A Horizontal Categorification of Gelfand Duality

A Horizon tal Categoriﬁcation of Gel’fand Dualit y Paolo Bertozzini ∗ @, Rob erto Co n ti ∗ ‡ , Wic harn Lewkeeratiyutkul ∗ § @ e-mail: paolo. th@gmail. com ‡ Mathematics, Scho ol of Mathematic al and Physic al Scienc es, University of Newc ast le, Cal laghan, N S W 2308, Austr alia e-mail: Robert o.Conti@ne wcastle.edu.au § Dep artment of Mathematics, F aculty of Scienc e, Chulalongkorn University, Bangkok 10330, Thailand e-mail: Wich arn.L@chu la.ac.th 22 July 20 10 This p ap er is de dic ate d to J . E. R ob erts, the “pione er” of C*-c ate gories. Abstract In th e setting of C*-categories , we p ro vide a d eﬁnition of sp ectrum of a commutative full C*-category as a one-dimensional unital F ell bundle o ver a suitable groupoid ( equiv- alence relation) and pro ve a categori cal Gel’fand dualit y theorem g eneralizing the usual Gel’fand d ualit y b et w een the categories of commutative u nital C*-algebras and com- pact H ausdorﬀ spaces. Although many of the individual ingredients that app ear along the w ay are well kno wn, t he somehow uncon ven tional w ay w e “glue” them together seems to shed some new light on the sub ject. MSC-2000: 46L87, 46M15, 46L08, 46M20, 16D90, 18F99. Keywo rds: C*-category , F ell Bundle, Duality , N on-comm ut ative Geometry . 1 In tro duction There is no need to explain wh y the notions of g eometry a nd space are fundamental b oth in mathematics and in physics. Typically , a rigor ous wa y to enco de a t leas t some basic geometrical co n tent in to a mathematical framework mak es use o f the notion of a top ological space, i.e. a set equipped with a topo logical structure. Although b eing just a pr e limina ry step in the pr ocess of developing a mo r e sophistica ted appa ratus, this w ay of thinking has bee n very fruitful fo r b oth a bstract and co ncrete purp oses. In a very imp ortan t developmen t, I. M. Gel’fand lo oked not a t the topolog ical space itse lf but rather at the space of all contin uous functions on it, and realized that these seem- ingly diﬀerent structures ar e in fact essentially the same. In slig h tly mor e pr ecise terms, he found a basic example of anti-equiv alence b et ween certain categor ies of spaces and al- gebras (see for exa mple [Bl, Theorems I I.2.2.4, I I.2 .2.6] or [L, Section 6]). Since on the ∗ Pa rtially supp orte d by the Thai Researc h F und: gran t n. RSA4780022. 1 analytic side C ( X ; C ) is a sp ecial type of a Banach algebr a called a C* - algebra, the study of p ossibly non-commutativ e C*-a lgebras has b een often regar de d a s a go od fra mew or k for “non-commutativ e top ology”. The duality aspect ha s been later enfor ced b y the Serre-Swan equiv alence [K, Theor em 6.18 ] betw een vector bundles and suitable mo dules (see also [F GV] for a Hermitian version of the theorem and [T1, T2, W] for ge ner alizations in volving Hilbert bundles). By then, br eak- through results have contin ued to emer ge b oth in geometry a nd functional ana lysis, based on Gel’fand’s o r iginal in tuition, for ab out four deca des. In connection with physical ideas , L. Crane-D. Y etter [CY] and J. Baez-J . Dolan [BD] have recently propos e d a pr ocess o f categoriﬁcation of mathematical structures, in which sets and functions are replaced by ca teg ories a nd functor s. F rom this pers p ective, in this paper, w e wish to discuss a categ oriﬁcation of the no tion o f space e x tending and mer ging together Gel’fa nd duality and Serr e-Sw a n equiv alence. On one side of the extended duality we ha ve a horizontal categoriﬁcation (a ter minology that we intro duce d in [BCL2, Section 4.2 ]) of the notion of commutativ e C*-algebr a, namely a commutativ e C*-categ ory , o r co mm utative C*-a lgebroid (see deﬁnition 2.1), whilst the corres p onding r eplacemen t of spa ces, the spa ceoids (see deﬁnition 3.2), a r e supp osed to parametrize their sp ectra. Spa ceoids could be describ ed in several diﬀer en t alb eit equiv alent wa ys. In this pap er we have decided to fo cus on a characterizatio n ba sed on the notion of F ell bundle. Originally F ell bundles were introduced in connection with the study of representations o f lo cally compact g roups, but we argue tha t they come to life natura lly on the ba sis of purely to p ologica l principles. Rather sur prisingly , to the b est o f our knowledge, the no tions of commutativ e C * -category and its sp e ctrum hav e no t b een discuss ed b efore, despite the fact that (mostly hig hly no n- commutativ e) C*-c ategories have b een somehow intensively exploited over the last 30 years in several are as of r esearch, including Mack ey induction, super selection structur e in quantum ﬁeld theory , a bstract g roup dualit y , subfactors and the Baum-Connes conjecture. A t any rate, we make frequent contact with the related notio ns that can b e found in the literature, hoping that our appr oac h sheds new light on the sub ject by appro ac hing the matter fro m a kind o f unco n ven tional viewp oin t. Of course, once we hav e a r unning deﬁnition, it seems quite challenging in the next step to lo ok for s o me na tur al o ccurre nc e o f the notion of spaceoid in other contexts. F or instance, we are not aw a r e of any c o nnection with the p o werful concepts that hav e b een int ro duced in algebraic top ology to date. Also, the app earance of bundles in the structure o f the sp ectrum suggests an in triguing co nnection to lo cal gaug e theory but we hav e not develop ed these ideas yet. Some of o ur consideratio ns have been motiv ated b y a categor ical a pproach to non-commutativ e geo metry [BCL2], and it is rewarding that some of its relev ant to ols (e.g., Serre-Swan theorem, Morita equiv alence) a ppear naturally in our context. More s tr ucture is exp ected to emer g e when our categories are equipp ed with a diﬀerentiable structur e . In the ca s e of usual s pa ces, in the setting of A. Co nnes ’ non-commutativ e g eometry [C], this has been achiev ed by means of a Dira c op erator, and then axiomatize d using the concept of sp ectral triple. Here b elow we present a shor t de s cription of the conten t of the pap er. In section 2 we men tion, mainly for the purp ose of ﬁxing our notation, some ba sic deﬁnitions on C*-catego ries. Section 3 op ens recalling the notion of a F ell bundle in the case of in volutiv e inv erse base ca tegories and then pr oceeds to introduce the deﬁnition of the c ategory of spaceoids that will even tually subsume that of compact Ha usdorﬀ spaces in our duality theorem. The cons truction o f a small commutativ e full C*-ca tegory starting from a spaceoid is undertaken in s e ction 4, while the sp ectral a nalysis o f a commutativ e full C*-ca tegory is 2 the sub ject of the more technical section 5 wher e a sp ectrum functor from the categ o ry of full co mm utative C*-categ o ries to our categor y of spaceoids is deﬁned. Section 6 presents the ma in result of this pap er in the form of a duality b et ween a cer tain category of co mmutative full C*- c ategories and the ca teg ory of their sp ectra (spaceoids). A categoriﬁe d v er sion of Gel’fand transform is in tro duced a nd used to prov e a Gel’fand sp ectral reconstructio n theore m for full commutativ e C*-catego ries. Similarly a catego riﬁed ev alu- ation transform is deﬁned for the purpo se of pro ving the representativit y of the sp ectrum functor. Section 7 is devoted to examples a nd applica tions. Here we mention several na tur al e x am- ples o f commutativ e full C*-categories and we pro duce explicit constructions of spa ceoids, either rea ssem bling the Hermitian line bundles o btained in [BCL3] a s spectra of imprim- itivit y Hilbert C*-bimo dules, or (in a way completely independent from C*-categories ) as asso ciated line bundles for a categor iﬁed version of T -torso rs. Among the several pos si- ble future a pplica tions of such ca tegoriﬁed Gel’fand duality , we describe in some detail a categoriﬁe d c o n tinuous functional c a lculus. The pap er e nds with an outlo ok. While in the usual Gel’fand duality theory a sp ectrum is just a compact top ological space , in the situa tio n under considera tio n it comes up equipp ed with a natur al bundle s tr ucture. In particular , the sp ectrum of a comm utative full C*-category can b e identiﬁed with a kind of “group oid of Hermitian line bundles” 1 that can b e co n venien tly describ ed using the language of F ell bundles (or equiv alently as a contin uous ﬁeld o f one-dimensiona l C*-c a tegories). Along the wa y , we a lso discuss several categorica l v e r sions of some well-kno wn conc e pts like the Gel’fand transform that w e think are of indep endent interest. Notice that a no tion of F ourier tra nsform in the setting of co mpact group oids has b een discusse d by M. Amini [A]. Our dualit y is reminiscent o f a n in teresting but widely ignored duality r esult of A. T a k a- hashi [T1, T2]. T ak ahashi’s duality can b e ess en tially understo o d as a duality of catego ries equipp e d with a partially deﬁned w eakly asso c iativ e tensor pro duct and a (weak) in volution, although he do es not explicitly examine such natural str uctures o n his categor ies of Hilb ert bundles a nd Hilber t C*-mo dules. 