A Tannaka Theorem for Proper Lie Groupoids

A T annak a Theorem for Prop er Lie Group oids Giorgio T ren tinaglia ∗ Abstrat By replaing the ategory of smo oth v etor bundles o v er a manifold with the ategory of what w e all smo oth Eulidean elds, whi h is a prop er enlargemen t of the former, and b y onsidering smo oth a- tions of Lie group oids on smo oth Eulidean elds, w e are able to pro v e a T annak a dualit y theorem for prop er Lie group oids. The notion of smo oth Eulidean eld w e in tro due here is the smo oth, nite dimen- sional analogue of the usual notion of on tin uous Hilb ert eld. In tro dution Classial T annak aKre   n dualit y theory leads to the result that a ompat group an b e reonstruted from a purely disrete, algebrai ob jet, namely the ring of its on tin uous nite dimensional represen tations or, more pre- isely , the algebra of its represen tativ e funtions. Compare [ 2℄. The same theory an b e eien tly reast in ategorial terms. This alternativ e p oin t of view on T annak a dualit y stems from Grothendie k's theory of motiv es in algebrai geometry [18, 6, 5℄. In this approa h one starts b y onsidering, for an arbitrary lo ally ompat group G , the ategory formed b y the on- tin uous represen tations of G on nite dimensional v etor spaes, endo w ed with the symmetri monoidal struture arising from the usual tensor pro d- ut of represen tations, and then one tries to reo v er G as the group of all tensor preserving natural endomorphisms of the standard forgetful funtor whi h assigns ea h G -mo dule the underlying v etor spae. See for instane [9℄. When G is a ompat Lie group, in partiular, it follo ws that G an b e reonstruted in this w a y up to isomorphism, as the C ∞ manifold struture of a Lie group is determined b y the underlying top ology . It is natural to ask for a generalization of the aforesaid dualit y theory to the realm of Lie group oids, in whi h prop er group oids are exp eted to pla y ∗ During the preparation of this pap er, the author w as partially supp orted b y a gran t of the foundation F ondazione Ing. Aldo Gini . 1 the same role as ompat groups. When trying to extend T annak a theory from Lie groups to Lie group oids, ho w ev er, one is rst of all onfron ted with the problem of  ho osing a suitable notion of represen tation for the latter. No w, the notion of smo oth or, equiv alen tly , on tin uous nite dimensional represen tation has an ob vious naiv e generalization to the Lie group oid set- ting: the represen tations of a Lie group oid G ould b e dened to b e the Lie group oid homomorphisms G → GL ( E ) of G in to the linear group oids asso i- ated with smo oth v etor bundles o v er its base manifold. Unfortunately , this naiv e generalization turns out to b e inadequate for the purp ose of arrying forw ard T annak a dualit y to Lie group oids, fr [19 , 11 , 8℄. This state of aairs fores us to in tro due a dieren t notion of represen tation for Lie group oids. It seems reasonable to require that the new notion should b e as lose as p ossible to the naiv e notion realled ab o v e, and that moreo v er in the ase of groups one should reo v er the usual notion of smo oth represen tation on a nite dimensional v etor spae. In this pap er w e w ork out the problems raised in the preeding para- graph. 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 Eulidean elds o v er X . Our notion of smo oth Eulidean eld is the analogue, in the smo oth and nite dimensional setting w e onne ourselv es to, of the notion of on tin uous Hilb ert eld in tro dued b y Dixmier and Douady [7℄. The ategory of smo oth Eulidean elds o v er X is, for ev ery paraompat manifold X , a prop er enlargemen t of the at- egory of smo oth v etor bundles o v er X . One an straigh tforw ardly dene a notion of represen tation of a Lie group oid on a smo oth Eulidean eld; su h represen tations form, for ea h Lie group oid G , a symmetri monoidal ategory whi h is onneted to the ategory of smo oth Eulidean elds o v er the base manifold of G b y a anonial forgetful funtor. F rom this fun- tor w e obtain, b y generalizing the onstrution men tioned at the b eginning, a group oid. This reonstruted group oidthe T annakian group oid of G , as w e all itomes equipp ed with a natural andidate for a dieren tiable struture on its spae of arro ws, namely a sheaf of algebras of on tin uous real v alued funtions stable under omp osition with arbitrary smo oth fun- tions of sev eral v ariables. A spae endo w ed with su h a struture onstitutes what w e all a C ∞ -spae. There is a anonial homomorphism of G in to its T annakian group oid, whi h pro v es to b e also a morphism of C ∞ -spaes. No w, our dualit y result (Theorem 8.9 ) an b e stated as follo ws: Theorem F or a prop er Lie group oid G , the anonial homomorphism of G in to its T annakian group oid is an isomorphism of b oth group oids and C ∞ -spaes. It follo ws that the T annakian group oid itself is a Lie group oid, isomorphi to G . 2 Our argumen t is omplemen tary to the pro of of the lassial T annak a dualit y theorem. Most eorts are direted to w ards sho wing ho w the lassial theo- rem implies the surjetivit y of the ab o v e-men tioned anonial isomorphism and then to w ards establishing the laim ab out the C ∞ -spae struture. By on trast, the fat that the anonial homomorphism is injetiv e is a diret onsequene of a theorem of N. T. Zung [20℄; Zung's theorem ma y in fat b e regarded as a P eter W eyl theorem for prop er Lie group oids. Compare [19℄. Man y of the reasonings leading to our dualit y theorem, although of ourse not all of them, also apply to the represen tations of prop er Lie group oids 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 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 this demand w as the theory of smo oth tensor sta ks. Smo oth v etor bundles and smo oth Eulidean elds are t w o examples of a smo oth tensor sta k. Ea h smo oth tensor sta k giv es rise to a orresp onding notion of represen tation for Lie group oids, and 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. What this group oid lo oks lik e will dep end v ery m u h, in general, on the initial  hoie of a smo oth tensor sta k, as w e p oin ted out already in the ourse of this in tro dution. A kno wledgements. The problem of pro ving a T annak a dualit y the- orem for prop er Lie group oids w as originally suggested to the author b y I. Mo erdijk, who also made sev eral useful remarks on an earlier draft of this pap er. Besides, the author w ould lik e to thank M. Craini and N. T. Zung for helpful disussions. Con ten ts 1 Prop er Lie Group oids . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 The Language of T ensor Categories . . . . . . . . . . . . . . . . . . 9 3 Smo oth T ensor Sta ks . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 F oundations of Represen tation Theory . . . . . . . . . . . . . . . . 21 5 Smo oth Eulidean Fields . . . . . . . . . . . . . . . . . . . . . . . . 26 6 Constrution of Equiv arian t Maps . . . . . . . . . . . . . . . . . . . 32 7 C ∞ Fibre F untors . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 8 Pro of of the Reonstrution Theorem . . . . . . . . . . . . . . . . . 41 3 1 Prop er Lie Group oids The presen t setion is in tro dutory . Its purp ose is to reall some ba kground notions and to x some notation that w e will b e using throughout the pap er. The reader is advised to onsult [14 , 1, 3, 20℄ for referene; other soures inlude [17℄ and [4℄. The term group oid refers to a small ategory where ev ery arro w is in v ertible. A Lie group 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 ob jets and manifold of arro ws, and a list of smo oth maps alled struture maps. The basi items in this list are the soure map s : G (1) → G (0) and the target map t : G (1) → G (0) , whi h 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 in v erse map i : G (1) → G (1) , for whi h the familiar algebrai la ws m ust b e satised. T erminolo gy and Notation: The p oin ts x = s ( g ) and x ′ = t ( g ) are resp. alled the soure and the target of the arro w 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 ′ , and w e use the abbreviation G | x for the isotrop y or v ertex 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 paraompa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 op en o v ers). 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 study , 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. 4 Note that w e do not require the manifold of arro ws G (1) to b e Hausdor or paraompat. A tually , this manifold is neither Hausdor nor seond oun table in man y examples of in terest. 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 mor e T erminolo gy: The manifold G (0) is usually alled the base 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 (1.1) 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) . Example: Let G b e a Lie group ating smo othly (from the left) on a manifold M . Then the ation (or translation) group oid G ⋉ M is dened to b e 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 maps 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 (1.2) ( g ′ , x ′ )( g , x ) = ( g ′ g , x ) . There is a similar onstrution M ⋊ G asso iated with righ t ations. A homomorphism of Lie group 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 . A homomorphism ϕ : G → H is said to b e a Morita equiv alene when G (1) ( s , t )   ϕ (1) / / H (1) ( s , t )   G (0) × G (0) ϕ (0) × ϕ (0) / / H (0) × H (0) (1.3) is a pullba k diagram in the ategory of C ∞ manifolds and the map (1.4) t H ◦ pr 2 : G (0) ϕ (0) × s H H (1) → H (0) is a surjetiv e submersion. 5 There is also a notion of top ologial group 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, sine 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 . 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) through x is the subset (1.5) 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 to G x . The follo wing fats hold, fr [ 14℄ p. 115: (a) G ( x, x ′ ) is a losed submanifold of G (1) (b) G x = G | x is a Lie group () the G -orbit through x is an immersed submanifold of G (0) and the target map t : G x → G x determines a prinipal G x -bundle o v er it (the set G x an ob viously b e iden tied with the homogeneous spae G x /G x , and 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). 1.1. C ∞ -Sp a es. Reall that a funtionally strutured spae is a top ologial spae X endo w ed with a sheaf F of real algebras of on tin uous real v alued funtions on X (funtional struture). Compare for instane [1℄, p. 297. There is an ob vious notion of morphism for su h spaes. 1 Let F b e an arbitrary funtional struture on a top ologial spae X . W e shall let F ∞ denote the sheaf of on tin uous real v alued funtions on X generated b y the presheaf (1.6) U 7→  f ( a 1 | U , . . . , a d | U ) : f : R d → R of lass C ∞ , a 1 , . . . , a d ∈ F ( U )  . Here the expression f ( a 1 | U , . . . , a d | U ) stands of ourse for the funtion u 7→ f  a 1 ( u ) , . . . , a d ( u )  on U . The pair ( X , F ∞ ) onstitutes a funtionally stru- tured spae to whi h w e shall refer as a C ∞ funtional ly strutur e d sp a e or, in short, C ∞ -sp a e. More orretly , a C ∞ -spae is a funtionally strutured spae ( X , F ) su h that F = F ∞ . 2 Observ e that smo oth manifolds an b e 1 Algebrai geometers w ould sa y that a morphism of funtionally strutured spaes is a (on tin uous) mapping induing a morphism of (lo ally) ringed spaes. 