﬀerent area s of research suc h as conformal ﬁeld theory (G. Segal [Se]) and top ologica l quan tum ﬁeld theory (M. A tiy ah [At]). A very in teresting r elation b etw een axiomatiza tions of these 59 See also the very detailed discussion on the blog “ n -category caf´ e”: J. Baez, Inﬁnitely Categoriﬁed Calculus, 09 F ebruary 2007. 60 The texts b y R. Haag [H], H. Araki [A], D. Kastler [K2] a nd the recen t b o ok [BBIM] con tain detailed int ro ductions to sup erselection theory in algebraic quantu m ﬁeld theory . 61 F or a complete li st of al l relev an t pap ers and a recen t “philosophical” ov erview of the sub ject see H. Halvorson-M. M ¨ u ger [HM] and also R. Brunetti-K. F reden hagen [BrF]. 62 A large li terature is of course av ailable on monoidal categories, tensor categories (see for example M. M ¨ uger [Mu] for a survey) and their application to the theory of “quan tum groups” (see for example R. Street [St] for a clear introduction) as we ll as many othe r diﬀeren t sub jects, but it is outside the scope of this survey to ent er into further details on these topics. 30 top ological qua ntum ﬁeld theories (in their “e x tended” functorial version [BD1]) and the Haag-Ka stler axioms for algebraic quantum ﬁeld theory has b een pro po sed in the r ecent work by U. Schreib er [Sch]. C. Isha m has been the pioneer in sugg esting to consider to po i as basic structures for the construction of alternative qua ntum theories in which ordinary set theor etic concepts (in- cluding rea l/complex n umbers and classical t wo v alued lo gic) are repla ced b y mor e gener al top os theor etic notions. His resear ch with J . Butterﬁeld [BI1, BI2, BI3, BI4, BI5, BI6, BHI] and more rece ntly with A. D¨ o ring [DI1, DI2, DI3, DI4 , DI5, Dor1, Dor2] has p ola rized the attent ion tow ards a p oss ible usa ge o f topo s theo ry in qua n tum mec hanics 63 and quan tum gravit y , a n idea tha t has inﬂuenced several other author s working on quantum gravity . S. Abramsky - B. Co eck e [AbC1, AbC2, AbC3, AbC4, Ab1, Ab3, Co1, Co2, Co 3, Co4, Co5, Ab7] with their collab or ators R. Duncan-B . Edwards-D. Pa vlovic-E. Paquette-S. Perdrix- J. Vicary [AbD, CPav , CPa1 , CPa2, CP P1, CPP2, CD1, CD2, CPPe, CE, CPV, Vi1, Vi2] are actively dev eloping a c ategorica l axiomatizatio n for quantum mechanics based on symmetric monoidal ca tegories with intriguing links to knot theory , logic and c o mputer science [Ab2, Ab4, Ab5, Ab6]. Related works on categ orical quantum logic in the setting o f dagger -monoidal o r dag ger categorie s with kernels a re carried on by C. Heunen-B. J a cobs [He1 , He2 , HeJ]. N. Landsman [La1, La2, La4] in his study of qua n tization and o f the rela tion betw een clas - sical Poisson geometry and o pe r ator algebras of q ua nt um systems, has b een constantly exploiting techniques from catego ry theory (group oids, Morita equiv alence). His recent works M. C a sp ers-C. Heunen-N. Landsman-B. Spitters [HLS1, CHLS, HLS3, HLS4], fur- ther elab ora te on the C. Isham-J. Butterﬁeld-A. D¨ oring prop osal to base ph ysics on top os theory , opening the wa y to reconsider algebra ic q uantum theor y as a “classical theory” living in a suitable “spectr al to p o s ” and prop osing an extens io n of general cov aria nce in terms of geometr ic morphisms b etw een top oi [HLS2]. J. Baez has b een one of the ﬁr st pioneers and the mo st prominent adv o cate in the develop- men t, with J. Dolan, of higher categor ical structures [BD1] (“op etopic” n -ca tegories [BD2, Ba2], categ oriﬁcation [BD3, BD4], 2-Hilb ert spaces [Ba1]) and in the usage of categori- cal metho ds in quan tum physics a nd in quan tum g ravit y [Ba3, B a4, Ba6]. J.Baez and his collab ora to rs and students, T. Bartles- A. Cra ns-A. Hoﬀnung- L. La ng ford-A. Lauda- J. Morton-M. Neuchl-C. Rogers-U. Schreiber- M. Sh ulman-M. Stay-D. Stevenson-C. W alker hav e b een eleb orating huge p ortions of “higher algebra e xtensions” of mathematics [BSh] (braided mo noidal 2-ca tegories [BN], 2-Lie a lgebras [BC], 2- tangles [BLa n1, BLan2], 2- groups [BLa , BSt, BBFW], 2-bundles [Bart], group oidiﬁcatio n [BHW]) and their applica- tions to ph ysics: higher ga uge theories [Ba5, Bar t, BSc1, BSc2, BCSS], higher symplectic geometry [BHR, BRo], qua nt um computation [BS] and “combinatorial” quan tum mec han- ics [Mor1]. A new emerg ing ﬁeld o f “catego rical quan tum gravity” is dev eloping (see the w orks by L. Cr ane [Cr1, Cr 2, Cr3, Cr4], J. Baez [Ba3, B a4, Ba6] a nd, for a catego rical appr o ach via top ological quantum ﬁeld theory co bo rdism, also J. Morton [Mor 2, Mor 3, Mo r4]. 5.2 Categorical Cov ariance. Cov aria nc e of physical theor ie s has been alwa ys disc us sed in the limited do ma in o f groups acting on spaces. 63 See the recen t pap ers b y C . Heunen-K. Landsman-B. Spitters [HLS1, HLS2 , HLS3, HLS4 ] and the preprint b y C. Flori [ Fl]. 31 • Aristotles’ physics is based on the cov ariance group S O 3 ( R ) of rotatio ns in R 3 that was the s uppo sed symmetry group of a three dimensional vector space with the center of the Ear th at the o rigin. • Galilei’s relativity principle require s as cov ariance gr o up the Galilei gro up, which is the ten parameter s symmetry group of the Newtonian s pa ce-time (i.e. a family E t of three dimensiona l E uclidean spa ces par ametrized by elements t in a one dimensio nal Euclidean space T ) gener ated by 3 space tra nslations, 1 time translation, 3 rotations and 3 bo o sts. • Poincar´ e cov ar iance g roup consists o f the semidirect pro duct of Lorentz group L with the g roup o f transla tions in R 4 and it is the symmetry g roup o f the four dimensional Minko wski space (a n aﬃne four dimensional s pa ce mo deled o n R 4 with metr ic of signature ( − + ++)). • Einstein co v ariance gr oup is the group of diﬀeomorphisms of a four dimensional Lorentzian manifold (note that in this case the metric and the c a usal structure is not preserved). Diﬀerent obser vers ar e “related” through transfor mations in the g iven cov arianc e group. # There is no deep physical or op era tional r eason to think that only groups (or qua nt um groups) might be the right mathematical structure to ca pture the “transla tion” betw een diﬀerent observers and actually , in our opinion, ca tegories provide a muc h more suitable en viro nmen t in whic h also the discussio n of “partial translations” betw