2 The dua lity presented in this pap er is essen tially a version of the former, where we ex ploit a str ict rea lization of these tensor pro ducts and in volutions, considering comm utative full C*-categor ies and spa ceoids instead of Hilb ert C*-mo dules and Hilber t bundles. Most of the results presented here have b een announced in our survey pap er [BCL2] and hav e b een pr esen ted in several seminars in Thaila nd, Australia, Italy , UK since May 200 6. Note added in pro of. When the prese nt work was under pr eparation, we b e c ame aware of s ome related results in T. Timmer mann’s PhD disserta tion [Ti] where, in the context of Hopf algebraic qua n tum gr oupoids, a very genera l non-commutativ e Pontry agin duality theory is developed by means o f pseudomultiplicativ e unitarie s in C* - modules; a nd als o in V. Deac o n u-A. Kumjian-B. Ramaza n [DKR], wher e a no tion of Ab elian F ell bundle (which contains o ur co mm utative C*-categ o ries as a sp ecial case) is intro duced and a s tructure theorem for them (in terms of “twisted coverings o f groupo ids”) is pr o ved. In the framework of T -duality , a Pont ryagin t yp e duality b et ween commutativ e pr incipal bundles a nd g e r bes has been prop osed b y C. Daenzer [D]; while a generalization of Pon tr y ag in dualit y for lo cally compact Ab elian group bundles has b een provided by G. Go ehle [G]. 1 T o b e more precise, the sp ectrum is uniquely asso ciated to a connected group oid (actually an equiv a- lence relation) of Hermitian l ine bundles, ov er a giv en compact Hausdorﬀ space X , with comp ositions and inv olutions provided b y a strict realization of ﬁb erwise tensor pro ducts and ﬁb erwise duals (the sp ect rum being the ∗ -category whose class of arrows is obtained as union of the Hom sets of this group oid). 2 With more precision, T ak ahashi’s result can be f ormally expressed as a dualit y betw een (a we ak inv olutive form of ) “2-fold-categories” (also called double catego ries in the literature). 3 2 Category A of full comm utativ e C*-categories The notion of C*-category , in tro duced b y J. Rob erts (se e P . Ghez-R. Lima-J. Roberts [GLR] and als o P . Mitchener [M2]) has bee n extensively used in algebra ic quantum ﬁeld theory: Deﬁnition 2.1. A C*-c ate gory is a c ate gory C such tha t: the sets C AB := Hom C ( B , A ) ar e c omplex Banach sp ac es; the c omp ositions ar e biline ar maps such t hat k xy k ≤ k x k · k y k , for al l x ∈ C AB , y ∈ C B C ; ther e is an involut ive ant il ine ar c ontr avariant functor ∗ : C → C , acting identic al ly on the obje cts, such that k x ∗ x k = k x k 2 , for al l x ∈ C B A and su ch that x ∗ x is a p ositive element in the C*-algebr a C AA , for every x ∈ C B A (i.e. x ∗ x = y ∗ y for some y ∈ C AA ). In a C*-category C , the “diagonal blo cks” C AA := Hom C ( A, A ) are unital C*-algebras and the “oﬀ-diag onal blo cks” C AB := Hom C ( B , A ) are unital Hilb ert C*-bimo dules o n the C*-algebr as C AA and C B B . W e say that C is full if all the bimo dules C AB are imprimitivity bimo dules. In practice, ev er y full C*-category is uniquely asso ciated to a “strictiﬁcation” of a sub-equiv alence r e lation of the Picard-Rieﬀel gr o upoid o f unital C*-a lgebras, 3 where the term sub-equiv alence relation of a category deno tes a sub category that is itself an equiv alence relatio n. It is also very useful to see a C*-categor y a s an in volutive catego ry ﬁber ed ov er the equiv alence rela tion of its ob jects: in this way , a (full) C*-categ ory b e- comes a special ca se o f a (satura ted) unital F ell bundle over an in volutiv e (discrete) base category as describ ed in deﬁnition 3.1 b elow. W e say that C is one - dimensional if all the bimo dules C AB are one-dimensio nal and hence Hilber t spaces. Clearly , all o ne-dimensional C*-catego ries ar e full. The ﬁrst problem that we have to f ace is how to select a suitable full s ubcategor y A of “co m- m utative” full C*-catego ries playing the role of horizontal categ oriﬁcation of the category of commutativ e unita l C*- algebras. Since we are working in a c o mpletely s tr ict categor ical environmen t, o ur choice is to deﬁne a C*-ca tegory C to b e commutativ e if all its dia g onal blo c ks C AA are commutativ e C*- a lgebras. If C , D ∈ A ar e t w o full commutativ e small C * -categories (with the same cardinality of the set of ob jects), a morphism in the ca tegory A is a n ob ject bijective ∗ -functor Φ : C → D . F or la ter usage, recall from [GLR, Deﬁnition 1 .6 ] and [M2, Section 4] that a clo s ed t wo- sided ideal I in a C*-ca tegory C is alwa ys a ∗ -ideal and that the quotient C / I has a natural structure as a C*-ca teg ory with a natural quotient functor π : C → C / I . W e ha ve this “ﬁrst is omorphism theorem”, whose pr oof is standa rd. Theorem 2.2. L et Φ : C → D b e a ∗ -fu n ctor b etwe en C*-c ate gories. The ker nel of Φ deﬁne d by ker Φ := { x ∈ C | Φ ( x ) = 0 } is a close d two-side d ide al in C and ther e exists a unique ∗ - fu n ctor ˇ Φ : C / ker Φ → D such that ˇ Φ ◦ π = Φ . The functor ˇ Φ is faithful and it is ful l if and only if Φ is ful l. Recall (see [GLR, Deﬁnition 1.8 ]) that a representa tion of a C*-ca tegory C is a ∗ -functor Φ : C → H with v alues in the C*-categ ory H of b ounded linear maps b et ween H ilb e r t spac e s . W e end this sec tion with a simple obser v a tion, whos e pro of is omitted. Lemma 2 .3. A one-dimensional C*-c ate gory C , admits at le ast one ∗ -functor γ : C → C . 3 By this we mean that, in a full C*-categ ory C , the family of Hom C ( · , · ) spaces is i tself a strict sub category of 1-arr o ws in the weak 2-C*-category of Hilb ert C*-bi modules (with Rieﬀel tensor pr oducts and duals as compositions and inv olutions), that “pro jects ont o” a sub-equ iv alence relation of the Pi card- Rieﬀel groupoid (see [BCL3] for details on these categories). 4 3 Category T of full top ological spaceoids W e now pr o ceed to the ident iﬁcation of a go od categor y T o f “ spaceoids” pla ying the role of horizontal ca tegoriﬁcation of the catego ry of con tinuous maps b etw een compact Hausdorﬀ top ological spa ces. Making use of Gel’fand duality (see e.g. [L, Section 6]) for the diago na l blo c ks C AA and (Hermitian) Serr e-Sw a n equiv alence (see e.g. [BCL2, Section 2.1.2] and references therein) for the oﬀ-dia g onal blo cks C AB of a commutativ e full C*-categ ory C , we see that the s pectrum o f C identiﬁes a sub-equiv alence relatio n embedded in the Picar d group oid of Hermitian line bundles o ver the Gel’fand spectra of the dia gonal C* - algebras C AA . Finally , reass em bling such blo c k- data, we reco gnize that, globa lly , the sp ectrum o f a commutativ e full C*-categ o ry can b e descr ibed as a v er y specia l kind o f a F ell bundle that we call a full to pologica l spaceoid. F ell bundles ov er top ological gro ups were ﬁrst introduced b y J. F ell [FD, Section II.16] and later generalized to the ca se o f group oids b y S. Y amagami (see A. Kumjian [Ku] a nd refer ences therein) and to the case of inv erse s e migroups by N. Sieb en (see R. Exel [E, Sectio n 2]). These no tio ns admit a natural extensio n to that of a F ell bundle ov er an inv o lutiv e inverse category 4 that we describe in deﬁnition 3.1 below. F or the deﬁnition o f a B a nac h bundle, we refer to J. F ell-R. Doran [FD , Section I.13]. W e r ecall, fr om A. Kumjian [Ku, Section 2], that a F ell bundle ov er a group oid is a Ba na c h bundle ( E , π , X ) ov er a top ological group oid X whos e total space E is equipp ed with a contin uous inv olution ∗ : E → E , denoted here by ∗ : e 7→ e ∗ , and with a co ntin uous multiplication ◦ : E 2 → E , denoted here by ◦ : ( e 1 , e 2 ) 7→ e 1 e 2 , deﬁned on the set E 2 := { ( e 1 , e 2 ) | ( π ( e 1 ) , π ( e 2 )) ∈ X 2 } , where X n denotes the family of n -paths in the group oid X , satisfying the following prop erties: π ( e 1 e 2 ) = π ( e 1 ) π ( e 2 ), for all ( e 1 , e 2 ) ∈ E 2 , for all ( x 1 , x 2 ) ∈ X 2 , the multiplication map is bilinear when res tricted to the sets E x 1 × E x 2 , whe r e E x := π − 1 ( x ), ( e 1 e 2 ) e 3 = e 1 ( e 2 e 3 ), for all ( e 1 , e 2 , e 3 ) ∈ E such that ( π ( e 1 ) , π 2 ( e 2 ) , π ( e 3 )) ∈ X 3 , (1) k e 1 e 2 k ≤ k e 1 k · k e 2 k , for a ll ( e 1 , e 2 ) ∈ E 2 , π ( e ∗ ) = π ( e ) ∗ , for all e ∈ E , wher e π ( e ) ∗ denotes the inv erse of π ( e ) in X , for a ll x ∈ X , the inv olutio n ma p is conjugate-linear when res tricted to the se t E x , ( e ∗ ) ∗ = e , for all e ∈ E , ( e 1 e 2 ) ∗ = e ∗ 2 e ∗ 1 , fo r all ( e 1 , e 2 ) ∈ E 2 , (2) k e ∗ e k = k e k 2 , for all e ∈ E , (3) the element e ∗ e is p ositive in the C*- a lgebra E π ( e ∗ e ) , fo r all e ∈ E . This deﬁnition ca n b e recasted in the follo wing concis e a nd slightly g eneralized for m, that we s y stematically adopt in the se quel. Deﬁnition 3. 1 . A unital F el l bund l e ( E , π , X ) over an involut ive inverse c ate gory X is a Banach bund le that is also an involut iv e c ate gory E ﬁb er e d over t he involutive c ate gory X with c ontinuous ﬁb erwise biline ar c omp ositions and ﬁb erwise c onjugate-line ar involutions satisfying pr op ert ie s (1) , (2) and (3) as ab ove. We say that the F el l bund le is r ank-one if E x is one-dimensional for al l x ∈ X . 