2 A more general notion of  C ∞ -ring w as in tro dued b y Mo erdijk and Rey es in the on text of smo oth innitesimal analysis [ 16 , 17 ℄. What w e are onsidering here is a sp eial instane of that notion, namely a C ∞ -ring of on tin uous funtions on a top ologial spae. F or simpliit y , w e  ho ose to w ork within the sub ategory of su h C ∞ -rings. 6 dened as top ologial spaes endo w ed with a C ∞ funtional struture lo ally isomorphi to that of smo oth funtions on R n . C ∞ -Spaes ha v e, in general, b etter ategorial prop erties than smo oth manifolds. Note that the latter form, within C ∞ -spaes, a full sub ategory . 1.2. C ∞ -Gr oup oids. Let us start b y observing that if ( X , F ) is a C ∞ -spae then so is ( S, F | S ) for an y subspae S of X , where F | S = i S ∗ F denotes the in v erse image of F along the inlusion i S : S ֒ → X . [Reall that, for an arbitrary on tin uous mapping f : S → T in to a funtionally strutured spae ( T , T ) , f ∗ T denotes the funtional sheaf on S formed b y the funtions whi h are lo ally the pullba k along f of funtions in T .℄ W e note next that if ( X , F ) and ( Y , G ) are t w o funtionally 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 ) . It follo ws that ( F ∞ ⊗ G ∞ ) ∞ is a C ∞ funtional struture on X × Y turning this in to the pro dut of ( X , F ∞ ) and ( Y , G ∞ ) in the ategory of C ∞ -spaes. The pre- eding onsiderations imply that the ategory of C ∞ -spaes is losed under bred pro duts (pullba ks). Notie that when X and Y are smo oth manifolds and S ⊂ X is a submanifold one reo v ers the orret manifold strutures, so that all these onstrutions for C ∞ -spaes agree with the usual ones on manifolds whenev er the latter mak e sense. W e shall use the term C ∞ -gr oup oid to indiate a group oid whose sets of ob jets and arro ws are ea h endo w ed with the struture of a C ∞ -spae so that all the maps arising from the group oid struture (soure, target, omp osition, unit setion, in v erse) are morphisms of C ∞ -spaes. The base spae X is alw a ys a smo oth manifold in pratie, with C ∞ funtional struture giv en b y the sheaf of smo oth funtions on X . Ev ery Lie group oid is, in partiular, an example of a C ∞ -group oid. A Lie (or top ologial or C ∞ ) group oid G is said to b e 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). When G is a prop er Lie group oid o v er a manifold M , ev ery G - orbit is in fat a losed submanifold of M . Normalized Haar systems on prop er Lie group oids are the analogue of Haar probabilit y measures on ompat Lie groups. W e no w reall the de- nition and the onstrution of Haar systems on prop er Lie group oids. Our exp osition is based on [3℄. Let G b e a Lie group oid o v er a manifold M . 1.3 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 supp orted b y the resp etiv e soure bre G x = G ( x, - ) = s − 1 ( x ) , satisfying the follo wing onditions: 7 i) R ϕ d µ x > 0 for all nonnegativ e real ϕ ∈ C c ∞ ( G x ) with ϕ 6 = 0 ; ii) for ea h ϕ ∈ C c ∞ ( G (1) ) the funtion Φ on M dened b y setting (1.7) Φ( x ) = Z G x ϕ | G x dµ x is of lass C ∞ ; iii) (righ t in v ariane) for all g ∈ G ( x, y ) and ϕ ∈ C c ∞ ( G x ) one has (1.8) Z G y ϕ ◦ τ g d µ y = Z G x ϕ d µ x where τ g : G ( y , - ) → G ( x, - ) denotes righ t translation h 7→ hg . The existene of p ositiv e Haar systems on a Lie group oid G an b e es- tablished when G is prop er. 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 [3℄ or Chapter 6 of [14℄; note the use of paraom- patness). Righ t translations determine isomorphisms T G ( x, - ) ≈ t ∗ g | G ( x, - ) for all x ∈ M . These isomorphisms an b e used to indue, on the soure bres G ( x, - ) , Riemann metris whose asso iated v olume densities pro vide the desired system of measures. 1.4 Denition A normalize d Haar system on G is a family of p ositiv e Radon measures { µ x } ( x ∈ M ), ea h one with supp ort onen trated in the resp etiv e soure bre G x , enjo ying 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 (1.9) C ∞ ( G x ) ⊂ L 1 ( µ x ) (b) Condition ii) , resp etiv ely iii) of the preeding denition holds for an arbitrary smo oth funtion ϕ on G (1) , resp etiv ely G x () The follo wing nor- malization ondition is satised: i*) R d µ x = 1 for ev ery x ∈ M . Ev ery prop er Lie group oid admits normalized Haar systems. 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 of the group oid su h that the soure map 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 an arbitrary p ositiv e Haar system  hosen in adv ane. The system of p ositiv e measures µ x ≡ ( c ◦ t ) ν x will ha v e the desired prop erties. 8 1.5. Zung's the or em. 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 a 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. Note that if N is a submanifold of M and 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 . F rom this remark it follo ws that for ea h submanifold N the subset of all p oin ts at whi h N is a slie forms an op en subset of N . If a submanifold S of M is a slie then the in tersetion s − 1 ( S ) ∩ t − 1 ( S ) is transv ersal, so that the restrition G | S is a Lie group oid o v er S ; moreo v er, G · S is an in v arian t op en subset of M . F or the pro of of the follo wing result, w e refer the reader to [20℄. Theorem (N. T. Zung) Let G b e a prop er Lie group oid. Let x b e a base p oin t whi h is not mo v ed b y the tautologial ation of G on its o wn base. Then there exists a on tin uous linear represen tation G → GL ( V ) of the isotrop y group G = G | x 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 one an nd an isomorphism of Lie group oids G | U ≈ G ⋉ V whi h mak es x orresp ond to zero. R emark: Consider t w o slies S, S ′ in M with, let us sa y , dim S ≦ dim S ′ . Supp ose g ∈ G ( S, S ′ ) . Put x = s ( g ) ∈ S and x ′ = t ( g ) ∈ S ′ . It is not diult to see that there is a smo oth target setion τ : B → G (1) dened o v er some op en neigh b ourho o d B of x ′ in S ′ su h that τ ( x ′ ) = g and the omp osite map s ◦ τ indues a submersion of B on to an op en neigh b ourho o d of x in S . Th us, when G is prop er, it follo ws from the preeding theorem that for ea h p oin t x ∈ M there are 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 in M for whi h there exists 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 the origin of V to x . 2 The Language of T ensor Categories This setion onsists of t w o parts. The rst one on tains a rather onise aoun t of some basi standard ategorial notions, a detailed exp osition of whi h an b e found for example in [ 12, 6, 5 ℄. In the seond part, and preisely from 2.2 on w ards, w e establish a ouple of fundamen tal prop ositions for later use in the pro of of our reonstrution theorem (Theorem 8.9 ). A tensor strutur e on a ategory C onsists of the follo wing data: (2.1) a bifuntor ⊗ : C × C − → C , a distinguished ob jet 1 ∈ Ob( C ) 9 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 (2.2) satisfying MaLane's  oherene onditions (f. for example MaLane [12℄, pp. 157 . and esp eially p. 180 for a detailed aoun t). A tensor  ate gory is a ategory endo w ed with a tensor struture. In the terminology of [12℄, the presen t notion orresp onds to that of  symmetri monoidal ategory. The natural isomorphism α , resp. γ is alled the asso iativit y , resp. omm utativit y onstrain t; λ and ρ are alled the (tensor) unit onstrain ts. In pratie, w e shall deal exlusiv ely with omplex tensor ategories. Reall that a k -linear ategory , where k is an y n um b er eld, is a ategory C whose hom-sets are ea h endo w ed with a struture of v etor spae o v er k with resp et to whi h omp osition of arro ws is bilinear. One also sa ys that C is a ategory endo w ed with a k -linear struture. A k -linear tensor ategory is a tensor ategory endo w ed with a k -linear struture su h that the bifuntor ⊗ is bilinear. F rom no w on, in this pap er,  tensor ategory will mean  additiv e C -linear tensor ategory. Th us, in partiular, there will b e a zero ob jet and for all ob jets R, S there will b e a diret sum R ⊕ S . Let C , C ′ b e tensor ategories. A tensor funtor of C in to C ′ is obtained b y atta hing, to an ordinary funtor F : C → C ′ , t w o isomorphisms τ R,S : F ( R ) ⊗ F ( S ) ∼ → F ( R ⊗ S ) (natural in R, S ) and υ : 1 ′ ∼ → F ( 1 ) , (2.3) alled tensor funtor onstrain ts, whi h are required to satisfy ertain ondi- tions expressing their ompatibilit y with the A CU onstrain ts of the tensor ategories C and C ′ . The reader is referred to lo . it. for a disussion of these onditions. Reall that a funtor of k -linear ategories is said to b e linear if the indued maps of hom-sets are k -linear. A linear funtor b et w een additiv e k -linear ategories will preserv e zero ob jets and diret sums. W e agree that an assumption of linearit y on the funtor F : C → C ′ is part of our denition of the notion of tensor funtor. Let F , F ′ b e tensor funtors of C in to C ′ . A natural transformation λ : F → F ′ is said to b e tensor pr eserving if the follo wing diagrams omm ute: F ( R ) ⊗ F ( S ) τ R,S   λ ( R ) ⊗ λ ( S ) / / F ′ ( R ) ⊗ F ′ ( S ) τ ′ R,S   1 ′ υ   1 ′ υ ′   F ( R ⊗ S ) λ ( R ⊗ S ) / / F ′ ( R ⊗ S ) F ( 1 ) λ ( 1 ) / / F ′ ( 1 ) . (2.4) 10 The olletion of all tensor preserving natural transformations F → F ′ will b e denoted b y Hom ⊗ ( F , F ′ ) . Note that an y natural transformation of F in to F ′ is neessarily additiv e i.e. satises λ ( R ⊕ S ) = λ ( R ) ⊕ λ ( S ) . 2.1. T ensor*  ate gories. By an an ti-in v olution on a tensor ategory C w e mean an an ti-linear tensor funtor (2.5) ∗ : C → C , R 7→ R ∗ with the prop ert y that there exists a tensor preserving natural isomorphism (2.6) ι R : R ∗∗ ∼ → R with ι ( R ∗ ) = ι ( R ) ∗ . By xing one su h isomorphism one and for all, one obtains a mathematial struture whi h shall here b e referred to as a tensor*  ate gory. A morphism of tensor* ategories, or tensor* funtor, is obtained b y atta hing, to an ordinary (linear) tensor funtor F , a tensor preserving natural isomorphism (2.7) ξ R : F ( R ) ∗ ∼ → F ( R ∗ ) su h that the follo wing diagram omm utes: F ( R ) ∗∗ ∼ = ∗ / / ∼ = & & N N N N N N F ( R ∗ ) ∗ ∼ = / / F ( R ∗∗ ) F ( ∼ = ) x x p p p p p p F ( R ) . (2.8) A morphism of tensor* funtors, b etter kno wn as a self-onjugate tensor preserving natural transformation, is dened to b e a tensor preserving natural transformation making the follo wing diagram omm utativ e: F ( R ) ∗ ξ R   λ ( R ) ∗ / / F ′ ( R ) ∗ ξ ′ R   F ( R ∗ ) λ ( R ∗ ) / / F ′ ( R ∗ ) . (2.9) W e write Hom ⊗ , ∗ ( F , F ′ ) for referene to su h transformations. 