math.CT 2008-09-22 0

Tannaka duality for proper Lie groupoids

The main contribution of this thesis is a Tannaka duality theorem for proper Lie groupoids. This result is obtained by replacing the category of smooth vector bundles over the base manifold of a Lie groupoid with a larger category, the category of sm…

Authors: Giorgio Trentinaglia

Tannaka duality for proper Lie groupoids
Giorgio T ren tinaglia T ANNAKA DUALITY F OR PROPER LIE GROUPOIDS † (PhD Thesis, Utreht Universit y , 2008) † This w ork w as nanially supp orted b y Utre h t Univ ersit y , the Univ ersit y of P adua , and a gran t of the foundation F ondazione Ing. Aldo Gini  2 Abstrat: The main on tribution of this thesis is a T annak a dualit y theorem for prop er Lie group oids. This result is obtained b y replaing the ategory of smo oth v etor bundles o v er the base manifold of a Lie group oid with a larger ategory , the ategory of smo oth Eulidean elds, and b y onsidering smo oth ations of Lie group oids on smo oth Eulidean elds. The notion of smo oth Eulidean eld that is in tro dued here is the smo oth, nite dimensional ana- logue of the familiar notion of on tin uous Hilb ert eld. In the seond part of the thesis, ordinary smo oth represen tations of Lie group oids on smo oth v etor bundles are systematially studied from the p oin t of view of T annak a dualit y , and v arious results are obtained in this diretion. Keyw ords: prop er Lie group oid, represen tation, tensor ategory , T annak a dualit y , sta k AMS Sub jet Classiations: 58H05, 18D10 A  kno wledgemen ts: I w ould lik e to thank m y sup ervisor, I. Mo erdijk, for ha ving suggested the resear h problem out of whi h the presen t w ork to ok shap e and for sev eral useful remarks, and also M. Craini and N. T. Zung , for their in terest and for helpful on v ersations. Con ten ts T able of Con ten ts 3 In tro dution 5 F rom Lie groups to Lie group oids . . . . . . . . . . . . . . . . . . . 5 Historial p ersp etiv e on T annak a dualit y . . . . . . . . . . . . . . 7 What is new in this thesis . . . . . . . . . . . . . . . . . . . . . . . 9 Outline  hapter b y  hapter . . . . . . . . . . . . . . . . . . . . . . . 10 Some p ossible appliations . . . . . . . . . . . . . . . . . . . . . . . 19 I Lie Group oids, Classial Represen tations 21 1 Generalities ab out Lie Group oids . . . . . . . . . . . . . . . . 21 2 Classial Represen tations . . . . . . . . . . . . . . . . . . . . . 24 3 Normalized Haar Systems . . . . . . . . . . . . . . . . . . . . 31 4 The Lo al Linearizabilit y Theorem . . . . . . . . . . . . . . . 32 5 Global Quotien ts . . . . . . . . . . . . . . . . . . . . . . . . . 36 I I The Language of T ensor Categories 39 6 T ensor Categories . . . . . . . . . . . . . . . . . . . . . . . . . 39 7 T ensor F untors . . . . . . . . . . . . . . . . . . . . . . . . . . 44 8 Complex T ensor Categories . . . . . . . . . . . . . . . . . . . 46 9 Review of Groups and T annak a Dualit y . . . . . . . . . . . . . 48 10 A T e hnial Lemma on Compat Groups . . . . . . . . . . . . 50 I I I Represen tation Theory Revisited 55 11 The Language of Fibred T ensor Categories . . . . . . . . . . . 55 12 Smo oth T ensor Sta ks . . . . . . . . . . . . . . . . . . . . . . 60 13 F oundations of Represen tation Theory . . . . . . . . . . . . . 63 14 Homomorphisms and Morita In v ariane . . . . . . . . . . . . . 65 IV General T annak a Theory 71 15 Sta ks of Smo oth Fields . . . . . . . . . . . . . . . . . . . . . 71 16 Smo oth Eulidean Fields . . . . . . . . . . . . . . . . . . . . . 78 17 Constrution of Equiv arian t Maps . . . . . . . . . . . . . . . . 81 18 Fibre F untors . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3 4 CONTENTS 19 Prop erness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 20 Reonstrution Theorems . . . . . . . . . . . . . . . . . . . . 98 V Classial Fibre F untors 109 21 Basi Denitions and Prop erties . . . . . . . . . . . . . . . . . 110 22 T ame Submanifolds of a Lie Group oid . . . . . . . . . . . . . 114 23 Smo othness, Represen tativ e Charts . . . . . . . . . . . . . . . 126 24 Morphisms of Fibre F untors . . . . . . . . . . . . . . . . . . . 134 25 W eak Equiv alenes . . . . . . . . . . . . . . . . . . . . . . . . 137 VI Classial T annak a Theory 143 26 The Classial En v elop e of a Prop er Group oid . . . . . . . . . 144 27 Prop er Regular Group oids . . . . . . . . . . . . . . . . . . . . 150 28 Classial Reexivit y: Examples . . . . . . . . . . . . . . . . . 153 Bibliograph y 156 Index 160 In tro dution Although a rigorous form ulation of the problem with whi h this do toral thesis is onerned will b e p ossible only after the en tral ideas of T annak a dualit y theory ha v e b een at least briey disussed, I an nev ertheless start with some ommen ts ab out the general on text where su h a problem tak es its appropriate plae. Roughly sp eaking, m y study aims at a b etter under- standing of the relationship that exists b et w een a giv en Lie group oid and the orresp onding ategory of represen tations. First of all, for the b enet of non-sp eialists, I w an t to explain the reasons of m y in terest in the theory of Lie group oids (a preise denition of the notion of Lie group oid an b e found in 1 of this thesis) b y dra wing atten tion to the prinipal appliations that justify the imp ortane of this theory; in the seond plae, I in tend to undertak e a ritial examination of the onept of represen tation in order to on vine the reader of the naturalness of the notions I will in tro due b elo w. F rom Lie groups to Lie group oids Group oids mak e their app earane in div erse mathematial on texts. As the name `group oid' suggests, this notion generalizes that of group. In order to explain ho w and to mak e the denition more plausible, it is b est to start with some examples. The reader is ertainly familiar with the notion of fundamen tal group of a top ologial spae. The onstrution of this group presupp oses the  hoie of a base p oin t, and an y t w o su h  hoies giv e rise to the same group pro vided there exists a path onneting the base p oin ts (for this reason one usually assumes that the spae is path onneted). Ho w ev er, instead of onsidering only paths starting and ending at the same p oin t, one migh t more generally allo w paths with arbitrary endp oin ts; t w o su h paths an still b e omp osed as long as the one starts where the other ends. One obtains a w ell-dened as- so iativ e partial op eration on the set of homotop y lasses of paths with xed endp oin ts, for whi h the (lasses of ) onstan t paths are b oth left and righ t neutral elemen ts. Observ e that ea h path has a t w o-sided in v erse, namely the path itself with rev erse orien tation. In geometry , groups are usually groups of transformationsor symme- triesof some ob jet or spae. If g is an elemen t of a group G ating on a 5 6 INTR ODUCTION spae X and x is a p oin t of X , one ma y think of the pair ( g , x ) as an arro w going from x to g · x ; again, t w o su h arro ws an b e omp osed in an ob vious w a y , b y means of the group op eration of G , pro vided one starts where the other ends. Comp osition of arro ws is an asso iativ e partial op eration on the set G × X , whi h eno des b oth the m ultipliation la w of the group G and the G -ation on X . In the represen tation theory of groups, the linear group GL ( V ) asso iated with a nite dimensional v etor spae V pla ys a fundamen tal role. If a v etor bundle E o v er a spae X is giv en instead of a single v etor spae V , one an onsider the set GL ( E ) of all triples ( x, x ′ , λ ) onsisting of t w o p oin ts of X and a linear isomorphism λ : E x ∼ → E x ′ b et w een the bres o v er these p oin ts. As in the examples ab o v e, an elemen t ( x, x ′ , λ ) of this set an b e view ed as an arro w going from x to x ′ ; su h an arro w an b e omp osed with another one as long as the latter has the form ( x ′ , x ′′ , λ ′ ) . Arro ws of the form ( x, x, id ) are b oth left and righ t neutral elemen ts for the resulting asso iativ e partial op eration, and ea h arro w admits a t w o-sided in v erse. By abstration from these and similar examples, one is led to onsider small ategories where ev ery arro w is in v ertible. Su h ategories are referred to as group oids. More expliitly , a group oid onsists of a spae X of base p oin ts (also alled ob jets), a set G of arro ws, endo w ed with soure and target pro jetions s , t : G → X , and an asso iativ e partial omp osition la w G s × t G → G (dened for all pairs of arro ws ( g ′ , g ) with the prop ert y that the soure of g ′ equals the target of g ), su h that in orresp ondene with ea h p oin t x of X there is a (neessarily unique) neutral or unit arro w, often itself denoted b y x , and ev ery arro w is in v ertible. The notion of Lie group oid generalizes that of Lie group. Mu h the same as a Lie group is a group endo w ed with a smo oth manifold struture ompat- ible with the m ultipliation la w and with the op eration of taking the in v erse, a Lie group oid is a group oid where the sets X and G are endo w ed with a smo oth manifold struture that mak es the v arious maps whi h arise from the group oid struture smo oth. F or instane, in ea h of the examples ab o v e one obtains a Lie group oid when the spae X of base p oin ts is a smo oth manifold, G is a Lie group ating smo othly on X and E is a smo oth v etor bundle o v er X ; these Lie group oids are resp etiv ely alled the fundamental gr oup oid of the manifold X , the tr anslation gr oup oid asso iated with the smo oth ation of G on X and the line ar gr oup oid asso iated with the smo oth v etor bun- dle E . There is also a more general notion of C ∞ -strutur e d gr oup oid, ab out whi h w e shall sp end a few w ords later on in the ourse of this in tro dution, whi h w e in tro due in our thesis in order to desrib e ertain group oids that arise naturally in the study of T annak a dualit y theory . In the ourse of the seond half of the t w en tieth en tury the notion of group oid turned out to b e v ery useful in man y bran hes of mathematis, although this notion had in fat already b een in the air sine the earliest a- Historial p ersp etiv e on T annak a dualit y 7 omplishmen ts of quan tum me hanisthink, for example, of Heisen b erg's formalism of matriesor, more ba k in time, sine the rst in v estigations in to lassiation problems in geometry . No w ada ys, the theory of Lie group- oids onstitutes the preferred language for the geometrial study of foliations [27℄; the same theory has appliations to nonomm utativ e geometry [8, 5℄ and quan tization deformation theory [21℄, as w ell as to sympleti and P oisson geometry [36, 9 , 15 ℄. Another soure of examples omes from the study of orbifolds [25℄; this sub jet is onneted with the theory of sta ks, whi h origi- nated in algebrai geometry from Grothendie k's suggestion to use group oids as the righ t notion to understand mo duli spaes. When trying to extend represen tation theory from Lie groups to Lie group- oids, one is rst of all onfron ted with the problem of dening a suitable notion of represen tation for the latter. As far as w e are onerned, w e w ould lik e to generalize the familiar notion of (nite dimensional) Lie group repre- sen tation, b y whi h one generally means a homomorphism G → GL ( V ) of a Lie group G in to the group of automorphisms of some nite dimensional v etor spae V , so that as man y onstrutions and results as p ossible an b e adapted to Lie group oids without essen tial  hanges; in partiular, w e w ould lik e to arry o v er T annak a dualit y theory (see the next subsetion) to the realm of Lie group oids. The notion of Lie group represen tation realled ab o v e has an ob vious naiv e extension to the group oid setting. Namely , a represen tation of a Lie group oid G an b e dened as a Lie group oid homomorphism G → GL ( E ) (smo oth funtor) in to the linear group oid asso iated with some smo oth v etor bundle E o v er the manifold of ob jets of G . An y su h represen tation assigns ea h arro w x → x ′ of G a linear isomorphism E x ∼ → E x ′ in su h a w a y that omp osition of arro ws is resp eted. In our dissertation w e will use the term `lassial represen tation' to refer to this notion. Unfortunately , lassial represen tations pro v e to b e ompletely inadequate for the ab o v e-men tioned purp ose of arrying forw ard T annak a dualit y to Lie group oids; w e shall sa y something more ab out this matter later. The preeding onsideration leads us to in tro due a dieren t notion of represen tation for Lie group oids. In doing this, ho w ev er, w e adhere to the p oin t of view that the latter should b e as lose as p ossible to the notion of lassial represen tationin partiular the new theory should extend the theory of lassial represen tationsand that moreo v er in the ase of groups one should reo v er the usual notion of represen tation realled ab o v e. Historial p ersp etiv e on T annak a dualit y It has b een kno wn for a long time, and preisely sine the pioneer w ork of Pontryagin and van Kamp en in the 1930's, that a omm utativ e lo ally ompat group an b e iden tied with its o wn bidual. Reall that if G is su h 8 INTR ODUCTION a group then its dual is the group formed b y all the  haraters on G , that is to sa y the on tin uous homomorphisms of G in to the m ultipliativ e group of omplex n um b ers of absolute v alue one, the group op eration b eing giv en b y p oin t wise m ultipliation of omplex funtions; one ma y regard the latter group as a top ologial groupin fat, a lo ally ompat oneb y taking the top ology of uniform on v ergene on ompat subsets. There is a anonial pairing b et w een G and this dual, giv en b y p oin t wise ev aluation of  haraters at elemen ts of G , whi h indues a on tin uous homomorphism of G in to its o wn bidual. Then one an pro v e that the latter orresp ondene is atually an isomorphism of top ologial groups; see for instane Dixmier (1969) [13 ℄, R udin (1962) [31℄, or the b o ok b y Cheval ley (1946) [ 6℄. When one tries to generalize this dualit y result to non-Ab elian lo ally ompat groups, su h as for instane Lie groups, it b eomes eviden t that the whole ring of represen tations m ust b e onsidered b eause  haraters are no longer suien t to reapture the group. Ho w ev er, it is still an op en problem to form ulate and pro v e a general dualit y theorem for nonomm utativ e Lie groups: ev en the ase of simple algebrai groups is not w ell understo o d, de- spite the enormous aum ulating kno wledge on their irreduible represen ta- tions. The situation is quite the opp osite when the group is  omp at, b eause the dual ob jet G ∨ of a ompat group G is disrete and so b elongs to the realm of algebra: in this ase, there is a go o d dualit y theory due to H. Peter, H. W eyl and T. T annaka, whi h w e no w pro eed to reall. The early dualit y theorems of T annaka (1939) [ 34℄ and Kr ein (1949) [20 ℄ onen trate on the problem of reonstruting a ompat group from the ring of isomorphism lasses of its represen tations. Owing to the ideas of Grothendie k [32℄, these results an no w ada ys b e form ulated within an ele- gan t ategorial framew ork. Although w e do not in tend to en ter in to details no w, these ideas are impliit in what w e are ab out to sa y . 1. One starts b y onsidering the ategory R 0 ( G ) of all on tin uous nite dimensional represen tations of the ompat group G : the ob jets of R 0 ( G ) are the pairs ( V ,  ) onsisting of a nite dimensional real v etor spae V and a on tin uous homomorphism  : G → GL ( V ) ; the morphisms are preisely the G -equiv arian t linear maps. 2. There is an ob vious funtor ω of the ategory R 0 ( G ) in to that of nite dimensional real v etor spaes, namely the forgetful funtor ( V ,  ) 7→ V . The natural endomorphisms of ω form a top ologial algebra End( ω ) , when one endo ws End( ω ) with the oarsest top ology making ea h map λ 7→ λ ( R ) on tin uous as R ranges o v er all ob jets of R 0 ( G ) . 3. The subset T ( G ) of this algebra, formed b y the elemen ts ompatible with the tensor pro dut op eration on represen tations, in other w ords the natural endomorphisms λ of ω su h that λ ( R ⊗ R ′ ) = λ ( R ) ⊗ λ ( R ′ ) and λ ( 1 ) = id , pro v es to b e a ompat group. 4. (T annaka) The anonial map π : G → End( ω ) , dened b y setting What is new in this thesis 9 π ( g )( R ) =  ( g ) for ea h ob jet R = ( V ,  ) of R 0 ( G ) , establishes an iso- morphism of top ologial groups b et w een G and T ( G ) . What is new in this thesis W e are no w ready to giv e a short summary of the original on tributions of the presen t study . Within the realm of Lie group oids, prop er group oids pla y the same role as ompat groups; for example, all isotrop y groups of a prop er Lie group oid are ompat (the isotrop y group at a base p oin t x onsists of all arro ws g with s ( g ) = t ( g ) = x ). The main result of our resear h is a T annaka duality the or em for pr op er Lie gr oup oids, whi h tak es the follo wing form. T o b egin with, w e onstrut, for ea h smo oth manifold X , a ategory whose ob jets w e all smo oth elds over X ; our notion of smo oth eld is the analogue, in the smo oth and nite dimensional setting in whi h w e are in ter- ested, of the familiar notion of on tin uous Hilb ert eld in tro dued b y Dixmier and Douady in the early 1960's [14℄ (see also Bos [2 ℄ or Kali²nik [19 ℄ for more reen t w ork related to on tin uous Hilb ert elds). The ategory of smo oth elds is a prop er enlargemen t of the ategory of smo oth v etor bundles. Lik e for v etor bundles, one an dene a notion of Lie group oid represen tation on a smo oth eld in a ompletely standard w a y . Giv en a Lie group oid G , su h represen tations and their ob vious morphisms form a ategory that is related to the ategory of smo oth elds o v er the base manifold M of G b y means of a forgetful funtor of the former in to the latter ategory . T o this funtor one an assign, b y generalizing the onstrution explained ab o v e in the ase of groups, a group oid o v er M , to whi h w e shall refer as the T annakian gr oup oid asso iate d with G , to b e denoted b y T ( G ) , endo w ed with a natural andidate for a smo oth struture on the spae of arro ws ( C ∞ -strutur e d gr oup oid). As for groups, there is a anonial homomorphism π of G in to T ( G ) that turns out to b e ompatible with this C ∞ -struture. Our T annak a dualit y theorem for prop er Lie group oids reads as follo ws: Theorem Let G b e a prop er Lie group oid. The C ∞ -struture on the spae of arro ws of the T annakian group oid T ( G ) is a gen uine manifold struture so that T ( G ) is a Lie group oid. The anonial homomorphism π is a Lie group oid isomorphism G ∼ = T ( G ) . The main p oin t here is to pro v e the surjetivit y of the homomorphism π ; the fat that π is injetiv e is a diret appliation of a theorem of N.T. Zung. A tually , the reasonings leading to our dualit y theorem also hold, for the most part, for the represen tations of a prop er Lie group oid on v etor bundles. Sine from the v ery b eginning of our resear h w e w ere equally in terested in studying su h represen tations, w e found it on v enien t to pro vide a general theoretial framew ork where the div erse approa hes to the represen tation 10 INTR ODUCTION theory of Lie group oids ould tak e their appropriate plae, so as to state our results in a uniform language. The outome of su h demand w as the theory of `smo oth tensor staks'. Smo oth v etor bundles and smo oth elds are t w o examples of smo oth tensor sta ks. Ea h smo oth tensor sta k giv es rise to a orresp onding notion of represen tation for Lie group oids; then, for ea h Lie group oid one obtains, b y the same general pro edure outlined ab o v e, a orresp onding T annakian group oid, whi h will dep end v ery m u h, in general, on the initial  hoie of a smo oth tensor sta k (for example, T annak a dualit y fails in the on text of represen tations on v etor bundles). Our remaining on tributions are mainly onerned with the study of T annakian group oids arising from represen tations of prop er Lie group oids on v etor bundles. Sine in this ase the reonstruted group oid ma y not b e isomorphi to the original one, the problem of whether the aforesaid standard C ∞ -struture on the spae of arro ws of the T annakian group oid turns the latter group oid in to a Lie group oid b eomes onsiderably more in teresting and diult than in the ase of represen tations on smo oth elds. Our prin- ipal result in this diretion is that the answ er to the indiated question is armativ e for all prop er regular group oids. In onnetion with this result w e pro v e in v ariane of the solv abilit y of the problem under Morita equiv alene. Finally , w e pro vide examples of lassi al ly r eexive prop er Lie group oids, i.e. prop er Lie group oids for whi h the group oid reonstruted from the repre- sen tations on v etor bundles is isomorphi to the original one; ho w ev er, our list is v ery short: failure of reexivit y is the rule rather than the exeption when one deals with represen tations on v etor bundles. Outline  hapter b y  hapter In order to help the reader nd their o wn w a y through the dissertation, w e giv e here a detailed aoun t of ho w the material is organized. ∗ ∗ ∗ In Chapter I w e reall basi notions and fats onerning Lie group oids. The initial setion is mainly ab out denitions, notation and on v en tions to b e follo w ed in the sequel. The seond setion on tains relativ ely more in teresting material: after briey realling the familiar notion of a represen tation of a Lie group oid on a v etor bundle (lassial represen tation), w e supply a onrete example, 1 whi h motiv ates our in tro duing the notion of represen tation on a smo oth eld in Chapter IV , sho wing that it is in general imp ossible to distinguish t w o Lie group oids from one another just on the basis of kno wledge of the 1 W e diso v ered this oun terexample indep enden tly , though it turned out later that the same had already b een around for some time [23 ℄. Outline  hapter b y  hapter 11 orresp onding ategories of represen tations on v etor bundles; more preisely , w e shall expliitly onstrut a prinipal T 2 -bundle o v er the irle (where T k denotes the k -torus), together with a homomorphism on to the trivial T 1 -bundle o v er the irle, su h that the ob vious pull-ba k of represen tations along this homomorphism yields an isomorphism b et w een the ategories of lassial represen tations of these t w o bundles of Lie groups. In Setion 3 w e review the notion of a (normalize d) Haar system on a Lie group oid; this is the analogue, for Lie group oids, of the notion of (probabilit y) Haar measure on a group. Lik e probabilit y Haar measures, normalized Haar systems an b e used to obtain in v arian t funtions, metris et. b y means of the usual a v eraging te hnique. The p ossibilit y of onstruting equiv arian t maps lies at the heart of our pro of that the homomorphism π men tioned ab o v e is surjetiv e for ev ery prop er Lie group oid. Setion 4 in tro dues the reader to a relativ ely reen t result obtained b y N.T. Zung ab out the lo al struture of prop er Lie group oids; this general re- sult w as rst onjetured b y A. W einstein in his famous pap er ab out the lo- al linearizabilit y of prop er regular group oids [37℄ (where the result is pro v ed preisely under the additional assumption of regularit y). Zung's lo al lin- earizabilit y theorem states that ea h prop er Lie group oid G is, lo ally in the viinit y of an y giv en G -in v arian t p oin t of its base manifold, isomorphi to the translation group oid asso iated with the indued linear ation of the iso- trop y group of G at the p oin t itself on the resp etiv e tangen t spae. As a onsequene of this, ev ery prop er Lie group oid is lo ally Morita equiv alen t to the translation group oid asso iated with some ompat Lie group ation. The lo al linearizabilit y of prop er Lie group oids aoun ts for the injetivit y of the homomorphism π . Finally , in Setion 5, w e pro v e a statemen t relating the global struture up to Morita equiv alene of a prop er Lie group oid and the existene of globally faithful represen tations: preisely , w e sho w that a prop er Lie group oid admits a globally faithful represen tation on a smo oth v etor bundle if and only if it is Morita equiv alen t to the translation group oid of a ompat Lie group ation. Although this result is not elsewhere used in our w ork, w e presen t a pro of of it here b eause w e b eliev e that the same te hnique, applied to represen tations on smo oth elds, ma y b e used to obtain non trivial information ab out the global struture of arbitrary prop er Lie group oids (sine ev ery su h group oid trivially admits globally faithful represen tations on smo oth elds). ∗ ∗ ∗ Chapter I I is mainly onerned with the ba kground notions needed in order to form ulate preisely the reonstrution problem in full generalit y . The for- mal ategorial framew ork within whi h this problem is most on v enien tly stated in the language of tensor ategories and tensor funtors. Setion 6 in tro dues the piv otal notion of a tensor  ate gory: this will b e, for us, an additiv e k -linear ategory C ( k = real or omplex n um b ers) 12 INTR ODUCTION endo w ed with a bilinear bifuntor ( A, B ) 7→ A ⊗ B : C × C → C alled a tensor pr o dut, a distinguished ob jet 1 alled the tensor unit and v arious natural isomorphisms alled A CU  onstr aints whi h, roughly sp eaking, mak e the pro dut ⊗ asso iativ e and omm utativ e with neutral elemen t 1 . The notion of rigid tensor ategory is also briey realled: this is a tensor ategory with the prop ert y that ea h ob jet R admits a dual, that is an ob jet R ′ for whi h there exist morphisms R ′ ⊗ R → 1 and 1 → R ⊗ R ′ ompatible with one another in an ob vious sense; the ategory of nite dimensional v etor spaesor, more generally , smo oth v etor bundles o v er a manifoldis an example. In Setion 7 w e review the notions of a tensor funtor (morphism of tensor ategories) and a tensor pr eserving natur al tr ansformation (morphism of tensor funtors): one obtains a tensor funtor b y atta hing, to an ordinary funtor F , (natural) isomorphisms F ( A ) ⊗ F ( B ) ∼ = F ( A ⊗ B ) and 1 ∼ = F ( 1 ) , alled tensor funtor  onstr aints, ompatible with the A CU onstrain ts of the t w o tensor ategories in v olv ed; a tensor preserving natural transformation of tensor funtors is simply an ordinary natural transformation λ su h that λ ( A ⊗ B ) = λ ( A ) ⊗ λ ( B ) and λ ( 1 ) = id up to the ob vious iden tiations pro vided b y the tensor funtor onstrain ts. If an ob jet R admits a dual R ′ in the ab o v e sense, then λ ( R ) is an isomorphism for an y tensor preserving λ (a tensor preserving funtor will preserv e duals whenev er they exist). A fundamen tal example of tensor funtor is the pull-ba k of smo oth v etor bundles along a smo oth mapping of manifolds. Setion 8 hin ts at the relationship b et w een real and omplex theory: to men tion one example, in the ase of groups one an either onsider linear represen tations on real v etor spaes and then tak e the group of all tensor preserving natural automorphisms of the standard forgetful funtor or, alter- nativ ely , onsider linear represen tations on omplex v etor spaes and then tak e the group of all self-onjugate tensor preserving natural automorphisms; these t w o groups, of ourse, will turn out to b e the same. W e indiate ho w these ommen ts ma y b e generalized to the abstrat ategorial setting w e ha v e just outlined to the reader. Setion 9 is dev oted to a onise exp osition, without an y am bition to ompleteness, of the algebrai geometer's p oin t of view on T annak a dualit y . In fat, man y fundamen tal asp ets of the algebrai theory are omitted here; w e refer more demanding readers to Saave dr a (1972) [ 32 ℄, Deligne and Milne (1982) [12℄ and Deligne (1990) [11 ℄. W e though t it neessary to inlude this exp osition with the in ten t of pro viding adequate grounds for understanding ertain questions reaised in Chapter V. Con trary to the rest of the  hapter, Setion 10 is en tirely based on our o wn w ork. In this setion w e pro v e a k ey te hnial lemma whi h w e exploit later on, in Setion 20, to establish the surjetivit y of the en v elop e homo- morphism π (see ab o v e) for all prop er Lie group oids; this lemma redues the Outline  hapter b y  hapter 13 latter problem to that of  he king that a ertain extendabilit y ondition for morphisms of represen tations is satised. The pro of of our result mak es use of the lassial T annak a dualit y theorem for ompat (Lie) groups, though for the rest it is purely algebrai and it do es not repro due an y kno wn argumen t. ∗ ∗ ∗ In Chapter I I I, w e in tro due our abstrat systematization of represen tation theory . Our ideas to ok shap e gradually , during the attempt to mak e the treat- men t of v arious inequiv alen t approa hes to the represen tation theory of Lie group oids uniform. A ollateral b enet of this abstration eort w as a gain in simpliit y and formal elegane, along with a general b etter understanding of the mathematial features of the theory itself. W e b egin with the desription of a ertain ategorial struture, that w e shall all br e d tensor  ate gory, whi h p ermits to mak e sense of the notion of `Lie group oid ation' in a natural w a y . Smo oth v etor bundles and smo oth elds pro vide examples of su h a struture. A bred tensor ategory C ma y b e dened as a orresp ondene that assigns a tensor ategory C ( X ) to ea h smo oth manifold X and a tensor funtor f ∗ : C ( X ) → C ( Y ) to ea h smo oth mapping f : Y → X , along with a oheren t system of tensor preserving natural isomorphisms ( g ◦ f ) ∗ ∼ = f ∗ ◦ g ∗ and id ∗ ∼ = Id . Most notions needed in represen tation theory an b e dened purely in terms of the bred tensor ategory struture, pro vided this enjo ys some additional prop erties whi h w e no w pro eed to summarize. In Setion 11, w e mak e from the outset the assumption that C is a pr estak, in other w ords that the ob vious presheaf U 7→ Hom C ( U ) ( E | U , F | U ) is a sheaf on X for all ob jets E , F of the ategory C ( X ) . W e also require C to b e smo oth, that is to sa y , roughly sp eaking, that for ea h X there is an isomorphism of omplex algebras End( 1 X ) ≃ C ∞ ( X ) , where 1 X denotes the tensor unit in C ( X ) . Let C ∞ X denote the sheaf of smo oth funtions on X . F or ea h smo oth presta k C one an asso iate to ev ery ob jet E of the ategory C ( X ) a sheaf of C ∞ X -mo dules, Γ E , to b e alled the she af of smo oth se tions of E . The latter op eration yields a funtor of C ( X ) in to the ategory of shea v es of C ∞ X -mo dules. One has a natural transformation Γ E ⊗ C ∞ X Γ E ′ → Γ ( E ⊗ E ′ ) , whi h need not b e an isomorphism, and an isomorphism C ∞ X ≃ Γ ( 1 X ) of C ∞ X -mo dules, that b eha v e m u h as usual tensor funtor onstrain ts do. The ompatibilit y of the op eration E 7→ Γ E with the pullba k along a smo oth map f : Y → X is measured b y a anonial natural morphism of shea v es of C ∞ Y -mo dules f ∗ ( Γ E ) → Γ ( f ∗ E ) . F or ea h p oin t x of X , there is a funtor whi h assigns, to ev ery ob jet E of the ategory C ( X ) , a omplex v etor spae E x to b e referred to as the br e of E at x ; a lo al smo oth setion ζ ∈ Γ E ( U ) , dened o v er an op en neigh b ourho o d U of x , will determine a v etor ζ ( x ) ∈ E x to b e referred to as the value of ζ at x . 14 INTR ODUCTION In order to sho w that Morita equiv alenes ha v e the usual prop ert y of induing a ategorial equiv alene b et w een the ategories of represen tations, w e further need to imp ose the ondition that C is a stak. This ondition, examined in Setion 12 , means that when one is giv en an op en o v er { U i } of a (paraompat) manifold M , along with a family of ob jets E i ∈ Ob C ( U i ) and a o yle of isomorphisms θ ij : E i | U i ∩ U j ∼ → E j | U i ∩ U j , there m ust b e some ob jet E in C ( M ) whi h admits a family of isomorphisms E | U i ∼ → E i ∈ C ( U i ) ompatible with { θ ij } . Naiv ely sp eaking, one an glue ob jets in C together. When C is a smo oth sta k, the ategory C ( M ) will essen tially on tain the ategory of all smo oth v etor bundles o v er M as a full sub ategory . In Setion 13, w e la y do wn the foundations of the represen tation theory of Lie group oids relativ e to a typ e T , for an arbitrary smo oth sta k of tensor ategories T . A r epr esentation of typ e T of a Lie group oid G is a pair ( E ,  ) onsisting of an ob jet E of the ategory T ( M ) (where M is the base of G ) and an arro w  : s ∗ E → t ∗ E in the ategory T ( G ) (where s , t : G → M are the soure resp. target map of G ) su h that u ∗  = id E (where u : M → G denotes the unit setion) and m ∗  = p 1 ∗  ◦ p 2 ∗  (where m , p 1 , p 2 : G s × t G → G resp etiv ely denote m ultipliation, rst and seond pro jetion). With the ob vious notion of morphism, represen tations of t yp e T of a Lie group oid G form a ategory R T ( G ) . This ategory inherits an additiv e linear tensor struture from the base ategory T ( M ) , making the forgetful funtor ( E ,  ) 7→ E a strit linear tensor funtor of R T ( G ) in to T ( M ) . The latter funtor will b e denoted b y ω T ( G ) and will b e alled the standar d br e funtor of typ e T asso iated with G . Ea h homomorphism of Lie group oids φ : G → H indues a linear tensor funtor φ ∗ : R T ( H ) → R T ( G ) that w e all the pul lb ak along φ . One has tensor preserving natural isomorphisms ( ψ ◦ φ ) ∗ ∼ = φ ∗ ◦ ψ ∗ . In Setion 14 w e sho w that for ev ery Morita equiv alene φ : G → H the pullba k funtor φ ∗ is an equiv alene of tensor ategories. ∗ ∗ ∗ Chapter IV is the ore of our dissertation. This is the plae where w e desrib e the general dualit y theory for Lie group oids in the abstrat framew ork of Chapters I II I I and where w e pro v e our most imp ortan t results, ulminating in the ab o v e-men tioned reonstrution theorem for prop er Lie group oids. Setion 15 on tains a detailed desription of in what t yp e of Lie group oid represen tations one should b e in terested, from our p oin t of view, when dealing with dualit y theory of Lie group oids. Namely , w e sa y that a t yp e T is a stak of smo oth elds if it meets a n um b er of extra requiremen ts, alled `axioms', whi h w e no w pro eed to summarize. Our rst axiom sa ys that the anonial morphisms Γ E ⊗ C ∞ X Γ E ′ → Γ ( E ⊗ E ′ ) and f ∗ ( Γ E ) → Γ ( f ∗ E ) (fr. the summary of Ch. I I I, 11 ) are sur- jetiv e; this axiom on v eys information ab out the smo oth setions of E ⊗ E ′ Outline  hapter b y  hapter 15 and f ∗ E and it implies that the bre at x of an ob jet E is spanned, as a v etor spae, b y the v alues ζ ( x ) as ζ ranges o v er all germs of lo al smo oth setions of E at x . Next, reall that an y arro w a : E → E ′ in T ( X ) indues a morphism of shea v es of C ∞ X -mo dules Γ a : Γ E → Γ E ′ and a bundle of linear maps { a x : E x → E ′ x } ; these are m utually ompatible, in an ob vious sense. Our seond and third axioms ompletely  haraterize the arro ws in T ( X ) in terms of their eet on smo oth setions and the bundles of linear maps they indue; namely , an arro w a : E → E ′ v anishes if and only if a x v anishes for all x , and ev ery pair formed b y a morphism of C ∞ X -mo dules α : Γ E → Γ E ′ and a ompatible bundle of linear maps { λ x : E x → E ′ x } giv es rise to a (unique) arro w a : E → E ′ su h that α = Γ a or, equiv alen tly , λ x = a x for all x . Then there is an axiom requiring the existene of lo al Hermitian metris on the ob jets of T ( X ) . A Hermitian metri on E is an arro w E ⊗ E ∗ → 1 induing a p ositiv e denite Hermitian sesquilinear form on ea h bre E x ; the axiom sa ys that for an y paraompat M , ea h ob jet of T ( M ) admits Hermit- ian metris. This assumption has man y useful onsequenes: for example, it implies v arious on tin uit y priniples for smo oth setions and a fundamen tal extension prop ert y for arro ws. The remaining t w o axioms imp ose v arious niteness onditions on T : roughly sp eaking, nite dimensionalit y of the bres of an arbitrary ob jet E and lo al niteness of the sheaf of mo dules Γ E . More preisely , one axiom anonially iden ties T ( ⋆ ) , as a tensor ategory , with the ategory of nite dimensional v etor spaeswhere ⋆ denotes the one-p oin t manifoldso that, for instane, the funtor E 7→ E x b eomes a tensor funtor of T ( X ) in to the ategory of su h spaes; the other axiom requires the existene, for ea h p oin t x , of an op en neigh b ourho o d U su h that Γ E ( U ) is spanned, as a C ∞ ( U ) -mo dule, b y a nite set of setions of E o v er U . In Setion 16 , w e in tro due our fundamen tal example of a sta k of smo oth elds (whi h is to pla y a role in our reonstrution theorem for prop er Lie group oids in 20), to whi h w e refer as the t yp e E ∞ of smo oth Eulide an elds. The notion of smo oth Eulidean eld o v er a manifold X generalizes that of smo oth v etor bundle o v er X in that the dimension of the bres is allo w ed to v ary dison tin uously o v er X or, in other w ords, the sheaf of smo oth setions is no longer a lo ally free C ∞ X -mo dule. Our theory of smo oth Eulidean elds ma y b e regarded as the oun terpart, in the smo oth setting, of the w ell-established theory of on tin uous Hilb ert elds [14℄. In Setion 17 w e pro v e v arious results ab out the equiv arian t extension of morphisms of Lie group oid represen tations whose t yp e is a sta k of smo oth elds; in om bination with the te hnial lemma of 10, these extension re- sults allo w one to establish the surjetivit y of the en v elop e homomorphism π asso iated with represen tations on an arbitrary sta k of smo oth elds. The pro ofs are based on the usual a v eraging te hniquewhi h mak es sense for 16 INTR ODUCTION an y prop er Lie group oid b eause of the existene of normalized Haar system- sand, of ourse, on the axioms for sta ks of smo oth elds. In Setions 18 19, w e delv e in to the formalism of bre funtors with v al- ues in an arbitrary sta k of smo oth elds. A br e funtor, with v alues in a sta k of smo oth elds F , is a faithful linear tensor funtor ω of some addi- tiv e tensor ategory C in to F ( M ) , for some xed paraompat manifold M to b e referred to as the b ase of ω . This notion is obtained b y abstrating the fundamen tal features, whi h allo w one to mak e sense of the onstru- tion of the T annakian group oid, from the onrete example pro vided b y the standard forgetful funtor asso iated with the represen tations of t yp e F of a Lie group oid o v er M . T o an y bre funtor ω with base M , one an assign a group oid T ( ω ) o v er M to whi h w e refer as the T annakian gr oup oid as- so iated with ω onstruted, lik e in the ase of groups, b y taking all tensor preserving natural automorphisms of ω . The set of arro ws of T ( ω ) omes naturally equipp ed with a top ology and a smo oth funtional strutur e that is a sheaf R ∞ of algebras of on tin uous real v alued funtions on T ( ω ) losed under omp osition with arbitrary smo oth funtions R d → R ; the notion of smo oth funtional struture is analogous to that of C ∞ -ring, fr [28, 29℄. In Setion 20, w e reap the fruits of all our previous w ork and pro v e sev- eral statemen ts of fundamen tal imp ortane ab out the T annakian group oid T ( G ) asso iated with the standard forgetful funtor ω ( G ) on the ategory of represen tations of an arbitrary prop er Lie group oid G . (W e are still dealing with a situation where the t yp e is an arbitrary sta k of smo oth elds.) Reall that there is a anonial homomorphism π : G → T ( G ) dened b y setting π ( g )( E ,  ) =  ( g ) , whi h, as previously men tioned, turns out to b e surjetiv e for prop er G ; the pro of of this theorem is based on the results of Setions 10 and 17. Moreo v er, when G is prop er, the T annakian group oid T ( G ) b eomes a top ologial group oid and π a homomorphism of top ologial group oids: then w e sho w that injetivit y of π implies that π is an isomorphism of top ologial group oids and that this in turn implies that the ab o v e-men tioned funtional struture on T ( G ) is atually a Lie group oid struture for whi h π b eomes an isomorphism of Lie group oids. A ordingly , w e sa y that a Lie group oid G is r eexive relativ e to a ertain t yp eif π indues a homeomorphism b et w een the spaes of arro ws of G and T ( G ) . Our main theorem, whi h onludes the setion, states that ev ery prop er Lie group oid is reexiv e relativ e to the t yp e E ∞ of smo oth Eulidean elds. The injetivit y of π for this partiular t yp e of represen tations is an easy onsequene of Zung's lo al linearizabilt y result for prop er Lie group oids. ∗ ∗ ∗ Besides establishing a T annak a dualit y theory for prop er Lie group oids, the w ork desrib ed ab o v e also leads to results onerning the lassial theory of represen tations of Lie group oids on v etor bundles. Chapter V onen trates Outline  hapter b y  hapter 17 on what an b e said ab out the latter ase exlusiv ely from the abstrat stand- p oin t of the theory of bre funtors outlined in 18 19 . The main ob jets of study here are ertain bre funtors, whi h will b e referred to as lassi-  al br e funtors, enjo ying formal prop erties analogous to those p ossessed b y the standard forgetful funtor asso iated with the ategory of lassial represen tations of a Lie group oid. The distintiv e features of lassial bre funtors are the rigidit y of the do- main tensor ategory C and the t yp e b eing equal to the sta k of smo oth v etor bundles. Setion 21 ollets some general remarks ab out su h bre funtors and some basi denitions. F or an y lassial bre funtor ω , the T annakian group oid T ( ω ) pro v es to b e a C ∞ -strutured group oid o v er the base M of ω ; this means that all struture maps of T ( ω ) are morphisms of funtionally strutured spaes with resp et to the C ∞ -funtional struture R ∞ on T ( ω ) in tro dued in  18. One an dene, for ev ery C ∞ -strutured group oid T , an ob vious notion of C ∞ -represen tation on a smo oth v etor bundle; su h rep- resen tations form a tensor ategory R ∞ ( T ) . Ev ery ob jet R of the domain ategory C of a lassial bre funtor ω determines a C ∞ -represen tation ev R , whi h w e all evaluation at R , of the T annakian group oid T ( ω ) on the v etor bundle ω ( R ) . The op eration R 7→ ev R pro vides a tensor funtor of C in to the ategory of C ∞ -represen tations of T ( ω ) , the evaluation funtor asso iated with ω . Setion 22 is preliminary to Setion 23 . It is dev oted to a disussion of the te hnial notion of a tame submanifold whi h w e in tro due in order to dene represen tativ e  harts in the subsequen t setion. All the reader needs to kno w ab out tame submanifolds is that these are partiular submanifolds of Lie group oids with the prop ert y that whenev er a Lie group oid homomorph- ism establishes a bijetiv e orresp ondene b et w een t w o of them, the indued bijetion is atually a dieomorphism and that Morita equiv alenes preserv e tame submanifolds. The fat that T ( ω ) is a C ∞ -strutured group oid for ev ery lassial ω p oses the question of whether T ( ω ) is atually a Lie group oid. In Setion 23 w e start ta kling this issue b y pro viding a neessary and suien t riterion, whi h pro v es to b e on v enien t enough to use in pratie, for the answ er to the latter question b eing p ositiv e for a giv en ω . This riterion is expressed in terms of the notion of a r epr esentative hart, that is a pair (Ω , R ) onsisting of an op en subset Ω of T ( ω ) and an ob jet R of the domain ategory C of ω su h that the ev aluation represen tation at R indues a homeomorphism b et w een Ω and a tame submanifold of the linear group oid GL ( ω R ) ; then T ( ω ) is a Lie group oid if, and only if, represen tativ e  harts o v er T ( ω ) and (Ω , R ⊕ S ) is a represen tativ e  hart for ev ery represen tativ e  hart (Ω , R ) and for ev ery ob jet S of C . Setion 24 in tro dues a notion of morphism for (lassial) bre funtors. Roughly sp eaking, a morphism of ω in to ω ′ , o v er a smo oth mapping f : 18 INTR ODUCTION M → M ′ of the base manifolds, is a tensor funtor of C ′ in to C ompatible with the pullba k of v etor bundles along f ; ev ery morphism ω → ω ′ o v er f indues a homomorphism of C ∞ -strutured group oids T ( ω ) → T ( ω ′ ) o v er f . Setion 25 is dev oted to the study of we ak e quivalen es of (lassial) bre funtors: w e dene them as those morphisms o v er a surjetiv e submersion whi h ha v e the prop ert y of b eing a ategorial equiv alene. As an appliation of the riterion of 23, w e sho w that if ω is w eakly equiv alen t to ω ′ , then T ( ω ) is a Lie group oid if and only if T ( ω ′ ) is; when this is the ase, the Lie group oids T ( ω ) and T ( ω ′ ) turn out to b e Morita equiv alen t. ∗ ∗ ∗ In Chapter VI, w e apply the general abstrat theory of the preeding  hapter to the motiv ating example pro vided b y the standard forgetful funtor on the ategory of lassial represen tations of a prop er Lie group oid G . The T annak- ian group oid asso iated with the latter lassial bre funtor will b e denoted b y T ∞ ( G ) ; in fat, this onstrution an b e extended to a funtor - 7→ T ∞ ( - ) of the ategory of Lie group oids in to the ategory of C ∞ -strutured group oids so that the en v elop e homomorphism π ( - ) b eomes a natural transformation ( - ) → T ∞ ( - ) . W e will fo us our atten tion on the follo wing t w o problems: in the rst plae, w e w an t to understand whether the T annakian group oid T ∞ ( G ) is a Lie group oid, let us sa y for G prop er; seondly , w e are in ter- ested in examples of lassi al ly r eexive Lie group oids, that is to sa y Lie group oids G for whi h the en v elop e homomorphism π is an isomorphism of top ologial group oids b et w een G and T ∞ ( G ) (reall that, under the assump- tion of prop erness, it is suien t that π is injetiv e). In Setion 26, w e ollet what w e kno w ab out the rst of the t w o ab o v e- men tioned problems in the general ase of an arbitrary prop er Lie group oid. Namely , w e sho w that the ondition, in the riterion for smo othness of 23 , that (Ω , R ⊕ S ) should b e a represen tativ e  hart for ev ery represen tativ e  hart (Ω , R ) and ob jet S , is alw a ys satised b y the standard forgetful funtor on the ategory of lassial represen tations of a prop er Lie group oid G so that T ∞ ( G ) is a (prop er) Lie group oid if and only if one an nd enough represen tativ e  harts; if this is the ase, then the en v elop e map π is a full submersion of Lie group oids whose asso iated pullba k funtor π ∗ establishes an isomorphism of the orresp onding ategories of lassial represen tations in v erse to the ev aluation funtor of 21. Setion 27 proseutes the study initiated in the previous setion b y pro- viding a pro of of the fat that T ∞ ( G ) is a Lie group oid for ev ery prop er regular group oid G . W e onjeture that the same statemen t holds true for ev ery prop er G , that is ev en without the regularit y assumption. Setion 28 on tains a list of examples of lassially reexiv e (prop er) Lie group oids; sine, as 2 exemplies, most Lie group oids fail to b e lassially Some p ossible appliations 19 reexiv e, this list annot b e v ery long. T o b egin with, translation group oids asso iated with ompat Lie group ations are eviden tly lassially reexiv e. Next, w e observ e that an y étale Lie group oid whose soure map is prop er is neessarily lassially reexiv e b eause, for su h group oids, one an mak e sense of the regular represen tation. Finally , orbifold group oidsb y whi h w e mean prop er eetiv e group oidsare lassially reexiv e b eause the stan- dard ation on the tangen t bundle of the base manifold yields a globally faithful lassial represen tation. Some p ossible appliations The study of lassial bre funtors in Chapter V w as originally motiv ated b y the example treated in Chapter VI, namely the standard forgetful funtor asso iated with the ategory of lassial represen tations of a Lie group oid. Ho w ev er, examples of lassial bre funtors an also b e found b y lo oking in to dieren t diretions. T o b egin with, one ould onsider represen tations of Lie algebr oids [27, 10, 16℄. Reall that a represen tation of a Lie algebroid g o v er a manifold M is a pair ( E , ∇ ) onsisting of a v etor bundle E o v er M and a at g -onnetion ∇ on E , that is, a bilinear map Γ( g ) × Γ( E ) → Γ( E ) (global setions), C ∞ ( M ) -linear in the rst argumen t, Leibnitz in the seond and with v anish- ing urv ature. Su h represen tations naturally form a tensor ategory . Another example of the same sort is pro vided b y the singular foliations in tro dued b y I. Androulidakis and G. Sk andalis [1℄. Here one is giv en a lo ally nite sheaf F of mo dules of v etor elds o v er a manifold M , losed under the Lie bra k et; this is to b e though t of as induing a `singular' foliation of M , in that F is no longer neessarily lo ally free and so the dimension of the lea v es ma y jump. Again, one an onsider pairs ( E , ∇ ) formed b y a v etor bundle E o v er M and a morphism of shea v es ∇ : F ⊗ Γ E → Γ E enjo ying formal prop erties analogous to those dening a at onnetion. In his pap er ab out the lo al linearizabilit y of prop er Lie group oids [38℄, N.T. Zung p oses the question of whether a spae, whi h is lo ally isomorphi to the orbit spae of a ompat Lie group ation, is neessarily the orbit spae M / G asso iated with a prop er Lie group oid G o v er a manifold M . Of ourse, this question is not stated v ery preisely; its rigorous form ulation, as far as w e an see, should b e giv en in the follo wing terms. Let us all a C ∞ -strutured spae ( X , F ∞ ) a gener alize d orbifold if the spae X is Hausdor, paraom- pat and lo ally isomorphi, as a funtionally strutured spae, to an orbit spae asso iated with some linear ompat Lie group ationin other w ords, lo ally isomorphi to a spae of the form ( V /G, C ∞ V /G ) for some represen tation G → GL ( V ) of a ompat Lie group G on a nite dimensional v etor spae V . The theory of funtionally strutured spaes suggests the righ t notion of smo oth map of generalized orbifolds and hene the righ t notion of isomorph- 20 INTR ODUCTION ism. Zung's theorem implies that the orbit spae ( M / G , C ∞ M / G ) of a prop er Lie group oid G o v er a manifold M is a generalized orbifold: then the question is whether an arbitrary generalized orbifold is atually of this preise form. Classial bre funtors mak e their natural app earane in onnetion with an y giv en generalized orbifold X . (Con v en tionally , w e will refer to the C ∞ -struture of X , when neessary , b y means of the notation C ∞ X .) Let V ∞ ( X ) denote the ategory of lo ally free shea v es of C ∞ X -mo dules (of lo- ally nite rank), endo w ed with the standard linear tensor struture; one ma y refer to the ob jets of this ategory as ve tor bund les over X . Cho ose a lo ally nite o v er { U i } of X b y op en subsets U i su h that for ea h i there is an isomorphism V i /G i ≈ U i ; w e regard the maps φ i : V i → U i as xed one and for all, and w e assume, for simpliit y , that the V i all ha v e the same dimension. Letting M b e the disjoin t union ` V i , one has an ob vious lassial bre funtor ω X M = ω X { V i ,φ i } o v er M sending ea h ob jet E of the ategory V ∞ ( X ) to the smo oth v etor bundle ⊕ i φ i ∗ E o v er M . The T annakian group oid T ∞ ( X ) = T ( ω X M ) is a C ∞ -strutured group oid with the prop ert y that the ob vious map φ : M → X indues an isomorph- ism of funtionally strutured spaes b et w een M / T ∞ ( X ) and X ; th us, the study of this group oid migh t b e relev an t to the ab o v e-men tioned problem. Similarly , the study of the T annakian group oids asso iated with the other examples migh t lead to in teresting information ab out the underlying geomet- rial ob jets, at least when the situation in v olv es some kind of prop erness. In this onnetion, it is natural to hop e for a general result relating the domain ategory of a lassial bre funtor with the ategory of C ∞ -represen tations of the orresp onding T annakian group oid, for example via the standard ev al- uation funtor desrib ed in 21. A w ell-kno wn onjeture, whi h has b een raising some in terest reen tly [17 , 19 ℄, states that ev ery prop er étale Lie group oid is Morita equiv alen t to the translation group oid asso iated with some ompat Lie group ation or, equiv alen tly , that ev ery su h group oid admits a globally faithful lassial represen tation (fr. Ch. I, 5). This onjeture is related to the question of whether prop er étale Lie group oids are lassially reexiv e (w e ha v e already observ ed that the answ er is armativ e in the eetiv e ase, see Ch. VI , 28). It is kno wn that for ea h group oid G of this kind, there exist a prop er ee- tiv e Lie group oid ˜ G and a submersiv e epimorphism G → ˜ G ; the k ernel of this homomorphism is neessarily a bundle of nite groups B em b edded in to G , hene, one gets an exat sequene of Lie group oids 1 → B ֒ → G → ˜ G → 1 where B and ˜ G are b oth lassially reexiv e. These onsiderations strongly suggest that one should in v estigate ho w the prop ert y of reexivit y b eha v es with resp et to Lie group oid extensions. Chapter I Lie Group oids and their Classial Represen tations The presen t  hapter is essen tially in tro dutory: w e regard all the material thereof as w ell-kno wn. Our purp ose is, rst of all, to x some notational on v en tions and some standard terminology onerning Lie group oids; this is done in 1. Next, in  2, w e pro vide a detailed disussion of a onrete example whi h is to serv e as motiv ation for the approa h w e will adopt in Chapters I I IIV. In 34 w e treat the t w o fundamen tal pillars on to whi h our main result holds: Haar systems and Zung's linearizabilit y theorem; w e deided to inlude a presen tation of these topis here b eause w e found it diult to pro vide adequate referenes for them. The  hapter ends with a digression on the problem of represen ting a prop er Lie group oid as a global quotien t arising from a smo oth ompat Lie group ation. 