4 By inv o lutiv e cat egory we mean a category X equipp ed with an inv olution i.e. an ob ject preserving con trav ariant functor ∗ : X → X such that ( x ∗ ) ∗ = x for all x ∈ X . If X has a topology we also require composition and inv olution to be contin uous. X is an inv olutive inv erse categor y if xx ∗ x = x f or all x ∈ X . 5 Note that a C*-catego ry C ca n always b e see n a s a F ell bundle over the maximal equiv alenc e relation Ob C × Ob C of its ob jects, with ﬁb ers C AB , for all ( A, B ) ∈ Ob C × Ob C . Co n versely , a F ell bundle whose base is s uc h an equiv alenc e r elation c a n always b e seen as a C*-ca tegory . Deﬁnition 3. 2. A top olo gic al sp ac e oi d ( or simply a sp ac e oid, for short) ( E , π , X ) is a unital r ank-one F el l bun d le over the pr o duct involutive top olo gic al c ate gory X := ∆ X × R O wher e ∆ X := { ( p, p ) | p ∈ X } is the minimal e quivalenc e r elation of a c omp act Hausdorﬀ sp ac e X and R O := O × O is t he max imal e quivalenc e r elation of a discr ete sp ac e O . With a sligh t a buse of notation, the arrows of the ba s e in volutive ca teg ory X of a full spaceoid will s imply b e denoted by p AB := (( p, p ) , ( A, B )) ∈ ∆ X × R O . Note tha t, since a co nstan t ﬁnite-rank Banach bundle over a lo cally compact Hausdor ﬀ space is lo cally trivial [FD, Remark I.13 .9] and hence a vector bundle, a topo logical spaceoid is a Hermitian line bundle over X and is a dis join t unio n of the Hermitian line bundles ( E AB , π | E AB , X AB ), with X AB := ∆ X × { AB } and E AB := π − 1 ( X AB ). F urthermore, a topolog ical spaceoid can b e seen as a one- dimensional C*- c ategory that is a copro duct (in the ca tegory o f s mall C*- categories) of the “co n tinuous ﬁeld” of the one- dimensional C* -categories E p := π − 1 ( X p ), wher e X p := { ( p, p ) } × R O , fo r all p ∈ X . A m orphism of spaceoids 5 ( f , F ) : ( E 1 , π 1 , X 1 ) → ( E 2 , π 2 , X 2 ) is a pair ( f , F ) where: • f := ( f ∆ , f R ) with f ∆ : ∆ 1 → ∆ 2 being a contin uous map o f top ological spac es and f R : R 1 → R 2 an iso morphism of equiv a le nce relations; • F : f • ( E 2 ) → E 1 is a ﬁb erwise linear con tin uous ∗ -functor such that π 1 ◦ F = π f 2 , where ( f • ( E 2 ) , π f 2 , X 1 ) deno tes the standa rd f - pull-bac k 6 of ( E 2 , π 2 , X 2 ). T op ological spaceo ids cons titute a catego ry if comp ositions and identities ar e given by ( g , G ) ◦ ( f , F ) := ( g ◦ f , F ◦ f • ( G ) ◦ Θ E 3 g,f ) and ι ( E , π , X ) := ( ι X , ι π X ) , where Θ E 3 g,f : ( g ◦ f ) • ( E 3 ) → f • ( g • ( E 3 )) is the natural isomorphis ms b et ween s tandard pull- backs g iv en by Θ E 3 g,f ( x 1 , e 3 ) := ( x 1 , ( f ( x 1 ) , e 3 )), for all ( x 1 , e 3 ) ∈ ( g ◦ f ) • ( E 3 ), and f • ( G ), thanks to the functoria l prop erties of pull-backs, is deﬁned on the standa rd pull-bac k b y f • ( G )( x 1 , ( x 2 , e 3 )) := ( x 1 , G ( x 2 , e 3 )), for all ( x 1 , ( x 2 , e 3 ))) ∈ f • ( g • ( E 3 )). 4 The section functor Γ Here w e are g oing to deﬁne a section functor Γ : T → A that to ev ery spaceoid ( E , π , X ), with X := ∆ X × R O , a ssocia tes a commutativ e full C*-categ ory Γ( E ) as follows: • O b Γ( E ) := O ; • for all A, B ∈ Ob Γ( E ) , Hom Γ( E ) ( B , A ) := Γ( X AB ; E ), where Γ( X AB ; E ) denotes the set of contin uo us sections σ : ∆ X × { ( A, B ) } → E , σ : p AB 7→ σ AB p ∈ E p AB of the restriction ( E AB , π | E AB , X AB ) o f ( E , π , X ) to the base space X AB . 5 Morphisms of spaceoids can be seen as examples of J. Baez notion of spans (in this case, a span of the F ell bundles of the spaceoids). 6 Recall that f • ( E 2 ) := { ( p AB , e ) ∈ X 1 × E 2 | f ( p AB ) = π 2 ( e ) } with f ◦ π f 2 = π 2 ◦ f π 2 , where π f 2 and f π 2 are deﬁned on f • ( E 2 ) by π f 2 ( p AB , e ) := p AB and f π 2 ( p AB , e ) := e . I f E 2 is a F ell bundle o v er X 2 , f • ( E 2 ) is a F ell bundle ov er X 1 . 6 • for all σ ∈ Hom Γ( E ) ( B , A ) and ρ ∈ Hom Γ( E ) ( C, B ): σ ◦ ρ : p AC 7→ ( σ ◦ ρ ) AC p := σ AB p ◦ ρ 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 o perations ta k en in the to tal space E of the F ell bundle. In the following, since for all σ ∈ Γ( E ) = U AB Γ( E ) AB , the discrete indeces AB are alr eady implicit in the sp eciﬁcation of the section σ ∈ Γ( E ) AB , we will simply use the shorter no tation σ p := σ AB p to denote the ev alua tion of the section σ at the p oin t p AB ∈ X . By construction, the commutativ e C*-ca tegory Γ( E ) so obtained has sections of Hermitian line bundles o n a compac t Hausdor ﬀ space as Ho m spaces, a nd thus it is full. W e ex tend now the deﬁnition of Γ to the mor phisms of T . Let ( f , F ) be a morphism in T from ( E 1 , π 1 , X 1 ) to ( E 2 , π 2 , X 2 ). Given σ ∈ Γ( E 2 ), we consider the unique sectio n f • ( σ ) : X 1 → f • ( E 2 ) such that f π 2 ◦ f • ( σ ) = σ ◦ f and the co mposition F ◦ f • ( σ ). In this wa y we have a map Γ ( f , F ) : Γ( E 2 ) → Γ( E 1 ) , Γ ( f , F ) : σ 7→ F ◦ f • ( σ ) , ∀ σ ∈ Γ( E 2 ) . Prop osition 4.1. F or any morphism ( E 1 , π 1 , X 1 ) ( f , F ) − − − − → ( E 2 , π 2 , X 2 ) in the c ate gory T , the map Γ ( f , F ) : Γ( E 2 ) → Γ( E 1 ) is a morphism in the c ate gory A . The p air of maps Γ : ( E , π , X ) 7→ Γ( E ) and Γ : ( f , F ) 7→ Γ ( f , F ) gives a c ontr avariant functor fr om the c ate gory T of s p ac e oids to the c ate gory A of smal l ful l c ommutative C*-c ate gories. Pr o of. Let ( E 1 , π 1 , X 1 ) ( f , F ) − − − − → ( E 2 , π 2 , X 2 ) and ( E 2 , π 2 , X 2 ) ( g, G ) − − − → ( E 3 , π 3 , X 3 ) b e t wo com- po sable morphisms in the catego r y T and le t ( E , π , X ) ( ι X , ι π X ) − − − − − → ( E , π , X ) b e the iden tit y morphism o f ( E , π , X ). T o complete the pro of we must show that Γ ( g, G ) ◦ ( f , F ) = Γ ( f , F ) ◦ Γ ( g, G ) , Γ ( ι X , ι π X ) = ι Γ( E ) , and these ar e obtained by tedious but straig htforward calcula tions. 5 The sp ectrum functor Σ This section is devoted to the constructio n of a s pectrum functor Σ : A → T that to every co mm utative full C * -category C asso ciates its sp ectral spaceoid Σ( C ). Letting C b e a C*-catego ry , we denote b y R C the topolog ically discrete ∗ -category C / C ≃ R Ob C and b y C R C := ρ • ( C ) the one-dimens io nal C*-catego ry pull- bac k of C (cons idered as a C*-catego ry with o nly one ob ject • ) under the constant map ρ : R C → { •} . Note that, via the canonica l pro jection ∗ -functor C → R C := C / C , fro m the de ﬁning pr operty of pull-backs ther e is a bijective map ω 7→ e ω be tw een the set of C -v alued ∗ -functors [ C ; C ] and the o b ject-preserving elements in the set o f C R C -v alued ∗ -functors [ C ; C R C ]. By deﬁnition, tw o ∗ -functors ω 1 , ω 2 in [ C ; C ] are unitarily equi v alen t , see P . Mitc hener [M1, Section 2], if there exists a “unitary” natural transfor ma tion A 7→ ν A ∈ T b etw een them. Note that the set I ω := { x ∈ C | ω ( x ) = 0 } , which is also equa l to { x ∈ C | ω ( x ∗ x ) = 0 } , is an ideal in C and I ω 1 = I ω 2 if (and o nly if ) the eq uiv a lence classes [ ω 1 ] and [ ω 2 ] coincide. W e a lso need the following lemmas whose routine pro ofs are o mitted: 7 7 Note that, for ω ∈ [ C ; C ] and A, B ∈ Ob C , we de note by ω AB the restriction of ω to C AB . 7 Lemma 5.1. If ω , ω ′ ∈ [ C ; C ] ar e unitarily e quivalent, ther e is a u nique map ψ : R C → T such that ω ′ AB = ψ AB · ω AB for al l AB ∈ R C given by ψ AB = ν B ν − 1 A , wher e ν is a u nitary natur al t r ansformation fr om ω to ω ′ , and the map ψ : AB 7→ ψ AB is a ∗ -fun ct or: ψ AB ψ B C = ψ AC , ψ AB = ψ − 1 B A , ψ AA = 1 C . (5.1) Conversely, given a ∗ - functor ψ ∈ [ R C ; T ] , two ∗ -functors ω, ω ′ such that ω ′ AB = ψ AB ω AB ar e u nitarily e quivalent. Lemma 5.2 . Eve ry obje ct pr eserving ∗ -aut omo rphism γ of the C*-c ate gory C R C is given by the multiplic ation by an element ψ ∈ [ R C ; T ] i.e. γ ( x ) = ψ AB · x for al l x ∈ ( C R C ) AB . Prop osition 5. 3. Two ∗ -functors ω , ω ′ ∈ [ C ; C ] ar e unitarily e quivalent if and only if ω AA = ω ′ AA for al l A ∈ Ob C . Pr o of. By lemma 5.1, if [ ω ] = [ ω ′ ], then ω ′ AA = ψ AA · ω AA = ω AA , fo r all ob jects A . Let ω , ω ′ ∈ [ C ; C ] and suppos e that ω AA = ω ′ AA , for all A ∈ Ob C . Consider the corresp onding ob ject-pr eserving C R C -v alued ∗ -functors e ω , e ω ′ ∈ [ C ; C R C ]. Note that Ker( e ω ) = I ω = I ω ′ = Ker( e ω ′ ) and hence, ω AB , e ω AB are no nzero if and only if ω ′ AB , e ω ′ AB are no nzero. If ω AB is nonzero for all AB ∈ R C , by theore m 2.2 we hav e tw o ∗ -is omorphisms C / Ker( ω ) ˆ ω − → C R C ˆ ω ′ ← − − C / Ker( ω ′ ) . F rom lemma 5 .2 there is a ψ ∈ [ R C ; T ] such that ˆ ω ′ = ψ · ˆ ω and hence als o ω ′ = ψ · ω so that the pr oposition follows from lemma 5.1. In o rder to complete the pro of, notice that the image s of ˆ ω and ˆ ω ′ coincide a nd then a pply the a bov e a rgumen t to the co nnected comp onen ts of the ima ge categor y . Given x ∈ C AA for so me ob ject A , b y ev a lua tion in x w e mean the map ev x : [ C ; C ] → C deﬁned by ω 7→ ω ( x ). Prop osition 5.4. The set [ C ; C ] of C - value d ∗ -functors ω : C → C , with the we akest top olo gy making al l evaluations c ontinu ous, is a c omp act Hausdorﬀ top olo gic al sp ac e. Pr o of. Note that for all ω ∈ [ C ; C ] and for all x ∈ C AB , | ω ( x ) | = q ω ( x ) ω ( x ) = p ω ( x ∗ x ) = p ω AA ( x ∗ x ) ≤ p k x ∗ x k = p k x k 2 = k x k , bec ause ω AA is a sta te ov er the C*-algebr a C AA . Hence [ C ; C ] is a subspac e o f the compact Hausdorﬀ space Q x ∈ C D k x k , where D k x k is the closed ball in C of radius k x k , and it is ea sy to chec k that it is closed. Let Sp b ( C ) := { [ ω ] | ω ∈ [ C ; C ] } denote the base sp ectrum of C , deﬁned as the set of unitary equiv alence class e s of ∗ -functors in [ C ; C ]. It is a co mpact space with the quotient top ology induced by the map ω 7→ [ ω ]. T o show that Sp b ( C ) is Hausdo r ﬀ it is enough to note that, by prop osition 5.3, if [ ω ] 6 = [ ω ′ ], there exists at least one ob ject A such that ω AA 6 = ω ′ AA and so ther e exists a t lea st one ev alua tion ev x with x ∈ C AA such that ev x ( ω ) 6 = ev x ( ω ′ ). Since, for x ∈ C AA , ev x induces a w ell-deﬁned map on the quotien t spa ce Sp b ( C ) by [ ω ] 7→ ω ( x ), the r esult follows. Prop osition 5.5. L et C b e a ful l c ommu tative C*-c ate gory. F or al l A ∈ O b C , ther e ex ist s a natur al bije ctive map, b etwe en t he b ase sp e ctrum of C and t he usual Gel’fand sp e ct rum Sp( C AA ) of the C*-algebr a C AA , given by the r estriction | AA : ω 7→ ω | C AA . In p articular, fo r al l obje cts A ∈ Ob C , one has Sp b ( C ) | AA = Sp( C AA ) ≃ Sp b ( C AA ) . 