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 also maps 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 ∗ 11 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 V e c C with V ∗∗ = V . Example: ve tor bund les. By a diret generalization of the preeding onstrution one obtains the tensor* ategory V e c C ∞ ( M ) of smo oth omplex v etor bundles (of lo ally nite rank) o v er a smo oth manifold M . W e shall iden tify V e c C ∞ ( ⋆ ) , where ⋆ denotes the one-p oin t manifold, with the tensor* ategory V e c C in tro dued ab o v e. Notie that the pullba k of v etor bundles along a smo oth mapping f : N → M determines an ob vious tensor* funtor f ∗ of V e c C ∞ ( M ) in to V e c C ∞ ( N ) . Let C b e a tensor* ategory . By a  real struture on an ob jet R ∈ Ob( C )  w e mean an isomorphism µ : R ∼ → R ∗ in C su h that the omp osite µ ∗ · µ equals the iden tit y on R mo dulo the anonial iden tiation R ∗∗ ∼ = R pro vided b y (2.6 ). Let R ( C ) denote the ategory whose ob jets are the pairs ( R, µ ) onsisting of an ob jet R ∈ Ob( C ) together with a real struture µ on R and whose morphisms a : ( R, µ ) → ( S, ν ) are the morphisms a : R → S in C su h that ν · a = a ∗ · µ . Note that R ( C ) is naturally an R -linear ategory; further, there is an ob vious indued tensor struture on it, whi h turns it 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, 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 real struture µ : U ∼ → U ∗ on a omplex v etor spae U determines the real eigenspae U µ ⊂ U of all µ -in v arian t v etors. There is an analogous equiv alene b et w een V e c R ∞ ( M ) and R  V e c C ∞ ( M )  , for ea h smo oth manifold M . Notie that an y tensor* funtor F : C → D indues an ob vious R -linear tensor funtor R ( F ) : R ( C ) → R ( D ) . F or an y tensor* funtors F , G : C → D , the map λ 7→ ˜ λ where ˜ λ ( R, µ ) ≡ λ ( R ) pro vides a bijetion (2.10) Hom ⊗ , ∗ ( F , G ) ∼ → Hom ⊗  R ( F ) , R ( G )  b et w een the self-onjugate tensor preserving transformations F → G and the tensor preserving transformations R ( F ) → R ( G ) . Indeed, b y exploiting the additivit y of the tensor* ategory C , one an onstrut a funtor C → R ( C ) whi h pla ys the same role as the funtor that assigns a omplex v etor spae the underlying real v etor spae: one  ho oses, for ea h pair R, S of ob jets of C , a diret sum R ⊕ S ; then the ob vious isomorphism R ⊕ R ∗ ≈ ( R ⊕ R ∗ ) ∗ is a real struture on R ⊕ R ∗ . Observ e that the funtor R ( C ) → C , ( R, µ ) 7→ R has an analogous in terpretation. One therefore sees that the formalism of tensor* ategories is essen tially equiv alen t to that of real tensor ategories. 12 The next results are original. They will b e put to use only in the nal setion of this pap er, in the pro of of the reonstrution theorem. 2.2. T erminolo gy. Let C b e a tensor* ategory and F : C → V e c C a tensor* funtor with v alues in to (nite dimensional) omplex v etor spaes. Let H b e a top ologial group, and supp ose a homomorphism of monoids is giv en (2.11) π : H − → End ⊗ , ∗ ( F ) . W e sa y that π is  ontinuous if for ev ery ob jet R ∈ Ob( C ) the indued represen tation (2.12) π R : H − → End( F ( R )) dened b y setting π R ( h ) = π ( h )( R ) is on tin uous. 2.3 Prop osition Let C , F , H and π b e as in 2.2, with π on tin uous, and supp ose, in addition, that H is a ompat Lie group. Assume that the follo wing ondition is satised: (*) for ea h pair of ob jets R, S ∈ Ob( C ) , and for ea h H -equiv arian t homomorphism A : F ( R ) → F ( S ) , there exists some arro w R a − → S in C with A = F ( a ) . Then the homomorphism π is surjetiv e; in partiular, the monoid End ⊗ , ∗ ( F ) is a group. Pr o of Put K = Ker π ⊂ H . This is a losed normal subgroup, b eause it oinides with the in tersetion T Ker π R o v er all ob jets R of C . On the quotien t G = H/K there is a unique (ompat) Lie group struture su h that the quotien t homomorphism H → G b eomes a Lie group homomorphism. Ev ery π R an b e indieren tly though t of as a on tin uous represen tation of H or a on tin uous represen tation of G , and ev ery linear map A : F ( R ) → F ( S ) 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 R is also smo oth. W e laim there exists an ob jet R 0 of C su h that the orresp onding π R 0 is faithful as a represen tation of G . Indeed, b y the ompatness of the Lie group G , w e an nd R 1 , . . . , R ℓ ∈ Ob( C ) with the prop ert y that (2.13) Ker π R 1 ∩ · · · ∩ Ker π R ℓ = { e } , where e denotes the unit of G ; ompare [2℄, p. 136. Then, if w e set R 0 = R 1 ⊕ · · · ⊕ R ℓ , the represen tation π R 0 will b e faithful b eause of the existene of an ob vious isomorphism of G -mo dules (2.14) F ( R 1 ⊕ · · · ⊕ R ℓ ) ≈ F ( R 1 ) ⊕ · · · ⊕ F ( R ℓ ) . 13 No w, it follo ws that the G -mo dule F ( R 0 ) is a generator for the tensor* ategory R ep C ( G ) of all 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 0 ) ⊗ k ⊗ ( F ( R 0 ) ∗ ) ⊗ ℓ , see for instane [2℄, p. 137. Sine ea h π ( h ) is, b y assumption, self-onjugate and tensor preserving, this tensor p o w er will b e naturally isomorphi to F  R 0 ⊗ k ⊗ ( R 0 ∗ ) ⊗ ℓ  as a G -mo dule and hene for ea h ob jet V of R ep C ( G ) there will b e some ob jet R of C su h that V em b eds in to F ( R ) as a submo dule. Next, onsider an arbitrary natural transformation λ ∈ End( F ) . Let R b e an ob jet of the ategory C , and let V ⊂ F ( R ) b e a submo dule. The  hoie of a omplemen t to V in F ( R ) determines an endomorphism of mo dules P V : F ( R ) → V ֒ → F ( R ) whi h, b y the assumption (*) , omes from some endomorphism of R in C . This implies that the linear op erators λ ( R ) and P V on the spae F ( R ) omm ute with one another and, onsequen tly , that λ ( R ) maps the subspae V in to itself. W e will usually omit an y referene to R and write simply λ V for the linear map that λ ( R ) indues on V b y restrition. Note nally that, giv en another submo dule W ⊂ F ( S ) and an equiv arian t map B : V → W , one alw a ys has (2.15) B · λ V = λ W · B . T o pro v e this iden tit y , one rst extends B to an equiv arian t map F ( R ) → F ( S ) and then in v ok es (*) as b efore. Let F G denote the tensor* funtor R ep C ( G ) − → V e c C that assigns ea h G -mo dule the underlying v etor spae. As our next step, w e will dene an isomorphism of omplex algebras (2.16) θ : End( F ) ∼ → End( F G ) so that the follo wing diagram omm utes H pro j.   π / / End( F ) ≃ θ   G π G / / End( F G ) , (2.17) where π G ( g ) is, for ea h g ∈ G , the natural transformation of F G in to it- self that assigns left m ultipliation b y g on V to ea h G -mo dule V . F or ea h G -mo dule V there exists an ob jet R of C together with an em b ed- ding V ֒ → F ( R ) , so w e ould dene θ ( λ )( V ) as the restrition λ V of λ ( R ) to V . Of ourse, it is neessary to  he k that this do es not dep end on the  hoies in v olv ed. Let t w o ob jets R, S ∈ Ob( C ) b e giv en along with t w o 14 equiv arian t em b eddings of V in to F ( R ) , F ( S ) . Sine it is alw a ys p ossible to em b ed ev erything equiv arian tly in to F ( R ⊕ S ) without aeting the indued λ V , it is no loss of generalit y to assume R = S . No w, it follo ws from the remark (2.15 ) ab o v e that the t w o em b eddings atually determine the same linear endomorphism of V . This sho ws that θ is w ell-dened. ( 2.15 ) also implies that θ ( λ ) ∈ End( F G ) . On the other hand put, for µ ∈ End ( F G ) and R ∈ Ob( C ) , µ F ( R ) = µ ( F ( R ) ) . Then µ F ∈ End ( F ) and θ ( µ F ) = µ , b eause of the existene of em b eddings of the form V ֒ → F ( R ) and b eause of the naturalit y of µ . This sho ws that θ is surjetiv e, and also injetiv e sine λ ( R ) = θ ( λ )( F ( R )) . Finally , it is straigh tforw ard to  he k that ( 2.17 ) omm utes. In order to onlude the pro of it will b e enough to  he k that θ indues a bijetion b et w een End ⊗ , ∗ ( F ) and End ⊗ , ∗ ( F G ) , for then our laim that π is surjetiv e will follo w immediately from the omm utativit y of (2.17 ) and the lassial T annak aKre   n dualit y theorem for ompat groups (whi h sa ys that π G establishes a bijetion b et w een G and End ⊗ , ∗ ( F G ) ; see for example [9℄ or [2℄ for a pro of ). This an safely b e left to the reader. q.e.d. The argumen t w e used in the foregoing pro of to onstrut a generator tells us something in teresting ev en in the nonompat ase. 2.4 Prop osition Let C and F b e as in 2.2 . Let G b e a Lie group, not neessarily ompat, and let π : G → End( F ) b e a faithful on- tin uous homomorphism. Then there exists an ob jet R 0 ∈ Ob( C ) su h that Ker π R 0 is a disrete subgroup of G or, equiv alen tly , su h that the represen tation (2.18) π R 0 : G → GL ( F ( R 0 )) is faithful in some op en neigh b ourho o d of the unit of G . Pr o of By indution. q.e.d. 3 Smo oth T ensor Sta ks In this setion w e shall in tro due the language of smo oth sta ks of tensor* ategories or, in short, smo oth tensor sta ks. W e prop ose this language as a new foundational framew ork for the represen tation theory of group oids. No w ada ys, man y onise aoun ts of the general theory of sta ks are a v ail- able; our o wn exp osition is based on [6℄ and [13 ℄. A fairly extensiv e treatmen t of the algebrai geometri theory an b e found in [10 ℄. 15 Over al l Conventions: The apital letters X , Y , Z denote C ∞ manifolds and the orresp onding lo w er-ase letters x, x ′ , . . . , y et. denote p oin ts on these manifolds. ` C ∞ X ' stands for the sheaf of smo oth funtions on X ; w e sometimes omit the subsript. W e refer to shea v es of C ∞ X -mo dules also as shea v es of mo dules o v er 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. 3.1. Fibr e d tensor  ate gories. Fibred tensor ategories shall b e denoted b y apital Gothi t yp e v ariables. A bred tensor ategory C assigns, to ea h smo oth manifold X , a tensor* ategory (3.1) C ( X ) =  C ( X ) , ⊗ X , 1 X , ∗ X  or  C ( X ) , ⊗ , 1 , ∗  for shortw e omit subsripts when they are retriev able from the on textand, to ea h smo oth map X f − → Y , a tensor* funtor (3.2) f ∗ : C ( Y ) − → C ( X ) whi h w e all 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 tensor ategory pro vides self-onjugate tensor preserving natural isomorphisms (3.3) ( δ : f ∗ ◦ g ∗ ∼ → ( g ◦ f ) ∗ ε : Id C ( X ) ∼ → id X ∗ whi h altogether are 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   p p p p p p p p p p p p p p p p p p p p p p p p p p ε · f ∗ o o ( g f ) ∗ h ∗ δ / / ( hg f ) ∗ f ∗ f ∗ id Y ∗ . δ o o (3.4) These are the only primitiv e data w e need to in tro due in order to b e able to sp eak ab out smo oth tensor sta ks and represen tations of Lie group oids. The latter onepts anand willb e dened  anoni al ly, i.e. purely in terms of the bred tensor ategory struture. 3.2. T ensor pr estaks. Let C b e an arbitrary bred tensor ategory . Let i U : U ֒ → X denote the inlusion of an op en subset. W e shall put, for ev ery ob jet E and morphism a of the ategory C ( X ) , E | U = i U ∗ E and a | U = i U ∗ a . More generally , w e shall mak e use of the same abbreviations for the inlusion i S : S ֒ → X of an arbitrary submanifold. 16 F or ea h pair of ob jets E , F ∈ Ob C ( X ) , let H om C X ( E , F ) denote the presheaf of omplex v etor spaes o v er X dened as (3.5) U 7→ Hom C ( U ) ( E | U , F | U ) where the restrition map orresp onding to an op en inlusion j : V ֒ → U is giv en [ob viously , up to anonial isomorphism℄ b y a 7→ j ∗ a . No w, the requiremen t that C is a presta k means exatly that ev ery su h presheaf is in fat a sheaf. This en tails, in partiular, that one an glue an y family of ompatible lo al morphisms o v er X . One sp eial ase, namely the sheaf Γ E = H om C X ( 1 , E ) , to whi h w e shall refer as the she af of se tions of E , will b e of ma jor in terest for us. F or an y op en subset U , the elemen ts of the v etor spae Γ E ( U ) shall b e referred to as se tions of E over U . Sine a morphism a : E → F in C ( 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 (3.6) Γ = Γ X : C ( X ) − → { shea v es of C X - mo dules } , where C X denotes the onstan t sheaf o v er X of v alue C . This funtor is ertainly linear. Moreo v er, there is a anonial w a y to mak e it a pseudo-tensor funtor of the tensor ategory  C ( X ) , ⊗ X , 1 X  in to the ategory of shea v es of C X -mo dules (with the familiar tensor struture): 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 Γ E ( U ) × Γ F ( U ) → Γ ( E ⊗ F )( U ) , ( a, b ) 7→ a ⊗ b [mo d ∼ = ℄ (3.7) (whi h are bilinear with resp et to lo ally onstan t o eien ts), and a unit onstrain t υ : C X → Γ X 1 ma y b e dened as follo ws: (3.8) 8 > > < > > : lo cally constan t complex v alued functions on U 9 > > = > > ; − → Γ 1 ( U ) , z 7→ z · id | U ; it is easy to  he k that these morphisms of shea v es mak e the same diagrams whi h  haraterize tensor funtor onstrain ts omm ute. One also has a natural morphism of shea v es of mo dules o v er X (3.9) ( 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 (3.9) is a natural isomorphism for an y tensor presta k. In fat, (3.9) is an isomorphism of pseudo-tensor funtors viz. it is ompatible with the natural transformations (3.7) and (3.8). 