1 Generalities ab out Lie Group oids The term group oid refers to a small ategory where ev ery arro w is in v ertible. A Lie gr oup oid an b e appro ximately desrib ed as an in ternal group oid in the ategory of smo oth manifolds. T o onstrut a Lie group oid G one has to giv e a pair of manifolds of lass C ∞ G (0) and G (1) , resp etiv ely alled manifold of obje ts and manifold of arr ows, and a list of smo oth maps alled strutur e maps. The basi items in this list are the sour  e map s : G (1) → G (0) and the tar get map t : G (1) → G (0) ; these ha v e to meet the requiremen t that the bred pro dut G (2) = G (1) s × t G (1) exists in the ategory of C ∞ -manifolds. Then one has to giv e a  omp osition map c : G (2) → G (1) , a unit map u : G (0) → G (1) and an inverse map i : G (1) → G (1) , for whi h the familiar algebrai la ws m ust b e satised. T erminology and Notation: The p oin ts x = s ( g ) and x ′ = t ( g ) are resp. alled the sour  e and the tar get of the arr ow g . W e let G ( x, x ′ ) denote the set of all the arro ws whose soure is x and whose target is x ′ ; w e shall use 21 22 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS the abbreviation G | x for the isotr opy or vertex group G ( x, x ) . Notationally , w e will often iden tify a p oin t x ∈ G (0) and the orresp onding unit arro w u ( x ) ∈ G (1) . It is ostumary to write g ′ · g or g ′ g for the omp osition c ( g ′ , g ) and g − 1 for the in v erse i ( g ) . Our desription of the notion of Lie group oid is still inomplete. It turns out that a ouple of additional requiremen ts are needed in order to get a reasonable denition. Reall that a manifold M is said to b e p ar a omp at if it is Hausdor and there exists an asending sequene of op en subsets with ompat losure · · · ⊂ U i ⊂ U i ⊂ U i +1 ⊂ · · · su h that M = ∞ ∪ i =0 U i . A Hausdor manifold is paraompat if and only if it p ossesses a oun table basis of op en subsets. An y op en o v er of a paraompat manifold admits a lo ally nite renemen t. An y paraompat manifold admits partitions of unit y of lass C ∞ (sub ordinated to an op en o v er; f. for instane Lang [22 ℄). In order to mak e the bred pro dut G (1) s × t G (1) meaningful as a manifold and for other purp oses related to our studies, w e shall inlude the follo wing additional onditions in the denition of Lie group oid: 1. The soure map s : G (1) → G (0) is a submersion with Hausdor bres; 2. The manifold G (0) is paraompat. Note that w e do not require that the manifold of arro ws G (1) is Hausdor or paraompat; atually , this manifold is neither Hausdor nor seond oun t- able in man y examples of in terest. The denition here diers from that in Mo erdijk and Mr£un [27℄ in that w e additionally require that the manifold G (0) is paraompat. The rst ondition implies at one that the domain of the omp osition map is a submanifold of the Cartesian pro dut G (1) × G (1) and that the target map is a submersion with Hausdor bres; th us, the soure bres G ( x, - ) = s − 1 ( x ) and the target bres G ( - , x ′ ) = t − 1 ( x ′ ) are losed Hausdor submanifolds of G (1) . A Lie group oid G is said to b e Hausdor if the manifold G (1) is Hausdor. Some more T erminology: The manifold G (0) is usually alled the b ase of the group oid G ; one also sa ys that G is a group oid o v er the manifold G (0) . W e shall often use the notation G x = G ( x, - ) = s − 1 ( x ) for the bre of the soure map o v er a p oin t x ∈ G (0) . More generally , w e shall write (1) G ( S, S ′ ) =  g ∈ G (1) : s ( g ) ∈ S & t ( g ) ∈ S ′  , G | S = G ( S, S ) and G S = G ( S, - ) = G ( S, G (0) ) = s − 1 ( S ) for all subsets S, S ′ ⊂ G (0) . A homomorphism of Lie gr oup oids is a smo oth funtor. More preisely , a homomorphism ϕ : G → H onsists of t w o smo oth maps ϕ (0) : G (0) → H (0) and ϕ (1) : G (1) → H (1) , ompatible with the group oid struture in the sense that s ◦ ϕ (1) = ϕ (0) ◦ s , t ◦ ϕ (1) = ϕ (0) ◦ t and ϕ (1) ( g ′ · g ) = ϕ (1) ( g ′ ) · ϕ (1) ( g ) . Lie group oids and their homomorphisms form a ategory . 1. GENERALITIES ABOUT LIE GR OUPOIDS 23 There is also a notion of top olo gi al gr oup oid: this is just an in ternal group oid in the ategory of top ologial spaes and on tin uous mappings. In the on tin uous ase the denition is m u h simpler and one need not w orry ab out the domain of denition of the omp osition map. With the ob vious notion of homomorphism, top ologial group oids onstitute a ategory . 2 Example Ev ery smo oth manifold M an b e regarded as a Lie group oid b y taking M itself as the manifold of arro ws and the iden tit y map id : M → M as the unit setion. Alternativ ely , one an form the p air gr oup oid over M ; this is the Lie group oid whose manifold of arro ws is M × M and whose soure and target map are the t w o pro jetions. 3 Example An y Lie group G an b e regarded as a Lie group oid o v er the one-p oin t manifold b y taking G itself as the manifold of arro ws. 4 Example: line ar gr oup oids If E is a real or omplex smo oth v etor bundle (of lo ally nite rank) o v er a manifold M , one an form the line ar gr oup oid GL ( E ) asso iate d with E . This is dened as the group oid o v er M whose arro ws x → x ′ are the linear isomorphisms E x ∼ → E x ′ b et w een the bres of E o v er the p oin ts x and x ′ . There is an ob vious smo oth struture turning GL ( E ) in to a Lie group oid. 5 Example: ation gr oup oids Let G b e a Lie group ating smo othly (from the left) on a manifold M . Then one an dene the ation (or tr anslation ) gr oup oid G ⋉ M as the Lie group oid o v er M whose manifold of arro ws is the Cartesian pro dut G × M , whose soure and target map are resp etiv ely the pro jetion on to the seond fator ( g , x ) 7→ x and the ation ( g , x ) 7→ g x and whose omp osition la w is the op eration (6) ( g ′ , x ′ )( g , x ) = ( g ′ g , x ) . There is a similar onstrution M ⋊ G asso iated with righ t ations. Let G b e a Lie group oid and let x b e a p oin t of its base manifold G (0) . The orbit of G (or G -orbit ) thr ough x is the subset (7) G x def = G · x def = t  G x  = { x ′ ∈ G (0) |∃ g : x → x ′ } . Note that the isotrop y group G | x ats from the the righ t on the manifold G x ; this ation is learly free and transitiv e along the bres of the restrition of the target map t to G x . The follo wing result holds (see [ 27 ℄ p. 115): 8 Theorem Let G b e a Lie group oid and let x, x ′ ∈ G (0) . Then 1. G ( x, x ′ ) is a losed submanifold of G (1) ; 2. G | x is a Lie group; 3. the G -orbit through x is an immersed submanifold of G (0) ; 24 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS 4. the target map t : G x → G x pro v es to b e a prinipal G | x -bundle. It is w orth while sp ending a ouple of w ords ab out the manifold struture that is asserted to exist on the G -orbit through x . The set G x an ob viously b e iden tied with the homogeneous spae G x / ( G | x ) . No w, it an b e pro v ed that there exists a (p ossibly non-Hausdor ) manifold struture on this quotien t spae, su h that the quotien t map turns out to b e a prinipal bundle. W e sa y that a Lie (or top ologial) group oid G is pr op er if G is Hausdor and the om bined souretarget map ( s , t ) : G (1) → G (0) × G (0) is prop er (in the familiar sense: the in v erse image of a ompat subset is ompat). The manifold of arro ws G (1) of a prop er Lie group oid G is alw a ys para- ompat. Indeed, b y the denition of Lie group oid, the base M of G is a paraompat manifold and therefore there exists an in v ading sequene · · · ⊂ U i ⊂ U i ⊂ U i +1 ⊂ · · · ⊂ M of pre-ompat op en subsets; the in- v erse images Γ i = G | U i = ( s , t ) − 1 ( U i × U i ) form an analogous sequene inside the (Hausdor ) manifold G (1) . Let x 0 b e a p oin t of M . W e kno w the orbit S = G x 0 is an immersed submanifold of M (preisely , there exists a unique manifold struture on S su h that t : G x 0 → S is a prinipal righ t G | x 0 -bundle and the inlusion S ֒ → M an immersion). No w, it follo ws from the prop erness of G that S is atually a submanifold of M . T o see this, x a p oin t s 0 ∈ S . Sine there exists a lo al equiv arian t  hart G ( x 0 , W ) ≈ W × G | x 0 where W is b oth an op en neigh b orho o d of s 0 in S and a submanifold of M , it will b e enough to pro v e the existene of an op en ball B ⊂ M at s 0 su h that S ∩ B ⊂ W . T o do this, tak e a sequene of op en balls B i shrinking to s 0 : the dereasing sequene Σ i = G ( x 0 , B i ) − G ( x 0 , W ) of losed subsets of the manifold G ( x 0 , - ) is on tained in the ompat subset G ( x 0 , B 1 ) and therefore, sine T Σ i = ∅ , there exists some i su h that G ( x 0 , B i ) ⊂ G ( x 0 , W ) . 2 Classial Represen tations In this setion w e in tro due the ostumary notion of represen tation of a Lie group oid on a smo oth v etor bundle and w e explain, b y means of a oun terexample, wh y this notion is inadequate for the purp ose of building a p ossible T annak a dualit y theory for prop er Lie group oids. Let G b e a Lie group oid and let M b e its base. W e let R ∞ ( G ; C ) denote the ategory of all C -linear lassi al r epr esentations of G . The ob jets of this ategory are the pairs ( E ,  ) onsisting of a smo oth omplex v etor bundle E (of lo ally nite rank) o v er M and a Lie group oid homomorphism G ( s , t )    / / GL ( E ) ( s , t )   M × M id × id / / M × M ; (1) 2. CLASSICAL REPRESENT A TIONS 25 the arro ws, let us sa y those a : ( E ,  ) → ( F , ς ) , are the morphisms of v etor bundles a : E → F su h that the square E x a x    ( g ) / / E x ′ a x ′   F x ς ( g ) / / F x ′ (2) omm utes for all x, x ′ ∈ M and g ∈ G ( x, x ′ ) . There is an en tirely analogous notion of R -linear lassial represen tation of G , where real v etor bundles are used instead of omplex ones. One obtains a orresp onding ategory R ∞ ( G ; R ) . Insofar as a partiular  hoie of o eien ts is not relev an t to the sub jet matter of a disussion, w e shall write simply R ∞ ( G ) and suppress an y further referene to o eien ts. Lie group oids annot alw a ys b e distinguished from one another just on the basis of kno wledge of the resp etiv e ategories of lassial represen tations; this onsideration motiv ates our approa h to T annak a dualit y as desrib ed in Chapter IV. W e are going to substan tiate our assertion b y means of a oun terexample whi h w e diso v ered indep enden tly in 2005: only reen tly A. Henriques p oin ted out to us that the same oun terexample w as already kno wn in the on text of orbispae theory , see Lü k and Oliv er (2001) [ 23℄. Reall that a Lie bund le (also kno wn as bund le of Lie gr oups ) is a Lie group oid whose soure and target map oinide. Fix a Lie group H and  ho ose an automorphism χ ∈ Aut( H ) . There is a general pro edureompletely analogous to the onstrution of Möbius bands, Klein b ottles et similia b y means of whi h one an obtain a lo ally trivial Lie bundle G = G H ; χ → S 1 with bre H o v er the unit irle. Put G (1) = ( R × H ) / ∼ where ∼ is the equiv alene relation (3) ( t, h ) ∼ ( t ′ , h ′ ) ⇔ t ′ − t = ℓ ∈ Z and h ′ = χ ℓ ( h ) . The bundle bration G (1) → S 1 (= soure map of G = target map of G ) is dened as the unique map that mak es the square R × H / / quot. pro j.   R t 7→ e 2 πit   G (1) / / _ _ _ _ S 1 (4) omm ute. In terms of represen tativ es of elemen ts of G (1) , the omp osition la w c : G (1) × S 1 G (1) → G (1) an b e dened b y setting (5) [ t ′ , h ′ ] · [ t, h ] = [ t ′ , h ′ · χ k ( h )] , where k = t ′ − t ∈ Z and the square bra k et notation indiates that w e are taking equiv alene lasses. This op eration turns G → S 1 in to a bundle of groups o v er the irle, with bre H . 26 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS Consider the op en o v er of S 1 determined b y the lo al exp onen tial parametrizations (0 , 1) ∼ → U and ( − 1 2 , 1 2 ) ∼ → V . One has t w o orresp ond- ing m utually ompatible trivializing  harts for G (1) o v er S 1 , namely (6) τ U : G (1) | U ∼ → U × H and τ V : G (1) | V ∼ → V × H : the former sends g ∈ G (1) | U to the pair ( e 2 π it , h ) with [ t, h ] = g and 0 < t < 1 , the latter sends g ∈ G (1) | V to the pair ( e 2 π it , h ) with [ t, h ] = g and − 1 2 < t < 1 2 . These  harts determine the dieren tiable struture. Notie, b y the w a y , that the transition map b et w een them, namely (7) τ U ◦ τ V − 1 : ( U ∩ V ) × H ∼ → ( U ∩ V ) × H , is giv en b y the iden tit y o v er W × H and b y ( w ′ , h ) 7→ ( w ′ , χ ( h )) o v er W ′ × H , if one lets (0 , 1 2 ) ∼ → W and ( 1 2 , 1) ∼ → W ′ denote the t w o onneted omp onen ts of the in tersetion U ∩ V . W e start b y studying the omplex lassial represen tations of the Lie bundle G H ; χ , whi h are te hnially easier to handle. The analogous result for real represen tations will b e dedued as a orollary . Fix a lassial represen tation ( E ,  ) ∈ Ob R ∞ ( G ; C ) on a smo oth omplex v etor bundle E of rank ℓ o v er S 1 . Sine U and V are on tratible op en subsets of S 1 , the v etor bundle E will b e trivial o v er ea h of them i.e. there will exist smo oth v etor bundle isomorphisms (8) E | U ∼ → U × C ℓ and E | V ∼ → V × C ℓ . These will form a trivializing atlas for E o v er S 1 , whose unique transition mapping will b e giv en b y , let us sa y , (9) Q : W → GL ( ℓ ; C ) and Q ′ : W ′ → GL ( ℓ ; C ) . A ordingly , the Lie bundle GL ( E ) o v er S 1 (that is, b y abuse of notation, the restrition of the linear group oid GL ( E ) to the diagonal S 1 ֒ → S 1 × S 1 ) will b e desrib ed b y trivializing  harts of the follo wing form (10) GL ( E ) | U ∼ → U × GL ( ℓ ; C ) and GL ( E ) | V ∼ → V × GL ( ℓ ; C ) , whose transition map ( U ∩ V ) × GL ( ℓ ; C ) ∼ → ( U ∩ V ) × GL ( ℓ ; C ) will send w ∈ W to A 7→ Q ( w ) AQ ( w ) − 1 and w ′ ∈ W ′ to A 7→ Q ′ ( w ′ ) AQ ′ ( w ′ ) − 1 . In this situation one an write do wn orresp onding lo al expressions for  , namely  U ( u, h ) =  u, A U ( u, h )  o v er U and  V ( v , h ) =  v , A V ( v , h )  o v er V with A U : U × H → GL ( ℓ ; C ) a smo oth family of represen tations of H et., whi h mak e the follo wing squares G (1) | U  | U / / τ U ≈   GL ( E ) | U ≈ U   G (1) | V  | V / / τ V ≈   GL ( E ) | V ≈ V   U × H  U / / _ _ _ U × GL ( ℓ ; C ) V × H  V / / _ _ _ V × GL ( ℓ ; C ) (11) 2. CLASSICAL REPRESENT A TIONS 27 omm ute. If w e tak e their restritions to W , W ′ resp etiv ely , w e obtain W × H  V / / _ _ _ W × GL ( ℓ ; C ) W ′ × H  V / / _ _ _ W ′ × GL ( ℓ ; C ) G (1) | W  | W / / τ U ≈   τ V ≈ O O GL ( E ) | W ≈ U   ≈ V O O G (1) | W ′  | W ′ / / τ U ≈   τ V ≈ O O GL ( E ) | W ′ ≈ U   ≈ V O O W × H  U / / _ _ _ W × GL ( ℓ ; C ) W ′ × H  U / / _ _ _ W ′ × GL ( ℓ ; C ) (12) and hene, making use of the expliit expression (7) for the transition map τ U ◦ τ V − 1 , w e are led to the follo wing relations: for all h ∈ H (13) ( A U ( w , h ) = Q ( w ) A V ( w , h ) Q ( w ) − 1 for all w ∈ W A U  w ′ , χ ( h )  = Q ′ ( w ′ ) A V ( w ′ , h ) Q ′ ( w ′ ) − 1 for all w ′ ∈ W ′ . F rom no w on, w e assume that H is ompat. W e also x t w o p oin ts w 0 ∈ W and w ′ 0 ∈ W ′ . There is a on tin uous path γ U : [0 , 1] → U from w 0 to w ′ 0 . This giv es a on tin uous map (14) [0 , 1] × H γ U × id − − − → U × H A U − − → GL ( ℓ ; C ) whi h is learly a homotop y of represen tations of H onneting A U ( w 0 , - ) to A U ( w ′ 0 , - ) . Then, as remark ed in Note 30 , there will b e an in v ertible matrix R ∈ GL ( ℓ ; C ) su h that (15) A U ( w 0 , - ) = RA U ( w ′ 0 , - ) R − 1 . A seond path γ V : [0 , 1] → V onneting w 0 to w ′ 0 will analogously yield a matrix S ∈ GL ( ℓ ; C ) su h that (16) A V ( w 0 , - ) = S A V ( w ′ 0 , - ) S − 1 . Making the appropriate substitutions in (13), w e nally nd an in v ertible matrix P ∈ GL ( ℓ ; C ) su h that (17) A U  w 0 , χ ( h )  = P A U ( w 0 , h ) P − 1 for all h ∈ H . Next, w e further sp eialize do wn to the ase where H is ab elian and onneted. Motiv ated b y Eq. (17 ), w e fo us our atten tion on those matrix represen tations A : H → GL ( ℓ ; C ) su h that (18) ∃ P ∈ GL ( ℓ ; C ) for whi h A ( χ ( h )) = P A ( h ) P − 1 . By S h ur's Lemma, ev ery irreduible matrix represen tation of an Ab elian Lie group m ust b e one-dimensional (f. for instane Brö  k er and tom Die k p. 69) 28 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS and therefore, b eause of the ompatness of H , neessarily a har ater i.e. a Lie group homomorphism of H in to the 1-torus T 1 . Sine ev ery represen tation of a ompat Lie group is a diret sum of irreduible ones ( ibid. p. 68), it is no loss of generalit y to assume Eq. ( 18 ) to b e of the follo wing form (19)    ( α 1 ◦ χ )( h ) · · · 0 . . . . . . . . . 0 · · · ( α ℓ ◦ χ )( h )    = P    α 1 ( h ) · · · 0 . . . . . . . . . 0 · · · α ℓ ( h )    P − 1 , where α 1 , . . . , α ℓ : H → T 1 are  haraters of H . The t w o omplex diagonal matries o urring in Eq. (19 ) m ust ha v e the same  harateristi p olynomial p ( h, X ) ∈ C [ X ] . Th us, if w e put (20) β j = α j ◦ χ : H → T 1 and F ij =  h ∈ H : α i ( h ) = β j ( h )  , w e an in partiular express H as a nite union F 11 ∪ · · · ∪ F 1 ℓ of losed subsets. No w, it follo ws b y a standard indutiv e argumen t that one of them, let us sa y F 11 , m ust ha v e nonempt y in terior; therefore, the t w o  haraters α 1 and β 1 oinide on all of H , b eause a homomorphism of onneted Lie groups is determined b y its dieren tial at the neutral elemen t ( ibid. p. 24). Canelling the t w o orresp onding linear fators in p ( h, X ) w e obtain (21)  X − β 2 ( h )  · · ·  X − β ℓ ( h )  =  X − α 2 ( h )  · · ·  X − α ℓ ( h )  . Then, arguing b y indution on the degree of the p olynomial, w e onlude that there is a p erm utation σ on ℓ letters su h that α i = β σ ( i ) = α σ ( i ) ◦ χ for all i = 1 , . . . , ℓ . No w, onsider for instane α 1 . W rite σ as a pro dut of disjoin t yles and onsider the yle  1 , σ (1) , . . . , σ r (1)  where r ≧ 0 and σ r +1 (1) = 1 . Then w e ha v e α 1 = α σ (1) ◦ χ =  α σ ( σ (1)) ◦ χ  ◦ χ = α σ 2 (1) ◦ χ 2 = · · · = α σ r (1) ◦ χ r =  α σ ( σ r (1)) ◦ χ  ◦ χ r = α σ r +1 (1) ◦ χ r +1 = α 1 ◦ χ r +1 . Therefore α 1 is an example of a  harater α : H → T 1 with the sp eial prop ert y (22) ∃ r ≧ 0 su h that α = α ◦ χ r +1 . Finally , let us tak e H = T 2 = T 1 × T 1 to b e the 2-torus. Fix an arbitrary ℓ ∈ Z , and onsider the map (23) χ ℓ : T 2 → T 2 dened b y the rule ( s, t ) 7→ ( s, s ℓ t ) . This is an automorphism of the Lie group T 2 , with in v erse χ − ℓ . An y 2- harater α : T 2 → T 1 an b e written as the pro dut α ( s, t ) = µ ( s ) ν ( t ) of the t w o 1- haraters µ ( s ) = α ( s, 1) and ν ( t ) = α (1 , t ) . If w e assume that α enjo ys the prop ert y (22) then w e get µ ( s ) ν ( t ) = α ( s, t ) = α  s, s ℓ ( r +1) t  = µ ( s ) ν ( s ) ℓ ( r +1) ν ( t ) and therefore ν ( s ) ℓ ( r +1) = 1 for all s ∈ T 1 . No w, if ℓ 6 = 0 then ν m ust b e trivial, b eause r + 1 > 0 . It follo ws that (24) α ( s, t ) = µ ( s ) do es not dep end on t . 2. CLASSICAL REPRESENT A TIONS 29 25 Prop osition Fix an y in teger 0 6 = ℓ ∈ Z and let G T 2 ; χ ℓ → S 1 b e the lo ally trivial Lie bundle with bre T 2 o v er the irle, onstruted as explained ab o v e b y using χ ℓ ∈ Aut(T 2 ) as t wisting automorphism. Then there exists an em b edding of Lie bundles o v er the irle S 1 × T 1   ϕ / /   G T 2 ; χ ℓ   S 1 × S 1 id × id / / S 1 × S 1 (26) with the prop ert y that ev ery lassial represen tation ( E ,  ) in R ∞ ( G T 2 ; χ ℓ ) pulls ba k to a trivial represen tation ( E ,  ◦ ϕ ) of S 1 × T 1 . Pro of Dene the em b edding ϕ as follo ws. Giv en ( x, z ) ∈ S 1 × T 1 , send it to the equiv alene lass [ t, (1 , z )] , no matter what t y ou  ho ose as long as e 2 π it = x . With resp et to either of the t w o  harts τ U and τ V of Eq. (6), the lo al expression of this em b edding is simply ( x, z ) 7→ ( x ; 1 , z ) . No w, let ( E ,  ) b e a C -linear represen tation of G and let w 0 ∈ W b e the p oin t w e seleted in the ourse of the disussion ab o v e. In the  hart τ U the isotrop y group G | w 0 and the torus T 2 are iden tied b y the indued Lie group isomorphism G | w 0 ≈ T 2 . The subgroups ϕ ( { w 0 } × T 1 ) ⊂ G | w 0 and { 1 } × T 1 ⊂ T 2 orresp ond to one another under this isomorphism; moreo v er, the homomorphism  w 0 : G | w 0 → GL ( E w 0 ) is giv en the matrix represen tation A = A U ( w 0 , - ) : T 2 → GL ( ℓ ; C ) of Eq. ( 18 ). Therefore, sine from Eq. (24) w e kno w that { 1 } × T 1 is on tained in Ker A , w e onlude that the image ϕ ( { w 0 } × T 1 ) is on tained in Ker  w 0 . By the standard homotop y argumen t of Note 30 w e nally get ϕ ( { x } × T 1 ) ⊂ Ker  x for all x ∈ S 1 . This ompletes the pro of in the C -linear ase. Finally , let R = ( E ,  ) b e an y R -linear lassial represen tation of G . It will b e enough to tak e the omplexiation R ⊗ C = ( E ⊗ C ,  ⊗ C ) and observ e that Ker  x = Ker  x ⊗ C = Ker (  ⊗ C ) x for all x . q.e.d. Consider the map R × T 2 → S 1 × T 1 giv en b y ( t ; z , z ′ ) 7→ ( e 2 π it , z ) . This indues an epimorphism of Lie bundles o v er S 1 ψ : G T 2 ; χ ℓ − → T 1  T 1 def = S 1 × T 1  (27) whose k ernel is preisely the image of the em b edding ϕ of the preeding prop osition. The latter map yields an iden tiation of forgetful funtors R ∞ (T 1 ) forg. fun.   ψ ∗ ≃ / / R ∞ ( G T 2 ; χ ℓ ) forg. fun.   V ∞ (S 1 ) V ∞ (S 1 ) (28) 30 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS dened as ψ ∗ ( E ,  ) def = ( E , ψ ◦  ) . One easily reognizes that the funtor ψ ∗ is an isomorphism of ategories. Indeed, its in v erse ψ ∗ an b e onstruted expliitly b y means of the familiar univ ersal prop ert y of the quotien t (whi h in the presen t ase follo ws immediately from Prop osition 25 ), namely G T 2 ; χ ℓ ψ    / / GL ( E ) T 1 ψ ∗  9 9 s s s s s s (29) for ev ery ( E ,  ) ∈ Ob R ∞ ( G T 2 ; χ ℓ ) , so that ( E , ψ ∗  ) is an ob jet of R ∞ (T 1 ) (one ob viously sets ψ ∗ ( a ) = a for all morphisms a ). The existene of the iden tiation of ategories ( 28 ) sho ws in a v ery on- vining w a y that, in general, a ategory of lassial represen tations do es not pro vide enough information to reo v er the Lie group oid from whi h it origi- nates; this is true indep enden tly of the reip e one migh t in v en t for a p ossible reonstrution theory . Note also that this failure already o urs under ir- umstanes where the Lie group oid is a v ery reasonable one (ompat, ab elian and onneted). Of ourse, what w e are sa ying do es not exlude the p ossi- bilit y that in some sp eial ases the reonstrution ma y b e feasible; w e shall giv e a few elemen tary examples of this sort later on in 28. 30 Note (Compare also Brö  k er and tom Die k [ 4℄ p. 84) Let G b e a Lie group and let  t : G → GL ( V ) b e a family of represen tations  t dep ending on tin uously on g ∈ G and t ∈ [0 , 1] , in other w ords, a homotop y of represen tations. W e laim that if G is ompat, the represen tations  0 and  1 are isomorphi i.e. there exists some A ∈ GL ( V ) whi h onjugates  0 in to  1 . T o b egin with, let G ∨ denote the set of isomorphism lasses of irreduible G -mo dules. F or ea h γ ∈ G ∨ , selet a represen tativ e V γ . Then for ev ery t ∈ [0 , 1] one an deomp ose the G -mo dule V t = ( V ,  t ) in to a diret sum V t ≈ ⊕ γ ∈ G ∨ n t γ V γ in whi h the in teger n t γ = m ultipliit y of V γ in V t = R χ t χ γ , where χ γ is the  harater of V γ and χ t = P γ ∈ G ∨ n t γ χ γ is the  harater of V t , dep ends on tin uously on t and is therefore onstan t. This pro v es the laim. More generally , one has that if f t : G → H is an y homotop y of homo- morphisms of a  omp at Lie group G in to a Lie group H then f 0 and f 1 are onjugate: see Conner and Flo yd (1964) [7℄ Lemma 38.1. Their result is a onsequene of the follo wing theorem of Mon tgomery and Zippin (1955) (whi h an b e found in [30 ℄ p. 216): Theorem Let G b e a Lie group and F a ompat subgroup of G . Then there exists an op en set O in G , F ⊂ O , with the prop ert y that if H is a ompat subgroup of G and H ⊂ O , then there is a g in G su h that g − 1 H g ⊂ F . 3. NORMALIZED HAAR SYSTEMS 31 Moreo v er giv en an y neigh b orho o d W of e , O an b e so  hosen that for ev ery H ⊂ O the desired g an b e seleted in W . Compare Bredon (1972) [3 ℄ I I.5.6. 3 Normalized Haar Systems Normalized Haar systems on prop er Lie group oids are the analogue of Haar probabilit y measures on ompat Lie groups. In the presen t setion w e review the basi denitions and giv e some details ab out the onstrution of Haar systems on prop er Lie group oids; an exp osition of this material an also b e found in Craini [10 ℄. Let G b e a Lie group oid o v er a manifold M . 1 Denition A p ositive Haar system on G is a family of p ositiv e Radon measures { µ x } ( x ∈ M ), ea h one with supp ort onen trated in the resp etiv e soure bre G x = G ( x, - ) = s − 1 ( x ) , satisfying the follo wing onditions: i) R ϕ d µ x > 0 for all nonnegativ e ϕ ∈ C ∞ c ( G x ) , ϕ 6 = 0 ; ii) for ev ery ϕ ∈ C ∞ c ( G (1) ; C ) , the funtion Φ : M → C dened b y (2) Φ( x ) def = Z G x ϕ | G x dµ x is of lass C ∞ ; iii) right invarian e: for arbitrary g ∈ G ( x, x ′ ) and ϕ ∈ C ∞ c ( G x ) , (3) Z G x ′ ϕ ◦ τ g d µ x ′ = Z G x ϕ d µ x where τ g : G ( x ′ , - ) → G ( x, - ) denotes righ t translation h 7→ hg . In this denition the term `p ositiv e' refers to the rst ondition whereas the term `smo oth' is o asionally used to emphasize the seond ondition. The existene of p ositiv e (smo oth) Haar systems on a Lie group oid G an b e established if G is pr op er. (Reall that G is prop er if it is Hausdor and the map ( s , t ) : G → M × M is prop er in the usual sense.) One w a y to do this is the follo wing. One starts b y xing a Riemann metri on the v etor bundle g → M , where g is the Lie algebroid of G (fr. Craini [ 10 ℄ or Mo erdijk and Mr£un [27℄, Chapter 6; note the use of paraompatness). Righ t translations determine isomorphisms T G ( x, - ) ≈ t ∗ g | G ( x, - ) for all x ∈ M . These an b e used to indue, on the soure bres G ( x, - ) , Riemann metris whose asso iated v olume forms pro vide the desired system of measures. P ositiv e Haar systems are not en tirely adequate for our purp oses. W e will nd the follo wing notion more useful: 32 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS 4 Denition A normalize d Haar system on G is a family of p ositiv e Radon measures { µ x } ( x ∈ M ), ea h with supp ort onen trated in the resp etiv e soure bre G ( x, - ) , with the follo wing prop erties: (a) all smo oth funtions on G ( x, - ) are in tegrable with resp et to µ x , that is to sa y (5) C ∞  G ( x, - ); C  ⊂ L 1 ( µ x ; C ) ; (b) Conditions ii) and iii) of the preeding denition hold for an arbitrary smo oth funtion ϕ on G (1) , resp etiv ely G ( x, - ) ; () the follo wing normalit y ondition is satised: i ′ ) R d µ x = 1 , for all x ∈ M . Ev ery prop er Lie group oid admits normalized (smo oth) Haar systems. F or su h a group oid G , one an pro v e this b y using a ut-o funtion, namely a p ositiv e, smo oth funtion c on the base M , su h that the soure map s restrits to a prop er map on supp ( c ◦ t ) and R ( c ◦ t ) d ν x = 1 for all x ∈ M , where { ν x } is a xed p ositiv e (smo oth) Haar system on G . The system of p ositiv e measures µ x = ( c ◦ t ) ν x has the desired prop erties. Observ e that if E ∈ Ob V ∞ ( M ) is a smo oth v etor bundle of lo ally nite rank o v er the base of G and ψ : G → E is a smo oth mapping su h that for ea h x ∈ M the bre G ( x, - ) is mapp ed in to the v etor spae E x , then the in tegral (6) Ψ( x ) def = Z ψ x d µ x mak es sense and denes a smo oth setion of E . This follo ws easily from the prop erties listed in Denition 4, b y w orking in lo al o ordinates. 4 The Lo al Linearizabilit y Theorem Let G b e a Lie group oid and let M b e its base manifold. W e sa y that a submanifold N of M is a sli e at the p oin t z ∈ N if the orbit immersion G z ֒ → M is transv ersal to N at z . A submanifold S of M will b e alled a slie if it is a slie at all of its p oin ts. The follo wing remark will b e used v ery often: Let N b e a submanifold of M and let g ∈ G N ≡ s − 1 ( N ) ; then N is a slie at z = s ( g ) if and only if the in tersetion G N ∩ t − 1 ( z ′ ) , z ′ = t ( g ) is transv ersal at g . Indeed, from the equalities T g G N = T z N ⊕ T g G z and T g t − 1 ( z ′ ) = T z G z ′ ⊕ W = T z G z ⊕ W , where W is a linear subspae of T g G z , it follo ws immediately that (1) T g G N + T g t − 1 ( z ′ ) =  T z N + T z G z  ⊕ T g G z . 4. THE LOCAL LINEARIZABILITY THEOREM 33 By virtue of this fat, one obtains that for ea h submanifold N of M , the subset of all p oin ts at whi h N is a slie is an op en subset of N . In order to asertain it, x a p oin t z b elonging to this subset. Sine the in tersetion of G N with the bre t − 1 ( z ) m ust b e transv ersal at u ( z ) ∈ G ( z , z ) , there will b e a neigh b ourho o d Γ N of u ( z ) in G N su h that for all g ∈ Γ N the in tersetion G N ∩ t − 1 ( t g ) is transv ersal at g . No w, if S is an op en neigh b ourho o d of z in N su h that u ( S ) ⊂ Γ N , one has that S is a slie. Let R , S b e m utually transv ersal submanifolds of a manifold N : then R ∩ S is a submanifold of N , of dimension r + s − n . Next, let p : Y → X b e a submersion, let S b e an y submanifold of Y and x s 0 ∈ S . Put x 0 = p ( s 0 ) . Then S in tersets the bre p − 1 ( x 0 ) transv ersally at s 0 if and only if the restrition p | S : S → X is submersiv e at that p oin t; from this, it immediately follo ws that when the in tersetion S ∩ p − 1 ( x 0 ) is transv ersal at s 0 , there exists a neigh b ourho o d A of s 0 in S su h that at all p oin ts a ∈ A the in tersetion S ∩ p − 1 ( x ) , x = p ( a ) is also transv ersal. In order to  he k the previous laim, it is not restritiv e to assume that Y = X × Z is a Cartesian pro dut and that p = pr is the pro jetion on to the rst fator. Setting s 0 = ( x 0 , z 0 ) , one obtains for the tangen t spaes the piture (2) T s 0 S + T z 0 Z ⊂ T s 0 ( X × Z ) = T x 0 X ⊕ T z 0 Z pr ∗ − − → T x 0 X , from whi h it is eviden t that T s 0 S on tains a linear subspae W su h that pr ∗ ( W ) = T x 0 X if and only if the inlusion (2) is an equalit y . 3 Note If a submanifold S of M is a slie then the in tersetion s − 1 ( S ) ∩ t − 1 ( S ) is transv ersal and the restrition G | S is a Lie group oid o v er S . Indeed, let us x g ∈ G ( z , z ′ ) with z , z ′ ∈ S . Sine (4) T g s − 1 ( S ) + T g t − 1 ( z ′ ) ⊂ T g s − 1 ( S ) + T g t − 1 ( S ) , one immediately obtains the transv ersalit y at g of the in tersetion writ- ten ab o v e. The target map t will indue a submersion of s − 1 ( S ) on to an op en subset of M and this submersion will in turn indue a submersion of s − 1 ( S ) ∩ t − 1 ( S ) on to S . 5 Note Let S b e a slie; then G · S is an op en subset of M . T o v erify this it will b e enough to sho w that giv en an y p oin t z ∈ S there exists a neigh b ourho o d U of z in M su h that s − 1 ( S ) ∩ t − 1 ( u ) 6 = ∅ for all u ∈ U . This is true b eause the in tersetion s − 1 ( S ) ∩ t − 1 ( z ) is nonempt y and transv ersal. Then U ⊂ G · S , from whi h the inlusion G · z ⊂ G · U ⊂ G · S nally follo ws. Theorem (N.T. Zung) Let G b e a prop er Lie group oid and let X b e its base manifold. Let x 0 ∈ X b e a p oin t whi h is not mo v ed b y the tautologial ation of G on its o wn base. 34 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS Then there exists a on tin uous linear represen tation G → GL ( V ) of the isotrop y group G ≡ G | x 0 on a nite dimensional v etor spae V , su h that for some op en neigh b ourho o d U of x 0 one an nd an isomorphism of Lie group oids G | U ≈ G ⋉ V whi h mak es x 0 orresp ond to 0 . Pro of See Zung's pap er [38℄. q.e.d. W e w an t to giv e a areful pro of of the statemen t that an y prop er Lie group oid is lo ally Morita equiv alen t to the translation group oid asso iated with a (linear) ompat Lie group ation; this will of ourse follo w from Zung's theorem. The latter statemen t is a k ey ingredien t in the pro of of our  main reonstrution theorem  , Theorem 20.28 . Let us b egin with a te hnial observ ation ab out slies. Let S , T b e t w o slies in M . Let g 0 ∈ G ( S, T ) ; put s 0 ≡ s ( g 0 ) ∈ S and t 0 ≡ t ( g 0 ) ∈ T . T o x ideas, supp ose dim S ≦ dim T . Then w e laim that there exists a smo oth setion τ : B → G (1) to the target map of G , dened o v er some op en neigh b ourho o d B of t 0 in T , su h that τ ( t 0 ) = g 0 and the omp osite map s ◦ τ indues a submersion of B on to an op en neigh b ourho o d of s 0 in S . T o b egin with, let us notiein generalthat if one is giv en a ouple of smo oth submersions Y p ← − X q − → Z with dim Y ≧ dim Z then for ea h p oin t x ∈ X there exists a smo oth p -setion π : U → X , dened o v er some op en neigh b ourho o d U of p ( x ) , su h that π ( p ( x )) = x and the omp osite q ◦ π : U → N is a submersion on to an op en neigh b ourho o d of q ( x ) in Z ; this is seen b y means of an ob vious argumen t based on elemen tary linear algebra: there exists a omplemen tary subspae F to Ker T x p in T x X su h that F + Ker T x q = T x X . No w, the in tersetion (6) X ≡ s − 1 ( S ) ∩ t − 1 ( T ) ⊂ G (1) is transv ersal, b eause for all g ∈ G ( s, t ) with s ∈ S and t ∈ T , s − 1 ( S ) will in terset t − 1 ( t ) and hene a fortiori t − 1 ( T ) transv ersally at g (sine S is a slie). Moreo v er, the soure map s : G → M restrits to a submersion of X on to S , forsine T is a sliethe submanifold t − 1 ( T ) is transv ersal to ev ery s -bre it in tersets and therefore the restrition s : t − 1 ( T ) → M is a submersion. Symmetrially , the indued mapping t | X : X → T will b e sub- mersiv e. Th us w e an apply the foregoing general remark ab out submersions to get a smo oth target setion τ with the desired prop erties. 7 Corollary Let G b e a prop er Lie group oid o v er a manifold M . Then for ea h p oin t x 0 ∈ M there exist a nite dimensional linear represen tation G → GL ( V ) of a ompat Lie group G , and a G -in v arian t op en neigh b ourho o d U of x 0 in M along with a Morita equiv alene ι : G ⋉ V ֒ → G | U , su h that ι (0) : V ֒ → U is an em b edding of manifolds mapping 0 to x 0 . 4. THE LOCAL LINEARIZABILITY THEOREM 35 Pro of By prop erness, w e an nd a slie S ⊂ M su h that S ∩ G · x 0 = { x 0 } . Then G | S is a prop er Lie group oid for whi h the p oin t x 0 is in v arian t. By Zung's theorem, w e an assume that there exists an isomorphism of Lie group oids G ⋉ V ≈ G | S , 0 7→ x 0 , for some linear ompat Lie group ation G → GL ( V ) . W e on tend that G ⋉ V ≈ G | S ֒ → G | U , where U is the op en subset G · S ⊂ M , is the Morita equiv alene ι w e are lo oking for. It will b e suien t to pro v e that the surjetiv e mapping V × U G | U → U , ( v , g ) 7→ t ( g ) is a submersion. This learly follo ws from the preeding observ ation ab out slies when w e tak e T ≡ U . q.e.d. W e onlude this setion with some remarks relating the group oids G | S and G | T indued on t w o dieren t slies S and T . Supp ose dim S ≦ dim T . Let s 0 ∈ S and t 0 ∈ T b e t w o p oin ts lying on the same G -orbit. Then i) for some op en neigh b ourho o ds B ⊂ T of t 0 and A ⊂ S of s 0 there exists a Morita equiv alene G | B ։ G | A mapping t 0 to s 0 and induing a submersion of B on to A ; ii) for some op en neigh b ourho o d A ⊂ S of s 0 there exists an em b edding of Lie group oids G | A ֒ → G | T mapping s 0 to t 0 and induing a slie em b edding A ֒ → T (ie an em b edding whose image is a slie); iii) if in partiular dim S = dim T then the Lie group oids G | S and G | T are lo ally isomorphi ab out the p oin ts s 0 and t 0 . Let us v erify the assertion i) . Cho ose an y g 0 ∈ G ( s 0 , t 0 ) . By the te hnial observ ations preeding Corollary 7, w e an nd a smo oth target setion τ : B → G (1) so that s ◦ τ is a submersion on to an op en neigh b ourho o d A ⊂ S of s 0 . The latter map an b e lifted to (8) G | B → G A , h 7→ τ ( t h ) − 1 · h · τ ( s h ) ; this form ula sets up the required Morita equiv alene. In an en tirely analogous manner assertion ii) an b e pro v ed b y onsidering a suitable smo oth soure setion σ : A → G (1) su h that t ◦ σ is a slie em b edding of A in to T and then b y lifting this em b edding to one of Lie group oids (9) G | A ֒ → G | T , g 7→ σ ( t g ) · g · σ ( s g ) − 1 . 10 Note Let σ : U → G (1) b e a lo al bisetion. Supp ose S ⊂ U is a slie. Then T ≡ t  σ ( S )  is also a slie; moreo v er, there exists a Lie group oid isomorphism G | S ≈ − → G | T whi h lifts the map t ◦ σ . Let us pro v e that T is a slie. Put V = t  σ ( U )  . Fix a p oin t s 0 ∈ S and let t 0 ≡ t ( σ ( s 0 )) . Then (11) t  σ ( G · s 0 ∩ U )  = G · t 0 ∩ V 36 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS and therefore, sine t ◦ σ is a dieomorphism of U on to V , the orbit G · s 0 in tersets the submanifold S transv ersally at s 0 if and only if G · t 0 in tersets T transv ersally at t 0 ; our laim follo ws. Next, observ e that t ◦ σ is ertainly a dieomorphism of S on to T , whi h an b e liftedvia σ , as in ( 9)to a Lie group oid isomorphism with the exp eted prop erties. 5 Global Quotien ts The material presen ted in this setion is not diretly relev an t to the problem disussed in the thesis; if the reader wishes to do so, he ma y go diretly to the next  hapter. As b efore, w e la y no laim to originalit y . 1 Lemma Let H b e a prop er Lie group oid, ating without isotrop y on its o wn base F (i.e. all isotrop y groups of H are assumed to b e trivial). Then the orbit spae F / H has a unique manifold struture su h that the quotien t map q : F → F / H is a submersion. Pro of The mapping ( s , t ) : H → F × F is an injetiv e immersion. Indeed, for a xed h ∈ H ( f , f ′ ) , f , f ′ ∈ F , the tangen t map (2) T h H T h ( s , t ) − − − − − → T ( f ,f ′ ) ( F × F ) ∼ = T f F ⊕ T f ′ F equals the linear map T h s ⊕ T h t ; therefore (3) Ker T h ( s , t ) = Ker T h s ∩ Ker T h t = T h H ( f , f ′ ) = 0 (fr. for example [27℄, pr o of of Thm. 5.4, p. 117; b y the trivialit y of the isotrop y groups of H , the latter tangen t spae m ust b e zero). Moreo v er, b eause of prop erness, ( s , t ) : H → F × F is also a losed map, hene in fat an em b edding of smo oth manifolds. It follo ws that the equiv alene relation R = Im ( s , t ) = { ( f , f ′ ) |∃ h : f → f ′ in H} is a submanifold of F × F ; the pro jetion on to the seond fator learly restrits to a submersion of R on to F . Therefore, b y Go demen t's Theorem ( se e [33℄, p. 92), there is a manifold struture on the quotien t spae F /R = F / H , making the quotien t map q : F → F / H a submersion. q.e.d. This lemma applies when a prop er Lie group oid G with base M ats fr e ely from the left on a manifold F along some smo oth mapping p : F → M . By denition, this means that the orresp onding ation group oid H ≡ G ⋉ F has trivial isotrop y groups. In order to onlude that there exists a smo oth manifold struture on the quotien t spae F / G , for whi h the pro jetion F → F / G is submersiv e, one needs to  he k that the group oid G ⋉ F is also prop er. So, let C ⊂ F × F b e an y ompat subset and put C 1 = pr 1 ( C ) ⊂ F ; sine F is a Hausdor manifold, the in v erse image ( s H , t H ) − 1 ( C ) will b e a losed subset of the manifold G × F and hene of the ompat set (4) ( s G , t G ) − 1  ( p × p )( C )  × C 1 ⊂ G × F , 5. GLOBAL QUOTIENTS 37 where p × p denotes the smo oth map ( f , f ′ ) 7→ ( p ( f ) , p ( f ′ )) . No w, supp ose that a Lie group K ats smo othly on F from the righ t, in su h a w a y that p : F → M turns out to b e a prinipal K -bundle. Assume that this ation omm utes with the giv en left ation of G . Then there is a w ell-dened indued righ t ation of K on the quotien t manifold F / G . This is in fat a smo oth ation b eause of an elemen tary prop ert y of submersions (see e.g. p. 147 b elo w): the ation map F / G × K → F / G has to b e smo oth b eause up on omp osing it with the submersion F × K → F / G × K one obtains a smo oth map, namely F × K → F → F / G . The next result should probably b e regarded as folklore. Its statemen t, along with the k ey idea for the pro of presen ted here, w as suggested to me b y I. Mo erdijk as early as the b eginning of 2006. 5 Theorem Supp ose a prop er Lie group oid G admits a global faithful represen tation on a smo oth v etor bundle. Then G is Morita equiv alen t to the translation group oid asso iated with a ompat, onneted Lie group ation. Pro of Let  : G ֒ → GL ( E ) b e a faithful represen tation on alet us sa y , realsmo oth v etor bundle E o v er the base M of G . By prop erness of G , w e an nd a  -in v arian t metri 1 on E , whi h w e x one and for all. Then let F = F r( E ) p − → M b e the orthonormal fr ame bund le of E (relativ e to the  hosen in v arian t metri): reall that the bre F x ab o v e x is the spae of all linear isometries f : R d ∼ → E x , where d is the rank of E x . The total spae F of this bre bundle is a paraompat Hausdor manifold; moreo v er, the bration p is a prinipal bundle for the anonial righ t ation of the orthogonal group K = O ( d ) on F (dened b y f k = f ◦ k ). Sine  ats on E b y isometries, the Lie group oid G will at on the manifold F from the leftvia the represen tation  , that is b y the rule g f =  ( g ) ◦ f along the bundle map p . Clearly , the t w o ations omm ute. Next, let the double ation group oid G ⋉ F ⋊ K b e the Lie group oid o v er the manifold F that is obtained as follo ws. Its manifold of arro ws is ( G ⋉ F ) × K , viz. the submanifold of the Cartesian pro dut ( G × F ) × K onsisting of all triples ( g , f , k ) with s ( g ) = p ( f ) . The soure map sends the arro w ( g , f , k ) to f and the target map to g f k . The omp osition of arro ws is dened to b e ( g ′ , f ′ , k ′ ) · ( g , f , k ) = ( g ′ g , f , k k ′ ) . Then the iden tit y arro w at f is ( p ( f ) , f , id ) and the in v erse m ust b e giv en b y ( g , f , k ) − 1 = ( g − 1 , g f k , k − 1 ) . All these struture maps are ob viously smo oth. No w, w e laim that there are Morita equiv alenes (6) G ˜ p ← − − − − G ⋉ F ⋊ K ˜ q − − − − → F / G ⋊ K 1 This an b e pro v ed in a standard w a y , v ery m u h lik e in the ase of groups, b y using Haar systems as a substitute for Haar measures. Cfr. Prop osition 17 .17 . 