8 Pr o of. By prop osition 5 .3, the cor respondence [ ω ] 7→ ω AA is well deﬁned. W e sho w that the map [ ω ] 7→ ω AA is injective. Giv en ω , ω ′ ∈ [ C ; C ] with ω AA = ω ′ AA , w e know from [BCL3, P r opositio n 2 .30], that ω B B ( x ) = ω AA ( φ AB ( x )), for all x ∈ C B B , for all B ∈ Ob C , where φ AB : C B B → C AA is the canonica l isomorphis m a ssocia ted to the imprimitivity bimo dule C B A . It follows that ω B B = ω AA ◦ φ AB = ω ′ AA ◦ φ AB = ω ′ B B , for all B ∈ Ob C and, by pr opositio n 5.3, we s ee that [ ω ] = [ ω ′ ]. W e show that the function [ ω ] 7→ ω AA is surjective. Let ω o ∈ Sp( C AA ). Deﬁne J to b e the inv olutive ideal in C gener a ted by Ker( ω o ). One can see that J ∩ C AA = J AA = Ker( ω o ). Since the quotient ∗ -functor π : C → C / J is bijectiv e on the ob jects and C is full, C / J is full. Since C AA / J AA is o ne-dimensional, the quotien t C*-categor y C / J is one-dimensio nal. If γ : C / J → C is a C -v a lued ∗ -functor as in lemma 2.3, γ ◦ π restricted to C AA m ust b e ω o since it v anishes o n J ∩ C AA . Theorem 5.6. L et C b e a ful l c ommutative C*-c ate gory. F or every A ∈ Ob C , the bije ctive map | AA : Sp b ( C ) → Sp( C AA ) given by [ ω ] 7→ ω AA is a home omorphism b etwe en Sp b ( C ) and the Gel’fand sp e ctrum Sp( C AA ) of the unital C*-algebr a C AA . Pr o of. Since b oth Sp b ( C ) and Sp( C AA ) ar e compact Hausdorﬀ space s , and the map | AA is bijectiv e, it is enough to show tha t | AA : Sp b ( C ) → Sp ( C AA ) is con tinuous. Since Sp b ( C ) is equipp e d with the quotien t topo logy induced b y the pr o jection map π : [ C ; C ] → Sp b ( C ), the map | AA is c on tinuous if and only if | AA ◦ π : [ C ; C ] → Sp ( C AA ) is contin uous . The spaces [ C ; C ] and Sp( C AA ) are equipp ed with the w ea kest top ology ma king the ev a luation maps contin uous. It follows that the co n tinuit y of | AA ◦ π is equiv alent to the contin uity o f ev x = ev x ◦ | AA ◦ π : [ C ; C ] → C for a ll x ∈ C AA . Since ev x : [ C ; C ] → C is con tinuous, the result is es tablished. Let X C := ∆ C × R C be the direct pro duct equiv alence relation of the compact Hausdorﬀ ∗ -catego ry ∆ C := ∆ Sp b ( C ) and the top ologically disc r ete ∗ -catego ry R C := C / C ≃ R Ob C . With a slight abuse of notation, we wr ite AB ∈ R C for the arr o w C AB / C AB in R C and deno te ([ ω ] , AB ) = ([ ω ] , C AB / C AB ) ∈ X C simply by ω AB . W e deﬁne E C as the disjoint union ov er [ ω ] ∈ ∆ C of the quotients C / I ω . In formulae: E C ω := C I ω , E C := ] ω ∈ ∆ C E C ω = ] ω AB ∈ X C E C ω AB , π C : E C → X C , π C : e 7→ ω AB , ∀ e ∈ E C ω AB , where E C ω AB := C AB / I ω AB , with I ω AB := I ω ∩ C AB . Prop osition 5. 7. The tr iple ( E C , π C , X C ) is natur al ly e quipp e d with the structure of a u nital r ank-one F el l bund le over the t op olo gic al involutive inverse c ate gory X C . Pr o of. Deﬁne o n E C the top ology who se fundamen tal system of neig h b orho ods are the sets U O, x 0 , ε e 0 := { e ∈ E C | π C ( e ) ∈ O , ∃ x ∈ C : ˆ x ( π C ( e )) = e, k ˆ x − ˆ x 0 k ∞ < ε } , where e 0 ∈ E C , O is ope n in X C , ε > 0, x 0 ∈ C with ˆ x 0 ( π C ( e 0 )) = e 0 and where ˆ x denotes the Gel’fand transform of x de ﬁned in section 6.1. This top ology e n tails that a net ( e µ ) is co n vergent to the po in t e in E C if and only if the net π C ( e µ ) conv erg es to π C ( e ) in X C and, for all p ossible Gel’fand tr ansforms ˆ x 0 “passing” through e 0 , ther e exis ts a net of Gel’fand tr ansforms ˆ x µ , “passing” through e µ , that uniformly converges, on every neighborho o d of π C ( e 0 ), to ˆ x 0 . With such a topolo gy the (partial) ope r ations on E C i.e. sum, sca lar m ultiplication, pro duct, inv olution, inner pro duct (and hence norm) b ecome contin uous and ( E C , π C , X C ) b ecomes a Banach bundle. 9 Since every sub-equiv ale nc e relation of X C is a disjoin t union of “grids” { [ ω ] } × R C whose inv erse image under π C is the one-dimensional C*-c ategory C / Ker( ω ), ( E C , π C , X C ) is a rank-one unital F e ll bundle over the equiv ale nce relation X C and hence a spaceo id. T o a co mm utative full C*-categ ory C we hav e asso ciated a top ological sp ectral spaceoid Σ( C ) := ( E C , π C , X C ). W e extend now the deﬁnition of Σ to the morphisms of A . Let Φ : C → D be a n ob ject-bijective ∗ -functor b et ween t wo small commutativ e full C*-catego ries with spa ceoids Σ( C ) , Σ( D ) ∈ T and deﬁne a morphism Σ Φ : Σ( D ) ( λ Φ , Λ Φ ) − − − − − → Σ( C ) in the category T as follows. W e se t λ Φ : X D ( λ Φ ∆ ,λ Φ R ) − − − − − → X C where λ Φ R : R D → R C is the iso mo rphism of eq uiv alence relations given b y λ Φ R ( AB ) := Φ − 1 ( A )Φ − 1 ( B ), for AB ∈ R D , and where λ Φ ∆ : ∆ D → ∆ C (since ω 7→ ω ◦ Φ is con tin uous and preserves equiv alence b y unitary natura l transfor ma tions) is the well-deﬁned co ntin uous map given by λ Φ ∆ ([ ω ]) := [ ω ◦ Φ] ∈ ∆ C , fo r all [ ω ] ∈ ∆ D . Since, for ω ∈ [ D ; C ], the ∗ - functor Φ : C → D induces a contin uous ﬁeld of quotient ∗ -functors Φ ω : C / I Φ ◦ ω → D / I ω betw een one- dimensional C*-categor ies, we can deﬁne 8 Λ Φ : ( λ Φ ) • ( E C ) → E D as the disjo in t union of the ∗ -functors Φ ω , for [ ω ] ∈ ∆ D and note that it is a contin uous ﬁb erwise linea r ∗ -functor. Prop osition 5.8. F or any m orphism C Φ − → D in A , the map Σ( D ) Σ Φ − − → Σ( C ) is a morphism of sp e ctr al sp ac e oids. The p air of maps Σ : C 7→ Σ( C ) and Σ : Φ 7→ Σ Φ gives a c ont r avari- ant functor Σ : A → T , fr om the c ate gory A of obj e ct-bije ctive ∗ -functors b etwe en smal l c ommutative ful l C*-c ate gories to t he c ate gory T of sp ac e oids. Pr o of. W e hav e to prov e that Σ is antim ultiplicative and pres erv es the identities. If Φ : C 1 → C 2 and Ψ : C 2 → C 3 are t wo ∗ -functors in A , by deﬁnition, Σ Ψ ◦ Φ = ( λ Ψ ◦ Φ , Λ Ψ ◦ Φ ) =  λ Φ ◦ λ Ψ , Λ Ψ ◦ ( λ Ψ ) • (Λ Φ ) ◦ Θ E C 1 λ Φ ,λ Ψ  = ( λ Φ , Λ Φ ) ◦ ( λ Ψ , Λ Ψ ) = Σ Φ ◦ Σ Ψ . Also, if ι C : C → C is the identit y functor of the C*-categor y C , then the morphism Σ ι C = ( λ ι C , Λ ι C ) is the identit y morphism of the spa ceoid Σ( C ). 6 Horizon tal Categoriﬁcation of Gel’fand Dualit y 6.1 Gel’fand T ransf orm F or a g iv en C*- category C in A , we deﬁne a horizo n tally catego r iﬁed version of Gel’ fand transform as G C : C → Γ(Σ( C )) given by G C : x 7→ ˆ x where ˆ x [ ω ] := x + I ω AB , for all x ∈ C B A . Clearly G C : C → Γ(Σ( C )) is an ob ject bijective ∗ -functor. Lemma 6. 1. L et C b e a c ommutative C*-c ate gory and C o a sub c ate gory of C which is a ful l C*-c ate gory such that C o AA = C AA for al l A ∈ Ob C = Ob C o . Then C o AB = C AB for al l A, B ∈ Ob C . Pr o of. By the fullness of the bimo dule A C o B there is a sequence of pairs u j , v j ∈ A C o B such that ι B = P ∞ j =1 u ∗ j v j . W e hav e x = xι B = x P ∞ j =1 u ∗ j v j = P ∞ j =1 ( xu ∗ j ) v j ∈ A C o B for all x ∈ A C B , b ecause xu ∗ j ∈ A C A = A C o A and so ( xu ∗ j ) v j ∈ A C o B for a ll j . Theorem 6.2. The Ge l’fand tr ansform G C : C → Γ(Σ( C )) of a c ommutative ful l C*-c ate gory C is a ful l faithful isometric ∗ -functor. 8 Note that ( λ Φ ) • ( E C ) i s the di sjoin t union of the cont inuo us ﬁeld of one-dimensional C*-categ ories C / I Φ ◦ ω . 10 Pr o of. The pro of follows from the fa ct that the Gel’fand transform G C , when r e stricted to any “diago nal” co mmutative unital C*-a lg ebra C AA can b e “naturally identiﬁed” with the usual Gel’fand transform of C AA via the ho meomorphism [ ω ] 7→ ω | AA (see pr opositio n 5.5 and theor em 5.6). T o prov e the faithfulness of G C , let x ∈ C B A with b x = 0. W e ha ve d x ∗ x = b x ∗ b x = 0 so that the usual Gel’fand transform of x ∗ x ∈ C AA is zero a nd, by Gel’fand is omorphism theor em applied to the C*- a lgebra C AA , we hav e x ∗ x = 0 and hence x = 0. The “image” G C ( C ) of G C is a sub category of the comm utative full C*- category Γ(Σ( C )) that is clearly a commutativ e full C*- category on its own. By lemma 6 .1, the ∗ -functor G C is full as lo ng as G C ( C AA ) = Γ(Σ( C )) AA , for all o b jects A ∈ Ob C and this fo llo ws a gain by the usua l Gel’fand iso morphism theorem applied to the C*- algebra C AA . F or the isometry of G C we note that for all x ∈ C B A , since the Gel’fand tra nsform restricted to the C*-a lgebra C AA is isometric, we have k G C ( x ) k 2 = k d x ∗ x k = k x ∗ x k = k x k 2 . 