17 3.3. Fibr es of an obje t. Note that for X = ⋆ (where ⋆ is the one-p oin t manifold) one ma y regard (3.6 ) as a pseudo-tensor* funtor of C ( ⋆ ) in to omplex v etor spaes. W e put, for all ob jets E ∈ Ob C ( ⋆ ) , (3.10) 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 forthoming denition of the general notion of smo oth tensor presta k, w e imp ose the follo wing requiremen t: the morphism of shea v es ( 3.8 ) is an isomorphism for X = ⋆ . Then there is a anonial isomorphism (3.11) C ∼ = 1 ∗ of omplex v etor spaes. F or an y ob jet E ∈ Ob C ( X ) , w e dene the br e of E at x to b e the 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 our ordinary notation (3.2) for the pull-ba k, x ∗ E b elongs to C ( ⋆ ) and w e an mak e use of the notation (3.10). Similarly , whenev er a : E → F is a morphism in C ( 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 pseudo-tensor* funtors, it itself ma y b e regarded as a pseudo-tensor* funtor. If in partiular w e apply this to a lo al setion ζ ∈ Γ E ( U ) and mak e use of the anonial iden tiation ( 3.11 ), w e get, for u ∈ U , a linear map (3.12) 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 to b e alled the value of ζ at u . One has the in tuitiv e form ula (3.13) a u ( ζ ( u )) =  ( Γ a )( U )( ζ )  ( u ) . Notie nally that the v etors ζ ( u ) ⊗ η ( u ) and ( ζ ⊗ η )( u ) orresp ond to one another via the anonial map E u ⊗ C F u → ( E ⊗ X F ) u . 3.4. Smo oth tensor pr estaks. Let 1 X denote the tensor unit of C ( X ) , and let x b e a p oin t of the manifold X . Under the assumption (3.11), one an use the omp osite linear isomorphism C ∼ = ( 1 ⋆ ) ∗ ∼ = ( x ∗ 1 X ) ∗ ≡ ( 1 X ) x to dene a anonial homomorphism of omplex algebras (3.14) End C ( X ) ( 1 X ) − → { complex functions on X } , e 7→ ˜ e b y putting ˜ e ( x ) = the omplex n um b er 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 ) x → ( 1 X ) x . 18 W e shall sa y that a tensor presta k C is smo oth if it v eries ( 3.11) and if the homomorphism (3.14 ) determines a bijetiv e orresp ondene b et w een End C ( X ) ( 1 X ) and the subalgebra of all smo oth funtions on X : (3.15) End( 1 X ) ∼ = C ∞ ( X ) . When a tensor presta k C is smo oth, it is p ossible to endo w ea h spae Hom C ( X ) ( E , F ) with a anonial C ∞ ( X ) -mo dule struture ompatible with the m ultipliation b y lo ally onstan t funtions, sine Hom( E , F ) has an ob vious End( 1 X ) -mo dule struture. A ordingly , H om C X ( E , F )( U ) inherits a anonial struture of C ∞ ( U ) -mo dule for ev ery op en subset U ⊂ X , whi h turns H om C X ( E , F ) in to a sheaf of C ∞ X -mo dules. This is true, in partiular, of the sheaf of smo oth setions Γ X E . It is also readily seen that ea h morphism a : E → F indues a morphism Γ X a : Γ X E → Γ X F of shea v es of C ∞ X - mo dules. Th us, one gets a C ∞ ( X ) -linear funtor (3.16) Γ X : C ( X ) − → { shea ve s of C ∞ X - mo dules } . The op erations (3.7 ) and (3.8) ma y no w b e used to dene morphisms of shea v es of C ∞ X -mo dules (3.17) ( τ : Γ X E ⊗ C ∞ X Γ X F → Γ X ( E ⊗ F ) υ : C ∞ X → Γ X 1 . The morphism τ = τ E ,F is natural in the v ariables E , F and, along with υ , turns (3.16) in to a pseudo-tensor funtor of C ( X ) in to the ategory of shea v es of C ∞ X -mo dules. This is loser than (3.6 ) to b eing a gen uine tensor funtor, in that the morphism υ is no w 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 C ( Y ) , a orresp ondene of lo al smo oth setions (3.18) ( Γ Y F )( V ) − → Γ X ( f ∗ F ) ( U ) , η 7→ η ◦ f an b e obtained b y dening η ◦ f as the follo wing omp osite arro w: (3.19) 1 X | U ∼ = ( f ∗ 1 Y ) | U ∼ = f U ∗ ( 1 Y | V ) f U ∗ ( η ) − − − − → f U ∗ ( F | V ) ∼ = ( f ∗ F ) | U . F or U xed, and V v ariable, the maps (3.18 ) onstitute an indutiv e system indexed o v er the inlusions V ⊃ V ′ ⊃ f ( U ) and hene they yield, on passing to the limit, a morphism of shea v es of C ∞ X -mo dules (3.20) f ∗ ( Γ Y F ) − → Γ X ( f ∗ F ) , 19 where f ∗ ( Γ Y F ) is the ordinary pullba k of shea v es of mo dules o v er a smo oth manifold. The morphism (3.20) is natural in F . It is also a morphism of pseudo-tensor funtors, in other w ords it is tensor preserving. Notie that the v etors η ( f ( x )) ∈ F f ( x ) and ( η ◦ f )( x ) ∈ ( f ∗ F ) x orresp ond to one another via the anonial isomorphism of v etor spaes (3.21) ( f ∗ F ) x = ( x ∗ f ∗ F ) ∗ ∼ = ( f ( x ) ∗ F ) ∗ = F f ( x ) . 3.5. Flat maps. It will b e on v enien t to regard op en o v erings of a manifold as smo oth maps. W e sa y that a smo oth map 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 think of p as o difying the op en o v ering of X giv en b y the onneted omp onen ts of X ′ . A renemen t of X ′ p − → X is obtained b y omp osing p ba kw ards with another at mapping X ′′ p ′ − → X ′ . If p is at then for an y smo oth map f : Y → X the pullba k (3.22) Y × X X ′ = { ( y , x ′ ) : f ( y ) = p ( x ′ ) } exists in the ategory of C ∞ manifolds and pr 1 : Y × X X ′ → Y is also at. When f itself is at this onstrution yields a ommon renemen t. F or an y at map p : X ′ → X , put (3.23) X ′′ def = X ′ × X X ′ = { ( x ′ 1 , x ′ 2 ) : p ( x ′ 1 ) = p ( x ′ 2 ) } and let p 1 , p 2 : X ′′ → X ′ b e the pro jetions. Also put (3.24) X ′′′ def = X ′ × X X ′ × X X ′ = { ( x ′ 1 , x ′ 2 , x ′ 3 ) : p ( x ′ 1 ) = p ( x ′ 2 ) = p ( x ′ 3 ) } and let p 12 : X ′′′ → X ′′ b e the map ( x ′ 1 , x ′ 2 , x ′ 3 ) 7→ ( x ′ 1 , x ′ 2 ) ; the maps p 23 and p 13 shall b e dened lik ewise. 3.6. Smo oth tensor staks. A desen t datum for a smo oth tensor presta k C relativ e to a at mapping p : X ′ → X is a pair ( E ′ , θ ) onsisting of an ob jet E ′ of the ategory C ( X ′ ) and an isomorphism θ : p 1 ∗ E ′ ∼ → p 2 ∗ E ′ in the ategory C ( X ′′ ) su h that p 13 ∗ ( θ ) = p 12 ∗ ( θ ) ◦ p 23 ∗ ( θ ) [mo d ∼ = ℄. A morphism a ′ : ( E ′ , θ ) → ( F ′ , ξ ) of desen t data relativ e to p is an arro w a ′ : E ′ → F ′ ompatible, in the ob vious sense, with the isomorphisms θ and ξ . Desen t data for C relativ e to p and their morphisms form a ategory D es C ( X ′ /X ) . There is an ob vious funtor (3.25) C ( X ) − → D es C ( X ′ /X ) , E 7→ ( p ∗ E , φ E ) where φ E is the anonial isomorphism p 1 ∗ ( p ∗ E ) ∼ = ( p ◦ p 1 ) ∗ E = ( p ◦ p 2 ) ∗ E ∼ = p 2 ∗ ( p ∗ E ) . If this anonial funtor is an equiv alene of ategories for ev ery at mapping p : X ′ → X , one sa ys that C is a sta k. 20 F or our purp oses, the ondition that the funtor (3.25 ) b e an equiv alene of ategories for ev ery at map X ′ → X an b e relaxed to some exten t. In fat, it is suien t to require it of all at maps X ′ → X o v er a Hausdor, paraompat X . 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'. 3.7. L o  al ly trivial obje ts. Let C b e a smo oth tensor presta k. An ob jet E ∈ Ob C ( X ) will b e alled trivial if there exists some V ∈ Ob C ( ⋆ ) for whi h one an nd an isomorphism E ≈ c X ∗ V in C ( X ) where c X : X → ⋆ denotes the ollapse map. An y su h pair ( V , ≈ ) will b e said to onstitute a trivialization of E . F or an arbitrary manifold X , let V C ( X ) denote the full sub ategory of C ( X ) formed b y the lo ally trivial ob jets of lo ally nite rank: E ∈ Ob C ( X ) is an ob jet of V C ( X ) if and only if one an o v er X with op en subsets U su h that there exists in ea h C ( U ) a trivialization E | U ≈ 1 U ⊕ · · · ⊕ 1 U . The op eration X 7→ V C ( X ) determines a bred tensor sub ategory of C . In fat, one gets a smo oth tensor presta k V C whi h is a parasta k resp. a sta k if so is C . The tensor* ategory V C ( X ) losely relates to that of smo oth omplex v etor bundles o v er X . Clearly , ev ery ob jet E ∈ V C ( X ) yields a v etor bundle ˜ E = { ( x, e ) : x ∈ X , e ∈ E x } o v er X . The op eration E 7→ ˜ E denes a faithful tensor* funtor of V C ( X ) in to V e c C ∞ ( X ) whi h is an equiv alene of tensor* ategories when C is a parasta k and X is paraompat or when C is a sta k. 4 F oundations of Represen tation Theory In this setion, w e dev elop the represen tation theory of Lie group oids within the framew ork desrib ed in 3. A p euliarit y of the notion of represen tation w e shall b e onsidering here is its dep endene on a `t yp e': the onstrution of our theory neessitates the preliminary  hoie of an arbitrary smo oth tensor sta k T (the t yp e). W e shall assume that su h a  hoie has b een made and w e shall regard T as xed throughout the setion. The denitions b elo w are diretly inspired b y [5℄. Let G b e a Lie group oid o v er a manifold M . W e start b y onstruting the ategory of represen tations of t yp e T  of G . Dene the ob jets of R T ( G ) , or briey , R ( G ) , to b e the pairs ( E ,  ) onsisting of an ob jet E of T ( M ) and an arro w  in T ( G ) s ∗ E  − → t ∗ E , where s , t : G → M denote the soure resp. target map of G , su h that 21 the appropriate onditions for  to b e an ation, in other w ords for  to b e ompatible with the group oid struture, are satised (mo dulo the appropriate anonial isomorphisms): i) u ∗  = id E , where u : M → G denotes the unit setion of G ; ii) c ∗  = p 0 ∗  · p 1 ∗  , where G (2) = G s × t G denotes the manifold of omp os- able arro ws, c : G (2) → G the omp osition map and p 0 , p 1 : G (2) → G the left resp. righ t pro jetion. Observ e that the onditions i) and ii) imply that the arro w s ∗ E  − → t ∗ E is in v ertible in the ategory T ( G ) . W e shall refer to the ob jets of R ( G ) also as G -ations ( of typ e T ). Dene the morphisms of G -ations ( E ,  ) → ( E ′ ,  ′ ) to b e the arro ws a : E → E ′ in T ( M ) whi h fulll the ondition (4.1) t ∗ a ·  =  ′ · s ∗ a . The ategory R ( G ) , endo w ed with the C -linear struture inherited from T ( M ) , is learly additiv e. 4.1. T ensor* strutur e. F or an y G -ations R = ( E ,  ) and S = ( F , σ ) w e put R ⊗ S = ( E ⊗ F ,  ⊗ σ ) where  ⊗ σ stands for (4.2) 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 itself is a G -ation. F urther, if ( E ,  ) a − → ( E ′ ,  ′ ) and ( F , σ ) b − → ( F ′ , σ ′ ) are morphisms of G -ations then so will b e a ⊗ b : R ⊗ S → R ′ ⊗ S ′ . W e dene the tensor unit of R ( G ) as the pair ( 1 M , ∼ = ) , where 1 M denotes the tensor unit of T ( M ) and ∼ = stands for (4.3) s ∗ 1 M ∼ = 1 G ∼ = t ∗ 1 M . The A CU onstrain ts for the tensor pro dut on the base ategory T ( M ) pro vide analogous onstrain ts for the tensor pro dut on R ( G ) . There is of ourse also an inherited anonial an ti-in v olution. The forgetful funtor (4.4) F T G : R T ( G ) − → T ( M ) , ( E ,  ) 7→ E (or F G , for short) is C -linear and faithful. By onstrution, it is a strit tensor* funtor of R ( G ) in to T ( M ) . W e shall refer to this funtor as the standar d for getful funtor ( of typ e T ) asso iated with G . 22 4.2. Pul lb ak along a homomorphism. 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 T ( H ) . Consider the morphismwhi h w e still denote b y ϕ ∗ σ , allo wing some notational abusedened as follo ws: (4.5) s G ∗ ( f ∗ F ) ∼ = ϕ ∗ s H ∗ F ϕ ∗ σ − − − → ϕ ∗ t H ∗ F ∼ = t G ∗ ( f ∗ F ) . The iden tities f ◦ s G = s H ◦ ϕ et. aoun t, of ourse, for the existene of the anonial isomorphisms o urring in (4.5 ). It is routine to  he k that the pair ( f ∗ F , ϕ ∗ σ ) onstitutes an ob jet of the ategory R T ( 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 T ( G ) . Hene w e get a funtor (4.6) ϕ ∗ : R T ( H ) − → R T ( G ) , whi h w e agree to all the inverse image (or pul l-b ak ) along ϕ . The onstrain ts asso iated with the tensor funtor f ∗ (4.7) ( 1 M ∼ → f ∗ 1 N f ∗ F ⊗ f ∗ F ′ ∼ → f ∗ ( F ⊗ F ′ ) funtion as isomorphisms of G -ations 1 ∼ → ϕ ∗ ( 1 ) and ϕ ∗ ( S ) ⊗ ϕ ∗ ( S ′ ) ∼ → ϕ ∗ ( S ⊗ S ′ ) for all S, S ′ ∈ R ( H ) with S = ( F , σ ) and S ′ = ( F ′ , σ ′ ) . A fortiori, these isomorphisms are natural and pro vide appropriate tensor funtor on- strain 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 base maps. 