een obser vers can b e descr ib e d. W ork is in progr ess o n these issues [B]. The substitution of g roups with c ategories (or graphs), as the basic cov ariance struc- ture of theories, s ho uld be a key ingredient for all the appro aches based on de ductio n of ph ysics from o p er ationally founded principles of information theor y (see C. Rovelli [Ro1] and A. Grin baum [Gri1, Gri2, Gri3]) and, in the context of quantum gravit y , a lso for theories based on the formalism of quantum casua l his tories (see for example E . Hawkins- H. Sahlmann-F.Makopo ulou [HMS] a nd F. Mar kopoulou [Ma3]). As an example of the r elev ance o f the idea of categorical cov ariance, we mention several new w orks b y R. Br unetti-K. F r edenhagen-R. V er ch [BFV], R. Brunetti-G. Ruzzi [Br R] and R. Br unetti-M. P orr mann-G. Ruzzi [BPR] that, following the fundamental idea of J. Dimo ck [Dim1, Dim2], aim a t a g eneralizatio n of H. Araki-R. Haag -D. Kastler alg e- braic quantum ﬁeld theory axio matization 64 , that is suitable for an Einstein cov ar iant background. Similar ideas are also used in the non- c ommut ative versions of the a xioms recently prop os ed by M. Pasc hke a nd R. V er ch [PV1, PV2]. 5.3 Non-comm utativ e Space-Time. There are three main rea sons for the in tro duction of no n-commutativ e spa ce-time struc- tures in ph ysics and for the deep interest develop e d b y ph ysicists for “non-commutativ e geometry” (not only A. Connes ’one): 64 See H. Araki’s and R. Haa g’s b o oks [A, H] and also K. F redenha gen-K.-H. Rehren-H. Seiler [FRS] for a discussi on and con textualization of algebraic quan tum ﬁeld theory . 32 • The aw areness that quantum eﬀects (Heisen be rg uncertaint y principle), coupled to the general rela tiv is tic eﬀect of the energy- momentum tenso r on the curv ature of space-time (Einstein equation), entail that at very sma ll scales the space - time man- ifold s tructure might b e “unphysical”. • The belie f that mo diﬁcation to the short scale structur e of space-time might help to resolve the proble ms of “ ultraviolet divergences” in quan tum ﬁeld theory (that arise, by Heisen b erg uncertaint y , from the ar bitrary high momen tum as so ciated with arbitrar y small length scale s ) a nd o f “singular ities” in gener al rela tivity . • The in tuition that in order to include the remaining physical forces (n uclear and electromagne tic) in a “geometriz ation” progr am, g oing beyond the one realized for gravit y by A. Einstein’s gener al r elativity , it migh t be necessary to mak e use of geometrical environmen ts more sophisticated than thos e provided by usual Rieman- nian/Lore ntzian geometry . The ﬁrst one to conjecture that, a t small sca le s , space- time modele d by “manifolds” might not b e an op erationally deﬁned concept was B. Riemann himself. A. Einstein immedi- ately recognize d the need to int ro duce “q ua nt um” modiﬁca tions to general relativity and M. Bronstein realized that the sp eciﬁc pro blems pose d by a cov ar iant quan tization of gen- eral relativity were calling for a rejectio n of the usual space-time mo deled via Riemannian geometry . Recent ly a more complete argument has be e n put forward by S. Doplicher- K. F redenhagen-J. Rob erts [DFR1, DFR2] and by many other in several v ariants. J. Wheeler [Wh1] in tro duced the well-known “space-time-foam” term to deﬁne the h y- po thetical geo metrical structure that sho uld sup ersede smo o th diﬀerentiable manifolds at small scales. Non-commutativ e g eometries are a natural c andidate to replace ordinary Lorentzian smo oth manifolds as the arena of ph ysics and provide a rigor ous (although incomplete, yet) formaliza tion of the notion of space-time “fuzziness” . The notion o f non-commutativ e space-time originated from a n idea of W. Heisen b erg 65 that was develop ed by H. Snyder [Sn]. Mor e recently S. Doplicher-K. F redenhagen- J. Rob erts [DFR1, DFR2, Do2, Do3] describ ed a new version of Poincar´ e cov a riant non- commutativ e spac e . An algebra ic quantum ﬁeld theory on such non-c ommut ative spaces is currently under active developmen t (see S. Doplicher [Do4] fo r a recent review) and there are so me ho p es to get in such cases a theo r y that is free fr om divergences. Many other “v ariants” of non- commut ative s pace-time (mostly obta ined by “defo rmation” of Minko wski space- time or a s “homogeneuo s spac e s ” of a “defor med” Poincar´ e gr oup) and non-commutativ e ﬁeld theor y on them are now under in vestigation in theor etical physics (see for e xample J. Mador e [Mad], B. Cer chiai-G. Fio r e-J. Madore [CFM], G. Fiore [Fio1, Fio2, Fio3], G. Fiore-J.W ess [FW], H. Gr osse-G. Lechner [GLe] and r eferences therein), but it is b eyond our s cop e here to enter the details o f their description. On the pa th o f complete “ g eometrization o f matter” envisioned by B. Riemann a nd W. K. Cliﬀor d, A. Einstein has b een one of the few to s tress the conceptual need for a geo metr ical trea tment of the nuclear and electro magnetic forces alo ng side with gravit y . T. Kaluza and O. Klein’s theor y of uniﬁcation of e le ctromagnetism with gr avit y via “extra- dimensional” Lorent zian manifolds was clearly going in this direction, but it has g ained some popular it y o nly r ecently , with the introduction of sup er s trings that, fo r reas ons of int ernal consistency , r equire the existence of (compactiﬁed) extra dimensio ns and whose 65 He comm unicated it in a letter to R. P eierls who shared the suggestion with W. P auli and R. Oppen- heimer. 