38 CHAPTER I. LIE GR OUPOIDS, CLASSICAL REPRESENT A TIONS from the double ation group oid. This will sho w that G is Morita equiv alen t to the ation group oid F / G ⋊ K , as on tended. P erhaps it is go o d to sp end a ouple of w ords to state the form ulas for right ation group oids; these are obtained b y regarding a giv en righ t ation of a Lie group H on a manifold X as a left ation of the opp osite group. Th us ( x, h ) 7→ x , resp. 7→ x · h is the soure, resp. target map, and ( x ′ , h ′ ) · ( x, h ) = ( x, hh ′ ) is the omp osition. W e start with the onstrution of the equiv alene to the left (7) ˜ p : G ⋉ F ⋊ K − → G . As the notation ˜ p suggests, this equiv alene is to b e giv en b y the surjetiv e submersion p : F → M on base manifolds; as to arro ws, w e put ˜ p ( g , f , k ) = g . It is immediate to  he k that ˜ p denes a Lie group oid homomorphism of G ⋉ F ⋊ K on to G . All one needs to do no w in order to sho w that ˜ p is a Morita equiv alene is to solv e, within the ategory of smo oth manifolds, the univ ersal problem stated in the left-hand diagram b elo w: X ( f ,f ′ ) & & # # G G G G G g # # X ( f ,f ′ ) & & # # G G G G G ( q ◦ f , k ) % % G ⋉ F ⋊ K ˜ p / /   G   G ⋉ F ⋊ K ˜ q / /   F / G ⋊ K   F × F p × p / / M × M F × F q × q / / F / G × F / G . (8) It will b e enough to notie that the map X → K , x 7→ κ ( x ) , whi h assigns the linear isometry κ ( x ) = f ′ ( x ) − 1 ◦  ( g ( x )) ◦ f ( x ) to ea h x , is of lass C ∞ . Then w e an dene the dashed arro w in the aforesaid diagram to b e x 7→ ( g ( x ) , f ( x ) , κ ( x )) . This is learly the unique p ossible solution. W e turn our atten tion no w to the other equiv alene (9) ˜ q : G ⋉ F ⋊ K − → F / G ⋊ K . This is giv en b y q on ob jets and b y ˜ q ( g , f , k ) = ( q ( f ) , k ) on arro ws. Clearly , the map ˜ q so dened is a homomorphism of Lie group oids. Sine the base mapping q : F → F / G is kno wn to b e a surjetiv e submersion b y Lemma 1 , in order to sho w that ˜ q is a Morita equiv alene it will b e enough to solv e the righ t-hand univ ersal problem of (8). W e observ e that from the prop erness of G and the faithfulness of  it follo wssee for instane Corollary 23.10 b elo wthat the image  ( G ) ⊂ GL ( E ) is a submanifold; moreo v er, it an b e sho wnfr. Lemma 26.3, for examplethat  : G ≈ − →  ( G ) is atually a dieomorphism. No w, the map X → GL ( E ) , x 7→ γ ( x ) , that sends x to the isometry γ ( x ) = f ′ ( x ) ◦ k ( x ) ◦ f ( x ) − 1 , is learly smo oth and fators through the submanifold  ( G ) . Then w e ma y use the fat that  is a dieomorphism of G on to  ( G ) and dene the dashed arro w as x 7→   − 1 ( γ ( x )) , f ( x ) , k ( x )  ; this is of ourse a smo oth orresp ondene. q.e.d. Chapter I I The Language of T ensor Categories With the exeption of 10, the presen t  hapter oers an in tro dution to the ategorial setting of the mo dern theory of T annak a dualit y originating from the ideas of A. Grothendie k and N. Saa v edra Riv ano; fr [ 32 , 12, 11, 18℄. In Setion 10 w e pro v e a k ey te hnial lemma whi h will b e used in the pro of of our reonstrution theorem in 20; sine this lemma deals with a fairly abstrat ategorial situation, w e though t it w as more appropriate to inlude it in this  hapter. 6 T ensor Categories A tensor strutur e on a ategory C onsists of the follo wing data: (1) a bifuntor ⊗ : C × C − → C , a distinguished ob jet 1 ∈ Ob( C ) and a list of natural isomorphisms, alled A CU  onstr aints: α R,S,T : R ⊗ ( S ⊗ T ) ∼ → ( R ⊗ S ) ⊗ T , γ R,S : R ⊗ S ∼ → S ⊗ R , λ R : R ∼ → 1 ⊗ R and ρ R : R ∼ → R ⊗ 1 (2) satisfying MaLane's  oher en e  onditions (fr for example MaLane (1971), pp. 157 . and esp eially p. 180 for a detailed exp osition). A tensor  ate gory is a ategory endo w ed with a tensor struture. In the terminology of [24 ℄, the presen t notion orresp onds to that of  symmetri monoidal ategory. The natural isomorphism α resp. γ is alled the asso iativity resp.  ommutativity onstrain t; λ and ρ are the tensor unit onstrain ts. In order to state MaLane's Coherene Theorem for tensor ategories, it will b e on v enien t to in tro due the onepts of  anonial m ulti-funtor  and 39 40 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES  anonial transformation  . These will onstitute resp etiv ely the ob jets and the morphisms of a ategory C an ( C ) . A multi-funtor on C is a funtor Φ : C I → C for some nite set I . The ardinalit y | I | = Card I will b e alled the  -ariet y  of Φ . The  anoni al m ulti-funtors are, roughly sp eaking, those obtained as formal iterates of ⊗ , p ossibly in v olving 1 . The adjetiv e `formal' here means that a `anonial m ulti-funtor' is not just a ertain t yp e of m ulti-funtor, in that one should regard the partiular indutiv e onstrution, b y whi h a anonial m ulti-funtor is obtained, as part of the dening data; w e do not w an t to go in to details here: the in terested reader ma y onsult [24℄. The reursiv e rules for generating anonial m ulti-funtors are listed b elo w: i) the unique 0 -ary anonial m ulti-funtor is 1 : C ∅ = { ⋆ } → C , ⋆ 7→ 1 ; ii) the iden tit y: C { ⋆ } → C is anonial; iii) if Φ : C I → C and Ψ : C J → C are anonial then so is Φ ⊗ Ψ : C I ⊔ J → C where I ⊔ J indiates disjoin t union; iv) if I σ − → J is a bijetion of nite sets and Φ : C I → C is anonial then Φ σ : C J → C I → C is also anonial. Canonial m ulti-funtors are the ob jets of C an ( C ) . As to anonial natural transformations, they are reursiv ely generated as follo ws: a) the iden tit y id : Φ → Φ is anonial; if η : Φ → Φ ′ , with Φ , Φ ′ : C I → C , and θ : Ψ → Ψ ′ , with Ψ , Ψ ′ : C J → C , are anonial transformations of anonial m ulti-funtors, then so is η ⊗ θ : Φ ⊗ Ψ → Φ ′ ⊗ Ψ ′ (natural transformations of m ulti-funtors C I ⊔ J → C ); if I σ − → J is a bijetion of sets then θ σ : Φ σ → Ψ σ is also anonial; b) α Φ , Ψ , X :  Φ ⊗ (Ψ ⊗ X)  σ ∼ →  (Φ ⊗ Ψ) ⊗ X  τ and its in v erse α Φ , Ψ , X − 1 are anonial transformations, where σ , τ are the bijetions I ⊔ ( J ⊔ K ) → I ⊔ J ⊔ K ← ( I ⊔ J ) ⊔ K ; ) γ Φ , Ψ : Φ ⊗ Ψ ∼ → [Ψ ⊗ Φ] σ (along with its in v erse) is anonial, where I ⊔ J σ ← − J ⊔ I is the ob vious bijetion; d) λ Φ : Φ ∼ → ( 1 ⊗ Φ) σ and ρ Φ : Φ ∼ → (Φ ⊗ 1 ) τ (along with their in v erses) are anonial, where ∅ ⊔ I σ − → I τ ← − I ⊔ ∅ are the ob vious bijetions. It is lear that all anonial transformations are isomorphisms. MaL ane's Coher en e The or em for  symmetri monoidal ategories  ( tensor ategories  in our terminology) an no w b e stated as follo ws: Theorem The ategory C an ( C ) is a preorder. That is to sa y , for an y anonial m ulti-funtors Φ and Ψ there is at most one anonial natural transformation Φ → Ψ . Pro of See [MaLane℄, xi.1 p. 253. q.e.d. 6. TENSOR CA TEGORIES 41 This theorem sa ys that an y diagram of anonial m ulti-funtors and anonial natural transformations one an p ossibly onstrut will omm ute. When one is giv en su h a diagram, let us sa y of m ulti-funtors C I → C , one ma y  ho ose an iden tiation { 1 , . . . , i } ∼ → I and denote a generi ob jet of C I b y ( R 1 , . . . , R i ) , R 1 , . . . , R i ∈ Ob( C ) . Ev aluating the giv en diagram at this i -tuple of ob jetsso that Φ θ − → Ψ b eomes Φ( R 1 , . . . , R i ) θ ( R 1 ,...,R i ) − − − − − − → Ψ( R 1 , . . . , R i ) , for instaneone obtains a omm utativ e diagram in C . 3 Note (See also Saave dr a, 1.3.3.1) Let ( C , ⊗ , 1 ) b e a tensor ategory . Then End C ( 1 ) is a omm utativ e ring. T o see this, observ e that the tensor unit onstrain t 1 ∼ = 1 ⊗ 1 establishes a anonial isomorphism of rings b et w een End( 1 ) and End( 1 ⊗ 1 ) . No w, if e, e ′ ∈ End( 1 ) then ee ′ ∼ = (1 ⊗ e )( e ′ ⊗ 1) = e ′ ⊗ e = ( e ′ ⊗ 1)(1 ⊗ e ) ∼ = e ′ e in this isomorphism and hene ee ′ = e ′ e . Note that this pro of only uses the oherene iden tit y λ 1 = ρ 1 for the tensor unit onstrain ts; the omm utativit y onstrain t pla ys no role. Rigid tensor ategories A tensor ategory ( C , ⊗ ) is said to b e lose d, whenev er one an exhibit a bifuntor hom : C op × C − → C , alled `in ternal hom' and denoted b y ( X , Y ) 7→ Y X ≡ hom( X, Y ) , along with natural transformations (in the v ariable Y ) η X Y : Y → ( Y ⊗ X ) X and ε X Y : Y X ⊗ X → Y , satisfying the triangular iden tities for an adjuntion C  X ⊗ T , Y  ∼ → C  X , hom( T , Y )  (in the v ariables ( X , Y ) ∈ C op × C ) b et w een the funtors  - ⊗ T  and  hom( T , - )  and making Y X ′ ⊗ X id ⊗ a   Y a ⊗ id / / Y X ⊗ X ε   ( Y ⊗ X ) X ( id ⊗ a ) id / / ( Y ⊗ X ′ ) X Y X ′ ⊗ X ′ ε / / Y Y η O O η / / ( Y ⊗ X ′ ) X ′ id a O O (4) omm ute for ev ery arro w a : X → X ′ . Supp ose no w that an `in ternal hom' bifuntor and natural transformations η , ε with these prop erties ha v e b een xed. Then there is an ob vious arro w (5) δ S,T X,Y : X S ⊗ Y T → ( X ⊗ Y ) S ⊗ T , namely the unique solution d to the equation ε ◦ ( d ⊗ id ) = ( ε ⊗ ε ) ◦ ∼ = , 42 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES where ∼ = is the unique anonial isomorphism. Beause of (4 ), the arro w δ m ust b e natural in all v ariables. By the same reason, the solution (6) ι X : X → X ∨∨ (where w e put X ∨ ≡ hom( X, 1 ) , to b e alled the dual of X ) to the equation ε ◦ ( ι X ⊗ id ) = ε ◦ ∼ = is natural in X . A dieren t  hoie of in ternal hom bifuntor and natural transformations η and ε will yield the same natural arro ws δ and ι up to isomorphism: th us it mak es sense to all a losed tensor ategory rigid when these natural arro ws are isomorphisms. One an also form ulate this notion in terms of duals, sine for a rigid tensor ategory one has the iden tiation (7) hom( X , Y ) ≈ X ∨ ⊗ Y , f. Deligne (1990), [11℄ 2.1.2. Let ( C , ⊗ ) b e a rigid tensor ategory . The on tra v arian t funtor X 7→ X ∨ , f 7→ t f is an equiv alene b et w een C and its opp osite ategory C op (b eause it is involutive , ie its omp osite with itself is naturally isomorphi to the iden tit y , sine rigidit y implies that (6) is a natural isomorphism). This giv es in partiular a bijetion b et w een the hom-sets f 7→ t f : Hom C ( X , Y ) ∼ → Hom C ( Y ∨ , X ∨ ) . One also has an in ternal isomorphism Y X ∼ → X ∨ Y ∨ , namely the omp osite Y X ≈ ← − X ∨ ⊗ Y id ⊗ ι Y − − − − → X ∨ ⊗ Y ∨∨ ≈ − → Y ∨∨ ⊗ X ∨ ≈ − → X ∨ Y ∨ . F or ev ery ob jet of C there is an arro w X X ∼ → X ∨ ⊗ X ε − → 1 . If w e apply the funtor Hom C ( 1 , · ) to this, w e obtain the tr a e map (8) T r X : End C ( X ) → End C ( 1 ) . The r ank of X is dened as T r X (1 X ) . There are the form ulas T r X ⊗ X ′ ( f ⊗ f ′ ) = T r X ( f )T r X ′ ( f ′ ) , T r 1 ( f ) = f . (9) 6. TENSOR CA TEGORIES 43 A tensor ategory ( C , ⊗ ) is said to b e additive if the ategory C is endo w ed with an additiv e struture su h that the bifuntor ⊗ is biadditiv e, that is additiv e in ea h v ariable separately . Moreo v er, if the hom-sets C ( A, B ) are endo w ed with a real (or omplex) v etor spae struture in su h a w a y that omp osition of arro ws and the bifuntor ⊗ are bilinear, then w e sa y that ( C , ⊗ ) is a line ar tensor ategory . 10 Example Let V e c C b e the ategory of v etor spaes o v er C of nite dimension. Then this is an ab elian rigid tensor ategory , and all the preeding denitions ha v e their usual meaning. 11 Example Let M b e a smo oth manifold. Let C = V ∞ ( M ; C ) b e the ategory of smo oth omplex v etor bundles of lo ally nite rank o v er M . The diret sum op eration ( E , F ) 7→ E ⊕ F mak es it in to an additiv e C - linear ategory , although in general not an ab elian one, sine a map of v etor bundles o v er M need not ha v e a k ernel, for instane. W e shall iden tify the ategory of nite dimensional v etor spaes o v er C with V ∞ ( ⋆ ; C ) where ⋆ is the one-p oin t manifold. The ategory V ∞ ( M ; C ) is endo w ed with a anonial rigid tensor stru- ture, obtained from the rigid tensor struture of V e c C b y means of the general pro edure desrib ed in Lang 2001 [ 22 ℄ p. 58, as follo ws. Reall that a m ulti- funtor Φ : V e c C × · · · × V e c C n times − → V e c C (where ase n = 0 orresp onds to the  hoie of an ob jet Φ( · ) ∈ Ob( V e c C ) , and w e allo w Φ to b e on tra v arian t in some v ariables), su h that the indued mappings L ( V 1 , W 1 ) × · · · × L ( V n , W n ) → L (Φ( V 1 , . . . , V n ) , Φ( W 1 , . . . , W n )) are of lass C ∞ , determines a orresp onding m ulti-funtor Φ : V ∞ ( M ; C ) × · · · × V ∞ ( M ; C ) − → V ∞ ( M ; C ) with the same v ariane and satisfying the follo wing prop erties: i) for ev ery x ∈ M , the b er ab o v e x is (12) Φ( E 1 , . . . , E n ) x = { x } × Φ( E 1 x , . . . , E n x ) ≈ Φ( E 1 x , . . . , E n x ); ii) for arbitrary morphisms of v etor bundles a i : E i → F i , i = 1 , . . . , n , Φ( a 1 , . . . , a n ) x orresp onds to Φ( a 1 x , . . . , a n x ) up to the anonial iden- tiations (12 ); 44 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES iii) If the v etor bundles E i ≈ M × E i are trivial, then these trivializations ≈ i determine a trivialization Φ( E 1 , . . . , E n ) ≈ M × Φ( E 1 , . . . , E n ) in a anonial w a y; in partiular, in the ase n = 0 , Φ( - ) ≈ M × Φ( - ) (the standard notation is then Φ( - ) = Φ( - ) ). A natural transformation λ : Φ → Ψ of m ulti-funtors with the same v ari- ane indues a natural transformation λ : Φ → Ψ , su h that λ ( E 1 , . . . , E n ) x orresp onds to λ ( E 1 x , . . . , E n x ) up to the iden tiations (12). Observ e that λ ◦ µ = λ ◦ µ and id = id . W e an apply these onstrutions to the m ultifuntors and natural trans- formations whi h dene the rigid tensor struture of V e c C , in order to obtain a similar struture on V ∞ ( M ; C ) . 7 T ensor F untors Let C , D b e tensor ategories. A tensor funtor : C − → D onsists of the data ( F , τ , υ ) , where F : C − → D is a funtor, τ is a natural isomorphism of bifuntors τ R,S : F ( R ) ⊗ F ( S ) ∼ → F ( R ⊗ S ) su h that the diagrams F R ⊗ ( F S ⊗ F T ) α   id ⊗ τ / / F R ⊗ F ( S ⊗ T ) τ / / F ( R ⊗ ( S ⊗ T ) ) F ( α )   ( F R ⊗ F S ) ⊗ F T τ ⊗ id / / F ( R ⊗ S ) ⊗ F T τ / / F (( R ⊗ S ) ⊗ T ) and F ( R ) ⊗ F ( S ) τ   γ / / F ( S ) ⊗ F ( R ) τ   F ( R ⊗ S ) F ( γ ) / / F ( S ⊗ R ) omm ute, and υ : 1 ∼ → F ( 1 ) is an isomorphism in D su h that F ( R ) F ( λ ) / / λ   F ( 1 ⊗ R ) F ( R ) F ( ρ ) / / ρ   F ( R ⊗ 1 ) 1 ⊗ F ( R ) υ ⊗ id / / F 1 ⊗ F ( R ) τ O O F ( R ) ⊗ 1 id ⊗ υ / / F ( R ) ⊗ F 1 τ O O 7. TENSOR FUNCTORS 45 omm ute. (Comm utativit y of one square implies omm utativit y of the other, b eause of the symmetry of the monoidal struture.) No w supp ose that C and D are losed tensor ategories. Let F : C − → D b e a tensor funtor. (W e shall usually omit writing do wn the full triple of data.) Then there is a anonial arro w p R S : F ( S R ) → F S F R , namely the unique solution p to the problem F ( S R ) ⊗ F R τ   p ⊗ id / / F S F R ⊗ F R ε   F ( S R ⊗ R ) F ( ε ) / / F S. This arro w is natural in the v ariables R, S . A rigid funtor is a tensor fun- tor b et w een losed tensor ategories su h that this natural arro w is an iso- morphism. If C and D are b oth rigid, then a tensor funtor F : C − → D is automatially rigid. 1 Example Let f : M → N b e a C ∞ -mapping of smo oth manifolds. This map indues the b ase hange or pul lb ak funtor f ∗ : V ∞ ( N ) − → V ∞ ( M ) . Reall that for x ∈ M the b er ( f ∗ F ) x oinides with { x } × F f ( x ) , sine f ∗ F is b y onstrution a subset of M × F . F or ev ery funtor of sev eral v ariables Φ as in the last example of Setion 6, w e ha v e a anonial natural isomorphism (2) f ∗ Φ( E 1 , . . . , E n ) ≈ Φ( f ∗ E 1 , . . . , f ∗ E n ) . It follo ws at one from the existene of these anonial natural isomorphisms that f ∗ an b e regarded as a tensor funtor (with resp et to the standard tensor struture desrib ed in the last example of the preeding setion). It is also lear from ( 2) that this tensor funtor is rigid. (Of ourse, rigidit y of the pullba k funtor follo ws also indiretly from rigidit y of the ategories V ∞ ( M ) , V ∞ ( N ) .) 3 Denition Let λ : F → G b e a natural transformation of tensor funtors. λ is said to b e tensor-pr eserving, or a morphism of tensor funtors, whenev er the diagrams F R ⊗ F S τ   λ ( R ) ⊗ λ ( S ) / / GR ⊗ GS τ   1 υ   id / / 1 υ   F ( R ⊗ S ) λ ( R ⊗ S ) / / G ( R ⊗ S ) F 1 λ ( 1 ) / / G 1 omm ute. The olletion of all su h λ 's will b e denoted b y Hom ⊗ ( F , G ) . 46 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES 8 Complex T ensor Categories An anti-involution on a C -linear tensor ategory C = ( C , ⊗ ) is an an ti-linear tensor funtor (1) ∗ : C → C , R 7→ R ∗ for whi h there exists a tensor preserving natural isomorphism (2) ι R : R ∗∗ ∼ → R with ι ( R ∗ ) = ι ( R ) ∗ . By xing one su h isomorphism, one obtains a mathematial struture whi h w e all  omplex tensor  ate gory. A morphism of omplex tensor ategories, or  omplex tensor funtor, is obtained b y atta hing, to an ordinary C -linear tensor funtor F , a tensor preserving natural isomorphism (3) ξ R : F ( R ) ∗ ∼ → F ( R ∗ ) su h that the follo wing diagram omm utes F ( R ) ∗∗ ∼ = ∗ / / ∼ = ' ' O O O O O O O F ( R ∗ ) ∗ ∼ = / / F ( R ∗∗ ) F ∼ = w w o o o o o o o F R . (4) 5 Example: the  ate gory of ve tor sp a es If V is a omplex v etor spae, w e let V ∗ denote the spae obtained b y retaining the additiv e struture of V but  hanging the salar m ultipliation in to z v ∗ = ( z v ) ∗ ; the star here indiates that a v etor of V is to b e regarded as one of V ∗ . Sine an y linear map f : V → W will map V ∗ linearly in to W ∗ , w e an also regard f as a linear map f ∗ : V ∗ → W ∗ . Moreo v er, the unique linear map of V ∗ ⊗ W ∗ in to ( V ⊗ W ) ∗ sending v ∗ ⊗ w ∗ 7→ ( v ⊗ w ) ∗ is an isomorphism, and omplex onjugation sets up a linear bijetion b et w een C and C ∗ . This turns v etor spaes in to a omplex tensor ategory with V ∗∗ = V . 6 Example: the  ate gory of ve tor bund les over a manifold By using the pro edure desrib ed in Example 6.11 one an transp ort the omplex tensor struture of the preeding example to the ategory V ∞ ( M ; C ) of smo oth omplex v etor bundles (of lo ally nite rank) o v er a manifold M . Consider a omplex tensor ategory ( C , ⊗ , ∗ ) . By a sesquiline ar form on an ob jet R ∈ Ob( C ) w e mean an y arro w b : R ⊗ R ∗ → 1 . A sesquilinear form b on the ob jet R will b e said to b e Hermitian when the sesquilinear form ˜ b on R , dened as the omp osite (7) R ⊗ R ∗ ∼ = R ∗∗ ⊗ R ∗ ∼ = ( R ⊗ R ∗ ) ∗ b ∗ − − → 1 ∗ ∼ = 1 , 8. COMPLEX TENSOR CA TEGORIES 47 oinides with b itself, i.e. ˜ b = b . Note that one alw a ys has the equalit y ˜ ˜ b = b . Clearly , in the examples ab o v e one reo v ers the familiar notions. Supp ose no w that our omplex tensor ategory is rigid. Then for ea h ob jet R w e an nd another ob jet R ′ , along with arro ws e R : R ′ ⊗ R → 1 and d R : 1 → R ⊗ R ′ , su h that the follo wing omp ositions are iden tities: (8) R ∼ = 1 ⊗ R d R ⊗ R − − − − → R ⊗ R ′ ⊗ R R ⊗ e R − − − − → R ⊗ 1 ∼ = R R ′ ∼ = R ′ ⊗ 1 R ′ ⊗ d R − − − − − → R ′ ⊗ R ⊗ R ′ e R ⊗ R ′ − − − − − → 1 ⊗ R ′ ∼ = R ′ . W e mak e the assumption that for ea h ob jet R w e ha v e seleted one su h triple ( R ∨ , e R , d R ) . Then for ea h R w e obtain a w ell-dened isomorphism q R : R ∨ ∗ ∼ → ( R ∗ ) ∨ , namely the unique arro w q su h that (9) R ∨ ∗ ⊗ R ∗ q ⊗ R ∗ − − − → R ∗ ∨ ⊗ R ∗ e R ∗ − − → 1 equals R ∨ ∗ ⊗ R ∗ ∼ = ( R ∨ ⊗ R ) ∗ ( e R ) ∗ − − − → 1 ∗ ∼ = 1 . W e sa y that a sesquilinear form b on R is nonde gener ate, when the arro ws b - : R → R ∗ ∨ and b - : R ∗ → R ∨ , dened as the unique solutions to (10) R ⊗ R ∗ b - ⊗ R ∗ − − − − → R ∗ ∨ ⊗ R ∗ e R ∗ − − → 1 equals b and b equals R ⊗ R ∗ R ⊗ b - − − − → R ⊗ R ∨ ∼ = R ∨ ⊗ R e R − − → 1 , are isomorphisms. If b is Hermitian then b - is an isomorphism if and only if so is b - . Indeed, the diagrams R ∗ b -   ( ˜ b - ) ∗ / / R ∗ ∨ ∗ R ∗∗ ∼ =   ( ˜ b ) - ∗ / / R ∨ ∗ q R ≈   R ∨ ∼ = / / R ∨ ∗∗ ≈ ( q R ) ∗ O O R b - / / R ∗ ∨ (11) omm ute for an arbitrary sesquilinear form b on R . Let ( C , ⊗ , ∗ ) b e a omplex tensor ategory . By a des ent datum on an ob jet R ∈ Ob( C ) w e mean an isomorphism µ : R ∼ → R ∗ su h that the omp osition R µ ≈ R ∗ µ ∗ ≈ R ∗∗ ∼ = R equals id R . W e let R C denote the ategory whose ob jets are the pairs ( R, µ ) onsisting of an ob jet R of C and a desen t datum µ on R and whose morphisms a : ( R, µ ) → ( R ′ , µ ′ ) are the morphisms a : R → R ′ su h that µ ′ · a = a ∗ · µ . Note that R C is naturally an R -linear ategory; moreo v er, there is an ob vious indued tensor struture, whi h turns R C in to an R -linear tensor ategory . As an example of this onstrution, observ e that one has an ob vious equiv- alene of real tensor ategories b et w een V e c R and R ( V e c C ) : in one diretion, 48 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES to an y real v etor spae V one an assign the pair ( C ⊗ V , z ⊗ v 7→ z ⊗ v ) ; on v ersely , an y desen t datum µ : U ∼ → U ∗ on a omplex v etor spae U determines the real subspae U µ ⊂ U of µ -in v arian t v etors. More generally , one has analogous equiv alenes of real tensor ategories b et w een V ∞ ( M ; R ) and R  V ∞ ( M ; C )  , R ∞ ( M ; R ) and R  R ∞ ( M ; C )  and so on. Notie that an y omplex tensor funtor F : C → D will indue a linear tensor funtor R F : R C → R D . By using the fat that the isomorphism R ⊕ R ∗ ≈ ( R ⊕ R ∗ ) ∗ is a desen t datum on R ⊕ R ∗ for ea h R , one an easily sho w that setting ˆ λ ( R, µ ) = λ ( R ) denes a bijetion (12) Hom ⊗ , ∗ ( F , G ) ∼ → Hom ⊗ ( R F , R G ) , λ 7→ ˆ λ b et w een the self- onjugate tensor preserving transformations F → G and the tensor preserving transformations R F → R G , for an y omplex tensor funtors F , G : C → D . 9 Review of Groups and T annak a Dualit y Throughout the presen t setion, k is a xed eld. W e let V e c k denote the ategory of nite dimensional v etor spaes o v er k ; this is a rigid ab elian linear tensor ategory with End( 1 ) = k . All k -algebras are understo o d to b e omm utativ e. Let G = Sp ec A b e an ane group s heme o v er k , ie a group ob jet in the ategory Sch ( k ) of (ane) s hemes o v er k (s hemes endo w ed with a morphism G → Sp ec k , in other w ords with A a k -algebra). This means that w e ha v e morphisms of s hemes: m ultipliation G × k G → G , unit elemen t Sp ec k → G , in v erse G → G (o v er k ), satisfying the usual group la ws; equiv alen tly , one is giv en morphisms of k -algebras ∆ : A → A ⊗ k A , ε : A → k and σ : A → A (the om ultipliation, ounit and oin v erse maps) su h that the follo wing axioms hold: oasso iativit y , oiden tit y A ∆   ∆ / / A ⊗ A id ⊗ ∆   A ≈ # # F F F F F F F F F ∆ / / A ⊗ A ε ⊗ id   A ⊗ A ∆ ⊗ id / / A ⊗ A ⊗ A k ⊗ A and oin v erse A ε   ∆ / / A ⊗ A ( σ , id )   k   / / A. If A is a nitely generated k -algebra, w e sa y that G is algebr ai or that it is an algebr ai gr oup . One denes a  o algebr a o v er k to b e a v etor spae C o v er k endo w ed with linear maps ∆ : C → C ⊗ k C and ε : C → k satisfying the 9. REVIEW OF GR OUPS AND T ANNAKA DUALITY 49 oasso iativit y and oiden tit y axioms. A (right)  omo dule o v er a oalgebra C is a v etor spae V o v er k together with a linear map ρ : V → V ⊗ C su h that the follo wing diagrams omm ute V ρ   ρ / / V ⊗ C ρ ⊗ ∆   V ≈ # # G G G G G G G G G ρ / / V ⊗ C id ⊗ ε   V ⊗ C ρ ⊗ id / / V ⊗ C ⊗ C V ⊗ k F or example, ∆ denes a C -omo dule struture on C itself. An ane group s heme G = Sp ec A an b e regarded as a funtor G : k - alg − → grou ps of k -algebras with v alues in to groups (f. also W aterhouse 1979 [35 ℄): G ( R ) = Hom k - a l g ( A, R ) , for ev ery k -algebra R, so in partiular, when R = k , G ( k ) = Hom k - a l g ( A, k ) = Hom S c h ( k ) (Sp ec k , G ) is the set of losed k -rational p oin ts of G . The group struture on G ( R ) is obtained as follo ws: for s, t ∈ G ( R ) , the pro dut s · t , the neutral elemen t and the in v erse s − 1 are resp etiv ely dened as A ∆ − → A ⊗ k A s ⊗ k t − − → R ⊗ k R m ult. − − − → R, A ε − → k unit − − → R, A σ − → A s − → R. Let C b e a rigid ab elian k -linear tensor ategory , and let ω : C − → V e c k b e an exat faithful k -linear tensor funtor. Then one an dene a funtor Aut ⊗ ( ω ) : k - alg − → groups , as follo ws. F or R a k -algebra, there is a anonial tensor funtor φ R : V e c k − → Mod R , V 7→ V ⊗ k R in to the ategory of R -mo dules (this is an ab elian tensor ategory with End( 1 ) = R , but in general it will not b e rigid b eause not all R -mo dules will b e reexiv e). If F , G : C − → V e c k are tensor funtors, then w e an dene Hom ⊗ ( F , G ) to b e the funtor of k -algebras Hom ⊗ ( F , G )( R ) = Hom ⊗ ( φ R ◦ F, φ R ◦ G ) . Th us Aut ⊗ ( ω )( R ) onsists of families ( λ X ) , X ∈ Ob( C ) where λ X is an R - linear automorphism of ω ( X ) ⊗ k R su h that λ X 1 ⊗ X 2 = λ X 1 ⊗ λ X 2 , λ 1 is the iden tit y mapping of R , and ω ( X ) ⊗ R ω ( a ) ⊗ i d   λ X / / ω ( X ) ⊗ R ω ( a ) ⊗ i d   ω ( Y ) ⊗ R λ Y / / ω ( Y ) ⊗ R 50 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES omm utes for ev ery arro w a : X → Y in C . In the sp eial ase where C = R ( G ; k ) for some ane group s heme G o v er k , and ω is the forgetful funtor R ( G ; k ) − → V e c k , it is lear that ev ery elemen t of G ( R ) denes an elemen t of Aut ⊗ ( ω )( R ) . One has the follo wing result 1 Prop osition The natural transformation G → Aut ⊗ ( ω ) (of funtors of k -algebras with v alues in to groups) is an isomorphism. 2 Theorem Let C b e a rigid ab elian tensor ategory su h that End( 1 ) = k , and let ω : C − → V e c k b e an exat faithful k -linear tensor funtor. Then i) the funtor Aut ⊗ ( ω ) of k -algebras is represen table b y an ane group s heme G ; ii) ω denes an equiv alene of tensor ategories C − → R ( G ; k ) . 3 Denition A neutr al T annakian  ate gory o v er k is a rigid ab elian k - linear tensor ategory C for whi h there exists an exat faithful k -linear tensor funtor ω : C − → V e c k . An y su h funtor is said to b e a br e funtor for C . 10 A T e hnial Lemma on Compat Groups Throughout the presen t setion, let V e c denote the omplex tensor ategory of omplex v etor spaes of nite dimension (see Note 8.5). Let C b e an arbitrary additiv e omplex tensor ategory . Let F : C → V e c b e a omplex tensor funtor. Moreo v er, let H b e a top ologial group. Supp ose w e are giv en a homomorphism of monoids (1) π : H → End ⊗ , ∗ ( F ) . W e shall sa y that π is  ontinuous if for ev ery ob jet R ∈ Ob( C ) the indued represen tation (2) π R : H → End  F ( R )  dened b y h 7→ π R ( h ) ≡ π ( h )( R ) is on tin uous. 3 Prop osition (T e hnial Lemma.) Let C , F and H b e as ab o v e. Supp ose in addition that H is a ompat Lie group. Finally , let π : H → End ⊗ , ∗ ( F ) b e a on tin uous homomorphism. Assume the follo wing ondition holds: 10. A TECHNICAL LEMMA ON COMP A CT GR OUPS 51 (*) for an y ouple of ob jets R, S ∈ Ob( C ) and for ea h homomorphism A : F ( R ) → F ( S ) of the orresp onding H -mo dulesin other w ords, for ea h C -linear map A su h that the diagram F ( R ) A   π R ( h ) / / F ( R ) A   F ( S ) π S ( h ) / / F ( S ) (4) omm utes ∀ h ∈ H there is an arro w R a − → S su h that A = F ( a ) . Then π is surjetiv e; in partiular, End ⊗ , ∗ ( F ) = Aut ⊗ , ∗ ( F ) is neessarily a group. Pro of Put K def = Ker π ⊂ H . This is a losed normal subgroup, b eause it oinides with the in tersetion T Ker π X o v er all ob jets X of C . On the quotien t G def = H /K there is a unique (ompat) Lie group struture su h that the quotien t homomorphism H ։ G is a Lie group homomorphism. Ev ery π X an indieren tly b e though t of as a on tin uos represen tation of H or a on tin uous represen tation of G , and ev ery linear map A : F ( X ) → F ( Y ) is a morphism of G -mo dules if and only if it is a morphism of H -mo dules. Being on tin uous, ev ery π X is also smo oth. W e laim there exists an ob jet R of C su h that the orresp onding π R is faithful as a represen tation of G . This an b e seen in a ompletely standard w a y , f. for instane Br ö ker and tom Die k (1985), pp. 136137; nonetheless, in the presen t more abstrat situation it will b e useful to ha v e a lo ok at the argumen t in detail an yw a y . The `No etherian' prop ert y of the ompat Lie group G allo ws us to nd X 1 , . . . , X ℓ ∈ Ob( C ) with the prop ert y that (5) Ker π X 1 ∩ · · · ∩ Ker π X ℓ = { e } as represen tations of G , where e denotes the neutral elemen t. Then, setting R def = X 1 ⊕ · · · ⊕ X ℓ , the represen tation π R will b e faithful b eause of the existene of an isomorphism of G -mo dules (6) F ( X 1 ⊕ · · · ⊕ X ℓ ) ≈ F ( X 1 ) ⊕ · · · ⊕ F ( X ℓ ) . (The existene of su h isomorphisms follo ws from the remark that a natural transformation of additiv e funtors is additiv e: for instane, when ℓ = 2 , F X F i X   π ( h )( X ) / / F X F i X   F X ⊕ F Y ≈   ⇒ π X ( h ) ⊕ π Y ( h ) / / F X ⊕ F Y ≈   F ( X ⊕ Y ) π ( h )( X ⊕ Y ) / / F ( X ⊕ Y ) F ( X ⊕ Y ) π X ⊕ Y ( h ) / / F ( X ⊕ Y ) F Y F i Y O O π ( h )( Y ) / / F Y F i Y O O 52 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES sho ws that the anonial isomorphism F ( X ) ⊕ F ( Y ) ≈ F ( X ⊕ Y ) is also an isomorphism of H -mo dules or, equiv alen tly , G -mo dules.) It follo ws that the G -mo dule F ( R ) is a tensor generator for the omplex tensor ategory R ( G ; C ) of on tin uous nite dimensional omplex G -mo dules. Indeed, ev ery irreduible su h G -mo dule V em b eds as a submo dule of some tensor p o w er F ( R ) ⊗ k ⊗ ( F ( R ) ∗ ) ⊗ ℓ (see for instane Br ö ker and tom Die k, 1985 ); sine b y assumption ea h π ( h ) is a self-onjugate tensor preserving natural transformation, this tensor p o w er will b e naturally isomorphi to F  R ⊗ k ⊗ ( R ∗ ) ⊗ ℓ  as a G -mo dule and hene, as a onsequene of the existene of the G -mo dule isomorphisms ( 6 ), for ea h ob jet V of R ( G ; C ) there will b e some ob jet X of C su h that V em b eds in to F ( X ) as a submo dule. Next, onsider an arbitrary natural transformation λ ∈ End( F ) . Let X b e an ob jet of C and let V ⊂ F X b e a submo dule. The  hoie of a omplemen t to V in F X determines a mo dule endomorphism P : F X → V ֒ → F X whi h, b y ondition (*), omes from some endomorphism X p − → X ∈ C . Therefore F X P   λ ( X ) / / F X P   F X λ ( X ) / / F X (7) omm utes and, onsequen tly , λ ( X ) maps V in to itself. I will usually omit X from the notation and simply write λ V : V → V for the linear map that λ ( X ) indues on V b y restrition. Giv en an y other submo dule W ⊂ F Y and an y mo dule homomorphism B : V → W , the diagram V B   λ V / / V B   W λ W / / W (8) is neessarily omm utativ e. T o pro v e this, extend the giv en homomorphism B : V → W to a homomorphism B ′ : F X → F Y (for instane, b y  ho osing a omplemen t to V in F X and then b y taking the omp osite map F X → V B − → W ֒ → F Y ) and then argue as b efore, b y in v oking the assumption (*). Next, w e dene an isomorphism of omplex algebras (9) θ : End( F ) ∼ → End( ω G ) so that the follo wing diagram omm utes H pr   π / / End( F ) θ   G π G / / End( ω G ) , (10) 10. A TECHNICAL LEMMA ON COMP A CT GR OUPS 53 where ω G : R ( G ; C ) → V e c is the standard forgetful funtor (whi h to an y G -mo dule asso iates the underlying omplex v etor spae) and π G ( g ) is the natural transformation  7→ π G ( g )(  ) ≡  ( g ) . Giv en a mo dule V , there exists an ob jet X of C together with an em b edding V ֒ → F X , so w e ma y dene θ ( λ )( V ) to b e the restrition of λ ( X ) to V (this mak es sense in view of the ab o v e remarks). Of ourse, it is neessary to  he k that θ is w ell-dened. Supp ose w e are giv en t w o ob jets X , Y ∈ Ob( C ) , along with G -mo dule em b eddings of V in to F X , F Y resp etiv ely . Sine it is alw a ys p ossible to em b ed ev erything equiv arian tly in to F ( X ⊕ Y ) and sine doing this do es not aet the indued λ V 's, it will b e no loss of generalit y to assume that X = Y . Let W , W ′ ⊂ F X b e the submo dules orresp onding to the t w o dieren t em b eddings of V in to F X . Then b y our remark ( 8) there is a omm utativ e diagram V ≈ / / W λ W / / ≈   W ≈   ≈ − 1 / / V V ≈ / / W ′ λ W ′ / / W ′ ≈ − 1 / / V , (11) whi h sho ws that the t w o dieren t em b eddings preisely determine the same linear endomorphism of V . Clearly , (8) implies that θ ( λ ) ∈ End( ω G ) . F or µ ∈ End( ω G ) and X ∈ C , put µ F ( X ) = µ ( F X ) ; then µ F ∈ End( F ) and θ ( µ F ) = µ , b eause of the existene of em b eddings V ֒ → F X and b eause of naturalit y of µ : hene θ is surjetiv e. The latter map is also injetiv e sine λ ( X ) = θ ( λ )( F X ) . It is straigh tforw ard to  he k that the diagram (10) omm utes. No w, to onlude the pro of, it will b e enough to sho w that θ indues a bijetion b et w een End ⊗ , ∗ ( F ) and End ⊗ , ∗ ( ω G ) = T ( G ) , b eause then from (10 ) w e get at one the follo wing omm utativ e square H pr   π / / End ⊗ , ∗ ( F ) θ ≈   G π G / / T ( G ) , (12) where the map on the b ottom is a bijetion (b y the lassial T annak a dualit y theorem for ompat groups), whene surjetivit y of π is eviden t. F or instane, supp ose λ ∈ End ⊗ ( F ) and let V and W b e G -mo dules that admit equiv arian t em b eddings V ֒ → F X and W ֒ → F Y for some X , Y ∈ Ob( C ) . Sine w e are dealing with nite dimensional spaes, V ⊗ W ֒ → F X ⊗ F Y ∼ = F ( X ⊗ Y ) will b e also an em b edding of G -mo dules. Then, b y the denition of θ and the assumption that λ is tensor preserving, w e see 54 CHAPTER I I. THE LANGUA GE OF TENSOR CA TEGORIES that the diagram F ( X ⊗ Y ) λ ( X ⊗ Y ) / / F ( X ⊗ Y ) V ⊗ W ?  O O λ V ⊗ λ W / / V ⊗ W ?  O O (13) m ust omm ute. This sho ws that θ ( λ )( V ⊗ W ) = θ ( λ )( V ) ⊗ θ ( λ )( W ) . The rev erse diretion is straigh tforw ard. q.e.d. The argumen t that w e used ab o v e in order to nd the tensor generator R admits the follo wing generalization to the non-ompat ase. Let C and F b e as in the statemen t of the preeding prop osition. 14 Prop osition Let G b e a Lie group. Supp ose that (15) π : G − → Aut( F ) is a faithful on tin uous homomorphismin other w ords, a on tin uous homomorphism su h that for ea h g 6 = e ∈ G there exists an ob jet X in C with π X ( g ) 6 = id F X . Then there exists an ob jet R ∈ Ob( C ) for whi h Ker π R is a disrete subgroup of G or, equiv alen tly , for whi h the on tin uous represen tation (16) π R : G → GL ( F R ) is faithfuli.e. injetiv eon some op en neigh b ourho o d of e . Pro of Let X b e an arbitrary ob jet of C . Then K def = Ker π X is a losed Lie subgroup of G . The onneted omp onen t K e of e in K is also a losed Lie subgroup of G ; in partiular, the inlusion map K e ֒ → G is an em b edding of Lie groups (that is, a Lie subgroup and an em b edding of manifolds). So, if Y is another ob jet, the on tin uous represen tation π Y : G → GL ( F Y ) indues b y restrition a on tin uous represen tation of K e . The k ernel D def = K e ∩ Ker π Y is a losed Lie subgroupin partiular, a losed submanifoldof K e again. Th us, either dim D < dim K e or D = K e , b eause K e is onneted. Sine π is faithful, when dim K e > 0 w e an alw a ys nd some ob jet Y su h that D $ K e . Then it follo ws that for ea h X ∈ Ob( C ) one an alw a ys nd another ob jet Y su h that the submanifold Ker π X ⊕ Y has dimension stritly smaller than the dimension of Ker π X , unless dim Ker π X = 0 . Hene an indutiv e argumen t using additivit y of the ategory C will yield an ob jet R su h that dim Ker π R = 0 i.e. Ker π R is disrete, as on tended. q.e.d. Chapter I I I Represen tation Theory Revisited In the presen t  hapter w e in tro due our language of smo oth staks of (addi- tive, r e al or  omplex) tensor  ate gories, or briey smo oth (r e al or  omplex) tensor staks. W e prop ose this language as the general foundational frame- w ork for the theory of represen tations of Lie group oids. Some gener al  onventions. W e use the expressions `smo oth' and `of lass C ∞ ' as synon yms. The apital letters X , Y and Z stand for manifolds of lass C ∞ , the orresp onding lo w er-ase letters x, x ′ , . . . , y , et. denote p oin ts on these manifolds. C ∞ X indiates the sheaf of smo oth funtions on X (w e usually omit the subsript). Shea v es of C ∞ X -mo dules will also b e referred to as she aves of mo dules over X . F or pratial purp oses, w e need to onsider manifolds whi h are p ossibly neither Hausdor nor paraompat. 11 The Language of Fibred T ensor Categories Fibr e d tensor  ate gories. Fibred tensor ategories will b e denoted b y means of apital Gothi t yp e v ariables. Of ourse, as in  8, w e ha v e to distinguish b et w een the notions of real and omplex bred tensor ategory . W e do the omplex v ersion; the real ase is en tirely analogous. A bred omplex tensor ategory T assigns, to ea h smo oth manifold X , an additiv e omplex tensor ategory (1) T ( X ) =  T ( X ) , ⊗ X , 1 X , ∗ X  or  T ( X ) , ⊗ , 1 , ∗  for shortomitting subsripts when they are lear from the on textand, to ea h smo oth mapping X f − → Y , a omplex tensor funtor (2) f ∗ : T ( Y ) − → T ( X ) alled  pull-ba k along f  . Moreo v er, for ea h pair of omp osable smo oth maps X f − → Y g − → Z and for ea h manifold X , an y bred omplex tensor 55 56 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED ategory pro vides self-onjugate tensor preserving natural isomorphisms (3) ( δ : f ∗ ◦ g ∗ ∼ → ( g ◦ f ) ∗ ε : Id ∼ → id X ∗ . These are altogether required to mak e the follo wing diagrams omm ute f ∗ g ∗ h ∗ δ · h ∗   f ∗ δ / / f ∗ ( hg ) ∗ δ   id X ∗ f ∗ δ   f ∗ f ∗ ε   v v v v v v v v v v v v v v v v v v v v ε · f ∗ o o ( g f ) ∗ h ∗ δ / / ( hg f ) ∗ f ∗ f ∗ id Y ∗ . δ o o (4) This is all of the mathematial data w e need to in tro due in order to sp eak ab out smo oth tensor sta ks and, later on, represen tations of Lie group oids. All the required onepts anand willb e dened in terms of the giv en ategorial struture T , i.e.  anoni al ly. W e no w explain ho w. Smo oth tensor presta ks Throughout the presen t subsetion w e let P denote a bred omplex tensor ategory , xed one and for all. Notation. F or i U : U ֒ → X the inlusion of an op en subset, w e shall put E | U = i U ∗ E and a | U = i U ∗ a for an y ob jet E and morphism a of the ategory P ( X ) . (More generally , w e shall adopt this abbreviation for the inlusion i S : S ֒ → X of an y submanifold. ) F or an y pair of ob jets E , F ∈ Ob P ( X ) , w e let H om P X ( E , F ) denote the presheaf of omplex v etor spaes o v er X dened b y (5) U 7→ Hom P ( U ) ( E | U , F | U ) , with the ob vious restrition maps a 7→ j ∗ a orresp onding to the inlusions j : V ֒ → U of op en subsets. (T o b e preise, restrition along j sends a to the unique morphism E | V → F | V whi h orresp onds to j ∗ a up to the anonial isomorphisms j ∗ ( E | U ) ∼ = E | V and j ∗ ( F | U ) ∼ = F | V of (3).) No w, the requiremen t that P b e a pr estak means exatly that an y su h presheaf is in fat a she af; in partiular, it en tails that one an glue an y family of ompatible lo al morphisms o v er X . T w o sp eial ases will b e of partiular in terest to us: the sheaf Γ E = H om P X ( 1 , E ) , to b e referred to as the she af of smo oth se tions of E ∈ Ob P ( X ) , and the sheaf E ∨ = H om P X ( E , 1 ) , to b e referred to as the she af dual of E . F or an y op en subset U , the elemen ts of Γ E ( U ) will b e of ourse referred to as the smo oth se tions of E o v er U ; it is p erhaps useful to p oin t out that it mak es sense, for smo oth setions o v er U , to tak e linear om binations with omplex o eien ts, b eause Γ E ( U ) has a anonial v etor spae struture. 11. THE LANGUA GE OF FIBRED TENSOR CA TEGORIES 57 Sine a morphism a : E → F in P ( X ) yields a morphism Γ a : Γ E → Γ F of shea v es of omplex v etor spaes o v er X (b y omp osing 1 | U → E | U a | U − − → F | U ), w e obtain a anonial funtor (6) Γ = Γ X : P ( X ) − → { shea v es o f C X - mo dules } , where C X denotes the onstan t sheaf o v er X of v alue C . (Note that a sheaf of omplex v etor spaes o v er a top ologial spae X is exatly the same thing as a sheaf of C X -mo dules.) This funtor is ertainly linear. Moreo v er, there is an eviden t w a y to mak e it a pseudo-tensor funtor of the tensor ategory  P ( X ) , ⊗ X , 1 X  in to the ategory of shea v es of C X -mo dules (with the standard tensor struture). In detail, a natural transformation τ E ,F : Γ X E ⊗ C X Γ X F → Γ X ( E ⊗ F ) arises, in the ob vious manner, from the lo al pairings (7) Γ E ( U ) × Γ F ( U ) − → Γ ( E ⊗ F )( U ) ( 1 | U a − → E | U , 1 | U b − → F | U ) 7→ 1 | U ∼ = 1 | U ⊗ 1 | U a ⊗ b − − → E | U ⊗ F | U ∼ = ( E ⊗ F ) | U (whi h are bilinear with resp et to lo ally onstan t o eien ts), and a morph- ism υ : C X → Γ X 1 an b e easily dened as follo ws (8) 8 > > < > > : lo cally constan t complex v alued functions o n U 9 > > = > > ; − → Γ 1 ( U ) t : U → C 7→ t · 1 U : 1 | U → 1 | U (where 1 U = id : 1 | U → 1 | U is the unit y onstan t setion); the op eration of m ultipliation b y t in ( 7) and (8) is w ell-dened b eause t is a omplex onstan t, at least lo ally . It is easy to  he k that these morphisms of shea v es mak e all the diagrams in the denition of a tensor funtor omm ute. Note that for X = ⋆ , where ⋆ is the one-p oin t manifold, one has the standard iden tiation { shea v es of C ⋆ - mo dules } = { complex v ector spaces } of omplex tensor ategories. One ma y therefore regard, for X = ⋆ , the funtor (6 ) as a linear pseudo-tensor funtor (9) P ( ⋆ ) − → { complex v ector spaces } . It will b e on v enien t to ha v e a short notation for this; making the ab o v e iden tiation of ategories expliit, w e put, for all ob jets E ∈ Ob P ( ⋆ ) , (10) E ∗ = ( Γ ⋆ E )( ⋆ ) (so this is a omplex v etor spae), and do the same for morphisms. No w, as a part of the denition of the general notion of smo oth tensor sta k, we ask that the fol lowing  ondition b e satise d: the morphism of shea v es (8) is an isomorphisms for X = ⋆ . Let us reord an immediate onsequene of this requiremen t: there is a  anoni al isomorphism of omplex v etor spaes (11) C ∼ → 1 ∗ . 58 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED 12 Note When dealing with the ase of bred omplex tensor ategories, one also has a natural morphism of shea v es of mo dules o v er X (13) ( Γ X E ) ∗ − → Γ X ( E ∗ ) dened b y means of the an ti-in v olution and the ob vious related anonial isomorphisms. Sine ζ ∗∗ = ζ (up to anonial isomorphism), it follo ws at one that ( 13) is a natural isomorphism for an arbitrary omplex tensor presta k; in fat, (13) is an isomorphism of pseudotensor funtors viz. it is ompatiblein the sense of 7with the natural transformations ( 7) and (8). Beause of these onsiderations, w e will not need to w orry ab out omplex struture in our subsequen t disussion of axioms in 15. Notation. (Fibres of an ob jet) Besides the fundamen tal notion of  sheaf of smo oth setions  w e are no w able to in tro due a seond one, that of  bre at a p oin t  . Namely , giv en an ob jet E ∈ Ob P ( X ) , w e dene the br e of E at x to b e the nite dimensional omplex v etor spae E x = ( x ∗ E ) ∗ ; w e use the same name for the p oin t x and for the (smo oth) mapping ⋆ → X, ⋆ 7→ x , so that x ∗ is just the ordinary notation (2) for the pull-ba k, x ∗ E b elongs to P ( ⋆ ) and w e an apply our notation (10). Similarly , whenev er a : E → F is a morphism in P ( X ) , w e let a x : E x → F x denote the linear map ( x ∗ a ) ∗ . Sine - 7→ ( - ) x is b y onstrution the omp osite of t w o omplex pseudo-tensor funtors, it ma y itself b e regarded as a omplex pseudo-tensor funtor. If in partiular w e apply this to a lo al smo oth setion ζ ∈ Γ E ( U ) and mak e use of the anonial iden tiation (11 ), w e get, for u in U , a linear map (14) C ∼ → ( 1 ⋆ ) ∗ ∼ = ( u ∗ 1 | U ) ∗ ( u ∗ ζ ) ∗ − − − → ( u ∗ E | U ) ∗ ∼ = ( u ∗ E ) ∗ = E u , whi h orresp onds to a v etor ζ ( u ) ∈ E u (the image of the unit y 1 ∈ C ) to b e alled the value of ζ at u . One has the in tuitiv e form ula (15) a u · ζ ( u ) = [ Γ a ( U ) ζ ]( u ) . Notie also that the v etors ζ ( u ) ⊗ η ( u ) and ( ζ ⊗ η )( u ) orresp ond to one another in the anonial linear map E u ⊗ F u → ( E ⊗ F ) u (w e ma y state this lo osely b y sa ying they are equal). W e ha v e not explained y et what w e mean when w e sa y that a tensor presta k is  smo oth  . This w as not neessary b efore b eause all w e ha v e said so far do es not dep end on that sp ei prop ert y . Ho w ev er, from this preise momen t w e b egin to dev elop systematially onepts whi h, ev en in order to b e dened, presupp ose the smo othness of the tensor presta k, so it b eomes neessary to ll the gap. Consider the tensor unit 1 ∈ Ob P ( X ) and let x b e an y p oin t. There is a anonial isomorphism C ∼ = 1 x analogous to (11 ), namely the omp osite 11. THE LANGUA GE OF FIBRED TENSOR CA TEGORIES 59 C ∼ = ( 1 ⋆ ) ∗ ∼ = ( x ∗ 1 ) ∗ = 1 x . This iden tiation allo ws us to dene a  anoni al homomorphism of omplex algebras (16) End P ( X ) ( 1 ) − → { functions X → C } , e 7→ ˜ e b y putting ˜ e ( x ) = the omplex onstan t su h that the linear map salar m ultipliation b y ˜ e ( x )  (of C in to itself ) orresp onds to e x : 1 x → 1 x under the linear isomorphism C ∼ = 1 x . W e shall sa y that the tensor presta k P is smo oth if the homomorphism (16) determines a one-to-one orresp ondene on to the subalgebra of smo oth funtions on X (17) End P ( X ) ( 1 ) ∼ = C ∞ ( X ) . A rst onsequene of the smo othness of P is the p ossibilit y to endo w ea h spae Hom P ( X ) ( E , F ) with a C ∞ ( X ) -mo dule struture, anonial and ompatible with the already dened op eration of m ultipliation b y lo ally onstan t funtions. Indeed, the natural ation (18) End P ( X ) ( 1 ) × Hom P ( X ) ( E , F ) − → Hom P ( X ) ( E , F ) , ( e, a ) 7→ E ∼ = 1 ⊗ E e ⊗ a − − → 1 ⊗ F ∼ = F turns Hom P ( X ) ( E , F ) in to a left End P ( X ) ( 1 ) -mo dule, hene w e an use the iden tiation of C -algebras (17) to mak e Hom P ( X ) ( E , F ) a C ∞ ( X ) -mo dule; in short, the mo dule m ultipliation an b e written as ( ˜ e, a ) 7→ e ⊗ a . A ordingly , H om P X ( E , F )( U ) = Hom P ( U ) ( E | U , F | U ) inherits a anonial struture of C ∞ ( U ) -mo dule, for ea h op en subset, and one v eries at one that this mak es H om P X ( E , F ) a sheaf of C ∞ X -mo dules. Of ourse, the remark applies in partiular to an y sheaf of `smo oth' setions Γ X E , partly justifying the terminology; moreo v er, one readily sees that an y morphism a : E → F in the ategory P ( X ) indues a morphism Γ X a : Γ X E → Γ X F of shea v es of C ∞ X -mo dules. So w e get a C ∞ ( X ) -linear funtor (19) P ( X ) − → { sheav es of C ∞ X - mo dules } , still denoted b y Γ X . (Notie that b oth ategories ha v e Hom -sets enri hed with a C ∞ ( X ) -mo dule struture 1 . The C ∞ ( X ) -linearit y of the funtor amoun ts b y denition to the C ∞ ( X ) -linearit y of all the maps Hom P ( X ) ( E , F ) → Hom C ∞ X ( Γ X E , Γ X F ) , a 7→ Γ X a . ) If one also tak es in to aoun t the tensor struture then the pro ess of  upgrading  the funtor (6 ) an b e pursued further b y observing that the op erations desrib ed in (7), (8) ma y no w b e used to dene morphisms of shea v es of C ∞ X -mo dules (20) ( τ : Γ X E ⊗ C ∞ X Γ X F → Γ X ( E ⊗ F ) , υ : C ∞ X → Γ X 1 ; 1 Su h that the omp osition of morphisms is C ∞ ( X ) -bilinear. 60 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED the morphism τ = τ E ,F is natural in the v ariables E , F and, along with υ , mak es (19) a pseudo-tensor funtor of the tensor ategory P ( X ) in to the tensor ategory of shea v es of C ∞ X -mo dules. This is loser than (6) to b eing a tensor funtor, in that the morphism υ is eviden tly an isomorphism of shea v es of C ∞ X -mo dules. Consider next a smo oth mapping of manifolds f : X → Y . Supp ose that U ⊂ X and V ⊂ Y are op en subsets with f ( U ) ⊂ V , and let f U denote the indued mapping of U in to V . F or an y ob jet F of the ategory P ( Y ) , w e obtain a orresp ondene of lo al smo oth setions (21) ( Γ Y F )( V ) − → Γ X ( f ∗ F )( U ) , η 7→ η ◦ f b y putting η ◦ f equal b y denition to the omp osite (22) 1 | U ∼ = ( f ∗ 1 ) | U ∼ = f ∗ U ( 1 | V ) f ∗ U ( η ) − − − → f ∗ U ( F | V ) ∼ = ( f ∗ F ) | U . One easily v eries that for U xed and V v ariable, the maps ( 21 ) form an indutiv e system indexed o v er the inlusions of neigh b ourho o ds V ⊃ V ′ ⊃ f ( U ) , and that ev en tually they indue a morphism of shea v es of C ∞ X -mo dules (23) f ∗ ( Γ Y F ) − → Γ X ( f ∗ F ) , where f ∗ ( Γ Y F ) is the ordinary pull-ba k in the sense of shea v es of mo dules o v er smo oth manifolds. It is also lear that the morphism (23) is natural in F , and also a morphism of pseudo-tensor funtors (in other w ords, it is tensor preserving). T o onlude, let us giv e some motiv ation for the notation  η ◦ f  . There is an ob vious anonial isomorphism of v etor spaes (24) ( f ∗ F ) x = ( x ∗ f ∗ F ) ∗ ∼ = ( f ( x ) ∗ F ) ∗ = F f ( x ) . No w, w e ha v e the t w o v etors η ( f ( x )) ∈ F f ( x ) and ( η ◦ f )( x ) ∈ ( f ∗ F ) x , and y ou an easily  he k that they orresp ond to one another in the ab o v e isomorphism. W e an state this lo osely as (25) ( η ◦ f )( x ) = η ( f ( x )) . The last expression eviden tly justies our notation. 