6.2 Ev aluation T ra nsform Given a top ological spaceo id ( E , π , X ), we deﬁne a horizo n tally catego riﬁed version of ev al- uation transform E E : ( E , π , X ) ( η E , Ω E ) − − − − − → Σ(Γ( E )) as fo llo ws: • η E R : R O → R Γ( E ) is the c a nonical isomorphism R O = R Ob Γ( E ) ≃ Γ( E ) / Γ( E ), e x plicitly: η E R ( AB ) := Γ( E ) AB / Γ( E ) AB , ∀ AB ∈ R O that is , ac c o rding to the running notation, written as an identit y map η E R ( AB ) = AB ∈ R Γ( E ) . • η E ∆ : ∆ X → ∆ Γ( E ) is g iv en b y η E ∆ : p 7→ [ γ p ◦ ev p ] ∀ p ∈ ∆ X , where the ev a luation map ev p : Γ( E ) → ⊎ ( AB ) ∈ R O E p AB given by ev p : σ 7→ σ p is a ∗ - functor with v alues in a one-dimensional C* -category that determines 9 a unique p oint [ γ p ◦ ev p ] ∈ ∆ Sp b (Γ( E )) . • Ω E : ( η E ) • ( E Γ( E ) ) → E is deﬁned by Ω E : ( p AB , σ + I η E ( p AB ) ) 7→ σ p , ∀ σ ∈ Γ( E ) AB , ∀ p AB ∈ X . In particular, with such deﬁnit ions we can pro ve that the sp e c trum f unctor is represe ntative: Theorem 6.3. The evaluation tr ansform E E : ( E , π , X ) → Σ(Γ( E )) , for al l sp ac e oids ( E , π , X ) , is an isomorphism in the c ate gory of sp ac e oids. Pr o of. Note tha t ( E AA , π , X ) is natura lly isomorphic to the trivial C -bundle ov er X and th us there is a n isomo r phism of the C*- a lgebras Γ( E ) AA and C ( X ) that “preser v es” ev aluations. Clearly the ma p ζ A : ∆ X → Sp(Γ( E ) AA ) given b y ζ A ( p ) := | AA ◦ η E ∆ ( p ) = γ p AA ◦ ev p AA , coincides with the usual Gel’fand ev aluation homeomorphis m for the diagonal C*-alg ebra Γ( E ) AA and hence, by prop osition 5 .5, η E ∆ = | − 1 AA ◦ ζ A is also a homeomor phism. F or every element e ∈ E , we hav e π ( e ) ∈ ∆ X × R O and, since a spaceo id is actually a vector bundle, it is alwa ys p ossible to ﬁnd a sec tio n σ ∈ Γ( E ) such that σ π ( e ) = e . F or any such sectio n we consider the elemen t σ + I η E ( π ( e )) ∈ Γ( E ) / I η E ( π ( e )) =: E Γ( E ) η E ( π ( e )) (note that the elemen t doe s not dep end on the c ho ic e of σ ∈ Γ( E ) suc h that σ π ( e ) = e ) and in this wa y we hav e a map Θ : E → E Γ( E ) by Θ : e 7→ σ + I η E ( π ( e )) . The map Θ uniquely induces a function Ξ E : E → ( η E ) • ( E Γ( E ) ) with the s tandard η E -pull-back of E Γ( E ) given b y 9 By lemma 2.3, there is alw ays a C -v alued ∗ -functor γ p : E p → C and by proposition 5.3 any t wo compositions of e v p with such ∗ -f unc tors are unitarily equiv alent because the y coinc ide on the diagona l C*-algebras E p AA . 11 Ξ E ( e ) := ( π ( e ) , Θ( e )). B y direct computation the map Ξ E is an “a lgebraic is omorphism” of F ell bundles 10 whose inverse is Ω E . The contin uity of Ω E is equiv alent to that o f e Ω E : E Γ( E ) → E , e Ω E ( σ + I η E ( p AB ) ) := σ p , with σ ∈ Γ( E ) AB . Given a net j → σ j + I η E ( p j AB j ) in E Γ( E ) conv er ging to the po in t σ + I η E ( p AB ) in the top ology deﬁned in pr oposition 5 .7, without loss o f g eneralit y we c a n assume that j → σ j is uniformly conv ergent to σ in a neighbo rhoo d U of η E ( p AB ). This means that, for all ǫ > 0, k σ j ([ ω ] AB ) − σ ([ ω ] AB ) k < ǫ for [ ω ] AB ∈ U for all suﬃciently large j . Since R Γ( E ) is discrete, the net AB j is even tually equa l to AB and since η E is a homeo morphism, p j AB even tually lies in any neighbor ho o d of p AB and hence the net e Ω E ( σ j + I η E ( p j AB j ) ) = ( σ j ) p j conv er ges to e Ω( σ + I η E ( p AB ) ) = σ p in the Ba nac h bundle top ology of E . Since Ω E is an isometry , it follows from [FD, Prop osition 13.17] that its in verse is contin uous to o and hence the ev alua tio n tr ansform E E := ( η E , Ω E ) is an isomo rphism o f spa ceoids. 6.3 Dualit y Theorem 6.4. The p air of functors (Γ , Σ) pr ovides a duality b etwe en the c ate gory T of morphisms b etwe en sp ac e oids t ha t ar e obje ct -bij e ctive on the discr ete p art of the obje cts and the c ate gory A of obje ct - bij e ctive ∗ -functors b etwe en smal l c ommutative ful l C*-c ate gories. Pr o of. T o see that the ma p G : C 7→ G C (that to every C ∈ O b A asso ciates the Gel’fand transform of C ) is a natural isomorphism b et ween the identit y endofunctor I A : A → A and the functor Γ ◦ Σ : A → A we ha ve to show tha t, given an ob ject-bijective ∗ -functor Φ : C 1 → C 2 , the iden tity Γ Σ Φ ( G C 1 ( x )) = G C 2 (Φ( x )) holds for a n y x ∈ C 1 , i.e. the commutativit y of the dia gram: C 1 G C 1 / / Φ   Γ(Σ( C 1 )) Γ Σ Φ   C 2 G C 2 / / Γ(Σ( C 2 )) , that fo llows from this direct computation: Γ Σ Φ ( G C 1 ( x )) [ ω 2 ] = Λ Φ  ( λ Φ ) • ( ˆ x ) [ ω 2 ]  = Λ Φ  [ ω 2 ] A 2 B 2 , ˆ x ( λ Φ ([ ω 2 ] A 2 B 2 ))  = Λ Φ  [ ω 2 ] A 2 B 2 , x + I λ Φ ([ ω 2 ] A 2 B 2 )  =  [ ω 2 ] A 2 B 2 , Φ( x ) + I [ ω 2 ] A 2 B 2  = G C 2 (Φ( x )) [ ω 2 ] . T o see that the map E : E 7→ E E (that to every spa ceoid ( E , π , X ) assoc ia tes its ev alua tion transform E E ) is a natural isomorphism b et ween the iden tit y endo functor I T : T → T and the functor Σ ◦ Γ : T → T we must prov e, for any given morphism of spa ceoids ( f , F ) fro m ( E 1 , π 1 , X 1 ) to ( E 2 , π 2 , X 2 ), the co mmutativit y of the diagram: ( E 1 , π 1 , X 1 ) E E 1 =( η E 1 , Ω E 1 ) / / ( f , F )   Σ(Γ( E 1 )) Σ Γ ( f, F ) =( λ Γ ( f, F ) , Λ Γ ( f, F ) )   ( E 2 , π 2 , X 2 ) E E 2 =( η E 2 , Ω E 2 ) / / Σ(Γ( E 2 )) . 10 By this we mean that Ξ E : E → ( η E ) • ( E Γ( E ) ) is a ﬁber preserving map b et ween bundles, ov er the same base space X , that is also a bij ec tiv e ﬁber wise linear ∗ -functor betw een the total spaces. 12 The pro of amounts to showing the equalities λ Γ ( f, F ) ◦ η E 1 = η E 2 ◦ f , Ω E 1 ◦ ( η E 1 ) • (Λ Γ ( f, F ) ) ◦ Θ 1 = F ◦ f • (Ω E 2 ) ◦ Θ 2 , (6.1) where Θ 1 := Θ E Γ( E 2 ) λ Γ ( f, F ) ,η E 1 , Θ 2 := Θ E Γ( E 2 ) η E 2 ,f . Since for every p oint p AB ∈ X 1 , w e ha ve λ Γ ( f, F ) ◦ η E 1 ( p AB ) = ([ γ p ◦ ev p ◦ Γ ( f , F ) ] , f R ( AB )) and η E 2 ◦ f ( p AB ) = ([ γ f ( p ) ◦ ev f ( p ) ] , f R ( AB )), the ﬁrs t equation is a consequence of prop osition 5.3. The second eq uation is then proved by a lengthy but elementary ca lculation. Remark 6 .5. Final ly, note that, although for simplicity we only describ e d a sp e ctr al the ory for c ommutative ful l C*-c ate gories, it is p erfe ctly viable and ther e ar e no substantial obstacles to the development of a sp e ctr al the ory for c ommutative ful l “n on-unital” C*-c ate gories 11 (as deﬁne d by P. Mitchener [M2]). In this c ase the b ase sp e ctru m is only lo c al ly c omp act and we have to de al with a lo c al ly c omp act version of top olo gic al sp ac e oids ( s o, for example, only se ctions “going to zer o at inﬁnity” ar e c onsider e d in the deﬁnition of the se ction functor). 6.4 Horizon tal Categoriﬁcation The usual Gel’fa nd-Na ˘ ımark dualit y theorem is easily re co vered fro m our result iden tifying a compact Hausdorﬀ topolo g ical space X with the trivial spaceoid T X with total space X X × C and base categor y X X := ∆ X × R O X where O X := { X } is a discrete space with only one p oint X ; and s imilarly , identifying a unital commutativ e C*-alg ebra A with the full commutativ e C*-catego ry C A with one ob ject via Hom C A := A and O b C A := { A } . More pr ecisely , th e duality (Γ , Σ) b et ween the categorie s T and A is a “ho rizon tal categor iﬁc a tion” of the usual Gel’fand-Na ˘ ımark duality in the sense sp eciﬁed by the following result whose pro of is absolutely elementary: Theorem 6.6. L et T (1) denote the ful l sub c ate gory of T c onsisting of those trivial sp ac e oids T X := X X × C , wher e X X := ∆ X × R O X , O X := { X } and X is a c omp act Hausdorﬀ sp ac e. L et A (1) denote the ful l sub c ate gory of A c onsisting of those ful l c ommutative smal l one- obje ct C*-c ate gories C A with morphi sms Hom C A := A , obje cts Ob C A := { A } and c omp osition involution and norm induc e d fr om those in t he c ommutative unital C*-a lgebr a A . The natur al duality (Γ , Σ) b etwe en the c ate gories T , A r estricts t o a duality (Γ (1) , Σ (1) ) b etwe en the c ate gories T (1) , A (1) i.e. the fol lowing p air of diagr ams of functors is c ommutative: T (1)  _   Γ (1) / A (1) Σ (1) o  _   T Γ / A , Σ o wher e Γ (1) and Σ (1) denote the re strictions of the functors Γ and Σ and the vertic al arr ows denote the inclusion functors of the re sp e ctive c ate gories. The c ate gory T 1 of c ontinuous maps b etwe en c omp act Hausdorﬀ sp ac es is isomorphic to the c ate gory T (1) via the functor T : T 1 → T (1) deﬁne d as fol lows: • t o every c omp act Hausdorﬀ sp ac e X , T asso ciates the sp ac e oid T X := ( E X , π X , X X ) that is the t rivi al bund le with ﬁb er C over the sp ac e X X := ∆ X × { ( X, X ) } , 11 Strictly speaking t hese ar e not categories, since t hey ar e lacking iden tities, but they othe rwise satisfy all the other prop erties listed in the deﬁnition of a C*-category . 