33 treatment of gr avit y is manifestly no n-background-indep endent (in the sense required by general relativity). T o date, the most succes s ful achiev ement in the direction of “ geometrizatio n of ph ysical int erac tions”, has been obtained b y A. Connes [C2, CLo, C3, CM2] (s e e also the works by A. Chamseddine-A. Co nnes [CC1, CC2 , CC3, CC4, Ch], A. Chamseddine-A. Connes- M. Marcolli [CCMa] a nd J . Barre tt [Ba r] for a Lor entzian version) who has pr omoted the view that the complexity o f the standard mo del in pa rticle physics sho uld b e reco n- sidered as revealing the features of the non-co mm utative geometry of spa ce-time. The progra m describ es (for the moment only at the cla ssical level) ho w g ravit y and all the other fundamental in terac tio ns of pa rticle physics a rise as a kind of gravitational ﬁeld on a non-c ommut ative space-time given by a spec tr al triple ov er a C* - algebra that is a tensor pro duct o f the algebr a of continous function on a 4-dimensional o rientable s pin-manifold and a ﬁnite dimensional r e a l C*-algebr a. 5.4 Sp ectral Space-Time. What we c a ll here “sp ectral space-time” is the idea that space- time (commu tative or not) has to be “reco nstructed a posterio ri”, from other oper ationally deﬁned degrees o f free do m, in a sp ectral wa y . The o rigin of such “pr egeometrica l philosophy” is less clear. Space-time as a “relational” a posterio ri en tity o riginate from ideas of G.W. Leibniz, G. Berkeley , E. Mach. Although pregeometrical sp ecula tions, in western philoso ph y , pr obably da te as far bac k as Pythagor as, their ﬁrst mo de r n inca rnation probably starts with J. Wheeler ’s “preg eome- tries” [Wh2, MLP] and “it fr om bit” [Wh3] prop osals . R. Gero ch [Ge], with his Einstein algebras, was the ﬁrs t to sugges t a “transition” fro m spaces to algebr a s in o rder to s olve the proble m of “sing ularities” in genera l relativity . The fundamental idea that space-time can be recov ered from the sp eciﬁca tion of suitable states of the sys tem, has b een the s ub ject of s c attered sp eculations in alg e br aic qua nt um ﬁeld theory in the past b y A. Ocneanu 66 , S. Doplicher [Do1], U. Bannier [Ba n] and, in the “mo dular lo calization prog ram” (see R. Brunetti-D. Guido-R. Lo ngo [BGL] and references therein), has b een conjecture d by N. Pinamonti [Pi]. Extremely imp ortant r igorous results including a complete recons truction of Mink owski space-time [SuW] have been achiev ed in the “ geometric mo dular a ction” prog ram b y D. Buchholz-S. J. Summers (see D. Buchholz-S. J . Summers [B S1, BS2 ], D. Buchholz- M. Flo rig-S. J. Summers [BFS], D. Buchholz-O. Dreyer-M. Florig-S. J. Summers [BDFS ], for details and S. J. Summers [Su2] fo r a n excellent revie w a nd additiona l references). More recently the idea has gained imp o rtance in the lig ht of attempts to recons truct quan- tum ph ysics fro m op era tionally founded quantum information (among others , J. Bub- R. Clifton- H. Halvorson [BCH], A. Grinbaum [Gri1, Gri2, Gr i3, Gr i4] and espe cially C. Rovelli’s s uggestion [Ro 8 , section 5.6.4]), but in its full genera lity , the reco struction of space-time is still an unso lved problem. # This is probably b eca use only now the Araki-Haag -Kastler a xiomatization ha s b een suitably extended to incorp or ate general cov aria nc e (R. Brunetti-K. F redenhage n- R. V erch [B FV]), but there are, in our o pinion, other fundamen tal issues that need to be addressed in a completely unconv entional wa y a nd that are r elated to the “philoso phical interpretation” of states a nd observ ables in the theor y in “atemp oral- cov aria nt” context (following ideas of C. Isham and co llab orato r s [I2, I3, 66 As rep orted in A. Jadczyk [Ja] . 34 IL1, IL2, ILSS], C. Rov elli a nd collab or ators [Ro8, Ro1, RR, MP R], J. Hartle [Hart], L. Hardy [Har1, Har2, Har3], J. Dowling-S. J ay Olson [DJO1, DJO2]). That ess ent ial infor mation ab out the underlying space-time is already contained in the algebra o f observ ables of the system (and its Hilb ert space repr e sentation) is clea rly indi- cated b y R. F eynman-F. Dyson [Dy] reconstruction of Maxwell equa tions (and hence of the Poincar´ e group of symmetries) from the comm utation relations of ordina r y non-relativistic quantum mechanics of a free particle, an a rgument recently r e vised and extended to non- commutativ e co nﬁguration spaces b y T. Ko pf-M. Pasc hke [P 1, KP3]. In a sligh tly diﬀer ent context, in their discussion of the construction of the quantum th eor y of spin particles on a (compact Riemannian ma nifold), J. F r¨ ohlich-O. Gr a ndjean-A. Rec k- nagel [FGR 1, FGR2, F GR3, F GR4], ha ve be e n considering several imp ortant unsolved asp ects of the r elationship b etw een the underlying conﬁg uration space of a physical sys- tem and the actual non-c ommut ative geometry exhibited at the level of it s algebra of observ ables (phase-spa ce). The solution of these problems is still fundamental in the con- struction of a theory of sp ectr al space-time and quant um gravit y ba sed on algebras o f observ ables and their states. W e will hav e more to say a bo ut this problem in the ﬁnal section 5.5.2. # That non-co mmutative geo metry provides a s uitable environmen t for the imple- men tation of sp ectra l reconstructio n of space-time from states and obser v ables in quantum physics has b een the main mo tiv a ting idea o f one us (P .B.) since 1990 and it is still an op en work in pr ogress [B1]. 5.5 Quan tum Grav ity . Quantum gravit y is the disc ipline of theoretical physics that deals with the interpla y betw een quant um ph ysics a nd general r e la tivity . The need for resea r ch in this direction was actually recognized by A. Einstein since the birth of gener al r elativity and several peo ple started to w ork on it fro m 1930. Unfortunately , a fter man y y ears of research by so me of the b e st scientists, we d o not hav e y et an established theory , let a lone a mathematically sound frame for thes e q uestions. F ollowing closely C. Isham’s excellent reviews [I1, I4], here b elow we try to summarize the several appro aches to quantum gravit y: 67 a) Quantizat ions of general relativit y . Approaches o f this kind, try to make use of a “ standard v ersion” o f quantum me- chanics to substitute (a mo diﬁed) g eneral relativity with a quantized version. – Canonical quan tization (initiated b y P . Dirac-P . Bergma nn, developed b y R. Ar nowitt-S. Deser -C. Misner and J. Wheeler -B. DeWitt and recently re- vived by A. Sen-A. Ash tek ar and L. Smolin-L. Cr ane-C. Ro velli-R. Gambini and others) is probably the ﬁrst non-per turbative propo sal: it trie s to ﬁnd suit- able canonic a l v ariables to describ e the dynamics of c la ssical gener a l rela tivit y and to per form a quantization on them. After a per io d of stagnation, this approach has been revived under the name of lo op quan tum gra vit y and it is curr ently the most elabor ate non-per turbative (background-indep endent) progra m in quantum g ravit y (see C. Rovelli [Ro8, Ro2] for a n intro duction and also T. Thiemann [Th1, Th2, Th3, Th4 ]). 