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 an isomorphism b et w een ϕ ∗ ( ψ ∗ T ) and ( ψ ◦ ϕ ) ∗ T in the ategory R ( G ) . Hene w e get an isomorphism of tensor funtors (4.8) ϕ ∗ ◦ ψ ∗ ≃ − → ( ψ ◦ ϕ ) ∗ . 4.3. Natur al tr ansformations. Reall that a transformation 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 f 0 ( x ) τ ( x ) − − → f 1 ( x ) for all x ∈ M and (4.9) ϕ 1 ( g ) · τ ( x ) = τ ( x ′ ) · ϕ 0 ( g ) 23 for all g ∈ G (1) , g : x → x ′ . Supp ose an ation S = ( F , σ ) ∈ R ( H ) is giv en. Then w e an apply τ ∗ to the isomorphism s ∗ F σ − → t ∗ F to obtain an isomorphism f ∗ 0 F → f ∗ 1 F in the ategory T ( M ) (4.10) f ∗ 0 F ∼ = τ ∗ s ∗ F τ ∗ σ − − → τ ∗ t ∗ F ∼ = f ∗ 1 F . By expressing (4.9) as an iden tit y b et w een suitable smo oth maps, one sees that (4.10 ) is atually an isomorphism of G -ations (b et w een ϕ ∗ 0 S and ϕ ∗ 1 S ). Th us, w e obtain an isomorphism of tensor funtors ϕ ∗ 0 ≃ ϕ ∗ 1 . 4.4. Morita e quivalen es. W e observ e next that the in v erse image funtor ϕ ∗ : R ( H ) → R ( G ) asso iated with a Morita equiv alene ϕ : G → H is an equiv alene of tensor ategories. 3 Clearly , this is tan tamoun t to sa ying that ϕ ∗ is a ategorial equiv alene. Although the pro edure to obtain a quasi-in v erse ϕ ! follo ws a w ell-kno wn pattern, w e review it for the reader's on v eniene. In fat, w e kno w of no adequate standard referene for this preise argumen t. The ondition that the map (1.4) b e a surjetiv e submersion will of ourse b e satised when ϕ (0) 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 ma y b e redued to the sp eial ase where ϕ (0) is preisely a surjetiv e submersion. T o this end, onsider the w eak pullba k (see [ 14℄, pp. 123132) P χ   ψ / / G ϕ   H τ * 2 H . (4.11) 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 pro v e that ψ ∗ and χ ∗ are ategorial equiv alenes then, sine b y (4.8) and the remarks on tained in 4.3 w e ha v e natural isomorphisms (4.12) χ ∗ ≃ − → ( ϕ ◦ ψ ) ∗ ≃ ← − ψ ∗ ◦ ϕ ∗ , the same will b e true of ϕ ∗ . F rom no w on, w e w ork under the h yp othesis that the Morita equiv alene ϕ determines a surjetiv e submersion f : M → N on the base manifolds. 3 Reall that a tensor funtor Φ : C → D is said to b e a tensor equiv alene in ase there exists a tensor funtor Ψ : D → C for whi h there are tensor preserving natural isomorphisms Ψ ◦ Φ ≃ Id C and Φ ◦ Ψ ≃ Id D . 24 This b eing the ase, there exists an op en o v er of the manifold N = ∪ i ∈ I V i 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 ( G ) b e giv en. F or ea h i ∈ I one an tak e the pullba k E i ≡ s i ∗ E ∈ T ( V i ) . Fix a ouple of indies i, j ∈ I . Then, sine (1.3) is a pullba 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 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 , (4.13) 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 θ ij = g ij ∗  [mo d ∼ = ℄ (4.14) in the ategory T ( V ij ) . Next, from the ob vious remark that for an arbitrary third index k ∈ I one has 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 , and from the m ultipliativ e axiom ii) 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 ∈ T  ` 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 T is a smo oth parasta k, there exist an ob jet ϕ ! E of T ( N ) and a system of isomorphisms θ i : ( ϕ ! E ) | i ≡ ( ϕ ! E ) | V i ∼ → E i in T ( V i ) ompatible with { θ ij } in the sense that θ j | i def = θ j | V ij = θ ij · θ i | V ij = θ ij · θ i | j . [mo d ∼ = ℄ (4.15) 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 25 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 . (4.16) 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 T ( H i,i ′ ) σ i,i ′ = ( t H | i,i ′ ) ∗ θ i − 1 · g i,i ′ ∗  · ( s H | i,i ′ ) ∗ θ i . [mo d ∼ = ℄ (4.17) By taking in to aoun t the equalit y of mappings (4.18) 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 ′ ) and the iden tities ( 4.14), (4.15 ) and (4.17 ), one sees that σ i,i ′ | j,j ′ = σ j,j ′ | i,i ′ in T ( H ij,i ′ j ′ ) . Hene the morphisms σ i,i ′ glue together in to a unique σ . F or an y morphism a : R → R ′ in the ategory R ( G ) , w e obtain a morph- ism ϕ ! a : ϕ ! R → ϕ ! R ′ b y setting b i = s i ∗ a and b y observing that (4.19) θ ′ ij · b i | j = b j | i · θ ij in T ( V ij ) . In this w a y w e get a funtor of R ( G ) in to R ( H ) . The onstrution of the isomorphisms ϕ ∗ ◦ ϕ ! ≃ I d R ( G ) and ϕ ! ◦ ϕ ∗ ≃ I d R ( H ) is left as an exerise. 5 Smo oth Eulidean Fields In order to get our reonstrution theory to w ork eetiv ely , w e need to mak e further h yp otheses on the t yp e. W e shall sa y that a smo oth tensor sta k F is Eulide an or, for brevit y , that F is a Eulide an stak, if it satises the follo wing axiomati onditions ( 5.1 5.7): 5.1. Axiom: tensor pr o dut and pul lb ak. The anonial natural morphisms (3.17) and (3.20 ) ( Γ E ⊗ C ∞ X Γ E ′ → Γ ( E ⊗ E ′ ) f ∗ ( Γ Y F ) → Γ X ( f ∗ F ) are surjetiv e (= epimorphisms of shea v es). 26 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 is an ob jet of F ( X ) . Let us onsider the ev aluation map Γ E ( U ) → E x , ζ 7→ ζ ( x ) dened in 3.3 for a generi op en neigh b ourho o d U of the p oin t x . When U v aries, these maps are eviden tly m utually ompatible, hene on passing to the indutiv e limit they determine a linear map (5.1) ( 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 . It follo ws from the axiom that for an y sta k of smo oth elds the ev aluation of germs at a p oin t is a surjetiv e map. Hene the v alues ζ ( x ) span the bre E x . 5.2. Axiom: riterion for vanishing. Let a : E → E ′ b e a morphism in F ( X ) . Supp ose that a x : E x → E ′ x is zero ∀ x ∈ X . Then a = 0 . As a rst, immediate onsequene, one gets that an arbitrary setion ζ ∈ Γ E ( U ) v anishes if and only if all the v alues ζ ( u ) are zero as u ranges o v er U . Th us, smo oth setions are  haraterized b y their v alues. F urthermore, b y om bining this axiom with the former, it follo ws that the funtor Γ X : F ( X ) → { shea ves of C ∞ X - mo dules } is faithful. Ea h morphism a : E → F in F ( X ) determines a family of linear maps { a x : E x → F x } and a morphism of shea v es of C ∞ X -mo dules α ≡ Γ a : Γ E → Γ F . The link b et w een these t w o piees of data is pro vided b y the ev aluation maps (5.1). Namely , for ev ery x , the stalk homomorphism α x and the linear map a x are ompatible: the diagram ( Γ E ) x ev al.     α x / / ( Γ F ) x ev al.     E x a x / / F x (5.2) omm utes. In general, w e sa y that a morphism of shea v es of mo dules α : Γ E → Γ F and a family of linear maps { a x : E x → F x } are ompatible if (5.2) omm utes for all x . Let us all a morphism α of shea v es of mo dules r epr esentable if there exists a family of linear maps ompatible with α . 5.3. Axiom: r epr esentable morphisms. F or ea h represen table morphism α : Γ E → Γ F there exists an arro w a : E → F in F ( X ) with Γ a = α . This axiom will pla y a role in 6 , where w e need it in order to onstrut morphisms of represen tations b y means of brewise in tegration. 27 W e sa y that a form φ : E ⊗ E ∗ → 1 in the ategory F ( X ) is a metri ( on E ) when for ev ery p oin t x the indued form on the bre E x (5.3) E x ⊗ C E x ∗ → ( E ⊗ E ∗ ) x φ x − → 1 x ∼ = C is p ositiv e denite Hermitian. 5.4. Axiom: lo  al metris. An y ob jet E of the ategory F ( X ) supp orts enough lo al metris; that is to sa y , the op en subsets U su h that one an nd a metri on the restrition E | U o v er X . In general, one an assume only lo al metris to exist. Global metris an b e onstruted from lo al ones pro vided smo oth partitions of unit y o v er the manifold X are a v ailable. Let φ b e a metri on E . By a φ -orthonormal fr ame ( for E ) ab out a p oint x ∈ 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 ∈ U the v etors ζ 1 ( u ) , . . . , ζ d ( u ) are orthonormal in E u and (5.4) Span { ζ 1 ( x ) , . . . , ζ d ( x ) } = E x . W e note that orthonormal frames for E exist ab out ea h p oin t x at whi h the bre E x is nite dimensional. Indeed, b y Axiom 5.1 , o v er some neigh b our- ho o d V of x one an 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 for E x . Sine for all v ∈ V the v etors ζ 1 ( v ) , . . . , ζ d ( v ) 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 ( v ) = 0 , the on tin uous funtion V × S 2 d − 1 → R , ( v ; s 1 , t 1 , . . . , s d , t d ) 7→     d P k =1 ( s k + it k ) ζ k ( v )     m ust ha v e a p ositiv e minim um at v = x , hene a p ositiv e lo w er b ound on a suitable neigh b ourho o d U of x , so that ζ 1 ( u ) , . . . , ζ d ( 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 order to obtain an orthonormal frame o v er U . Consider an em b edding e : E ′ ֒ → E of ob jets of 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 . 4 4 It follo ws immediately from Axiom 5.2 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 es not enjo y 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. 28 Supp ose there exists a global metri φ on the ob jet E . Also assume that E ′ is a lo ally trivial ob jet of lo ally nite rank. Then e admits a osetion, i.e. there exists a morphism p : E → E ′ with p ◦ e = id . T o pro v e this, note rst of all that the metri φ indues a metri φ ′ on E ′ . F or ea h p oin t x there exists a φ ′ -orthonormal frame ζ ′ 1 , . . . , ζ ′ d ∈ [ Γ E ′ ]( U ) for E ′ ab out x , sine E ′ x is nite dimensional. Let ζ i b e the omp osite (5.5) E | U ∼ = E | U ⊗ 1 U ∼ = E | U ⊗ 1 | U ∗ E | U ⊗ ζ i ∗ − − − − − → E | U ⊗ E | U ∗ φ | U − − − − − → 1 U , where ζ i ≡ [ Γ e ( U )]( ζ ′ i ) . Dene p U as ( ζ ′ 1 ⊕ · · · ⊕ ζ ′ d ) · ( ζ 1 ⊕ · · · ⊕ ζ d ) (or- thogonal pro jetion on to E ′ | U ). Our laim follo ws from Axiom 5.2. By using the last remark, and one more the existene of lo al orthonor- mal frames, one an sho w that if the dimension of the bres of an ob jet E of F ( X ) is nite and lo ally onstan t o v er X then E is lo ally trivial of lo ally nite rank. 5.5 Lemma (Let F b e a Eulidean sta k.) Let X b e a paraompat manifold and i S : S ֒ → X a losed submanifold. Let E , F b e ob jets of F ( X ) and supp ose that E ′ ≡ E | S is lo ally free of lo ally nite rank o v er S . Put F ′ = F | S . Then ev ery morphism a ′ : E ′ → F ′ in F ( S ) an b e extended to a morphism a : E → F in F ( X ) . Pr o of Fix a p oin t s ∈ S . There exists an op en neigh b ourho o d A of s in S su h that there is a trivialization E ′ | A ≈ 1 A ⊕ · · · ⊕ 1 A o v er A . Let ζ ′ 1 , . . . , ζ ′ d ∈ Γ E ′ ( A ) b e the orresp onding frame of lo al setions. Also, let U b e an y op en subset of X su h that U ∩ S = A . After taking U and A smaller ab out s if neessary , it is no loss of generalit y to assume, b y Axiom 5.1 , that there are lo al setions ζ 1 , . . . , ζ d ∈ Γ E ( U ) with ζ ′ k = ζ k ◦ i S , k = 1 , . . . , d . The v alues ζ k ( x ) , k = 1 , . . . , d m ust b e 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 . 