aa r X i v : . [ m a t h . C T ] S e p Giorgio TrentinagliaTANNAKA DUALITY FORPROPER LIE GROUPOIDS † (PhD Thesis, Utre ht University, 2008) † This work was (cid:28)nan ially supported by Utre ht University, the Universityof Padua, and a grant of the foundation (cid:16)Fondazione Ing. Aldo Gini(cid:17)Abstra t: The main ontribution of this thesis is a Tannaka duality theoremfor proper Lie groupoids. This result is obtained by repla ing the ategory ofsmooth ve tor bundles over the base manifold of a Lie groupoid with a larger ategory, the ategory of smooth Eu lidean (cid:28)elds, and by onsidering smootha tions of Lie groupoids on smooth Eu lidean (cid:28)elds. The notion of smoothEu lidean (cid:28)eld that is introdu ed here is the smooth, (cid:28)nite dimensional ana-logue of the familiar notion of ontinuous Hilbert (cid:28)eld. In the se ond partof the thesis, ordinary smooth representations of Lie groupoids on smoothve tor bundles are systemati ally studied from the point of view of Tannakaduality, and various results are obtained in this dire tion.Keywords: proper Lie groupoid, representation, tensor ategory, Tannakaduality, sta kAMS Subje t Classi(cid:28) ations: 58H05, 18D10A knowledgements: I would like to thank my supervisor, I. Moerdijk, forhaving suggested the resear h problem out of whi h the present work tookshape and for several useful remarks, and also M. Craini and N. T. Zung,for their interest and for helpful onversations.ontentsTable of Contents 3Introdu tion 5From Lie groups to Lie groupoids . . . . . . . . . . . . . . . . . . . 5Histori al perspe tive on Tannaka duality . . . . . . . . . . . . . . 7What is new in this thesis . . . . . . . . . . . . . . . . . . . . . . . 9Outline hapter by hapter . . . . . . . . . . . . . . . . . . . . . . . 10Some possible appli ations . . . . . . . . . . . . . . . . . . . . . . . 19I Lie Groupoids, Classi al Representations 21Ÿ1 Generalities about Lie Groupoids . . . . . . . . . . . . . . . . 21Ÿ2 Classi al Representations . . . . . . . . . . . . . . . . . . . . . 24Ÿ3 Normalized Haar Systems . . . . . . . . . . . . . . . . . . . . 31Ÿ4 The Lo al Linearizability Theorem . . . . . . . . . . . . . . . 32Ÿ5 Global Quotients . . . . . . . . . . . . . . . . . . . . . . . . . 36II The Language of Tensor Categories 39Ÿ6 Tensor Categories . . . . . . . . . . . . . . . . . . . . . . . . . 39Ÿ7 Tensor Fun tors . . . . . . . . . . . . . . . . . . . . . . . . . . 44Ÿ8 Complex Tensor Categories . . . . . . . . . . . . . . . . . . . 46Ÿ9 Review of Groups and Tannaka Duality . . . . . . . . . . . . . 48Ÿ10 A Te hni al Lemma on Compa t Groups . . . . . . . . . . . . 50III Representation Theory Revisited 55Ÿ11 The Language of Fibred Tensor Categories . . . . . . . . . . . 55Ÿ12 Smooth Tensor Sta ks . . . . . . . . . . . . . . . . . . . . . . 60Ÿ13 Foundations of Representation Theory . . . . . . . . . . . . . 63Ÿ14 Homomorphisms and Morita Invarian e . . . . . . . . . . . . . 65IV General Tannaka Theory 71Ÿ15 Sta ks of Smooth Fields . . . . . . . . . . . . . . . . . . . . . 71Ÿ16 Smooth Eu lidean Fields . . . . . . . . . . . . . . . . . . . . . 78Ÿ17 Constru tion of Equivariant Maps . . . . . . . . . . . . . . . . 81Ÿ18 Fibre Fun tors . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 CONTENTSŸ19 Properness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Ÿ20 Re onstru tion Theorems . . . . . . . . . . . . . . . . . . . . 98V Classi al Fibre Fun tors 109Ÿ21 Basi De(cid:28)nitions and Properties . . . . . . . . . . . . . . . . . 110Ÿ22 Tame Submanifolds of a Lie Groupoid . . . . . . . . . . . . . 114Ÿ23 Smoothness, Representative Charts . . . . . . . . . . . . . . . 126Ÿ24 Morphisms of Fibre Fun tors . . . . . . . . . . . . . . . . . . . 134Ÿ25 Weak Equivalen es . . . . . . . . . . . . . . . . . . . . . . . . 137VI Classi al Tannaka Theory 143Ÿ26 The Classi al Envelope of a Proper Groupoid . . . . . . . . . 144Ÿ27 Proper Regular Groupoids . . . . . . . . . . . . . . . . . . . . 150Ÿ28 Classi al Re(cid:29)exivity: Examples . . . . . . . . . . . . . . . . . 153Bibliography 156Index 160ntrodu tionAlthough a rigorous formulation of the problem with whi h this do toralthesis is on erned will be possible only after the entral ideas of Tannakaduality theory have been at least brie(cid:29)y dis ussed, I an nevertheless startwith some omments about the general ontext where su h a problem takesits appropriate pla e. Roughly speaking, my study aims at a better under-standing of the relationship that exists between a given Lie groupoid andthe orresponding ategory of representations. First of all, for the bene(cid:28)t ofnon-spe ialists, I want to explain the reasons of my interest in the theoryof Lie groupoids (a pre ise de(cid:28)nition of the notion of Lie groupoid an befound in Ÿ1 of this thesis) by drawing attention to the prin ipal appli ationsthat justify the importan e of this theory; in the se ond pla e, I intend toundertake a riti al examination of the on ept of representation in order to onvin e the reader of the naturalness of the notions I will introdu e below.From Lie groups to Lie groupoidsGroupoids make their appearan e in diverse mathemati al ontexts. As thename `groupoid' suggests, this notion generalizes that of group. In order toexplain how and to make the de(cid:28)nition more plausible, it is best to startwith some examples.The reader is ertainly familiar with the notion of fundamental group ofa topologi al spa e. The onstru tion of this group presupposes the hoi e ofa base point, and any two su h hoi es give rise to the same group providedthere exists a path onne ting the base points (for this reason one usuallyassumes that the spa e is path onne ted). However, instead of onsideringonly paths starting and ending at the same point, one might more generallyallow paths with arbitrary endpoints; two su h paths an still be omposedas long as the one starts where the other ends. One obtains a well-de(cid:28)ned as-so iative partial operation on the set of homotopy lasses of paths with (cid:28)xedendpoints, for whi h the ( lasses of) onstant paths are both left and rightneutral elements. Observe that ea h path has a two-sided inverse, namely thepath itself with reverse orientation.In geometry, groups are usually groups of transformations(cid:22)or symme-tries(cid:22)of some obje t or spa e. If g is an element of a group G a ting on a5 INTRODUCTIONspa e X and x is a point of X , one may think of the pair ( g, x ) as an arrowgoing from x to g · x ; again, two su h arrows an be omposed in an obviousway, by means of the group operation of G , provided one starts where theother ends. Composition of arrows is an asso iative partial operation on theset G × X , whi h en odes both the multipli ation law of the group G andthe G -a tion on X .In the representation theory of groups, the linear group GL ( V ) asso iatedwith a (cid:28)nite dimensional ve tor spa e V plays a fundamental role. If a ve torbundle E over a spa e X is given instead of a single ve tor spa e V , one an onsider the set GL ( E ) of all triples ( x, x ′ , λ ) onsisting of two points of X and a linear isomorphism λ : E x ∼ → E x ′ between the (cid:28)bres over these points.As in the examples above, an element ( x, x ′ , λ ) of this set an be viewed asan arrow going from x to x ′ ; su h an arrow an be omposed with anotherone as long as the latter has the form ( x ′ , x ′′ , λ ′ ) . Arrows of the form ( x, x, id ) are both left and right neutral elements for the resulting asso iative partialoperation, and ea h arrow admits a two-sided inverse.By