67 See also Appendix C in C. Rov elli’s bo ok [Ro8] for a detailed history of the sub ject. 35 – Co v arian t quantiza tion (initiated b y L. Ro senfeld, M. Fierz-W. Pauli and developed by B. DeWitt, R. F eynma n, G. ’t Ho oft) is a background-dep endent per turbative approach (usually the one pr eferred by par ticle physicists) in whic h the no n-Minko wskian part o f the metric tensor is conside r ed as a class ical ﬁeld propaga ting on a ﬁxed Minko wski space and qua ntized a s any other s uch ﬁe ld. The pro of of non-renorma lizability o f general relativity in th is setting has some- how stopp ed any further attempts in this dir ection for cing res earchers to take the stand that general relativity is no t a fundamen tal theory and prompting the developmen t of supergr avit y and later str ing theory (see the approa ch es listed in item c) her e be low). This appro ach is also receiving rene wed attention b ecause of imp or ta n t results in the asymptotic safet y scenario originally suggested by S. W einberg [W ei] a nd dev elop ed by M. Niedermaie r-M. Reuter [NR] and R. Percacci [Per]. – P ath integral quan tization (initiated by C. Misner-J . Wheeler , developed b y S. Ha wking-J. Hartle) is a non-p erturbative pro p o s al that is c hara c terized by its use of the for malism of F eynman funct ional in tegrals for quantization. In its ﬁrs t incarnation, Eucl i dean quan tum gra vit y , the theory was p er forming a path quantization of a Riemannian v ersion o f genera l relativity and it was motiv a ted by semiclassical studies b y S. Ha wking o n the thermodyna mic prop er ties of black-holes (quantum ﬁeld theory on curved space- times). Discretized versions of functional in tegra l quantization (see [Lol] for a review) hav e been or iginally based on Reg ge calculus propo sed b y T. Regge [Reg], but recen tly the approach has been revived in a L orentzian version k nown a s causal dynamical triangulations that has achiev ed extremely go od r esults in the reconstructio n of some of the features o f general relativity (such as the four dimensionalit y o f spac e - time) in the “large s c a le limit” (see J. Am b jørn- J. Jurkie w ic z-R. Loll [AJL1, AJL2 ] a nd references therein). – Co v arian t canonical quan tization is a non-p ertur bative a pproach based either on the usage o f ﬁeld quantization via R. Peierls bra ck ets [Pei, BEMS] (se e B. DeWitt [D W] for details) or on co v ariant quantization on phas e s pace [ABR]. – Precanonical qua n tum gra vit y is a non-per tur bative cov ariant appr oach based on T. De Donder-H. W eyl [DeD , W ey] Hamiltonia n formulation of ﬁeld theory that is studied by I. K anatchik ov (se e [Kan] and references therein). – Aﬃne quan tum gra vit y , developed by J. Klauder [Kl1, Kl3, Kl4, Kl5 , Kl6, Kl7, Kl8, Kl9 ], is based on a non-ca nonical (aﬃne) quantization that makes heavy use of coherent states and pr o jection o p er ator metho ds [K l0, K l2] for dealing with quantum co nt raints. b) Relativi zations of quan tum mecha nics. In this cas e w e a re for cing as muc h as p ossible of the formalism required by general cov aria nce on qua ntum mechanics (even tually mo difying it if necessary). Although the prop osa l is very natural, there are almo s t no developed progr ams following this approach, probably b ecause traditionally “quantization” has alwa ys b een the sta n- dard route; – F ollowing seminal ideas by P . Dirac [Dir], J.-M. Souria u [So u] a nd G. Es po sito- G. Gion ti-C. Stornaiolo [E GS], C. Ro velli [Ro4, Ro5, Ro 6, Ro7, Ro8] has de- veloped a cov ar iant form ulation of class ical and quan tum mec hanics that is appropria te for the needs of qua n tum rela tivit y . 36 – K. F redenhage n-R. Haag [FH], a nd more recently R. Brunetti-K. F redenhagen- R. V erch [BFV] hav e been s tudying the problem in the context of algebr aic quantum ﬁeld theor y . – A few resea rchers, among them B. Mielnik [Mi] and mo r e rec e ntly A. Ash tek ar- T. Schilling [AS], C. Br o dy-L. Hugston [BH1, BH2], hav e been tr ying to mo dify the usua l phase-s pace of quantum mec hanics (the K¨ a hler manifold g iven b y the pro jective s pace o f a separable Hilbe rt spac e with the F ubini-Study metric) in or der to allow mo re “geo metrical v ariabilit y” in the hop e to facilitate the confrontation with g eneral relativity . – The “consistent histories formulation” of quant um mec hanics elabor ated by R. Griﬃths [Gr], R. Omnes [Om1, O m2], M. Gell-Mann-J. Har tle [Har t], and more r ecently the “history pro jection o per ator theor y” develop ed b y C Isham- N. Linden-N. Sa vvidou-S. Shreck enberg [I2, I3, IL1, I L 2, ILSS], provides another cov aria n t gener a lization of qua n tum mec hanics that is suitable fo r qua n tum gravit y [IS1, IS2 , Sav1 , Sav2, Sav3, Sav4 , Sav5, Sav6, Sav7 ]. – Some pr op osal to modify quantum mechanics in a “ relational” or “ cov aria nt wa y” starting with H. Everett-J. Wheeler and more rece ntly with C. Rov- elli [Ro1, Ro 4], C. Rovelli-M. Smerlak [RS ] or with the use of ca teg ories/ to po i (L. Cr ane [Cr 1, Cr2], J . Butterﬁeld-C. Isha m [BI3, BI4, BI5, I5, I6], C. Isham- A. D¨ oring [D I1, DI2, DI3, DI4, DI5, Dor 1, Dor2], C. Flori [Fl]) in order to make it suitable for quantization of genera l rela tivit y (either in the case of lo op quantum gravit y progra m o f other more radical appro aches) can be cons idered also in this catego ry . c) General rel ativit y as an emergen t theory . Here quantu m mechanics and quan tum ﬁeld theory are consider ed as basic and general relativity is obtained as an appr oximation from a fundamental theory . These kind of approa ches pionee r ed b y A. Sa kharov’s “induced gravity” ha ve always been the most fashionable among particle physicists and a re now gaining momentum a lso among “relativists” . – String theory in a ll o f its v ariants is the mo st p opula r approach to qua nt um gravit y . W e r efer to M. Green-J. Sch w arz- E. Witten [GSW], J . Polc hinski [Pol ] and K. Baker-M. Ba ker-J. Sc hw arz [BBS] as standard reference s . – Analog gra vit y and other mo dels o f general relativity ba sed on quantum solid state ph ysics, acoustic, h ydro dynamics. F or a review, see f or example G. V olovik [V o1, V o2, V o3] and C. B a rcel´ o-S. Lib erati-M. Visser [BL V]. – Hˇ ora v a gra vity [Hor] is a non-relativistic quantum ﬁeld theory o f gr avitons in 3 + 1-dimensio ns wher e Lorentz inv arianc e and r elativity emerge only