12 Smo oth T ensor Sta ks It will b e on v enien t to regard the op en o v erings of a manifold X as smo oth mappings on to X . This an b e made preise as follo ws. Borro wing some standard terminology from algebrai geometers, w e shall sa y that a smo oth mapping p : X ′ → X is at, if it is surjetiv e and it restrits to an op en em b edding p U ′ : U ′ ֒ → X on ea h onneted omp onen t U ′ of X ′ ; w e ma y 12. SMOOTH TENSOR ST A CKS 61 think of p as o difying a ertain op en o v ering of X , indexed b y the set of onneted omp onen ts of X ′ . A r enement of X ′ p − → X will b e obtained b y omp osing p ba kw ards with another at mapping X ′′ p ′ − → X ′ . The funda- men tal prop ert y of at mappings is that they an b e pulled ba k, preserving atness, along an y smo oth map: preisely , for an y Y f − → X the pull-ba k (1) Y × X X ′ =  ( y , x ′ ) : f ( y ) = p ( x ′ )  will mak e sense in the ategory of C ∞ -manifolds and the rst pro jetion pr 1 : Y × X X ′ → Y will b e a at mapping. P artiularly relev an t is the ase where f is also a at mapping, leading to the standard ommon renemen t for f and p . Some standard abbreviations. F or an y at mapping p : X ′ → X , let (2) X ′′ = X ′ × X X ′ =  ( x ′ 1 , x ′ 2 ) : p ( x ′ 1 ) = p ( x ′ 2 )  , with the t w o pro jetions p 1 , p 2 : X ′′ → X ′ ; and the triple bred pro dut (3) X ′′′ = X ′ × X X ′ × X X ′ =  ( x ′ 1 , x ′ 2 , x ′ 3 ) : p ( x ′ 1 ) = p ( x ′ 2 ) = p ( x ′ 3 )  with its pro jetions p 12 , p 23 , p 13 : X ′′′ → X ′′ resp. giv en b y ( x ′ 1 , x ′ 2 , x ′ 3 ) 7→ ( x ′ 1 , x ′ 2 ) and so forth. A des ent datum for a smo oth omplex tensor presta k P , r elative to the at mapping p : X ′ → X , will b e a pair ( E ′ , θ ) onsisting of an ob jet E ′ ∈ P ( X ′ ) and an isomorphism θ : p 1 ∗ E ′ ∼ → p 2 ∗ E ′ in P ( X ′′ ) , su h that p 13 ∗ ( θ ) = p 12 ∗ ( θ ) ◦ p 23 ∗ ( θ ) up to the anonial isos. A morphism of desen t data, let us sa y of ( E ′ , θ ) in to ( F ′ , ξ ) , will b e a morphism a ′ : E ′ → F ′ in P ( X ′ ) ompatible with θ and ξ in the sense that p 2 ∗ ( a ′ ) ◦ θ = ξ ◦ p 1 ∗ ( a ′ ) . Desen t data of t yp e P and relativ e to X ′ p − → X (and their morphisms) form a ategory D es P ( X ′ /X ) . There is an ob vious funtor (4) P ( X ) − → D es P ( X ′ /X ) , E 7→ ( p ∗ E , φ E ) , a 7→ p ∗ a dened b y letting φ E b e the anonial isomorphism p 1 ∗ ( p ∗ E ) ∼ = ( p ◦ p 1 ) ∗ E = ( p ◦ p 2 ) ∗ E ∼ = p 2 ∗ ( p ∗ E ) . Whenev er the funtor ( 4 ) is an equiv alene of at- egories for ev ery at mapping of manifolds p : X ′ → X , one sa ys that the presta k P is a stak. 5 Note Dep ending on one's purp oses, the ondition that the funtors (4) b e equiv alenes of ategories for all at mappings X ′ → X an b e w eak ened to some exten t. F or example, one ould ask it to b e satised just for all at X ′ → X o v er a Hausdor, paraompat X . In fat, the latter ondition will pro v e to b e suien t for all our purp oses: no relev an t asp et of the theory seems to dep end on the stronger requiremen t. W e prop ose to use the term  parasta k  for the w eak er notion; w e will often b e slopp y and use `sta k' as a synon ym to `parasta k'. 62 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED Lo ally trivial ob jets Let S b e an y smo oth tensor presta k. An ob jet E ∈ Ob S ( X ) will b e alled trivial if there exists some V ∈ Ob S ( ⋆ ) for whi h one an nd an isomorphism E α ≈ c X ∗ V in S ( X ) , where c X : X → ⋆ denotes the ollapse map. An y su h pair ( V , α ) will b e said to onstitute a trivialization of E . F or an arbitrary manifold X , let V S ( X ) denote the full sub ategory of S ( X ) formed b y the lo ally trivial ob jets of lo ally nite rank; more expliitly , E ∈ Ob S ( X ) will b e an ob jet of V S ( X ) pro vided one an o v er X with op en subsets U su h that E | U trivializes in S ( U ) b y means of a trivialization of the form ( 1 ⊕ · · · ⊕ 1 , α ) or, equiv alen tly , su h that in S ( U ) there exists an isomorphism E | U ≈ 1 U ⊕ · · · ⊕ 1 U . It follo ws at one from the bilinearit y of ⊗ , the trivialit y of 1 and the linearit y of f ∗ that the op eration X 7→ V S ( X ) determines a bred (additiv e, omplex) tensor sub ategory of S . Hene X 7→ V S ( X ) inherits a bred tensor struture from S . It is easy to see that one gets in fat a smo oth tensor presta k V S ; moreo v er, it is ob vious that V S is a parasta k resp. a sta k if su h is S . The omplex tensor ategory V S ( X ) v ery losely relates to that of smo oth omplex v etor bundles o v er X . Let us mak e this preise. Clearly , ev ery ob jet E ∈ V S ( X ) yields a smo oth omplex v etor bundle o v er X : just put ˜ E = { ( x, e ) : x ∈ X, e ∈ E x } , with the lo al trivializing  harts obtained from lo al trivializations E | U α ≈ 1 U ⊕ · · · ⊕ 1 U , α = ( α 1 , . . . , α d ) b y setting ( u, e ) =  u ; α 1 ,u ( e ) , . . . , α d,u ( e )  ∈ U × C d . Sine an y morphism a : E → E ′ in V S ( X ) an b e lo ally desrib ed in terms of matrix expressions with smo oth o eien ts, setting ˜ a · ( x, e ) = ( x, a x · e ) denes a morphism of smo oth v etor bundles ˜ a : ˜ E → ˜ E ′ . It is an exerise to sho w that the assignmen t E 7→ ˜ E denes a faithful omplex tensor funtor of V S ( X ) in to smo oth omplex v etor bundles. Under extremely mild h yp otheses, this funtor will atually pro v e to b e an equiv alene of omplex tensor ategories; this will happ en, for example, when S is a parasta k and X is paraompat, or when S is sta k. In onlusion, w e see that for S a smo oth tensor (para-)sta k (and X a reasonable manifold), the ategory S ( X ) will essen tially inludeas a full tensor sub ategoryall smo oth v etor bundles o v er X . One arriv es at the same results, alternativ ely , b y onsidering the funtor Γ X and the ategory of lo ally free shea v es of C ∞ X -mo dules of lo ally nite rank. This last remark an b e summarized in the diagram V S ( X )  u Γ X ( ( P P P P P P - 7→ f ( - ) ≃ / / V ∞ ( X ) I i Γ X v v n n n n n n { shea ves of C ∞ X - mo dules } (6) (omm utativ e up to anonial natural isomorphism). The smo oth tensor sta k V ∞ is therefore, in a v ery preise sense, the smallest p ossible. 13. F OUND A TIONS OF REPRESENT A TION THEOR Y 63 13 F oundations of Represen tation Theory W e dev elop our theory of represen tations relativ e to a  t yp e  . This an b e an y smo oth omplex tensor parasta k S , in the sense of Note 12.5. One a t yp e S has b een xed, one an asso iate to an y Lie group oid a mathematial ob jet alled  bre funtor  . This is done as follo ws. Let G b e a Lie group oid, let us sa y , with base M . W e are going to onstrut a ategory R S ( G ) , along with a funtor ω S ( G ) of R S ( G ) in to S ( M ) that w e shall all the  standard bre funtor  of G (of t yp e S ). An ob jet of the ategory R ( G ) = R S ( G ) (ev ery time w e lik e w e an omit writing the t yp e S , as this is xed) is dened to b e a pair ( E ,  ) with E an ob jet of S ( M ) and  a morphism in S ( G ) (1)  : s ∗ E → t ∗ E (where s , t : G → M denote the soure, resp. target map of G ), su h that the appropriate onditions for  to b e an ationin other w ords, for it to b e ompatible with the group oid strutureare satised, namely: i) p u ∗  q = id E , where u : M → G denotes the unit setion. (Here and in the sequel w e adopt the devie of putting orners around a morphism to indiate the morphismwhi h one, will alw a ys b e lear from the on textthat orresp onds to it up to some anonial iden tiations; for instane, the last equalit y , sp elled out expliitly , means that the diagram u ∗ s ∗ E ∼ = an. " " D D D D D D D u ∗  / / u ∗ t ∗ E ∼ = an. } } z z z z z z z E (2) omm utes, where w e use the iden tiations u ∗ s ∗ E ∼ = ( s ◦ u ) ∗ E = id M ∗ E ∼ = E et. pro vided b y the bred tensor struture onstrain ts asso iated with S ); ii) if w e let G (2) = G s × t G denote the manifold of omp osable arro ws of G , c : G (2) → G , ( g ′ , g ) 7→ g ′ g the omp osition of arro ws and p 0 , p 1 : G (2) → G the t w o pro jetions ( g ′ , g ) 7→ g ′ , 7→ g on to the rst and seond fator resp etiv ely , w e ha v e the iden tit y p c ∗  q = p p ∗ 0  q · p p ∗ 1  q ; that is to sa y , aording to our on v en tion, w e ha v e the omm utativit y of the follo wing diagram in the ategory S ( G (2) ) : c ∗ s ∗ E j j j j j j j j c ∗  / / c ∗ t ∗ E T T T T T T T T p 1 ∗ s ∗ E p 1 ∗  ' ' O O O O O O O O O O O O O p 0 ∗ t ∗ E 7 7 p 0 ∗  o o o o o o o o o o o o o p 1 ∗ t ∗ E p 0 ∗ s ∗ E (3) 64 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED (whi h in v olv es the anonial iden tiations c ∗ s ∗ E ∼ = ( s ◦ c ) ∗ E = ( s ◦ p 1 ) ∗ E ∼ = p 1 ∗ s ∗ E et. pro vided b y the struture onstrain ts of S ). W e shall also write  c ∗  = p ∗ 0  · p ∗ 1  (mo d ∼ = )  . This onludes the desription of the ob jets of R S ( G ) ; w e shall all them r epr esentations of G , or G -ations (of t yp e S ). As morphisms of G -ations a : ( E ,  ) → ( E ′ ,  ′ ) w e tak e all those morphisms a : E → E ′ in S ( M ) whi h mak e the follo wing square omm utativ e s ∗ E s ∗ a    / / t ∗ E t ∗ a   s ∗ E ′  ′ / / t ∗ E ′ . (4) W e endo w the ategory R S ( G ) with the linear struture of S ( M ) . Then the forgetful funtor (5) ω S ( G ) : R S ( G ) − → S ( M ) , ( E ,  ) 7→ E is linear and faithful. W e all it the standar d br e funtor of G (of typ e S ). Observ e that the linear ategory R S ( G ) is additive. Indeed, x an y ob jets R 0 , R 1 ∈ R ( G ) , let us sa y R i = ( E i ,  i ) , and  ho ose a represen tativ e E 0 i 0 ֒ → E 0 ⊕ E 1 i 1 ← ֓ E 1 for the diret sum in S ( M ) . Then, sine the linear funtors s ∗ , t ∗ ha v e to preserv e diret sums (f. MaL ane (1998), p. 197), there will b e a unique `univ ersal' isomorphism in S ( G ) s ∗ ( E 0 ⊕ E 1 ) = s ∗ E 0 ⊕ s ∗ E 1  0 ⊕  1 − − − − − → t ∗ E 0 ⊕ t ∗ E 1 = t ∗ ( E 0 ⊕ E 1 ) . One  he ks that the pair R 0 ⊕ R 1 = ( E 0 ⊕ E 1 ,  0 ⊕  1 ) is a G -ation, that R 0 i 0 ֒ → R 0 ⊕ R 1 i 1 ← ֓ R 1 are morphisms of G -ations, and that they yield a diret sum in R ( G ) . The pro ess to obtain a n ull represen tation is en tirely analogous, starting from a n ull ob jet in S ( M ) . 6 Lemma F or an arbitrary G -ation ( E ,  ) ∈ R S ( G ) , the morphism  : s ∗ E → t ∗ E is neessarily an isomorphism in S ( G ) . Pro of Let C b e an y ategory . Dene t w o arro ws a, a ′ to b e `equiv alen t', and write a ∼ a ′ , if they are isomorphi as ob jets of the arro w ategory A r ( C ) (in other w ords, if there exist isomorphisms b et w een their domains and o domains whi h transform the one arro w in to the other). Then the follo wing assertions hold: a) for an y funtor F : C → D , a ∼ a ′ implies F a ∼ F a ′ ; b) the existene of a natural iso F ∼ → G implies F a ∼ Ga for ev ery a ; ) if a ∼ a ′ and a is left (resp. righ t) in v ertible, then the same is true of a ′ ; d) ba ∼ id implies that a is left in v ertible and b righ t in v ertible. 14. HOMOMORPHISMS AND MORIT A INV ARIANCE 65 Let i : G → G , g 7→ g − 1 b e the in v erse, and onsider the t w o maps ( i , id ) , ( i d , i ) : G → G (2) giv en b y g 7→ ( g − 1 , g ) , 7→ ( g , g − 1 ) resp etiv ely . Then one has the follo wing equiv alenes of arro ws in the ategory S ( G ) id s ∗ E = s ∗ id E a) ∼ s ∗ u ∗  b) ∼ ( u ◦ s ) ∗  = [ c ◦ ( i , id )] ∗  b) ∼ ( i , id ) ∗ c ∗  a) ∼ ( i , id ) ∗ p c ∗  q (3) = ( i , id ) ∗ ( p p ∗ 0  q · p p ∗ 1  q ) = ( i , id ) ∗ p p ∗ 0  q · ( i , id ) ∗ p p ∗ 1  q , hene ( i , id ) ∗ p p ∗ 1  q is left in v ertible in S ( G ) , b y d) . Sine this is in turn equiv alen t to ( i , id ) ∗ p 1 ∗  ∼ [ p 1 ◦ ( i , id )] ∗  = id G ∗  ∼  ,  itself will b e left in v ertible in S ( G ) , b y ) . An analogous reasoning will establish the righ t in v ertibilit y of  . It follo ws that  is in v ertible. q.e.d. Next, w e disuss the standard tensor struture on the ategory R ( G ) . This struture mak es R ( G ) an additiv e linear tensor ategory . The standard bre funtor ω = ω ( G ) turns out to b e a strit tensor funtor of R ( G ) in to S ( M ) , in the sense that the iden tities ω ( R ⊗ S ) = ω ( R ) ⊗ ω ( S ) and ω ( 1 ) = 1 hold, so that they an b e tak en resp etiv ely as the natural onstrain ts τ and υ in the denition of tensor funtor. W e start with the onstrution of the bifuntor ⊗ : R ( G ) × R ( G ) → R ( G ) . F or t w o arbitrary represen tations R, S ∈ R ( G ) , let us sa y R = ( E ,  ) and S = ( F , σ ) , w e put R ⊗ S = ( E ⊗ F , p  ⊗ σ q ) , wherefollo wing the usual on v en tion p  ⊗ σ q stands for the omp osite morphism (7) s ∗ ( E ⊗ F ) ∼ = s ∗ E ⊗ s ∗ F  ⊗ σ − − − − → t ∗ E ⊗ t ∗ F ∼ = t ∗ ( E ⊗ F ) . It is easy to reognize that the pair R ⊗ S is itself a G -ation, i.e. an ob jet of the ategory R ( G ) ; moreo v er, if ( E ,  ) a − → ( E ′ ,  ′ ) and ( F , σ ) b − → ( F ′ , σ ′ ) are morphisms in R ( G ) then so is a ⊗ b : R ⊗ S → R ′ ⊗ S ′ . W e dene the tensor unit of R ( G ) to b e the pair ( 1 M , p id q ) , where 1 M the tensor unit of S ( M ) and p id q is the omp osite anonial isomorphism (8) s ∗ 1 M ∼ = 1 G ∼ = t ∗ 1 M . The A CU natural onstrain ts α , γ , λ , ρ for the tensor struture of the base ategory S ( M ) will pro vide analogous onstrain ts for the tensor pro dut w e just in tro dued on R ( G ) . (F or example, onsider three represen tations R, S, T ∈ R ( G ) and let E , F , G ∈ S ( M ) b e the resp etiv e supp orts; then the isomorphism α E ,F ,G : E ⊗ ( F ⊗ G ) ∼ → ( E ⊗ F ) ⊗ G is also an isomorphism α R,S,T : R ⊗ ( S ⊗ T ) ∼ → ( R ⊗ S ) ⊗ T in R ( G ) .) A fortiori, the oherene diagrams for su h `inherited' onstrain ts will omm ute. 14 Homomorphisms and Morita In v ariane W e no w pro eed to study the op eration of taking the in v erse image of a represen tation along a homomorphism of Lie group oids. Then w e onen trate 66 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED on the sp eial ase of Morita equiv alenes; in order to giv e a satisfatory treatmen t of these, it will b e neessary to analyze natural transformations of Lie group oid homomorphisms rst. Let ϕ : G → H b e a homomorphism of Lie group oids and let M f − → N b e the smo oth map indued b y ϕ on the base manifolds. Supp ose ( F , σ ) ∈ R S ( H ) . Consider the morphismwhi h w e also denote b y ϕ ∗ σ , sligh tly abusing notationdened as follo ws: (1) s G ∗ ( f ∗ F ) ∼ = ϕ ∗ s H ∗ F ϕ ∗ σ − − − → ϕ ∗ t H ∗ F ∼ = t G ∗ ( f ∗ F ) ; the equalities f ◦ s G = s H ◦ ϕ et. aoun t, of ourse, for the existene of the anonial isomorphisms o urring in ( 1 ). It is straigh tforw ard to  he k that the pair ( f ∗ F , ϕ ∗ σ ) onstitutes an ob jet of the ategory R S ( G ) and that if ( F , σ ) b − → ( F ′ , σ ′ ) is a morphism of H -ations then f ∗ b is a morphism of ( f ∗ F , ϕ ∗ σ ) in to ( f ∗ F ′ , ϕ ∗ σ ′ ) in R S ( G ) . Hene w e get a funtor (2) ϕ ∗ : R S ( H ) − → R S ( G ) , whi h w e agree to all the inverse image or pul l-b ak (of represen tations) along ϕ . It is fairly easy to  he k that the onstrain ts (3) ( υ : 1 M ∼ → f ∗ 1 N τ F ,F ′ : f ∗ F ⊗ f ∗ F ′ ∼ → f ∗ ( F ⊗ F ′ ) , asso iated with the tensor funtor f ∗ , an also funtion as isomorphisms of G -ations, υ : 1 ∼ → ϕ ∗ ( 1 ) and τ S,S ′ : ϕ ∗ ( S ) ⊗ ϕ ∗ ( S ′ ) ∼ → ϕ ∗ ( S ⊗ S ′ ) , for all S, S ′ ∈ R ( H ) with, let us sa y , S = ( F, σ ) and S ′ = ( F ′ , σ ′ ) . A fortiori, these isomorphisms are natural and they pro vide appropriate tensor funtor onstrain ts for ϕ ∗ , th us making ϕ ∗ a tensor funtor of the tensor ategory R ( H ) in to the tensor ategory R ( G ) . Let G ϕ − → H ψ − → K b e t w o omp osable homomorphisms of Lie group oids and let X ϕ 0 − → Y ψ 0 − → Z denote the resp etiv e maps on bases. Note that for an arbitrary ation T = ( G, τ ) ∈ R ( K ) the anonial isomorphism ϕ 0 ∗ ψ 0 ∗ G ∼ = ( ψ 0 ◦ ϕ 0 ) ∗ G = ( ψ ◦ ϕ ) 0 ∗ G is atually a morphism ϕ ∗ ( ψ ∗ T ) ∼ → ( ψ ◦ ϕ ) ∗ T in the ategory R ( G ) . Hene w e get an isomorphism of tensor funtors (4) ϕ ∗ ◦ ψ ∗ ∼ = − → ( ψ ◦ ϕ ) ∗ . It is w orth while remarking that ϕ ∗ ts in the follo wing diagram R S ( H ) ω S ( H )   ϕ ∗ / / R S ( G ) ω S ( G )   S ( N ) f ∗ / / S ( M ) , (5) 14. HOMOMORPHISMS AND MORIT A INV ARIANCE 67 whose omm utativit y is to b e in terpreted as an equalit y of omp osite tensor funtors th us, in v olving also the onstrain ts. The notion from Lie group oid theory w e w an t to dualize next is that of natural transformation; this omes ab out esp eially when one onsiders Morita equiv alenes, as w e shall see so on. Reall that a tr ansformation τ : ϕ 0 → ϕ 1 (b et w een t w o Lie group oid homomorphisms ϕ 0 , ϕ 1 : G → H ) is a smo oth mapping τ of the base manifold M of G in to the manifold of arro ws of H , su h that τ ( x ) : f 0 ( x ) → f 1 ( x ) ∀ x ∈ M and the familiar diagram f 0 ( x ) ϕ 0 ( g )   τ ( x ) / / f 1 ( x ) ϕ 1 ( g )   f 0 ( x ′ ) τ ( x ′ ) / / f 1 ( x ′ ) (6) is omm utativ e for all g ∈ G (1) , g : x → x ′ . Supp ose an ation S = ( F , σ ) ∈ R S ( H ) is giv en. Then one an apply τ ∗ to the isomorphism σ : s ∗ F ≈ − → t ∗ F to obtain an isomorphism f ∗ 0 F ≈ − → f ∗ 1 F in the ategory S ( M ) (7) f ∗ 0 F ∼ = τ ∗ s ∗ F τ ∗ σ − − → τ ∗ t ∗ F ∼ = f ∗ 1 F , whi h ma y b e denoted b y the sym b ol σ ◦ τ . (Here one uses the iden tities f 0 = s H ◦ τ and f 1 = t H ◦ τ .) By expressing (6) as an iden tit y b et w een suitable smo oth maps, one an  he k that σ ◦ τ is atually an isomorphism of G -ations b et w een ϕ ∗ 0 S and ϕ ∗ 1 S : in detail, onsider the maps ( τ ◦ t , ϕ 0 ) and ( ϕ 1 , τ ◦ s ) , of G (1) (manifold of arro ws) in to H (2) ≡ H s × t H (mani- fold of omp osable arro ws), resp etiv ely giv en b y g 7→ ( τ ( t g ) , ϕ 0 ( g )) and g 7→ ( ϕ 1 ( g ) , τ ( s g )) ; the omm utativit y of ( 6 ) implies that up on omp os- ing these maps with m ultipliation c : H (2) → H one gets the same result, c ◦ ( τ ◦ t , ϕ 0 ) = c ◦ ( ϕ 1 , τ ◦ s ) ; from the latter iden tit y it is easy to see that (7) is a morphism in R S ( G ) . Then the rule ( F , σ ) 7→ σ ◦ τ denes a nat- ural isomorphismin fat, a tensor preserving oneb et w een the funtors ϕ ∗ 0 , ϕ ∗ 1 : R S ( H ) → R S ( G ) ; w e will use the notation (8) τ ∗ : ϕ ∗ 0 ∼ − → ϕ ∗ 1 , τ ∗ ∈ Iso ⊗ ( ϕ ∗ 0 , ϕ ∗ 1 ) . W e are no w ready to disuss Morita equiv alenes. Reall that a homo- morphism ϕ : G → H is said to b e a Morita e quivalen e in ase G ( s , t )   ϕ / / H ( s , t )   M × M f × f / / N × N (9) is a pullba k diagram in the ategory of C ∞ manifolds and the mapping (10) t ◦ pr 2 : M f × s H → N , 68 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED whi h, lo osely sp eaking, sends f ( x ) h − → y to y , is a surjetiv e submersion. Our main goal in this setion is to sho w that the pull-ba k funtor ϕ ∗ : R ( H ) → R ( G ) asso iated with a Morita equiv alene ϕ is an equiv alene of tensor ategories. 2 Clearly , it will b e enough to sho w that ϕ ∗ is a ategorial equiv alene (in the familiar sense): this means that w e ha v e to lo ok for a funtor ϕ ! : R ( G ) → R ( H ) su h that natural isomorphisms ϕ ! ◦ ϕ ∗ ≃ Id R ( H ) and ϕ ∗ ◦ ϕ ! ≃ Id R ( G ) exist. Notie that the ondition that the map (10 ) should b e a surjetiv e sub- mersion will of ourse b e satised when f itself is a surjetiv e submersion. As a rst step, w e sho w ho w the task of onstruting a quasi-in v erse for the pullba k funtor ϕ ∗ asso iated with an arbitrary Morita equiv alene ϕ ma y b e redued to the sp eial ase where f is preisely a surjetiv e submersion. T o this end, onsider the we ak pul lb ak (see [ 27 ℄, pp. 123132) P χ   ψ / / G ϕ   H Id / / τ & . H . (11) Let P b e the base manifold of the Lie group oid P . It is w ell-kno wn (ibid. p. 130) that the Lie group oid homomorphisms ψ and χ are Morita equiv a- lenes with the prop ert y that the resp etiv e base maps ψ (0) : P → M and χ (0) : P → N are surjetiv e submersions. No w, if w e sueed in pro ving that ψ ∗ and χ ∗ are ategorial equiv alenes then, sine b y ( 4) and (8) ab o v e w e ha v e a natural isomorphism (atually , a tensor preserving one) (12) χ ∗ ≈ − → ( ϕ ◦ ψ ) ∗ ∼ = ← − ψ ∗ ◦ ϕ ∗ , the same will b e true of ϕ ∗ . F rom no w on w e will w ork under the h yp othesis that the giv en Morita equiv alene ϕ (9) determines a surjetiv e submersion f : M ։ N on base manifolds. This b eing the ase, there exists an op en o v er N = ∪ i ∈ I V i of the manifold N b y op en subsets V i su h that for ea h of them one an nd a smo oth setion s i : V i ֒ → M to f . W e x su h a o v er and su h setions one and for all. Let an arbitrary ob jet R = ( E ,  ) ∈ R S ( G ) b e giv en. F or ea h i ∈ I one an tak e the pull-ba k E i ≡ s i ∗ E ∈ S ( V i ) . Fix a ouple of indies i, j ∈ I . Then, sine (9) is a pull-ba k diagram, for ea h y ∈ V i ∩ V j there is exatly one arro w g ( y ) : s i ( y ) → s j ( y ) su h that ϕ ( g ( y )) = y . More preisely , let y 7→ g ( y ) = g ij ( y ) b e the smo oth mapping dened as the unique solution to 2 Reall that a tensor funtor Φ : C → D is said to b e a tensor e quivalen e in ase there exists a tensor funtor Ψ : D → C along with tensor preserving natural isomorphisms Ψ ◦ Φ ≃ Id C and Φ ◦ Ψ ≃ Id D . 14. HOMOMORPHISMS AND MORIT A INV ARIANCE 69 the follo wing univ ersal problem (in the C ∞ ategory) V ij ( s i ,s j ) ' ' g ij % % J J J J J J u | V ij # # G ( s , t )   ϕ / / H ( s , t )   M × M f × f / / N × N , (13) where u : N → H denotes the unit setion and V ij ≡ V i ∩ V j . Then, putting E i | j = E i | V i ∩ V j and E j | i = E j | V i ∩ V j , one ma y pull the ation  ba k along the map g ij so as to get an isomorphism θ ij : E i | j ∼ → E j | i in the ategory S ( V ij ) : (14) E i | j ∼ = ( s ◦ g ij ) ∗ E ∼ = g ij ∗ s ∗ E g ij ∗  − − − − → g ij ∗ t ∗ E ∼ = ( t ◦ g ij ) ∗ E ∼ = E j | i or, as an iden tit y up to anonial isomorphisms, θ ij = g ij ∗  .  mo d ∼ =  (15) (Note that the fat that  is an isomorphism in the ategory S ( G ) , that is to sa y Lemma 13.6 , is used in an essen tial w a y .) Next, from the ob vious remark that for an arbitrary third index k ∈ I one has g ik ( y ) = g j k ( y ) g ij ( y ) ∀ y ∈ V ij k ≡ V i ∩ V j ∩ V k (or b etter g ik | j = c ◦ ( g j k | i , g ij | k ) , where g ik | j denotes the restrition of g ik to V ij k et.), and from the m ultipliativ e axiom (13 .3) for  , it follo ws that the system of isomorphisms { θ ij } onstitutes a o yle or desen t datum for the family { E i } i ∈ I ∈ S  ` i ∈ I V i  , relativ e to the at mapping ` i ∈ I V i → N . Sine N is a paraompat manifold and S is a smo oth parasta k, there exists some ob jet ϕ ! E of S ( N ) along with isomorphisms θ i : ( ϕ ! E ) | i ≡ ( ϕ ! E ) | V i ≈ − → E i in S ( V i ) , ompatible with { θ ij } in the sense that, mo dulo the iden tiation ( ϕ ! E ) i | V ij ∼ = ( ϕ ! E ) j | V ij , one has the iden tit y θ j | i = θ j | V ij = θ ij · θ i | V ij = θ ij · θ i | j .  mo d ∼ =  (16) F or simpliit y , let us put F ≡ ϕ ! E . Our next step will b e to dene a morphism σ = ϕ !  : s H ∗ F → t H ∗ F , whi h is to pro vide the H -ation on F . F or ea h pair V i , V i ′ w e in tro due the abbreviation H i,i ′ ≡ H ( V i , V i ′ ) ; w e also write H ij,i ′ j ′ ≡ H ( V ij , V i ′ j ′ ) . Then the subsets H i,i ′ ⊂ H (1) form an op en o v er of the manifold H (1) . No w, let g i,i ′ : H i,i ′ → G b e the smo oth map obtained b y solving the follo wing univ ersal problem H i,i ′ ( s , t )   g i,i ′ & & N N N N N N N inlusion # # V i × V i ′ s i × s i ′ - - G ( s , t )   ϕ / / H ( s , t )   M × M f × f / / N × N . (17) 70 CHAPTER I I I. REPRESENT A TION THEOR Y REVISITED W e an use this map to dene a morphism σ i,i ′ : ( s H ∗ F ) | i,i ′ → ( t H ∗ F ) | i,i ′ in the ategory S ( H i,i ′ ) , as follo ws: (18) ( s H ∗ F ) | i,i ′ ∼ = ( s H | i,i ′ ) ∗ ( F | i ) ( s H | i,i ′ ) ∗ θ i − − − − − − − → ( s H | i,i ′ ) ∗ E i ∼ = g i,i ′ ∗ s G ∗ E g i,i ′ ∗  − − − − → g i,i ′ ∗ t G ∗ E ∼ = ( t H | i,i ′ ) ∗ E i ′ ( t H | i,i ′ ) ∗ θ i − 1 − − − − − − − − → ( t H | i,i ′ ) ∗ ( F | i ′ ) ∼ = ( t H ∗ F ) | i,i ′ or, in the form of an iden tit y mo dulo anonial iden tiations, σ i,i ′ = ( t H | i,i ′ ) ∗ θ i − 1 · g i,i ′ ∗  · ( s H | i,i ′ ) ∗ θ i .  mo d ∼ =  (19) Starting from the equalit y of mappings (20) g i,i ′ | j,j ′ = ( g j ′ i ′ ◦ t H | ij,i ′ j ′ ) g j,j ′ | i,i ′ ( g j i ◦ s H | ij,i ′ j ′ ) (note that g j ′ i ′ = i G ◦ g i ′ j ′ where i G is the in v erse map of G ) and the mo d ∼ =  iden tities (15 ), (16) and (19 ), one an  he k that σ i,i ′ | j,j ′ = σ j,j ′ | i,i ′ in S ( H ij,i ′ j ′ ) ; hene the morphisms σ i,i ′ glue together in to a unique morphism σ = ϕ !  of S ( H (1) ) , with the prop ert y that σ | i,i ′ = σ i,i ′ . Next, supp ose w e are giv en a morphism a : R → R ′ in R S ( G ) , where R ′ = ( E ′ ,  ′ ) , let us sa y . Then w e an obtain a morphism ϕ ! a : ϕ ! R → ϕ ! R ′ , where ϕ ! R = ( ϕ ! E , ϕ !  ) et., b y rst letting b i = s i ∗ a and the observing that (21) θ ′ ij · b i | j = b j | i · θ ij in S ( V ij ) (b eause of the denition of θ ij = θ R ij and θ ′ ij = θ R ′ ij and b eause a is a G -equiv arian t morphism). In this w a y w e get a funtor of R S ( G ) in to R S ( H ) . The onstrution of the isomorphisms ϕ ∗ ◦ ϕ ! ≃ Id R ( G ) and ϕ ! ◦ ϕ ∗ ≃ Id R ( H ) is left as an exerise, to b e done along the same lines. Chapter IV General T annak a Theory In the preeding  hapter w e laid do wn the foundations of Represen tation Theory in the abstrat setting of smo oth tensor sta ks. The assumptions on the t yp e S w ere quite mild there, nothing more than just smo othness and the prop ert y of b eing a sta k. Ho w ev er, in order to get our reonstrution theory to w ork eetiv ely , w e need to imp ose further restritions on the t yp e S . W e will all a smo oth tensor sta k a stak of smo oth elds when it meets su h additional requiremen ts. The additional prop erties whi h  haraterize sta ks of smo oth elds are in tro dued in 15. The sta k of smo oth v etor bundles is an example. In the subsequen t setion w e pro vide another fundamen tal example, the sta k of smo oth (Eulide an) elds, whi h will pla y a ma jor role in the a hiev emen t of our T annak a dualit y theorem for prop er Lie group oids in 20. This sta k is a non trivial extension of the sta k of smo oth v etor bundles, but its denition is as simple. 15 Sta ks of Smo oth Fields The expression  sta k of smo oth elds  will b e emplo y ed to indiate a smo oth (real or omplex) tensor sta k 1 for whi h the axiomati onditions listed b elo w are satised. When dealing sp eially with sta ks of smo oth elds w e shall prefer them to b e represen ted b y the letter F , whi h is more suggestiv e than the usual S . The axioms Our rst axiom is ab out the tensor pro dut and pull-ba k op erations. Roughly sp eaking, it states that the setions of a tensor pro dut or a pull-ba k are exatly what one w ould exp et them to b e on the basis of the standard 1 In aordane with the philosoph y of Note 12 .5 , w e use the w ord `sta k' but w e really mean `parasta k'. 71 72 CHAPTER IV. GENERAL T ANNAKA THEOR Y denition of tensor pro dut and pull-ba k of shea v es of C ∞ -mo dules; ho w- ev er, for su h setions the relation of equalit y ma y b e oarser, in the sense that more setions ma y b e regarded as b eing iden tial. 1 Axiom I (tensor pro dut & pull-ba k) The  anoni al natur al morph- isms (11.20 ) and (11 .23) ( Γ E ⊗ C ∞ X Γ E ′ → Γ ( E ⊗ E ′ ) f ∗ ( Γ Y F ) → Γ X ( f ∗ F ) ar e surjetiv e (= epimorphisms of she aves). Th us, ev ery lo al smo oth setion of E ⊗ E ′ will p ossess, in the viinit y of ea h p oin t, an expression as a nite linear om bination, with smo oth o eien ts, of setions of the form ζ ⊗ ζ ′ . Similarly , giv en an y partial smo oth setion of f ∗ F , it will b e p ossible to express it lo ally as a nite linear om bination, with o eien ts in C ∞ X , of setions of the form η ◦ f . Supp ose E ∈ F ( X ) . Let us go ba k for a momen t to the map Γ E ( U ) → E x , ζ 7→ ζ ( x ) w e dened in 11 (for ea h op en neigh b ourho o d U of the p oin t x ). These maps are eviden tly ompatible with the restrition to a smaller op en neigh b ourho o d of x , hene on passing to the indutiv e limit they will determine a linear map (2) ( Γ E ) x → E x , ζ 7→ ζ ( x ) of the stalk of Γ E at x in to the bre of E at the same p oin t. W e all this map the evaluation (of germs) at x . Notie, b y the w a y , that the iden tit y (3) ( αζ )( x ) = α ( x ) ζ ( x ) holds for all germs of smo oth setions ζ ∈ ( Γ E ) x and of smo oth funtions α ∈ C ∞ X,x . It follo ws from Axiom i (pull-ba k) that for any stak of smo oth elds, the evaluation of germs at a p oint is a surje tive line ar map. Indeed, the stalk ( Γ E ) x oinides, as a v etor spae, with the spae of global setions of x ∗ ( Γ E ) (reall that ( Γ E ) x = lim − → U ∋ x Γ E ( U ) = x − 1 ( Γ E )( ⋆ ) , atually as a C ∞ X,x -mo dule), and the bre E x is dened as the spae of global setions of Γ ( x ∗ E ) ; it is immediate to reognize that the ev aluation of germs is just the map of global setions indued b y (11.23 ). The seond axiom sa ys that a dierene b et w een an y t w o morphisms an b e deteted b y lo oking at the linear maps they indue on the bres. 4 Axiom I I (v anishing) L et a : E → E ′ b e a morphism in F ( X ) . Supp ose that a x : E x → E ′ x is zer o ∀ x ∈ X . Then a = 0 . 15. ST A CKS OF SMOOTH FIELDS 73 As a rst, immediate onsequene, an arbitrary setion ζ ∈ Γ E ( U ) will v anish if and only if all its v alues ζ ( u ) will b e zero as u ranges o v er U : th us, one sees that smo oth se tions ar e har aterize d by their values; in tuitiv ely , one an think of the elemen ts of Γ E ( U ) as setionsin the usual senseof the `bundle' of bres { E u } . F urthermore, b y om bining Axioms i i and i , it follo ws that the funtor Γ X : F ( X ) → { shea v es of C ∞ X - mo dules } is faithful. This is an easy onse- quene of the surjetivit y of the ev aluation of germs at a p oin t; the argumen t w e prop ose no w will also b e preparatory to the next axiom. F or ea h morphism a : E → F in F ( X ) , onsider the `bundle' of linear maps { a x : E x → F x } and the morphism α = Γ a : Γ E → Γ F of shea v es of C ∞ X -mo dules. W e start b y asking what relation there is b et w een these data. The link b et w een the t w o is ob viously pro vided b y the ab o v e anonial ev aluation maps of the stalks on to the bres ( Γ E ) x ։ E x : it is lear that the stalk homomorphism α x and the linear map a x ha v e to b e ompatible, in the sense that the follo wing square should omm ute ( Γ E ) x ev al.     α x / / ( Γ F ) x ev al.     E x a x / / F x . (5) In general, w e shall sa y that a morphism of shea v es of mo dules α : Γ E → Γ F and a `bundle' of linear maps { a x : E x → F x } are  omp atible, whenev er the diagram (5) omm utes for all x ∈ X . Notie that, in view of the preeding axioms, ompatibilit y implies that the morphism of shea v es and the bundle of linear maps determine ea h other unam biguously . (Indeed, in one diretion, the morphism α learly determines the maps a x through the omm utativit y of (5). Con v ersely , the omm utativit y of ( 5) for all x en tails that for an y smo oth setion ζ ∈ Γ E ( U ) one has the form ula [ α ( U ) ζ ]( x ) = a x  ζ ( x )  , and therefore, if α and β are b oth ompatible with { a x } , it follo ws b y Axiom i i that α ( U ) ζ = β ( U ) ζ for all ζ and hene that α = β .) In partiular, from Γ a = Γ b it will follo w that a x = b x for all x and therefore that a = b . Let us all a morphism of shea v es of mo dules α : Γ E → Γ F r epr esentable, if it admits a ompatible bundle of linear maps { a x : E x → F x } . Our next axiom, whi h omplemen ts the preeding one b y pro viding a general riterion for the existene of morphisms in F ( X ) , states that the olletion of su h morphisms is as big as p ossible: 6 Axiom I I I (morphisms) F or every r epr esentable α : Γ E → Γ F , ther e exists a morphism a : E → F in F ( X ) suh that Γ a = α . This axiom will not b e used an yhere in the presen t setion. It will pla y a role only in 17, where it is needed in order to onstrut morphisms of represen tations b y means of brewise in tegration. 74 CHAPTER IV. GENERAL T ANNAKA THEOR Y W e annot y et dedue, from the axioms w e ha v e in tro dued so far, ertain v ery in tuitiv e prop erties that are surely reasonable for a smo oth setion; for instane, if a setionor, more generally , a morphismv anishes o v er a dense op en subset of its domain of denition, it w ould b e natural to exp et it to b e zero ev erywhere. Analogously , if the v alue of a setion is non zero at a p oin t then it should b e non zero at all nearb y p oin ts. The next axiom yields su h prop erties, among man y other onsequenes. W e shall sa y that a Hermitianor, in the real ase, symmetriform φ : E ⊗ E ∗ → 1 in F ( X ) is a Hilb ert metri on E , when for ev ery p oin t x the indued form φ x on the bre E x (7) E x ⊗ E x ∗ an. − − → ( E ⊗ E ∗ ) x φ x − − → 1 x ∼ = C is a Hilb ert metri (in the familiar sense, viz. p ositiv e denite). 8 Axiom IV (metris) A ny obje t E ∈ Ob F ( X ) supp orts lo  al metris; that is to say, the op en subsets U suh that one  an nd a Hilb ert metri on E | U  over X . In general, one an only assume lo  al metris to exist, think e.g. of smo oth v etor bundles; ho w ev er, as for v etor bundles, global metris an b e on- struted from lo al ones as so on as smo oth partitions of unit y are a v ailable on the manifold X (e.g. when X is paraompat). Let E ∈ Ob F ( X ) and let φ b e a Hilb ert metri on E . By a φ -orthonormal fr ame for E ab out a p oin t x of X w e mean a list of setions ζ 1 , . . . , ζ d ∈ Γ E ( U ) , dened o v er a neigh b ourho o d of x , su h that for all u in U the v etors ζ 1 ( u ) , . . . , ζ d ( u ) are orthonormal in E u (with resp et to φ u ) and (9) Span  ζ 1 ( x ) , . . . , ζ d ( x )  = E x . Orthonormal fr ames for E exist ab out e ah p oint x for whih the br e E x is nite dimensional. Indeed, o v er some neigh b ourho o d N of x w e an rst of all nd lo al smo oth setions ζ 1 , . . . , ζ d with the prop ert y that the v etors ζ 1 ( x ) , . . . , ζ d ( x ) form a basis of the spae E x (Axiom i ). Sine for all n ∈ N the v etors ζ 1 ( n ) , . . . , ζ d ( n ) are linearly dep enden t if and only if there is a d -tuple of omplex n um b ers ( z 1 , . . . , z d ) with | z 1 | 2 + · · · + | z d | 2 = 1 and d P i =1 z i ζ i ( n ) = 0 , the on tin uous funtion N × S 2 d − 1 → R , ( n ; s 1 , t 1 , . . . , s d , t d ) 7→     d P ℓ =1 ( s ℓ + i t ℓ ) ζ ℓ ( n )     m ust ha v e a minim um c > 0 at n = x , hene a lo w er b ound c 2 on a suitable neigh b ourho o d U of x so that the ζ i ( u ) m ust b e linearly indep enden t for all u ∈ U . A t this p oin t it is enough to apply the GramS hmidt pro ess in 15. ST A CKS OF SMOOTH FIELDS 75 order to obtain an orthonormal frame ab out x . This elemen tary observ ation (existene of orthonormal frames) will pro v e to b e v ery useful. Let us start to illustrate its imp ortane with some basi appliations. Consider an emb e dding e : E ′ ֒ → E in the ategory F ( X ) , that is to sa y , a morphism su h that the linear map e x : E ′ x ֒ → E x is injetiv e for all x . 2 Supp ose there exists a global metri φ on the ob jet E ; also assume that E ′ ∈ Ob V F ( X ) is lo ally trivial of (lo ally) nite rank. Then e admits a  o-se tion, i.e. ther e exists a morphism p : E → E ′ with p ◦ e = id (so e is a setion in the ategorial sense). T o pro v e this, note rst of all that the metri φ will indue a metri φ ′ on E ′ ֒ → E . Fix an y p oin t x ∈ X . Sine E ′ x is nite dimensional, there exists a φ ′ -orthonormal frame for E ′ ab out x , let us sa y ζ ′ 1 , . . . , ζ ′ d ∈ Γ E ′ ( U ) . Put ζ i = Γ e ( U ) ζ ′ i ∈ Γ E ( U ) , let φ U b e the metri indued on E | U , and onsider (10) ζ i : E | U ∼ = E | U ⊗ 1 U ∼ = E | U ⊗ 1 | U ∗ E | U ⊗ ζ ∗ i − − − − → E | U ⊗ E | U ∗ φ U − → 1 U . Dene p U : E | U → E ′ | U as the omp osite of E | U ζ 1 ⊕ ··· ⊕ ζ d − − − − − − → 1 ⊕ · · · ⊕ 1 and 1 ⊕ · · · ⊕ 1 ζ ′ 1 ⊕ ··· ⊕ ζ ′ d − − − − − − → E ′ | U . Note that ( p U ) u : E u → E ′ u is the orthogonal pro jetion, with resp et to φ u , on to E ′ u ֒ → E u : it follo ws b y Axiom i i that p U do es not atually dep end on U or the other  hoies in v olv ed, so that w e get a w ell-dened morphism p : E → E ′ , b y the presta k prop ert y; moreo v er, w e ha v e p ◦ e = id for similar reasons. Another appliation: let E ∈ Ob F ( X ) , and supp ose that the dimension of the bres is (nite and) lo ally onstan t o v er X ; then E ∈ Ob V F ( X ) i.e. E is lo  al ly trivial, of lo  al ly nite r ank. Indeed, x an arbitrary p oin t x . By Axiom iv , there exists an op en neigh b ourho o d U of x su h that E | U supp orts a metri φ U . Sine E x is nite dimensional, it is no loss of generalit y to assume that a φ U -orthonormal system ζ 1 , . . . , ζ d ∈ Γ E ( U ) an b e found; one an also assume dim E u = d onstan t o v er U . T ak e e def = ζ 1 ⊕ · · · ⊕ ζ d : E ′ def = 1 ⊕ · · · ⊕ 1 ֒ → E | U and p : E | U → E ′ as ab o v e. It is immediate to see that e and p are brewise in v erse to one another. 11 Lemma Let X b e a paraompat manifold and let S i S ֒ → X b e a losed submanifold. Let F b e a sta k of smo oth elds. Let E , F ∈ Ob F ( X ) , and supp ose that E ′ = E | S b elongs to V F ( S ) , i.e. is lo ally free, of lo ally nite rank. Then ev ery morphism a ′ : E ′ → F ′ in F ( S ) an b e extended to a morphism a : E → F in F ( X ) , i.e. a ′ = a | S for su h an a . 2 It follo ws immediately from Axiom i i that an em b edding is a monomorphism. The on v erse need not b e true b eause the funtor E 7→ E x do esn't ha v e an y exatness prop- erties. F or example, let a b e a smo oth funtion on R su h that a ( t ) = 0 if and only if t = 0 . Then a , regarded as an elemen t of End( 1 ) , is b oth mono and epi in F ( R ) while a 0 = 0 : C → C is neither injetiv e nor surjetiv e. 76 CHAPTER IV. GENERAL T ANNAKA THEOR Y Pro of Fix a p oin t s ∈ S . Then there exists an op en neigh b ourho o d A of s in S su h that o v er A w e an nd a trivialization ( d summands) (12) E ′ | A ≈ 1 A ⊕ · · · ⊕ 1 A . Let ζ ′ 1 , . . . , ζ ′ d ∈ Γ E ′ ( A ) b e the setions orresp onding to this trivialization (so for instane ζ ′ 1 is the omp osite 1 S | A ∼ = 1 A 1st ֒ → 1 A ⊕ · · · ⊕ 1 A ≈ E ′ | A ). Also, let U b e an y op en subset of X su h that U ∩ S = A . No w, b y Axiom i (pull-ba k ase), taking smaller U and A ab out s if neessary , it is no loss of generalit y to assume that there exist lo al setions ζ 1 , . . . , ζ d ∈ Γ E ( U ) with ζ ′ k = ζ k ◦ i S , k = 1 , . . . , d . T o see this, observ e that lo ally ab out s ea h ζ ′ k is a nite linear om bination P j α j,k ( ζ j,k ◦ i S ) with ζ j,k ∈ Γ E ( U ) and α j,k ∈ C ∞ ( A ) , b y the ited axiom; hene if U is  hosen on v enien tly , let us sa y sa y so that there exists a dieomorphism of U on to a pro dut A × R n , the o eien ts α j,k will extend to some smo oth funtions ˜ α j,k ∈ C ∞ ( U ) and ζ k = P j ˜ α j,k ζ j,k will meet our requiremen ts. W e ha v e already observ ed ( 11.24 ) that there is a anonial isomorphism of v etor spaes ( i ∗ S E ) s ∼ = E i ( s ) whi h mak es ( ζ k ◦ i S )( s ) orresp ond to ζ k ( x ) , where w e put x = i S ( s ) . Hene the v alues ζ k ( x ) , k = 1 , . . . , d are linearly indep enden t in the bre E x , b eause the same is true of the v alues ζ ′ k ( s ) , k = 1 , . . . , d in E ′ s (the trivializing isomorphism (12 ) ab o v e yields a linear isomorphism ( E ′ ) s ≈ C d whi h, as one an easily  he k, mak es ζ ′ k ( s ) or- resp ond to the k -th standard basis v etor of C d ). This implies that if U is small enough then the morphism ζ = ζ 1 ⊕ · · · ⊕ ζ d : 1 U ⊕ · · · ⊕ 1 U → E | U is an em b edding and admits a osetion p : E | U → 1 U ⊕ · · · ⊕ 1 U , b y Axiom iv (existene of lo al metris). Next, set η ′ k = Γ a ′ ( A ) ζ ′ k ∈ Γ F ′ ( A ) . As remark ed earlier in the pro of, it is no loss of generalit y to assume that there exist partial setions η 1 , . . . , η d in Γ F ( U ) with η ′ k = η k ◦ i S . Again, these setions an b e om bined in to a morphism η : 1 U ⊕ · · · ⊕ 1 U → F | U ( d -fold diret sum). Finally , w e an tak e the omp osite E | U p − → 1 U ⊕ · · · ⊕ 1 U | {z } d summands η − → F | U . It is immediate to  he k that the restrition of this morphism to the sub- manifold A ֒ → U oinides with a ′ | A , up to the anonial iden tiations ( E | U ) | A ∼ = E ′ | A and ( F | U ) | A ∼ = F ′ | A . Let us summarize briey what w e ha v e done so far: starting from an arbitra y p oin t s ∈ S , w e ha v e found an op en neigh b ourho o d U = U s of x = i S ( s ) in X , along with a morphism a s : E | U → F | U whose restrition to A = U ∩ S agrees with a ′ | A . This means that w e ha v e solv ed our problem lo ally . T o onlude the pro of, onsider the op en o v er of X formed b y the op en subsets { U s : s ∈ S } and the omplemen t U = ∁ X S . (Here w e use, of 15. ST A CKS OF SMOOTH FIELDS 77 ourse, the losedness of S .) Sine X is a paraompat manifold, w e an nd a smo oth partition of unit y { θ i : i ∈ I } ∪ { θ } sub ordinated to this op en o v er. Thenb y the presta k prop ert ythe sum a def = P i ∈ I θ i a s i orresp onds to a w ell-dened morphism E → F in F ( X ) , learly extending a ′ . q.e.d. The last t w o axioms imp ose v arious niteness requiremen ts, b oth on the bres and on the sheaf of smo oth setions of an ob jet. T o b egin with, there is a sto  k of onditions w e shall imp ose on F in order that the ategory F ( ⋆ ) ma y b e equiv alen t, as a tensor ategory , to the ategory of v etor spaes of nite dimension. W e gather these onditions in to what w e all the  dimension axiom: 13 Axiom V (dimension) It is r e quir e d of the  anoni al pseudo-tensor funtor (11.9 ) : F ( ⋆ ) → { v ector spaces } that a) it is fully faithful; b) it fators thr ough the sub  ate gory whose obje ts ar e the nite dimen- sional ve tor sp a es, in other wor ds E ∗ (11.10 ) is nite dimensional for al l E ∈ F ( ⋆ ) ; ) it is a genuine tensor funtor, i.e. (11 .7) and (11.8) b e  ome iso- morphisms of she aves for X = ⋆ . In partiular, for ea h ob jet V ∈ F ( ⋆ ) there exists a trivialization of V , i.e. an isomorphism V ≈ 1 ⊕ · · · ⊕ 1 ( nite diret sum). The n um b er of opies of 1 in an y su h deomp osition determines the dimension of an ob jet. Moreo v er, it follo ws from this axiom, and preisely from ) , that the funtor `bre at x ', E 7→ E x is a  omplex tensor funtor. (In general, it is only a  omplex pseudo-tensor funtor, see 11.) An ob jet E of F ( X ) is lo  al ly nite, if Γ E is a lo ally nitely generated C ∞ X -mo dule. In other w ords, E is lo ally nite if the manifold X admits a o v er b y op en subsets U su h that there exist lo al setions ζ 1 , . . . , ζ d ∈ Γ E ( U ) with the prop ert y (14) Γ E | U = C ∞ U { ζ 1 , . . . , ζ d } . (The expression on the righ t-hand side has a lear meaning as a presheaf of setions o v er U ; sine it is alw a ys p ossible to assume U paraompat, this presheaf is in fat a sheaf, as one an easily see b y means of partitions of unit y .) The ondition on U amoun ts to the existene of an epimorphism of shea v es of mo dules (15) C ∞ U ⊕ · · · ⊕ C ∞ U | {z } d summands ։ Γ E | U . 78 CHAPTER IV. GENERAL T ANNAKA THEOR Y 16 Axiom VI (lo al niteness) L et X b e a smo oth manifold. Every obje t E ∈ Ob F ( X ) is lo  al ly nite. The presen t axiom, lik e Axiom i i i ab o v e, will pla y a role in the pro of of the `A v eraging Lemma' only , in 17. 16 Smo oth Eulidean Fields Our next goal in this setion is to elab orate a onrete mo del for the axioms w e just prop osed. Of ourse, in order to b e useful, su h a mo del ough t to on tain m u h more than just v etor bundles: in fat, w e in tend to exploit it later on, in 20, to pro v e a general reonstrution theorem for prop er Lie group oids. W e rst in tro due a somewhat w eak er notion whi h, ho w ev er, is of some in terest on its o wn. 1 Denition By a smo oth Hilb ert eld w e mean an ob jet H onsisting of (a) a family { H x } of Hilb ert spaes, indexed o v er the set of p oin ts of a manifold X , and (b) a sheaf Γ H of C ∞ X -mo dules of lo al setions of { H x } , sub jet to the follo wing onditions: i)  ζ ( x ) : ζ ∈ ( Γ H ) x  , where ( Γ H ) x indiates the stalk at x , is a dense linear subspae of H x ; ii) for ea h op en subset U , and for all setions ζ , ζ ′ ∈ Γ H ( U ) , the funtion h ζ , ζ ′ i on U dened b y u 7→  ζ ( u ) , ζ ′ ( u )  turns out to b e smo oth. W e refer to the manifold X as the b ase of H ; w e an also sa y that H is a smo oth Hilb ert eld over X . Some explanations are p erhaps in order. By a  lo al setion of { H x }  w e mean here an elemen t of the pro dut Q x ∈ U H x of all the spaes o v er some op en subset U of X . The denition establishes in partiular that for ea h op en subset U the set of setions Γ H ( U ) is a submo dule of the C ∞ ( U ) -mo dule of all the setions of { H x } o v er U . Γ H will b e alled the she af of smo oth se tions of H and the elemen ts of Γ H ( U ) will b e aordingly referred to as the smo oth se tions of H over U . This terminology , o v erlapping with that of 11, has b een in tro dued in ten tionally and will b e justied so on. Next, w e need a suitable notion of morphism. There are v arious p ossibil- ities here. W e  ho ose the notion whi h seems to t our purp oses b etter: a bundle of b ounded linear maps induing a morphism of shea v es of mo dules. Preisely , let E and F b e smo oth Hilb ert elds o v er X . A morphism of E in to F is a family of b ounded linear maps { a x : E x → F x } , indexed o v er the set of p oin ts of X , su h that for ea h op en subset U ⊂ X and for all ζ ∈ Γ E ( U ) the setion o v er U giv en b y u 7→ a u · ζ ( u ) b elongs to Γ F ( U ) . Smo oth Hilb ert elds o v er X and their morphisms form a ategory whi h will b e denoted b y H ∞ ( X ) . W e w an t to turn the op eration X 7→ H ∞ ( X ) in to 16. SMOOTH EUCLIDEAN FIELDS 79 a bred (omplex) tensor ategory H ∞ , in the sense of 11 . This bred tensor ategory will pro v e to b e a smo oth tensor parasta k (but not a sta k: this is the reason wh y w e w ork with the w eak er notion of parasta k) satisfying some of the axioms, althoughof oursenot all of them: for this reason, H ∞ onstitutes a soure of in teresting examples. Let us start with the denition of the tensor struture on the ategory H ∞ ( X ) of smo oth Hilb ert elds. W e shall onern ourselv es with the tensor pro dut of Hilb ert elds in a momen t; b efore doing that ho w ev er w e review the tensor pro dut of Hilb ert spaes. Let V b e a omplex v etor spae. W e denote b y V ∗ the spae ob- tained b y retaining the additiv e struture of V while  hanging the salar m ultipliation in to z v ∗ = ( z v ) ∗ ; the star here indiates that a v etor of V is to b e regarded as one of V ∗ . If φ : E ⊗ E ∗ → C and ψ : F ⊗ F ∗ → C are sesquilinear forms then w e an om bine them in to a sesquilinear form on the tensor pro dut E ⊗ F (2) ( E ⊗ F ) ⊗ ( E ⊗ F ) ∗ ∼ = ( E ⊗ E ∗ ) ⊗ ( F ⊗ F ∗ ) φ ⊗ ψ − − − → C ⊗ C ∼ = C . If w e ompute this form on the generators of E ⊗ F w e get (3) h e ⊗ f , e ′ ⊗ f ′ i = h e, e ′ i h f , f ′ i . Supp ose no w that b oth φ and ψ are Hilb ert spae inner pro duts. Then this form ula sho ws that the form (2) is Hermitian. Moreo v er, if w e express an arbitrary elemen t w of E ⊗ F as a linear om bination k P i =1 ℓ P j =1 a i,j e i ⊗ f j with e 1 , . . . , e k , resp. f 1 , . . . , f ℓ orthonormal in E , resp. F , w e see from (3) that a i,j = h w , e i ⊗ f j i = 0 for all i, j implies w = 0 . Hene the form is non degenerate. The same expression an b e used to sho w that the form is p ositiv e denite: h w , w i = P i,i ′ P j,j ′ a i,j a i ′ ,j ′ δ j,j ′ i,i ′ = P i,j | a i,j | 2 ≧ 0 . The spae E ⊗ F an b e ompleted with resp et to the pre-Hilb ert inner pro dut (2) to a Hilb ert spae alled the  Hilb ert tensor pro dut  of E and F . W e agree that from no w on, when E and F are Hilb ert spaes, the sym b ol E ⊗ F will denote the Hilb ert tensor pro dut of E and F . It is equally easy to see that if a : E → E ′ and b : F → F ′ are b ounded linear maps of Hilb ert spaes then their tensor pro dut extends b y on tin uit y to a b ounded linear map of E ⊗ F in to E ′ ⊗ F ′ that w e still denote b y a ⊗ b . Moreo v er, the anonial isomorphisms of v etor spaes u ⊗ ( v ⊗ w ) 7→ ( u ⊗ v ) ⊗ w et. extend b y on tin uit y to unitary isomorphisms E ⊗ ( F ⊗ G ) ∼ → ( E ⊗ F ) ⊗ G et. of Hilb ert spaes. Supp ose no w that E and F are Hilb ert elds o v er X . Consider the bundle of tensor pro duts { E x ⊗ F x } . F or arbitrary lo al setions ζ ∈ Γ E ( U ) and 80 CHAPTER IV. GENERAL T ANNAKA THEOR Y η ∈ Γ F ( U ) , w e let ζ ⊗ η denote the setion of { E x ⊗ F x } giv en b y u 7→ ζ ( u ) ⊗ η ( u ) . The la w (4) U 7→ C ∞ ( U )  ζ ⊗ η : ζ ∈ Γ E ( U ) , η ∈ Γ F ( U )  denes a sub-presheaf of the sheaf of lo al setions of { E x ⊗ F x } . (W e use expressions of the form C ∞ ( U ) { · · · } to indiate the C ∞ ( U ) -mo dule spanned b y a olletion of setions o v er U .) Let E ⊗ F denote the Hilb ert eld o v er X onsisting of the bundle { E x ⊗ F x } and the sheaf (of setions of this bundle) generated b y the presheaf (4 ), in other w ords, the smallest subsheaf of the sheaf of lo al setions of { E x ⊗ F x } on taining (4 ). W e all E ⊗ F the tensor pr o dut of E and F . Observ e that for all morphisms E α − → E ′ and F β − → F ′ of Hilb ert elds o v er X , the bundle of b ounded linear maps { a x ⊗ b x } yields a morphism α ⊗ β of E ⊗ F in to E ′ ⊗ F ′ . Another op eration whi h applies to Hilb ert spaes is onjugation. This op eration sends a Hilb ert spae E to the onjugate v etor spae E ∗ endo w ed with the Hermitian pro dut h v ∗ , w ∗ i = h w , v i . W e no w arry onjugation of Hilb ert spaes o v er to a funtorial onstrution on Hilb ert elds. Let E b e a Hilb ert eld o v er X . W e get the onjugate eld E ∗ b y taking the bundle { E x ∗ } of onjugate spaes, along with the lo al smo oth setions of E regarded as lo al setions of { E x ∗ } . If α = { a x } : E → F is a morphism of Hilb ert elds o v er X then, sine a linear map a x : E x → F x also maps E x ∗ linearly in to F x ∗ , w e get a morphism α ∗ = { a x ∗ } : E ∗ → F ∗ . Observ e that the orresp ondene α 7→ α ∗ is anti- linear. Note also that E ∗∗ = E . The rest of the onstrution (tensor unit, the v arious onstrain ts . . . ) is ompletely ob vious. One obtains a omplex tensor ategory , that is easily reognized to b e additiv e as a C -linear ategory . It remains to onstrut the omplex tensor funtor f ∗ : H ∞ ( Y ) → H ∞ ( X ) asso iated with a smo oth map f : X → Y , and to dene the onstrain ts (11 .3). Let H b e a Hilb ert eld o v er Y . The pul l-b ak of H along f , denoted b y f ∗ H , is the Hilb ert eld o v er X whose desription is as follo ws: the underlying bundle of Hilb ert spaes, indexed b y the p oin ts of X , is  H f ( x )  ; the sheaf of smo oth setions is generatedas a subsheaf of the sheaf of all lo al setions of the bundle  H f ( x )  b y the presheaf (5) U 7→ C ∞ X ( U )  η ◦ f : η ∈ Γ H ( V ) , V ⊃ f ( U )  . Sine this is a presheaf of C ∞ X -mo dules (of setions), it follo ws that Γ ( f ∗ H ) is a sheaf of C ∞ X -mo dules (of setions). Moreo v er, it is lear that for an y morphism β : H → H ′ of Hilb ert elds o v er Y , the family of b ounded linear maps { b f ( x ) } denes a morphism f ∗ β : f ∗ H → f ∗ H ′ of Hilb ert elds o v er X . Observ e that f ∗ H ⊗ f ∗ H ′ and f ∗ ( H ⊗ H ′ ) are exatly the same smo oth Hilb ert eld o v er X , essen tially b eause ( η ⊗ η ′ ) ◦ f = ( η ◦ f ) ⊗ ( η ′ ◦ f ) ; also C ∞ X = f ∗ C ∞ Y . These iden tities an funtion as tensor funtor 17. CONSTR UCTION OF EQUIV ARIANT MAPS 81 onstrain ts. Similarly f ∗ ( H ∗ ) = ( f ∗ H ) ∗ an b e tak en as a onstrain t, so w e get a omplex tensor funtor f ∗ : H ∞ ( Y ) → H ∞ ( X ) . Sine the iden tities f ∗ ( g ∗ H ) = ( g ◦ f ) ∗ H and id X ∗ H = H hold, the op eration X 7→ H ∞ ( X ) is a strit bred omplex tensor ategory . Note that the `sheaf of setions'dened abstratly only in terms of the presta k struture of H ∞ , as explained in 11turns out to b e preisely the `sheaf of smo oth setions' whi h w e in tro dued in the ab o v e denition as one of the t w o onstituen t data of a smo oth Hilb ert eld. Ho w ev er, note that the bre H x (in the sense of 11) will b e in general only a dense subspae of the Hilb ert spae H x (this is the reason wh y w e use t w o distint notations); of ourse, H x = H x whenev er H x is nite dimensional. Let E ∞ ( X ) b e the full sub ategory of H ∞ ( X ) onsisting of all ob jets E whose sheaf of setions is lo ally nitely generated o v er X , in the sense of Axiom vi . E ∞ ( X ) is a omplex tensor sub ategory i.e. it is losed under ⊗ , ∗ and it on tains the tensor unit: indeed, Γ E ⊗ C ∞ Γ E ′ , whi h is a lo ally nitely generated sheaf of mo dules o v er X b eause su h are Γ E and Γ E ′ , surjets (as a sheaf ) on to Γ ( E ⊗ E ′ ) , b y Axiom i , so the latter will b e lo ally nite to o, as on tended. Moreo v er, the pull-ba k funtor f ∗ : H ∞ ( Y ) → H ∞ ( X ) arries E ∞ ( Y ) in to E ∞ ( X ) . W e obtain a smo oth substa k E ∞ ⊂ H ∞ of additiv e omplex tensor ategories; it is lear that E ∞ satises Axioms i  vi . The ob jets of the sub ategory E ∞ ( X ) ⊂ H ∞ ( X ) will b e referred to as smo oth Eulide an elds o v er X . 