abstra tion from these and similar examples, one is led to onsidersmall ategories where every arrow is invertible. Su h ategories are referredto as groupoids. More expli itly, a groupoid onsists of a spa e X of (cid:16)basepoints(cid:17) (also alled obje ts), a set G of (cid:16)arrows(cid:17), endowed with sour e andtarget proje tions s , t : G → X , and an asso iative partial omposition law G s × t G → G (de(cid:28)ned for all pairs of arrows ( g ′ , g ) with the property that thesour e of g ′ equals the target of g ), su h that in orresponden e with ea hpoint x of X there is a (ne essarily unique) (cid:16)neutral(cid:17) or (cid:16)unit(cid:17) arrow, oftenitself denoted by x , and every arrow is invertible.The notion of Lie groupoid generalizes that of Lie group. Mu h the sameas a Lie group is a group endowed with a smooth manifold stru ture ompat-ible with the multipli ation law and with the operation of taking the inverse,a Lie groupoid is a groupoid where the sets X and G are endowed with asmooth manifold stru ture that makes the various maps whi h arise from thegroupoid stru ture smooth. For instan e, in ea h of the examples above oneobtains a Lie groupoid when the spa e X of base points is a smooth manifold, G is a Lie group a ting smoothly on X and E is a smooth ve tor bundle over X ; these Lie groupoids are respe tively alled the fundamental groupoid ofthe manifold X , the translation groupoid asso iated with the smooth a tionof G on X and the linear groupoid asso iated with the smooth ve tor bun-dle E . There is also a more general notion of C ∞ -stru tured groupoid, aboutwhi h we shall spend a few words later on in the ourse of this introdu tion,whi h we introdu e in our thesis in order to des ribe ertain groupoids thatarise naturally in the study of Tannaka duality theory.In the ourse of the se ond half of the twentieth entury the notion ofgroupoid turned out to be very useful in many bran hes of mathemati s,although this notion had in fa t already been in the air sin e the earliest a -istori al perspe tive on Tannaka duality 7 omplishments of quantum me hani s(cid:22)think, for example, of Heisenberg'sformalism of matri es(cid:22)or, more ba k in time, sin e the (cid:28)rst investigationsinto lassi(cid:28) ation problems in geometry. Nowadays, the theory of Lie group-oids onstitutes the preferred language for the geometri al study of foliations[27℄; the same theory has appli ations to non ommutative geometry [8, 5℄ andquantization deformation theory [21℄, as well as to symple ti and Poissongeometry [36, 9, 15℄. Another sour e of examples omes from the study oforbifolds [25℄; this subje t is onne ted with the theory of sta ks, whi h origi-nated in algebrai geometry from Grothendie k's suggestion to use groupoidsas the right notion to understand moduli spa es.When trying to extend representation theory from Lie groups to Lie group-oids, one is (cid:28)rst of all onfronted with the problem of de(cid:28)ning a suitablenotion of representation for the latter. As far as we are on erned, we wouldlike to generalize the familiar notion of ((cid:28)nite dimensional) Lie group repre-sentation, by whi h one generally means a homomorphism G → GL ( V ) ofa Lie group G into the group of automorphisms of some (cid:28)nite dimensionalve tor spa e V , so that as many onstru tions and results as possible an beadapted to Lie groupoids without essential hanges; in parti ular, we wouldlike to arry over Tannaka duality theory (see the next subse tion) to therealm of Lie groupoids.The notion of Lie group representation re alled above has an obviousnaive extension to the groupoid setting. Namely, a representation of a Liegroupoid G an be de(cid:28)ned as a Lie groupoid homomorphism G → GL ( E ) (smooth fun tor) into the linear groupoid asso iated with some smooth ve torbundle E over the manifold of obje ts of G . Any su h representation assignsea h arrow x → x ′ of G a linear isomorphism E x ∼ → E x ′ in su h a waythat omposition of arrows is respe ted. In our dissertation we will use theterm ` lassi al representation' to refer to this notion. Unfortunately, lassi alrepresentations prove to be ompletely inadequate for the above-mentionedpurpose of arrying forward Tannaka duality to Lie groupoids; we shall saysomething more about this matter later.The pre eding onsideration leads us to introdu e a di(cid:27)erent notion ofrepresentation for Lie groupoids. In doing this, however, we adhere to thepoint of view that the latter should be as lose as possible to the notionof lassi al representation(cid:22)in parti ular the new theory should extend thetheory of lassi al representations(cid:22)and that moreover in the ase of groupsone should re over the usual notion of representation re alled above.Histori al perspe tive on Tannaka dualityIt has been known for a long time, and pre isely sin e the pioneer workof Pontryagin and van Kampen in the 1930's, that a ommutative lo ally ompa t group an be identi(cid:28)ed with its own bidual. Re all that if G is su h INTRODUCTIONa group then its dual is the group formed by all the hara ters on G , thatis to say the ontinuous homomorphisms of G into the multipli ative groupof omplex numbers of absolute value one, the group operation being givenby pointwise multipli ation of omplex fun tions; one may regard the lattergroup as a topologi al group(cid:22)in fa t, a lo ally ompa t one(cid:22)by taking thetopology of uniform onvergen e on ompa t subsets. There is a anoni alpairing between G and this dual, given by pointwise evaluation of hara tersat elements of G , whi h indu es a ontinuous homomorphism of G into itsown bidual. Then one an prove that the latter orresponden e is a tuallyan isomorphism of topologi al groups; see for instan e Dixmier (1969) [13℄,Rudin (1962) [31℄, or the book by Chevalley (1946) [6℄.When one tries to generalize this duality result to non-Abelian lo ally ompa t groups, su h as for instan e Lie groups, it be omes evident that thewhole ring of representations must be onsidered be ause hara ters are nolonger su(cid:30) ient to re apture the group. However, it is still an open problemto formulate and prove a general duality theorem for non ommutative Liegroups: even the ase of simple algebrai groups is not well understood, de-spite the enormous a umulating knowledge on their irredu ible representa-tions. The situation is quite the opposite when the group is ompa t, be ausethe dual obje t G ∨ of a ompa t group G is dis rete and so belongs to therealm of algebra: in this ase, there is a good duality theory due to H. Peter,H. Weyl and T. Tannaka, whi h we now pro eed to re all.The early duality theorems of Tannaka (1939) [34℄ and Krein (1949) [20℄ on entrate on the problem of re onstru ting a ompa t group from thering of isomorphism lasses of its representations. Owing to the ideas ofGrothendie k [32℄, these results an nowadays be formulated within an ele-gant ategori al framework. Although we do not intend to enter into detailsnow, these ideas are impli it in what we are about to say.1. One starts by onsidering the ategory R ( G ) of all ontinuous (cid:28)nitedimensional representations of the ompa t group G : the obje ts of R ( G ) are the pairs ( V, ̺ ) onsisting of a (cid:28)nite dimensional real ve tor spa e V anda ontinuous homomorphism ̺ : G → GL ( V ) ; the morphisms are pre iselythe G -equivariant linear maps.2. There is an obvious fun tor ω of the ategory R ( G ) into that of (cid:28)nitedimensional real ve tor spa es, namely the forgetful fun tor ( V, ̺ ) V .The natural endomorphisms of ω form a