a s approximations in the long sca le limit. – Emergent Gra vit y : inspired b y the partial achiev ement s of “ analog grav- it y” a new clus ter are a of r esearch in gravit y , seen as an emergent la rge-sca le phenomenon, is ga ining mo mentum (se e for example the pap ers b y F. Girelli- S. Liber ati-L. Sindoni [GLS1 , GLS2, GLS3 , GLS4, GLS5]). Some of the more recent developmen ts o f the “path integral” approach to quantum gravity such “group ﬁeld theory” by D. Oriti [Or1, Or2, Or3] or “causal dynamical tri- angulations” by J. Amb jørn-J . Jurkie wic z-R. Loll [AJL1, AJL2] as well as some mo re radica l prop osa ls to o btain spa ce (but not time!) and gravit y as emergent fro m a qua n tum substra tum such a s “internal q uantu m gravit y” by 37 O. Dreyer [D1, D2, D3, D5, D4, D6], “quant um causal history” and “quan- tum g raphity” b y F. Mar kopoulou and collab o rators [Ma1, Ma2, Ma 3, Ma 4, HMS, KM, KMS, KMSe], “causal sets” by R. Sorkin [So1, So2, So3] can also be consided in this catego ry . d) Quan tum me c hanics as an emergent theory (without mo diﬁcation of g eneral relativity). V ery few p eople hav e b een trying this ro ad, probably b ecause every one is ex pec ting that a classica l theo ry (as general re la tivity is) should b e sub ject to qua n tum mo d- iﬁcations in the small distances regime, there ar e anyw ay some incomplete ideas in this dir ection: – G. t’Ho oft [tH1, tH2, tH3, tH4, tH5 , tH6] is prop os ing mo dels to r eplace quan- tum mechanics with a classical fundamen tal deterministic theor y . – The dev elopments of g eometro dynamics , as describ ed in the recent review by D. Giulini [Giu], sugg e s t the pos s ibilit y to recover a t lea st some of the prop erties of matter from pure g eometry . – The theory of geons (H. Hadley [Ha1, Ha2, Ha3]), tries to sim ulate the quan tum behaviour of elementary particles starting with lo calize d geometrical s tructures on the Lor ent zian manifolds of gener a l rela tivit y . – L. Smolin [Sm2] has recently co nsidered the p os s ibilit y that quan tum mec hanics might aris e as a sto chastic theory induced by non-lo c a l v a riables. – E. Prug ov ecki [P r1, Pr2, Pr3] also propo sed an approach to quantum mec hanics through sto chastic pro cesses in a g eneral relativistic geometric a l setting. – The theo ry of g ravitational induced collapse of the quantum w av e function by R. Penrose (see [Pe] and reference s therein) c an b e considered in this catego ry . e) Pregeome trical approac hes (suggested by J. Wheeler ) are alternative approaches that require at least some ba sic modiﬁca tions of general relativity and quantum me- chanics that might both “emerg e” b y some deeper dynamic of degr ees of freedom not necessarily related to an y macrosc opic geo metrical entit y . Most of these theories are at least par tially background-indep endent (dep ending on the amoun t of “residual” geometrical structure used to deﬁne their kinema tic). The ma in problems arising in pregeometric a l theories is usually the descr iption of an appr opriate dynamic and the recov ery fro m it of some “ approximate” descr iption of gener al re lativity and ordinary quantum physics in the “mac roscopic” limit. The prop o sals that can b e listed in this category are extremely heterogeneous and they might range from “gener alizations” of other more conserv ative approaches: – algebra ic quantum gr avit y: a generalization of lo op quantum g ravit y recently developed by K. Giesel- T. Thiemann [GT1, GT2, GT3, GT4], – group ﬁeld theory quantum gravity: a p ow erful extension of the path integral approach to quantum g ravit y pro po sed by D. Oriti [Or1, Or3], to mor e radical pa ths (that w e co llect here just for the beneﬁt of the in terested reader): – twistor theor y (R. Penrose [PeR, Pe]), – quantum co de (D. Fink elstein [Fi1, Fi2]), 38 – causal sets (R. Sor kin [So1, So2, So3]), – causalo ids (L. Hardy [Har 1, Har2, Har3, Har4, Har5, Har6]), – computational appro ach (S. Lloyd [Ll1, Ll2]), – internal quantum gravit y (O. Dreyer [D1, D2, D3, D5, D4, D6]), – quantum causal history (F. Markopo ulou [Ma1, Ma2, Ma3, Ma4], E. Hawkins- F. Markop oulou-H. Sahlman [HMS], D. Kr ibs-F. Makopoulou [KM]); quantum graphity (T. K onopk a-F. Markop oulou-L. Smolin-S. Severini [KMS, KMSe]), – abstract diﬀerential geometry (A. Ma llios [Mal1, Ma l2, Ma l3], J . Raptis [Ra1, Ra2, Ra3, Ra4, Ra 5, Ra6], A . Mallios - J. Raptis [MR1, MR2, MR3 , MR4], A. Mallios-E . Rosing e r [MRo]), – categor ical approaches (J. Baez [Ba 3, Ba4, Ba6], L. Crane [Cr1, Cr2, Cr3, Cr4], J. Morton [Mo r3], J. Butterﬁeld-C. Isham [BI3, BI4, BI5], C. Isham [I6, I7, I8, I9], A. D¨ oring -C. Isha m [DI1, DI2, DI3, DI4, DI5, Dor1, Dor2]), C. Flo ri [Fl], – non-commutativ e g eometry approa ches: 68 ∗ via deriv a tions on non-co mmutative (gro upo id) algebr a s: J. Madore [Mad], M. Heller-Z. Odrzyg ozdz-L. Pysia k-W. Sas in [HOS, HS1, HS2, HS3, HS4, HS5, HS6, HPS1, HP S2, HPS3, HPS4, HOP S1, HOPS2, HOPS3, HOP S4], ∗ via deformation quantization (Moy al-W eyl): P . Aschieri and c o llab ora- tors [As1, As2, ADMW, ADMSW, ABDMSW], ∗ via qua nt um groups: S. Ma jid [Ma j3, Ma j4, Ma j5, Ma j6], ∗ via A. Co nnes’ non-commutativ e ge o metry: M. Pasc hke [P2], A. Connes-M. Marco lli [CM2]. Since w e are he r e mainly in terested in A. Connes’ non-comm utative geometry , we are going to conclude by e x amining a bit more in detail the situa tio n as regar ds its p ossible applications to quantum g ravit y . 5.5.1 A. Connes’ Non-com m utativ e Geom etry and Gra vi t y It is often claimed that non-commutativ e geo metry will b e a key ingredient (a kind of quantum version of Riemannian geo metry) for the for mulation of a fundamental theory of quantum g ravit y (see for example L. Smolin [Sm1] and P . Martinetti [Mart2]) and actually non-commutativ e g eometry is o ften listed a mong the current alternative appro aches to quantum gr avity . In reality , with the only notable exceptions o f the e xtremely interesting progr ams out- lined in M. Paschk e [P 2] and in A. Connes-M. Mar c olli [CM2], a foundationa l approach to quantum ph ysics based on A. Connes’ non- commut ative geometr y has never bee n pro - po sed. So far , mo st o f the current applica tions of A. Connes’ non- c o mm utative geometry to (quantu m) gravit y hav e been limited to: • the study of some “q uantized” example: C. Rov elli [Ro 3], F, Besnar d [Be1], • the use of its mathematical framework for the reformulation of classical (Euc lide a n) general relativity: D. K astler [K4], A. Chamseddine-G. F elder-J. F r¨ o hlich [CFF], W. Kalau-M. W alze [KW], C. Rovelli-G. Landi [LR1, LR2, Lan2], 68 See also the recen t pa p ers by B. Booss - Bav nbek-G. Esp osito-M. Lesch [BEL] and F. M¨ ul ler- Hoissen [ M-H] for more detailed and alternativ e surveys on noncommutat ive geometry in gra vity . 