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 , as observ ed ab o v e. Set η ′ k = [ Γ a ′ ( A )]( ζ ′ k ) ∈ [ Γ F ′ ]( A ) . As b efore, it is no loss of generalit y to assume that there are setions η 1 , . . . , η d ∈ Γ F ( U ) with η ′ k = η k ◦ i S . These an b e om bined in to a morphism η : 1 U ⊕ · · · ⊕ 1 U → F | U ( d -fold diret sum). Then one an tak e the omp osition (5.6) E | U p − → 1 U ⊕ · · · ⊕ 1 U 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 . One onludes the pro of b y using a partition of unit y o v er X . q.e.d. 29 5.6. Axiom: dimension. It is required of the anonial pseudo-tensor* funtor F ( ⋆ ) − → { complex v ector spaces } ( 3.10 ) that i) it is fully faithful; ii) it fators through the sub ategory whose ob jets are the nite dimen- sional v etor spaes, in other w ords the v etor spae E ∗ (3.10 ) is nite dimensional for all E in F ( ⋆ ) ; iii) it is a gen uine tensor* funtor, i.e. ( 3.7) and (3.8) are isomorphisms of shea v es for X = ⋆ . It follo ws from this axiom that the funtor E 7→ E x is a true tensor* funtor (in general it is only a pseudo-tensor* funtor). W e shall sa y that an ob jet E in F ( X ) is lo  al ly nite if the sheaf 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 for ea h of them there is an epimorphisms of shea v es of mo dules (5.7) C ∞ U ⊕ · · · ⊕ C ∞ U epi − − − − − → ( Γ E ) | U . 5.7. Axiom: lo  al niteness. F or ev ery manifold X , all the ob jets of the ategory F ( X ) are lo ally nite. 5.8. Example: smo oth Hilb ert elds. By a smo oth Hilb ert eld w e mean an ob jet H onsisting of a family { H x } of omplex Hilb ert spaes indexed b y the set of p oin ts of a manifold X and 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→ h ζ ( u ) , ζ ′ ( u ) i is smo oth. W e refer to the manifold X as the base of H ; w e will also sa y that H is a smo oth Hilb ert eld o v er X . Let H and H ′ b e smo oth Hilb ert elds o v er a manifold X . A morphism of H in to H ′ is a family of b ounded linear maps { a x : H x → H ′ x } indexed b y the set of p oin ts of X su h that for ea h op en subset U and for ea h ζ ∈ Γ H ( U ) the setion o v er U giv en b y u 7→ a u · ζ ( u ) b elongs to Γ H ′ ( U ) . Smo oth Hilb ert elds o v er X and their morphisms form a ategory whi h w e shall denote b y H il b ∞ ( X ) . Supp ose H and G are Hilb ert elds o v er a manifold X . Consider the bundle of tensor pro duts { H x ⊗ G x } . F or an y pair of setions ζ ∈ [ Γ H ]( U ) 30 and η ∈ [ Γ G ]( U ) w e let ζ ⊗ η denote the setion of the bundle { H x ⊗ G x } dened o v er U b y u 7→ ζ ( u ) ⊗ η ( u ) . The orresp ondene (5.8) U 7→ C ∞ ( U )  ζ ⊗ η : ζ ∈ [ Γ H ]( U ) , η ∈ [ Γ G ]( U )  denes a sub-presheaf of the sheaf of setions of { H x ⊗ G x } . [Here C ∞ ( U ) {· · · } stands for the C ∞ ( U ) -mo dule spanned b y {· · · } .℄ Let H ⊗ G denote the Hilb ert eld o v er X giv en b y the bundle { H x ⊗ G x } together with the sheaf of setions generated b y the presheaf ( 5.8). W e all H ⊗ G the tensor pro dut of H and G . Observ e that for all morphisms H α − → H ′ and G β − → G ′ 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 H ⊗ G in to H ′ ⊗ G ′ . One gets the onjugate eld H ∗ of a Hilb ert eld H b y taking the bundle { H x ∗ } of onjugate spaes along with the lo al setions of H regarded as lo al setions of { H x ∗ } . With the ob vious tensor unit and the ob vious A CU onstrain ts, these op erations turn H il b ∞ ( X ) in to a tensor* ategory . It remains to dene a tensor* funtor f ∗ : H ilb ∞ ( Y ) → H ilb ∞ ( X ) for ea h smo oth map f : X → Y , along with suitable bred tensor ategory onstrain ts. Let G b e a Hilb ert eld o v er Y . The pull-ba k of G along f , to b e denoted b y f ∗ G , is the smo oth Hilb ert eld o v er X whose asso iated bundle of Hilb ert spaes is { G f ( x ) } and whose asso iated sheaf of setions is generated b y the follo wing presheaf of setions of the bundle { G f ( x ) } : (5.9) U 7→ C ∞ X ( U )  η ◦ f : η ∈ [ Γ G ]( V ) , V ⊃ f ( U )  . F or ev ery morphism β : G → G ′ 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 ∗ G → f ∗ G ′ of Hilb ert elds o v er X . The op eration G 7→ f ∗ G denes a strit tensor* funtor of H il b ∞ ( Y ) in to H il b ∞ ( X ) , in other w ords one has the iden tities f ∗ ( G ⊗ G ′ ) = f ∗ G ⊗ f ∗ G ′ , f ∗ ( 1 Y ) = 1 X and f ∗ ( G ∗ ) = ( f ∗ G ) ∗ . Finally , the iden tities of tensor* funtors ( g ◦ f ) ∗ = f ∗ ◦ g ∗ and id X ∗ = Id pro vide the required bred tensor ategory onstrain ts. The bred tensor ategory X 7→ H ilb ∞ ( X ) is a smo oth tensor sta k satisfying Axioms 5.1 , 5.2 and 5.4. Ho w ev er, as it do es not satisfy the other axioms, it is not an example of a Eulidean sta k. 31 5.9. Example: smo oth Eulide an elds. Let E uc ∞ ( X ) denote the full sub- ategory of H il b ∞ ( X ) onsisting of all E whose asso iated sheaf of setions Γ E is lo ally nite. W e refer to the ob jets of this sub ategory as smo oth Eulidean elds (o v er X ). Observ e that E uc ∞ ( X ) is a tensor* sub ategory of H il b ∞ ( X ) . Indeed, sine the smo oth tensor sta k of smo oth Hilb ert elds satises Axiom 5.1 , the lo ally nite C ∞ X -mo dule Γ E ⊗ C ∞ X Γ E ′ surjets on to the C ∞ X -mo dule Γ ( E ⊗ E ′ ) . F or similar reasons, for an y map f : X → Y the pullba k funtor f ∗ : H ilb ∞ ( Y ) → H ilb ∞ ( X ) m ust arry E uc ∞ ( Y ) in to E uc ∞ ( X ) . The smo oth tensor sta k X 7→ E uc ∞ ( X ) also satises Axioms 5.3, 5.6 and 5.7 and is therefore Eulidean. 6 Constrution of Equiv arian t Maps Let F denote an arbitrary Eulidean sta k. F is to b e regarded as xed throughout the en tire setion. 6.1 Lemma Let G b e a (lo ally) transitiv e Lie group oid, and let X b e its base manifold. T ak e an arbitrary represen tation ( E ,  ) ∈ R F ( G ) . Then E is lo ally trivial in F ( X ) . Pr o 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 } . Let x : ⋆ → X denote the map ⋆ 7→ x . By Axiom 5.6 , there is a trivialization for x ∗ E in F ( ⋆ ) . W e pull  ba k to U along the smo oth map g , and observ e that there is a unique fatorization of t ◦ g through ⋆ (ollapse c : U → ⋆ follo w ed b y x : ⋆ → X ). Sine  is an isomorphism, E | U = i U ∗ E = ( s ◦ g ) ∗ E ∼ = g ∗ s ∗ E g ∗  − − − → g ∗ t ∗ E ∼ = ( t ◦ g ) ∗ E = = ( x ◦ c ) ∗ E ∼ = c ∗ ( x ∗ E ) ≈ c ∗ ( 1 ⊕ · · · ⊕ 1 ) = 1 U ⊕ · · · ⊕ 1 U pro vides a trivialization for E | U in F ( U ) . q.e.d. Let i S : S ֒ → X b e an in v arian t immersed submanifold. The pullba k G | S of G along i S is w ell-dened and is a Lie subgroup oid of G . 5 [Observ e that 5 In general, a Lie subgroup oid is a Lie group oid homomorphism ( ϕ, f ) su h that b oth ϕ and f are injetiv e immersions; ompare for instane [ 15 ℄. 32 G | S = G S = s G − 1 ( S ) .℄ In the sp eial ase of an orbit immersion, G | S will b e transitiv e o v er S . Then the lemma sa ys that for an y ( E ,  ) ∈ Ob R ( G ) the restrition E | S is a lo ally trivial ob jet of F ( S ) . 6.2. A lternative desription of r epr esentations. The notion of represen tation with whi h w e ha v e b een w orking 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 s ∗ E  − → t ∗ E of G . Ea h arro w g determines a linear map  ( g ) : E s ( g ) → E t ( g ) dened via the omm utativit y of the diagram [ g ∗ s ∗ E ] ∗ [ g ∗  ] ∗   [ ∼ = ] ∗ / / [ s ( g ) ∗ E ] ∗ def E s ( g )  ( g )      [ g ∗ t ∗ E ] ∗ [ ∼ = ] ∗ / / [ t ( g ) ∗ E ] ∗ def E t ( g ) (6.1) [the notation (3.10 ) is in use℄. It is routine to  he k that the onditions i) and ii) in the denition of a represen tation (b eginning of 4) imply that the orresp ondene g 7→  ( g ) is m ultipliativ e, i.e. that  ( g ′ g ) =  ( g ′ ) ◦  ( g ) and  ( x ) = id for all p oin ts x of the base manifold X . Fix an arbitrary arro w g 0 . 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 (3.18), ζ will determine a setion ζ ◦ s ∈ Γ G ( s ∗ E )( G U ) at whi h the morphism of shea v es of mo dules Γ  an b e ev aluated so as to get a setion of t ∗ E o v er G U . No w, Axiom 5.1 implies that there exists an op en neigh b ourho o d Γ of g 0 in G U o v er whi h the latter setion 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) . In sym b ols, (6.2)  Γ  (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 to get (6.3)  ( g ) · ζ ( s g ) = d P i =1 r i ( g ) ζ ′ i ( t g ) . By Axiom 5.3, an y m ultipliativ e op eration g 7→  ( g ) , lo ally of the form (6.3), omes from a represen tation of G on E . 33 6.3. Pr eliminary extension. Supp ose G prop er hereafter. Fix a p oin t x 0 ∈ X , and let G 0 denote the isotrop y group at x 0 . It is eviden t from (6.3 ) that (6.4)  0 : G 0 → GL ( E 0 ) , g 7→  ( g ) is a on tin uous represen tation of the ompat Lie group G 0 on the nite dimensional v etor spae E 0 (the bre of E at x 0 ). Supp ose another G -ation ( F , σ ) is giv en, along with some G 0 -equiv arian t linear map A 0 : E 0 → F 0 . Let S 0 ֒ → X b e the orbit through x 0 . Our remarks ab out Morita equiv alenes in 4 sa y there exists a unique morphism A ′ : ( E ′ ,  ′ ) → ( F ′ , σ ′ ) in R ( G ′ ) [the primes here signify that w e are taking the orresp onding restritions to S 0 ℄ su h that ( A ′ ) 0 = A 0 . In fat, for ev ery p oin t z ∈ S 0 and arro w g ∈ G ( x 0 , z ) , one has (6.5) ( A ′ ) z = σ ( g ) · A 0 ·  ( g ) − 1 : E z → F z . By Lemma 6.1, E ′ is a lo ally trivial ob jet of F ( S 0 ) . Then Lemma 5.5 yields a global morphism a : E → F extending A ′ and hene, a fortiori, A 0 . W e pro eed to a v erage out this a to mak e it G -equiv arian t, as follo ws. 6.4. A ver aging op er ators. Fix an arbitrary (righ t in v arian t, normalized) Haar system µ = { µ x } on the (prop er) Lie group oid G . W e shall onstrut, for ea h pair of G -ations R = ( E ,  ) and S = ( F , σ ) , a linear op erator (6.6) Av = Av µ : Ho m F ( M ) ( E , F ) → Hom R ( G ) ( R, S ) (a v eraging op erator), with the prop ert y that Av ( a ) = a whenev er a already b elongs to Hom R ( G ) ( R, S ) . More generally , if S is an in v arian t submanifold o v er whi h a restrits to an equiv arian t morphism, Av ( a ) | S = a | S . W e start from a v ery simple remark. Supp ose setions ζ ∈ Γ E ( U ) and η 1 , . . . , η n ∈ Γ F ( U ) are giv en su h that η 1 , . . . , η n are lo al generators for Γ F o v er U . Then for ea h g 0 ∈ G U there exists an op en neigh b ourho o d Γ ⊂ G U of g 0 along with smo oth funtions φ 1 , . . . , φ n on Γ su h that (6.7) σ ( g ) − 1 · a t ( g ) ·  ( g ) · ζ ( s g ) = n P j =1 φ j ( g ) η j ( s g ) for all g ∈ Γ . T o see this, note thatas observ ed in ( 6.3 )there are an op en neigh b ourho o d Γ of g 0 in G U and lo al smo oth setions ζ ′ 1 , . . . , ζ ′ m of E o v er U ′ = t (Γ) su h that  ( g ) ζ ( s g ) = m P i =1 r i ( g ) ζ ′ i ( t g ) for some 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 0 − 1 , b y using the h yp othesis that the η j 's are 34 generators w e an also assume Γ to b e so small that for ea h i = 1 , . . . , m there exist 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 ) for ea h g ∈ Γ . This pro v es ( 6.7). Put α = Γ a . W e an use the last remark to obtain a new morphism of shea v es of mo dules ˜ α : Γ E → Γ F , as follo ws. Let ζ b e a lo al setion of E dened o v er an op en subset U so small that b y Axiom 5.7 there exists a system η 1 , . . . , η n of lo al generators for Γ F o v er U . F or ea h g 0 ∈ G 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 )  satisfying (6.7). Then  ho ose a smo oth partition of unit y o v er G U { θ i : i ∈ I } sub ordinated to the Γ( g 0 ) , and put (6.8) ˜ 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 ) . Some arbitrary  hoies are in v olv ed here, so one has to mak e sure that this is a go o d denition. If w e lo ok at ( 6.7 ) for x = s ( g ) xed, w e reognize that the op eration g 7→ σ ( g ) − 1 · a t ( g ) ·  ( g ) · ζ ( x ) denes a smo oth mapping on the manifold G x with v alues in the nite dimensional v etor spae F x . F or ea h v ∈ E x , there is some lo al setion ζ ab out x su h that ζ ( x ) = v , so one is allo w ed to tak e the in tegral (6.9) κ x ( v ) = Z G x σ ( g ) − 1 · a t ( g ) ·  ( g ) · v d µ x ( g ) . This denes, for ea h base p oin t x , a linear map κ x : E x → F x . No w, [ ˜ 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 ) h σ ( g ) − 1 · a t ( g ) ·  ( g ) · ζ ( s g ) i d µ u ( g ) = κ u · ζ ( u ) . It follo ws from Axiom 5.2 that the setion ˜ α ( U )( ζ ) in (6.8) do es not dep end on an y of the auxiliary  hoies w e made in order to dene it (as the κ u don't). W e dene Av ( a ) as the unique morphism ˜ a : E → F with Γ (˜ a ) = ˜ α . [Its existene follo ws from the preeding omputation and Axiom 5.3, its uniqueness from Axiom 5.2 .