topologi al algebra End( ω ) , whenone endows End( ω ) with the oarsest topology making ea h map λ λ ( R ) ontinuous as R ranges over all obje ts of R ( G ) .3. The subset T ( G ) of this algebra, formed by the elements ompatiblewith the tensor produ t operation on representations, in other words thenatural endomorphisms λ of ω su h that λ ( R ⊗ R ′ ) = λ ( R ) ⊗ λ ( R ′ ) and λ ( ) = id , proves to be a ompa t group.4. (Tannaka) The anoni al map π : G → End( ω ) , de(cid:28)ned by settinghat is new in this thesis 9 π ( g )( R ) = ̺ ( g ) for ea h obje t R = ( V, ̺ ) of R ( G ) , establishes an iso-morphism of topologi al groups between G and T ( G ) .What is new in this thesisWe are now ready to give a short summary of the original ontributions ofthe present study.Within the realm of Lie groupoids, proper groupoids play the same roleas ompa t groups; for example, all isotropy groups of a proper Lie groupoidare ompa t (the isotropy group at a base point x onsists of all arrows g with s ( g ) = t ( g ) = x ). The main result of our resear h is a Tannaka dualitytheorem for proper Lie groupoids, whi h takes the following form.To begin with, we onstru t, for ea h smooth manifold X , a ategorywhose obje ts we all smooth (cid:28)elds over X ; our notion of smooth (cid:28)eld is theanalogue, in the smooth and (cid:28)nite dimensional setting in whi h we are inter-ested, of the familiar notion of ontinuous Hilbert (cid:28)eld introdu ed by Dixmierand Douady in the early 1960's [14℄ (see also Bos [2℄ or Kali²nik [19℄ for morere ent work related to ontinuous Hilbert (cid:28)elds). The ategory of smooth(cid:28)elds is a proper enlargement of the ategory of smooth ve tor bundles. Likefor ve tor bundles, one an de(cid:28)ne a notion of Lie groupoid representation ona smooth (cid:28)eld in a ompletely standard way. Given a Lie groupoid G , su hrepresentations and their obvious morphisms form a ategory that is relatedto the ategory of smooth (cid:28)elds over the base manifold M of G by means ofa forgetful fun tor of the former into the latter ategory. To this fun tor one an assign, by generalizing the onstru tion explained above in the ase ofgroups, a groupoid over M , to whi h we shall refer as the Tannakian groupoidasso iated with G , to be denoted by T ( G ) , endowed with a natural andidatefor a smooth stru ture on the spa e of arrows ( C ∞ -stru tured groupoid). Asfor groups, there is a anoni al homomorphism π of G into T ( G ) that turnsout to be ompatible with this C ∞ -stru ture.Our Tannaka duality theorem for proper Lie groupoids reads as follows:Theorem Let G be a proper Lie groupoid. The C ∞ -stru ture on thespa e of arrows of the Tannakian groupoid T ( G ) is a genuine manifoldstru ture so that T ( G ) is a Lie groupoid. The anoni al homomorphism π is a Lie groupoid isomorphism G ∼ = T ( G ) .The main point here is to prove the surje tivity of the homomorphism π ; thefa t that π is inje tive is a dire t appli ation of a theorem of N.T. Zung.A tually, the reasonings leading to our duality theorem also hold, for themost part, for the representations of a proper Lie groupoid on ve tor bundles.Sin e from the very beginning of our resear h we were equally interested instudying su h representations, we found it onvenient to provide a generaltheoreti al framework where the diverse approa hes to the representation0 INTRODUCTIONtheory of Lie groupoids ould take their appropriate pla e, so as to state ourresults in a uniform language. The out ome of su h demand was the theoryof `smooth tensor sta ks'. Smooth ve tor bundles and smooth (cid:28)elds are twoexamples of smooth tensor sta ks. Ea h smooth tensor sta k gives rise toa orresponding notion of representation for Lie groupoids; then, for ea hLie groupoid one obtains, by the same general pro edure outlined above, a orresponding Tannakian groupoid, whi h will depend very mu h, in general,on the initial hoi e of a smooth tensor sta k (for example, Tannaka dualityfails in the ontext of representations on ve tor bundles).Our remaining ontributions are mainly on erned with the study ofTannakian groupoids arising from representations of proper Lie groupoidson ve tor bundles. Sin e in this ase the re onstru ted groupoid may not beisomorphi to the original one, the problem of whether the aforesaid standard C ∞ -stru ture on the spa e of arrows of the Tannakian groupoid turns thelatter groupoid into a Lie groupoid be omes onsiderably more interestingand di(cid:30) ult than in the ase of representations on smooth (cid:28)elds. Our prin- ipal result in this dire tion is that the answer to the indi ated question isa(cid:30)rmative for all proper regular groupoids. In onne tion with this result weprove invarian e of the solvability of the problem under Morita equivalen e.Finally, we provide examples of lassi ally re(cid:29)exive proper Lie groupoids, i.e.proper Lie groupoids for whi h the groupoid re onstru ted from the repre-sentations on ve tor bundles is isomorphi to the original one; however, ourlist is very short: failure of re(cid:29)exivity is the rule rather than the ex eptionwhen one deals with representations on ve tor bundles.Outline hapter by hapterIn order to help the reader (cid:28)nd their own way through the dissertation, wegive here a detailed a ount of how the material is organized. ∗ ∗ ∗ In Chapter I we re all basi notions and fa ts on erning Lie groupoids.The initial se tion is mainly about de(cid:28)nitions, notation and onventionsto be followed in the sequel.The se ond se tion ontains relatively more interesting material: afterbrie(cid:29)y re alling the familiar notion of a representation of a Lie groupoid ona ve tor bundle ( lassi al representation), we supply a on rete example,1whi h motivates our introdu ing the notion of representation on a smooth(cid:28)eld in Chapter IV, showing that it is in general impossible to distinguishtwo Lie groupoids from one another just on the basis of knowledge of the1We dis overed this ounterexample independently, though it turned out later that thesame had already been around for some time [23℄.utline hapter by hapter 11 orresponding ategories of representations on ve tor bundles; more pre isely,we shall expli itly onstru t a prin ipal T -bundle over the ir le (where T k denotes the k -torus), together with a homomorphism onto the trivial T -bundle over the ir le, su h that the obvious pull-ba k of representationsalong this homomorphism yields an isomorphism between the ategories of lassi al representations of these two bundles of Lie groups.In Se tion 3 we review the notion of a (normalized) Haar system on a Liegroupoid; this is the analogue, for Lie groupoids, of the notion of (probability)Haar measure on a group. Like probability Haar measures, normalized Haarsystems an be used to obtain invariant fun tions, metri s et . by meansof the usual averaging te hnique. The possibility of onstru ting equivariantmaps lies at the heart of our proof that the homomorphism π mentionedabove is surje tive for every proper Lie groupoid.Se tion 4 introdu es the reader to a relatively re ent result obtained byN.T. Zung about the lo al stru ture of proper Lie groupoids; this general re-sult was (cid:28)rst onje tured by A. Weinstein in his famous paper about the lo- al linearizability of proper regular groupoids [37℄ (where the result is provedpre isely under the additional assumption of regularity). Zung's lo al lin-earizability theorem states that ea h proper Lie groupoid G is, lo ally in thevi inity of any given G -invariant point of its base manifold, isomorphi tothe