17 Constrution of Equiv arian t Maps Let F denote an arbitrary sta k of smo oth elds, to b e regarded as xed throughout the presen t setion. The next lemma is to b e used in om bination with Lemma 15 .11. 1 Lemma Let G b e a (lo ally) transitiv e Lie group oid, and let X b e its base manifold. Consider an y represen tation ( E , ρ ) ∈ R F ( G ) . Then E ∈ V F ( X ) i.e. E is a lo ally trivial ob jet of F ( X ) . Pro of Lo al transitivit y means that the mapping ( s , t ) : G → X × X is a submersion. Fix a p oin t x ∈ X . Sine ( x, x ) lies in the image of the map ( s , t ) , the latter admits a lo al smo oth setion U × U → G o v er some op en neigh b ourho o d of ( x, x ) . Let us onsider the `restrition' g : U → G of this setion to U ≡ U × { x } : g will b e a smo oth map for whi h the iden tities s ( g ( u )) = u and t ( g ( u ) ) = x hold for all u ∈ U . Let ⋆ x − → X denote the map ⋆ 7→ x . W e ha v e already notied that, b y the `dimension' Axiom (15.13 ), there is an isomorphism x ∗ E ≈ 1 ⊕ · · · ⊕ 1 (a trivialization) in F ( ⋆ ) . No w, it will b e enough to pull ρ ba k to U along the smo oth map g and observ e that there is a fatorization of the map t ◦ g 82 CHAPTER IV. GENERAL T ANNAKA THEOR Y as the ollapse c : U → ⋆ follo w ed b y x : ⋆ → X in order to onlude that there is also a trivialization E | U ≈ 1 U ⊕ · · · ⊕ 1 U in F ( U ) . Indeed, sine ρ is an isomorphism, one an form the follo wing long in v ertible  hain E | U = i ∗ U E = ( s ◦ g ) ∗ E ∼ = g ∗ s ∗ E g ∗ ρ − − → g ∗ t ∗ E ∼ = ( t ◦ g ) ∗ E = = ( x ◦ c ) ∗ E ∼ = c ∗ ( x ∗ E ) ≈ c ∗ ( 1 ⊕ · · · ⊕ 1 ) = 1 U ⊕ · · · ⊕ 1 U (reall that the pull-ba k c ∗ preserv es diret sums). q.e.d. Let i : S ֒ → X b e an invariant immersed submanifold, viz. one whose image i ( S ) is an in v arian t subset under the `tautologial' ation of G on its o wn base. The pull-ba k of G along i mak es sense and pro v es to b e a Lie subgroup oid 3 ι : G | S ֒ → G of G . (Observ e that G | S = G S = s − 1 G ( S ) .) In the sp eial ase of an orbit immersion, G | S will b e a transitiv e Lie group oid o v er S . Then the lemma sa ys that for an y ( E , ρ ) ∈ Ob R ( G ) the pull-ba k i ∗ S E is a lo ally trivial ob jet of F ( S ) , b eause the transitiv e Lie group oid R ( G | S ) ats on i ∗ S E via ι ∗ S ρ . In partiular, when the orbit S ֒ → X is a submanifold, w e an also write E | S = i ∗ S E ∈ V F ( S ) . 2 Note The notion of Lie group oid represen tation w e ha v e b een w orking with so far is ompletely in trinsi. W e w ere able to pro v e all results b y means of purely formal argumen ts, in v olving only manipulations of omm utativ e diagrams. F or the purp oses of the presen t setion, ho w ev er, w e ha v e to  hange our p oin t of view. Let G b e a Lie group oid. Consider a represen tation ( E , ρ ) ∈ Ob R ( G ) , s ∗ E ρ − → t ∗ E . Ea h arro w g determines a linear map ρ ( g ) : E s ( g ) → E t ( g ) dened via the omm utativit y of the diagram [ g ∗ s ∗ E ] ∗ [ g ∗ ρ ] ∗   [ ∼ = ] ∗ / / [ s ( g ) ∗ E ] ∗ def. E s ( g ) ρ ( g )      [ g ∗ t ∗ E ] ∗ [ ∼ = ] ∗ / / [ t ( g ) ∗ E ] ∗ def. E t ( g ) (3) where the notation (11.10 ) is used. It is routine to  he k that the o yle onditions (13.2) and (13.3) in the denition of represen tation imply that the orresp ondene g 7→ ρ ( g ) is m ultipliativ e i.e. that ρ ( g ′ g ) = ρ ( g ′ ) ◦ ρ ( g ) and ρ ( x ) = id for ea h p oin t of the base manifold X . Next, onsider an y arro w g 0 . Also, let ζ ∈ Γ E ( U ) b e a setion dened o v er a neigh b ourho o d of s ( g 0 ) in X . Reall that aording to (11 .21) ζ will determine the setion ζ ◦ s ∈ Γ G ( s ∗ E )( G U ) , dened o v er the op en subset G U = s − 1 ( U ) of the manifold of arro ws G (1) ; the morphism of shea v es of mo dules Γ ρ an b e ev aluated at ζ ◦ s : [ Γ ρ ( G U )]( ζ ◦ s ) ∈ Γ ( t ∗ E )( G U ) . Axiom 3 In general, a  Lie sub gr oup oid  is a Lie group oid homomorphism ( ϕ, f ) su h that b oth ϕ and f are injetiv e immersions. 17. CONSTR UCTION OF EQUIV ARIANT MAPS 83 (15 .1) implies that there exists an op en neigh b ourho o d Γ ⊂ G U of g 0 o v er whi h [ Γ ρ ( G U )]( ζ ◦ s ) an b e expressed as a nite linear om bination, with o eien ts in C ∞ (Γ) , of setions of the form ζ ′ i ◦ t with ζ ′ i , i = 1 , · · · , d dened o v er t (Γ) . Expliitly , (4)  Γ ρ (Γ)  ( ζ ◦ s | Γ ) = d P i =1 r i ( ζ ′ i ◦ t ) | Γ with r 1 , . . . , r d ∈ C ∞ (Γ) and ζ ′ 1 , . . . , ζ ′ d ∈ ( Γ E )( t (Γ)) . This equalit y an b e ev aluated at g ∈ Γ in the abstrat sense of (11 .14), also taking (3) in to aoun t, to get a more in tuitiv e expression (5) ρ ( g ) · ζ ( s g ) = d P i =1 r i ( g ) ζ ′ i ( t g ) . T o summarize: an y G -ation ( E , ρ ) determines an op eration g 7→ ρ ( g ) whi h assigns a linear isomorphism E x ρ ( g ) − − → E x ′ to ea h arro w x g − → x ′ in su h a w a y that the omp osition of arro ws is resp eted; moreo v er, the op eration enjo ys a `smo othness prop ert y' whose te hnial form ulation is syn thesized in Equation (5 ). Con v ersely , it is y et another exerise to reognize that su h data determine an ation of G on E , b y Axiom ( 15 .6). Therefore w e see that for the represen tations whose t yp e is a sta k of smo oth elds the in trinsi denition of 13 is equiv alen t to a more onrete denition in v olving an op eration g 7→ ρ ( g ) and a `smo othness ondition' expressed p oin t wise. Let G b e a Lie group oid o v er a manifold X . Consider an y represen tation ( E , ρ ) ∈ Ob R ( G ) . Fix an arbitrary p oin t x 0 ∈ X . Using the remarks of the preeding note, the fat that the bre E 0 def = E x 0 is a nite dimensional v etor spae, b y Axiom (15.13 ), and the fat that the ev aluation map (15 .2) ( Γ E ) 0 → E 0 , ζ 7→ ζ ( x 0 ) is surjetiv e, one sees at one that the op eration (6) ρ 0 : G 0 → GL ( E 0 ) , g 7→ ρ ( g ) is a smo oth represen tation of the Lie group G = G 0 (= the isotrop y group at x 0 ) on the nite dimensional v etor spae E 0 . No w, supp ose w e are giv en a G -equiv arian t linear map A : E 0 → F 0 , for some other G -ation ( F , σ ) . Let S ֒ → X b e the orbit through x 0 ; just to x ideas, assume it is a submanifold. The theory of Morita equiv alenes of 14 sa ys that there exists a unique morphism A ′ : ( E | S , ρ | S ) → ( F | S , σ | S ) in R ( G | S ) su h that ( A ′ ) 0 = A , up to the standard anonial iden tiations. A tually , for an y p oin t z ∈ S and an y arro w g ∈ G ( x 0 , z ) one has (7) ( A ′ ) z = σ ( g ) · A · ρ ( g ) − 1 : E z → F z . 84 CHAPTER IV. GENERAL T ANNAKA THEOR Y Set E ′ = E | S . As remark ed earlier, sine the group oid G | S is transitiv e it follo ws that the ob jet E ′ is lo ally trivial, b y Lemma 1. If the submanifold S ֒ → X is in addition losed then, sine base manifolds of Lie group oids are alw a ys paraompat, Lemma 15.11 will yield a morphism a : E → F extending A ′ and hene, a fortiori, A . The a v eraging op erator W e are no w ready to desrib e an  a v eraging te hnique  whi h is of en tral imp ortane in our w orkas the reader will see. W e explain in detail ho w, starting from an y (righ t-in v arian t) Haar system µ = { µ x } on a prop er Lie group oid G o v er a manifold M , one an onstrut, for ea h pair of represen- tations R = ( E , ρ ) , S = ( F , σ ) ∈ R ( G ) (of t yp e F ), a linear op erator (8) Av µ : Hom F ( M ) ( E , F ) → Hom R ( G ) ( R, S ) alled the  a v eraging op erator (of t yp e F )  asso iated with µ , with the prop ert y that Av µ ( a ) = a whenev er a already b elongs to the subspae Hom R ( G ) ( R, S ) ⊂ Hom F ( M ) ( E , F ) . This onstrution will b e ompatible with the restrition to an in v arian t submanifold of the base: namely , if N ⊂ M is an y su h submanifold then, letting ν denote the Haar system indued b y µ on the subgroup oid G | N = G N ι N ֒ → G (what w e are sa ying mak es sense b eause N is in v arian t), the follo wing diagram will omm ute Hom F ( M ) ( E , F ) i ∗ N   Av µ / / Hom R ( G ) ( R, S ) ι ∗ N   Hom F ( N ) ( E | N , F | N ) Av ν / / Hom R ( G | N ) ( ι ∗ N R, ι ∗ N S ) . (9) Th us, in partiular, if a restrits to an in v arian t morphism o v er N then Av µ ( a ) | N = a | N . Sine µ will b e xed throughout the presen t disussion, w e abbreviate Av µ ( a ) in to ˜ a from no w on. W e start from a v ery simple remark, v alid ev en without assuming G to b e prop er. Supp ose that ζ ∈ Γ E ( U ) and η 1 , . . . , η n ∈ Γ F ( U ) are setions o v er some op en subset of M , and moreo v er that η 1 , . . . , η n are lo al generators for Γ F o v er U ; then for ea h g 0 ∈ G U = s − 1 ( U ) there exists an op en neigh b our- ho o d g 0 ∈ Γ ⊂ G U , along with smo oth funtions φ 1 , . . . , φ n ∈ C ∞ (Γ) , su h that the iden tit y (10) σ ( g ) − 1 · a t ( g ) · ρ ( g ) · ζ ( s g ) = n P j =1 φ j ( g ) η j ( s g ) holds in the bre F s ( g ) for all g ∈ Γ . T o see this, reall that, aording to Note 2, there are an op en neigh b ourho o d Γ of g 0 in G U and lo al smo oth 17. CONSTR UCTION OF EQUIV ARIANT MAPS 85 setions ζ ′ 1 , . . . , ζ ′ m of E o v er U ′ = t (Γ) , su h that ρ ( g ) ζ ( s g ) = m P i =1 r i ( g ) ζ ′ i ( t g ) for some smo oth funtions r 1 , . . . , r m ∈ C ∞ (Γ) . F or i = 1 , . . . , m , put η ′ i = Γ a ( U ′ )( ζ ′ i ) ∈ Γ F ( U ′ ) . Sine Γ − 1 is a neigh b ourho o d of g − 1 0 w e an assumeagain b y Note 2 , using the h yp othesis that the η j 's are genera- tors Γ to b e so small that for ea h i = 1 , . . . , m there exist smo oth fun- tions s 1 ,i , . . . , s n,i ∈ C ∞ (Γ − 1 ) with σ ( g − 1 ) η ′ i ( t g ) = n P j =1 s j,i ( g − 1 ) η j ( s g ) ∀ g ∈ Γ . Hene for all g ∈ Γ w e get σ ( g ) − 1 · a t ( g ) · ρ ( g ) · ζ ( s g ) = σ ( g − 1 ) · a t ( g ) · m P i =1 r i ( g ) ζ ′ i ( t g ) = = m P i =1 r i ( g ) σ ( g − 1 ) η ′ i ( t g ) = n P j =1  m P i =1 r i ( g ) s j,i ( g − 1 )  η j ( s g ) , whi h is (10) with φ j ( g ) = m P i =1 r i ( g ) s j,i ( g − 1 ) , j = 1 , . . . , n . Let α = Γ a ∈ Hom C ∞ ( Γ E , Γ F ) . W e an use the last remark to obtain a morphism ˜ α : Γ E → Γ F of shea v es of mo dules o v er M , in the follo wing w a y . Let ζ b e a lo al smo oth setion of E , dened o v er an op en subset U ⊂ M so small that there exists a system η 1 , . . . , η n of lo al generators for F o v er U (su h a system an alw a ys b e found lo ally , b eause F satises Axiom (15.16 )). F or ea h g 0 ∈ G U = s − 1 ( U ) , selet an op en neigh b ourho o d Γ( g 0 ) , along with smo oth funtions φ g 0 1 , . . . , φ g 0 n ∈ C ∞  Γ( g 0 )  , as in ( 10). Sine the manifold of arro ws of G , andonsequen tlyits op en submanifold G U , is paraompat (w e are assuming G prop er no w; f.  1), there will b e a smo oth partition of unit y { θ i } , i ∈ I on G U sub ordinated to the op en o v er { Γ( g ) } , g ∈ G U . Then w e put (11) ˜ α ( U ) ζ = n P j =1 Φ j η j , where Φ j ( u ) = Z G u P i ∈ I θ i ( g ) φ i j ( g ) d µ u ( g ) (note that the in tegrand P i ∈ I θ i φ i j is a smo oth funtion on G U and hene Φ j ∈ C ∞ ( U ) , j = 1 , . . . , n ). Of ourse, man y arbitrary  hoies are in v olv ed here, so one has to mak e sure that this denition is not am biguous (ho w ev er, as so on as ( 11 ) is kno wn to b e indep enden t of all these  hoies, it will ertainly dene a morphism of shea v es of mo dules o v er M ). One an do this, in t w o steps, b y in tro duing indep enden tly a ertain bundle of linear maps { λ x : E x → F x } o v er M rst and then  he king that [ ˜ α ( U ) ζ ]( u ) = λ u  ζ ( u )  for all u ∈ U . Sine the righ t-hand term in the last equalit y will not dep end on an y  hoie, Axiom (15 .4) will imply at one that ˜ α ( U ) ζ is a w ell-dened setion of F o v er U . The same equalit y will furthermore yield the onlusion that ˜ α ∈ Hom C ∞ M ( Γ E , Γ F ) is equal to Γ ˜ a for a unique ˜ a ∈ Hom F ( M ) ( E , F ) , b y Axiom (15.6). It should b e lear ho w to pro eed no w, but let us arry out 86 CHAPTER IV. GENERAL T ANNAKA THEOR Y the details an yw a y , for ompleteness. If w e lo ok at (10) with s ( g ) = x xed, w e immediately reognize that the map (12) G x → F x , g 7→ σ ( g ) − 1 · a t ( g ) · ρ ( g ) · ζ ( x ) , of the manifold G x = s − 1 ( x ) in to the nite dimensional v etor spae F x , is of lass C ∞ and hene on tin uous. Sine for ea h v ∈ E x there is some lo al setion ζ of E ab out x su h that v = ζ ( x ) , b y Axiom (15.1), w e an write do wn the in tegral (13) a µ ( x ) · v def = Z G x σ ( g ) − 1 · a t ( g ) · ρ ( g ) · v d µ x ( g ) for ea h v ∈ E x . Clearly v 7→ a µ ( x ) · v denes a linear map of E x in to F x , so w e get our bundle of linear maps  a µ ( x ) : E x → F x  . It remains to  he k, for an arbitrary u ∈ U , the equalit y [ ˜ α ( U ) ζ ]( u ) = a µ ( u ) · ζ ( u ) with ˜ α ( U ) ζ giv en b y (11 ). The omputation is straigh tforw ard: [ ˜ α ( U ) ζ ]( u ) = n P j =1 Φ j ( u ) η j ( u ) = n P j =1 Z G u P i ∈ I θ i ( g ) φ i j ( g ) d µ u ( g ) η j ( u ) = Z G u P i ∈ I θ i ( g ) n P j =1 φ i j ( g ) η j ( s g ) d µ u ( g ) = Z G u P i ∈ I θ i ( g )  σ ( g ) − 1 · a t ( g ) · ρ ( g ) · ζ ( s g )  d µ u ( g ) = a µ ( u ) · ζ ( u ) . In onlusion, w e dene Av µ ( a ) as the unique morphism ˜ a : E → F ∈ F ( M ) su h that Γ ˜ a = g ( Γ a ) . The linearit y of a 7→ Av µ ( a ) follo ws no w from (13 ), the relation [ ˜ α ( U ) ζ ]( u ) = a µ ( u ) · ζ ( u ) and the faithfulness of a 7→ Γ a . It remains to sho w that Av µ ( a ) b elongs to Hom R ( G ) ( R, S ) and that Av µ ( a ) equals a when a already b elongs to Hom R ( G ) ( R, S ) ; although the alulation is ompletely standard, w e review it b eause of its imp ortane. In order to pro v e that ˜ a ≡ Av µ ( a ) is a morphism of G -ations, it will b e enough (b y Axiom 15.4) to  he k the iden tit y ˜ a t ( g ) ◦  ( g ) = σ ( g ) ◦ ˜ a s ( g ) or equiv alen tly , letting x = s ( g ) and x ′ = t ( g ) , the iden tit y a µ ( x ′ ) ◦  ( g ) = σ ( g ) ◦ a µ ( x ) for ea h arro w g ; the orresp onding omputation reads as follo ws: a µ ( x ′ ) ◦  ( g ) = Z G ( x ′ , - ) σ ( g ′ ) − 1 a t ( g ′ )  ( g ′ )  ( g ) d µ x ′ ( g ′ ) b y (13 ) = Z G ( x, - ) σ ( g ) σ ( h ) − 1 a t ( h )  ( h ) d µ x ( h ) b y righ t-in v ariane = σ ( g ) ◦ a µ ( x ) b y (13 ) again. 17. CONSTR UCTION OF EQUIV ARIANT MAPS 87 Next, whenev er a is an elemen t of Hom R ( G ) ( R, S ) , the omputation a µ ( x ) = Z G ( x, - ) σ ( g ) − 1 a t ( g )  ( g ) d µ x ( g ) b y (13) = Z G ( x, - ) a x d µ x ( g ) b eause a ∈ Hom R ( G ) ( R, S ) = a x b eause µ is normalized pro v es the iden tit y ˜ a = a . Appliations F or the reader's on v eniene and for future referene, it will b e useful to ollet the onlusions of the previous subsetion in to a single statemen t. As ev er, F will denote an arbitrary sta k of smo oth elds, for example the sta k of smo oth v etor bundles or the sta k of smo oth Eulidean elds. 14 Prop osition (A v eraging Lemma) Let G b e a prop er Lie group oid o v er a manifold M , and let µ b e a righ t-in v arian t Haar system on G . Then for an y giv en G -ations R = ( E ,  ) and S = ( F , σ ) of t yp e F , ea h morphism a : E → F in the ategory F ( M ) determines a (unique) morphism ˜ a = Av µ ( a ) : R → S ∈ R F ( G ) through the requiremen t that for ea h x ∈ M the map ˜ a x : E x → F x should b e giv en b y the form ula ˜ a x ( v ) = Z G x σ ( g ) − 1 · a t ( g ) ·  ( g ) · v d µ x ( g ) .  ∀ v ∈ E x  (15) In partiular, ˜ a = a for all G -equiv arian t a . W e will no w deriv e a series of useful orollaries, whi h en ter as k ey ingredien ts in man y pro ofs throughout 20. 16 Corollary (Isotrop y Extension Lemma) Let G b e a prop er Lie group oid o v er a manifold M , and let x 0 ∈ M b e an y p oin t. Let R = ( E ,  ) and S = ( F , σ ) b e G -ations of t yp e F and put E 0 ≡ E x 0 and F 0 ≡ F x 0 . Moreo v er, let A : E 0 → F 0 b e a G -equiv arian t linear map, where G ≡ G 0 denotes the isotrop y group of G at x 0 . Then there exists a morphism a : R → S in R F ( G ) su h that a 0 ≡ a x 0 = A . Pro of Apply Lemma 15 .11 and then the A v eraging Lemma to the morphism A S : ( E | S ,  | S ) → ( F | S , σ | S ) ∈ R F ( G | S ) (7 ), where S = G · x 0 . The orollary will follo w from the form ula (15 ) written at x = x 0 . q.e.d. 88 CHAPTER IV. GENERAL T ANNAKA THEOR Y 17 Corollary (Existene of In v arian t Metris) Let G b e a prop er Lie group oid o v er a manifold M . Let R = ( E ,  ) ∈ R ( G ) b e a represen tation. Then there exists a metri m : R ⊗ R ∗ → 1 in R ( G ) . Pro of Cho ose an y metri φ : E ⊗ E ∗ → 1 in F ( M ) (su h metris exist b eause F satises Axiom 15.8 and M is paraompat); also x an y righ t- in v arian t Haar system µ on G . By applying the a v eraging op erator w e obtain a morphism ˜ φ = Av µ ( φ ) : R ⊗ R ∗ → 1 in R ( G ) . W e on tend that ˜ φ is an in v arian t metri on R . It sues to pro v e that for ea h x ∈ M the indued form ˜ φ x : E x ⊗ E x ∗ → C is a Hilb ert metri (i.e. Hermitian and p ositiv e denite). F orm ula ( 15 ) reads ˜ φ x ( v , w ) = Z G x   ( g ) v ,  ( g ) w  φ d µ x ( g ) ,  ∀ v , w ∈ E x  (18) whene our laim is eviden t. q.e.d. Let R = ( E ,  ) b e an y G -ation. By a G -invariant setion of E , dened o v er an in v arian t submanifold N of the base M of G , w e mean an y setion ζ ∈ Γ( N ; E | N ) whi h is at the same time a morphism 1 → R | N in R ( G | N ) . 19 Corollary (In v arian t Setions) Let S b e a losed in v arian t submani- fold of the base M of a prop er Lie group oid G . Let R = ( E ,  ) ∈ R ( G ) b e a represen tation. Then ea h G -in v arian t setion ξ of E o v er S an b e extended to a global G -in v arian t setion; in other w ords, there exists some G -in v arian t Ξ ∈ Γ( M ; E ) su h that Ξ | S = ξ . Pro of Apply Lemma 15.11 and the A v eraging Lemma. q.e.d. In general, w e shall sa y that a partial funtion ϕ : S → C , dened on an arbitrary subset S ⊂ M , is smo oth when for ea h x ∈ M one an nd an op en neigh b ourho o d B of x in M and a smo oth funtion B → C that restrits to ϕ o v er B ∩ S . 20 Corollary (In v arian t F untions) Let S b e an y in v arian t subset of the base manifold M of a prop er Lie group oid G . Supp ose ϕ : S → R is a smo oth in v arian t funtion (i.e. ϕ ( g · s ) = ϕ ( s ) for all g ). Then there exists a smo oth in v arian t funtion Φ : M → R extending ϕ outside S . Pro of Apply the A v eraging Lemma to an y smo oth funtion extending ϕ outside S (su h an extension an b e obtained b y means of a partition of unit y o v er M , b eause of the smo othness of ϕ ). q.e.d. 18. FIBRE FUNCTORS 89 18 Fibre F untors Let F b e a sta k of omplex smo oth elds, to b e regarded as xed one and for all. Let M b e a paraompat smo oth manifold. 1 Denition By a br e funtor (of typ e F ) over M , or with b ase M , w e mean a faithful omplex tensor funtor (2) ω : C − → F ( M ) , of some additiv e omplex tensor ategory C , with v alues in to F ( M ) . W e do not assume C to b e rigid. When a bre funtor ω is assigned o v er M , one an onstrut a group oid T ( ω ) ha ving the p oin ts of M as ob jets. Under reasonable assumptions, it is p ossible to mak e T ( ω ) a top ologial group oid o v er the (top ologial) spae M ; the  hoie of a top ology is ditated b y the idea that the ob jets of C should giv e rise to  on tin uous represen tations  of T ( ω ) and that, vie v ersa, on tin uit y of these represen tations should b e enough to  haraterize the top ologial struture. An impro v emen t of the same idea leads one to study a ertain funtional strutur e on T ( ω ) , in the sense of Br e don (1972), p. 297, and the imp ortan t related problem of determining suien t onditions for this funtional struture to b e ompatible with the group oid op erations. Another fundamen tal issue here is to understand whether one gets in fat a manifold strutur e 4 making T ( ω ) a Lie group oid o v er M ; if this pro v es to b e the ase, w e sa y that the bre funtor ω is smo oth. Some notation is needed rst of all. Let x b e a p oin t of M . If x also denotes the (smo oth) map ⋆ → M , ⋆ 7→ x , one an onsider the omplex tensor funtor `bre at x ' whi h w as in tro dued in 11 (3) F ( M ) → { v ector spaces } , E 7→ E x def = ( x ∗ E ) ∗ . Let ω x b e the omp osite omplex tensor funtor (4) C ω − → F ( M ) ( - ) x − − → { v ector spaces } , R 7→ ω x ( R ) def =  ω ( R )  x . Dene the  omplex, resp. r e al, T annakian gr oup oid of ω in the follo wing w a y: for x, x ′ ∈ M , put (5) ( T ( ω ; C )( x, x ′ ) def = Iso ⊗ ( ω x , ω x ′ ) T ( ω ; R )( x, x ′ ) def = Iso ⊗ , ∗ ( ω x , ω x ′ ) . (Reall that the righ t-hand side of the seond equal sign denotes the set of all the self-onjugate tensor preserving natural isomorphisms ω x ∼ → ω x ′ , that 4 A manifold an b e dened as a top ologial spae endo w ed with a funtional struture lo ally lo oking lik e the struture of smo oth real v alued funtions on some R d . 90 CHAPTER IV. GENERAL T ANNAKA THEOR Y is to sa y , the subset of Iso ⊗ ( ω x , ω x ′ ) onsisting of those λ whi h mak e the follo wing square omm utativ e for ea h ob jet R ∈ Ob( C ) : ω x ( R ) ∗ an. ∼ =   λ ( R ) ∗ / / ω x ′ ( R ) ∗ an. ∼ =   ω x ( R ∗ ) λ ( R ∗ ) / / ω x ′ ( R ∗ ) .) (6) Setting ( λ ′ λ )( R ) = λ ′ ( R ) ◦ λ ( R ) and x ( R ) = id , one obtains t w o group oids o v er the set of p oin ts of M , with in v erse giv en b y λ − 1 ( R ) = λ ( R ) − 1 . W e ma y also express (5 ) in short b y writing T ( ω ; C ) = Aut ⊗ ( ω ) and T ( ω ; R ) = Aut ⊗ , ∗ ( ω ) . Let us in v estigate the relationship b et w een the omplex tannakian group- oid T ( ω ; C ) and its subgroup oid T ( ω ; R ) rst. As a on v enien t notational devie, w e omit writing ω when w e simply refer to the set of arro ws of the tannakian group oid; th us for instane T ( C ) is the set of arro ws of the group- oid T ( ω ; C ) . W e dene a map T ( C ) → T ( C ) , λ 7→ λ , whi h w e all omplex onjugation, b y setting λ ( R ) = λ ( R ∗ ) ∗ ; more preisely , λ ( R ) is dened b y imp osing the omm utativit y of ω x ( R ∗ ) ∗ λ ( R ∗ ) ∗   ∼ = / / ω x ( R ∗∗ ) ω x ( ∼ = ) / / ω x ( R ) λ ( R )      ω x ′ ( R ∗ ) ∗ ∼ = / / ω x ′ ( R ∗∗ ) ω x ′ ( ∼ = ) / / ω x ′ ( R ) . (7) It is straigh tforw ard to  he k that λ ∈ Hom ⊗ ( ω x , ω x ′ ) implies λ ∈ Hom ⊗ ( ω x , ω x ′ ) and that λ 7→ λ is a group oid homomorphism of T ( ω ; C ) in to itself, iden tial on ob jets; this endomorphism is moreo v er in v olutiv e viz. λ = λ . Then w e an  haraterize the arro ws b elonging to the subgroup- oid T ( ω ; R ) as the xed p oin ts of the in v olution λ 7→ λ : (8) T ( R ) = { λ ∈ T ( C ) : λ = λ } . Next, w e endo w the set T = T ( C ) or T ( R ) with a top ology . In order to do this, w e need to in tro due the notion of  metri  in F ( M ) . Let E b e an ob jet of F ( M ) . A metri on E , or supp orte d by E , is a Hermitian form φ : E ⊗ E ∗ → 1 in F ( M ) su h that for all x ∈ M the indued Hermitian form φ x on the bre E x (9) E x ⊗ E x ∗ ∼ = ( E ⊗ E ∗ ) x φ x − → 1 x ∼ = C is p ositiv e denite (and hene turns E x in to a omplex Hilb ert spae of nite dimension). W e start b y dening a olletion R of omplex v alued funtions on T , whi h w e ma y all the  represen tativ e funtions  . (Whenev er w e need to 18. FIBRE FUNCTORS 91 distinguish b et w een T ( C ) and T ( R ) , w e an write R ( C ) or R ( R ) as the ase ma y b e.) Cho ose an ob jet R ∈ Ob( C ) , and let φ b e a metri on the ob jet ω ( R ) of F ( M ) . Also x a pair of global smo oth setions ζ , ζ ′ ∈ Γ ( ω R )( M ) . Y ou get a omplex funtion (10) r R,φ,ζ ,ζ ′ : T → C , λ 7→  λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ )  φ ≡ φ t ( λ )  λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ )  . Then put (11) R =  r R,φ,ζ ,ζ ′ : R ∈ Ob( C ) , φ metri on ω ( R ) in F ( M ) , ζ , ζ ′ ∈ Γ ( ω R )( M )  . W e endo w T with the oarsest top ology making all the funtions in R on tin- uous. F rom no w on in our disussion T ( C ) and T ( R ) will alw a ys b e regarded as top ologial spaes, with this top ology . Observ e that T ( R ) turns out to b e a subspae of T ( C ) ; more expliitly , the top ology on T ( R ) indued b y R ( R ) oinides with the top ology indued from T ( C ) along the inlusion T ( R ) ⊂ T ( C ) . W e no w w an t to establish a few fundamen tal algebrai prop erties of the olletion R of omplex v alued funtions on T . W e are going to sho w that R is a omplex algebra of funtions, and moreo v er that R ( R ) is losed under taking the omplex onjugate. Both assertions are immediate onsequenes of the follo wing iden tities: i) F or all smo oth funtions a ∈ C ∞ ( M ) , (12) ( a ◦ s ) r R,φ,ζ ,ζ ′ = r R,φ,aζ ,ζ ′ and ( a ◦ t ) r R,φ,ζ ,ζ ′ = r R,φ,ζ , aζ ′ ; in partiular, if c ∈ C is onstan t, r R,φ,cζ ,ζ ′ = c r R,φ,ζ ,ζ ′ = r R,φ,ζ , cζ ′ . ii) If w e let τ denote the metri on ω ( 1 ) orresp onding to the trivial metri 1 ⊗ 1 ∗ ∼ = 1 ⊗ 1 ∼ = 1 on the ob jet 1 of F ( M ) , and 1 ∈ Γ ( ω 1 )( M ) orresp ond to the unit y setion of 1 ∈ F ( M ) under the iso υ : 1 ∼ → ω ( 1 ) , then (13) `unit y onstan t funtion' = r 1 ,τ , 1 , 1 . iii) F or an y  hoie of a diret sum R ֒ → R ⊕ S ← ֓ S in C , (14) r R,φ,ζ ,ζ ′ + r S,ψ ,η,η ′ = r R ⊕ S ,φ ⊕ ψ ,ζ ⊕ η ,ζ ′ ⊕ η ′ , where ζ ⊕ η ∈ Γ ( ω ( R ⊕ S ))( M ) et. are obtained b y setting ω ( R ) ⊕ ω ( S ) = ω ( R ⊕ S ) . 92 CHAPTER IV. GENERAL T ANNAKA THEOR Y iv) Allo wing the ob vious (anonial) iden tiations, (15) r R,φ,ζ ,ζ ′ r S,ψ ,η,η ′ = r R ⊗ S ,φ ⊗ ψ ,ζ ⊗ η ,ζ ′ ⊗ η ′ . (F or instane, ζ ⊗ η here denotes really the global setion of ω ( R ⊗ S ) orresp onding to the true ζ ⊗ η in the iso τ R,S : ω ( R ) ⊗ ω ( S ) ∼ → ω ( R ⊗ S ) .) v) Allo wing again some lo ose notation, (16) r R,φ,ζ ,ζ ′ = r R ∗ ,φ ∗ ,ζ ∗ ,ζ ′∗ ◦  λ 7→ λ  . In partiular, sine the omplex onjugation  λ 7→ λ  restrits to the iden tit y on T ( R ) , it follo ws that r R,φ,ζ ,ζ ′ = r R ∗ ,φ ∗ ,ζ ∗ ,ζ ′∗ in R ( R ) . Notie that from the fat that R ( R ) is losed under omplex onjugation it follo ws immediately that the real and imaginary parts of an y funtion in R ( R ) will b elong to R ( R ) as w ell. Th us, if w e let R [ R ] ⊂ R ( R ) denote the subset of all the real v alued funtions, w e an express R ( R ) = C ⊗ R [ R ] as the omplexiation of a real funtional algebra. F or the rest of the setionand for the purp oses of the presen t thesisw e will only b e in terested in studying the real tannakian group oid T ( ω ; R ) . So from no w on w e forget ab out T ( ω ; C ) and simply write T ( ω ) for T ( ω ; R ) . There is one further piee of struture w e w an t to onsider on T ( ω ) , b esides the top ology . Let the sheaf of on tin uous (real v alued) funtions on an arbitrary to- p ologial spae T b e denoted b y C 0 T . Then reall that aording to Br e don (1972), a  funtionally strutured spae is a top ologial spae T , endo w ed with a sheaf of real algebras of on tin uous funtions on T in other w ords, a subsheaf of algebras of C 0 T . A morphism of su h funtionally strutured spaes is then dened as a on tin uous mapping su h that the pullba k of on tin uous funtions along the mapping is ompatible with the funtional strutures. F or more details, w e refer the reader to lo . it., p. 297. W e adopt this p oin t of view in order to obtain a natural surrogate on T ( R ) of the no- tion of  smo oth funtion  , dra wing on the in tuition that the represen tativ e funtions should b e regarded as the protot yp e  smo oth funtions  . It is ob vious that if w e start from the idea that the (real) represen tativ e funtions are  smo oth then so w e ha v e to regard an y funtion obtained b y omp osing them with a smo oth funtion f : R d → R . Dene R ∞ to b e the sheaf, of on tin uous real v alued funtions on the spae T = T ( R ) , generated b y the presheaf (17) Ω 7→  f ( r 1 | Ω , . . . , r d | Ω ) : f : R d → R of lass C ∞ , r 1 , . . . , r d ∈ R [ R ]  . In other w ords, R ∞ is the smallest subsheaf of C 0 T on taining (17) as a sub- presheaf. The expression f ( r 1 | Ω , . . . , r d | Ω ) denotes of ourse the funtion λ 7→ 18. FIBRE FUNCTORS 93 f  r 1 ( λ ) , . . . , r d ( λ )  , λ ∈ Ω . Sine (17) is eviden tly a presheaf of R -algebras of on tin uous funtions on T , R ∞ will b e a sheaf of su h algebras and hene the pair ( T , R ∞ ) will onstitute a funtionally strutured spae. Of ourse, w e w ould lik e to sa y that the funtional struture R ∞ on T is ompatible with the group oid struture of T ( ω ) . This means that the struture maps of T ( ω ) should b e all morphisms of funtionally strutured spaes, the base M b eing regarded as su h a spae b y means of its o wn sheaf of smo oth real v alued funtions; in partiular, the struture maps should b e all on tin uous. What w e are sa ying is not v ery preise, of ourse, unless w e turn the spae of omp osable arro ws itself in to a funtionally strutured spae. Let us b egin b y observing that if ( X , F ) and ( Y , G ) are an y funtion- ally strutured spaes then so is their Cartesian pro dut endo w ed with the sheaf F ⊗ G lo ally generated b y the funtions ( ϕ ⊗ ψ )( x, y ) = ϕ ( x ) ψ ( y ) . Then one an rep eat the foregoing pro edure to obtain, on X × Y , a sheaf ( F ⊗ G ) ∞ of lass C ∞ , i.e. losed under omp osition with arbitrary smo oth funtions as in (17). An y subspae S ⊂ X × Y ma y b e nally regarded as a funtionally strutured spae b y endo wing it with the indued sheaf ( F ⊗ G ) ∞ | S def = i S ∗ [( F ⊗ G ) ∞ ] , where i S denotes the inlusion mapping of S in to X × Y . (Reall that if f : S → T is an y on tin uous mapping in to a funtionally strutured spae ( T , T ) then f ∗ T is the funtional sheaf on S asso iated with the presheaf U 7→ lim − → V ⊃ f ( U ) T ( V ) .  Notie that in ase X and Y are smo oth manifolds and S ⊂ X × Y is a submanifold, one reo v ers the orret funtional strutures: ( C ∞ X ⊗ C ∞ Y ) ∞ = C ∞ X × Y and C ∞ X × Y | S = C ∞ S . It is therefore p erfetly reasonable to endo w the spae of omp osable arro ws T (2) = T s × t T with the funtional struture R (2) , ∞ def = ( R ∞ ⊗ R ∞ ) ∞ | T (2) and to all the omp osition map c : T (2) → T smo oth whenev er it is a morphism of su h funtionally strutured spae in to ( T , R ∞ ) . Later on w e will sho w that T ( ω ) is atually a funtionally strutured group oid in the t w o ases of ma jor in terest for us, namely when ω is the standard bre funtor ω ( G ) asso iated with a prop er Lie group oid (20) or when ω is a  lassial  bre funtor (21 ). Ho w ev er, w e an already v ery easily  he k the smo othness (in partiular, the on tin uit y) of some of the struture maps: (a) The soure map s : T → M . First of all observ e that for an arbitrary a ∈ C ∞ ( M ) w e ha v e a ◦ s ∈ R , b y (12 ) and (13 ). Let U ⊂ M b e op en. F or ea h u ∈ U there exists f u ∈ C ∞ ( M ) with supp f u ⊂ U and f u ( u ) = 1 . Sine f u ◦ s ∈ R , the subset ( f u ◦ s ) − 1 ( C 6 =0 ) ⊂ T m ust b e op en. No w ( f u ◦ s ) − 1 ( C 6 =0 ) = s − 1  f u − 1 ( C 6 =0 )  ⊂ s − 1 ( U ) , so s − 1 ( U ) an b e expressed as a union of op en subsets of T and therefore it is op en. This sho ws that s is on tin uous; sine a ◦ s ∈ R [ R ] whenev er a is real v alued, it also follo ws that 94 CHAPTER IV. GENERAL T ANNAKA THEOR Y s is a morphism of funtionally strutured spaes. (b) The target map t : T → M . The disussion here is en tirely analogous, starting from the other iden tit y a ◦ t = r 1 ,τ , 1 ,a 1 ∈ R . () The unit setion u : M → T . This time let r = r R,φ,ζ ,ζ ′ ∈ R b e giv en; w e m ust sho w that r ◦ u ∈ C ∞ ( M ) . This is trivial b eause ( r ◦ u ) ( x ) =  x ( R ) · ζ ( x ) , ζ ′ ( x )  φ =  ζ ( x ) , ζ ′ ( x )  φ = h ζ , ζ ′ i φ ( x ) . Finally , let us remark that, as a onsequene of the existene of metris on an y ob jet of F ( M ) (b eause F is a sta k of smo oth elds and M admits partitions of unit y), the sp a e T of arr ows of T ( ω ) is always Hausdor. Indeed, let µ 6 = λ ∈ T . W e an assume s ( µ ) = x = s ( λ ) and t ( µ ) = x ′ = t ( λ ) otherwise w e are immediately done b y using the Hausdorness of M and the on tin uit y of either the soure or the target map. Then there exists R ∈ Ob( C ) with µ ( R ) 6 = λ ( R ) . Cho ose an y metri φ on ω ( R ) (there is at least one): sine φ x ′ is in partiular non-degenerate on E x ′ , there will b e glob al again, b eause of the existene of partitions of unit ysetions ζ , ζ ′ ∈ Γ ( ω R )( M ) with z µ =  µ ( R ) · ζ ( x ) , ζ ′ ( x ′ )  φ 6 =  λ ( R ) · ζ ( x ) , ζ ′ ( x ′ )  φ = z λ . Let D µ , D λ ⊂ C b e disjoin t op en disks ab out z µ , z λ resp etiv ely . Then, setting r = r R,φ,ζ ,ζ ′ , the in v erse images r − 1 ( D µ ) and r − 1 ( D λ ) will b e disjoin t op en neigh b ourho o ds of µ and λ in T . 19 Prop erness W e shall sa y that a metri φ on the ob jet ω ( R ) , R ∈ Ob( C ) of F ( M ) is ω -invariant, when there exists a Hermitian form m : R ⊗ R ∗ → 1 in C su h that φ oinides with the indued Hermitian form (1) ω ( R ) ⊗ ω ( R ) ∗ ∼ = ω ( R ⊗ R ∗ ) ω ( m ) − − − → ω ( 1 ) ∼ = 1 . W e express this in short b y writing φ = ω ∗ m . Note that b y the faithfulness of ω there is at most one su h m . 2 Denition A bre funtor ω : C − → F ( M ) will b e alled pr op er if i) the on tin uous mapping ( s , t ) : T → M × M is prop er, and ii) for ev ery ob jet R ∈ Ob( C ) , the ob jet ω ( R ) of F ( M ) supp orts an ω -in v arian t metri. W e an express the seond ondition more suintly b y sa ying that  there are enough ω -in v arian t metris  . 3 Example Let ω b e the standard funtor ω ( G ) : R ( G ) − → F ( M ) , of t yp e F , asso iated with a prop er Lie group oid G o v er M . Then ω is a prop er bre funtor. 19. PR OPERNESS 95 In order to see this, observ e (fr. also 20 ) that there is an ob vious homo- morphism of group oids (4) π : G − → T ( G ) def = T ( ω ( G )) , iden tial on the base, alled the  F -en v elop e homomorphism  of G and de- ned b y setting π ( g )( R ) =  ( g ) for ea h ob jet R = ( E ,  ) of R ( G ) ; the notation  ( g ) w as in tro dued in  17. The mapping π (1) : G (1) → T (1) is on- tin uous. Indeed, if w e x an y represen tativ e funtion r = r R,φ,ζ ,ζ ′ ∈ R , let us sa y with R = ( E ,  ) , and a small op en subset Γ ⊂ G on whi h w e ha v e, for  ating on ζ , the sort of expression  ( g ) · ζ ( s g ) = ℓ P i =1 r i ( g ) ζ ′ i ( t g ) , r i ∈ C ∞ (Γ) w e deriv ed in 17, then for all g ∈ Γ w e obtain ( r ◦ π )( g ) =  π ( g )( R ) · ζ ( s g ) , ζ ′ ( t g )  φ = ℓ P i =1 r i ( g )  h ζ ′ i , ζ i φ ◦ t  ( g ) . Therefore, w e onlude that r ◦ π ∈ C ∞ ( G ) and hene, in partiular, that r ◦ π is on tin uous. Note that in fat this argumen t sho ws that the map π is a morphism of funtionally strutured spaes, of ( G , C ∞ G ) in to ( T , R ∞ ) . W e will pro v e in  20 that the en v elop e mapping π is also surjetiv e; the prop erness of ( s , t ) : T → M × M is no w a trivial onsequene of this fat and the prop erness of ( s , t ) : G → M × M . The existene of enough in v arian t metris w as established in 17 as a orollary to the A v eraging Lemma. Ba k to general notions, it turns out that in order to  haraterize the top ology of T the ω -in v arian t metris are (for ω prop er) as go o d as the generi, `not neessarily in v arian t' ones. More exatly , let R ′ ⊂ R b e the set of all the represen tativ e funtions r R,φ,ζ ,ζ ′ with φ = ω ∗ m an ω -in v arian t metri on ω ( R ) . Note that R ′ is a subalgebra of R , losed under omplex onjugation; this follo ws from the iden tities pro v ed ab o v e, b y observing that ω ∗ m ⊗ ω ∗ n = ω ∗ ( m ⊗ n ) and so on. Then w e laim that 5 Lemma The top ology on T is also the oarsest making all the fun- tions in R ′ on tin uous. Pro of Reall that the top ology on T w as dened as the oarsest making all the funtions b elonging to R on tin uous. W e ha v e already observ ed that R ′ is an algebra of on tin uous omplex funtions on T , losed under onjugation. Moreo v er, it separates p oin ts, b eause of the existene of enough ω -in v arian t metris, f. the argumen t used to pro v e Hausdorness of T . Heneforth, for ev ery op en subset Ω ⊂ T with ompat losure Ω , the in v olutiv e subalgebra 96 CHAPTER IV. GENERAL T ANNAKA THEOR Y R ′ Ω ⊂ C 0 (Ω; C ) , formed b y the restritions to Ω of elemen ts of R ′ , is sup- norm dense in C 0 (Ω) and a fortiori in R Ω = { r | Ω : r ∈ R } , as a onsequene of the StoneW eierstrass theorem. This remark applies in partiular to Ω = T | U × U ′ where U and U ′ are op en subsets of M with ompat losure. (Here is where w e use the prop erness of T ( s , t ) − − → M × M .) Note that the subset T | U × U ′ is also op en in the spae T ′ =  T ( R ) with the top ology generated b y R ′  b eause T ′ ( s , t ) − − → M × M is learly still on tin uous. Sine the subsets T | U × U ′ o v er T , w e are no w redued to sho wing that the iden tit y mappings T | U × U ′ = − → T ′ | U × U ′ are homeomorphisms. T o simplify the notation, w e reform ulate our laim as follo ws: giv en a subset Ω ⊂ T ( R ) , op en in b oth T and T ′ and with ompat losure in T , sho w that the iden tit y mapping Ω ′ = − → Ω is on tin uous (here Ω ′ denotes of ourse the op en subset, view ed as a subspae of T ′ ). Notie that the top ology on Ω generated b y the olletion of funtions R Ω = { r | Ω : r ∈ R } oinides with the subspae top ology indued from T . Then, let r ∈ R b e xed; sine Ω is ompat in T , the restrition r | Ω will b e, as remark ed at the b eginning, a uniform limit of on tin uous funtions on Ω ′ and hene itself a on tin uous funtion on Ω ′ . q.e.d. W e shall mak e impliit use of the lemma throughout the rest of the presen t subsetion. Another easy , although imp ortan t observ ation is that all λ ∈ T ( R ) will at unitarily under an y ω -in v arian t metri. More preisely , for an y ob jet R ∈ Ob( C ) and an y ω -in v arian t metri φ on ω ( R ) , the linear isomorphism λ ( R ) will preserv e the inner pro dut h , i φ : (6)  λ ( R ) · v , λ ( R ) · v ′  φ = h v , v ′ i φ . W e use this observ ation to pro v e the follo wing 7 Prop osition Let ω b e a prop er bre funtor. Then T ( ω ) is a (Hausdor, prop er) top ologial group oid. Pro of W e m ust sho w that the in v erse and omp osition maps of T ( ω ) are on tin uous. a) Con tin uit y of the in v erse map i : T → T . It m ust b e pro v ed that the omp osite r ◦ i is on tin uous on T , for an y r = r R,φ,ζ ,ζ ′ ∈ R with φ an ω -in v arian t metri on ω ( R ) . This is immediate, b eause ( r R,φ,ζ ,ζ ′ ◦ i )( λ ) =  λ ( R ) − 1 · ζ ( t λ ) , ζ ′ ( s λ )  φ =  ζ ( t λ ) , λ ( R ) · ζ ′ ( s λ )  φ =  λ ( R ) · ζ ′ ( s λ ) , ζ ( t λ )  φ = r R,φ,ζ ′ ,ζ ( λ ) , 19. PR OPERNESS 97 in view of (6 ). b) Con tin uit y of omp osition c : T s × t T → T (the domain of the map b eing top ologized as a subspae of the artesian pro dut T × T ). W e mak e a te hnial observ ation rst. Fix λ ∈ T , let us sa y λ : x → x ′ . Let R ∈ Ob C and let φ b e an y ω -in v arian t metri on E = ω ( R ) . Fix a lo al φ -orthonormal system ζ ′ 1 , . . . , ζ ′ d ∈ Γ ( ω R )( U ′ ) for E ab out x ′ as in (15 .9); hene, in partiular, (8) E x ′ = Span { ζ ′ 1 ( x ′ ) , . . . , ζ ′ d ( x ′ ) } . Sine M is paraompat, it is no loss of generalit y to assume that for ev ery i = 1 , . . . , d ζ ′ i is the restrition to U ′ of a global setion ζ i of ω ( R ) . Let ζ ∈ Γ ( ω R )( M ) b e another global setion. Consider an op en neigh b ourho o d Ω of λ in T su h that t (Ω) ⊂ U ′ . Also let Φ i ∈ C 0 (Ω; C ) ( i = 1 , . . . , d ) b e a list of on tin uous omplex funtions on Ω . Then the norm funtion (9) µ 7→     µ ( R ) · ζ ( s µ ) − d P i =1 Φ i ( µ ) ζ ′ i ( t µ )     is ertainly on tin uous on Ω : indeed, its square is   µ ( R ) ζ ( s µ )   2 − 2 X i ℜ e h Φ i ( µ )  µ ( R ) ζ ( s µ ) , ζ ′ i ( t µ )  i +     d P i =1 Φ i ( µ ) ζ ′ i ( t µ )     2 =   ζ ( s µ )   2 − 2 X i ℜ e h Φ i ( µ )  µ ( R ) ζ ( s µ ) , ζ i ( t µ )  i + d P i =1   Φ i ( µ )   2 (b eause µ ( R ) is unitary ( 6) and the v etors ζ ′ i ( t µ ) , i = 1 , . . . , d form an orthonormal system in E t ( µ ) ). No w, mak e Φ i ( µ ) =  µ ( R ) ζ ( s µ ) , ζ i ( t µ )  in (9) and ev aluate the funtion y ou get at µ = λ : the result will b e zero, b eause the v etors ζ i ( x ′ ) , i = 1 , . . . , d onstitute an orthonormal b asis of E x ′ . Hene, b y the just observ ed on tin uit y , for ea h ε > 0 there will b e an op en neigh b ourho o d of λ in T , let us all it Ω ε ( λ ) , o v er whi h the follo wing estimate holds (10)     µ ( R ) · ζ ( s µ ) − d P i =1 r R,φ,ζ ,ζ i ( µ ) ζ i ( t µ )     < ε . With this preliminary observ ation at hand it is easy to sho w on tin uit y of the omp osition of arro ws. Indeed, onsider an arbitrary ob jet R ∈ Ob C , an arbitrary ω -in v arian t metri φ on ω ( R ) , and arbitrary global setions ζ , η ∈ Γ ( ω R )( M ) . W e ha v e to  he k the on tin uit y of the funtion (11) ( µ ′ , µ ) 7→ ( r R,φ,ζ ,η ◦ c )( µ ′ , µ ) =  µ ′ ( R ) · µ ( R ) · ζ ( s µ ) , η ( t µ ′ )  φ on the spae of omp osable arro ws T (2) . Let x λ − → x ′ λ ′ − → x ′′ b e an arbitrary pair of omp osable arro ws, whi h w e regard as xed. Cho ose a lo al φ -orthonormal 98 CHAPTER IV. GENERAL T ANNAKA THEOR Y system ab out x ′ as b efore. Then, b y the estimate (10 ) and our remark (6) that µ ′ ( R ) is unitary , for all ( µ ′ , µ ) lose enough to ( λ ′ , λ ) , let us sa y for µ ∈ Ω ε ( λ ) , the funtion (11 ) will dier from the funtion d P i =1 r R,φ,ζ ,ζ i ( µ )  µ ′ ( R ) · ζ i ( s µ ′ ) , η ( t µ ′ )  φ = d P i =1 r R,φ,ζ ,ζ i ( µ ) r R,φ,ζ i ,η ( µ ′ ) up to ε k η k , where k η k is a p ositiv e b ound for the norm of η in a neigh b our- ho o d of x ′′ . This pro v es the desired on tin uit y , b eause the last funtion is ertainly on tin uous on T × T and hene on T (2) . q.e.d. 20 Reonstrution Theorems When applying the formal apparatus of  18 to the standard bre funtor ω F ( G ) asso iated with a Lie group oid G , w e prefer to use the alternativ e notation T F ( G ) for the real T annakian group oid T  ω F ( G ); R  and refer to the latter as the (r e al) F -envelop e of G . If expliit men tion of t yp e is not neessary , w e normally just write T ( G ) . The F -envelop e homomorphism asso iated with a Lie group oid G is the group oid homomorphism π : G → T ( G ) , or, more p edan tially , (1) π F ( G ) : G − → T F ( G ) dened b y the form ula π ( g )( E ,  ) def =  ( g ) . (Ha ving a lo ok at Note 17.2 one more time migh t b e useful at this p oin t.) The study of prop erties of the en v elop e homomorphism π ( G ) for prop er G will onstitute our main onern in this setion. Let M / G b e the top ologial spae obtained b y endo wing the set of orbits {G · x | x ∈ M } with the quotien t top ology indued b y the orbit map (2) o : M → M / G (the map sending a p oin t x to the resp etiv e G -orbit o ( x ) = G · x ). Note that the map o is op en: indeed, if U ⊂ M is an op en subset then so is o − 1 ( o ( U )) = t ( s − 1 ( U )) b eause t is an op en mapatually , a submersion. F urthermore, M / G is a lo ally ompat Hausdor spae. Indeed, supp ose G ( x, x ′ ) empt y . Prop erness of G , applied to some sequene of balls B i × B i ′ shrinking to the p oin t ( x, x ′ ) , will yield op en balls B , B ′ ⊂ M at x, x ′ su h that ( s , t ) − 1 ( B × B ′ ) is empt y , in other w ords, su h that o ( B ) ∩ o ( B ′ ) = ∅ , as on tended. In partiular, ev ery orbit G · x = o − 1 { o ( x ) } is a losed subset of M . 3 Theorem Let F b e an y sta k of smo oth elds. Let G b e a prop er Lie group oid. Then the F -en v elop e homomorphism π F ( G ) : G → T F ( G ) is full (i.e. surjetiv e, as a mapping of the spaes of arro ws). 