13 • t o every c ontinu ous map f : X → Y b etwe en c omp act Hausdorﬀ sp ac es, T asso ciates the morphism of sp ac e oids T ( f ) : T X → T Y deﬁne d by T ( f ) := ( T ( f ) X , T ( f ) E ) whe r e T ( f ) X : p X X 7→ f ( p ) Y Y , for al l p ∈ X , and T ( f ) E : T ( f ) • X ( E Y ) → T X denotes the c anonic al isomorphism b etwe en trivial line bu n d les over X . 12 The c ate gory A 1 of u nital ∗ -homomorphisms of un ital c ommutative C*-algebr as is isomor- phic to the c ate gory A (1) via the functor C : A 1 → A (1) that to every unital c ommu ta- tive C*-algebr a A asso ciates the C*-c ate gory C A and that to every unital ∗ -homomorphism φ : A → B asso ciates the ∗ -functor C ( φ ) : C A → C B given on arr ows by C ( φ )( x ) := φ ( x ) , for al l x ∈ A , and on obje cts by C ( φ ) o : A 7→ B . The fun ctors C ◦ Γ and Γ (1) ◦ T ar e natu r al ly e qu iv alent via t he n atu r al tr ansformation that to every X asso ciates the c anonic al isomorphism b etwe en C C ( X ; C ) and Γ( T X ) . The functors T ◦ Σ and Σ (1) ◦ C ar e natur al ly e quivalent via t he natur al tr ansformation that to every A asso ciates the c anonic al isomorphism b etwe en T Sp( A ) and Σ( C A ) . 7 Examples and Applications Commutativ e full C* -categories are abunda n t, just to mention a few examples: • E v er y Abe lia n unital C*-algebr a A gives a comm utative full C*-categor y C A with only one ob ject (as a lready men tioned in s ubsection 6.4). • E xamples of co mm utative full C*- c ategories with tw o ob jects ca n b e obtained via the follo wing construction (see L. Brown-P . Green- M. Rieﬀel [BGR]) of the “linking C*-catego ry ” L ( M ) of an imprimitivity Hilb ert C*-bimo dule A M B ov er tw o commu- tative un ital C*-a lgebras A , B . Let A M B be an imprimitivity C*-bimo dule ov er unital commutativ e C*- a lgebras A , B . Denote by B M + A its Rieﬀel dual and by ι : M → M + the canonical bijective map suc h that ι ( a · x · b ) = b ∗ · ι ( x ) · a ∗ for all a ∈ A , b ∈ B , x ∈ M (see for example [BCL3 , Pr opositio n 2 .19] for mor e de ta ils). The linking C*-ca tegory L ( M ) of M is the full commut ative C*-categor y with tw o ob jects A, B and morphisms given by L ( M ) AA := A , L ( M ) B B := B , L ( M ) AB := M , L ( M ) B A := M + where the only non-e le men tary opera tions a re the inv olutions o f e le ments o f M and M + , giv en by x ∗ := ι ( x ) and y ∗ := ι − 1 ( y ) for all x ∈ M , y ∈ M + and the comp ositions b et ween elements of M and M + that are given via their resp ectiv e A -v alued and B -v alued inner pro ducts as fo llows: x ◦ y := A h x | y ∗ i a nd y ◦ x := h y ∗ | x i B , fo r all x ∈ M , y ∈ M + . Making use of the canonical isomor phis ms of imprimitivity bimo dules ( M ⊗ N ) ⊗ T ≃ ( M ⊗ N ⊗ T ) ≃ M ⊗ ( N ⊗ T ) and ( M ⊗ N ) + ≃ N + ⊗ M + for the deﬁnition o f the comp ositions via tensor pro ducts and “co n trac tio ns”, we can generalize the pr evious constructio n of linking C* -category to the case of an a rbi- trary (ﬁnite) collection of ob jects. In practice, given a (ﬁnite) family of commutativ e unital C*- algebras A 1 , . . . , A n and a family of imprimitivit y Hilbert C*-bimo dules A 1 M A 2 , . . . , A n − 1 M A n , the “ linking C*-catego ry” L ( A 1 M A 2 , . . . , A n − 1 M A n ) is the full co mm utative C*-categ o ry with n o b jects B 1 , . . . , B n , whe r e L ( A 1 M A 2 , . . . , A n − 1 M A n ) B j B k :=      A j M A j +1 ⊗ · · · ⊗ A k − 1 M A k , for j < k , A j , for j = k , ( A k M A k +1 ⊗ · · · ⊗ A j − 1 M A j ) + , for k < j , 12 Remem ber that the pull-back of the trivial line bundle T Y under the homeomorphism T ( f ) X is a trivial line bundle on X X . 14 The examples in the next item a re rather natural, esp ecially for tho se who are familiar with the Doplicher-Rob erts abstract duality theor y for co mpact groups. • Le t G be a compact gro up, and consider the C*-categor y Rep( G ) with ob jects the unitary re presen ta tions of G on Hilb ert spa c es and ar ro ws their intertwiners (actually , Rep( G ) is a W*- category). Then the full s ub categor y of Rep( G ) whos e ob jects are the m ultiplicit y-free representations is commutativ e. Moreov er, a category of multiplicit y- free re pr esen ta tions is full whenever all the ob jects are equiv alent. Let A b e a unital C*-algebr a. A c ategory of nondegenera te ∗ -representations of A on Hilber t spaces is comm utative whenev er all the ob jects a re m ultiplicity-free (see e.g. [Ar, Chapter 2]). In addition, such a commutativ e W*-catego ry is full when- ever all the ob jects are equiv a len t. If A is commu tative with metrizable sp ectrum, a categor y o f nondegener ate repr esen tatio ns of A on s eparable Hilb ert spaces tha t is bo th co mm utative and full can b e interpreted in terms of a family of eq uiv alent ﬁnite Borel mea sures on the sp ectrum of A [Ar, Theor ems 2.2.2 and 2.2.4 ]. This fact can b e generalized to GCR algebr as [Ar , Cha pter 4]. Let A b e a unital C*-alge br a, and cons ider the C* -category End( A ) with ob jects the unital ∗ -e ndo morphisms of A and Bana ch spaces of ar ro ws ( ρ, σ ) = { x ∈ A | xρ ( a ) = σ ( a ) x, ∀ a ∈ A } . The ca teg ory Aut( A ), i.e . the full sub category of End( A ) with ob jects the unital ∗ -automor phisms of A , is clearly co mm utative, as ( ρ, ρ ) equals the center of A , for every automorphism ρ . 13 Again, a subc a tegory of Aut( A ) is full whenever all the ob jects are equiv a len t. • P . Mitchener [M2] a ssocia tes C*-categories C ∗ ( G ) and C ∗ r ( G ) to a discrete group oid G . It is easy to see that these categ o ries are commutativ e ex actly when all the s tabilizer subgroups of th e group oid G are Abelian (i.e. G AA is an Abelian gro up for all ob jects A of the group oid). In that case they are f ull if the groupo id G is tra nsitiv e (i.e. G AB 6 = ∅ , for every pair of o b jects A, B ∈ Ob G ). This exa mple a dmits an immediate g eneralization to the case o f involutiv e categ ories. Given an inv olutive category X , the set of C -v a lued maps on X with ﬁnite supp ort contained in any one of the sets X AB , with A, B ∈ Ob X , is the family of morphisms of a ∗ -catego ry C ∗ o ( X ), with ob jects O b X , whe r e the co mposition is the usual convolution of ﬁnite sequences and the inv olution is deﬁned via ( α x ) ∗ := α x ∗ . The ∗ -categor y C ∗ o ( X ) has a natural cont inuous left-regular action on L 2 ( X ) (that is the family of Hilber t spaces, indexed by A ∈ Ob X , o bta ined by completing ⊕ B ∈ Ob X C ∗ o ( X ) AB under the inner pr oduct h ( α x ) | ( β y ) i := P x,y α x ∗ · β y ) and its C* -completion in the induced op erator nor m is the C*-catego ry C ∗ r ( X ). T aking the C*- c ompletion of C ∗ o ( X ) under the supremum of all the C*-norms induced b y its contin uous representations we obtain the C*- category C ∗ ( X ). The ca teg ories C ∗ ( X ) and C ∗ r ( X ) are commutativ e whenever X AA is commutativ e for a ll ob jects A ∈ Ob X and they are full if a nd only if the category X is saturated in the following sense X AB ◦ X B C = X AC for all ob jects A, B , C ∈ Ob X . • Given any non-diago nal arrow x in a C*-c a tegory C , the C* -subcatego ry C ( x ) of C generated by x is full and comm utative, see theorem 7.1 and the r elated discussion (notice that C ( x ) might well b e a non-unital C* -category if x is not inv er tible). 13 Of course, this is still true for all ir reducible morphisms. 15 • The category of Hermitia n line bundles o ver a compa ct Hausdo r ﬀ space X with line bundle mo r phisms as a rrows is a C* -category , which turns o ut to b e full and co mm u- tative. W e now deal with speciﬁc examples and constructions of spa ceoids. Note that (although every topolo gical spaceoid is of co urse isomorphic to the sp ectrum o f a commutativ e full C*-catego ry) the examples mentioned here below hav e in principle no direct relation with C*-catego ries and arise from some well-known co nstructs in (algebraic) top ology . • As alr eady describ ed in detail in subsection 6.4, the most element ary examples of spaceoid are those asso ciated to every compact Hausdorﬀ top ological spa ces X via the trivia l Hermitian line bundles T X := ( E X , π X , X X ) o ver the top ological s pace X X := ∆ X × { ( X, X ) } with total space E X := X X × C and pro jection π X onto the ﬁrst facto r . • E v er y (p ossibly non-tr iv ial) Hermitian line bundle ( E , π , X ) ov e r a compact Hausdorﬀ top ological space X uniquely deter mines a spaceoid L ( E ) := ( E E , π E , X E ), calle d its “ linking spaceoid”, in the following canonical wa y . Deﬁne the bas e top ological inv olutive category as X E := ∆ X × R O with O := { A, B } , and as tota l space con- sider E E AA := (∆ X × { AA } ) × C , E E B B := (∆ X × { B B } ) × C , E E AB := E × { AB } , E E B A := E + × { B A } , wher e by ( E + , π + , X ) we deno te the Hermitian line bundle dual to ( E , π , X ) (this is the line bundle with ﬁb ers E + p := ( E p ) + given by the dual of the inner product vector spa ce E p ). Deﬁne o n the total space E E the ope rations o f inv olution as usual ﬁb erwise conjugatio n on E E AA and E E B B and by the canonical anti- linear ma p induced b e tween E E AB and E E B A by the natural ﬁb erwise anti-isomorphism betw een E a nd E + . Finally deﬁne the comp osition on the total space a s the usual ﬁber wise pr oduct