translation groupoid asso iated with the indu ed linear a tion of the iso-tropy group of G at the point itself on the respe tive tangent spa e. As a onsequen e of this, every proper Lie groupoid is lo ally Morita equivalentto the translation groupoid asso iated with some ompa t Lie group a tion.The lo al linearizability of proper Lie groupoids a ounts for the inje tivityof the homomorphism π .Finally, in Se tion 5, we prove a statement relating the global stru ture upto Morita equivalen e of a proper Lie groupoid and the existen e of globallyfaithful representations: pre isely, we show that a proper Lie groupoid admitsa globally faithful representation on a smooth ve tor bundle if and only if it isMorita equivalent to the translation groupoid of a ompa t Lie group a tion.Although this result is not elsewhere used in our work, we present a proof of ithere be ause we believe that the same te hnique, applied to representationson smooth (cid:28)elds, may be used to obtain nontrivial information about theglobal stru ture of arbitrary proper Lie groupoids (sin e every su h groupoidtrivially admits globally faithful representations on smooth (cid:28)elds). ∗ ∗ ∗ Chapter II is mainly on erned with the ba kground notions needed in orderto formulate pre isely the re onstru tion problem in full generality. The for-mal ategori al framework within whi h this problem is most onvenientlystated in the language of tensor ategories and tensor fun tors.Se tion 6 introdu es the pivotal notion of a tensor ategory: this willbe, for us, an additive k -linear ategory C ( k = real or omplex numbers)2 INTRODUCTIONendowed with a bilinear bifun tor ( A, B ) A ⊗ B : C × C → C alled atensor produ t, a distinguished obje t alled the tensor unit and variousnatural isomorphisms alled ACU onstraints whi h, roughly speaking, makethe produ t ⊗ asso iative and ommutative with neutral element . Thenotion of rigid tensor ategory is also brie(cid:29)y re alled: this is a tensor ategorywith the property that ea h obje t R admits a dual, that is an obje t R ′ forwhi h there exist morphisms R ′ ⊗ R → and → R ⊗ R ′ ompatible withone another in an obvious sense; the ategory of (cid:28)nite dimensional ve torspa es(cid:22)or, more generally, smooth ve tor bundles over a manifold(cid:22)is anexample.In Se tion 7 we review the notions of a tensor fun tor (morphism oftensor ategories) and a tensor preserving natural transformation (morphismof tensor fun tors): one obtains a tensor fun tor by atta hing, to an ordinaryfun tor F , (natural) isomorphisms F ( A ) ⊗ F ( B ) ∼ = F ( A ⊗ B ) and ∼ = F ( ) , alled tensor fun tor onstraints, ompatible with the ACU onstraints of thetwo tensor ategories involved; a tensor preserving natural transformation oftensor fun tors is simply an ordinary natural transformation λ su h that λ ( A ⊗ B ) = λ ( A ) ⊗ λ ( B ) and λ ( ) = id up to the obvious identi(cid:28) ationsprovided by the tensor fun tor onstraints. If an obje t R admits a dual R ′ in the above sense, then λ ( R ) is an isomorphism for any tensor preserving λ (a tensor preserving fun tor will preserve duals whenever they exist). Afundamental example of tensor fun tor is the pull-ba k of smooth ve torbundles along a smooth mapping of manifolds.Se tion 8 hints at the relationship between real and omplex theory: tomention one example, in the ase of groups one an either onsider linearrepresentations on real ve tor spa es and then take the group of all tensorpreserving natural automorphisms of the standard forgetful fun tor or, alter-natively, onsider linear representations on omplex ve tor spa es and thentake the group of all self- onjugate tensor preserving natural automorphisms;these two groups, of ourse, will turn out to be the same. We indi ate howthese omments may be generalized to the abstra t ategori al setting wehave just outlined to the reader.Se tion 9 is devoted to a on ise exposition, without any ambition to ompleteness, of the algebrai geometer's point of view on Tannaka duality.In fa t, many fundamental aspe ts of the algebrai theory are omitted here;we refer more demanding readers to Saavedra (1972) [32℄, Deligne and Milne(1982) [12℄ and Deligne (1990) [11℄. We thought it ne essary to in lude thisexposition with the intent of providing adequate grounds for understanding ertain questions reaised in Chapter V.Contrary to the rest of the hapter, Se tion 10 is entirely based on ourown work. In this se tion we prove a key te hni al lemma whi h we exploitlater on, in Se tion 20, to establish the surje tivity of the envelope homo-morphism π (see above) for all proper Lie groupoids; this lemma redu es theutline hapter by hapter 13latter problem to that of he king that a ertain extendability ondition formorphisms of representations is satis(cid:28)ed. The proof of our result makes use ofthe lassi al Tannaka duality theorem for ompa t (Lie) groups, though forthe rest it is purely algebrai and it does not reprodu e any known argument. ∗ ∗ ∗ In Chapter III, we introdu e our abstra t systematization of representationtheory. Our ideas took shape gradually, during the attempt to make the treat-ment of various inequivalent approa hes to the representation theory of Liegroupoids uniform. A ollateral bene(cid:28)t of this abstra tion e(cid:27)ort was a gainin simpli ity and formal elegan e, along with a general better understandingof the mathemati al features of the theory itself.We begin with the des ription of a ertain ategori al stru ture, that weshall all (cid:28)bred tensor ategory, whi h permits to make sense of the notion of`Lie groupoid a tion' in a natural way. Smooth ve tor bundles and smooth(cid:28)elds provide examples of su h a stru ture. A (cid:28)bred tensor ategory C maybe de(cid:28)ned as a orresponden e that assigns a tensor ategory C ( X ) to ea hsmooth manifold X and a tensor fun tor f ∗ : C ( X ) → C ( Y ) to ea h smoothmapping f : Y → X , along with a oherent system of tensor preservingnatural isomorphisms ( g ◦ f ) ∗ ∼ = f ∗ ◦ g ∗ and id ∗ ∼ = Id . Most notions neededin representation theory an be de(cid:28)ned purely in terms of the (cid:28)bred tensor ategory stru ture, provided this enjoys some additional properties whi h wenow pro eed to summarize.In Se tion 11, we make from the outset the assumption that C is apresta k, in other words that the obvious presheaf U Hom C ( U ) ( E | U , F | U ) is a sheaf on X for all obje ts E , F of the ategory C ( X ) . We also require C to be smooth, that is to say, roughly speaking, that for ea h X there is anisomorphism of omplex algebras End( X ) ≃ C ∞ ( X ) , where X denotes thetensor unit in C ( X ) .Let C ∞ X denote the sheaf of smooth fun tions on X . For ea h smoothpresta k C one an asso iate to every obje t E of the ategory C ( X ) a sheafof C ∞ X -modules, Γ E , to be alled the sheaf of smooth se tions of E . Thelatter operation yields a fun tor of C ( X ) into the ategory of sheaves of C ∞ X -modules. One has a natural transformation Γ E ⊗ C ∞ X Γ E ′ → Γ ( E ⊗ E ′ ) ,whi h need not be an isomorphism, and an isomorphism C ∞ X ≃ Γ ( X ) of C ∞ X -modules, that behave mu h as usual tensor fun tor onstraints do. The ompatibility of the operation E Γ E with the pullba k along a smoothmap f : Y → X is measured by a anoni al natural morphism of sheaves of C ∞ Y -modules f ∗ ( Γ E ) → Γ ( f ∗ E ) . For ea h point x of X , there is a fun torwhi h assigns, to every obje t E of the ategory C ( X ) , a omplex ve torspa e E x to be referred to as the (cid:28)bre of E at x ; a lo al smooth se tion ζ ∈ Γ E ( U ) , de(cid:28)ned over an open neighbourhood