20. RECONSTR UCTION THEOREMS 99 Pro of T o b egin with, let us pro v e that G ( x, x ′ ) empt y implies T ( G )( x, x ′ ) empt y . Put S = G x ∪ G x ′ and let ϕ : S → C b e the funtion whi h tak es the v alue 1 o v er the orbit G x and the v alue 0 o v er the orbit G x ′ ; ϕ is w ell-dened b eause G x ∩ G x ′ = ∅ . S is an in v arian t submanifold of M . Sine S is the union of t w o disjoin t losed subsets of M , it is also a losed submanifold. Moreo v er, ϕ is equiv arian t with resp et to the trivial represen tation of G , i.e. ϕ ( g · s ) = ϕ ( s ) . Corollary 17.20 sa ys that there is some smo oth in v arian t funtion Φ : M → C , extending ϕ , equiv alen tly , some smo oth funtion Φ on M , onstan t along the G -orbits and with Φ( x ) = 1 , Φ( x ′ ) = 0 . By setting b z def = Φ( z ) id , one gets an endomorphism b of the trivial represen tation with b x = id and b x ′ = 0 . No w, supp ose there exists some λ ∈ T ( G )( x, x ′ ) : then, b y the naturalit y of λ , one gets a omm utativ e square C id   λ / / C 0   C λ / / C whi h on tradits the in v ertibilit y of λ ( 1 ) . In order to nish the pro of of the theorem, it will b e suien t to pro v e surjetivit y of all isotrop y homomorphisms indued b y π , b eause G | x g - ≈   π x / / T ( G ) | x π ( g ) - ≈   G ( x, x ′ ) π x,x ′ / / T ( G )( x, x ′ ) omm utes for all g ∈ G ( x, x ′ ) . More expliitly , it will b e suien t to pro v e that π x : G | x → T ( G ) | x is an epimorphism of groups, for ev ery x ∈ M . This follo ws immediately from Prop osition 10 .3 and Corollary 17 .16. q.e.d. W e on tin ue to w ork with an arbitrary sta k of smo oth elds. 4 Denition A Lie group oid G will b e said to b e F -r eexive, or self-dual r elative to F , if its F -en v elop e homomorphism π F ( G ) : G → T F ( G ) is an isomorphism of top ologial group oids. It turns out, for prop er Lie group oids, that the requiremen t that the on tin uous mapping π (1) : G (1) → T ( G ) (1) should b e op en is sup eruous; more preisely , one has the follo wing statemen t: 5 Theorem Let G b e a prop er Lie group oid. Let F b e an y sta k of smo oth elds. Then G is F -reexiv e if and only if the homomorphism π F ( G ) is faithful (i.e. injetiv e, as a mapping of the spaes of arro ws). 100 CHAPTER IV. GENERAL T ANNAKA THEOR Y Pro of The assertion that injetivit y implies bijetivit y , or, to sa y the same thing dieren tly , that faithfulness implies full faithfulness, is an immediate onsequene of Theorem 3 ab o v e. As to the statemen t that the mapping π is op en, w e ha v e to sho w that whenev er Γ is an op en subset of G (1) and g 0 a p oin t of Γ , the image π (Γ) is a neigh b ourho o d of π ( g 0 ) in T ( G ) . T o x ideas, supp ose g 0 ∈ G ( x 0 , x ′ 0 ) . Let us start b y observing that, as in the pro of of Prop osition 10.3 , it is p ossible to nd a represen tation R = ( E ,  ) ∈ Ob R ( G ) whose asso iated x 0 -th isotrop y homomorphism  0 : G 0 → GL ( E 0 ) is injetiv e (same notation as in Eq. (17.6)); for su h an R , the map G ( x 0 , x ′ 0 ) → Lis( E x 0 , E x ′ 0 ) , g 7→  ( g ) is also injetiv e. W e regard R as xed one and for all. Moreo v er, w e  ho ose an arbitrary Hilb ert metri φ on E . As w e kno w from 15, there are lo al φ -orthonormal frames for E (6) ( ζ 1 , . . . , ζ d ∈ Γ E ( U ) ab out x 0 and ζ ′ 1 , . . . , ζ ′ d ∈ Γ E ( U ′ ) ab out x ′ 0 ; their ardinalit y turns out to b e the same b eause E x 0 ≈ E x ′ 0 . Sine M is paraompat, it is no loss of generalit y to assume that the ζ i and the ζ ′ i ′ are (restritions of ) global setions. Finally , w e selet an y ompatly supp orted smo oth funtions a, a ′ : M → C with supp a ⊂ U and supp a ′ ⊂ U ′ , su h that a ( x ) = 1 ⇔ x = x 0 and a ′ ( x ′ ) = 1 ⇔ x ′ = x ′ 0 . Let us put, for all 1 ≦ i, i ′ ≦ d ,  i,i ′ = r i,i ′ ◦ π def = r R,φ,ζ i ,ζ ′ i ′ ◦ π : G → C , [using notation (18.10)℄ (7) and for i = 0 and 0 ≦ i ′ ≦ d , resp. 0 ≦ i ≦ d and i ′ = 0 , 5 (8) (  0 ,i ′ = r 0 ,i ′ ◦ π def = a ◦ s G =  a ◦ s T ( G )  ◦ π : G → C , resp.  i, 0 = r i, 0 ◦ π def = a ′ ◦ t G =  a ′ ◦ t T ( G )  ◦ π : G → C . Also, put z i,i ′ =  i,i ′ ( g 0 ) ∈ C . W e laim that, as a onsequene of prop erness, there exist op en disks D i,i ′ ⊂ C en tred at z i,i ′ su h that (9) \ 0 ≦ i,i ′ ≦ d  i,i ′ − 1 ( D i,i ′ ) ⊂ Γ . Before w e go in to the pro of of this laim, let us sho w ho w the statemen t that π (Γ) is a neigh b ourho o d of π ( g 0 ) follo ws from (9 ). Sine, b y Theorem 3, π is 5 F or i = i ′ = 0 either  hoie will do; for d = 0 there are ob vious mo diations whi h w e lea v e to the reader. The only thing that really matters is that b oth a ◦ s and a ′ ◦ t should o ur in the in tersetion (9) at least one. 20. RECONSTR UCTION THEOREMS 101 surjetiv e as a mapping of G (1) in to T ( G ) (1) , w e ha v e \ r i,i ′ − 1  D i,i ′  = π π − 1  \ r i,i ′ − 1  D i,i ′   = π  \ π − 1 r i,i ′ − 1  D i,i ′   = π  \  i,i ′ − 1  D i,i ′   ⊂ π (Γ) . (b y the inlusion (9)) No w w e are done, b eause g 0 ∈ r i,i ′ − 1  D i,i ′  and r i,i ′ ∈ C 0 ( T ( G ) (1) ; C ) for all 0 ≦ i, i ′ ≦ d . In order to pro v e our laim (9 ), let us onsider, for ea h 0 ≦ i, i ′ ≦ d , a dereasing sequene of op en disks (10) · · · ⊂ D ℓ +1 i,i ′ ⊂ D ℓ i,i ′ ⊂ · · · ⊂ D 0 i,i ′ ⊂ C en tred at z i,i ′ and whose radius δ ℓ i,i ′ tends to zero. If w e mak e the inno uous assumption δ 0 i,i ′ = 1 then it will follo w from our h yp otheses on the funtions a, a ′ that the sets Σ ℓ def = \ 0 ≦ i,i ′ ≦ d r i,i ′ − 1  D ℓ i,i ′  − Γ  ℓ = 1 , 2 , . . .  (11) are losed subsets of the ompat spae G ( K , K ′ ) , where K = supp a and K ′ = supp a ′ . The sets Σ ℓ form a dereasing sequene. Their in tersetion ∞ ∩ ℓ =1 Σ ℓ has to b e empt y b eause of the faithfulness of g 7→  ( g ) on G ( x 0 , x ′ 0 ) and our h yp otheses on a , a ′ . Hene, b y ompatness, there will b e some ℓ su h that Σ ℓ = ∅ . This pro v es the laim, and therefore, the theorem. q.e.d. 12 Note (The pr esent r emark wil l b e use d nowher e else and ther efor e it may b e skipp e d without  onse quen es. Y ou should r e ad 2425 rst, anyway.) Observ e that whenev er G and H are Morita equiv alen t Lie group oids, one of them is F -reexiv e if and only if the other is. Indeed, b y naturalit y of the en v elop e transformation π F ( - ) : Id → T F ( - ) , one gets a omm utativ e square of top ologial group oid homomorphisms G ϕ Morita eq.   π ( G ) / / T ( G ) T ( ϕ )   H π ( H ) / / T ( H ) (13) in whi h b oth ϕ and T ( ϕ ) are fully faithful. It follo ws immediately that π ( G ) is fully faithful if and only if the same is true of π ( H ) . With a bit more w ork, it an b e sho wn that π ( G ) is an op en map if and only if π ( H ) is so (use the simplifying assumption that ϕ (0) : G (0) → H (0) is a surjetiv e submersion). 102 CHAPTER IV. GENERAL T ANNAKA THEOR Y By denition, a Lie group oid G is F -reexiv e if and only if one an solv e top ologially the problem of reonstruting G from its represen tations of t yp e F (that is to sa y one an reo v er G up to isomorphism of top ologial group oids from su h represen tations). In the ase of Lie groups, a top ologial solution pro vides a ompletely satisfatory answ er b eause the smo oth struture of an y Lie group is uniquely determined b y the top ology of the group itself. Ho w ev er, in the presen t more general on text it is not eviden t a priori that the notion of reexivit y w e in tro dued ab o v e is as strong as to settle the smo othness problem men tioned at the b eginning of  18, think e.g. of G = M a smo oth manifold. More preisely , w e onsider the follo wing question: do es reexivit y of G , in the foregoing purely top ologial sense, atually imply that the funtionally strutured spae ( T ( G ) (1) , R ∞ ) dened in 18 is a smo oth manifold and the en v elop e map π (1) : G (1) → T ( G ) (1) a dieomorphism? The answ er pro v es to b e armativ e, as w e shall no w see. Let G b e an arbitrary Lie group oid. Cho ose an arro w g 0 ∈ G ( x 0 , x ′ 0 ) and a represen tation R = ( E ,  ) of G rst of all. Then  ho ose an arbitrary metri φ on E and global setions ζ 1 , . . . , ζ d , resp. ζ ′ 1 , . . . , ζ ′ d , forming a lo al φ -orthonormal frame for E ab out x 0 , resp. x ′ 0 , as in the pro of of Theorem 5 . These data determine a smo oth mapping (14)  ζ 1 ...,ζ d ζ ′ 1 ,...,ζ ′ d : G (1) − → M × M × M ( d ; C ) , as follo ws: g 7→  s ( g ); t ( g );  1 , 1 ( g ) , . . . ,  i,i ′ ( g ) , . . . ,  d,d ( g )  (the funtions  i,i ′ are those dened in (7); M ( d ; C ) = End( C d ) is the spae of d × d omplex matries). If the en v elop e homomorphism π ( G ) : G → T ( G ) of the Lie group oid G is faithful, it follo ws from Lemma 10 .14 that for ev ery p oin t x of the base manifold M of G there exists a represen tation ( E ,  ) ∈ Ob R ( G ) su h that Ker  x is a disrete subgroup of the isotrop y group G x = G | x . Consequen tly , for an arbitrary arro w g 0 ∈ G ( x 0 , x ′ 0 ) there will exist ( E ,  ) ∈ Ob R ( G ) su h that the map G ( x 0 , x ′ 0 ) → Lis( E x 0 , E x ′ 0 ) , g 7→  ( g ) is injetiv e on some op en neigh b ourho o d of g 0 in G ( x 0 , x ′ 0 ) . Then the follo wing lemma applies: 15 Lemma Let G b e a Lie group oid. Fix an arro w g 0 ∈ G ( x 0 , x ′ 0 ) and let ( E ,  ) ∈ Ob R ( G ) b e a represen tation. Supp ose the map g 7→  ( g ) : G ( x 0 , x ′ 0 ) → Lis( E x 0 , E x ′ 0 ) is injetiv e on some op en neigh b ourho o d of g 0 in G ( x 0 , x ′ 0 ) . Then the smo oth mapping  ζ ζ ′ : G (1) → M × M × M ( d ; C ) (14 ) is an immersion at g 0 , for an y  hoie of a metri and of related orthonormal frames ζ = { ζ 1 , . . . , ζ d } , ζ ′ = { ζ ′ 1 , . . . , ζ ′ d } . Pro of Let M b e the base manifold of G . Fix op en balls U, U ′ ⊂ M , en tred at x 0 , x ′ 0 resp etiv ely and so small that the setions ζ 1 , . . . , ζ d , resp. ζ ′ 1 , . . . , ζ ′ d form a lo al orthonormal frame for E o v er U , resp. U ′ . Sine the soure map 20. RECONSTR UCTION THEOREMS 103 s of G is a submersion, one an alw a ys  ho ose U also so small that there exists a lo al trivialization Γ ≈ U × B pr − → U for s in a neigh b ourho o d Γ of g 0 in G (1) , where B is an op en eulidean ball. It is no loss of generalit y to assume t (Γ) ⊂ U ′ . Then w e obtain, for the restrition of the mapping  ζ ζ ′ =  ζ 1 ...,ζ d ζ ′ 1 ,...,ζ ′ d to Γ , a o ordinate expression of the follo wing form (16) U × B → U × U ′ × M ( d ; C ) , ( u, b ) 7→  u, u ′ ( u, b ) ,  ( u, b )  where  ( g ) ∈ M ( d ; C ) denotes the matrix {  i,i ′ ( g ) } 1 ≦ i,i ′ ≦ d . The dieren tial of the mapping (16 ) at, let us sa y , g 0 = ( x 0 , 0) reads   Id 0 ∗ D 2 u ′ ( x 0 , 0) ∗ D 2  ( x 0 , 0)   (17) and it is therefore injetiv e if and only if su h is the dieren tial of the partial map b 7→  u ′ ( x 0 , b ) ,  ( x 0 , b )  : B → U ′ × M ( d ; C ) at the origin of B . W e are no w redued to sho wing that the restrition  ζ ζ ′ : G ( x 0 , - ) − → M × GL ( d ) = { x 0 } × M × GL ( d ; C ) is an immersion at g 0 . Let G 0 = G | x 0 b e the isotrop y group at x 0 and  ho ose, in the viinit y of g 0 , a lo al (equiv arian t) trivialization G ( x 0 , S ) ≈ S × G 0 for the prinipal G 0 -bundle t x 0 : G ( x 0 , - ) → G x ; w e an assume that S is a submanifold of U ′ and that in this lo al  hart g 0 = ( x ′ 0 , e 0 ) , where e 0 stands for the neutral elemen t of G 0 . W e then obtain a new o ordinate expression for the restrition of  ζ ζ ′ to G ( x 0 , - ) , namely (18) S × G 0 → U ′ × GL ( d ; C ) , ( s, g ) 7→  s,  ( s, g )  . Sine its rst omp onen t is the inlusion of a submanifold, this map will b e an immersion at g 0 = ( x ′ 0 , e 0 ) pro vided the partial map g 7→  ( x ′ 0 , g ) is an immersion at e 0 . The latter orresp onds to the diagonal of the square G 0 g 0 - ≈    / / Aut( E x 0 ) ρ ( g 0 ) - ≈   G ( x 0 , x ′ 0 )  / / Lis( E x 0 , E x ′ 0 ) , so our problem redues to pro ving that the homomorphism  : G 0 → GL ( E x 0 ) is immersiv e. By h yp othesis, this is injetiv e in an op en neigh b ourho o d of e 0 and hene our laim follo ws at one. q.e.d. W e are no w ready to establish our previous laims ab out the funtional struture R ∞ on the T annakian group oid T ( G ) . Let G b e an y F -reexiv e Lie group oid ( F an arbitrary sta k of smo oth elds, as ev er). 104 CHAPTER IV. GENERAL T ANNAKA THEOR Y Fix an arro w λ 0 ∈ T ( G ) (1) . Our rst task will b e to nd some op en neigh b ourho o d Ω of λ 0 su h that (Ω , R ∞ Ω ) turns out to b e isomorphi, as a funtionally strutured spae, to a smo oth manifold ( X , C ∞ X ) . Sine w e are w orking under the h yp othesis that G is reexiv e, there is a unique g 0 ∈ G (1) su h that λ 0 = π ( g 0 ) . By Lemma 15 and the ommen ts preeding it, w e an nd, for a on v enien tly  hosen ( E ,  ) ∈ Ob R ( G ) , an op en neigh b ourho o d Γ of g 0 in G (1) su h that the smo oth map  ζ ζ ′ : G (1) → M × M × M ( d ; C ) (14) indues a dieomorphism of Γ on to a submanifold X ⊂ M × M × M ( d ; C ) . Notie that the same data whi h determine the map ( 14 ) also determine a map of funtionally strutured spaes (19) r ζ ζ ′ = r ζ 1 ...,ζ d ζ ′ 1 ,...,ζ ′ d : T ( G ) (1) − → M × M × M ( d ; C ) , λ 7→  s ( λ ); t ( λ ); { r i,i ′ ( λ ) } 1 ≦ i,i ′ ≦ d  , where w e put r i,i ′ = r R,φ,ζ i ,ζ ′ i ′ ∈ R (18.11 ). F rom the reexivit y of G again, it follo ws that the en v elop e map π indues a homeomorphism b et w een Γ and the op en subset Ω def = π (Γ) of T ( G ) (1) . The follo wing diagram Γ π | Γ ≈ homeo & & N N N N N N N N N N N N N N  ζ ζ ′ | Γ ≈ dieo / / X ⊂ M × M × M ( d ; C ) Ω r ζ ζ ′ | Ω 6 6 n n n n n n n n n n n n n n (20) is learly omm utativ e. W e on tend that the map r ζ ζ ′ | Ω pro vides the desired isomorphism of funtionally strutured spaes. Expliitly , this means that an arbitrary funtion f : X ′ → C b elongs to C ∞ ( X ′ ) if and only if its pullba k h = f ◦ r ζ ζ ′ b elongs to R ∞ (Ω ′ ) , for ea h xed pair of orresp onding op en subsets Ω ′ ⊂ Ω , X ′ ⊂ X . Note that sine the problem is lo al, w e an mak e the simplifying assumption Ω ′ = Ω , X ′ = X . Th us, supp ose f ∈ C ∞ ( X ) rst; b eause of the lo al  harater of the problem again, it is not restritiv e to assume that f admits a smo oth extension ˜ f ∈ C ∞  M × M × M ( d )  . Then h oinides with the restrition to Ω of a global funtion ˜ h = ˜ f ◦ r ζ ζ ′ : T (1) = T ( G ) (1) → C b elonging to R ∞ ( T (1) ) b eause (19 ) is a map of funtionally strutured spaes. Con v ersely , supp ose h = f ◦ r ζ ζ ′ ∈ R ∞ (Ω) . W e kno w, from Example 19.3 , that the en v elop e map π is a morphism of funtionally strutured spaes. Hene the omp osite h ◦ π will b elong to C ∞ (Γ) . Sine h ◦ π = f ◦ r ζ ζ ′ ◦ π = f ◦  ζ ζ ′ and  ζ ζ ′ | Γ is a dieomorphism of Γ on to X , it follo ws that f ∈ C ∞ ( X ) , as on tended. W e ha v e therefore pro v ed that if a Lie group oid G is F -reexiv e then the spae ( T F ( G ) (1) , R ∞ ) is atually a (Hausdor ) smo oth manifold. There is little w ork left to b e done b y no w: 21 Prop osition Let F b e an arbitrary sta k of smo oth elds and let G b e a Lie group oid. Supp ose G is F -reexiv e. 20. RECONSTR UCTION THEOREMS 105 Then the T annakian group oid T F ( G ) , endo w ed with its anonial funtional struture R ∞ , turns out to b e a Lie group oid; moreo v er, the F -en v elop e homomorphism (22) π F ( G ) : G ≈ − − → T F ( G ) turns out to b e an isomorphism of Lie group oids. Pro of W e kno w from the foregoing disussion that ( T (1) , R ∞ ) is a smo oth manifold. Then all w e ha v e to sho w no w, learly , is that the en v elop e map π : G (1) → T (1) is a dieomorphism. Equiv alen tly , w e ha v e to sho w that π is an isomorphism of funtionally strutured spaes b et w een ( G (1) , C ∞ G (1) ) and ( T (1) , R ∞ ) . This follo ws immediately , lo ally , from the omm utativit y of the triangles (20) and the previously established fat that b oth  ζ ζ ′ | Γ and r ζ ζ ′ | Ω are funtionally strutured spae isomorphisms on to ( X , C ∞ X ) . q.e.d. Let us pause for a momen t to summarize our urren t kno wledge of the F -en v elop e π F ( G ) : G → T F ( G ) of an arbitrary pr op er Lie group oid G . First of all, w e kno w that π ( G ) is faithful (Thm. 3). W e ha v e also asertained that T ( G ) is a top ologial group oid (Ex. 19.3 and Pr op. 19 .7). Moreo v er, it has b een established that π ( G ) is neessarily an isomorphism of top ologial group oids in ase π ( G ) is faithful (Thm. 5); whenev er this happ ens to b e true, one an ompletely solv e the reonstrution problem for G (Pr op. 21 ). No w observ e that faithfulness of π ( G ) is equiv alen t to the follo wing prop ert y: if g 6 = u ( x ) in the isotrop y group G | x then there exists a represen tation ( E ,  ) ∈ Ob R ( G ) su h that  ( g ) 6 = id ∈ Aut( E x ) . W e an therefore onlude b y sa ying that an arbitrary prop er Lie group oid an b e reo v ered from its represen tations of t yp e F if and only if su h represen tations are  enough  in the sense of the ab o v e-men tioned prop ert y . The nal part of the presen t setion will b e dev oted to sho wing that an y prop er Lie group oid admits enough represen tations of t yp e E ∞ (= smo oth Eulidean elds, fr. 16). By the foregoing remarks, this will immediately imply the general reonstrution theorem w e w ere striving for. Reall that our approa h via smo oth Eulidean elds is motiv ated b y the imp ossibilit y to obtain that result b y using represen tations of t yp e V ∞ (smo oth v etor bundles), as illustrated b y the examples disussed in  2. W e b egin with some preliminary remarks of a purely top ologial nature. Let G b e a prop er Lie group oid and let M denote the base manifold of G . Reall that a subset S ⊂ M is said to b e invariant when s ∈ S implies g · s ∈ S for all arro ws g ∈ G (1) . If S is an arbitraryviz., not neessarily in v arian tsubset of M , w e let G · S denote the satur ation of S , that is to sa y the smallest in v arian t subset of M on taining S , so that S is in v arian t if and only if G · S = S ; note that the saturation of an op en subset is also 106 CHAPTER IV. GENERAL T ANNAKA THEOR Y op en. No w let V b e an y op en subset with ompat losure: w e on tend that G · V = G · V . The diretion ` ⊂ ' of this equalit y is v alid ev en for a non-prop er Lie group oid; it follo ws for instane from the existene of lo al bisetions. T o  he k the opp osite inlusion, one an resort to the w ell-kno wn fat that the orbit spae 6 of a prop er Lie group oid is Hausdor and then use the ompatness of V ; in detail: sine the image of the ompat set V under the on tin uous mapping o : M → M / G is a ompat and hene losed subset of the Hausdor spae M / G , the in v erse image G · V = o − 1  o  V  m ust b e losed as w ell. Next, let U b e an in v arian t op en subset of M . F rom the equalit y w e ha v e just pro v ed, it follo ws immediately that U oinides with the union of all its op en in v arian t subsets V , V ⊂ U . Indeed, sine an y giv en p oin t u 0 ∈ U admits an op en neigh b ourho o d W with ompat losure on tained in U , one has u 0 ∈ G · W = V ⊂ V = G · W = G · W ⊂ G · U = U . The latter remark applies to the onstrution of G -in v arian t partitions of unit y on M ; for our purp oses it will b e enough to illustrate a sp eial ase of this onstrution. Consider an arbitrary p oin t x 0 ∈ M and let U b e an op en invariant neigh b ourho o d of x 0 . Cho ose another op en neigh b ourho o d V of x 0 , in v arian t and with losure on tained in U . The orbit G · x 0 and the set-theoreti omplemen t ∁ V are in v arian t disjoin t losed subsets of M , so Cor ol lary 17.20 pro vides us with an in v arian t funtion Φ ∈ C ∞ ( M ; R ) su h that Φ( x 0 ) = 1 and Φ = 0 outside V . W e are no w ready to establish a basi extension prop ert y enjo y ed b y the represen tations of t yp e E ∞ of prop er Lie group oids; our  main theorem  b elo w will b e essen tially a onsequene of this prop ert y and of Zung's results on lo al linearizabilit y . Our goal will b e a hiev ed b y means of an ob vious ut-o te hnique whi h is of ourse not a v ailable when one limits oneself to represen tations on v etor bundles. Sine throughout the subsequen t disussion the t yp e F = E ∞ is xed, w e agree to systematially suppress an y referene to t yp e. Let G b e an arbitrary pr op er Lie group oid and let M denote its base as usual. Let U ⊂ M b e a G -invariant op en neigh b ourho o d of a p oin t x 0 ∈ M , and supp ose w e are giv en a p artial represen tation ( E U ,  U ) ∈ R ( G | U ) . W e kno w from 17 that there is an indued Lie group represen tation (23)  U, 0 : G 0 − → GL ( E U, 0 ) of the isotrop y Lie group G 0 = G | x 0 on the v etor spae E U, 0 = ( E U ) x 0 . W e on tend that one an onstrut a global represen tation ( E ,  ) ∈ R ( G ) for whi h it is p ossible to exhibit an isomorphism of G 0 -spaes E 0 def = E x 0 ≈ E U, 0 . 6 The quotien t of M asso iated with the equiv alene x ∼ g · x . W e will indiate b y o the map (of M in to this quotien t) whi h sends x to its equiv alene lass. 20. RECONSTR UCTION THEOREMS 107 (The G 0 -spae struture on E 0 omes from the indued represen tation (24)  0 : G 0 − → GL ( E 0 ) , that on E U, 0 from (23 ).) T o b egin with, let us x an y in v arian t smo oth funtion a ∈ C ∞ ( M ) with a ( x 0 ) = 1 and supp a ⊂ U ; su h funtions alw a ys existas w e sa w b eforein view of the prop erness of G . Let V ⊂ M denote the op en subset onsisting of all x su h that a ( x ) 6 = 0 . One an dene the follo wing bundle { E x } of Eulidean spaes o v er M : (25) E x = ( E U,x if x ∈ V { 0 } otherwise. Let Γ E b e the smallest sheaf of setions of the bundle { E x } whi h on tains the follo wing presheaf (26) W 7→  aζ : ζ ∈ Γ ( E U )( U ∩ W )  . (Here of ourse aζ is to b e understo o d as the appropriate prolongation b y zero of the indiated setion; note that sine M admits partitions of unit y (25 ) atually equals Γ E .) One v eries immediately that these data dene a smo oth Eulidean eld E o v er M . Next, in tro due  b y putting (27)  ( g ) = (  U ( g ) for g ∈ G | V 0 otherwise. This la w m ust b e understo o d as desribing a bundle   ( g ) : ( s ∗ E ) g ∼ → ( t ∗ E ) g  of linear isomorphisms indexed o v er the manifold G . The ompati- bilit y of this family of maps with the omp osition of arro ws, amoun ting to the equalities  ( g ′ g ) =  ( g ′ )  ( g ) and  ( x ) = id , is lear. No w,  will b e an ation of G on E pro vided it is a morphism s ∗ E → t ∗ E of Eulidean elds o v er G : this is ob vious, b eause for suitable funtions r i ∈ C ∞ one has  ( g ) aζ ( s g ) = a ( s g )  ( g ) ζ ( s g ) = a ( t g ) ℓ P i =1 r i ( g ) ζ ′ i ( t g ) = ℓ P i =1 r i ( g ) aζ ′ i ( t g ) , in view of the G -in v ariane of a . Hene ( E ,  ) ∈ R ( G ) . Finally , the iden tit y E 0 = E x 0 def = E U,x 0 = E U, 0 pro vides the required G 0 -equiv arian t isomorphism. 28 Theorem (General Reonstrution Theorem, Main Theorem) Ea h prop er Lie group oid is E ∞ -reexiv e. Pro of Let G b e an y su h group oid and x a p oin t x 0 of its base manifold M . W e need to sho w the existene of a Eulidean represen tation ( E ,  ) ∈ Ob R ( G ) induing a faithful isotrop y represen tation  0 : G 0 ֒ → GL ( E 0 ) (24) 108 CHAPTER IV. GENERAL T ANNAKA THEOR Y (w e freely use the notation ab o v e). By the previously established extension prop ert y of Eulidean represen tations, it will b e enough to nd a partial rep- resen tation ( E U ,  U ) ∈ Ob R ( G | U ) dened o v er some in v arian t op en neigh- b ourho o d U of x 0 and with  U, 0 : G 0 ֒ → GL ( E U, 0 ) (23) injetiv e. It w as observ ed in 4 that Zung's Lo al Linearizabilit y Theorem yields the existene of (a) a smo oth represen tation G 0 → GL ( V ) on some (real) nite dimensional v etor spae (b) an em b edding of manifolds V i ֒ → M su h that 0 7→ x 0 and su h that U def = G · i ( V ) is an op en subset of M () a Morita equiv alene G 0 ⋉ V ι − → G | U induing V i ֒ → U at the lev el of base manifolds. Note that the isotrop y of G 0 ⋉ V at 0 equals G 0 and that the equiv alene ι indues an automorphism ι 0 ∈ Aut( G 0 ) (whi h an b e assumed to b e the iden tit y , just to x ideas). No w let Φ : G 0 ֒ → GL ( E ) b e an y faithful represen tation on a nite dimensional omplex v etor spae. One has an indued faithful represen tation e Φ of G 0 ⋉ V on V × E (fr. the end of 28). By the theory of 14, there exists some represen tation ( E U ,  U ) ∈ Ob R ( G | U ) su h that ι ∗ ( E U ,  U ) ≈ ( V × E , e Φ) ; this is preisely the one w e are lo oking for, b eause  U, 0 : G 0 ֒ → GL ( E U, 0 ) ≈ GL ( E ) m ust oinide with Φ . q.e.d. Chapter V Classial Fibre F untors In the presen t  hapter w e will again o up y ourselv es with the study of the abstrat notion of bre funtor. Ho w ev er, w e shall b e exlusiv ely in terested in bre funtors whi h tak e v alues in the ategory of smo oth v etor bundles o v er a manifold, in other w ords bre funtors of the form ω : C → V ∞ ( M ) or, equiv alen tly , of t yp e V ∞ . Moreo v er, sine in all examples of su h funtors w e ha v e in mind the tensor ategory C in v ariably turns out to b e rigid, w e shall mak e the assumption that C is rigid ev en though this is not indisp ensable; note that in this ase End ⊗ ( ω ) = Aut ⊗ ( ω ) ie λ tensor preserving implies λ in v ertible, see, for instane, [ 12℄ Prop. 1.13. W e shall use the adjetiv e `lassial' to refer to bre funtors of this sort. Setion 21 is dev oted to the study of some general prop erties of lassial bre funtors. T o start with, the T annakian group oid T ( ω ) asso iated with a lassial bre funtor ω pro v es to b e a C ∞ -strutured group oid, that is to sa y all the struture maps of T ( ω ) turn out to b e morphisms of fun- tionally strutured spaes; ompare 18. This allo ws us to in tro due the ategory R ∞ ( T ( ω )) of C ∞ -represen tations of the C ∞ -strutured group oid T ( ω ) , along with an  ev aluation funtor ev : C − → R ∞ ( T ( ω )) . The latter is in fat a tensor funtor, b y whi h the ategory C is put in relation to R ∞ ( T ( ω )) ; w e shall sa y more ab out this funtor in 26. Finally , w e observ e that a lassial bre funtor ω whi h admits enough ω -in v arian t metris (in the sense of Denition 19.2) is prop erin other w ords, so is the orresp onding map ( s , t ) : T ( ω ) → M × M . Setion 22 deals with the te hnial notion of tame submanifold, and is preliminary to 2325 . Ho w ev er, in order to read the latter setions a thor- ough understanding of 22 is not really neessary: it is atually enough to kno w what tame submanifolds are and the statemen ts of Prop ositions 22.5, 22.11 ; one ma y skip what remains of  22 at rst reading. Setion 23 pro vides, for the T annakian group oid T ( ω ) asso iated with a lassial bre funtor ω : C → V ∞ ( M ) , an alternativ e  haraterization of the 109 110 CHAPTER V. CLASSICAL FIBRE FUNCTORS prop ert y of smo othness in terms of what w e all represen tativ e  harts. Su h  harts arise from the ob jets of the ategory C , and their denition in v olv es tame submanifolds of linear group oids GL ( E ) o v er the manifold M . Setions 24 25 are dev oted to morphisms of bre funtors. F or ea h morphism b et w een t w o lassial bre funtors there exists a orresp onding homomorphism b et w een the asso iated T annakian group oids, whi h turns out to b e  smo oth ie a homomorphism of C ∞ -strutured group oids. In 25 w e in tro due, as a sp eial ase, the notion of w eak equiv alene; the alterna- tiv e  haraterization of smo othness pro vided in 23 is here put to w ork to sho w that the prop ert y of smo othness is, for lassial bre funtors, in v ari- an t under w eak equiv alene. Finally , the homomorphism asso iated with a w eak equiv alene of smo oth lassial bre funtors is pro v ed to b e a Morita equiv alene. 21 Basi Denitions and Prop erties In this setion w e study general prop erties of lassial bre funtors. Let us b egin b y giving a preise denition: 1 Denition W e shall all a bre funtor ω : C → F ( M ) lassi al if it meets the follo wing requiremen ts: i) the domain tensor ategory C is rigid; ii) for ev ery R ∈ Ob( C ) , ω ( R ) is a lo ally trivial ob jet of F ( M ) . Observ e that sine the t yp e F is a sta k of smo oth elds, ω ( R ) in ii) will atually b elong to Ob V F ( M ) ie it will b e a lo ally trivial ob jet of F ( M ) of lo ally nite rank (fr 11). Sine V F ( M ) is equiv alen t to the ategory V ∞ ( M ) of smo oth v etor bundles of lo ally nite rank o v er M (reall that the base M is alw a ys paraompat), it follo ws that the theory of lassial bre funtors essen tially redues to just one t yp e F = V ∞ . Beause of this, for the rest of the presen t  hapteratually , for the rest of the presen t w orkw e shall omit an y referene to t yp e. So, for instane, w e will write V ∞ ( M ) or V ∞ ( M ) at all plaes where w e w ould otherwise write F ( M ) . The piv otal fat of lassial bre funtor theory is that for su h bre funtors one has lo al form ulas analogous to (17.5). Namely , let ω : C → V ∞ ( M ) b e a lassial bre funtor. Let an ob jet R ∈ Ob( C ) and an arro w λ 0 ∈ T ≡ T ( ω ) (1) b e giv en. Cho ose, on E ≡ ω ( R ) , an arbitrary Hilb ert metri φ , whose existene is guaran teed b y the paraompatness of M . By the lo al trivialit y assumption on E , it will b e p ossible to nd a lo al φ - orthonormal frame ζ 1 ′ , . . . , ζ d ′ ∈ Γ E ( U ′ ) ab out x 0 ′ ≡ t ( λ 0 ) su h that E u ′ = Span  ζ 1 ′ ( u ′ ) , . . . , ζ d ′ ( u ′ )  for all u ′ ∈ U ′ . (Note that here one really needs lo al trivialit y of E within F , in the sense of  11, and not just the h yp othesis that Γ E is lo ally free as a sheaf of mo dules o v er M .) Then for an y giv en 21. BASIC DEFINITIONS AND PR OPER TIES 111 lo al setion ζ ∈ Γ E ( U ) , dened in a neigh b ourho o d U of x 0 ≡ s ( λ 0 ) , one gets, b y letting Ω ≡ s − 1 ( U ) ∩ t − 1 ( U ′ ) ⊂ T , λ ( R ) · ζ ( s λ ) = d P i ′ =1 r R,φ,ζ ,ζ i ′ ′ ( λ ) ζ i ′ ′ ( t λ ) , ( ∀ λ ∈ Ω) (2) where r R,φ,ζ ,ζ i ′ ′ ∈ R ∞ (Ω) denotesas in 18the represen tativ e funtion λ 7→  λ ( R ) · ζ ( s λ ) , ζ i ′ ′ ( t λ )  φ . W e shall immediately put this basi remark to w ork in the pro of of the follo wing 3 Prop osition F or ev ery lassial bre funtor ω : C → V ∞ ( M ) , the T annakian group oid T ( ω ) is a C ∞ -strutured group oid (with resp et to the standard C ∞ -struture R ∞ dened in 18). T ( ω ) is, in partiular, a top ologial group oid for ev ery lassial ω . Pro of Let us tak e an arbitrary represen tativ e funtion r = r R,φ,ζ ,ζ ′ : T → C on the spae T ≡ T ( ω ) (1) , as in (18.10). W e shall regard r as xed throughout the en tire pro of. T o b egin with, w e onsider the omp osition map T (2) = T s × t T c − → T . Our goal is to sho w that the funtion r ◦ c is a global setion of the sheaf R (2) , ∞ ≡ ( R ∞ ⊗ R ∞ ) ∞ | T (2) . (Review, if neessary , the disussion ab out funtionally strutured group oids in 18.) Fix an y pair of omp osable arro ws ( λ 0 ′ , λ 0 ) ∈ T (2) . There will b e some φ -orthonormal frame ζ 1 ′ , . . . , ζ d ′ ∈ Γ ( ω R )( U ′ ) ab out x 0 ′ ≡ t ( λ 0 ) , su h that Eq. (2) ab o v e holds for all λ ∈ Ω ′ ≡ t − 1 ( U ′ ) . Then, for ev ery pair ( λ ′ , λ ) b elonging to the op en subset Ω ′′ ≡ s − 1 ( U ′ ) s × t t − 1 ( U ′ ) ⊂ T (2) , one gets the iden tit y ( r ◦ c )( λ ′ , λ ) = r ( λ ′ ◦ λ ) =  λ ′ ( R ) · λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ ′ )  φ = d P i ′ =1 r R,φ,ζ ,ζ i ′ ′ ( λ ) r R,φ,ζ i ′ ′ ,ζ ′ ( λ ′ ) , b y (2) whi h expresses ( r ◦ c ) | Ω ′′ in the desired form, namely as an elemen t of R (2) , ∞ (Ω ′′ ) . Next, onsider the in v erse map T i − → T . Fix an y λ 0 ∈ T . In a neigh b our- ho o d U of x 0 = s ( λ 0 ) it will b e p ossible to nd a trivializing φ -orthonormal frame ζ 1 , . . . , ζ d ∈ Γ ( ω R )( U ) . One an write do wn (2 ) for ea h ζ i ( i = 1 , . . . , d ): λ ( R ) · ζ i ( s λ ) = d P i ′ =1 r R,φ,ζ i ,ζ ′ i ′ ( λ ) ζ ′ i ′ ( t λ ) . ( λ ∈ Ω = s − 1 ( U ) ∩ t − 1 ( U ′ ) . ) (4) Letting { r ′ i ′ ,i ( λ ) : 1 ≦ i ′ , i ≦ d } denote the in v erse of the matrix { r R,φ,ζ i ,ζ ′ i ′ ( λ ) : 1 ≦ i ′ , i ≦ d } for ea h λ (this mak es sense b eause λ ( R ) is a linear iso), w e see from the standard form ula in v olving the in v erse of the determinan t that 112 CHAPTER V. CLASSICAL FIBRE FUNCTORS r ′ i ′ ,i ∈ R ∞ (Ω) for all 1 ≦ i ′ , i ≦ d . If w e no w put a ′ i ′ = h ζ , ζ ′ i ′ i φ ∈ C ∞ ( U ′ ) for all i ′ = 1 , . . . , d and a i = h ζ i , ζ ′ i φ ∈ C ∞ ( U ) for all i = 1 , . . . , d , w e obtain the follo wing expression for ( r ◦ i ) | Ω ( r ◦ i )( λ ) = r ( λ − 1 ) =  λ ( R ) − 1 · ζ ( t λ ) , ζ ′ ( s λ )  φ = = d P i ′ =1 a ′ i ′ ( t λ )  λ ( R ) − 1 · ζ ′ i ′ ( t λ ) , ζ ′ ( s λ )  φ = = d P i ′ =1 d P i =1 a ′ i ′ ( t λ ) r ′ i ′ ,i ( λ ) a i ( s λ ) , whi h learly sho ws mem b ership of ( r ◦ i ) | Ω in R ∞ (Ω) . The smo othness of the remaining struture maps w as already pro v ed in 18 for an arbitrary bre funtor. q.e.d. By exploiting the ategorial equiv alene V ( M ) ≈ − → V ∞ ( M ) , E 7→ ˜ E (12 .6), one an mak e sense of the expression GL ( E ) for ev ery E ∈ Ob V ( M ) simply b y regarding GL ( E ) as short for GL ( ˜ E ) . If ω : C → V ( M ) is a lassial bre funtor, ea h ob jet R ∈ Ob( C ) will determine a homomorphism of funtionally strutured group oids (5) ev R : T ( ω ) − → GL ( ω R ) , λ 7→ λ ( R ) (note that if φ is an y Hilb ert metri on E = ω ( R ) , the funtions q φ,ζ ,ζ ′ : GL ( E ) (1) → C , µ 7→  µ · ζ ( s µ ) , ζ ′ ( t µ )  φ will pro vide suitable lo al o ordinate systems for the manifold GL ( E ) (1) ), whi h ma y b e though t of as a smo oth represen tation of T ( ω ) . It is w orth while men tioning the follo wing univ ersal prop ert y , whi h  har- aterizes the funtional struture (and top ology) w e endo w ed the T annakian group oid with. Let ω b e a lassial bre funtor. Then for an y funtionally strutured spae ( Z , F ) , a mapping f : Z → T = T ( ω ) (1) is a morph- ism of ( Z , F ) in to ( T , R ∞ ) (or simply , a on tin uous mapping of Z in to T ) if and only if su h is ev R ◦ f for ev ery R ∈ Ob C . The `only if ' diretion is lear b eause of the foregoing remarks ab out the smo othness of ev R . Con v ersely , onsider an y represen tativ e funtion r = r R,φ,ζ ,ζ ′ : T → C ; if q φ,ζ ,ζ ′ : GL ( ω R ) (1) → C is the smo oth funtion dened ab o v e then one has r ◦ f = q φ,ζ ,ζ ′ ◦ ev R ◦ f ∈ F ( Z ) , b eause b y assumption ev R ◦ f is a morph- ism of ( Z , F ) in to the smo oth manifold GL ( ω R ) (1) . The equiv alene is no w pro v en. In a manner en tirely analogous to 2 , one an dene the omplex tensor ategory R ∞ ( T ( ω ); C ) of all  smo oth represen tations of the funtionally strutured group oid T ( ω ) on smo oth omplex v etor bundles o v er the base manifold M of ω . Preisely , an y su h represen tation will onsist of a omplex v etor bundle E ∈ Ob V ∞ ( M ) and a homomorphism  : T ( ω ) → GL ( E ) of 21. BASIC DEFINITIONS AND PR OPER TIES 113 funtionally strutured group oids o v er M (  iden tial on M ). Then one has the omplex tensor funtor (6) ev : C − → R ∞ ( T ( ω ); C ) , R 7→ ( ] ω ( R ) , ev R ) (the so-alled  ev aluation funtor). The parallel with the situation depited in 9 leads us to form ulate the problem of determining whether or not the funtor (6) is in generalfor an arbitrary lassial bre funtora ategorial equiv alene. The answ er is kno wn to b e y es, atually in the strong form of an isomorphism of ategories, for a large lass of examples: see 26 , Prop osition (26 .21) and related ommen ts. W e onlude this in tro dutory setion with an observ ation ab out prop er lassial bre funtors (fr. 19). W e in tend to sho w that, in the lassial ase, existene of enough in v arian t metris is suien t to ensure prop erness and hene that the rst ondition of Denition 19.2 is atually redundan t for an y lassial bre funtor. Notie rst of all that ea h Hilb ert metri φ on a omplex v etor bundle E ∈ Ob V ∞ ( M ) determines a subgroup oid U ( E , φ ) ⊂ GL ( E ) , onsisting of all φ -unitary linear isomorphisms b et w een the bres of E ; more expliitly , the arro ws x → x ′ in U ( E , φ ) are the unitary isomorphisms of ( E x , φ x ) on to ( E x ′ , φ x ′ ) . Clearly , U ( E , φ ) is a prop er Lie group oid o v er the manifold M , em b edded in to GL ( E ) . When there is no danger of am biguit y ab out the metri, w e will just suppress φ from the notation. F rom our elemen tary remark (19.6) it follo ws that for an y ω -in v arian t Hilb ert metri φ on ω ( R ) ( R ∈ Ob C ) the ev aluation homomorphism ev R (5) m ust fator through the subgroup oid U ( ω R ) ֒ → GL ( ω R ) . Hene one ma y view ev R as a smo oth homomorphism (7) ev R : T ( ω ) − → U ( ω R ) , λ 7→ λ ( R ) . 8 Prop osition Let ω : C → V ∞ ( M ) b e a lassial bre funtor. Supp ose there are enough ω -in v arian t metris (fr 19, Denition 2 ). Then ω is prop er; in partiular, T ( ω ) is a prop er group oid. Pro of Let us assign, to ea h ob jet R ∈ Ob C , an ω -in v arian t metri φ R on ω ( R ) one and for all. W e shall simply write U ( ω R ) in plae of U ( ω ( R ) , φ R ) . Let K b e an arbitrary ompat subset of the base manifold M . W e ha v e to sho w that T | K = ( s , t ) − 1 ( K × K ) is a ompat subset of the top ologial spae T = T ( ω ) (1) . Consider the auxiliary spae (9) Z K def = Y R ∈ Ob C U ( ω R ) | K (pro dut of top ologial spaes) and observ e that Z K is ompat b eause the same is true of ea h fator U ( ω R ) | K . There is an ob vious on tin uous injetiv e 114 CHAPTER V. CLASSICAL FIBRE FUNCTORS map e : T | K ֒ → Z K giv en b y λ 7→ { λ ( R ) } R ∈ Ob C . W e laim that this map is atually a top ologial em b edding of T | K on to a losed subset of Z K : this will en tail the required ompatness of T | K . The map e is an emb e dding. This will b e implied at one b y the follo wing extension prop ert y of represen tativ e funtions: for ev ery r = r R,φ,ζ ,ζ ′ ∈ R (18 .11), there exists a on tin uous funtion h : Z K → C su h that r = h ◦ e on T | K . In order to obtain su h an extension of r , note simply that on T | K one has r R,φ,ζ ,ζ ′ = ( q φ,ζ ,ζ ′ ◦ π R ) ◦ e , where π R : Z K → U ( ω R ) | K is the R -th pro jetion and q φ,ζ ,ζ ′ is the (restrition to U ( ω R ) | K of ) the smo oth funtion GL ( ω R ) → C , µ 7→  µ · ζ ( s µ ) , ζ ′ ( t µ )  φ . The image of e is a lose d subset of Z K . It is suien t to observ e that the onditions expressing mem b ership of µ = { µ R } R ∈ Ob C ∈ Q U ( ω R ) | K in the image of e namely that s ( µ R ) = s ( µ S ) and t ( µ R ) = t ( µ S ) ∀ R, S ∈ Ob C , naturalit y of µ and its b eing tensor preserving and self-onjugateare ea h stated in terms of a h uge n um b er of iden tities whi h in v olv e only the o ordinates µ R = π R ( µ ) in a on tin uous w a y . q.e.d. 10 Note A v ery marginal ommen t ab out prop er lassial bre funtors, impro ving, in the lassial ase, Lemma 19.5: for any pr op er lassi al ω , the e quality R = R ′ holds. In order to see this, notie rst of all that if U is an y op en subset of M on whi h E | U ( E = ω ( R ) ) trivializes then w e an nd a ∈ Aut ( E | U ) su h that φ u ( v , v ′ ) = φ R,u ( v , a u · v ′ ) for all u ∈ U ( φ an arbitrary metri on E , φ R as in the pro of of the preeding prop osition, v , v ′ ∈ E u ). No w, if w e put ξ ′ U = a ( U ) ζ ′ U where ζ ′ U is the restrition to U of ζ ′ , w e get r R,φ,ζ ,ζ ′ = r R,φ R ,ζ ,ξ ′ U on t − 1 ( U ) ⊂ T . W e an use a partition of unit y o v er all su h U 's to obtain a global setion ξ ′ with the prop ert y that r R,φ,ζ ,ζ ′ = r R,φ R ,ζ ,ξ ′ ∈ R ′ . 22 T ame Submanifolds of a Lie Group oid Let G b e a Lie group oid o v er a manifold M . 1 Denition A submanifold Σ of the manifold of arro ws G (1) will b e said to b e prinip al if it an b e o v ered with lo al parametrizations (viz in v erses of lo al  harts or, equiv alen tly , op en em b eddings) of the form (2) ( Z × A ֒ → Σ ( z , a ) 7→ τ ( z ) · η ( a ) , where Z is a submanifold of M , τ : Z → G ( x, - ) is, for some p oin t x ∈ M , a smo oth setion to the target map of the group oid, η : H ֒ → G x is a Lie subgroup of the x -th isotrop y group G x of G and A is an op en subset of H su h that η restrits to an em b edding of A in to G x . 22. T AME SUBMANIF OLDS OF A LIE GR OUPOID 115 Note that the image Σ = τ ( Z ) · η ( A ) of a map of the form ( 2) is alw a ys a submanifold of G (1) and that the same map indues a smo oth isomorphism of Z × A on to Σ . So, in partiular, it mak es sense to use su h maps as lo al parametrizations. (Details an b e found in Note 6 b elo w.) Note also that an y prinipal submanifold of G (1) admits an op en o v er b y lo al parametrizations of t yp e (2) with the additional prop ert y that the Lie group H is onneted and A is an op en neigh b ourho o d in H of the neutral elemen t e . (Indeed, let σ ∈ Σ b e a giv en p oin t and  ho ose a lo al parametrization τ · η of the form (2). Supp ose σ = ( z , a ) ∈ Z × A in this lo al  hart. Replaing A with a − 1 A and τ with τ · η ( a ) aomplishes the redution to the situation where A is a neigh b ourho o d of e and σ = ( z , e ) ; in terseting with the onneted omp onen t of e in H nishes the job.) 3 Lemma Let ϕ : G → G ′ b e a Lie group oid homomorphism, induing an immersion f : M → M ′ at the lev el of manifolds of ob jets. Assume that Σ and Σ ′ are prinipal submanifolds of G and G ′ resp etiv ely , with the prop ert y that ϕ maps Σ injetiv ely in to Σ ′ . Then ϕ restrits to an immersion of Σ in to Σ ′ . Pro of Fix an y p oin t σ 0 ∈ Σ and let x 0 ≡ s ( σ 0 ) , z 0 ≡ t ( σ 0 ) . Cho ose lo al parametrizations τ · η : Z × A ֒ → Σ and τ ′ · η ′ : Z ′ × A ′ ֒ → Σ ′ of t yp e (2) with, let us sa y , σ 0 = ( z 0 , e ) ∈ Z × A and ϕ ( σ 0 ) = ( f ( z 0 ) , e ′ ) ∈ Z ′ × A ′ , where e , resp. e ′ is the neutral elemen t of the Lie subgroup η : H ֒ → G x 0 , resp. η ′ : H ′ ֒ → G ′ f ( x 0 ) . As remark ed ab o v e, the Lie groups H and H ′ an b e assumed to b e onneted. Let the domain of the rst parametrization shrink around the p oin t ( z 0 , e ) un til the smo oth injetion ϕ : Σ ֒ → Σ ′ admits a lo al represen tation relativ e to the  hosen parametrizations, namely Σ ϕ / / Σ ′ Z × A  ? τ · η O O ˜ ϕ / / _ _ _ _ Z ′ × A ′ .  ? τ ′ · η ′ O O ˜ ϕ will b e a smo oth injetiv e map, of the form ( z , a ) 7→  z ′ ( z , a ) , a ′ ( z , a )  . Note that z ′ ( z , a ) = f ( z ) so that, in partiular, f maps Z in to Z ′ ; this follo ws b y omparing the target of the t w o sides of the equalit y τ ′ ( z ′ ) · η ′ ( a ′ ) = ϕ ( τ ( z )) · ϕ ( η ( a )) . Sine the restrition of f to Z is an immersion of Z in to Z ′ , the mapping ˜ ϕ is immersiv e at ( z 0 , e ) if and only if the orresp onding partial map a 7→ a ′ ( z 0 , a ) is immersiv e at e ∈ A . No w, onsider the follo wing h uge omm utativ e 116 CHAPTER V. CLASSICAL FIBRE FUNCTORS diagram, where w e put x ′ 0 ≡ f ( x 0 ) and z ′ 0 ≡ f ( z 0 ) : G x 0 G x 0 ϕ / / G ′ f ( x 0 ) G ′ f ( x 0 ) G ( x 0 , z 0 ) τ ( z 0 ) − 1 · O O ϕ / / G ′ ( x ′ 0 , z ′ 0 ) ϕ ( τ ( z 0 )) − 1 · O O A ?  η O O { z 0 } × A  ? τ · η O O / / { z ′ 0 } × A ′  ? τ ′ · η ′ O O A ′ ?  η ′ O O [the retangle on the righ t omm utes b eause ϕ ( τ ( z 0 )) = ϕ ( σ 0 ) = τ ′ ( f ( z 0 )) = τ ′ ( z ′ 0 ) ℄. The omm utativit y of the outer retangle en tails that the b ottom map in this diagram, namely a 7→ a ′ ( z 0 , a ) , oinides with the restrition to A of a (neessarily unique) Lie group homomorphism ζ : H → H ′ ; the same map is therefore an immersion, b eause a Lie group homomorphism whi h is injetiv e in a neigh b ourho o d of e m ust b e immersiv e, see eg Brö  k er and tom Die k [4℄, p. 27. The pro of of the existene of the homomorphism of Lie groups ζ is deferred to Note 9 b elo w. q.e.d. 4 Denition A submanifold Σ of the arro w manifold of a Lie group oid G will b e said to b e tame if the follo wing onditions are satised: i) the soure map of G restrits to a submersion of Σ on to an op en subset of the base manifold M of G ; ii) for ea h p oin t x ∈ M , the orresp onding soure bre Σ( x, - ) ≡ Σ ∩ G ( x, - ) is a prinipal submanifold. Note that from the rst ondition it already follo ws that the soure bre Σ( x, - ) is a submanifold (of Σ and hene) of G (1) . 5 Prop osition Let ϕ : G → G ′ b e a Lie group oid homomorphism, induing an immersion f : M → M ′ at the lev el of base manifolds. Supp ose that Σ , resp. Σ ′ is a tame submanifold of G , resp. G ′ and that ϕ maps Σ injetiv ely in to Σ ′ . Then ϕ restrits to an immersion of Σ in to Σ ′ . Pro of Fix σ 0 ∈ Σ , and put x 0 = s ( σ 0 ) . Cho ose lo al parametrizations U × B ֒ → Σ at σ 0 ≈ ( x 0 , 0) ∈ U × B , and U ′ × B ′ ֒ → Σ ′ at ϕ ( σ 0 ) ≈ ( f ( x 0 ) , 0) ∈ U ′ × B ′ , lo ally trivializing the resp etiv e soure mapwhi h is a submersion b eause of Condition i) of Denition 4o v er the op en sub- sets U ⊂ M , U ′ ⊂ M ′ . (Here B and B ′ are op en balls.) This means, for instane, that the rst parametrization mak es the diagram U × B pr # # G G G G G G G G G   / / Σ s          U 22. T AME SUBMANIF OLDS OF A LIE GR OUPOID 117 omm ute. If the domain of the rst parametrization is made to b e on v e- nien tly small around the en ter ( x 0 , 0) , the mapping ϕ : Σ ֒ → Σ ′ will indue a smo oth and injetiv e lo al expression Σ ϕ / / Σ ′ U × B  ? O O / / _ _ _ U ′ × B ′  ? O O of the form ( x, b ) 7→ ( x ′ ( x, b ) , b ′ ( x, b )) = ( f ( x ) , b ′ ( x, b )) , so that, in partiular, f will map U in to U ′ . Sine f : U → U ′ is then an immersion b y assumption, the ab o v e lo al expression is an immersiv e map at ( x 0 , 0) if and only if the partial map b 7→ b ′ ( x 0 , b ) is immersiv e at 0 ∈ B . A t this p oin t w e an use Lemma 3 to onlude the pro of. q.e.d. In partiular, it follo ws that when a homomorphism ϕ of Lie group oids (let us sa y o v er the same manifold M and with f = id ) indues a homeo- morphism b et w een t w o tame submanifolds Σ and Σ ′ , then it restrits in fat to a dieomorphism of Σ on to Σ ′ . This will b e for us the most useful prop ert y of tame submanifolds, and w e shall mak e rep eated appliation of it in the subsequen t setions. A tually , the motiv ation for in tro duing the onept of tame submanifold w as preisely to ensure this kind of automati dieren tia- bilit y out of on tin uit y. 6 Note Let S = G m b e the m -th orbit. As a notational on v en tion, w e shall use the letter S when w e think of this orbit as a manifold, endo w ed with the unique dieren tiable struture that turns the target map (7) t : G ( m, - ) → S in to a prinipal bundle with bre the Lie group G m (ating on the manifold G ( m, - ) from the righ t, in the ob vious w a y); (7) is in partiular a bre bundle, whi h is in fat equiv arian tly lo ally trivial. The inlusion S ֒ → M is an injetiv e immersion, although not in general an em b edding of manifolds. See also Mo er dijk and Mr £un (2003), [27 ℄ pp. 115117. T o b egin with, w e sho w that the inlusion map is an em b edding of the manifold Z in to S . Of ourse, Z is a submanifold of M and w e ha v e the inlusion Z ⊂ G m , but from this fat w e annot a priori onlude that Z em b eds in to S , not ev en that the inlusion map Z ֒ → S is on tin uous; the reason wh y w e an do a w a y with this diult y is that o v er Z there exists, b y assumption, a smo oth setion τ to the target map G ( m, - ) → M . (Iniden tally , observ e that an y su h τ : Z → G ( m, - ) is an em b edding of manifolds. Clearly , it will b e enough to see that τ is an em b edding of Z in to G . Sine τ is a smo oth setion o v er Z to t : G → M , it is an injetiv e immersion; moreo v er, for an y op en subset U of M w e ha v e τ ( Z ∩ U ) = τ ( Z ) ∩ t − 1 ( U ) .) 