on E E AA , E E B B and, b et ween elemen ts in E E AB and E E B A , via the canon- ical contraction b et ween E and E + as e AB ◦ e ′ B A := e ′ ( e ) AA and e ′ B A ◦ e AB := e ′ ( e ) B B , for a ll e ∈ E p and e ′ ∈ E + p , with p ∈ X . • In p erfect para llel with the construction o f the linking C*-c a tegory for a family of Hilber t C* -bimodules, the pr e vious c o nstruction of the linking spa ceoid o f a Hermitian line bundle ca n b e g eneralized in o rder to deﬁne the “linking spac e oid” L ( E 1 , . . . , E n ) of a fa mily of (p ossibly non-tr ivial) Hermitian line bundles ( E 1 , π 1 , X ) , . . . , ( E n , π n , X ) ov er the same compact Hausdorﬀ s pace X . F or this purp ose, denoting L ( E 1 , . . . , E n ) := ( E E 1 ...E n , π E 1 ...E n , X E 1 ...E n ), we take the base to p ologica l ∗ -ca tegory a s X E 1 ...E n := ∆ X × R O , whe r e O := { A 1 , . . . , A n +1 } is a set of n + 1 elements. W e then deﬁne the to tal spa ce E E 1 ...E n of the linking spaceoid sp ecifying its blo c ks on the to pologica l spac e X E 1 ...E n A j A k = ∆ X × { A j A k } as follows: E E 1 ...E n A j A k :=                E j ⊗ · · · ⊗ E k − 1 , for j < k − 1 , E j for j = k − 1 , C × (∆ X × { A j A k } ) , for j = k , ( E j ) + for k = j − 1 , ( E k − 1 ) + ⊗ · · · ⊗ ( E j ) + for k < j − 1 , where ⊗ denotes the ﬁb erwise tensor pr oducts of line bundles ov er the same s pa ce X . On the tota l space E E 1 ...E n the ﬁb erwise inv olution and the c omposition ar e deﬁned making use of the canonical is omorphisms o f line bundles ( E j ⊗ E k ) + ≃ ( E k ) + ⊗ ( E j ) + and E i ⊗ ( E j ⊗ E k ) ≃ E i ⊗ E j ⊗ E k ≃ ( E i ⊗ E j ) ⊗ E k , for all i, j, k = 1 , . . . n , via the 16 contraction dualities b et ween E j and ( E j ) + for j = 1 , . . . , n and the tensor pro ducts E j × E k → E j ⊗ E k for a ll j, k = 1 , . . . , n . Examples of linking spaceoids of Her mitian line bundles, that stay in p erfect duality with those of the linking C* -categories of imprimitivit y Hilb ert C*-bimodules previ- ously des c r ibed, can be obta ined via a “biv ariant” notion o f Hermitian line bundle (i.e. a Hermitian line bundle on a ba se spa ce that is the gra ph f ⊂ X × Y of a home- omorphism f : X → Y b et ween tw o compact Hausdor ﬀ top ologica l s paces X , Y ) that is develop e d in mor e detail in our companio n work [BCL3]. • Fina lly , w e brieﬂy in tro duce here a nother natural wa y to pro duce spaceoids via “a ssoci- ated line bundles” to a suitable categor iﬁc a tion of T -torso rs. In more details: given an equiv alence relatio n R , consider the family [ R ; T ] of homomor phisms o f the g r oupoid R with v alues in the torus gro up T := { α ∈ C | | α | = 1 } . Clea rly [ R ; T ] is itself an Abelian gr oup with the ope ration of p oin twise m ultiplication be tw een homomor- phisms. F or a ny compac t Hausdorﬀ space X consider a [ R ; T ]-torso r ( T , π , X ). Since the set [ R ; C ] of C -v alue d homomor phisms has a natura l structure o f [ R ; T ]-spa ce with action given by p oin t wise multiplication, we can construct the “asso ciated bundle” T × [ R ; T ] [ R ; C ] ov e r the s pace X who se elements are equiv a lence classes [( φ, v )] of pa irs ( φ, v ) ∈ T × [ R ; C ] under the equiv alence relation ( φ, v ) ≃ ( ψ , w ) if and only if ther e exists g ∈ [ R ; T ] such that φ · g = ψ and v = g · w . E very such “ass ociated bundle” can be seen, simply b y rearrange men t of v aria bles, as a space o id over the base ∆ X × R . T o obtain the sp ectral spaceoid of a full commutativ e C*-categ ory C , it is suﬃcient to take T := [ C ; C ], the s e t o f C - v a lued ∗ -functors on C . Spec tr al s paceoids can b e easily “a ssem bled” star ting from the sp ectra of imprimitivit y C*-bimo dules dev elop ed (via Ser re-Sw an theorem) in [BCL3, Theor em 3.1 ]. In every full commutativ e C*-catego ry C and for every pair o f ob jects A, B ∈ Ob C , the sp ectrum o f the imprimitivity C*-bimodule C AB is a Hermitian line bundle ( E B A , π B A , R B A ) on the g raph 14 of a unique homeomorphism R B A : Sp( C AA ) → Sp( C B B ) betw een the Gel’fand sp ectra of the tw o unital co mmutative C*-a lg ebras C AA and C B B . Now the (necessarily disjoint) union of all the g raphs R B A ⊂ Sp( C AA ) × Sp( C B B ) of the homeomorphisms R B A , with A, B ∈ Ob C , ca n be seen as the graph o f a new relation  S A,B ∈ Ob C R B A  ⊂  S B ∈ Ob C Sp( C B B )  ×  S A ∈ Ob C Sp( C AA )  in the set S A ∈ Ob C Sp( C AA ) that is the “ disjoin t union” of the Gel’fand spectr a of the diagonal C*-algebras C AA with A ∈ Ob C . Since the homeo morphisms R B A are given by R B A = | B B ◦ | − 1 AA in terms of the res tric- tion homeomorphisms | AA : Sp b ( C ) → Sp( C AA ) deﬁned in prop ositions 5 .5 and theo r em 5.6, the relatio n S A,B ∈ Ob C R B A is an equiv alence relation. 15 F urthermore, the “disjo in t union” of all such Hermitian line bundles ( E B A , π B A , R B A ) b ecomes a new Her mitian line bun- dle U A,B ∈ Ob C ( E AB , π AB , R AB ) :=  S A,B ∈ Ob C E B A , S A,B ∈ Ob C π B A , S A,B ∈ Ob C R B A  with total spa ce S B ,A ∈ Ob C E B A and base space S A,B ∈ Ob C R B A . The ma p τ : [ ω ] AB 7→ ( ω AA , ω B B ) provides an iso morphism of topo lo gical inv o lutive cate- gories betw een ∆ C × R C and S AB ∈ Ob C R B A . Since, as Hilbert s pa ces, the ﬁb ers E ( ω AA ,ω BB ) 14 Note that here w e are using R BA to denote the homeomorphism R BA : Sp( C AA ) → Sp( C BB ) and togethe r its graph R BA ⊂ Sp( C AA ) × Sp( C BB ). More generally , we use the same letter R to denote a relation from the set A to the set B and its graph R ⊂ A × B . 15 Note that the new eq uiv alence relation S A,B ∈ Ob C R BA is a r elation b et ween elements of the union S A ∈ Ob C Sp( C AA ) of the Gel’fand sp ec tra not to be confused with the “coarse grained” equiv alence r elat ion { (Sp( C AA ) , Sp( C BB )) | A, B ∈ Ob C } betw een the Gel’fand spectra themselv es or equiv alently with the sub-equiv alence relation { R AB | A, B ∈ Ob C } of the groupoi d of homeomorphisms of compact Hausdorﬀ spaces. 17 are given by E ( ω AA ,ω BB ) := C AB / ( C AB · Ker( ω )) and s o coincide with the ﬁb ers of the sp ectral spaceoid Σ( C ) on the elemen ts [ ω ] AB , we see tha t S A,B ∈ Ob C E ( ω AA ,ω BB ) can be natura lly equipp e d with the str uctur e of C*- category a nd so the bundle U A,B ∈ Ob C ( E AB , π AB , R AB ) is a rank- one F ell bundle. Finally we see that the sp ectral space oid Σ( C ) coincides with the τ -pull-ba c k τ • ( S A,B ∈ Ob C E AB ) o f the rank -one F ell bundle U A,B ∈ Ob C ( E AB , π AB , R AB ). The cla ssical Gel’fand- Na ˘ ımark duality for co mmutative C*-a lgebras had a num ber of im- po rtan t a pplications and, in a pa r allel wa y , its extension for co mm utative full C*-catego ries describ ed here, will provide interesting “horizo n tal categor iﬁcations” of thos e applications. Among them there ar e, for exa mple: • a F ourie r tra nsform in the c o n text of Pontry ag in duality for commutativ e gr oupoids, • a contin uous functional ca lculus for b ounded linea r o perator s b et ween Hilb ert spaces, • a sp ectral theor em for b ounded linea r op erators be tw een Hilbe r t spaces. Most of these ideas will a c tua lly r equire a serious amount of additiona l work that deserves a separ ate detailed treatment elsewhere and so, in order to exemplify the “capa bilities” of our r esult, we limit ourselves to the development of a “ho rizon tally ca tegoriﬁed co n tinuous functional calculus” which is the mo st imm ediate and stra igh tfor ward o f the previo us topics . Let C b e a C*-catego ry , not necessar ily commu tative o r full, and let x ∈ C AB be o ne of its morphisms. Consider no w C ( x ), the (non-necess arily unital) C* -category generated b y x . By deﬁnition, this is the C*-suba lgebra of C AA generated by x if A = B , and the C*-sub category of C with tw o ob jects A and B and arr ow spaces C ( x ) AA = span { ( x ◦ x ∗ ) n | n = 1 , 2 , 3 , . . . } − , C ( x ) B B = span { ( x ∗ ◦ x ) n | n = 1 , 2 , 3 , . . . } − and C ( x ) AB = x ◦ C ( x ) B B = C ( x ) ∗ B A otherwise. Notice that C ( x )(= C ( x ∗ )) is alwa y s full. If A 6 = B the C*-categor y C ( x ) is alwa ys comm utative and for A = B it is commutativ e if and only if x is normal and in thes e t wo cases we can immedia tely apply o ur spec tr al results o n the horizo n tal catego r iﬁed Gel’fand transform to rea liz e that: every morphism in the ca tegory C ( x ) is uniquely describ ed by a contin uous section of a “blo c k” of the s p ectral spaceoid of C ( x ). In mor e detail we hav e: Theorem 7. 1 (Horizontally categoriﬁe d con tinuous functiona l calculus ) . L et x ∈ C AB b e an element of a C*-c ate gory C and let C ( x ) denote the (n on- ne c essarily unital) C*-c ate gory gener ate d by x inside C . If either the obje cts A and B ar e diﬀer ent, or A = B and the element x ∈ C AA is normal, then the C*-c ate gory C ( x ) is ful l and c ommu tative. In that c ase, for every c ontinuous se ction σ ∈ Γ(Σ( C ( x ))) AB of the blo ck AB of the sp e ct r al sp ac e oid of C ( x ) , t her e is an asso ciate d element σ ( x ) ∈ C ( x ) AB . Mor e over, the r esult ing m ap F x : σ 7→ σ ( x ) is an isometric ∗ -functor fr om the (p ossibly non-unital) C*-c ate gory Γ(Σ( C ( x ))) onto C ( x ) ⊂ C . Pr o of. Giv e n x ∈ C AB , we simply deﬁne F x : Γ(Σ( C ( x ))) → C as the ma p given b y the inv erse o f the Gel’fand transform G : C ( x ) → Γ(Σ( C ( x ))) i.e. F x := G − 1 C ( x ) . W e ca ll the ∗ - functor F x the contin uous functional calculus of x . Note that the previous result migh t open the w ay to obtaining a s p ectral representation (and hence a sp ectral theo rem) also for b o unded linea r op erators that are no t nor mal. In fact if T : H → H is an arbitra ry b o unded linear oper ator o n a Hilb ert space H , we can a lw ays regar d T as a morphism in an oﬀ-diagonal block o f the C*-ca tegory , with tw o ob jects, of bo unded linear op erators b et ween H A := H = : H B . 