U of x , will determine ave tor ζ ( x ) ∈ E x to be referred to as the value of ζ at x .4 INTRODUCTIONIn order to show that Morita equivalen es have the usual property ofindu ing a ategori al equivalen e between the ategories of representations,we further need to impose the ondition that C is a sta k. This ondition,examined in Se tion 12, means that when one is given an open over { U i } ofa (para ompa t) manifold M , along with a family of obje ts E i ∈ Ob C ( U i ) and a o y le of isomorphisms θ ij : E i | U i ∩ U j ∼ → E j | U i ∩ U j , there must be someobje t E in C ( M ) whi h admits a family of isomorphisms E | U i ∼ → E i ∈ C ( U i ) ompatible with { θ ij } . Naively speaking, one an glue obje ts in C together.When C is a smooth sta k, the ategory C ( M ) will essentially ontain the ategory of all smooth ve tor bundles over M as a full sub ategory.In Se tion 13, we lay down the foundations of the representation theoryof Lie groupoids relative to a type T , for an arbitrary smooth sta k of tensor ategories T . A representation of type T of a Lie groupoid G is a pair ( E, ̺ ) onsisting of an obje t E of the ategory T ( M ) (where M is the base of G ) and an arrow ̺ : s ∗ E → t ∗ E in the ategory T ( G ) (where s , t : G → M are the sour e resp. target map of G ) su h that u ∗ ̺ = id E (where u : M → G denotes the unit se tion) and m ∗ ̺ = p ∗ ̺ ◦ p ∗ ̺ (where m , p , p : G s × t G → G respe tively denote multipli ation, (cid:28)rst and se ond proje tion).With the obvious notion of morphism, representations of type T of a Liegroupoid G form a ategory R T ( G ) . This ategory inherits an additive lineartensor stru ture from the base ategory T ( M ) , making the forgetful fun tor ( E, ̺ ) E a stri t linear tensor fun tor of R T ( G ) into T ( M ) . The latterfun tor will be denoted by ω T ( G ) and will be alled the standard (cid:28)bre fun torof type T asso iated with G .Ea h homomorphism of Lie groupoids φ : G → H indu es a linear tensorfun tor φ ∗ : R T ( H ) → R T ( G ) that we all the pullba k along φ . One hastensor preserving natural isomorphisms ( ψ ◦ φ ) ∗ ∼ = φ ∗ ◦ ψ ∗ . In Se tion 14 weshow that for every Morita equivalen e φ : G → H the pullba k fun tor φ ∗ isan equivalen e of tensor ategories. ∗ ∗ ∗ Chapter IV is the ore of our dissertation. This is the pla e where we des ribethe general duality theory for Lie groupoids in the abstra t framework ofChapters II(cid:21)III and where we prove our most important results, ulminatingin the above-mentioned re onstru tion theorem for proper Lie groupoids.Se tion 15 ontains a detailed des ription of in what type of Lie groupoidrepresentations one should be interested, from our point of view, when dealingwith duality theory of Lie groupoids. Namely, we say that a type T is a sta kof smooth (cid:28)elds if it meets a number of extra requirements, alled `axioms',whi h we now pro eed to summarize.Our (cid:28)rst axiom says that the anoni al morphisms Γ E ⊗ C ∞ X Γ E ′ → Γ ( E ⊗ E ′ ) and f ∗ ( Γ E ) → Γ ( f ∗ E ) ( fr. the summary of Ch. III, Ÿ11) are sur-je tive; this axiom onveys information about the smooth se tions of E ⊗ E ′ utline hapter by hapter 15and f ∗ E and it implies that the (cid:28)bre at x of an obje t E is spanned, as ave tor spa e, by the values ζ ( x ) as ζ ranges over all germs of lo al smoothse tions of E at x .Next, re all that any arrow a : E → E ′ in T ( X ) indu es a morphismof sheaves of C ∞ X -modules Γ a : Γ E → Γ E ′ and a bundle of linear maps { a x : E x → E ′ x } ; these are mutually ompatible, in an obvious sense. Ourse ond and third axioms ompletely hara terize the arrows in T ( X ) in termsof their e(cid:27)e t on smooth se tions and the bundles of linear maps they indu e;namely, an arrow a : E → E ′ vanishes if and only if a x vanishes for all x ,and every pair formed by a morphism of C ∞ X -modules α : Γ E → Γ E ′ and a ompatible bundle of linear maps { λ x : E x → E ′ x } gives rise to a (unique)arrow a : E → E ′ su h that α = Γ a or, equivalently, λ x = a x for all x .Then there is an axiom requiring the existen e of lo al Hermitian metri son the obje ts of T ( X ) . A Hermitian metri on E is an arrow E ⊗ E ∗ → indu ing a positive de(cid:28)nite Hermitian sesquilinear form on ea h (cid:28)bre E x ; theaxiom says that for any para ompa t M , ea h obje t of T ( M ) admits Hermit-ian metri s. This assumption has many useful onsequen es: for example, itimplies various ontinuity prin iples for smooth se tions and a fundamentalextension property for arrows.The remaining two axioms impose various (cid:28)niteness onditions on T :roughly speaking, (cid:28)nite dimensionality of the (cid:28)bres of an arbitrary obje t E and lo al (cid:28)niteness of the sheaf of modules Γ E . More pre isely, one axiom anoni ally identi(cid:28)es T ( ⋆ ) , as a tensor ategory, with the ategory of (cid:28)nitedimensional ve tor spa es(cid:22)where ⋆ denotes the one-point manifold(cid:22)so that,for instan e, the fun tor E E x be omes a tensor fun tor of T ( X ) into the ategory of su h spa es; the other axiom requires the existen e, for ea hpoint x , of an open neighbourhood U su h that Γ E ( U ) is spanned, as a C ∞ ( U ) -module, by a (cid:28)nite set of se tions of E over U .In Se tion 16, we introdu e our fundamental example of a sta k of smooth(cid:28)elds (whi h is to play a role in our re onstru tion theorem for proper Liegroupoids in Ÿ20), to whi h we refer as the type E ∞ of smooth Eu lidean(cid:28)elds. The notion of smooth Eu lidean (cid:28)eld over a manifold X generalizesthat of smooth ve tor bundle over X in that the dimension of the (cid:28)bresis allowed to vary dis ontinuously over X or, in other words, the sheaf ofsmooth se tions is no longer a lo ally free C ∞ X -module. Our theory of smoothEu lidean (cid:28)elds may be regarded as the ounterpart, in the smooth setting,of the well-established theory of ontinuous Hilbert (cid:28)elds [14℄.In Se tion 17 we prove various results about the equivariant extension ofmorphisms of Lie groupoid representations whose type is a sta k of smooth(cid:28)elds; in ombination with the te hni al lemma of Ÿ10, these extension re-sults allow one to establish the surje tivity of the envelope homomorphism π asso iated with representations on an arbitrary sta k of smooth (cid:28)elds. Theproofs are based on the usual averaging te hnique(cid:22)whi h makes sense for6 INTRODUCTIONany proper Lie groupoid be ause of the existen e of normalized Haar system-s(cid:22)and, of ourse, on the axioms for sta ks of smooth (cid:28)elds.In Se tions 18(cid:21)19, we delve into the formalism of (cid:28)bre fun tors with val-ues in an arbitrary sta k of smooth (cid:28)elds. A (cid:28)bre fun tor, with values in asta k of smooth (cid:28)elds F , is a faithful linear tensor fun tor ω of some addi-tive tensor ategory C into F ( M ) , for some (cid:28)xed para ompa t manifold M to be referred to as the base of ω . This notion is obtained by abstra tingthe fundamental features, whi h allow one to make sense of the onstru -tion of the Tannakian groupoid, from the on rete example provided by thestandard forgetful fun tor asso iated with the representations of type F of aLie groupoid over M . To any (cid:28)bre fun tor ω with base M , one an assigna groupoid T ( ω ) over M to whi h we refer as the Tannakian groupoid as-so iated with ω onstru ted, like in the ase of groups, by taking all tensorpreserving natural automorphisms of ω . The set of arrows of T ( ω ) omesnaturally equipped with a topology and a smooth fun tional stru ture thatis a sheaf R ∞ of algebras