℄ It remains to sho w that Av µ is a pro jetion op erator on to Hom R ( G ) ( R, S ) . W e will lea v e the v eriation to the reader. Summing up 6.3 and 6.4, one gets 35 6.5 Prop osition Supp ose G is prop er, and let x 0 b e a base p oin t. F or ea h pair of G -ations R = ( E ,  ) and S = ( F , σ ) , and for ea h G 0 - equiv arian t linear map A 0 : E 0 → F 0 , there exists in R ( G ) a morphism a : R → S extending A 0 . By applying the a v eraging op erator to a randomly  hosen Hermitian metri, w e get the existene of in v arian t metris 6.6 Prop osition Let R = ( E ,  ) b e a represen tation of a prop er Lie group oid G . Then there exists a G -in v arian t metri on E , that is, a metri on E whi h is at the same time a morphism R ⊗ R ∗ → 1 in R ( G ) . By a  -in v arian t partial setion of E o v er an in v arian t submanifold S of the base of G w e mean a setion of E | S o v er S whi h is at the same time a morphism in R ( G | S ) . Lemma 5.5 in om bination with 6.4 yields 6.7 Prop osition Let S b e a losed in v arian t submanifold of the base of a prop er Lie group oid G . Let R = ( E ,  ) b e a represen tation of G . Then ea h  -in v arian t partial setion of E o v er S an b e extended to a global  -in v arian t setion of E . A funtion ϕ dened on an arbitrary subset S of a manifold X is alled smo oth when for ea h x ∈ X one an nd an op en neigh b ourho o d U of x in X and a smo oth funtion on U whi h agrees with ϕ on U ∩ S . 6.8 Prop osition Let S b e an in v arian t subset of the base manifold X of a prop er Lie group oid G . Supp ose ϕ is a smo oth in v arian t (viz. onstan t along the G -orbits) funtion on S . Then there exists a smo oth in v arian t funtion extending ϕ on all of X . Pr o of A v erage out an y smo oth extension of ϕ obtained b y means of a partition of unit y o v er X . q.e.d. 7 C ∞ Fibre F untors W e k eep on w orking with a generi Eulidean sta k F . Let M b e a paraom- pat smo oth manifold. 7.1 Denition By a br e funtor ( of typ e F ) over M , or with b ase M , w e mean a faithful tensor* funtor (7.1) ω : C − → F ( M ) dened on some tensor* ategory C . 36 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 base p oin ts. Under reasonable assumptions, it is p ossible to endo w T ( ω ) with a natural struture of top ologial group- oid; 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 to- p ology . An impro v emen t of the same idea leads one to in tro due a ertain C ∞ funtional struture on the spae of arro ws of T ( ω ) . (Reall 1.1 and 1.2.) When T ( ω ) is a C ∞ -group oid relativ e to this partiular C ∞ -struture, w e sa y that ω is a C ∞ br e funtor. In detail, these onstrutions read as follo ws. 7.2. The T annakian gr oup oid T ( ω ) . Let x b e a p oin t of M ; the same sym b ol will b e used to denote the orresp onding (smo oth) map ⋆ → M . Consider the tensor* funtor (fr 3.3 and 5.6) (7.2) F ( M ) − → { v ector spaces } , E 7→ E x . Let ω x denote the omp osite tensor* funtor (7.3) C ω − − → F ( M ) ( - ) x − − → { ve ctor spaces } , R 7→ ( ω ( R )) x . W e dene t w o group oids T ( ω ; C ) and T ( ω ; R ) o v er M b y putting (7.4) T ( ω ; C )( x, x ′ ) = Iso ⊗ ( ω x , ω x ′ ) and T ( ω ; R )( x, x ′ ) = Iso ⊗ , ∗ ( ω x , ω x ′ ) where x, x ′ ∈ M . (Reall that the righ t-hand term in the seond equalit y denotes the set of all self-onjugate tensor preserving natural isomorphisms.) By setting ( λ ′ · λ )( R ) = λ ′ ( R ) ◦ λ ( R ) and x ( R ) = id , in ea h ase w e obtain a struture of group oid o v er M . The relationship b et w een T ( ω ; C ) and its subgroup oid T ( ω ; R ) an b e laried b y in tro duing the onjugation in v o- lution of T ( ω ; C ) : this sends an arro w λ to the arro w λ dened b y setting λ ( R ) = λ ( R ∗ ) ∗ [up to ∼ = ℄. The elemen ts of T ( ω ; R ) are the xed p oin ts of the onjugation in v olution. The group oid T ( ω ; R ) shall b e referred to as the T annakian gr oup oid ( asso iate d with ω ). W e will abbreviate T ( ω ; R ) in to T ( ω ) . 7.3. R epr esentative funtions. Let R ∈ Ob( C ) b e arbitrary and let φ b e an y metri on ω ( R ) . F or ea h pair of global setions ζ , ζ ′ ∈ Γ ( ω R )( M ) w e in tro due the funtion (7.5) r R,φ,ζ ,ζ ′ : T ( ω ) → C , λ 7→  λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ )  φ def = φ t ( λ )  λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ )  . 37 W e put (7.6) R = { r R,φ,ζ ,ζ ′ : R ∈ Ob( C ) , φ metri on ω ( R ) , ζ , ζ ′ ∈ Γ ( ω R )( M ) } . W e all the elemen ts of R r epr esentative funtions. Observ e that R is a omplex algebra of funtions on T ( ω ) , losed under the op eration of taking the omplex onjugate. This implies that the real and imaginary parts of an y funtion of R also b elong to R . Th us, if w e let R [ R ] ⊂ R denote the subset of all real v alued funtions, w e ha v e R = C ⊗ R [ R ] . 7.4. T op olo gy and C ∞ -strutur e. W e endo w T ( ω ) with the smallest top ology making all represen tativ e funtions on tin uous. As a onsequene of the existene of metris on an y ob jet of F ( M ) , the top ologial spae T ( ω ) is neessarily Hausdor. The funtions in R [ R ] generate a funtional struture on the spae T ( ω ) . One an omplete this funtional struture to a C ∞ - struture R ∞ as explained in 1.1. W e remark that the soure map of the group oid T ( ω ) is a morphism of C ∞ -spaes relativ e to the C ∞ -struture R ∞ . The same statemen t is true of the target map and the unit setion. Ho w ev er, without stronger assumptions on the bre funtor ω w e are at presen t unable to sho w that T ( ω ) is a C ∞ - group oid relativ e to R ∞ . It migh t b e the ase that not ev ery bre funtor is C ∞ . W e will see later on that the standard forgetful funtor asso iated with a reexiv e group oid is alw a ys a C ∞ bre funtor. This is in fat the only ase of in terest in onnetion with the pro of of our reonstrution theorem. 7.5. Invariant metris. Let R ∈ Ob( C ) . W e sa y that a metri φ on ω ( R ) is ω -invariant if there is a Hermitian form m : R ⊗ R ∗ → 1 su h that φ oinides with the indued form (7.7) ω ( R ) ⊗ ω ( R ) ∗ ∼ = ω ( R ⊗ R ∗ ) ω ( m ) − − − → ω ( 1 ) ∼ = 1 . Note that, b y the faithfulness of ω , there is at most one su h m . 7.6 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 ( ω ) → M × M is prop er, and ii) for ev ery R ∈ Ob( C ) , the ob jet ω ( R ) 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. 7.7. Example. As an example of a prop er bre funtor, w e men tion [reall 4℄ the standard forgetful funtor F G : R ( G ) → F ( M ) asso iated with the represen tations of t yp e F of a prop er Lie group oid G o v er M . 38 T o b egin with, w e observ e that there is a homomorphism of group oids (7.8) G − → T ( F G ) whi h sends g to the natural transformation assigning ea h ob jet ( E ,  ) of the ategory R ( G ) the isomorphism  ( g ) [fr 6.2℄. This homomorphism is eviden tly a morphism of C ∞ -spaes and, in partiular, a on tin uous map. It will b e established in the next setion that (7.8 ) is a surjetion. The prop erness of ( s , t ) : T ( F G ) → M × M is then an immediate onsequene of the prop erness of ( s , t ) : G → M × M . The existene of enough in v arian t metris has b een pro v ed in the preeding setion (Prop osition 6.6). Let R ′ ⊂ R b e the set of all represen tativ e funtions of the form r R,φ,ζ ,ζ ′ where φ is an ω -in v arian t metri. Note that R ′ is a subalgebra of R losed under omplex onjugation. 7.8 Lemma Let ω b e a prop er bre funtor. Then the top ology in tro- dued in 7.4 oinides with the smallest top ology on T ( ω ) making all the elemen ts of R ′ on tin uous. Pr o of The algebra of on tin uous funtions R ′ separates p oin ts b eause of the existene of enough ω -in v arian t metris. Then, for ev ery op en subset Ω with ompat losure the in v olutiv e subalgebra R ′ Ω ⊂ C 0 ( Ω) formed b y the restritions to the losure Ω of elemen ts of R ′ is dense in the subspae R Ω = { r | Ω : r ∈ R } with resp et to the sup-norm, as a onsequene of the Stone W eierstrass theorem. The subsets of the form T ( ω ) | U × U ′ , where U and U ′ are op en subsets of M with ompat losure, are ertainly op en and of ompat losure, as w ell as op en relativ e to the top ology asso iated with R ′ . Let Ω b e an y one of these op en subsets. W e laim that b oth top ologies agree on Ω . Indeed, for ea h r ∈ R the restrition r | Ω m ust b e a uniform limit of funtions whi h are on tin uous for the R ′ top ology , and hene r itself m ust b e a on tin uous funtion for the R ′ top ology . q.e.d. 7.9. R emark. Ea h arro w λ of the group oid T ( ω ) ats as a unitary trans- formation with resp et to all ω -in v arian t metris. More expliitly , for ev ery R ∈ Ob( C ) and ω -in v arian t metri φ on ω ( R ) one has (7.9)  λ ( R ) v , λ ( R ) v ′  φ = h v , v ′ i φ . W e use this remark in the pro of of the follo wing 7.10 Prop osition Let ω b e a prop er bre funtor. Then T ( ω ) is a top ologial group oid. 39 Pr o of (a) Con tin uit y of the in v erse map i . By Lemma 7.8, it sues to pro v e that the omp osite r ◦ i is on tin uous for ev ery r = r R,φ,ζ ,ζ ′ with φ ω -in v arian t. This is lear, for b y Remark 7.9 r R,φ,ζ ,ζ ′ ◦ i = ( r R,φ,ζ ′ ,ζ ) . (b) Con tin uit y of the omp osition map c . W e start with a preliminary observ ation. Let R ∈ Ob( C ) . Let φ b e an y ω -in v arian t metri on ω ( R ) . F or an y giv en arro w λ : x → x ′ in T ( ω ) w e an x a lo al φ -orthonormal frame ζ ′ 1 , . . . , ζ ′ d of setions dened o v er some neigh b ourho o d U ′ of x ′ . [See 5.4.℄ Cho ose an op en neigh b ourho o d Ω of λ su h that t (Ω) ⊂ U ′ . Let ζ b e a global setion of ω ( R ) and let Φ i ( i = 1 , . . . , d ) b e arbitrary on tin uous funtions on Ω . The funtion (7.10) µ 7→     µ ( R ) · ζ ( s µ ) − d P i =1 Φ i ( µ ) ζ ′ i ( t µ )     is ertainly on tin uous; indeed, b y (7.9 ), its square is   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 . Up on making the substitution Φ i ( µ ) =  µ ( R ) ζ ( s µ ) , ζ ′ i ( t µ )  in (7.10 ), w e get a funtion v anishing at λ sine b y onstrution the v etors ζ ′ i ( x ′ ) onstitute an orthonormal basis. No w, w e ha v e to  he k the on tin uit y of all funtions of the form (7.11) ( µ ′ , µ ) 7→ ( r R,φ,ζ ,η ◦ c )( µ ′ , µ ) =  µ ′ ( R ) · µ ( R ) · ζ ( s µ ) , η ( t µ ′ )  φ with φ ω -in v arian t. Let x λ − → x ′ λ ′ − → x ′′ b e an y pair of omp osable arro ws. By the foregoing observ ation and (7.9), w e see that for ea h ǫ > 0 there is a neigh b ourho o d Ω ǫ of λ su h that for all omp osable ( µ ′ , µ ) with µ ∈ Ω ǫ the v alue of the funtion ( 7.11) at ( µ ′ , µ ) diers from 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 ′ ) b y C ǫ at most, where C is a p ositiv e b ound for the φ -norm of the setion η in a giv en neigh b ourho o d of x ′′ . q.e.d. 40 8 Pro of of the Reonstrution Theorem W e start with some results whi h hold for an arbitrary Eulidean sta k F . W e in tro due the shorthand T ( G ) for the T annakian group oid asso iated with the standard forgetful funtor (of t yp e F ) of a Lie group oid G . 