39 • attempts to use its mathematical framework “inside” some alrea dy established the- ories suc h a s s tring theor y (A. Connes-M. Douglas-A. Sch warz [CDS ], J. F r¨ o hlich- O. Gr andjean-A. Rec knagel [F GR3] and J. Bro dzki-V. Matha i-J. Rosenberg-R. Sz- ab o [BMRS]) and lo op gravit y (J. Aastrup-J. Gr imstrup-R. Nest-M. Pasc hke [AG1, A G2, AGN1, AGN2, AGN3, A GN4, AGN P], F. Girelli-E. Livine [GL]), • the formulation of Hamiltonian theo ries of g ravit y o n globally hyper bo lic cases, where only the “spacia l-slides” ar e descr ibed b y non-commutativ e g eometries: E. Hawkins [Haw], T. Ko pf-M. Pasc hke [KP 1, K P 2, Ko]. In a slig h tly diﬀerent dir ection, there a re some imp ortant a reas of resea rch that are so me- how connected to the problems of quantum gra vity and tha t s e em to sugg est a more prominent ro le of T omita- T akesaki mo dular theor y 69 in quantum ph ysics (a nd in pa r ticu- lar in the physics o f g ravit y): • Since the work of W. Unruh [U], it ha s b een conjectured the exis tence of a deep connection b etw een gr avit y (equiv alence principle), thermal physics (hence T omita- T akesaki and KMS-sta tes) and quantum ﬁeld theo ry; this idea has not b een fully exploited so far. This line o f thought is actually reinforced by the works on thermo - dynamical deriv a tio n of E instein equation b y T. Jacobson [J ac] (see also R. B r ustein- M. Ha da d [BH] and M. Parikh-S. Sark ar [PaS]) . • Starting from the w ork s of J. Bisognano- E. Wic hmann [BW1, BW2], G. Sewell [Sew] and more recen tly , H. J . Borchers [Bo1], there is mo un ting evidence that T omita- T akesaki mo dular theory should play a fundamen tal role in the “spectr a l rec o n- struction” of the space-time informatio n from the a lgebraic setting of states and observ ables. The mo st interesting r esults in this dir ection have been obtained s o far: - in the theory of “half-sided mo dular inclusions ” and mo dula r in tersections (see H.- J. Borchers [Bo2] and references ther ein, H. Araki-L . Zsido [AZ]); - in t he “ geometric mo dular action” program (see for mo re details D. Buc hholz- S. J . Summers [BS1, BS2 ], D. Buchholz-M. Flor ig-S. J . Summers [BFS ], D. Buchholz- O. Dreyer-M. Florig- S. J . Summer s [BDFS ], S. J. Summers-R. White [SuW]); - in “mo dular n uclearity” (see for more details R. Haag [H] and, for r ecent applicatio ns to the “form factor program”, D. Buchholz-G. Le ch ner [BL, Le1, Le2, Le3, Le4, Le5], D. Buchholz-S. J. Summers [BS3]); - in the “mo dula r lo calization pr ogra m” (see B. Schroer -H.-W. Wiesbr o ck [Sc1, Sc2, SW1, SW2], R. Brunetti-D. Guido-R. Longo [BGL], F. Lled´ o [Lle1], J. Mund- B. Sc hro er -J. Yngv anson [MSY] and N. Pinamo n ti [Pi]). • Starting with the co nstruction o f cyclic cocy cles from supe r symmetric quantum ﬁeld theories by A. Ja ﬀe- A. Lesniewski-K. O sterwalder [JLO 1, JLO2], there has alw ays 69 The ori ginal ideas about mo dular theory were developed by M. T omita [T o1, T o2]. W e r efer to the texts b y M. T ak esaki [T], B. Bl ac k adar [Bl] for a modern mathematical int ro duction and to O . Bratteli- D. Robinson [ BR], R. Haag [H] for a more physics oriente d presen tation. Excellen t updated reviews on the relev ance of mo dular theory in quant um ph ysics are given b y S. J. Summers [ Su1], H.- J. Borc hers [Bo2] and the recent works by D. Guido [G] and F. Ll ed´ o [Lle2] (but see also R. Longo [L1]). Outside the realm of operator algebras, T omita-T ak esaki theorem for classical statistical mec hanical systems has been discussed b y G. Galla vot ti-M. Pulvi r en ti [GP] and a strictly related corresp ondence betw een modular theory and Poisson geomet ry has been p oint ed out by A. W einstein [W]. 40 bee n a cons ta n t interest in the po ssible deep str uctural relationship b etw een sup er- symmetry , mo dula r theory of t yp e I I I von Neumann alg ebras and non-commutativ e geometry (see D. K a stler [K4] and A. Jaﬀe-O. Stoytc hev [J, JS]). Some deep r esults by R. Lo ng o [L3] established a bridge b et ween the theory o f super selections sectors and cyclic co c ycles obtained b y sup er-KMS states . The recent work by D. Buchholz- H. Grundling [BGr1, BGr2] opens ﬁnally a wa y to c onstruct super- K MS function- als and sp ectral tr iple s in algebr aic quan tum ﬁeld theor y (see S. Carpi-R. Hillier- Y. Kaw ahigas hi- R. Lo ngo [CHK L]). • In the co nt ext of C. Rov elli “thermal time hypo thesis” [Ro8] in qua ntum gravity , A. Connes-C. Rov elli [CR] (see also P . Ma rtinetti-C. Rovelli [MR] and P . Mar- tinetti [Mar t1, Mar t3]) hav e b een using T o mita-T akesaki modular theory in order to induce a macrosc o pic time evolution for a r elativistic quantum sys tem. • A. Connes-M. Mar colli [CM2] with the “co oling procedur e” ar e pro po sing to examine the o per ator alg ebra of observ ables of a qua nt um gravitational system, via mo dular theory , at “diﬀere n t temp e r atures” in o rder to ex tr act by “ s ymmetry breaking” an emerging geometry . # The idea that space- time migh t b e sp ectrally r econstructed, via non- c ommut ative geometry , from T o mita-T akesaki mo dular theory applied to the algebra of physical observ ables was elab or ated in 199 5 by one of the authors (P .B.) and indep endent ly (motiv a ted b y the po s sibility to obtain cyclic co cycles in alge br aic quantum ﬁeld theory from modular theor y) b y R. Longo [L2]. Since then this conjecture is still the main sub ject and motiv a tion of our inv estigation [B, BCL]. Similar speculations on the interplay b etw een modular theory and (s o me aspects of ) space- time geo metr y hav e been suggested b y S. Lord [Lo, Section VII.3 ] and by M. Paschk e- R. V erch [PV1, Section 6 ]. # One o f the authors (R.C.) has raised the somehow puzz ling question whether it is po ssible to reinterpret the one par ameter gr oup of modula r a utomorphisms as a renormaliz a tion (semi-)gr oup in ph ysics. The c o nnection with P . Cartier’s idea of a “universal Galois group” [Car], currently developed by A. Connes-M. Mar colli, is extremely intriguing. 