of ontinuous real valued fun tions on T ( ω ) losedunder omposition with arbitrary smooth fun tions R d → R; the notion ofsmooth fun tional stru ture is analogous to that of C ∞ -ring, fr [28, 29℄.In Se tion 20, we reap the fruits of all our previous work and prove sev-eral statements of fundamental importan e about the Tannakian groupoid T ( G ) asso iated with the standard forgetful fun tor ω ( G ) on the ategory ofrepresentations of an arbitrary proper Lie groupoid G . (We are still dealingwith a situation where the type is an arbitrary sta k of smooth (cid:28)elds.) Re allthat there is a anoni al homomorphism π : G → T ( G ) de(cid:28)ned by setting π ( g )( E, ̺ ) = ̺ ( g ) , whi h, as previously mentioned, turns out to be surje tivefor proper G ; the proof of this theorem is based on the results of Se tions 10and 17. Moreover, when G is proper, the Tannakian groupoid T ( G ) be omesa topologi al groupoid and π a homomorphism of topologi al groupoids: thenwe show that inje tivity of π implies that π is an isomorphism of topologi algroupoids and that this in turn implies that the above-mentioned fun tionalstru ture on T ( G ) is a tually a Lie groupoid stru ture for whi h π be omesan isomorphism of Lie groupoids. A ordingly, we say that a Lie groupoid G isre(cid:29)exive(cid:22)relative to a ertain type(cid:22)if π indu es a homeomorphism betweenthe spa es of arrows of G and T ( G ) . Our main theorem, whi h on ludes these tion, states that every proper Lie groupoid is re(cid:29)exive relative to the type E ∞ of smooth Eu lidean (cid:28)elds. The inje tivity of π for this parti ular typeof representations is an easy onsequen e of Zung's lo al linearizabilty resultfor proper Lie groupoids. ∗ ∗ ∗ Besides establishing a Tannaka duality theory for proper Lie groupoids, thework des ribed above also leads to results on erning the lassi al theory ofrepresentations of Lie groupoids on ve tor bundles. Chapter V on entratesutline hapter by hapter 17on what an be said about the latter ase ex lusively from the abstra t stand-point of the theory of (cid:28)bre fun tors outlined in ŸŸ18(cid:21)19. The main obje tsof study here are ertain (cid:28)bre fun tors, whi h will be referred to as lassi- al (cid:28)bre fun tors, enjoying formal properties analogous to those possessedby the standard forgetful fun tor asso iated with the ategory of lassi alrepresentations of a Lie groupoid.The distin tive features of lassi al (cid:28)bre fun tors are the rigidity of the do-main tensor ategory C and the type being equal to the sta k of smooth ve torbundles. Se tion 21 olle ts some general remarks about su h (cid:28)bre fun torsand some basi de(cid:28)nitions. For any lassi al (cid:28)bre fun tor ω , the Tannakiangroupoid T ( ω ) proves to be a C ∞ -stru tured groupoid over the base M of ω ; this means that all stru ture maps of T ( ω ) are morphisms of fun tionallystru tured spa es with respe t to the C ∞ -fun tional stru ture R ∞ on T ( ω ) introdu ed in Ÿ18. One an de(cid:28)ne, for every C ∞ -stru tured groupoid T , anobvious notion of C ∞ -representation on a smooth ve tor bundle; su h rep-resentations form a tensor ategory R ∞ ( T ) . Every obje t R of the domain ategory C of a lassi al (cid:28)bre fun tor ω determines a C ∞ -representation ev R ,whi h we all evaluation at R , of the Tannakian groupoid T ( ω ) on the ve torbundle ω ( R ) . The operation R ev R provides a tensor fun tor of C into the ategory of C ∞ -representations of T ( ω ) , the evaluation fun tor asso iatedwith ω .Se tion 22 is preliminary to Se tion 23. It is devoted to a dis ussion ofthe te hni al notion of a tame submanifold whi h we introdu e in order tode(cid:28)ne representative harts in the subsequent se tion. All the reader needs toknow about tame submanifolds is that these are parti ular submanifolds ofLie groupoids with the property that whenever a Lie groupoid homomorph-ism establishes a bije tive orresponden e between two of them, the indu edbije tion is a tually a di(cid:27)eomorphism and that Morita equivalen es preservetame submanifolds.The fa t that T ( ω ) is a C ∞ -stru tured groupoid for every lassi al ω poses the question of whether T ( ω ) is a tually a Lie groupoid. In Se tion 23we start ta kling this issue by providing a ne essary and su(cid:30) ient riterion,whi h proves to be onvenient enough to use in pra ti e, for the answer tothe latter question being positive for a given ω . This riterion is expressed interms of the notion of a representative hart, that is a pair (Ω , R ) onsistingof an open subset Ω of T ( ω ) and an obje t R of the domain ategory C of ω su h that the evaluation representation at R indu es a homeomorphismbetween Ω and a tame submanifold of the linear groupoid GL ( ω R ) ; then T ( ω ) is a Lie groupoid if, and only if, representative harts over T ( ω ) and (Ω , R ⊕ S ) is a representative hart for every representative hart (Ω , R ) andfor every obje t S of C .Se tion 24 introdu es a notion of morphism for ( lassi al) (cid:28)bre fun tors.Roughly speaking, a morphism of ω into ω ′ , over a smooth mapping f : M → M ′ of the base manifolds, is a tensor fun tor of C ′ into C ompatiblewith the pullba k of ve tor bundles along f ; every morphism ω → ω ′ over f indu es a homomorphism of C ∞ -stru tured groupoids T ( ω ) → T ( ω ′ ) over f . Se tion 25 is devoted to the study of weak equivalen es of ( lassi al) (cid:28)brefun tors: we de(cid:28)ne them as those morphisms over a surje tive submersionwhi h have the property of being a ategori al equivalen e. As an appli ationof the riterion of Ÿ23, we show that if ω is weakly equivalent to ω ′ , then T ( ω ) is a Lie groupoid if and only if T ( ω ′ ) is; when this is the ase, the Liegroupoids T ( ω ) and T ( ω ′ ) turn out to be Morita equivalent. ∗ ∗ ∗ In Chapter VI, we apply the general abstra t theory of the pre eding hapterto the motivating example provided by the standard forgetful fun tor on the ategory of lassi al representations of a proper Lie groupoid G . The Tannak-ian groupoid asso iated with the latter lassi al (cid:28)bre fun tor will be denotedby T ∞ ( G ) ; in fa t, this onstru tion an be extended to a fun tor -
7→ T ∞ ( - ) of the ategory of Lie groupoids into the ategory of C ∞ -stru tured groupoidsso that the envelope homomorphism π ( - ) be omes a natural transformation ( - ) → T ∞ ( - ) . We will fo us our attention on the following two problems:in the (cid:28)rst pla e, we want to understand whether the Tannakian groupoid T ∞ ( G ) is a Lie groupoid, let us say for G proper; se ondly, we are inter-ested in examples of lassi ally re(cid:29)exive Lie groupoids, that is to say Liegroupoids G for whi h the envelope homomorphism π is an isomorphism oftopologi al groupoids between G and T ∞ ( G ) (re all that, under the assump-tion of properness, it is su(cid:30) ient that π is inje tive).In Se tion 26, we olle t what we know about the (cid:28)rst of the two above-mentioned problems in the general ase of an arbitrary proper Lie groupoid.Namely, we show that the ondition, in the riterion for smoothness of Ÿ23,that (Ω , R ⊕ S ) should be a representative hart for every representative hart (Ω , R ) and obje t S , is always satis(cid:28)ed by the standard forgetful fun toron the ategory of lassi al representations of a proper Lie groupoid G sothat T ∞ ( G ) is a (proper) Lie groupoid if and only if one an (cid:28)nd enoughrepresentative harts; if this is the ase, then the envelope map π is a fullsubmersion of Lie groupoids whose asso iated pullba k fun tor π ∗ establishesan isomorphism of the