118 CHAPTER V. CLASSICAL FIBRE FUNCTORS No w, from the existene of τ it follo ws immediately that the inlusion t ◦ τ of Z in to S is a smo oth mapping; moreo v er, w e ha v e that this is atually an injetiv e immersion, b eause on omp osing it with S ֒ → M one obtains the em b edding Z ֒ → M . It only remains to notie that if U is op en in M then Z ∩ U oinides with Z ∩ W where W = t G ( m, U ) is op en in S . Next, w e sho w that 8 Lemma F or ev ery z 0 ∈ Z , there is a lo al trivialization of the prinipal bundle (7 ), of the form G ( m, W ) ≈ W × G m o v er an op en neigh b ourho o d W of z 0 in S , su h that its unit setion agrees with τ on Z ∩ W . (Reall that the unit setion of su h a lo al trivialization is the mapping that orresp onds to W ֒ → W × G m , w 7→ ( w , 1 m ) .) Pro of Sine Z em b eds as a submanifold of S , it is p ossible to nd an op en neigh b ourho o d W of z 0 in S dieomorphi to a pro dut of manifolds W ≈ ( W ∩ Z ) × B , z 0 ≈ ( z 0 , 0) , where B is an op en eulidean ball. Moreo v er, it is learly not restritiv e to assume that the prinipal bundle (7 ) an b e trivialized o v er W . Then, after ha ving xed one su h trivialization, w e an tak e the omp osite mapping W ≈ ( W ∩ Z ) × B pr − − → W ∩ Z τ − → G ( m, W ) ≈ W × G m pr − − → G m , whi h w e denote b y θ : W → G m , and use it to pro due an equiv arian t  hange of  harts and hene a new lo al trivialization for ( 7), namely W × G m ∼ → W × G m ≈ G ( m, W ) , ( w , g ) 7→ ( w , θ ( w ) g ) , whose unit setion is immediately seen to agree with τ on Z ∩ W . q.e.d. Our aim w as to pro v e that Σ = τ ( Z ) · η ( A ) is a submanifold of G and that τ · η is a smo oth isomorphism b et w een Z × A and Σ . Th us, x σ 0 ∈ Σ , an let z 0 = t ( σ 0 ) ; the latter is a p oin t of Z . Fix also a trivializing  hart for the prinipal bundle (7 ) as in the statemen t of Lemma 8; then W × G m ≈ dieo. / / G ( m, W ) ( Z ∩ W ) × A  ? em b ed. O O bijet. / / Σ ∩ G ( m, W )  ? set-th. inl. O O omm utes, where on the left w e ha v e the ob vious em b edding of manifolds, and the b ottom map is ( z , a ) 7→ τ ( z ) · η ( a ) , the restrition of τ · η . (The diagram omm utes preisely b eause the unit setion of the  hart agrees with τ o v er Z ∩ W .) It is then lear that Σ ∩ G ( m, W ) is a submanifold of 22. T AME SUBMANIF OLDS OF A LIE GR OUPOID 119 the op en neigh b ourho o d G ( m, W ) of σ 0 in G ( m, - ) , and that τ · η restrits to a dieomorphism of ( Z ∩ W ) × A on to this submanifold. Heneforth, Σ is a submanifold of G ( m, - ) and τ · η is a bijetiv e lo al dieomorphism b et w een Z × A and Σ . (Note that the statemen t that Z ֒ → S is an em b edding is really used here.) 9 Note Assume that a  ommutative r e tangle A     / / H ∃ ! ζ     η / / G ϕ   A ′   / / H ′   η ′ / / G ′ is given, wher e G , G ′ ar e Lie gr oups, ϕ is a Lie gr oup homomorphism, η : H ֒ → G and η ′ : H ′ ֒ → G ′ ar e Lie sub gr oups with H  onne te d, A ⊂ H , A ′ ⊂ H ′ ar e op en neighb ourho o ds of the unit elements e , e ′ of H , H ′ r esp e tively, and A → A ′ is a smo oth mapping. Then ther e exists a unique Lie gr oup homomorphism ζ : H → H ′ whih ts in the diagr am as indi ate d. Indeed, sine A is an op en neigh b ourho o d of e in H and H is onneted, A generates H as a group, see Br ö ker and tom Die k (1995), [ 4℄ p. 10. So ϕη ( A ) generates ϕη ( H ) , and therefore ϕη ( H ) ⊂ η ′ ( H ′ ) b eause ϕη ( A ) ⊂ η ′ ( A ′ ) ⊂ η ′ ( H ′ ) . Sine η ′ : H ′ → η ′ ( H ′ ) is a bijetiv e homomorphism of groups, there exists a unique group-theoreti solution ζ : H → H ′ to the problem η ′ ◦ ζ = ϕ ◦ η . The restrition of ζ to A oinides with the giv en smo oth map A → A ′ , th us ζ is smo oth in a neigh b ourho o d of e ; sine left translations are Lie group automorphisms, the omm utativit y of H ≈ h ·   ζ / / H ′ ≈ ζ ( h ) ·   H ζ / / H ′ sho ws that ζ is smo oth in the neigh b ourho o d of an y h ∈ H , and hene globally smo oth, in other w ords a Lie group homomorphism. T ameness and Morita equiv alene There is still one fundamen tal p oin t w e need to disuss, for the treatmen t of w eak equiv alenes of lassial bre funtors in Setion 25 b elo w. Namely , supp ose one is giv en a Morita equiv alene of Lie group oids ϕ : G → G ′ su h that at the lev el of manifolds of ob jets it is giv en b y a submersion ϕ : M → M ′ . Let Σ b e a subset of the manifold of arro ws of G , and assume that ev ery p oin t of Σ has an op en neigh b ourho o d Γ in G with (10) ϕ − 1 (Σ ′ ) ∩ Γ ⊂ Σ , 120 CHAPTER V. CLASSICAL FIBRE FUNCTORS where w e put Σ ′ = ϕ (Σ) ; note that this is equiv alen t to sa ying that ∀ γ ∈ Γ , γ ∈ Σ ⇔ ϕ ( γ ) ∈ Σ ′ . Then one has what follo ws 1. Σ is a submanifold of G if and only if Σ ′ is a submanifold of G ′ ; 2. Σ is a submanifold of G verifying Condition i) of Denition 4 if and only if the same is true of Σ ′ in G ′ ; 3. for every m ∈ M , the r estrition ϕ : Σ( m, - ) → Σ ′ ( ϕ ( m ) , - ) is an op en mapping b etwe en top olo gi al subsp a es of the manifolds G and G ′ ; 4. for every m ∈ M , the br e Σ( m, - ) is a prinip al submanifold of G if and only if its image ϕ (Σ( m, - )) is a prinip al submanifold of G ′ . Before w e start with the pro ofs, let us sho w ho w these statemen ts 1-4 ma y b e used to deriv e the follo wing main result 11 Prop osition Let ϕ : G − → G ′ b e a Morita equiv alene of Lie group oids induing a submersion at the lev el of base manifolds. Let Σ b e a subset of the manifold of arro ws of G whi h satises ondition ( 10) ab o v e, and put Σ ′ = ϕ (Σ) . Then Σ is a tame submanifold of G if and only if Σ ′ is a tame submanifold of G ′ . Pro of ( ⇐ ) Supp ose m ∈ M is giv en: w e m ust sho w that Σ( m, - ) is a prini- pal submanifold of G . Beause of Statemen t 3, ϕ (Σ( m, - )) is an op en subset of the subspae Σ ′ ( ϕ ( m ) , - ) ⊂ G ′ . Sine the latter is b y assumption a prinipal submanifold of G ′ , it follo ws that the op en subset ϕ (Σ( m, - )) is a prini- pal submanifold of G ′ as w ell, and hene, b y Statemen t 4, that Σ( m, - ) is a prinipal submanifold of G . ( ⇒ ) Fix m ′ ∈ M ′ . A ording to Statemen t 3, w e ha v e the op en o v ering Σ ′ ( m ′ , - ) = [ m ∈ ϕ − 1 ( m ′ ) ϕ (Σ( m, - )) , and ev ery op en set b elonging to this o v ering is a prinipal submanifold of G ′ , b y Statemen t 4 and the assumption. Hene the whole submanifold Σ ′ ( m ′ , - ) ⊂ G ′ is a prinipal submanifold of G ′ . q.e.d. No w w e ome to the pro ofs of Statemen ts 1 to 4: Pr oof of St a tement 1. Reall from Note 15, (16 ) b elo w that, up to dieomorphism, one has for the morphism ϕ a anonial deomp osition Γ ( s , t )   ≈ / / Γ ′ × B × C   pr / / Γ ′ ( s ′ , t ′ )   U × V ≈×≈ / / U ′ × B × V ′ × C pr × pr / / U ′ × V ′ 22. T AME SUBMANIF OLDS OF A LIE GR OUPOID 121 in a neigh b ourho o d Γ of ev ery p oin t of Σ , with Γ v erifying ondition ( 10). W e ha v e that Σ ′ ∩ Γ ′ is a submanifold of Γ ′ if and only if (Σ ′ ∩ Γ ′ ) × A is a submanifold of Γ ′ × A , where A = B × C . Th us, sine (Σ ′ ∩ Γ ′ ) × A = pr − 1 (Σ ′ ∩ Γ ′ ) orresp onds to ϕ − 1 (Σ ′ ∩ Γ ′ ) ∩ Γ = ϕ − 1 (Σ ′ ) ∩ Γ = Σ ∩ Γ in the dieomorphism Γ ≈ Γ ′ × B × C , this is in turn equiv alen t to sa ying that Σ ∩ Γ is a submanifold of Γ . Th us w e see that Σ is a submanifold of G if and only if Σ ′ is a submanifold of G ′ . Pr oof of St a tement 2. F rom the previous diagram, w e get that, up to dieomorphism, s : Γ → U orresp onds to s ′ × pr : Γ ′ × B × C → U ′ × B , so it restrits to a submersion Σ ∩ Γ → U if and only if s ′ × pr restrits to a submersion (Σ ′ ∩ Γ ′ ) × B × C → U ′ × B ; and this is in turn true if and only if s ′ : Σ ′ ∩ Γ ′ → U ′ is a submersion. Pr oof of St a tement 3. Fix a p oin t σ 0 ∈ Σ( m, - ) and an op en neigh b our- ho o d of that p oin t in G . Then from Note 15 b elo w, w e ha v e for the restrition of ϕ to Σ a anonial lo al deomp osition Σ ∩ Γ s   ≈ / / (Σ ′ ∩ Γ ′ ) × B × C s ′ × id   pr / / Σ ′ ∩ Γ ′ s ′   U ≈ / / U ′ × B pr / / U ′ at σ 0 = ( σ ′ 0 , 0 , 0) , where Γ an b e  ho osen as small as one lik es around σ 0 , simply b y taking a smaller Γ ′ = ϕ (Γ) at σ ′ 0 = ϕ ( σ 0 ) and reduing the radius of the op en balls B , C ; in partiular, Γ an b e  hosen so small that it ts in the previously assigned op en neigh b ourho o d of σ 0 in G . It is immediate to reognize that ϕ (Σ( m, - ) ∩ Γ) = Σ ′ ( ϕ ( m ) , - ) ∩ Γ ′ , where the latter is learly an op en subset of the subspae Σ ′ ( ϕ ( m ) , - ) of G ′ . Indeed, in the left-hand square of the preeding diagram, the s -bre ab o v e m ∈ U , namely (Σ ∩ Γ)( m, - ) = Σ( m, - ) ∩ Γ , orresp onds to the s ′ × pr -bre ab o v e ( ϕ ( m ) , 0) , namely (Σ ′ ∩ Γ ′ )( ϕ ( m ) , - ) × 0 × C . The latter is mapp ed b y the pro jetion pr on to (Σ ′ ∩ Γ ′ )( ϕ ( m ) , - ) = Σ ′ ( ϕ ( m ) , - ) ∩ Γ ′ , hene ϕ maps Σ( m, - ) ∩ Γ on to Σ ′ ( ϕ ( m ) , - ) ∩ Γ ′ , as on tended. Pr oof of St a tement 4. This will b e based on the follo wing lemma: 122 CHAPTER V. CLASSICAL FIBRE FUNCTORS 12 Lemma Let ϕ : G → G ′ b e a fully faithful homomorphism of Lie group oids and let ϕ : M → M ′ b e the map indued on base manifolds. Supp ose that Σ ⊂ G and Σ ′ = ϕ (Σ) ⊂ G ′ are submanifolds. Supp ose also that a omm utativ e diagram (13) Σ t   ≈ / / Σ ′ × C t ′ × id   pr / / Σ ′ t ′   V ≈ / / V ′ × C pr / / V ′ is giv en, where V ⊂ M and V ′ ⊂ M ′ are op en subsets, C is an op en ball and the ≈ 's are dieomorphisms su h that the top ro w oinides with ϕ (arro ws) and the b ottom one with ϕ (ob jets). Let σ 0 ∈ Σ b e a p oin t with σ 0 ≈ ( σ ′ 0 , 0) ∈ Σ ′ × C . Then Σ admits a lo al parametrization of t yp e (2 ) at σ 0 if and only if Σ ′ admits su h a parametrization at σ ′ 0 . Pro of Notation: let z 0 = t ( σ 0 ) ∈ V and z ′ 0 = t ′ ( σ ′ 0 ) = ϕ ( z 0 ) ∈ V ′ . Observ e that from (13) it follo ws that z 0 orresp onds to ( z ′ 0 , 0) in the dieomorphism V ≈ V ′ × C , b eause σ 0 orresp onds to ( σ ′ 0 , 0) in Σ ≈ Σ ′ × C . ( ⇐ ) Supp ose that Σ ′ admits a t yp e (2 ) lo al parametrization σ ′ · η ′ : Z ′ × A ′ ֒ → Σ ′ at σ ′ 0 ≈ ( z ′ 0 , e ′ ) ∈ Z ′ × A ′ . It is learly no loss of generalit y to assume that the whole Σ ′ is the image of this lo al parametrization. Z ′ = t ′ ( σ ′ ( Z ′ )) ⊂ t ′ (Σ ′ ) ⊂ V ′ is a submanifold, b eause so is Z ′ ⊂ M ′ . W rite the dieomorphism V ≈ V ′ × C as v 7→ ( ϕ ( v ) , c ( v )) and let Z ⊂ V b e the submanifold orresp onding to Z ′ × C . Dene σ : Z → Σ as σ ( z ) = ( σ ′ ( ϕ ( z )) , c ( z )) ∈ Σ ′ × C ≈ Σ , and η b y (14) G ( m, m ) ϕ ≈ / / G ′ ( m ′ , m ′ ) H ′ R 2 η d d I I I I I  , η ′ 9 9 t t t t t t t t t t so that σ is learly a smo oth t -setion t ( σ ( z )) ≈  t ′ × id  σ ′ ( ϕ ( z )) , c ( z )  =  t ′ ( σ ′ ( ϕ ( z ))) , c ( z )  = ( ϕ ( z ) , c ( z )) ≈ z with σ ( z 0 ) ≈  σ ′ ( ϕ ( z 0 )) , c ( z 0 )  = ( σ ′ 0 , 0) ≈ σ 0 , and η : H ֒ → G m is a Lie subgroup, where w e put H = H ′ . Let A = A ′ . It is immediate to alulate that the image of σ · η : Z × A ֒ → G is the whole Σ : th us w e ha v e onstruted a global parametrization of Σ at σ 0 . ( ⇒ ) In the other diretion, supp ose w e are giv en a lo al parametrization σ · η : Z × A ֒ → Σ of t yp e (2 ) su h that σ 0 ∈ Σ orresp onds to ( z 0 , e ) = ( t ( σ 0 ) , e ) ∈ Z × A . Clearly , Z = t ( σ ( Z )) ⊂ t (Σ) ⊂ V is a submanifold sine so is Z ⊂ M . 22. T AME SUBMANIF OLDS OF A LIE GR OUPOID 123 T o b egin with, observ e that it is not restritiv e to assume that the sub- manifold Z ⊂ V orresp onds to Z ′ × C under the dieomorphism V ≈ V ′ × C , where of ourse Z ′ = ϕ ( Z ) . Preisely , the dieomorphism Σ ≈ Σ ′ × C , that iden ties σ 0 with ( σ ′ 0 , 0) , allo ws one to  ho ose a smaller op en neigh b ourho o d ( σ ′ 0 , 0) ∈ Σ ′ 0 × C 0 ⊂ Σ ′ × C su h that Σ 0 ≈ Σ ′ 0 × C 0 is on- tained in the domain of the lo al  hart ( σ · η ) − 1 . F rom the omm utativit y of the diagram Z × A pr   Σ 0 ? _ ( σ · η ) − 1 op en em b. o o t   ≈ / / Σ ′ 0 × C 0 t ′ × id   Z t (Σ 0 ) ? _ inlusion o o ≈ / / t ′ (Σ ′ 0 ) × C 0 it follo ws at one that Z 0 = t (Σ 0 ) ⊂ Z is an op en subset su h that V ≈ V ′ × C indues a bijetion Z 0 ≈ Z ′ 0 × C 0 , where Z ′ 0 = t ′ (Σ ′ 0 ) . Sine it is ompatible with the aims of the presen t pro of to replae C with a smaller C 0 en tered at 0 , w e an w ork with the smaller lo al parametrization obtained b y restriting σ to the op en subset Z 0 of Z . Seondly , the t -setion σ : Z → Σ indues, b y means of the dieomorph- isms Z ≈ Z ′ × C and Σ ≈ Σ ′ × C , a smo oth mapping Z ′ × C → Σ ′ × C of the form ( z ′ , c ) 7→ ( σ ′ ( z ′ , c ) , c ) ; indeed ( z ′ , c ) ≈ z = t ( σ ( z )) ≈ ( t ′ × id )  σ ′ ( z ′ , c ) , c ( z ′ , c )  =  t ′ ( σ ′ ( z ′ , c )) , c ( z ′ , c )  , hene it follo ws t ′ ( σ ′ ( z ′ , c )) = z ′ and c ( z ′ , c ) = c . W e laim that it is no loss of generalit y to assume that it atually is of the form ( z ′ , c ) 7→ ( σ ′ ( z ′ ) , c ) , ie that σ ′ do es not really dep end on the v ariable c . Indeed, dene τ : Z → Σ as τ ( z ) =  σ ′ ( ϕ ( z ) , 0) , c ( z )  ∈ Σ ′ × C = Σ ; su h a map is also a smo oth t -setion t ( τ ( z )) ≈ ( t ′ × id )  σ ′ ( ϕ ( z ) , 0) , c  =  t ′ ( σ ′ ( z ′ , c )) , c  = ( ϕ ( z ) , c ) ≈ z with τ ( z 0 ) =  σ ′ ( z ′ 0 , 0) , 0  = σ ( z 0 ) = σ 0 . Then w e an apply Lemma 20 b elo w, the `Reparametrization Lemma', to obtain a new t yp e ( 2) lo al parametriza- tion of Σ at σ 0 , for whi h su h an assumption holds as w ell. Then w e an in tro due a smo oth t ′ -setion σ ′ : Z ′ → Σ ′ su h that σ ′ ( z ′ 0 ) = σ ′ 0 , b y setting σ ′ ( z ′ ) = σ ′ ( z ′ , 0) ; also, w e dene η ′ b y means of (14 ) and put H ′ = H and A ′ = A . Th us, from the simplifying assumption ab o v e, it follo ws that σ ′ ( ϕ ( z )) = ϕ ( σ ( z )) for ev ery z ∈ Z , and therefore that the image of σ ′ · η ′ : Z ′ × A ′ ֒ → G ′ oinides with ϕ (Im σ · η ) . But Im σ · η ⊂ Σ is an op en subset, and ϕ : Σ → Σ ′ = ϕ (Σ) is an op en mapping, whene Im σ ′ · η ′ is an op en subset of Σ ′ . This onludes the pro of. q.e.d. 124 CHAPTER V. CLASSICAL FIBRE FUNCTORS 15 Note Fix a p oin t σ 0 ∈ Σ . Sine f is a submersion, one an  ho ose op en neigh b ourho o ds U and V of s ( σ 0 ) and t ( σ 0 ) in M resp etiv ely , so small that, up to dieomorphism, f | U b eomes an op en pro jetion U ≈ U ′ × B pr − → U ′ ( U ′ is an op en subset of M ′ and B is an op en ball; moreo v er, w e shall assume that s ( σ 0 ) orresp onds to ( f ( s ( σ 0 )) , 0) in the dieomorphism U ≈ U ′ × B ), and f | V b eomes an op en pro jetion V ≈ V ′ × C pr − → V ′ ( V ′ is an op en subset of M ′ , and C is an op en ball; also, t ( σ 0 ) orresp onds to ( f ( t ( σ 0 )) , 0) in the dieomorphism V ≈ V ′ × B ). Sine ϕ is a Morita equiv alene, w e ha v e the follo wing pullba k in the ategory of dieren tiable manifolds of lass C ∞ G ( U, V ) ( s , t )   ϕ / / G ′ ( U ′ , V ′ ) ( s ′ , t ′ )   U × V f × f / / U ′ × V ′ whi h has therefore, up to dieomorphism, the follo wing asp et G ( U, V ) ≈ dieo. / / ( s , t )   G ′ ( U ′ , V ′ ) × B × C ( s ′ , t ′ ) × id × id   pr / / G ′ ( U ′ , V ′ ) ( s ′ , t ′ )   U × V ≈ × ≈ / / U ′ × B × V ′ × C pr × pr / / U ′ × V ′ , where the top omp osite arro w oinides with ϕ and the b ottom one with f × f . Next, tak e an op en neigh b ourho o d Γ of σ 0 in G su h that the relation (10 ) holds. Then the same relation is learly also satised b y an y smaller op en neigh b ourho o d of σ 0 in G , hene it is no loss of generalit y to assume that Γ is on tained in G ( U, V ) and that it orresp onds to a pro dut Γ ′ × B 0 × C 0 (with Γ ′ = ϕ (Γ) neessarily op en in G ′ ( U ′ , V ′ ) , b eause ϕ : G ( U, V ) → G ′ ( U ′ , V ′ ) is op en as it is lear from the latter diagram, and with B 0 ⊂ B , C 0 ⊂ C op en balls en tered at 0 of smaller radius) in the dieomorphism G ( U, V ) ≈ G ′ ( U ′ , V ′ ) × B × C . Then, b y our  hoie of Γ w e obtain a omm utativ e diagram Γ ≈ dieo. / / ( s , t )   Γ ′ × B 0 × C 0 ( s ′ , t ′ ) × id × id   pr / / Γ ′ ( s ′ , t ′ )   U 0 × V 0 ≈ × ≈ / / U ′ × B 0 × V ′ × C 0 pr × pr / / U ′ × V ′ (16) where the top omp osite arro w oinides with ϕ and the b ottom one with f × f . Finally , b y pasting the follo wing omm utativ e diagram U 0 × V 0 pr   ≈ × ≈ / / U ′ × B 0 × V ′ × C 0 pr   pr × pr / / U ′ × V ′ pr   V 0 ≈ / / V ′ × C 0 pr / / V ′ 22. T AME SUBMANIF OLDS OF A LIE GR OUPOID 125 to the former one along the ommon edge, w e obtain Γ ≈ / / t   Γ ′ × B 0 × C 0 t ′ × pr   pr / / Γ ′ t ′   V 0 ≈ / / V ′ × C 0 pr / / V ′ (17) and then, sine prop ert y ( 10) holds for Γ , Σ ∩ Γ ≈ / / t   (Σ ′ ∩ Γ ′ ) × B 0 × C 0 t ′ × pr   pr / / Σ ′ ∩ Γ ′ t ′   V 0 ≈ / / V ′ × C 0 pr / / V ′ . (18) Both in (17 ) and in (18), the top omp osite arro w oinides with the restri- tion of ϕ and the b ottom one with the restrition of f . Of ourse, one has analogous diagrams with soure maps replaing target maps. 19 Note Here w e shall state and pro v e the Lo al Reparametrization Lemma, whi h w as needed in the pro of of Lemma 12. 20 Lemma (Lo al Reparametrization) Let G ⇒ M b e a Lie group oid. Supp ose w e are giv en: a p oin t m ∈ M , a smo oth t -setion τ : Z → G ( m, - ) dened o v er a submanifold Z ⊂ M , a Lie subgroup η : H ֒ → G m and an op en neigh b ourho o d A of the unit e in H su h that the restrition of η is an em b edding. Let Σ = τ ( Z ) · η ( A ) b e the image of the mapping of t yp e (2) obtained from these data. Let σ 0 ≈ ( z 0 , e ) ∈ Z × A b e a giv en p oin t in Σ , and supp ose that σ : Z → Σ is an y other smo oth t -setion su h that σ ( z 0 ) = σ 0 = τ ( z 0 ) . Then there exists a smaller op en neigh b ourho o d Z 0 × A 0 of the p oin t ( z 0 , e ) in Z × A su h that σ · η : Z 0 × A 0 ֒ → Σ is still a lo al parametrization for Σ at σ 0 . Pro of If w e onsider the omp osite ( τ · η ) − 1 ◦ σ : Z → Σ → Z × A , w e get smo oth o ordinate maps z 7→ ( ζ ( z ) , α ( z )) ,  haraterized b y the equation σ ( z ) = τ ( ζ ( z )) · η ( α ( z )) . Comparing the target of the sides of this equation w e get ζ ( z ) = z . Th us σ is ompletely determined b y the smo oth mapping α : Z → A via the relation σ ( z ) = τ ( z ) · η ( α ( z )) . No w, w e  ho ose a smaller op en neigh b ourho o d A 0 ⊂ A of the unit e su h that A 0 · A 0 ⊂ A , whi h exists b y on tin uit y of the m ultipliation of H , and next an op en neigh b ourho o d Z 0 of z 0 in Z su h that α ( Z 0 ) ⊂ A 0 ; this is p ossible b eause α ( z 0 ) = e , whi h follo ws from σ ( z 0 ) = τ ( z 0 ) = τ ( z 0 ) · η ( e ) . It is then lear that σ · η maps Z 0 × A 0 in to Σ : indeed, ∀ ( z , a ) ∈ Z 0 × A 0 , 126 CHAPTER V. CLASSICAL FIBRE FUNCTORS σ ( z ) · η ( a ) = ( τ ( z ) · η ( α ( z ))) · η ( a ) = τ ( z ) · η ( α ( z ) · a ) , and this is learly an elemen t of τ ( Z 0 ) · η ( A 0 · A 0 ) ⊂ τ ( Z ) · η ( A ) = Σ . If again w e omp ose ( τ · η ) − 1 ◦ ( σ · η ) : Z 0 × A 0 → Σ → Z × A , w e get smo oth o ordinate maps ( z , a ) 7→  ζ ( z , a ) , α ( z , a )  ,  haraterized b y the rela- tion σ ( z ) · η ( a ) = τ ( ζ ( z , a )) · η ( α ( z , a )) . T aking the target yields ζ ( z , a ) = z , th us w e ha v e a smo oth mapping Z 0 × A 0 → Z × A of the form ( z , a ) 7→ ( z , α ( z , a ))  haraterized b y the equation σ ( z ) · η ( a ) = τ ( z ) · η ( α ( z , a )) . (So, in partiular, α ( z , e ) = α ( z ) and α ( z 0 , e ) = e .) T o onlude, it will b e enough to observ e that this mapping has in v ertible dieren tial at ( z 0 , e ) ∈ Z 0 × A 0 , b eause if that is the ase then the mapping indues a lo al dieomorphism of an op en neigh b ourho o d of ( z 0 , e ) in Z 0 × A 0 (whi h an b e assumed to b e Z 0 × A 0 itself, up to shrinking) on to an op en neigh b ourho o d of ( z 0 , e ) ∈ Z × A , so that if w e then omp ose ba k with τ · η w e see that σ · η is a dieomorphism of Z 0 × A 0 on to an op en subset of Σ . T o see the in v ertibilit y of the dieren tial, it will b e suien t to pro v e that the partial map a 7→ α ( z 0 , a ) has in v ertible dieren tial at e ∈ A 0 . But from the  haraterizing equation (setting z = z 0 ) α ( z 0 , a ) = η − 1 ( τ − 1 ( z 0 ) σ ( z 0 )) · a = η − 1 (1 m ) · a = a w e see at one that this dieren tial is in fat the iden tit y . q.e.d. 21 Note W e inlude here a disussion of tame submanifolds in onnetion with em b eddings of Lie group oids, parallel to the one onerning Morita equiv alenes. Supp ose one is giv en su h an em b edding, ie a Lie group oid homomorphism ι : G ֒ → G ′ su h that the mapping ι itself and the mapping i : M ֒ → M ′ indued on bases are em b eddings of manifolds. Let Σ b e a subset of G , and put Σ ′ = ι (Σ) ⊂ G ′ . The follo wing statemen ts hold i) Σ is a submanifold of G if and only if Σ ′ is a submanifold of G ′ , in whi h ase the restrition ι : Σ → Σ ′ is a dieomorphism; ii) Σ is a prinipal submanifold of G if and only if Σ ′ is a prinipal sub- manifold of G ′ ; iii) in ase i : M ֒ → M ′ is an op en em b edding, Σ is a tame submanifold of G if and only if Σ ′ is a tame submanifold of G ′ . Note that, as a sp eial ase, w e get in v ariane of tame submanifolds under isomorphisms of Lie group oids. 23 Smo othness and Represen tativ e Charts In 21 w e disussed some general prop erties of lassial bre funtors, whi h hold quite apart from the ev en tualit y that the anonial C ∞ -struture on the spae of arro ws of the T annakian group oid migh t pro v e not to b e a smo oth manifold struture. On the on trary , in the presen t setion w e turn 23. SMOOTHNESS, REPRESENT A TIVE CHAR TS 127 our atten tion sp eially to the problem of nding eetiv e riteria to deide whether a giv en lassial bre funtor is  smo oth in the sense illustrated at the b eginning of 18. Su h riteria will b e emplo y ed in  26; they in v olv e the te hnial notion of tame submanifold in tro dued in the preeding setion. T o motiv ate our denitions (whi h ma y app ear rather artiial at rst glane) let us onsider a smo oth lassial bre funtor ω o v er a manifold M . Reall that ω b eing smo oth means b y denition that the standard C ∞ - struture R ∞ on the spae T ( ω ) (1) turns T ( ω ) in to a Lie group oid o v er M ; ompare 18. Consider an y lassial represen tation  : T ( ω ) → GL ( E ) on a smo oth v etor bundle E ; w e kno w from Lemma 20.15 that if the map λ 7→  ( λ ) is injetiv e in the viinit y of λ 0 within the subspae T ( ω )( x 0 , x 0 ′ ) [ x 0 ≡ s ( λ 0 ) , x 0 ′ ≡ t ( λ 0 ) ℄ of T ( ω ) (1) , the same map m ust b e an immersion, in to the manifold of arro ws of GL ( E ) , of some op en neigh b ourho o d Ω ⊂ T of λ 0 and therefore it m ust indue, pro vided Ω is  hosen small enough, a dieomorphism of Ω on to a submanifold  (Ω) of GL ( E ) . When, in partiular,  = ev R for some R ∈ Ob( C ) , w e agree to write R (Ω) for the submanifold [of the manifold of arro ws of GL ( ω R ) ℄ that orresp onds to Ω , namely w e put (1) R (Ω) def = ev R (Ω) . It is not exeedingly diult to see that the submanifolds of GL ( E ) of the form  (Ω) , for all  and Ω su h that  indues a dieomorphism of Ω on to  (Ω) , are neessarily tame submanifolds of GL ( E ) , fr Lemma 26.3 b elo w. It will b e on v enien t to ha v e a name for the lo al dieomorphisms of the ab o v e-men tioned t yp e: 2 Denition W e shall all r epr esentative hart an y pair (Ω , R ) onsisting of an op en subset Ω of the spae of arro ws of T ( ω ) and an ob jet R ∈ Ob( C ) , su h that ev R : T ( ω ) → GL ( ω R ) restrits to a homeomorphism of Ω on to a tame submanifold R (Ω) of the linear group oid GL ( ω R ) . Note that this denition has b een form ulated so that it mak es sense for an arbitrary lassial bre funtor ω ; when ω is smo oth and (Ω , R ) is a represen tativ e  hart, the map λ 7→ λ ( R ) indues a dieomorphism of Ω on to the submanifold R (Ω) of GL ( ω R ) : this justies our denition. Observ e that if R and S are t w o isomorphi ob jets of C then (Ω , R ) is a represen tativ e  hart of T ( ω ) if and only if the same is true of (Ω , S ) (see Note 11 b elo w). Moreo v er, if (Ω , R ) is a represen tativ e  hart of T ( ω ) , the same is ob viously true of (Ω ′ , R ) for ea h op en subset Ω ′ ⊂ Ω . W e kno w from Lemma 10.14 that if a lassial bre funtor ω is smo oth then for ea h λ 0 there exists some R ∈ Ob( C ) su h that the map λ 7→ λ ( R ) is injetiv e in a neigh b ourho o d of λ 0 within the subspae T ( ω )( s λ 0 , t λ 0 ) of T ( ω ) (1) . No w, as remark ed b efore, this implies that λ 0 lies in the domain Ω of a represen tativ e  hart (Ω , R ) : th us w e see that for an y smo oth lassial bre funtor, the domains of represen tativ e  harts form an op en o v ering of the spae of arro ws of the orresp onding T annakian group oid. 128 CHAPTER V. CLASSICAL FIBRE FUNCTORS Next, let us onsider an arbitrary represen tativ e  hart (Ω , R ) of T ( ω ) , for a smo oth ω . Let S b e an arbitrary ob jet of C . By  ho osing diret sum represen tativ es on v enien tly , w e ma y supp ose that ω ( R ⊕ S ) = ω R ⊕ ω S . The ev aluation map ev R ⊕ S will yield a one-to-one orresp ondene b et w een Ω and the subspae ( R ⊕ S )(Ω) of GL ( ω R ⊕ ω S ) : indeed, sine λ ( R ⊕ S ) = λ ( R ) ⊕ λ ( S ) for all λ ∈ T ( ω ) , it is lear that the map λ 7→ λ ( R ⊕ S ) fators through the submanifold GL ( ω R ) × M GL ( ω S ) ֒ → GL ( ω R ⊕ ω S ) (fr Note 16 b elo w) as the map λ 7→  λ ( R ) , λ ( S )  (the latter is eviden tly injetiv e, b eause so is λ 7→ λ ( R ) , b y h yp othesis). W e on tend that ev R ⊕ S atually indues a homeomorphism of Ω on to the resp etiv e image; sine ev R ⊕ S is im- mersiv e (b y Lemma 20.15 ), our on ten tion will imply at one that ( R ⊕ S )(Ω) is a submanifold of GL ( ω R ⊕ ω S ) and that ev R ⊕ S yields a dieomorphism b et w een Ω and this submanifold. No w, let Ω ′ ⊂ Ω b e a giv en op en subset; x an y op en subset Λ ′ ⊂ GL ( ω R ) su h that R (Ω) ∩ Λ ′ = R (Ω ′ ) (su h Λ ′ exist b eause Ω and R (Ω) are homeomorphi via ev R ): then (3) ( R ⊕ S )(Ω) ∩  Λ ′ × M GL ( ω S )  = ( R ⊕ S )(Ω ′ ) , whi h pro v es our on ten tion. F rom the remarks that preede Denition 2 w e immediately onlude that the follo wing prop ert y is satised b y an y smo oth lassial bre funtor ω : when (Ω , R ) is a represen tativ e  hart of T ( ω ) , so m ust b e (Ω , R ⊕ S ) for ea h ob jet S ∈ Ob( C ) . The on v erse holds: 4 Prop osition Let ω b e a lassial bre funtor. Then ω is smo oth if and only if the follo wing t w o onditions are satised: i) the domains of represen tativ e  harts o v er the spae of arro ws of the T annakian group oid T ( ω ) , ie for ea h λ ∈ T ( ω ) there exists a represen tativ e  hart (Ω , R ) with λ ∈ Ω ; ii) if (Ω , R ) is a represen tativ e  hart of T ( ω ) then the same is true of (Ω , R ⊕ S ) for ev ery ob jet S ∈ Ob( C ) . Pro of W e ha v e already pro v ed that a smo oth lassial bre funtor satises onditions i) and ii) . Vie v ersa, supp ose these onditions are satised: the ruial p oin t no w is to sho w that an y represen tativ e  hart (Ω , R ) establishes an isomorphism of funtionally strutured spaes b et w een (Ω , R ∞ Ω ) and the submanifold X def = R (Ω) ⊂ GL ( ω R ) (endo w ed with the struture C ∞ X ). Sine ev R : T → GL ( ω R ) is a morphism of funtionally strutured spaes, it is lear that f ∈ C ∞ ( X ) implies f ◦ ev R ∈ R ∞ (Ω) (fr. the pro of of Prop osition 20.21). The on v erse impliation is less ob vious: w e will mak e use of the sp eial prop erties of tame submanifolds w e deriv ed in the preeding setion. Supp ose r = r S,ψ ,η,η ′ ∈ R ∞ (Ω) and let f b e the fun- tion on X su h that f ◦ ev R = r ; w e m ust sho w that f ∈ C ∞ ( X ) . Sine 23. SMOOTHNESS, REPRESENT A TIVE CHAR TS 129 f = q ψ, η,η ′ ◦ ev S ◦ ev R − 1 where q ψ, η,η ′ is the smo oth funtion on GL ( ω S ) giv en b y ν 7→  ν · η ( s ν ) , η ′ ( t ν )  ψ and ev R − 1 : X ≈ − → Ω is the in v erse map, it will b e enough to sho w that ev S ◦ ev R − 1 is a smo oth mapping of X in to GL ( ω S ) . Put E = ω ( R ) , F = ω ( S ) . Reall that GL ( E ) × M GL ( F ) is the pro dut of GL ( E ) and GL ( F ) in the ategory of Lie group oids o v er M (see Note 16 b elo w) and that therefore it omes equipp ed with t w o pro jetions pr E , pr F that are morphisms of Lie group oids o v er M . One an build the follo wing omm utativ e diagram ( R ⊕ S )(Ω) ≈ homeo     e R,S / / GL ( E ) × M GL ( F ) pr E   Ω ev R ⊕ S 8 8 q q q q q q q q q q q q q ev R / / X = R ( Ω)   submanifold / / GL ( E ) , (5) where e R,S is the smo oth em b edding whose omp osition with (6) GL ( E ) × M GL ( F ) ֒ → GL ( E ⊕ F ) = GL  ω ( R ⊕ S )  , ( µ, ν ) 7→ µ ⊕ ν equals the inlusion of ( R ⊕ S )(Ω) in to GL ( ω R ⊕ S ) . No w, (Ω , R ⊕ S ) is a represen tativ e  hart of T ( ω ) and hene ( R ⊕ S )(Ω) is a tame submanifold of GL ( ω R ⊕ S ) , so w e an apply Prop osition 22 .5 to onlude that the tran- sition homeomorphism in (5) is in fat a dieomorphism. This immediately implies the desired smo othness of the transition mapping ev S ◦ ev R − 1 : X → GL ( F ) , b eause of the omm utativit y of the follo wing diagram: ( R ⊕ S )(Ω)   e R,S smo oth / / GL ( E ) × M GL ( F ) pr F   X trans. dieo ≈ 7 7 p p p p p p p p p p p p p ev R − 1 / / Ω   ev S / / ev R ⊕ S O O GL ( F ) . (7) F rom ondition i) and what w e ha v e just pro v ed, w e see that ( T , R ∞ ) is a smo oth manifold and that ea h represen tativ e  hart (Ω , R ) indues a dieomorphism ev R | Ω of Ω on to R (Ω) . Moreo v er, sine on the domain of an y represen tativ e  hart (Ω , R ) the soure map of T ( ω ) is the omp osition of ev R | Ω with the restrition to R (Ω) of the soure map of GL ( ω R ) , w e also see that the soure map of T ( ω ) is a submersionb eause su h remains the soure map of GL ( ω R ) when restrited to the tame submanifold R (Ω) ⊂ GL ( ω R ) . Prop osition 21.3 allo ws us to nish the pro of. q.e.d. There is y et one useful remark onerning Condition ii) : under the h y- p othesis that (Ω , R ) is a represen tativ e  hart, the ev aluation map ev R ⊕ S es- tablishes, as in (3), a homeomorphism b et w een Ω and the subset ( R ⊕ S )(Ω) of the manifold GL ( ω R ⊕ S ) , wherefore the pair (Ω , R ⊕ S ) is a represen ta- tiv e  hart if and only if ( R ⊕ S )(Ω) is a tame submanifold of GL ( ω R ⊕ S ) . The usefulness of the last prop osition will b eome eviden t in the study of w eak equiv alenes of lassial bre funtors (fr Setion 25 ) and in the study of lassial bre funtors asso iated with prop er Lie group oids (Chapter VI). 130 CHAPTER V. CLASSICAL FIBRE FUNCTORS 8 Corollary Let ω : C → V ∞ ( M ) b e a lassial bre funtor satisfying onditions i) and ii) of the preeding prop osition. Then there exists a unique manifold struture on the spae of arro ws of the group oid T ( ω ) , that renders T ( ω ) a Lie group oid and ev R : T ( ω ) − → GL ( ω R ) a smo oth represen tation for ea h ob jet R . Equiv alen tly , the same mani- fold struture an b e  haraterized as the unique manifold struture for whi h an arbitrary mapping f : X → T is smo oth if and only if so is ev R ◦ f for all R . The orresp ondene R 7→  ω ( R ) , ev R  , a 7→ ω ( a ) de- termines a faithful tensor funtor ev of C in to R ∞ ( T ( ω )) , whi h mak es C ω $ $ I I I I I I I I I ev / / R ∞ ( T ( ω )) w w n n n n n n n n n V ∞ ( M ) (9) omm ute as a diagram of tensor funtors (where the unlab elled arro w is the standard forgetful funtor of 13). Pro of W e only need to  he k the assertions onerning the uniqueness of the smo oth struture. Th us, supp ose ev R smo oth ∀ R . F or on v eniene, let T ( ω ) ∗ denote the unkno wn manifold struture on the set T ( ω ) . Sine the top ology of T ( ω ) ∗ is neessarily ner than that of T ( ω ) , an op en subset of T ( ω ) m ust b e in partiular a tame submanifold of T ( ω ) ∗ . Therefore if (Ω , R ) is a represen tativ e  hart, the homomorphism of Lie group oids ev R : T ( ω ) ∗ → GL ( ω R ) restrits to a smo oth isomorphism of the op en subset Ω ⊂ T ( ω ) ∗ on to the (tame) submanifold R (Ω) of GL ( ω R ) . Th us, w e see that the iden tit y map is, lo ally in the domains of represen tativ e  harts, a dieomorphism b et w een T ( ω ) and T ( ω ) ∗ ; sine represen tativ e  harts o v er T ( ω ) , w e get T ( ω ) ∗ = T ( ω ) , as w as to b e pro v ed. q.e.d. F or the sak e of ompleteness, w e also reord the follo wing renemen t of Lemma 20 .15, whi h ma y b e regarded as a statemen t ab out the existene of represen tativ e  harts of a sp eial t yp e: 10 Corollary Let G b e a prop er Lie group oid o v er a manifold M . Assume that ( E ,  ) is a lassial represen tation of G , mapping a subset G ( x, x ′ ) injetiv ely in to Lis( E x , E x ′ ) . Then there exist op en balls B and B ′ in M , en tred at x and x ′ resp etiv ely , su h that the restrition  : G ( B , B ′ ) → GL ( E ) is an em b edding of manifolds. 23. SMOOTHNESS, REPRESENT A TIVE CHAR TS 131 Pro of T o b egin with, observ e that for an y giv en arro w g ∈ G ( x, x ′ ) and op en neigh b ourho o d Γ of g in G there is an op en ball P inside GL ( E ) , en tred at  ( g ) , su h that  − 1 ( P ) ⊂ Γ . T o see this, w e x a sequene · · · ⊂ P i +1 ⊂ P i ⊂ · · · ⊂ P 1 of op en balls inside GL ( E ) , en tred at  ( g ) and with lim i radius( P i ) = 0 , and then w e argue as in the pro of of Theorem 20.5. By Lemma 20 .15, ev ery g ∈ G ( x, x ′ ) admits an op en neigh b ourho o d Γ g in G su h that  indues a smo oth isomorphism b et w een Γ g and a submanifold of GL ( E ) . As observ ed ab o v e, one an then  ho ose an op en ball P g ⊂ GL ( E ) at  ( g ) su h that  − 1 ( P g ) ⊂ Γ g . No w, let Γ = S  − 1 ( P g ) . W e laim that  indues a smo oth isomorphism b et w een Γ and a submanifold of GL ( E ) . By onstrution,  restrits to an immersion of Γ in to GL ( E ) . If g ∈ G ( x, x ′ ) then  (Γ) ∩ P g =    − 1 ( P g )  is an op en subset of the submanifold  (Γ g ) ⊂ GL ( E ) , b eause  is a smo oth isomorphism of Γ g on to  (Γ g ) . Sine the op en balls P g o v er  (Γ) as g ranges o v er G ( x, x ′ ) ,  (Γ) is a submanifold of GL ( E ) . Moreo v er, sine  is a lo al smo oth isomorphism of Γ on to  (Γ) , it will b e also a global dieomorphism pro vided it is globally injetiv e o v er Γ : no w, if  ( γ ′ ) =  ( γ ) then γ ′ , γ ∈  − 1 ( P g ) ⊂ Γ g for some g and therefore γ ′ = γ b eause  is injetiv e o v er Γ g . Finally , one further appliation of the usual prop erness argumen t will yield op en balls B , B ′ ⊂ M at x, x ′ su h that G ( B , B ′ ) is on tained in Γ (this is an op en neigh b ourho o d of G ( x, x ′ ) in G ). q.e.d. Note that the preeding orollary en tails in partiular that the image  ( G ) is a submanifold of GL ( E ) for ev ery prop er Lie group oid G and faithful lassial represen tation ( E ,  ) of G . T e hnial notes 11 Note Supp ose one is giv en an isomorphism E ≈ F of v etor bundles o v er a manifold M . Then there is an indued isomorphism of Lie group oids o v er M (ie one that restrits to the iden tit y mapping on M ) (12) GL ( E ) ≈ − → GL ( F ) , giv en, for ea h ( x, x ′ ) ∈ M × M , b y the bijetion that mak es the linear isomorphisms α and β orresp ond to ea h other when they t in the diagram E x ≈ x   α / / E x ′ ≈ x ′   F x β / / F x ′ . (13) 132 CHAPTER V. CLASSICAL FIBRE FUNCTORS In partiular, if t w o ob jets R, S ∈ Ob( C ) are isomorphi, an y indued isomorphism ω ( ≈ ) : ω ( R ) ≈ ω ( S ) will in turn yield an isomorphism of the orresp onding linear group oids GL ( ω R ) ≈ GL ( ω S ) (iden tial on M ), su h that for ea h λ ∈ T ( ω ) the linear mappings λ ( R ) and λ ( S ) orresp ond to one anotherb eause of naturalit y of λ : ( ω R ) x ω ( ≈ ) x   ω x ( R ) ω x ( ≈ )   λ ( R ) / / ω x ′ ( R ) ω x ′ ( ≈ )   ( ω R ) x ′ ω ( ≈ ) x ′   ( ω S ) x ω x ( S ) λ ( S ) / / ω x ′ ( S ) ( ω S ) x ′ . (14) Th us, the latter isomorphism will transform ev R in to ev S : GL ( ω R ) O O ≈   T ( ω ) ev S , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y ev R 2 2 e e e e e e e e e e e e e e GL ( ω S ) . (15) It follo ws that if Ω ⊂ T is an y op en subset then R (Ω) is a tame submanifold of GL ( ω R ) if and only if S (Ω) is a tame submanifold of GL ( ω S ) (see, for instane, Note 22 .21) and that R (Ω) and S (Ω) are homeomorphi subsets; hene ev R will indue a homeomorphism b et w een Ω and R (Ω) if and only if ev S indues one b et w een Ω and S (Ω) . 16 Note Let G and H b e t w o Lie group oids o v er the manifold M . W e w an t to onstrut, pro vided this is p ossible, their pro dut in the ategory of Lie group oids o v er M . It ough t to b e a Lie group oid o v er M endo w ed with anonial pro jetions, satisfying the usual univ ersal prop ert y G K ψ / / ϕ / / ( ϕ,ψ ) / / _ _ _ _ _ _ G × M H pr 1 6 6 n n n n n n n n n n n n n n pr 2 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q H . (17) It m ust b e k ept in mind that all the arro ws in this diagram are morphisms of Lie group oids o v er M , ie they all indue the iden tit y map id : M → M at the base lev el. The onstrution of the pro dut o v er M an b e obtained as a sp eial ase of the so-alled strong bred pro dut onstrution for Lie group oids, fr. for example Mo er dijk and Mr £un (2003), [ 27 ℄ p. 123. Namely , w e regard the maps / /   G ( s , t )   (viz. G ( s , t )   ( s , t ) / / M × M ( pr 1 , pr 2 ) = id   H ( s , t ) / / M × M M × M id × id / / M × M et.) 23. SMOOTHNESS, REPRESENT A TIVE CHAR TS 133 as morphisms of lie group oids o v er M , where M × M is the pair group oid, and apply the strong bred pro dut onstrution to them: ( set of arro ws =  ( g , h ) ∈ G × H : ( s , t )( g ) = ( s , t )( h )  , set of ob jets =  ( m, m ′ ) ∈ M × M : m = m ′  ∼ = M . T ransv ersalit y riteria imply that this denes a Lie group oid G × M H o v er ∆( M ) ∼ = M whenev er, for instane, one of the t w o maps is a submersion. (T erminology: w e sa y that a Lie group oid G ⇒ M is lo ally transitiv e if the map ( s , t ) : G → M × M is a submersion. This app ears to b e reasonable, sine G is said to b e transitiv e if that map is a surjetiv e submersion.) More- o v er, if the trasv ersalit y ondition is satised, this onstrution giv es a bred pro dut with the familiar univ ersal prop ert y . Supp ose that G × M H mak es sense, ie that the transv ersalit y ondition is satised. W e remark that the univ ersal prop ert y ( 17) is a onsequene of the univ ersal prop ert y of the pullba k. Indeed, rst of all, the t w o pro jetions of the bred pro dut to its o wn fators are morphisms o v er M , as one sees diretly at one. Seondly , if ϕ : K → G and ψ : K → H are morphisms o v er M , then the follo wing diagram omm utes (preisely b y denition of morphism o v er M ) K ψ   ( s , t ) & & M M M M M M ϕ / / G ( s , t )   H ( s , t ) / / M × M and therefore there exists a unique morphism of Lie group oids ( ϕ, ψ ) : K → G × M H su h that diagram (17 ) omm utes, so w e need only v erify that ( ϕ, ψ ) is in fat a morphism o v er M . This follo ws at one from the omm utativit y of the diagram K ( s , t )   ( ϕ,ψ ) / / ϕ # # G G G G G G G G G G G × M H ( s , t )   pr 1 z z u u u u u u u u u u G ( s , t ) $ $ I I I I I I I I I I M × M id × id / / M × M . Observ ation. By onstrution, the manifold of arro ws of G × M H is a submanifold of the Cartesian pro dut G × H ; it follo ws that the subsets of the form Γ × Λ , for Γ ⊂ G and Λ ⊂ H op en, form a basis for the top ology of G × M H . (Of ourse, w e write Γ × Λ but w e mean (Γ × Λ) ∩ ( G × M H ) .) Th us, one sees immediately that, when the dieren tiable struture is dis- arded, the same onstrution yields the pro dut in the ategory of top o- logial group oids o v er M . No w, w e apply this general onstrution to the lo ally transitiv e Lie group oids GL ( E ) asso iated to v etor bundles E ∈ Ob V ∞ ( M ) . (These are 134 CHAPTER V. CLASSICAL FIBRE FUNCTORS lo ally transitiv e sine if E U ≈ U × E and E V ≈ V × F are lo al trivializa- tions of E , then up to dieomorphism the map ( s , t ) oinides lo ally with a pro jetion GL ( E )( U, V ) ≈ U × V × Lis( E , F ) pr − → U × V and is in partiular a submersion; note that this mak es sense ev en when Lis( E , F ) = ∅ .) 24 Morphisms of Fibre F untors A morphism of bre funtors, let us sa y one ( C , ω ) → ( C ′ , ω ′ ) , onsists of a smo oth map f : M → M ′ of the resp etiv e base manifolds together with a linear tensor funtor Φ ∗ : C ′ − → C and a tensor preserving isomorphism α C ′ ω ′   Φ ∗ / / C ω   V ∞ ( M ′ ) f ∗ / / α ) 1 V ∞ ( M ) , (1) where f ∗ = pullba k along f . In plae of the orret ( f , Φ ∗ , α ) , our preferred notation for morphisms of bre funtors will b e the inorret ( f ∗ , Φ ∗ ) , in order to emphasize the algebrai symmetry . Comp osition of morphisms is dened as (2) ( g ∗ , Ψ ∗ ) · ( f ∗ , Φ ∗ ) =  ( g ◦ f ) ∗ , Φ ∗ ◦ Ψ ∗  . Note that if in our denition w e required ( 1) to omm ute in the strit sense w e w ould get in to trouble b eause ( g ◦ f ) ∗ ∼ = f ∗ ◦ g ∗ are anonially isomorphi but not really iden tial tensor funtors. Lemmas 9 and 11 b elo w apply diretly to (1) to yield maps (3) Hom ⊗ ( ω x , ω y ) Lem. 9 − − − − → Hom ⊗  ω x ◦ Φ ∗ , ω y ◦ Φ ∗  = Hom ⊗  x ∗ ◦ ω ◦ Φ ∗ , y ∗ ◦ ω ◦ Φ ∗  ≈ (1 ) + Lem. 11 − − − − − − − − − → Hom ⊗  x ∗ ◦ f ∗ ◦ ω ′ , y ∗ ◦ f ∗ ◦ ω ′  ∼ = Lem. 11 − − − − − − → Hom ⊗  f ( x ) ∗ ◦ ω ′ , f ( y ) ∗ ◦ ω ′  = Hom ⊗  ω ′ f ( x ) , ω ′ f ( y )  . Moreo v er, sine ( λ ◦ µ ) · Φ ∗ = ( λ · Φ ∗ ) ◦ ( µ · Φ ∗ ) and id · Φ ∗ = id , these an b e pieed together in a funtorial w a y , so that they form a homomorphism of group oids T ( ω )   Φ / / T ( ω ′ )   M × M f × f / / M ′ × M ′ , (4) 24. MORPHISMS OF FIBRE FUNCTORS 135 whi h an b e  haraterized as the unique map making T ( ω ) ev Φ ∗ R ′   Φ / / T ( ω ′ ) ev R ′   GL ( ω Φ ∗ R ′ ) γ ◦ α − 1 ∗ / / GL ( ω ′ R ′ ) (5) omm ute for all R ′ ∈ Ob( C ′ ) , where the morphism γ is the pro jetion GL ( f ∗ ( ω ′ R ′ )) ∼ = ( f × f ) ∗ ( GL ( ω ′ R ′ )) → GL ( ω ′ R ′ ) and the isomorphism (6) α ∗ : GL ( f ∗ ω ′ R ′ ) ∼ → GL ( ω Φ ∗ R ′ ) omes from α R ′ : f ∗ ω ′ R ′ ∼ → ω Φ ∗ R ′ aording to Note 23 .11. It is also immediate from (5 ) that su h a solution Φ is neessarily a morphism of C ∞ -funtionally strutured spaes, so (4) pro v es to b e a homomorphism of C ∞ -funtionally strutured group oids. W e shall refer to Φ as the r e alization of the morphism ( f ∗ , Φ ∗ ) . This onstrution is funtorial with resp et to omp osition of morphisms of bre funtors, and therefore denes a funtor in to the ategory of C ∞ -strutured group oids, alled the r e alization funtor . 7 Prop osition Let ( C , ω ) , ( C ′ , ω ′ ) b e smo oth lassial bre funtors and ( f ∗ , Φ ∗ ) : ( C , ω ) → ( C ′ , ω ′ ) a morphism of bre funtors. Then the orresp onding realization is a homomorphism of Lie group oids. Pro of It follo ws from (5) that the omp osite ev R ′ ◦ Φ is smo oth for ev ery ob jet R ′ of C ′ . The map Φ is then smo oth b y the univ ersal prop ert y of the Lie group oid T ( ω ′ ) . q.e.d. Notes 8 Note In this note w e reall a ouple of elemen tary prop erties of tensor funtors and tensor preserving natural transformations. 9 Lemma Let F , G , S , T b e tensor funtors relating suitable tensor ategories. Then 1. the rule λ 7→ λ · S maps Hom ⊗ ( F , G ) in to Hom ⊗ ( F ◦ S , G ◦ S ) ; 2. the rule λ 7→ T · λ maps Hom ⊗ ( F , G ) in to Hom ⊗ ( T ◦ F , T ◦ G ) . 136 CHAPTER V. CLASSICAL FIBRE FUNCTORS Pro of (1) The natural transformation ( λ · S )( X ) = λ ( S X ) is a morphism of tensor funtors if su h is λ , b eause F S X ⊗ F S Y ∼ =   λ ( S X ) ⊗ λ ( S Y ) / / GS X ⊗ GS Y ∼ =   1 ∼ =   id / / 1 ∼ =   F ( S X ⊗ S Y ) F ∼ =   λ ( S X ⊗ S Y ) / / G ( S X ⊗ S Y ) G ∼ =   F 1 F ∼ =   λ ( 1 ) / / G 1 G ∼ =   F S ( X ⊗ Y ) λ ( S ( X ⊗ Y )) / / GS ( X ⊗ Y ) F S 1 λ ( S 1 ) / / GS 1 . (2) The same an b e said of ( T · λ )( X ) = T ( λ ( X )) , sine T F X ⊗ T F Y ∼ =   T λ ( X ) ⊗ T λ ( Y ) / / T GX ⊗ T GY ∼ =   1 ∼ =   id / / 1 ∼ =   T ( F X ⊗ F Y ) T ∼ =   T ( λ ( X ) ⊗ λ ( Y )) / / T ( GX ⊗ GY ) T ∼ =   T 1 T ∼ =   T ( id ) / / T 1 T ∼ =   T F ( X ⊗ Y ) T λ ( X ⊗ Y ) / / T G ( X ⊗ Y ) T F 1 T λ ( 1 ) / / T G 1 . q.e.d. Let ( C , ⊗ ) and ( V , ⊗ ) b e tensor ategories. Supp ose that F , F ′ , G, G ′ : C − → V are tensor funtors, and that F ≈ F ′ , G ≈ G ′ are tensor preserving natural isomorphisms. F or ev ery X ∈ Ob( C ) , there is an ob vious bijetiv e map a 7→ a ′ determined b y the omm utativit y of F X ≈   a / / GX ≈   F ′ X a ′ / / G ′ X . (10) Giv en a natural transformation λ ∈ Hom( F , G ) , w e put λ ′ ( X ) = λ ( X ) ′ . 11 Lemma The rule whi h to λ asso iates λ ′ determines a bijetiv e orresp ondene (12) Hom ⊗ ( F , G ) ∼ → Hom ⊗ ( F ′ , G ′ ) . Pro of Ob vious. q.e.d. 25. WEAK EQUIV ALENCES 137 25 W eak Equiv alenes 1 Denition A we ak e quivalen e 1 of bre funtors, sym b olially ( C , ω ) ≈ − → ( C ′ , ω ′ ) , is a morphism of bre funtors ( f ∗ , Φ ∗ ) : ( C , ω ) → ( C ′ , ω ′ ) satisfying the follo wing t w o onditions 1. the base mapping f : M → M ′ is a surjetiv e submersion; 2. the funtor Φ ∗ is a tensor equiv alene, ie there exist a tensor funtor Φ ∗ : C − → C ′ and tensor preserving natural isomorphisms  Φ ∗ ◦ Φ ∗ ≈ Id C Φ ∗ ◦ Φ ∗ ≈ Id C ′ . In order to onlude that Φ ∗ is a tensor equiv alene, it sues to kno w it to b e an ordinary ategorial equiv alene. Ev ery quasi-in v erse equiv alene Φ ∗ is then neessarily a linear funtor. (Details ma y b e found in Note 10.) W eak equiv alenes of bre funtors are stable under omp osition of morphisms of bre funtors, as dened in Setion 24. 2 Prop osition Let ( f ∗ , Φ ∗ ) : ( C , ω ) ≈ − → ( C ′ , ω ′ ) b e a w eak equiv alene of bre funtors. Then its realization diagram T ( ω )   Φ / / T ( ω ′ )   M × M f × f / / M ′ × M ′ (3) is a top ologial pullba k, ie a pullba k in the ategory of top ologial spaes, and Φ : T ( ω ) ։ T ( ω ′ ) is a surjetiv e op en mapping. Pro of Let T b e a top ologial spae, and supp ose giv en a problem T " " a % % & & T ( ω ) ev Φ ∗ R ′   Φ / / T ( ω ′ ) ev R ′   GL ( ω Φ ∗ R ′ )   γ ◦ α − 1 ∗ / / GL ( ω ′ R ′ )   M × M f × f / / M ′ × M ′ (4) 1 Note on terminology: W e shall reserv e the term `w eak equiv alene' for the on text of bre funtors. When dealing with Lie group oids, w e prefer to use the term `Morita equiv alene'. 138 CHAPTER V. CLASSICAL FIBRE FUNCTORS stated in the ategory of top ologial spaes and on tin uous mappings. There exists a unique set-theoreti solution a , b eause (3) is already kno wn to b e a set-theoreti pullba k (b y Note 10 again). Th us, w e m ust  he k that a is on tin uous. Note that ∀ R in C , ev R ◦ a is on tin uous if and only if ev Φ ∗ Φ ∗ R ◦ a is on tin uous, b eause of the isomorphism Φ ∗ Φ ∗ R ≈ R , see also the ommen ts in Note 23 .11. Therefore, if w e put R ′ = Φ ∗ R in (4), w e onlude at one that ev Φ ∗ R ′ ◦ a is on tin uous from the fat that the lo w er square of (4) is, b y denition, a top ologial pullba k. Next, observ e that if one has a top ologial pullba k X p   f / / Y q   M g / / N (5) along a submersiv e morphism g of smo oth manifolds, there is the follo wing lo al deomp osition up to dieomorphism X U p   f / / Y V q   U g / / V X U p   ≈ Y V × P q × id   pr / / Y V q   U ≈ V × P pr / / V , (6) where U ⊂ M is op en and so small that, up to dieomorphism, g | U is a pro jetion V × P → V = g ( U ) for some op en ball P ; of ourse, X U = p − 1 ( U ) et. (Note that in (6 ), U ≈ V × P is a dieomorphism whereas X U ≈ Y V × P is a homeomorphism.) It follo ws that f is a `top ologial submersion', in partiular an op en mapping; in addition, if g is surjetiv e then it is lear that f m ust b e also surjetiv e. This sho ws that the statemen t that Φ is an op en mapping follo ws from the statemen t that (25 .3) is a top ologial pullba k. q.e.d. Supp ose a top ologial pullba k (5) along a smo oth submersion is giv en, and let U ⊂ M b e an op en subset su h that g | U is, up to dieomorphism, a pro jetion U ≈ V × P pr − → V on to an op en subset V ⊂ N . Let A ⊂ X b e an op en subset, and put B = f ( A ) ; B ⊂ Y is op en b eause f is an op en mapping. W e shall b e in terested in the subspaes p ( A ) ⊂ M and q ( B ) ⊂ N ; note that g restrits to a on tin uous mapping of p ( A ) on to q ( B ) . Assume that A has the follo wing prop ert y: the  ommutative squar e A ∩ p − 1 ( U ) p   f / / B ∩ q − 1 ( V ) q   U g / / V (7) 25. WEAK EQUIV ALENCES 139 is a top olo gi al pul lb ak . Then there is a trivialization, analogous to ( 6), whi h sho ws that the smo oth iso U ≈ V × P indues a orresp ondene b et w een p ( A ) ∩ U = p  A ∩ p − 1 ( U )  and  q ( B ) ∩ V  × P = q  B ∩ q − 1 ( V )  × P . Th us, ∀ u ∈ U one has u ∈ p ( A ) ⇔ g ( u ) ∈ q ( B ) . Note also that p restrits to a homeomorphism of A ∩ p − 1 ( U ) on to p ( A ) ∩ U if and only if q restrits to a homeomorphism of B ∩ q − 1 ( V ) on to q ( B ) ∩ V . The t w o relev an t ases for the presen t disussion o ur, in the rst plae, when A = f − 1 ( f ( A )) , and seondly , when A ⊂ p − 1 ( U ) oinides with B × P in the trivialization (6). Fix an ob jet R ′ ∈ Ob( C ′ ) . Then the outer retangle of (4) is a top o- logial pullba knote that it oinides with (3); the lo w er square enjo ys the same prop ert y . Consequen tly , the upp er square, viz ( 24 .5), m ust b e a top ologial pullba k as w ell; moreo v er, sine the smo oth mapping γ ◦ α − 1 ∗ : GL ( ω Φ ∗ R ′ ) → GL ( ω ′ R ′ ) is a (surjetiv e) submersion, it is a pullba k of the form (6). Hene the preeding remarks apply , and w e get: 1. If (Ω ′ , R ′ ) is a r epr esentative hart of ( C ′ , ω ′ ) then (Φ − 1 (Ω ′ ) , Φ ∗ R ′ ) is a r epr esentative hart of ( C , ω ) . Sine diagram ( 24 .5) is a top ologial pullba k, Φ − 1 (Ω ′ ) ev Φ ∗ R ′   Φ / / Ω ′ ev R ′   Φ ∗ R ′ (Φ − 1 (Ω ′ )) γ ◦ α − 1 ∗ / / R ′ (Ω ′ ) is also a top ologial pullba k and therefore ev Φ ∗ R ′ indues a homeomorph- ism b et w een Φ − 1 (Ω ′ ) and its image Φ ∗ R ′ (Φ − 1 (Ω ′ )) , b eause ev R ′ , on the righ t, do es the same. Prop osition 22.11 implies that Φ ∗ R ′ (Φ − 1 (Ω ′ )) is a tame submanifold of GL ( ω Φ ∗ R ′ ) if and only if R ′ (Ω ′ ) is a tame submanifold of GL ( ω ′ R ′ ) , b eause γ ◦ α − 1 ∗ is a Morita equiv alene and Ω ′ = Φ(Φ − 1 (Ω ′ )) . 