8.1. The enveloping homomorphism. The anonial homomorphism (8.1) π G : G − → T ( G ) is dened b y means of the iden tit y π G ( g )( E ,  ) =  ( g ) . [Reall Example 7.7.℄ W e shall refer to π G as the enveloping homomorphism ( of typ e F ) of G . 8.2 Theorem Let G b e a prop er Lie group oid. Then the en v eloping homomorphism of G is a surjetion. Pr o of T o b egin with, w e pro v e that whenev er G ( x, x ′ ) is empt y , so m ust b e T ( G )( x, x ′ ) . Let ϕ : G x ∪ G x ′ → C b e the funtion whi h tak es the v alue one on the orbit G x and the v alue zero on the orbit G x ′ . This funtion is w ell-dened, b eause G ( x, x ′ ) is empt y . By Corollary 6.8, there is a global in v arian t smo oth funtion Φ extending ϕ . Being in v arian t, Φ determines an endomorphism a of the trivial represen tation 1 ∈ Ob R ( G ) su h that a z = Φ( z ) id for all z (th us, in partiular, a x = id and a x ′ = 0 ). No w, supp ose λ ∈ T ( G )( x, x ′ ) . Beause of the naturalit y of λ , the existene of the morphism a on tradits the in v ertibilit y of the linear map λ ( 1 ) . W e are therefore redued to pro ving that the indued isotrop y homo- morphisms π G | x : G | x → T ( G ) | x are surjetiv e. This is no w a diret onse- quene of Prop ositions 2.3 and 6.5 . q.e.d. 8.3 Denition W e sa y that a Lie group oid G is r eexive or self-dual ( r elative to F ) when its en v eloping homomorphism is an isomorphism of to- p ologial group oids. 8.4 Theorem Let G b e a prop er Lie group oid. Then in order that G ma y b e reexiv e it is enough that its en v eloping homomorphism b e injetiv e. Pr o of The on tin uit y of π G is ob vious, hene what w e really ha v e to sho w is that for ea h op en subset Γ of G and for ea h p oin t g 0 ∈ Γ the image π G (Γ) is a neigh b ourho o d of π G ( g 0 ) in T ( G ) . Let g 0 ∈ G ( x 0 , x 0 ′ ) . W e start b y observing that it is p ossible to nd a represen tation R = ( E ,  ) whose asso iated x 0 -th isotrop y homomorphism 41  0 : G | 0 → GL ( E 0 ) is injetiv e. [Compare the pro of of Prop osition 2.3 and also 4.1.℄ Fix an arbitrary metri φ on E and lo al φ -orthonormal frames ζ 1 , . . . , ζ d ab out x 0 and ζ ′ 1 , . . . , ζ ′ d ab out x 0 ′ . Cho ose an y ompatly supp orted smo oth funtion 0 ≦ a ≦ 1 , resp. 0 ≦ a ′ ≦ 1 with supp ort lying lose enough to x 0 , resp. x 0 ′ and su h that a ( z ) = 1 ⇔ z = x 0 , resp. a ′ ( z ) = 1 ⇔ z = x 0 ′ . Then put  i,i ′ def = r i,i ′ ◦ π G def = r R,φ,ζ i ,ζ ′ i ′ ◦ π G , and  ι,ι ′ def = r ι,ι ′ ◦ π G def = ( a ◦ s G )( a ′ ◦ t G ) with ι = 0 or ι ′ = 0 . Finally , let ω ι,ι ′ =  ι,ι ′ ( g 0 ) for 0 ≦ ι, ι ′ ≦ d . W e laim that there exist op en disks D ι,ι ′ , with D ι,ι ′ enirling the om- plex n um b er ω ι,ι ′ , whi h satisfy (8.2) \ 0 ≦ ι,ι ′ ≦ d  ι,ι ′ − 1 ( D ι,ι ′ ) ⊂ Γ . One this laim is pro v en, the statemen t that π G (Γ) is a neigh b ourho o d of π G ( g 0 ) will b e pro v en as w ell. Indeed, b y Theorem 8.2 w e ha v e \ r ι,ι ′ − 1 ( D ι,ι ′ ) = π G π G − 1  \ r ι,ι ′ − 1 ( D ι,ι ′ )  = π G  \  ι,ι ′ − 1 ( D ι,ι ′ )  where ea h r ι,ι ′ − 1 ( D ι,ι ′ ) is an op en neigh b ourho o d of π G ( g 0 ) in T ( G ) . In order to establish (8.2), w e x for ea h 0 ≦ ι, ι ′ ≦ d a dereasing sequene of op en disks en tred at ω ι,ι ′ (8.3) · · · ⊂ D ι,ι ′ p +1 ⊂ D ι,ι ′ p ⊂ · · · ⊂ D ι,ι ′ 1 ⊂ C with radius on v erging to zero. If w e agree that D ι,ι ′ 1 has radius 1 2 then Σ p def = \ r ι,ι ′ − 1  D ι,ι ′ p  − Γ ( p = 1 , 2 , . . . ) (8.4) is a losed subset of the ompat spae G ( K, K ′ ) where K = supp a and K ′ = supp a ′ . The in tersetion ∞ ∩ p =1 Σ p is empt y b eause of the injetivit y of the map G ( x 0 , x 0 ′ ) → Iso ( E x 0 , E x 0 ′ ) , g 7→  ( g ) and the  hoie of a, a ′ . Th us, there m ust b e some p su h that Σ p = ∅ . This pro v es the laim. q.e.d. In 7.7, w e remark ed on passing that π G is a morphism of C ∞ -spaes. This is in fat true for an arbitrary , not neessarily prop er Lie group oid G . One ma y w onder whether more an b e said when G is reexiv e. 42 Hereafter w e shall freely mak e use of some notation in tro dued in the on text of the preeding pro of. W e dene the smo oth mappings (8.5)  ζ 1 ,...,ζ d ζ ′ 1 ,...,ζ ′ d : G − → M × M × End( C d ) , g 7→  s ( g ); t ( g );  1 , 1 ( g ) , . . . ,  i,i ′ ( g ) , . . . ,  d,d ( g )  , where M is the base of G , and in tro due the abbreviations ζ ≡ ζ 1 , . . . , ζ d , ζ ≡ ζ ′ 1 , . . . , ζ ′ d . If the homomorphism π G is faithful, Lemma 2.4 implies that for ea h arro w g 0 there exists a represen tation R = ( E ,  ) su h that the map G ( x 0 , x 0 ′ ) − → Iso( E x 0 , E x 0 ′ ) , g 7→  ( g ) b eomes injetiv e when restrited to a suien tly small op en neigh b ourho o d of g 0 . 8.5 Lemma Supp ose the map G ( x 0 , x 0 ′ ) → Iso( E x 0 , E x 0 ′ ) , g 7→  ( g ) is injetiv e near g 0 . Then (8.5) is an immersion at g 0 . Pr o of Fix op en balls U and U ′ en tred at x 0 and x 0 ′ resp etiv ely , 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 ′ ). Up to a lo al dieomorphism, the map ( 8.5 ) has the follo wing form near g 0 , pro vided U is  hosen small enough: (8.6) U × R k → U × U ′ × End( C d ) , ( u, v ) 7→  u ; u ′ ( u, v ) ;  ( u, v )  , where  ( g ) denotes the matrix {  i,i ′ ( g ) } 1 ≦ i,i ′ ≦ d . Eviden tly , (8.6) is immersiv e at g 0 = ( x 0 , 0) if and only if the partial map v 7→  u ′ ( x 0 , v );  ( x 0 , v )  is immersiv e at zero. W e are therefore redued to sho wing that the restrition of (8.5) to G ( x 0 , - ) is immersiv e at g 0 . Let G b e the isotrop y group of G at x 0 . By  ho osing a lo al equiv ari- an t trivialization G ( x 0 , S ) ≈ S × G where S is a submanifold of U ′ passing through x 0 ′ , the restrition of (8.5) to G ( x 0 , - ) tak es the form (8.7) S × G → U ′ × End( C d ) , ( s, g ) 7→  s ;  ( s, g )  . This map is immersiv e at g 0 = ( x 0 ′ , e ) if and only if so is at e the partial map g 7→  ( x 0 ′ , g ) , where e is the unit of the group G . Th us, it sues to sho w that the isotrop y represen tation G → GL ( E x 0 ) indued b y  is immersiv e at e . By h yp othesis, this represen tation is injetiv e in an op en neigh b ourho o d of e and hene our laim follo ws at one. q.e.d. Let an arro w λ 0 ∈ T ( G ) b e giv en. W e on tend that there exists some op en neigh b ourho o d Ω of λ 0 su h that (Ω , R ∞ Ω ) is isomorphi, as a C ∞ -spae, to a smo oth manifold ( X , C ∞ X ) . Sine G is reexiv e, there is a unique g 0 ∈ G su h that λ 0 = π G ( g 0 ) . By Lemma 8.5 and the remarks preeding it, w e an nd some R for whi h 43 there exists an op en neigh b ourho o d Γ of g 0 in G su h that  ζ ζ ′ indues a dieomorphism of Γ on to a submanifold X of M × M × End( C d ) . Dene (8.8) r ζ 1 ...,ζ d ζ ′ 1 ,...,ζ ′ d : T ( G ) − → M × M × End( C d ) , λ 7→  s ( λ ); t ( λ ); r 1 , 1 ( λ ) , . . . , r i,i ′ ( λ ) , . . . , r d,d ( λ )  . This map is eviden tly a morphism of C ∞ -spaes. By the reexivit y of G , π G indues a homeomorphism b et w een Γ and the op en subset Ω ≡ π G (Γ) of T ( G ) . Clearly ,  ζ ζ ′ | Γ = r ζ ζ ′ | Ω ◦ π G | Γ and so r ζ ζ ′ | Ω yields a homeomorphism b et w een Ω and X . W e laim that the map r ζ ζ ′ | Ω is the desired isomorphism of C ∞ -spaes. (a) In one diretion, supp ose f ∈ C ∞ ( X ) . Beause of the lo al  harater of the laim, it is no loss of generalit y to assume that f admits a smo oth extension ˜ f ∈ C ∞  M × M × End( C d )  , th us f ◦ r ζ ζ ′ | Ω = ˜ f ◦ r ζ ζ ′ | Ω is eviden tly an elemen t of R ∞ (Ω) . (b) Con v ersely , let f : X → C b e a funtion su h that f ◦ r ζ ζ ′ | Ω b elongs to R ∞ (Ω) . Sine π G is a morphism of C ∞ -spaes, the omp osite f ◦ r ζ ζ ′ | Ω ◦ π G | Γ = f ◦  ζ ζ ′ | Γ will b elong to C ∞ (Γ) . As  ζ ζ ′ | Γ is a dieomorphism, it follo ws that f ∈ C ∞ ( X ) . The laim is pro v en. Summarizing our onlusions: 8.6 Prop osition Let G b e a reexiv e group oid (Denition 8.3 ). Then the en v eloping homomorphism π G is an isomorphism of C ∞ -spaes; it follo ws that the T annakian group oid T ( G ) is a Hausdor Lie group oid, isomorphi to G . W e shall no w turn our atten tion to a v ery deliate issue, namely the inje- tivit y of the en v eloping homomorphism. Clearly , π G is injetiv e if and only if G admits enough represen tations; this means that for ea h x ∈ M and g 6 = x in the x -th isotrop y group of G there is a represen tation ( E ,  ) su h that  ( g ) 6 = id ∈ Aut( E x ) . F or a generi Lie group oid G , this prop ert y dramatially dep ends on the t yp e of represen tations one is onsidering. W e laim that ea h prop er Lie group oid admits enough represen tations on smo oth Eulidean elds (fr  5.9). F or the rest of the setion, w e shall exlusiv ely deal with su h represen tations. 8.7. Cut-o funtions. 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 o v er a manifold M . Reall that a subset S ⊂ M is said to b e in v arian t when s ∈ S ⇒ g · s ∈ S 44 for all arro ws g . If S is an y subset of M , w e let G · S denote the saturation of S , that is to sa y the smallest in v arian t subset of M on taining S . The saturation of an op en subset is also op en. It is an easy exerise to sho w that G · V = G · V for all op en subsets V with ompat losure. It follo ws that if U is an in v arian t op en subset of M then U oinides with the union o v er all in v arian t op en subsets V whose losure is ompat and on tained in U . The last remark applies to the onstrution of G -in v arian t partitions of unit y o v er 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 in v arian t 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 b y Corollary 6.8 there exists an in v arian t smo oth funtion on M whi h tak es the v alue one at x 0 and v anishes outside V . 8.8. Extendability of pr op er Lie gr oup oid ations on smo oth Eulide an elds. Let G b e a prop er Lie group oid, with base M . Supp ose w e are giv en a partial represen tation ( E U ,  U ) of G | U on a smo oth Eulidean eld E U o v er U , where U is an in v arian t op en neigh b ourho o d of a p oin t x 0 in M . W e w an t to sho w that there exists a global represen tation ( E ,  ) of G on a smo oth Eulidean eld E su h that ( E U ) 0 ≡ ( E U ) x 0 and E 0 ≡ E x 0 are isomorphi G -mo dules, where G is the isotrop y group of G at x 0 . T o b egin with, w e x an y in v arian t smo oth funtion a ∈ C ∞ ( M ) with a ( x 0 ) = 1 and supp a ⊂ U (ut-o funtion). Let V denote the set of all x su h that a ( x ) 6 = 0 . Dene E x to b e the bre ( E U ) x if x ∈ V and { 0 } otherwise. Let Γ E b e the follo wing sheaf of setions of the bundle { E x } : (8.9) W 7→  prolongation of aζ b y zero : ζ ∈ Γ ( E U )( U ∩ W )  . These data dene a smo oth Eulidean eld E o v er M . Dene  ( g ) to b e  U ( g ) if g ∈ G | V and the zero map otherwise. The bundle of linear maps   ( g ) : ( s ∗ E ) g ∼ → ( t ∗ E ) g  will pro vide an ation of G on E as long as it is a morphism of smo oth Eulidean elds o v er G of s ∗ E in to t ∗ E . No w, b y the in v ariane of a and the lo al expression (6.3) for  U , one has  ( g )[ aζ ( s g )] = a ( s g )  ( g ) ζ ( s g ) = a ( t g ) d P i =1 r i ( g ) ζ ′ i ( t g ) = d P i =1 r i ( g )[ aζ ′ i ( t g )] , as desired. Finally , the iden tit y E 0 = ( E U ) x 0 (b y onstrution) is a G - equiv arian t isomorphism. 45 Putting Theorem 8.4, Prop osition 8.6, the onsiderations of 1.5 and those of the last subsetion together, w e onlude 8.9 Theorem (Reonstrution Theorem) Within the t yp e E uc ∞ of smo oth Eulidean elds, ev ery prop er Lie group oid is reexiv e, that is to sa y C ∞ -isomorphi to its T annakian group oid via the orresp onding en v eloping homomorphism. 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