orresponding ategories of lassi al representationsinverse to the evaluation fun tor of Ÿ21.Se tion 27 prose utes the study initiated in the previous se tion by pro-viding a proof of the fa t that T ∞ ( G ) is a Lie groupoid for every properregular groupoid G . We onje ture that the same statement holds true forevery proper G , that is even without the regularity assumption.Se tion 28 ontains a list of examples of lassi ally re(cid:29)exive (proper) Liegroupoids; sin e, as Ÿ2 exempli(cid:28)es, most Lie groupoids fail to be lassi allyome possible appli ations 19re(cid:29)exive, this list annot be very long. To begin with, translation groupoidsasso iated with ompa t Lie group a tions are evidently lassi ally re(cid:29)exive.Next, we observe that any étale Lie groupoid whose sour e map is properis ne essarily lassi ally re(cid:29)exive be ause, for su h groupoids, one an makesense of the regular representation. Finally, orbifold groupoids(cid:22)by whi h wemean proper e(cid:27)e tive groupoids(cid:22)are lassi ally re(cid:29)exive be ause the stan-dard a tion on the tangent bundle of the base manifold yields a globallyfaithful lassi al representation.Some possible appli ationsThe study of lassi al (cid:28)bre fun tors in Chapter V was originally motivatedby the example treated in Chapter VI, namely the standard forgetful fun torasso iated with the ategory of lassi al representations of a Lie groupoid.However, examples of lassi al (cid:28)bre fun tors an also be found by lookinginto di(cid:27)erent dire tions.To begin with, one ould onsider representations of Lie algebroids [27,10, 16℄. Re all that a representation of a Lie algebroid g over a manifold M isa pair ( E, ∇ ) onsisting of a ve tor bundle E over M and a (cid:29)at g - onne tion ∇ on E , that is, a bilinear map Γ( g ) × Γ( E ) → Γ( E ) (global se tions), C ∞ ( M ) -linear in the (cid:28)rst argument, Leibnitz in the se ond and with vanish-ing urvature. Su h representations naturally form a tensor ategory.Another example of the same sort is provided by the singular foliationsintrodu ed by I. Androulidakis and G. Skandalis [1℄. Here one is given alo ally (cid:28)nite sheaf F of modules of ve tor (cid:28)elds over a manifold M , losedunder the Lie bra ket; this is to be thought of as indu ing a `singular' foliationof M , in that F is no longer ne essarily lo ally free and so the dimensionof the leaves may jump. Again, one an onsider pairs ( E, ∇ ) formed by ave tor bundle E over M and a morphism of sheaves ∇ : F ⊗ Γ E → Γ E enjoying formal properties analogous to those de(cid:28)ning a (cid:29)at onne tion.In his paper about the lo al linearizability of proper Lie groupoids [38℄,N.T. Zung poses the question of whether a spa e, whi h is lo ally isomorphi to the orbit spa e of a ompa t Lie group a tion, is ne essarily the orbit spa e M/ G asso iated with a proper Lie groupoid G over a manifold M . Of ourse,this question is not stated very pre isely; its rigorous formulation, as far as we an see, should be given in the following terms. Let us all a C ∞ -stru turedspa e ( X, F ∞ ) a generalized orbifold if the spa e X is Hausdor(cid:27), para om-pa t and lo ally isomorphi , as a fun tionally stru tured spa e, to an orbitspa e asso iated with some linear ompa t Lie group a tion(cid:22)in other words,lo ally isomorphi to a spa e of the form ( V /G, C ∞ V/G ) for some representation G → GL ( V ) of a ompa t Lie group G on a (cid:28)nite dimensional ve tor spa e V . The theory of fun tionally stru tured spa es suggests the right notion ofsmooth map of generalized orbifolds and hen e the right notion of isomorph-0 INTRODUCTIONism. Zung's theorem implies that the orbit spa e ( M/ G , C ∞ M/ G ) of a properLie groupoid G over a manifold M is a generalized orbifold: then the questionis whether an arbitrary generalized orbifold is a tually of this pre ise form.Classi al (cid:28)bre fun tors make their natural appearan e in onne tionwith any given generalized orbifold X . (Conventionally, we will refer to the C ∞ -stru ture of X , when ne essary, by means of the notation C ∞ X .) Let V ∞ ( X ) denote the ategory of lo ally free sheaves of C ∞ X -modules (of lo- ally (cid:28)nite rank), endowed with the standard linear tensor stru ture; onemay refer to the obje ts of this ategory as ve tor bundles over X . Choose alo ally (cid:28)nite over { U i } of X by open subsets U i su h that for ea h i thereis an isomorphism V i /G i ≈ U i ; we regard the maps φ i : V i → U i as (cid:28)xedon e and for all, and we assume, for simpli ity, that the V i all have the samedimension. Letting M be the disjoint union ` V i , one has an obvious lassi al(cid:28)bre fun tor ω XM = ω X { V i ,φ i } over M sending ea h obje t E of the ategory V ∞ ( X ) to the smooth ve tor bundle ⊕ i φ i ∗ E over M .The Tannakian groupoid T ∞ ( X ) = T ( ω XM ) is a C ∞ -stru tured groupoidwith the property that the obvious map φ : M → X indu es an isomorph-ism of fun tionally stru tured spa es between M/ T ∞ ( X ) and X ; thus, thestudy of this groupoid might be relevant to the above-mentioned problem.Similarly, the study of the Tannakian groupoids asso iated with the otherexamples might lead to interesting information about the underlying geomet-ri al obje ts, at least when the situation involves some kind of properness. Inthis onne tion, it is natural to hope for a general result relating the domain ategory of a lassi al (cid:28)bre fun tor with the ategory of C ∞ -representationsof the orresponding Tannakian groupoid, for example via the standard eval-uation fun tor des ribed in Ÿ21.A well-known onje ture, whi h has been raising some interest re ently [17,19℄, states that every proper étale Lie groupoid is Morita equivalent to thetranslation groupoid asso iated with some ompa t Lie group a tion or,equivalently, that every su h groupoid admits a globally faithful lassi alrepresentation ( fr. Ch. I, Ÿ5). This onje ture is related to the question ofwhether proper étale Lie groupoids are lassi ally re(cid:29)exive (we have alreadyobserved that the answer is a(cid:30)rmative in the e(cid:27)e tive ase, see Ch. VI, Ÿ28).It is known that for ea h groupoid G of this kind, there exist a proper e(cid:27)e -tive Lie groupoid ˜ G and a submersive epimorphism G → ˜ G ; the kernel of thishomomorphism is ne essarily a bundle of (cid:28)nite groups B embedded into G ,hen e, one gets an exa t sequen e of Lie groupoids → B ֒ → G → ˜ G → where B and ˜ G are both lassi ally re(cid:29)exive. These onsiderations stronglysuggest that one should investigate how the property of re(cid:29)exivity behaveswith respe t to Lie groupoid extensions.hapter ILie Groupoids and their Classi alRepresentationsThe present hapter is essentially introdu tory: we regard all the materialthereof as well-known. Our purpose is, (cid:28)rst of all, to (cid:28)x some notational onventions and some standard terminology on erning Lie groupoids; thisis done in Ÿ1. Next, in Ÿ2, we provide a detailed dis ussion of a on reteexample whi h is to serve as motivation for the approa h we will adopt inChapters III(cid:21)IV. In ŸŸ3(cid:21)4 we treat the two fundamental pillars on to whi hour main result holds: Haar systems and Zung's linearizability theorem; wede ided to in lude a presentation of these topi s here be ause we found itdi(cid:30) ult to provide adequate referen es for them. The hapter ends with adigression on the problem of representing a proper Lie groupoid as a globalquotient arising from a smooth ompa t Lie group a tion.