18 8 Outlo ok W e hav e in tro duced comm utative C*-categories and started a program for their “topolog ical description” in terms o f their sp e ctra, here c alled spaceoids. In particular , w e have obtained a Gel’fand-t yp e theorem for full comm utative C*-categories. Although the statement of the main res ult (theore m 6 .4) looks extremely natural, o ur pr oofs mostly rely on a “ brute for ce” ex ploitation o f the underlying structure and more streamlined arguments are likely to b e fo und. Also , the result by itself is not as g eneral as p ossible and certainly it le aves ro om fo r extensio ns in several directions, still hop efully we hav e provided some insig h t a bout how to achieve them. F or instance, we hav e only c o nsidered the case of ∗ -functors betw e en (full, comm utative) C*-catego ries that are bijective on the ob jects. (Of cours e , this trivially includes morphisms betw een commutativ e C*- algebras). In the next step, one would lik e to treat the case of ∗ -functors that are not bijective on the ob jects. W e tend to b elieve that this should not require signiﬁcant modiﬁcatio ns of our treatment and p ossibly co uld be dealt with using relators (that we introduced in [BCL1]). 16 Perhaps a more imp ortant p oint would b e to r emo ve the condition of fullness . A t present we ha ve not discus sed the issue in detail, but certainly the information tha t w e hav e a lready acquired should s ig niﬁcan tly simplify the task. Also, along the way , we ha ve somehow taken adv a n tage of our prior knowledge of the Gel’fand and Serr e -Sw an theor ems. Ev en tually one would like to provide mor e intrinsic pro ofs directly in the framew ork o f C*-ca tegories (p ossibly unifying and ex tending b oth Gel’fand and Serr e- Swan theorems in a “strict ∗ -mo no idal” v ers ion of T ak aha s hi theorem [T1, T2]). In this resp ect, it lo oks pro mising to work dire c tly with mo dule categor ies. B esides, it is s omeho w disapp oin ting that to date, for X a nd Y compact Hausdorﬀ spaces, there seems to b e no av ailable g eneral classiﬁca tion re s ult for C ( X )- C ( Y )-bimo dules. Hilber t C*-bimo dules that are not- necessarily imprimitivity bimo dules sho uld deﬁnitely pla y a role when discussing a classiﬁcation r esult for generally non- comm utative C*-categ ories, po ssibly along the lines of a genera lization to C*-categ ories o f the Dauns- Hofmann the- orem for C*- algebras [DH]. One migh t a lso explor e p ossible co nnections with the no n- commutativ e Gel’fand sp ectral theorem of R. Cir elli-A. Mani` a-L. Pizzo cc hero [CMP] and the subs e quen t no n-comm uta tive Serre-Swan dua lit y by K. Kaw a m ura [Ka] a nd E. E lliott- K. Kaw a m ura [E K]. Similarly , it might b e very interesting to in vestigate the connections be- t ween our sp ectral spaceo ids and other s pectral notions such as lo cales and topoi alrea dy used in the sp ectral theor ems b y B. Banachewki-C. Mulvey [BM] a nd C. Heunen-K . Landsmann- B. Spitters [HLS ]. In the lo ng run, one would like to (deﬁne and) clas s ify commutativ e F ell bundles ov er suitable inv olutive categories. The notion of a F ell bundle could be ev en gener alized to that o f a ﬁber ed catego ry enriched ov e r ano ther ( ∗ -monoidal) ca tegory . Needless to say , one should a nalyze more closely the mathematical s tructure of s pa ceoids, int ro duce suitable topo logical inv ar ia n ts, study their symmetr ie s , . . . , and in vestigate r e- lations to other concepts that are widely used in other branches of mathematics , e.g. in algebraic top ology/g eometry as well as in g a uge theories. Some ge ometric structur es could bec ome apparent when cons idering the representation o f spa ceoids as con tinuous ﬁelds of (one-dimensional commutativ e) C*-ca teg ories as discussed by E. V a s selli in [V]. 16 F or this purpose it should b e enou gh to in tro duce a category of space oids in whic h the morphisms f : X 1 → X 2 betw een the tw o base ∗ -categories X 1 = ∆ 1 × R 1 and X 2 = ∆ 2 × R 2 are given by ∗ -r elat ors f := ( f ∆ , f R ) where now the ∗ -functor f R : R 2 → R 1 is acting in the “rev erse direction”. 19 The Gel’fand transform for general commutativ e C*-catego ries raises several questions (un- doubtedly it could b e deﬁned for more g eneral Ba nac h ca teg ories, leading to a w ide range of po ssibilities for fur ther studies). In pa rticular, a n immediate application would yield a F ourier transfo r m a nd accor dingly a reasona ble concr ete duality theory for commutativ e discrete gro upoids (see M. Amini [A] for another appr oac h that applies to compac t but-no t-necessarily-commutativ e-group oids, T. Timmermann [Ti] for a more abstr act setup and G. Go ehle [G] for a discussio n of dualit y for lo cally compact Ab elian group bundles). As far as we are concerned, our main motiv ation to w ork with C*-categ ories co mes from analyzing the ca tegorical structure of no n-comm utative g eometry (where morphisms of “non- commutativ e spac es” are given by bimodules) a nd one is naturally led to sp eculating a bout the pos sible evolution of the notion of spectr a a nd morphisms in A. Connes’ non-commutativ e geometry (cf. [BC L 1 , BC L 2, CCM]). In this dir ection, some of the ﬁr st questions tha t co me to mind a re: Is there a suitable notion o f sp ectral triple over a C*-categ ory? Is it p ossible to cons ider a horizo n tal ca tegoriﬁcation of a sp ectral triple? Of cours e this repres en ts only the star ting point for a m uch more ambitious program aiming at a “ v ertica l categoriﬁcatio n” of the notio n o f sp ectral triple 17 and from s e v era l front s (see for example [DTT] and also the very de ta iled discussion by J. Bae z [B] on the weblog “ The n -Categor y Caf´ e”) it is mounting the evidence that a suitable notion of non-commutativ e calculus nec e s sarily require a hig her (actually ∞ ) ca tegorical setting. In this resp ect, it seems reasona ble to lo ok for a Gel’fand theorem that applies to (strict) com- m utative higher categ ories (cf. [Ko]). A suitable deﬁnition of strict n -C*-ca tegories (cf. [Z ] for the ca se n = 2) and the pr oof of a categorical Gel’fand duality (a t lea st for “commutativ e” full str ict n -C*-ca tegories) are topics that hav e r ecen tly attracted our attention [BCLS]. Finally , in this line of thoughts, one could envisage p otential applica tions of a notion of Gromov-Hausdorﬀ distance (cf. [R]) for C*-c a tegories. Ac knowledgmen ts. W e ac knowledge the supp ort provided by the Thai R esea rch F und, gra nt n. RSA4780022. The main part of this work has b een d one in the tw o-year v isi ting p eriod of R . Conti t o the Dep artmen t of Mathematics of Chulal ongkorn Universit y . P . Bertozzini ackno wledges the “w eek ly hospitalit y ” oﬀered by the Dep artmen t of Mathematics in Chula longko rn Un iv ersity , where most of the work leading to this pu blica tion has b een done, from June 2005. He also desires to thank A. Carey and B. W ang at ANU in C anberra, R. Street, A. Da vy- dov and M. Batanin at Macquarie Universit y in Syd ney , K. Hibb ert and J. Link s at the Center for Mathematical Physics in Brisbane, A. V az F erreira at the Universi ty of Bologna, F. Cipriani at th e “P olitecnico di Milano”, G. Landi and L. Dabro wski a t SIS S A in T rieste, R. Longo , C. D’Antoni and L. Zsido at t he Univers ity o f “T or V ergata” in R ome, W. Szymanski at the U niv ersity of Newca stle in Australia, E. Beggs and T. Brzezinski at Sw ansea Universi ty , D. Ev ans at Cardiﬀ Un iv ersity , A. D¨ oring and B. Coeck e at the CLAP wo rkshop in I m p erial College, for the k in d hospitalit y , the muc h appreciated partial supp ort and most of all for the p ossibili ty to oﬀer th e tal ks/seminars where preliminary versio ns of the results contained in th is pap er hav e b een announced in Octob er 2006, Marc h/Octob er 2007 and May 2008. Finally we thank the tw o anonymous referees for a very careful reading of the manuscript and for suggesting severa l impro vemen ts. 17 The need for a notion of “higher sp ectral triple” has b een alr ead y advocated by U. Sc hreib er [ S]. 20 References [A] Amini M. (2007) . T annak a-Krein Duality for Compact Groupoids: I Represen tation Theory , A dv. Math. 214, n. 1, 78-91, arXiv:math.OA/ 0308259v1 ; II F ourier T r ansform, ar Xiv:math /0308260v 1 ; III Duality Theory , arXiv:math /0308261v 1 . [Ar] Arveson , W. (1976). An Invitation to C*-algebr as , GTM 39, Springer-V erl ag. 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A Horizontal Categorification of Gelfand Duality

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