5.5.2 A Prop osal for (Mo dular) Algebraic Quan tum Grav ity . Our ongoing resea rch pr o ject [B1, BCL, B3 ] 70 is a iming a t the constructio n of an alge- braic theory of quan tum gra vit y in which “non-commutativ e” space-time is spec tr ally reconstructed from T omita- T a kesaki modula r theory . What we prop ose is to develop a n appro ach to the foundations of qua nt um ph ysics techn ically based on a lgebraic quantum theory (op erato r algebras) and A. Connes’ non- commutativ e geometry . The resear ch is building on the experience alrea dy gained in our previous/curr ent mathematics research plans on “mo dula r sp ectr a l triples in non- commutativ e geometr y and ph ysics” [BCL] 71 and on “categ o rical non-commutativ e g e om- etry” a nd is conducted in the sta ndard of ma thematical rigo ur typical o f the tra dition of mathematical physics’ res earch in a lgebraic quantum ﬁeld theo ry [A, H]. In t he mathematical f ramework of A. Co nnes ’ non-comm utative g eometry , we ar e ad- dressing the pro blem o f the “sp ectral reconstruction” of “geometries” from the underlying 70 P . Be rtozzini, R. Cont i, W. Lewk eeratiyutkul, Mo dular Algebraic Q uan tum Gra vit y , w ork i n progress. 71 Pa rtially supported by the Thai Researc h F und TRF project RSA458 0030 . 41 op erational data deﬁned by “states” ov er “observ ables’ C*-a lgebras” of ph ysical sys tems. More sp eciﬁcally: # Building on our pr evious rese arch on “modular spectral triples” 72 and on recent results o n semi-ﬁnite sp ectra l triples dev elop ed by A. Carey-J. Phillips-A. Rennie- F. Sukho chev [CP R1, CPR2, CPR3, CPR4] 73 , we m ake use of T omita-T akesaki mo dular theor y o f op er ator a lgebras to ass o ciate, to suitable states ω over inv olu- tive normed alg ebras A , no n-commutativ e geometrical ob jects ( A ω , H ω , D ω ) that are only formally simila r to A. Connes’ sp ectral-tr iples and wher e the “Dirac o p e r - ator” D ω , that is usually taken as the mo dular Hamilto nian K ω = log ∆ ω , satisﬁes the modular inv arianc e prop erty ∆ it D ω ∆ − it = D ω (for some more details see the slides [B3, Pages 74 -77]). # W e are no w dev eloping 74 an “even t” interpretation of the for malism of states and ob- serv ables in algebraic quantum physics that is in line with C. Isham’s “histor y pro jec- tion op erator theory” [I2, I3, IL1, IL2, ILSS] and C. Rov elli’s “ relational/ relativistic quantum mechanics” [Ro1] (for a dditional details see the slides [B3, Pages 7 8-81]). # Making contact with our current resear ch pr o ject o n “catego rical non-co mmu tative geometry” and with other pr o jects in categorica l quantum g ravit y (J. B a ez [Ba4, Ba6] and L. Cra ne [Cr1, Cr 2]), we plan to generalize the diﬀeomorphism cov ar iance group of general relativity in a ca tegorical context and use it to “identify” the degrees of freedom related to the spa tio -tempo ral structure of t he physical s ystem (more details can be found in the slides [B3, Pages 82-84]). # T echniques from “deco herence/einselec tion” (H. Zeh [Ze], W. Zurek [Zu]), “e mer - gence/noise le ss subsystems” (O. Dreyer [D1, D2, D3], F. Markopoulo u [Ma1, Ma2, Ma3, KoM]), sup erselection (I. Ojima [O 1, O2, OT]) and the “co o ling ” pro ce dure developed by A. Connes - M. Marcolli [CM2] are exp ected to be r elev an t in or der to extra c t from our spec trally deﬁned non-commutativ e geometries, a macrosc opic space-time for the pair state/sys tem a nd its “classica l residue” . # Possible repr o duction of q uantu m geo metries a lready deﬁned in the context of lo op quantum gravity (T. Thiemann [Th1, Th4] and J. Aa strup-J. Grimstrup-R. Nest- M. Paschk e [AG1, AG2, A GN1, A GN2, AGN3, AGN4, AGN P]) or in S. Doplicher- J. Ro be rts-K. F r edenhagen mo dels [DFR1, DFR2, Do2, Do 3, Do4] will b e inv esti- gated. If partially s uccessful, the pro ject will ha ve a signiﬁcant fallout: a bac kground-indep endent powerful a pproach to “q uantum relativity” that is suitable for the purp os e of uniﬁcation of physics, geo metry a nd infor mation theory that lies ahea d. App endix: Some Recen t Dev elopmen ts The ﬁrs t version of this arXiv preprint w as wr itten in Nov em b er 20 07 and this s econd corrected and expanded version w as actually prepar ed in Nov em b er 2009. Now, in De- cember 2 011, after mor e than t wo years, a few notable development s o ccur red, but we decided, for this ﬁnal ar Xiv version, to “freeze” the bibliogr aphical references directly dis- cussed in the pap er to Octob er 200 9, limiting our revisio n of the main text to correction of 72 P . Bertozzini, R. Con ti, W. Lewke eratiyutkul, Mo dular Spectral T riples, in preparation. 73 See also M. Laca–S. Nesh vey ev [LN] and A. Carey-S. Nesh vey ev-R. Nest-A. Rennie [CN NR]. 74 P . Bertozzini, Algebraic F ormalism for Rov elli Quantum Theory , in preparation. 42 misprints and up dates only of thos e bibliogra phical sour ces already appea red in preprint befo re Nov em b er 2009 . As a par tial remedy , in this appendix w e provide, for the interested reader, (a very s elective choice of ) a few a dditional bibliog r aphical r eferences to recently a ppea red works mainly related to A. Connes’ s p ectr al triples in non-commutativ e diﬀer e n tial geometr y . On the “Riemannian version of spectr a l triples”: • Lord, Steven; Rennie Adam; V a rilly , Jo seph C., Riemannian Manifolds in Noncom- m utative Geometr y , a rXiv: 1109.2 196v1 . F or lo cally compact sp ectral triples : • Carey , Alan; Gayral, Victor; Rennie, Adam; Suk o chev, F edo r , Index Theo ry for Lo cally Compact Noncommutativ e Geo metries arXiv :1107. 0805v1 . On Lorentzian non- c ommut ative geometry : • V erch Rainer, Qua n tum Dirac Field on Moyal-Mink owski Spacetime - Illustrating Quantum Field Theory ov er Lorentzian Sp e c tral Geometry , arXi v:1106 .1138v1 . • F ranco, Nicolas, Lor e ntzian Approa ch to Noncommutativ e Geo metry , arXiv: 1108. 