2. L et Ω ⊂ T ( ω ) b e an op en subset and λ 0 ∈ Ω . F or any given obje t R ′ ∈ Ob( C ′ ) , ther e is a smal ler op en neighb ourho o d λ 0 ∈ Ω 0 ⊂ Ω suh that (Ω 0 , Φ ∗ R ′ ) is a r epr esentative hart of ( C , ω ) if and only if (Φ(Ω 0 ) , R ′ ) is a r epr esentative hart of ( C ′ , ω ′ ) . Let Λ b e an op en neigh b ourho o d of λ 0 (Φ ∗ R ′ ) in GL ( ω Φ ∗ R ′ ) su h that γ ◦ α − 1 ∗ | Λ is, up to dieomorphism, a pro jetion Λ ′ × P → Λ ′ = γ ◦ α − 1 ∗ (Λ) . Making the op en ball P , and th us Λ , smaller if neessary , w e nd an op en neigh b ourho o d Ω 0 ⊂ ev − 1 Φ ∗ R ′ (Λ) ∩ Ω of λ 0 su h that the homeomorphism ev − 1 Φ ∗ R ′ (Λ) ≈ ev − 1 R ′ (Λ ′ ) × P of (6) pro dues a de- omp osition Ω 0   Φ / / Φ(Ω 0 )   Λ γ ◦ α − 1 ∗ / / Λ ′ Ω 0   ≈ Φ(Ω 0 ) × P × id   pr / / Φ(Ω 0 )   Λ ≈ Λ ′ × P pr / / Λ ′ (8) 140 CHAPTER V. CLASSICAL FIBRE FUNCTORS Therefore, if w e put Σ = Φ ∗ R ′ (Ω 0 ) ⊂ Λ and Σ ′ = R ′ (Φ(Ω 0 )) ⊂ Λ ′ w e ha v e λ ∈ Σ ⇔ γ ◦ α − 1 ∗ λ ∈ Σ ′ for all λ ∈ Λ , and Prop osition 22 .11 implies that Σ is a tame submanifold of GL ( ω Φ ∗ R ′ ) if and only if Σ ′ is a tame submanifold of GL ( ω ′ R ′ ) , sine γ ◦ α − 1 ∗ is a Morita equiv alene. Clearly , these statemen ts imply that whenev er a w eak equiv alene of bre funtors ( C , ω ) ≈ − → ( C ′ , ω ′ ) is giv en, Condition i) of Prop osition 23.4 holds for ( C , ω ) if and only if it holds for ( C ′ , ω ′ ) . (As a onsequene of the fat that Φ is surjetiv e and op en: Fix λ ′ 0 = Φ( λ 0 ) . If (Ω , R ) is a  hart at λ 0 , then (Φ(Ω 0 ) , Φ ∗ R ) is a  hart at λ ′ 0 for some op en λ 0 ∈ Ω 0 ⊂ Ω ; on v ersely , if (Ω ′ , R ′ ) is a  hart at λ ′ 0 then (Φ − 1 (Ω ′ ) , Φ ∗ R ′ ) is a  hart at λ 0 .) On the other hand, they also imply in v ariane of Condition ii) of the same prop osition, p. 128 , as follo ws. Assume the ondition holds for ( C ′ , ω ′ ) : Let (Ω , R ) b e a  hart of ( C , ω ) and S ∈ Ob( C ) an ob jet. Cho ose a p oin t λ 0 ∈ Ω . There exists a neigh b ourho o d Ω 0 ⊂ Ω of λ 0 su h that (Φ(Ω 0 ) , Φ ∗ R ) , and on- sequen tly (Φ(Ω 0 ) , Φ ∗ R ⊕ Φ ∗ S ) , is a  hart of ( C ′ , ω ′ ) . Sine Ω 0 ⊂ Φ − 1 Φ(Ω 0 ) and Φ ∗ (Φ ∗ R ⊕ Φ ∗ S ) ≈ R ⊕ S , it follo ws that (Ω 0 , R ⊕ S ) is a  hart of ( C , ω ) . Sine λ 0 w as arbitrary , w e onlude that Ω an b e o v ered with op en sub- sets Ω 0 su h that (Ω 0 , R ⊕ S ) is a  hart, and therefore that (Ω , R ⊕ S ) is a  hart as w ell. Con v ersely , assume Condition 2 holds for ( C , ω ) : Let (Ω ′ , R ′ ) b e a  hart of ( C ′ , ω ′ ) and S ′ ∈ Ob( C ′ ) an ob jet. Fix a p oin t λ ′ 0 ∈ Ω ′ ; sine Φ is surjetiv e, ∃ λ 0 with λ ′ 0 = Φ( λ 0 ) . Sine (Φ − 1 (Ω ′ ) , Φ ∗ R ′ ) is a  hart, (Φ − 1 (Ω ′ ) , Φ ∗ R ′ ⊕ Φ ∗ S ′ ) and, onsequen tly , (Φ − 1 (Ω ′ ) , Φ ∗ ( R ′ ⊕ S ′ )) are  harts of ( C , ω ) as w ell. Hene there exists a neigh b ourho o d Ω 0 ⊂ Φ − 1 (Ω ′ ) of λ 0 su h that (Φ(Ω 0 ) , R ′ ⊕ S ′ ) is a  hart of ( C ′ , ω ′ ) . As b efore, sine λ ′ 0 w as arbitrary it follo ws that (Ω ′ , R ′ ⊕ S ′ ) is a  hart of ( C ′ , ω ′ ) . W e an ollet our onlusions in the follo wing 9 Prop osition Let ( f ∗ , Φ ∗ ) : ( C , ω ) − → ( C ′ , ω ′ ) b e a w eak equiv alene of bre funtors. Then ( C , ω ) is a smo oth lassial bre funtor if and only if so is ( C ′ , ω ′ ) . In this ase, ( f , Φ) : T ( ω ) − → T ( ω ′ ) is a Morita equiv alene of Lie group oids. Pro of That (24.4) is a pullba k in the ategory of manifolds of lass C ∞ follo ws b y the same argumen t used in the pro of of Prop osition 2, b eause of the univ ersal prop ert y of the T annakian group oid. q.e.d. 25. WEAK EQUIV ALENCES 141 Notes 10 Note List of elemen tary fats. 1. An y quasi-in v erse equiv alene Φ ∗ is automatially a linear funtor. Indeed, the map Hom C ( R, S ) → Hom C (Φ ∗ Φ ∗ R, Φ ∗ Φ ∗ S ) , a 7→ Φ ∗ Φ ∗ a is a linear bijetion, as it is lear from the omm utativit y of Φ ∗ Φ ∗ R Φ ∗ Φ ∗ a   ≈ R / / R a   Φ ∗ Φ ∗ S ≈ S / / S , and the funtor Φ ∗ is linear and, b eing a ategorial equiv alene, faithful, hene the equalit y Φ ∗ Φ ∗ ( αa + β b ) = α Φ ∗ Φ ∗ a + β Φ ∗ Φ ∗ b = Φ ∗ ( α Φ ∗ a + β Φ ∗ b ) implies the desired linearit y Φ ∗ ( αa + β b ) = α Φ ∗ a + β Φ ∗ b . 2. The realization Φ : T ( ω ) − → T ( ω ′ ) of a w eak equiv alene is a fully faithful morphism of group oids, in other w ords (3 ) is a set-theoreti pullba k. This an b e seen as follo ws. The tensor preserving isomorphism Φ ∗ ◦ Φ ∗ ≈ I d C giv es, aording to Lemma 24.9 p. 135 , a tensor preserving isomorphism (11) ω x ≈ ω x ◦ Φ ∗ ◦ Φ ∗ ≈ ω ′ f ( x ) ◦ Φ ∗ ; similarly , Φ ∗ ◦ Φ ∗ ≈ Id C ′ yields another su h isomorphism (12) ω ′ f ( x ) ≈ ω ′ f ( x ) ◦ Φ ∗ ◦ Φ ∗ . If no w w e apply Lemma 24 .11 p. 136 to these, w e onlude at one from the omm utativit y of the diagram Hom ⊗ ( ω x , ω y ) (11 ) ≈   Φ x,y / / Hom ⊗  ω ′ f ( x ) , ω ′ f ( y )  (12 ) ≈   s s h h h h h h h h h h h h h h h h h h h Hom ⊗  ω ′ f ( x ) Φ ∗ , ω ′ f ( y ) Φ ∗  / / Hom ⊗  ω ′ f ( x ) Φ ∗ Φ ∗ , ω ′ f ( y ) Φ ∗ Φ ∗  that the diagonal arro w is a surjetiv e and injetiv e map, and hene that Φ x,y is bijetiv e. (The omm utativit y of the t w o triangles follo ws from the omm utativit y of the t w o squares ω x (Φ ∗ Φ ∗ R ) λ Φ ∗ Φ ∗ R / / ω y (Φ ∗ Φ ∗ R ) ω ′ f ( x ) ( R ′ ) ω ′ f ( x ) ≈   λ R ′ / / ω ′ f ( y ) ( R ′ ) ω ′ f ( y ) ≈   ω x ( R ) ω x ≈ O O λ R / / ω y ( R ) ω y ≈ O O ω ′ f ( x ) (Φ ∗ Φ ∗ R ′ ) λ Φ ∗ Φ ∗ R ′ / / ω ′ f ( y ) (Φ ∗ Φ ∗ R ′ ) expressing naturalit y of λ, λ ′ resp etiv ely .) 142 CHAPTER V. CLASSICAL FIBRE FUNCTORS 13 Note Let X and Y b e top ologial spaes, and let M and N b e smo oth manifolds. Supp ose X p   f / / Y q   M g / / N (14) is a pullba k diagram in the ategory of top ologial spaes, where g is a smo oth mapping. 1. Giv en an op en subset B ⊂ Y , put A = f − 1 ( B ) . Then the on tin uous maps in (14 ) restrit to a omm utativ e diagram of top ologial spaes A p   f / / B q   p ( A ) g / / q ( B ) , (15) whi h is again a top ologial pullba k. Observ e that if the restrition q | B indues a homeomorphism of B on to q ( B ) , then p | A indues one b et w een A and p ( A ) . (This is a general prop ert y of pullba ks. Indeed, from C g / / p ′ id " " D q − 1   A p   f / / B q   C g / / D and from the equalities f p ′ p = f and p p ′ p = p , it follo ws that p ′ p = id , th us p is in v ertible.) 2. Giv en an op en subset U ⊂ M su h that V = g ( U ) is op en, p − 1 ( U ) f / / p   q − 1 ( V ) q   U g / / V (16) mak es sense and is learly also a top ologial pullba k. Chapter VI Study of Classial T annak a Theory of Lie Group oids In this onlusiv e  hapter w e are ideally going ba k to the p oin t where w e started from, namely the theory of lassial represen tations of Lie group oids exp ounded in 2. W e will try to see what an b e said ab out su h theory b y the ligh t of the general results of Chapters IVV. In partiular, w e will study in detail the standard lassial bre funtor asso iated with a Lie group oid. Reall that in  2 w e in tro dued the ategory R ∞ ( G ) of lassial represen tations R = ( E ,  ) of a Lie group oid G , along with the standard lassial bre funtor ω ∞ ( G ) dened as the forgetful funtor ( E ,  ) 7→ E of R ∞ ( G ) in to the ategory V ∞ ( M ) of smo oth v etor bundles of lo ally nite rank o v er the base M of G . Let us giv e a brief review of the items w e will b e in terested in, so as to x the tait notational on v en tions to b e follo w ed throughout the  hapter. Let T ∞ ( G ) denote the T annakian group oid T ( ω ∞ ( G ); R ) asso iated with the bre funtor ω ∞ ( G ) . Note that it do es not mak e an y dierene whether w e use real or omplex o eien ts in our theory , b eause ev en tually the group oid T ∞ ( G ) and the other related items disussed b elo w will b e exatly the same; in fat, all what w e are going to sa y holds for real as w ell as for omplex o eien ts: for simpliit y , w e assume real o eien ts whenev er w e need to write them do wn expliitly . Reall from 21 that the T annakian onstrution denes an op eration G 7→ T ∞ ( G ) ,  Lie group oids  − →  C ∞ -fun. strutured group oids  ; also note that the soure and target map of T ∞ ( G ) are submersions, in the sense that they admit lo al setions whi h are morphisms of funtionally strutured spaes: this follo ws from the existene of su h setions for G and the fat that the en v elop e homomorphism π ∞ (see b elo w) is a morphism of funtionally strutured spaes. Next, observ e that for ea h Lie group oid homomorphism ϕ : G → H the onstrutions of 24 ma y b e applied to the equation ω ∞ ( G ) ◦ ϕ ∗ = 143 144 CHAPTER VI. CLASSICAL T ANNAKA THEOR Y f ∗ ◦ ω ∞ ( H ) (iden tit y of tensor funtors), so as to yield a homomorphism of C ∞ -funtionally strutured group oids T ∞ ( ϕ ) : T ∞ ( G ) → T ∞ ( H ) . In spite of the la k of funtorialit y of the op eration ϕ 7→ ϕ ∗ , in other w ords in spite of ( ψ ◦ ϕ ) ∗ ∼ = ϕ ∗ ◦ ψ ∗ b eing anonially isomorphi but not equal, the orresp ondene ϕ 7→ T ∞ ( ϕ ) atually turns out to b e a funtor, i.e. the iden tities T ∞ ( ψ ◦ ϕ ) = T ∞ ( ψ ) ◦ T ∞ ( ϕ ) and T ∞ ( id ) = id hold. W e let π ∞ ( G ) or, when there is no am biguit y , π ∞ denote the en v elop e homomorphism G → T ∞ ( G ) dened b y π ∞ ( g )( E ,  ) =  ( g ) . The results of 20 onerning en v elop e homomorphisms an b e applied. In partiular, π ∞ ( G ) will b e a morphism of C ∞ -funtionally strutured group oids. The orresp ondene G 7→ π ∞ ( G ) determines, in fat, a natural transformation π ∞ ( - ) : ( - ) 7→ T ∞ ( - ) , that is to sa y the diagram b elo w omm utes for ea h Lie group oid homomorphism ϕ : G → H G ϕ   π ∞ ( G ) / / T ∞ ( G ) T ∞ ( ϕ )   H π ∞ ( H ) / / T ∞ ( H ) . The main result of the presen t  hapter, to b e pro v ed in 27, is: for G prop er and regular, the standard lassial bre funtor ω ∞ ( G ) is smo oth; in fat, T ∞ ( G ) is a prop er regular Lie group oid although, in general, not one equiv alen t to G . F urthermore, in  26 w e pro v e some partial results ab out the smo othness of the standard lassial bre funtor, that are v alid for arbitrary prop er Lie group oids; w e also remark that the ev aluation funtor ev : R ∞ ( G ) − → R ∞ ( T ∞ ( G )) , R = ( E ,  ) 7→ ( E , ev R ) is an isomorphism of tensor ategories for ea h prop er G (reall the denition of the ategory R ∞ ( T ∞ ( G )) in 21). Finally , in 28 w e giv e a few examples of lassially reexiv e (prop er) Lie group oids. 26 On the Classial En v elop e of a Prop er Lie Group oid Let G b e a Lie group oid. Reall from 21 that to ea h lassial represen tation R = ( E ,  ) of G one an asso iate a represen tation ev R : T ∞ ( G ) → GL ( E ) , giv en b y ev aluation at the ob jet R ∈ Ob R ∞ ( G ) : (1) T ∞ ( G )( x, x ′ ) ∋ λ 7→ λ ( R ) ∈ Lis( E x , E x ′ ) , 26. THE CLASSICAL ENVELOPE OF A PR OPER GR OUPOID 145 whi h mak es the follo wing triangle omm ute G  % % K K K K K K K K K K K π ∞ ( G ) / / T ∞ ( G ) ev R w w o o o o o o o o o o o GL ( E ) , (2) where π ∞ ( G ) denotes the en v elop e homomorphism π ∞ ( g )( E ,  ) =  ( g ) . Throughout the presen t setion w e shall b e in terested mainly in prop er Lie group oids. Therefore, from no w on w e assume that G is a prop er Lie group oid and w e regard this assumption as made one and for all. As ev er, M will denote the base manifold of G . When w e w an t to state a result that is true under less restritiv e assumptions on G , w e shall expliitly p oin t it out. W e are going to apply the general theory of represen tativ e  harts (23) to the standard lassial bre funtor ω ∞ ( G ) . 3 Lemma Let ( E ,  ) b e a lassial represen tation of a (not neessarily prop er) Lie group oid G . Supp ose w e are giv en an op en subset Γ of the manifold of arro ws of G , su h that the image Σ =  (Γ) is a submanifold of GL ( E ) and su h that  restrits to an op en mapping of Γ on to Σ . Then Σ is a tame submanifold of GL ( E ) , and the restrition of  to Γ is a submersion of Γ on to Σ . Moreo v er, when G is prop er then the assumption that  should restrit to an op en mapping of Γ on to Σ is sup eruous. Pro of W e pro v e the statemen t in the prop er ase rst, so without making the assumption that  is an op en map of Γ on to Σ . W e start b y observing that for ea h x 0 ∈ M the image   G ( x 0 , - )  is a prinipal submanifold of GL ( E ) and the mapping (4) G ( x 0 , - )  − →   G ( x 0 , - )  is a submersion. In partiular, the latter will b e an op en mapping and this fores the op en subset (5) Σ( x 0 , - ) =   G ( x 0 , - ) ∩ Γ  ⊂  G ( x 0 , - ) to b e a prinipal submanifold of GL ( E ) as w ell. Our argumen t is as follo ws. Fix g 0 in G ( x 0 , - ) and let λ 0 =  ( g 0 ) . Cho ose an op en subset V ⊂ M on taining x ′ 0 = t ( g 0 ) , small enough to ensure that the prinipal bundle G ( x 0 , - ) is trivial o v er Z = G x 0 ∩ V , ie that a lo al equiv arian t  hart G ( x 0 , Z ) ≈ Z × G 0 an b e found, where G 0 denotes the isotrop y group at x 0 ; it is no loss of generalit y to assume g 0 ≈ ( x ′ 0 , e ) in su h a  hart whi h w e no w use, along with the represen tation  , to obtain a smo oth setion z 7→ ( z , e ) ≈ g 7→  ( g ) to the target map of GL ( E ) o v er 146 CHAPTER VI. CLASSICAL T ANNAKA THEOR Y Z . Next, the isotrop y homomorphism G 0 → GL ( E ) 0 determined b y  at x 0 anonially fators through the quotien t Lie group obtained b y dividing out the k ernel, th us yielding a losed Lie subgroup H ֒ → GL ( E ) 0 . As usual, this Lie subgroup and the target setion ab o v e an b e om bined in to an em b edding of manifolds of t yp e ( 22.2), whi h ts in the follo wing square Z × G 0 id × pr   ≈ / / G ( x 0 , Z )    Z × H   (22 .2 ) / / GL ( E ) (6) and hene sim ultaneously displa ys  G ( x 0 , Z ) as a prinipal submanifold of GL ( E ) and, aording to the initial remarks of Setion 22 , the mapping  : G ( x 0 , Z ) →  G ( x 0 , Z ) as a submersion; sine the subset (7)  G ( x 0 , Z ) =  G ( x 0 , - ) ∩ t − 1 ( V ) ⊂  G ( x 0 , - ) is an op en neigh b orho o d of λ 0 in  G ( x 0 , - ) , w e an onlude. A t this p oin t, in order to pro v e that Σ is a tame submanifold of GL ( E ) w e need only v erify that the restrition Σ → M of the soure map of GL ( E ) is a submersion. So, x σ 0 ∈ Σ , sa y σ 0 =  ( g 0 ) with g 0 ∈ Γ . There exists a lo al smo oth soure setion γ : U → Γ through g 0 = γ ( s g 0 ) , hene w e an also nd a lo al smo oth soure setion σ =  ◦ γ : U → Σ through σ 0 . Finally , w e ome to the statemen t that  : Γ → Σ is a submersion. Fix g 0 ∈ Γ and let σ 0 =  ( g 0 ) . Sine b oth Γ and Σ are tame submanifolds, there exist lo al trivializations of the resp etiv e soure maps around the p oin ts g 0 ≈ ( x 0 , 0) and σ 0 ≈ ( x 0 , 0) , whi h yield a lo al expression for  | Γ 0 Γ 0 ≈    / / Σ 0 ≈   U × B / / _ _ _ V × C (8) of the form ( u, b ) 7→ ( u, c ( u, b )) , where U ⊂ V are op en subsets of M and B , C are Eulidean balls. The partial map b 7→ c ( x 0 , b ) is submersiv e at the origin b eause it is the lo al expression of (4 ). No w w e turn to the general ase where G is not neessarily prop er. Th us, assume that  restrits to an op en mapping of Γ on to Σ . As explained ab o v e, for an y giv en g 0 ∈ G ( x 0 , - ) there is a submanifold Z ⊂ M on tained in G x 0 although, in general, this is no longer of the form Z = G x 0 ∩ V su h that the subset G ( x 0 , Z ) ⊂ G ( x 0 , - ) is op en, the image  G ( x 0 , Z ) is a prinipal submanifold of GL ( E ) and the indued mapping  : G ( x 0 , Z ) →  G ( x 0 , Z ) is submersiv e. On the other hand, from the assumption that  : Γ → Σ is op en it follo ws that the restrition  : Γ( x 0 , - ) → Σ( x 0 , - ) m ust b e op en as w ell, b eause one has (9)   X( x 0 , - )  =  (X)( x 0 , - ) 26. THE CLASSICAL ENVELOPE OF A PR OPER GR OUPOID 147 for an y subset X ⊂ G (1) . Then, sine Γ ∩ G ( x 0 , Z ) = Γ( x 0 , - ) ∩ G ( x 0 , Z ) is an op en subset of Γ( x 0 , - ) , it is eviden t that (10) Σ( x 0 , Z ) =   Γ ∩ G ( x 0 , Z )  ⊂  G ( x 0 , Z ) is b oth an op en neigh b ourho o d of λ 0 in Σ( x 0 , - ) and an op en subset of the prinipal submanifold  G ( x 0 , Z ) of GL ( E ) . This means that Σ( x 0 , - ) is a prinipal submanifold of GL ( E ) . Moreo v er, from what w e said it is eviden t that  indues a submersion of Γ( x 0 , - ) on to Σ( x 0 , - ) . The rest of the pro of holds without mo diations. q.e.d. Note that the preeding lemma holds for real as w ell as for omplex o eien tsthat is, for ( E ,  ) in R ∞ ( G , R ) or in R ∞ ( G , C ) . Our main goal in the presen t setion is to sho w that the standard lassial bre funtor ω ∞ ( G ) asso iated with a prop er Lie group oid G alw a ys satises ondition ii) of Prop osition 23 .4. First of all, note that in order that (Ω , R ) ma y b e a represen tativ e  hart of T ∞ ( G ) , where Ω is an op en subset of the spae of arro ws of T ∞ ( G ) and R = ( E ,  ) ∈ Ob R ∞ ( G ) , it is suien t that ev R establishes a one-to-one orresp ondene b et w een Ω and a submanifold of GL ( E ) . F or if w e set Γ = ( π ∞ ) − 1 (Ω) , w e ha v e  (Γ) = R ( Ω) b eause of (2) and the surjetivit y of π ∞ ; then Lemma 3 implies that R (Ω) is a tame submanifold of GL ( E ) and that  : Γ → R (Ω) is a submersionso, in partiular, that the map ev R : Ω → R (Ω) is op en and hene a homeomorphism. Our laim ab out the ondition ii) of Prop osition 23 .4 essen tially follo ws from a simple general remark ab out submersions. Namely , supp ose that a omm utativ e triangle of the form X g      Y f ′ + + X X X X X X X X X X X X X X X f 3 3 f f f f f f f f f f f f f f f X ′ (11) is giv en, where X , X ′ and Y are smo oth manifolds, f is a submersion on to X , f ′ is a smo oth mapping and all w e kno w ab out g is that it is a set-theoreti solution whi h ts in the triangle. Then the map g is neessarily smo oth; in partiular, in ase f ′ is also a surjetiv e submersion, g is a dieomorphism if and only if it is a set-theoreti bijetion. T o see ho w this ma y b e used to pro v e ompatibilit y of  harts, supp ose w e are giv en an arbitrary represen tativ e  hart (Ω , R ) of T ∞ ( G ) to start with, where let us sa y R = ( E ,  ) , and an arbitrary lassial represen tation S = ( F , σ ) . Let Γ = ( π ∞ ) − 1 (Ω) , so that Γ is an op en submanifold of G . W e ha v e already observ ed that  indues a submersion of Γ on to the submanifold R (Ω) of GL ( E ) ; also, the homomorphism of Lie group oids (12) ( , σ ) : G − → GL ( E ) × M GL ( F ) 148 CHAPTER VI. CLASSICAL T ANNAKA THEOR Y an b e restrited to Γ to yield a smo oth mapping in to GL ( E ) × M GL ( F ) . W e get an instane of ( 11 ) b y in tro duing the follo wing map (13) s = ( ev R , ev S ) ◦ ev R − 1 : R (Ω) → GL ( E ) × M GL ( F ) (note that ev R : Ω → R (Ω) is in v ertible b eause w e assume (Ω , R ) to b e a represen tativ e  hart), whi h is then a smo oth setion to the pro jetion (14) GL ( E ) × M GL ( F ) → GL ( E ) and th us, in partiular, an immersion. No w, if s is indeed the em b edding of a submanifoldie if it is an op en map on to its imagethen w e are done, sine in that ase ( R, S )(Ω) = s ( R (Ω)) is a submanifold of GL ( E ) × M GL ( F ) and ( ev R , ev S ) a bijetiv e map on to it; equiv alen tly , ( R ⊕ S )(Ω) is a submanifold of GL ( E ⊕ F ) and ev R ⊕ S is a bijetion of Ω on to it. (Cf. Setion 23. As observ ed ab o v e, this is enough to onlude that (Ω , R ⊕ S ) is a represen tativ e  hart.) F or ea h op en subset Λ of GL ( E ) , (15) s  R (Ω) ∩ Λ  = s ( R (Ω)) ∩  Λ × GL ( F )  is in fat an op en subset of the subspae s ( R (Ω)) . W e an summarize what w e ha v e onluded so far as follo ws: 16 Prop osition Let G b e a prop er Lie group oid. Then the standard lassial bre funtor ω ∞ ( G ) is smo oth if and only if the spae of arro ws of the lassial T annakian group oid T ∞ ( G ) an b e o v ered with op en subsets Ω su h that for ea h of them one an nd some R = ( E ,  ) ∈ Ob R ∞ ( G ) with the prop ert y that ev R establishes a bijetion b et w een Ω and a submanifold R (Ω) of GL ( E ) . Moreo v er, in ase the latter ondition is satised then the en v elop e homomorphism π ∞ ( G ) : G − → T ∞ ( G ) will b e a surjetiv e submersion of Lie group oids. Pro of The rst assertion is already pro v en. The seond assertion follo ws from the (previously notied) fat that for ea h represen tativ e  hart (Ω , R ) the mapping  : Γ → R (Ω) is a submersion, where as usual R = ( E ,  ) and w e put Γ = ( π ∞ ) − 1 (Ω) . (Remem b er from the pro of of Prop. 23 .4 that ev R establishes a dieomorphism b et w een Ω and the submanifold R (Ω) of GL ( E ) .) q.e.d. Note that, for an y prop er Lie group oid G whose asso iated standard las- sial bre funtor ω ∞ ( G ) is smo oth, the preeding prop osition allo ws us to  haraterize the familiar Lie group oid struture on the T annakian group oid T ∞ ( G ) as the unique su h struture for whi h the en v elop e homomorphism π ∞ ( G ) b eomes a submersion. Indeed, assume that an unkno wn Lie group- oid struture, making π ∞ ( G ) a submersion, is assigned on the T annakian 26. THE CLASSICAL ENVELOPE OF A PR OPER GR OUPOID 149 group oid of G . Let T ∗ ( G ) indiate the T annakian group oid of G endo w ed with the unkno wn smo oth struture. No w, the iden tit y homomorphism of the T annakian group oid in to itself ts in the follo wing triangle T ∞ ( G ) id      G π ∗ , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y π ∞ 2 2 e e e e e e e e e e e e e e e e e e e T ∗ ( G ) (17) where π ∞ = π ∞ ( G ) = π ∗ are surjetiv e submersions. It follo ws that the iden tit y id : T ∞ ( G ) = T ∗ ( G ) is a dieomorphism. Under the assumption of prop erness, w e an also sa y something useful ab out ondition i) of Prop osition 23 .4: 18 Note Let G b e a prop er Lie group oid. Supp ose that for ea h iden tit y arro w x 0 of the T annakian group oid T ∞ ( G ) one an nd a represen tativ e  hart for T ∞ ( G ) ab out x 0 . Then w e on tend that the ondition i) of Prop o- sition 23.4 is satised b y the lassial bre funtor ω ∞ ( G ) . Let an arbitrary arro w λ 0 : x 0 → x ′ 0 of T ∞ ( G ) b e giv en. Beause of prop erness, w e ha v e λ 0 = π ∞ ( g 0 ) for some arro w g 0 : x 0 → x ′ 0 of G . Selet an y smo oth lo al bisetion σ : U → G (1) , dened o v er a neigh b ourho o d U of x 0 and with σ ( x 0 ) = g 0 , and let U ′ = t ( σ ( U )) . No w, let (Ω , R ) b e a represen tativ e  hart ab out x 0 , let us sa y with Ω ⊂ T ∞ ( G ) | U and R = ( E ,  ) . Notie that one has the follo wing omm utativ e square G | U ≈ σ -    / / GL ( E ) | U ≈ (  ◦ σ ) -   G ( U, U ′ )  / / GL ( E )( U, U ′ ) , (19) where σ - denotes the left translation dieomorphism g 7→ σ ( t ( g )) · g and, similarly , (  ◦ σ ) - denotes the dieomorphism µ 7→  ( σ ( t µ )) · µ . Let Γ = ( π ∞ ) − 1 (Ω) , so Γ ⊂ G | U is an op en subset. Then Γ σ = σ - (Γ) is an op en neigh b ourho o d of g 0 , Ω σ = ( π ∞ ◦ σ ) - (Ω) is an op en neigh b ourho o d of λ 0 and Γ σ = ( π ∞ ) − 1 (Ω σ ) . It follo ws that the subset (20) R (Ω σ ) =  (Γ σ ) = (  ◦ σ ) - (  (Γ)) = (  ◦ σ ) - ( R (Ω)) is a submanifold of GL ( E )( U, U ′ ) . Similarly , one sees that Ω σ bijets on to R (Ω σ ) via ev R . So (Ω σ , R ) is a represen tativ e  hart ab out λ 0 . The next, onlusiv e result pro vides, in the sp eial ase under exam, a p ositiv e answ er to the question raised in 21 ab out the ev aluation funtor b eing an equiv alene of ategories. 150 CHAPTER VI. CLASSICAL T ANNAKA THEOR Y 21 Prop osition Let G b e an y prop er Lie group oid. Then the ev aluation funtor ev : R ∞ ( G ) − → R ∞ ( T ∞ ( G )) , R = ( E ,  ) 7→ ( E , ev R ) is an isomorphism of ategories, ha ving the pullba k along the en v elop e homomorphism of G as in v erse. Pro of This an b e v eried diretly , sine the en v elop e homomorphism of a prop er Lie group oid is already kno wn to b e surjetiv e. q.e.d. 27 Smo othness of the Classial En v elop e of a Prop er Regular Group oid W e start b y realling a few basi denitions and prop erties. F or additional information, see Mo er dijk (2003) [26℄. Reall that a Lie group oid G o v er a manifold M is said to b e r e gular when the rank of the dieren tiable map t x : G ( x, - ) → M lo ally k eeps onstan t as the v ariable x ranges o v er M ; an equiv alen t ondition is that the an hor map of the Lie algebroid of G , let us all it ρ : g → T M , should ha v e lo ally onstan t rank (as a morphism of v etor bundles o v er M ). If G is regular then the image of the an hor map ρ is a subbundle F of the tangen t bundle T M ; in fat, F turns out to b e an in tegrable subbundle of T M and hene determines a foliation F of the base manifold M , alled the orbit foliation asso iated with the regular group oid G . Reall that a le af of a foliation F asso iated with an in tegrable subbundle F of T M is a maximal onneted immersed submanifold L of M with the prop ert y of b eing ev erywhere tangen t to F . The o dimension of L in M oinides with the o dimension of F in T M . Also reall that a tr ansversal for F is a submanifold T of M , ev erywhere transv ersal to F and of dimension equal to the o dimension of F . There alw a ys exist  omplete transv ersals, i.e. transv ersals that meet ev ery leaf of the foliation. Bund les of Lie gr oups, that is to sa y Lie group oids whose soure and target map oinide, form a v ery sp eial lass of regular Lie group oids. Prop er bundles of Lie groups are also alled bund les of  omp at Lie gr oups. 1 Lemma Let G b e a bundle of ompat Lie groups o v er a manifold M . Let R = ( E ,  ) b e a lassial represen tation of G . Then the image  ( G ) is a submanifold of GL ( E ) . Pro of By a result of W einstein [37 ℄, ev ery bundle of ompat Lie groups is lo  al ly trivial. This means that for ea h x ∈ M one an nd an op en neigh b orho o d U of x in M and a ompat Lie group G su h that there exists an isomorphism of Lie group oids o v er U (viz. a lo al trivialization) (2) G | U ≈ U × G . 27. PR OPER REGULAR GR OUPOIDS 151 A t the exp ense of replaing U with a smaller op en neigh b orho o d, one an also assume that there is a lo al trivialization E | U ≈ U × V , where V is some v etor spae of nite dimension; as explained in Note 23 .11, su h a trivialization will determine an isomorphism GL ( E | U ) ≈ U × GL ( V ) of Lie group oids o v er U . Then one an tak e the follo wing omp osite homomorphism U × G ≈ / /   G | U  | U / /   GL ( E | U ) ≈ / /   U × GL ( V ) pr / /   GL ( V )   U × U id / / U × U id / / U × U id / / U × U / / ⋆ × ⋆ . (3) This yields a smo oth family of represen tations of the ompat Lie group G on the v etor spae V , parametrized b y the onneted op en set U . W e will denote su h family b y  U : U × G → GL ( V ) . No w, it follo ws from the so-alled `homotop y prop ert y of represen tations of ompat Lie groups' (Note 2.30) that all the represen tations of the smo oth family  U are equiv alen t to ea h other; in partiular, they all ha v e the same k ernel K ⊂ G . Hene there exists a unique map f  U making the diagram U × G id × pr   id ×  U / / U × GL ( V ) U × ( G/K ) f  U 6 6 n n n n n n (4) omm ute. Note that the map f  U m ust b e smo oth, b eause id × pr is a sur- jetiv e submersion; of ourse, the same map is also a faithful represen tation of the bundle of ompat Lie groups U × ( G/K ) on the trivial v etor bundle U × V . Then Corollary 23 .10 implies that the image of f  U is a submanifold of U × GL ( V ) . The latter submanifold oinides, via the dieomorphism GL ( E ) | ∆ U ≈ U × GL ( V ) , with the in tersetion  ( G ) ∩ GL ( E ) | U . q.e.d. It is eviden t from the ab o v e pro of that the k ernel of the en v elop e homo- morphism π ∞ : G → T ∞ ( G ) m ust b e a (lo ally trivial) bundle of ompat Lie groups K , em b edded in to G . Th us, if U is a onneted op en subset of M and R = ( E ,  ) is a lassial represen tation su h that Ker  u = K| u at some p oin t u ∈ U , it follo ws from the aforesaid homotop y prop ert y that Ker  | U = K| U and thereforefrom the omm utativit y of (26 .1)that the ev aluation represen tation ev R is faithful on T ∞ ( G ) | U . F rom the latter remark, the disussion ab out smo othness in the preeding setion and Lemma 1 it follo ws immediately that the standard lassial bre funtor ω ∞ ( G ) asso iated with a bundle of ompat Lie groups G is smo oth. Indeed, let an arbitrary arro w λ 0 ∈ T ∞ ( G ) b e xed, let us sa y λ 0 ∈ T ∞ ( G ) | x 0 with x 0 ∈ M . T ak e an ob jet R ∈ Ob R ∞ ( G ) with the prop ert y that the restrition of the ev aluation represen tation ev R to T ∞ ( G ) | x 0 is faithful (this exists b y Prop. 10 .14) and then  ho ose an y onneted op en neigh b ourho o d 152 CHAPTER VI. CLASSICAL T ANNAKA THEOR Y U of x 0 in M . Then the pair  T ∞ ( G ) | U , R  onstitutes a represen tativ e  hart for ω ∞ ( G ) ab out λ 0 . More generally , let G b e a prop er Lie group oid with the prop ert y that for ea h x 0 ∈ M there exists an op en neigh b ourho o d U of x 0 in M su h that G | U is a bundle of ompat Lie groups. By adapting the ab o v e reip e for the onstrution of represen tativ e  harts ab out the arro ws b elonging to the isotrop y of T ∞ ( G ) and b y taking in to aoun t Note 26.18, w e see that ω ∞ ( G ) is smo oth also in the presen t ase. W e are going to generalize the latter remark to arbitrary prop er regular Lie group oids. The shortest w a y to do this is to apply the theory of w eak equiv alenes of  25. 5 Prop osition Let G b e a prop er regular Lie group oid. Then the standard lassial bre funtor ω ∞ ( G ) asso iated with G is smo oth. Reall that in view of Prop osition 26 .16 this an also b e expressed b y sa ying that there exists a (neessarily unique) Lie group oid struture on the T annakian group oid T ∞ ( G ) su h that the en v elop e homomorphism π ∞ ( G ) b eomes a smo oth submersion. Pro of Let M b e the base of G . Selet a omplete transv ersal T for the foliation of the manifold M determined b y the orbits of G . Note that T is in partiular a sli e, so the restrition G | T is a prop er Lie group oid em b edded in to G (b y Note 4.3). If i : T ֒ → M denotes the inlusion map then, b y the remarks at the end of 4, the em b edding of Lie group oids G | T     inlusion / / G   T × T   i × i / / M × M (6) is a Morita equiv alene. One ma y therefore nd another (prop er) Lie group oid K , along with Morita equiv alenes G | T M.e. ← − − − K M.e. − − − → G induing surjetiv e submersions at the lev el of base manifolds. The orresp onding morphisms of standard lassial bre funtors (7)  R ∞ ( G | T ) , ω ∞ ( G | T )  w.e. ← − −  R ∞ ( K ) , ω ∞ ( K )  w.e. − − →  R ∞ ( G ) , ω ∞ ( G )  are w eak equiv alenes. Hene, b y Prop osition 25.9, one is redued to sho wing that ω ∞ ( G | T ) is a smo oth bre funtor. No w, G | T is a prop er Lie group oid o v er T with the ab o v e-men tioned prop ert y of b eing, lo ally , just a bundle of ompat Lie groups. q.e.d. Let ProReg denote the ategory of prop er regular Lie group oids. One ma y summarize the onlusions of the presen t setion as follo ws: 28. CLASSICAL REFLEXIVITY: EXAMPLES 153 8 Theorem The lassial T annakian orresp ondene G 7→ T ∞ ( G ) in- dues an idemp oten t funtor (9) T ∞ ( - ) : ProReg − → ProReg ; moreo v er, en v elop e homomorphisms t together in to a natural transfor- mation (10) π ∞ ( - ) : Id − → T ∞ ( - ) . Op en Question. It is natural to ask whether this result an b e generalized to the whole ategory of prop er Lie group oids. 28 A few Examples of Classially Reexiv e Lie Group oids Reall that a Lie group oid G ⇒ X is said to b e étale if the soure and target maps s , t : G → X are étale maps, that is to sa y lo al isomorphisms of smo oth manifolds. An op en subset Γ ⊂ G will b e said to b e at if the soure and target map restrit to op en em b eddings of Γ in to X . A Lie group oid G will b e said to b e sour  e-pr op er or, for short, s -pr op er when the soure map of G is a prop er map. 1 Prop osition Let G b e a soure-prop er étale Lie group oid. Then G admits globally faithful lassial represen tations. Pro of The r e gular r epr esentation ( R,  ) of G exists and has lo ally nite rank. A ouple of remarks b efore starting. Let X b e the base of G . F or ev ery p oin t x of X , the s -b er s − 1 ( x ) is a nite set. Indeed, it is disrete, b eause if g ∈ s − 1 ( x ) then sine s is étale there exists a at op en neigh b orho o d Γ ⊂ G and therefore { g } = Γ ∩ s − 1 ( x ) is a neigh b orho o d of g in the s -b er. It is also ompat, b eause of s -prop erness. Put ℓ ( x ) = k s − 1 ( x ) k , the ardinalit y of this nite set. Then the b er R x of the v etor bundle R → X is b y denition the v etor spae (2) C 0 ( s − 1 ( x ); R ) ∼ = R ℓ ( x ) of R -v alued maps. This mak es sense b eause 3 Lemma The assignmen t x 7→ ℓ ( x ) denes a lo ally onstan t funtion on X , with v alues in to p ositiv e in tegers. Pro of of the lemma. Fix x ∈ X , and sa y s − 1 ( x ) = { g 1 , . . . , g ℓ } . F or ev ery i = 1 , . . . , ℓ , there exists a at op en neigh b orho o d Γ i ⊂ G of g i . Cho osing an op en ball B ⊂ T s (Γ i ) at x , w e an assume s : Γ i ∼ → B to b e an isomorphism ∀ i . Moreo v er, it is no loss of generalit y to assume the op en subsets Γ 1 , . . . , Γ ℓ 154 CHAPTER VI. CLASSICAL T ANNAKA THEOR Y to b e pairwise disjoin t. (As a onsequene of the fat that a nite union of op en balls in an y manifoldnot neessarily Hausdoris a Hausdor op en submanifold.) Then, ∀ i = 1 , . . . , ℓ and ∀ z ∈ B , the in tersetion s − 1 ( z ) ∩ Γ i onsists of a single p oin t g i ( z ) , and these p oin ts g 1 ( z ) , . . . , g ℓ ( z ) ∈ G are pairwise distint, b eause the Γ i are pairwise disjoin t. This sho ws ℓ ( z ) ≧ ℓ ( x ) ∀ z ∈ B . T o pro v e the on v erse inequalit y , it will sue to pro v e that ∃ N ⊂ B , a smaller ball at x , su h that s − 1 ( N ) ⊂ Γ = Γ 1 ∪ · · · ∪ Γ ℓ . Consider a dereasing sequene of losed balls C n +1 ⊂ C n ⊂ B shrinking to x , and the orresp onding dereasing sequene Σ n = s − 1 ( C n ) − Γ of losed subsets of the ompat subspae s − 1 ( C 1 ) ⊂ G ; there ∃ n su h that Σ n = ∅ , in other w ords s − 1 ( C n ) ⊂ Γ . This onludes the pro of of the lemma. Th us, it mak es sense to regard R → X as the set-theoreti supp ort of a R -linear v etor bundle of lo ally nite rank. The pro of of the lemma on tains also a reip e for the onstrution of lo al trivializations. Namely , let x ∈ X b e xed, and  ho ose an ordering s − 1 ( x ) = { g 1 , . . . , g ℓ } of the orresp onding b er; there exist an op en ball B ⊂ X en tered at x and disjoin t at op en neigh b orho o ds Γ 1 , . . . , Γ ℓ ⊂ G of g 1 , . . . , g ℓ su h that s − 1 ( B ) = Γ 1 ∪ · · · ∪ Γ ℓ . Then one gets a bijetion R | B ≈ B × R ℓ b y setting, for z ∈ B and f ∈ C 0 ( s − 1 ( z ); R ) , ( z , f ) 7→  z , f ( g 1 ( z )) , . . . , f ( g ℓ ( z ))  . (Cf. the notation used in the pro of of the lemma.) The transition map- pings are smo oth, b eause lo ally they are giv en b y onstan t p erm utations ( a 1 , . . . , a ℓ ) 7→  a τ (1) , . . . , a τ ( ℓ )  . The R -linear isomorphism  ( g ) ∈ Lis( R x , R y ) , asso iated with g ∈ G ( x, y ) , is dened b y `translation' f 7→  ( g )( f ) ≡ f ( - g ) . The resulting funtorial map  : G − → GL ( R ) is learly faithful; it is also smo oth, b eause in an y trivializing lo al  harts it lo oks lik e a lo ally onstan t p erm utation. q.e.d. If G is an y étale Lie group oid with base manifold X , there is a morphism of Lie group oids Ef : G − → Γ X , where Γ X is the étale Lie group oid (with base X ) of germs of smo oth isomorphisms U ∼ → V b et w een op en subsets of X . It sends g ∈ G to the germ of the lo al smo oth isomorphism asso iated with a at op en neigh b orho o d of g . An ee tive Lie group oid is an étale Lie group oid su h that Ef is faithful, in other w ords su h that ev ery g ∈ G is uniquely determined b y its `lo al ation' on the base manifold X . (Some of the simplest étale group oids, su h as for instane the trivial ones X × K , K a disrete group, are not eetiv e at all!) 28. CLASSICAL REFLEXIVITY: EXAMPLES 155 The lass of eetiv e Lie group oids is stable under w eak equiv alene among étale Lie group oids. (Cf. Mo er dijk and Mr £un (2003), [ 27 ℄ p. 137.) The follo wing onditions on a Lie group oid G are equiv alen t: 1. G is w eakly equiv alen t to a prop er eetiv e group oid; 2. G is w eakly equiv alen t to the Lie group oid asso iated with an orbifold. (Cf. ibid. p. 143.) The relev ane of this theorem in the presen t on text is that it tells that if one w an ts to study orbifolds through their asso iated Lie group oid and T annakian dualit y , it is suien t to pro v e the dualit y result for prop er eetiv e group oids. An y étale Lie group oid G ⇒ X has a anonial represen tation on the tangen t bundle T X → X , whi h asso iates to g ∈ G ( x, y ) the in v ertible R - linear map T x X → T y X of tangen t spaes giv en b y the tangen t map at x of the germ of lo al smo oth isomorphisms Ef ( g ) . In general, this represen tation need not b e faithful. Ho w ev er 4 Prop osition If G is a prop er eetiv e Lie group oid with base X , the anonial represen tation on the tangen t bundle T X is faithful. 1 Pro of If G ⇒ X is a prop er étale Lie group oid and x ∈ X , there exist a neigh b orho o d U ⊂ X of x and a smo oth ation of the isotrop y group G x = G | x on U , su h that the Lie group oid G | U ⇒ U is isomorphi to the ation group oid G x ⋉ U . I need to reall part of the pro of. (Cf. Mo er dijk and Mr £un (2003), [27℄ p. 142.) Let G x = { 1 , . . . , ℓ } . There are a onneted op en neigh b orho o d W ⊂ X of x and s -setions σ 1 , . . . , σ ℓ : W → G with σ i ( x ) = i ∈ G x ∀ i , su h that the maps f i = t ◦ σ i send W dieomorphially on to itself and satisfy f i ◦ f j = f ij for all i, j ∈ G x . Sine G is also eetiv e, the group homomorphism i 7→ f i , of G x in to the group Aut( W ; x ) of smo oth automorphisms of W that x the p oin t x , is injetiv e. No w, if M is a onneted manifold and H ⊂ Aut( M ) is a nite group of smo oth automorphisms of M , the group homomorphism whi h maps f ∈ H x = { f ∈ H | f ( x ) = x } to the tangen t map T x f ∈ Aut( T x M ) is injetiv e ∀ x ∈ M . ( Ibid. p. 36.) In the ase M = W and H = { f i | i ∈ G x } = H x , this sa ys preisely that the anonial represen tation of G on the tangen t bundle T X restrits to a faithful represen tation G x ֒ → Aut( T x X ) . q.e.d. Another simple example is oered b y ation group oids asso iated with om- pat Lie group ations. Preisely , let K b e a ompat Lie group ating smo othly on a manifold X , sa y from the left. W e denote b y K ⋉ X the Lie group oid o v er X whose 1 This w as p oin ted out to me b y I. Mo erdijk. 156 CHAPTER VI. CLASSICAL T ANNAKA THEOR Y manifold of arro ws is the Cartesian pro dut K × X , with the seond pro- jetion ( k , x ) 7→ x as soure map, the ation K × X → X as range map and ( k ′ , k · x ) · ( k , x ) = ( k ′ k , x ) as omp osition of arro ws. If V is a faithful K -mo dule (in other w ords a faithful represen tation  of the ompat Lie group K on a v etor spae V ), then w e get a faithful represen tation of the group oid K ⋉ X on the trivial v etor bundle X × V , dened b y ( k , x ) 7→  x, k · x,  ( k )  . Bibliograph y [1℄ I. Androulidakis and G. Sk andalis. The holonom y group oid of a singular foliation. Preprin t arXiv math.DG/0612370. [2℄ R. Bos. Con tin uous represen tations of group oids. Preprin t arXiv math.R T/0612639. [3℄ G. E. Bredon. Intr o dution to Comp at T r ansformation Gr oups . A a- demi Press, New Y ork, 1972. [4℄ T. Brö k er and T. tom Die k. R epr esentations of Comp at Lie Gr oups . Graduate T exts in Mathematis 98. Springer-V erlag, New Y ork, 1985. [5℄ A. Cannas da Silv a and A. W einstein. Ge ometri Mo dels for Non ommu- tative A lgebr as . Num b er 10 in Berk eley Mathematial Letures. Ameri- an Mathematial So iet y , Pro videne, 1999. [6℄ C. Chev alley . The ory of Lie Gr oups . Prineton Univ ersit y Press, 1946. [7℄ P . E. Conner and E. E. Flo yd. Dier entiable Perio di Maps . Springer- V erlag, 1964. [8℄ A. Connes. Non ommutative Ge ometry . A ademi Press, New Y ork, 1994. [9℄ A. Coste, P . Dazord, and A. W einstein. Group oïdes sympletiques. Publiations du Départemen t de Mathématiques, Nouv elle Série (A) 87, Univ. Claude-Bernarde, Ly on, 1987. P ages iii, 162. [10℄ M. Craini. Dieren tiable and algebroid ohomology , V an Est isomorph- ism, and  harateristi lasses. Comm. Math. Helvetii , 78:681721, 2003. [11℄ P . Deligne. Catégories tannakiennes. In P . Cartier, L. Illusie, N. M. Katz, et al., editors, The Gr othendie k F estshrift , v olume I I, pages 111194. Birkhäuser, Basel, 1991. [12℄ P . Deligne and J. S. Milne. T annakian ategories. In Ho dge Cyles, Mo- tives and Shimur a V arieties , Leture Notes in Mathematis 900, pages 101228. Springer-V erlag, 1982. 157 158 BIBLIOGRAPHY [13℄ J. Dixmier. L es C ∗ -A lgébr es et leurs R epr ésentations . Gauthier-Villars, P aris, 1969. [14℄ J. Dixmier and A. Douady . Champs on tin ues d'espaes Hilb ertiens et de C ∗ -algébres. Bul l. de la SMF , 19:227284, 1963. [15℄ J.-P . Dufour and N. T. Zung. Poisson strutur es and their normal forms . Num b er 242 in Progress in Mathematis. Birkhäuser, Basel, 2005. [16℄ S. Ev ens, J.-H. Lu, and A. W einstein. T ransv erse measures, the mo dular lass and a ohomology pairing for Lie algebroids. Quart. J. Math. Oxfor d , 50(2):417436, 1999. [17℄ A. Henriques and D. S. Metzler. Presen tations of noneetiv e orbifolds. T r ans. A m. Math. So iety , 356(6):24812499, 2004. [18℄ A. Jo y al and R. Street. An in tro dution to T annak a dualit y and quan- tum groups. In A. Carb oni, M. C. P edi hio, and G. Rosolini, edi- tors, Cate gory The ory, Pr o  e e dings, Como 1990 , n um b er 1448 in Leture Notes in Mathematis, pages 411492. Springer-V erlag, 1991. [19℄ J. Kali²nik. Represen tations of orbifold group oids. Preprin t arXiv math.DG/07090176 v1, Sept. 2007. [20℄ M. G. Krein. A priniple of dualit y for a biompat group and square blo  k algebra. Dokl. A kad. Nauk. SSSR , 69:725728, 1949. [21℄ N. P . Landsman. Mathemati al T opis b etwe en Classi al and Quantum Me hanis . Springer-V erlag, New Y ork, 1998. [22℄ S. Lang. F undamentals of Dier ential Ge ometry . Springer-V erlag, 2001. [23℄ W. Lü k and B. Oliv er. The ompletion theorem in K -theory for prop er ations of a disrete group. T op olo gy , 40(3):585616, 2001. [24℄ S. MaLane. Cate gories for the W orking Mathematiian . Springer- V erlag, 1971. [25℄ I. Mo erdijk. Orbifold as group oids: an in tro dution. Contemp or ary Mathematis , 310:205222, 2002. [26℄ I. Mo erdijk. Lie group oids, gerb es, and non-Ab elian ohomology . K - The ory , 28:207258, 2003. [27℄ I. Mo erdijk and J. Mr£un. Intr o dution to F oliations and Lie Gr oup oids . Cam bridge Univ ersit y Press, 2003. [28℄ I. Mo erdijk and G. E. Rey es. Rings of smo oth funtions and their lo al- izations, I. Journal of A lgebr a , 99(2):324336, Apr. 1986. BIBLIOGRAPHY 159 [29℄ I. Mo erdijk and G. E. Rey es. Mo dels for Smo oth Innitesimal A nalysis . Springer-V erlag, New Y ork, 1991. [30℄ D. Mon tgomery and L. Zippin. T op olo gi al T r ansformation Gr oups . In- tersiene Publishers, New Y ork, 1955. [31℄ W. Rudin. F ourier A nalysis on Gr oups . John Wiley & Sons, New Y ork, 1962. [32℄ N. Saa v edra Riv ano. Caté gories T annakiennes . Num b er 265 in Leture Notes in Mathematis. Springer-V erlag, 1972. [33℄ J.-P . Serre. Lie A lgebr as and Lie Gr oups . Leture Notes in Mathematis 1500. Springer-V erlag, Berlin, Heidelb erg, 1992. Original 1964 edition: W. A. Benjamin, New Y ork. [34℄ T. T annak a. Üb er den Dualitätssatz der ni h tk omm utativ en top o- logis hen Grupp en. Thoku Math. J. (First Series) , 45:112, 1939. [35℄ W. C. W aterhouse. Intr o dution to Ane Gr oup Shemes . Springer- V erlag, 1979. [36℄ A. W einstein. Sympleti group oids and P oisson manifolds. Bul l. A mer. Math. So . (New Series) , 16:101104, 1987. [37℄ A. W einstein. Linearization of regular prop er group oids. J. Inst. Math. Jussieu , 1(3):493511, 2002. [38℄ N. T. Zung. Prop er group oids and momen tum maps: linearization, an- it y , and on v exit y . Preprin t arXiv math.SG/0407208 v4, July 24, 2006. Index ation, se e represen tation ation group oid G ⋉ M , 23 , 155 A CU onstrain ts, 39 , 65 additiv e tensor ategory , 43 algebroid, 31, 150 an ti-in v olution, 46 arro ws, manifold of -, 21 asso iativit y onstrain t, 39 a v eraging te hnique, 84 base, 22, 89 bundle of Lie groups, se e Lie bundle C ∞ -represen tation, 24, 112 , 143 C ∞ -strutured group oid, 93 , 111 C an ( C ) (ategory of anonial m ulti- funtors), 40 anonial funtional struture, se e standard C ∞ -struture anonial isomorphism, se e on- strain t anonial m ulti-funtor, 40 anonial transformation, 40 lassial bre funtor, 110 lassial represen tation, se e C ∞ - represen tation losed tensor ategory , 41 oherene onditions, 39 , 44 , 46 , 56 oherene theorem, 40 omm utativit y onstrain t, 39 omplete transv ersal, 150 omplex tensor ategory , 46 , 65 omplex tensor funtor, 46 , 65 omp osition, 21 onstrain t, 39, 44 , 46 , 56 , 65 , 66 ut-o funtion, 32 , 106 D es C ( X ′ /X ) (ategory of desen t data), 61 desen t datum, 47 , 61 dimension axiom, 77 dual, 42 , 47 em b edding, 75 en v elop e, se e T annakian group oid en v elop e homomorphism π T ( G ) , π ∞ ( G ) , 95 , 98 , 144 , 152 equiv alene of tensor ategories, 68 ev (ev aluation funtor), 113 , 150 ev R (ev aluation represen tation), 112 , 144 ev aluation funtor ev , 113 , 150 ev aluation of germs, 72 ev aluation represen tation ev R , 112 , 144 bre, 58 bre funtor, 89 lassial, 110 o v er a manifold, se e base prop er, 94 , 113 smo oth, 89 , 127 , 140 , 148 , 152 bred tensor ategory , 55 bred tensor ategory onstrain ts, 56 at map, 60 funtional struture, 92 funtionally strutured group oid, 93 funtionally strutured spae, 92 G ⋉ M (ation group oid), 23 , 155 Γ E , Γ H (sheaf of setions), 56 , 78 GL ( E ) (linear group oid), 23 , 112 group oid, 21 Hausdor, 22 160 INDEX 161 lo ally transitiv e, 81, 133 o v er a manifold, se e base prop er, 24 reexiv e, 99 , 153 regular, 150 self-dual, se e reexiv e transitiv e, 81, 133 Haar system normalized, 32 p ositiv e, 31 Hausdor group oid, 22 Hermitian form, 46 H om C X ( E , F ) (sheaf hom), 56 homomorphism of group oids, 22 , 65 , 135 , 144 in ternal hom (bifuntor), 41 in v arian t submanifold, subset, 82 , 105 in v erse, 21 in v erse image, 66 isotrop y group, 22 leaf, 150 Lie algebroid, 31 , 150 Lie bundle, 25 Lie group oid, se e group oid linear group oid GL ( E ) , 23 , 112 linear tensor ategory , 43 lo al metri, 74 lo ally nite ob jet, sheaf, 77 lo ally transitiv e group oid, 81, 133 lo ally trivial ob jet, 62, 110 main theorem, 107 manifold of arro ws, ob jets, 21 metri, 74 , 90 ω -in v arian t, 94 , 113 Morita equiv alene, 34 , 37, 67 , 101 , 119 , 140 Morita equiv alen t, se e Morita equiv- alene morphism of bre funtors, 134 m ulti-funtor, 40, 43 nondegenerate form, 47 normalized Haar system, 32 ob jets, manifold of -, 21 ω T ( G ) , ω ∞ ( G ) (forgetful funtor), 64 , 94 , 143 ω -in v arian t metri, 94 , 113 orbit, 23 , 98 orbit foliation, 150 orbit map, spae, 98 orthonormal frame, 74 , 110 paraompat, 22 parasta k, 61 π T ( G ) , π ∞ ( G ) (en v elop e homomorph- ism), 95 , 98 , 144 p ositiv e Haar system, 31 presta k, 56 prinipal submanifold, 114 prop er bre funtor, 94 , 113 prop er group oid, 24 pullba k along a smo oth map, 45, 55, 80 of represen tations, 66 of smo oth Hilb ert elds, 80 R (olletion of represen tativ e fun- tions), 90 R ∞ (anonial funtional struture on the T annakian group oid), 92 R T ( G ) (ategory of t yp e T represen- tations), 63 R ∞ ( T ; k ) (ategory of smo oth repre- sen tations on v etor bundles), 24 , 112 , 143 rank, 42 renemen t, 61 reexiv e, 99 , 153 regular group oid, 150 represen tation C ∞ - or smo oth, 24, 112 , 143 lassial, se e C ∞ - or smo oth of t yp e T , 63 represen tativ e  hart, 127 , 147 represen tativ e funtion, 90 162 INDEX rigid tensor ategory , 42, 47 , 110 saturation, 105 setion, 56 , 78 self-onjugate, 48 , 89 self-dual, se e reexiv e sesquilinear form, 46 sheaf hom H om C X ( E , F ) , 56 sheaf of setions Γ E , Γ H , 56 , 78 slie, 32 smo oth Eulidean eld, 81 smo oth bre funtor, 89, 127 , 140 , 148 , 152 smo oth Hilb ert eld, 78 smo oth represen tation, se e C ∞ - represen tation smo oth setion, se e setion smo oth tensor parasta k, 61 smo oth tensor presta k, 59 smo oth tensor sta k, 61 soure, 21 sta k, 61 sta k of smo oth elds, 71 standard C ∞ -struture R ∞ , 92 , 111 standard bre funtor (lassial) ω ∞ ( G ) , 143 (of t yp e T ) ω T ( G ) , 64 , 94 struture maps, 21 T T ( G ) , T ∞ ( G ) (T annakian group oid asso iated with a Lie group- oid), 98 , 143 T ( ω ) (T annakian group oid asso i- ated with a bre funtor), 89 , 111 tame submanifold, 116 , 145 T annakian group oid (lassial) T ∞ ( G ) , 143 (of t yp e T ) T T ( G ) , 98 T ( ω ) , 89 , 111 target, 21 tensor ategory , 39 , 65 additiv e, 43 losed, 41 linear, 43 rigid, 42 , 47 , 110 tensor equiv alene, 68 , 137 tensor funtor, 44 , 65, 66 tensor funtor onstrain ts, 44 , 66 tensor parasta k, 61 tensor preserving, 45 tensor presta k, 56 smo oth, 59 tensor pro dut of Hilb ert spaes, 79 of smo oth Hilb ert elds, 80 tensor sta k, 61 tensor struture, se e tensor ategory tensor unit 1 , 39 tensor unit onstrain ts, 39 top ologial group oid, 23 trae, 42 transformation, 67 transitiv e group oid, 81, 133 translation group oid, se e ation group oid transv ersal (omplete), 150 trivial ob jet, 62 trivialization, 62 t yp e, 63 unit map, setion, 21 V C ( X ) (sub ategory of lo ally trivial ob jets), 62 V ∞ ( X ; k ) , V ∞ ( X ) (ategory of v e- tor bundles), 43 , 110 v alue, 58 v ertex group, se e isotrop y group w eak equiv alene, 137 , 152 w eak pullba k, 68

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Leave a Comment