0592v1 . V ariations of Connes’ reco nstructions theorem for almost commutativ e spe ctral triples: • ´ Ca´ ci´ c, Branimir, A Reconstruction Theor em for Almost-commutativ e Sp ectral T riples, arXiv :1101 .5908v3 . On sp e ctral characterization of is ometries: • Corneliss en, Gunther; de J ong, Jan Willem, The Spe c tral Length of a Map Betw een Riemannian Manifolds, arXiv :1007 .0907v 3 . Spec tr al triples on cro ssed pro ducts: • Bellissar d, Jean; Marc olli, Matilde; Reihani, Kamra n, Dyna mical Systems on Sp ec- tral Metric Spaces, arXi v:100 8.4617 v1 . • Hawkins, A ndrew; Sk alski, Adam; White, Stuar t; Zacharias, Joachim, Sp ectral T riples on Crossed Pro ducts Arising fro m E quicontin uous Actions, arXiv: 1103. 6199v3 . F urther works on application of non-commutativ e geometry to the standa rd mo de l and ph ysics ar e: • Chamseddine, Ali; Connes Alain (2010). Noncommutativ e Geometry as a F rame- work for Uniﬁca tio n o f all F undamental Interactions including Gravity . Part I F ortsch. Phys. 58, 553-6 00, arX iv:100 4.0464 v1 . • Chamseddine, Ali; Connes Alain, Space-Time from the Sp ectral Poin t of View, arXiv: 1008. 0985v1 . • Chamseddine, Ali; Connes Alain (2011). Noncommutativ e Geometric Spaces with Boundary: Spectr al Action, J. Ge om. Phys. 6 1 n. 1, 317- 3 32, arXiv :1008. 3980v 1 . 43 • v an den Dungen, Ko en; v an Suijlekom W alter, E lectro dynamics from Noncommuta- tive Geometry arXi v:110 3.2928 v1 . • Bo eijink, Jord; v an Suijlekom, W alter, The Noncommutativ e Geometr y of Y ang - Mills Fields, arXi v:1008 .5101v1 . • v an den Br o ek, Thijs; v an Suijlekom, W alter , Sup ers ymmetric Q CD and Noncom- m utative Geometr y arXiv: 1003.3 788v1 . • Bhowmic k, Jyotishman; D’Andrea, F rancesco ; Das, Bis warup; Dabrowski, Ludwik, Quantum Gauge Symmetries in Nonco mm utative Geometry , ar Xiv:1 112.36 22v1 . F or the study o f Co nnes’ s pec tral distance see the following pap er s and references therein: • Cagnache, E ric; D’Andrea , F ranc e sco; Martinetti, Pierre; W allet, Jean-Christo phe (2011). The Spectra l Distance on the Moyal Plane, J. Ge om. Phys. 61, 1881 -1897 , arXiv: 0912. 0906v3 . • Martinetti, Pierre; Mercati, Flavio; T o massini, Luca , Minimal Length in Qua n- tum Space and Integrations of the Line Element in Nonco mm utative Geometry arXiv: 1106. 0261v1 . • Martinetti, Pierr e; T omassini, Luca, No nc o mm utative Geometr y of the Mo yal Pla ne: T ranslation Isometries and Sp ectral Distance Bet ween Cohere n t States , arXiv: 1110. 6164v1 . Semi-ﬁnite and mo dular sp ectr al triples are treated in: • Carey , Alan; Phillips, John; Putnam, Ian; Rennie Adam (2011). F amilies o f Type II I KMS States on a Class of C*-algebra s Containing O n and Q N , J. F unct. Anal. 260 n. 6, 1637 -1681 , arXi v:100 1.0424 v1 . • Lai, Alan, On Type I I Noncommutative Geometry and the JLO Char acter, arXiv: 1003. 4226v1 . • Rennie, Adam; Senior, Roger, The Resolven t Co cycle in Twisted Cyclic Co homology and a Lo cal Index F ormula for the Po dles Sphere, arXiv :1111. 5862v1 . • Rennie, Adam; Sitarz, Andrz e j; Y amashita, Ma koto, Twisted Cyclic Cohomology and Mo dular F redholm Mo dules, arX iv:11 11.632 8v1 . • Kaad, Jens, On Mo dular Semiﬁnite Index Theory , arXiv: 1111. 6546v1 . On “Mo rita morphisms” of sp ectral triples, b eside Bram Mesland w ork (now alre ady cited in the main pap er): • Kaad, Jens; Lesch, Matthias, Sp ectral Flow and the Unbounded Kaspar ov Pro duct, arXiv: 1110. 1472v1 . As rega r ds non-commutativ e g eometrical approaches to (lo op) q uantum gravity: • Denicola, Do menic; Marcolli, Matilde; Zainy al-Y asry , Ahmad (20 10). Spin F oams and Noncommut ative Geometry . Classic al Q uantum Gr avity 27 n. 20, 2 0 5025 , 53 pp. arXiv: 1005. 1057v1 . • Gracia - Bondia J ose, Notes on ”Quantum Gra vity” and Non-co mm utative Geometry , arXiv: 1005. 1174v1 . 44 • Lai, Alan, The JLO Character for The Noncommutativ e Space o f Connections of Aastrup-Grimstrup- Nes t arXi v:1010 .5226 v1 . • Aastrup, Johannes; Grimstrup, Jesp er Møller ; Paschk e, Mario (2011 ). Q uantum Gravit y Coupled to Ma tter via Noncommutativ e Geometry , Classic al Qu antum Gr avity 28 n. 7, 0 7501 4, 10 pp., arX iv:10 12.071 3v1 . • Aastrup, Johannes; Grimstr up, J e spe r Møller; F rom Qua nt um Gr avit y to Quantum Field Theory via Noncommutativ e Geometry arXi v:1105 .0194 v1 . Our w ork on mo dular algebra ic quantum gravit y has received a mor e deta iled trea tmen t in the pap er: • Bertozzini, Paolo; Conti, Rob erto; Lewkeeratiyutkul, Wicharn (2010). Mo dular Theory , Non-co mm utative Geometry and Quantum Gravit y . S IGMA Symmetry In- te gr ab ility Ge om. Metho ds Appl. 6 pap er 067, 47 pp. ar Xiv:1 007.40 94v2 . F or studies r ecently app eared on the usa ge of T omita- T akesaki modula r theory in quantum ph ysics a nd lo op quant um gravity , that although not directly r elated to sp ectral tr iples in non-commutativ e geometry , migh t have deep impact on o ur approach to mo dular algebraic quantum gr avity se e the fo llowing preprints a nd the references ther ein: • Asselmeyer-Maluga, T o r sten; Kr ol, J erzy , Constructing a Quantum Field Theory from Spacetime, arX iv:110 7.3458v1 . • Kaminski, Diana, Algebras of Quantum V ariables for Loop Quan tum Gravity , I. Overview, arX iv:110 8.4577v1 . • Kostecki, Ry szard, Information Dynamics and New Geometric F oundations of Quan- tum Theory , arXiv: 1110.4 492v3 . References [AG1] Aastrup J., G rim strup J. ( 2006). Sp ectral T r iples of Holonomy Lo ops, Comm un. Math. Phys. 264 n. 3, 657-681, arXiv:h ep-th/05032 46v2 . [AG2] Aastrup J., Grimstrup J. (2007). Intersecting Connes Noncommutat ive Geometry with Quan- tum Gravit y , Internat. J. Mo dern Phys. A 22 n. 8- 9, 1589-1603, arXiv:he p-th/0601127 v1 . [AGN1] Aastrup J., Gri mstrup J., Nest R. (2009). On Spectral T ri ples in Quantum Gravit y I, Class. Quant. Gr av. 26 n. 6, 065011, arXiv:08 02.1783v1 . [AGN2] Aastrup J., Gri mstrup J., Nest R. (2009). On Sp ectral T ri ples in Quantum Gravit y II, J. No n- c ommut. Ge om. 3 n. 1, 47-81, arXiv:0802.1 784v1 . [AGN3] Aastrup J., Grimstrup J., Nest R. (2009). A New Spectral T riple Over a Space of Connections, Comm. Math. Phys. 290 n. 1, 389-398, arXiv: 0807.3664v1 . [AGN4] Aastrup J., Grim strup J., Nest R. (2009). H ol onom y Lo ops, Spectral T riples and Quan tum Gra vity , Class. Quant. Gr av. 26 n. 16, 165001, arXiv:0902.4191 v1 . [AGNP] Aastrup J., Grimstrup J., Nest R., P aschk e M. (2011). On Semi-Cl assical States of Quan- tum Gravit y and Noncommutativ e Geometry , Comm. Math. Phys. 302 n. 3, 675-696, arXiv:09 07.5510v1 . 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