Ÿ1 Generalities about Lie GroupoidsThe term groupoid refers to a small ategory where every arrow is invertible.A Lie groupoid an be approximately des ribed as an internal groupoid inthe ategory of smooth manifolds. To onstru t a Lie groupoid G one has togive a pair of manifolds of lass C ∞ G (0) and G (1) , respe tively alled manifoldof obje ts and manifold of arrows, and a list of smooth maps alled stru turemaps. The basi items in this list are the sour e map s : G (1) → G (0) and thetarget map t : G (1) → G (0) ; these have to meet the requirement that the (cid:28)bredprodu t G (2) = G (1) s × t G (1) exists in the ategory of C ∞ -manifolds. Then onehas to give a omposition map c : G (2) → G (1) , a unit map u : G (0) → G (1) andan inverse map i : G (1) → G (1) , for whi h the familiar algebrai laws must besatis(cid:28)ed.Terminology and Notation: The points x = s ( g ) and x ′ = t ( g ) are resp. alled the sour e and the target of the arrow g . We let G ( x, x ′ ) denote theset of all the arrows whose sour e is x and whose target is x ′ ; we shall use212 CHAPTER I. LIE GROUPOIDS, CLASSICAL REPRESENTATIONSthe abbreviation G| x for the isotropy or vertex group G ( x, x ) . Notationally,we will often identify a point x ∈ G (0) and the orresponding unit arrow u ( x ) ∈ G (1) . It is ostumary to write g ′ · g or g ′ g for the omposition c ( g ′ , g ) and g − for the inverse i ( g ) .Our des ription of the notion of Lie groupoid is still in omplete. It turnsout that a ouple of additional requirements are needed in order to get areasonable de(cid:28)nition.Re all that a manifold M is said to be para ompa t if it is Hausdor(cid:27)and there exists an as ending sequen e of open subsets with ompa t losure · · · ⊂ U i ⊂ U i ⊂ U i +1 ⊂ · · · su h that M = ∞ ∪ i =0 U i . A Hausdor(cid:27) manifold ispara ompa t if and only if it possesses a ountable basis of open subsets. Anyopen over of a para ompa t manifold admits a lo ally (cid:28)nite re(cid:28)nement. Anypara ompa t manifold admits partitions of unity of lass C ∞ (subordinatedto an open over; f. for instan e Lang [22℄).In order to make the (cid:28)bred produ t G (1) s × t G (1) meaningful as a manifoldand for other purposes related to our studies, we shall in lude the followingadditional onditions in the de(cid:28)nition of Lie groupoid:1. The sour e map s : G (1) → G (0) is a submersion with Hausdor(cid:27) (cid:28)bres;2. The manifold G (0) is para ompa t.Note that we do not require that the manifold of arrows G (1) is Hausdor(cid:27) orpara ompa t; a tually, this manifold is neither Hausdor(cid:27) nor se ond ount-able in many examples of interest. The de(cid:28)nition here di(cid:27)ers from that inMoerdijk and Mr£un [27℄ in that we additionally require that the manifold G (0) is para ompa t. The (cid:28)rst ondition implies at on e that the domain of the omposition map is a submanifold of the Cartesian produ t G (1) × G (1) andthat the target map is a submersion with Hausdor(cid:27) (cid:28)bres; thus, the sour e(cid:28)bres G ( x, - ) = s − ( x ) and the target (cid:28)bres G ( - , x ′ ) = t − ( x ′ ) are losedHausdor(cid:27) submanifolds of G (1) . A Lie groupoid G is said to be Hausdor(cid:27) ifthe manifold G (1) is Hausdor(cid:27).Some more Terminology: The manifold G (0) is usually alled the base ofthe groupoid G ; one also says that G is a groupoid over the manifold G (0) . Weshall often use the notation G x = G ( x, - ) = s − ( x ) for the (cid:28)bre of the sour emap over a point x ∈ G (0) . More generally, we shall write(1) G ( S, S ′ ) = (cid:8) g ∈ G (1) : s ( g ) ∈ S & t ( g ) ∈ S ′ (cid:9) , G| S = G ( S, S ) and G S = G ( S, - ) = G ( S, G (0) ) = s − ( S ) for all subsets S, S ′ ⊂ G (0) .A homomorphism of Lie groupoids is a smooth fun tor. More pre isely,a homomorphism ϕ : G → H onsists of two smooth maps ϕ (0) : G (0) → H (0) and ϕ (1) : G (1) → H (1) , ompatible with the groupoid stru ture in the sensethat s ◦ ϕ (1) = ϕ (0) ◦ s , t ◦ ϕ (1) = ϕ (0) ◦ t and ϕ (1) ( g ′ · g ) = ϕ (1) ( g ′ ) · ϕ (1) ( g ) .Lie groupoids and their homomorphisms form a ategory.1. GENERALITIES ABOUT LIE GROUPOIDS 23There is also a notion of topologi al groupoid: this is just an internalgroupoid in the ategory of topologi al spa es and ontinuous mappings. Inthe ontinuous ase the de(cid:28)nition is mu h simpler and one need not worryabout the domain of de(cid:28)nition of the omposition map. With the obviousnotion of homomorphism, topologi al groupoids onstitute a ategory.2 Example Every smooth manifold M an be regarded as a Lie groupoidby taking M itself as the manifold of arrows and the identity map id : M → M as the unit se tion. Alternatively, one an form the pair groupoid over M ; this is the Lie groupoid whose manifold of arrows is M × M and whosesour e and target map are the two proje tions.3 Example Any Lie group G an be regarded as a Lie groupoid over theone-point manifold by taking G itself as the manifold of arrows.4 Example: linear groupoids If E is a real or omplex smooth ve torbundle (of lo ally (cid:28)nite rank) over a manifold M , one an form the lineargroupoid GL ( E ) asso iated with E . This is de(cid:28)ned as the groupoid over M whose arrows x → x ′ are the linear isomorphisms E x ∼ → E x ′ between the(cid:28)bres of E over the points x and x ′ . There is an obvious smooth stru tureturning GL ( E ) into a Lie groupoid.5 Example: a tion groupoids Let G be a Lie group a ting smoothly (fromthe left) on a manifold M . Then one an de(cid:28)ne the a tion (or translation)groupoid G ⋉ M as the Lie groupoid over M whose manifold of arrows is theCartesian produ t G × M , whose sour e and target map are respe tively theproje tion onto the se ond fa tor ( g, x ) x and the a tion ( g, x ) gx andwhose omposition law is the operation(6) ( g ′ , x ′ )( g, x ) = ( g ′ g, x ) .There is a similar onstru tion M ⋊ G asso iated with right a tions.Let G be a Lie groupoid and let x be a point of its base manifold G (0) .The orbit of G (or G -orbit ) through x is the subset(7) G x def = G · x def = t (cid:0) G x (cid:1) = { x ′ ∈ G (0) |∃ g : x → x ′ } .Note that the isotropy group G| x a ts from the the right on the manifold G x ;this a tion is learly free and transitive along the (cid:28)bres of the restri tion ofthe target map t to G x . The following result holds (see [27℄ p. 115):8 Theorem Let G be a Lie groupoid and let x, x ′ ∈ G (0) . Then1. G ( x, x ′ ) is a losed submanifold of G (1) ;2. G| x is a Lie group;3. the G -orbit through x is an immersed submanifold of G (0) ;4 CHAPTER I. LIE GROUPOIDS, CLASSICAL REPRESENTATIONS4. the target map t : G x → G x proves to be a prin ipal G| x -bundle.It is worthwhile spending a ouple of words about the manifold stru ture thatis asserted to exist on the G -orbit through x . The set G x an obviously beidenti(cid:28)ed with the homogeneous spa e G x / ( G| x ) . Now, it an be proved thatthere exists a (possibly non-Hausdor(cid:27)) manifold stru ture on this quotientspa e, su h that the quotient map turns out to be a prin ipal bundle.We say that a Lie (or topologi al) groupoid G is proper if G is Hausdor(cid:27) andthe ombined sour e(cid:21)target map ( s , t ) : G (1) → G (0) × G (0) is proper (in thefamiliar sense: the inverse image of a ompa t subset is ompa t).The manifold of arrows G (1) of a proper Lie groupoid G is always para- ompa t. Indeed, by the de(cid:28)nition of Lie groupoid, the base M of G isa para ompa t manifold and therefore there exists an invading sequen e · · · ⊂ U i ⊂ U i ⊂ U i +1 ⊂ · · · ⊂ M of pre- ompa t open subsets; the in-verse images Γ i = G| U i = ( s , t ) − ( U i × U i ) form an analogous sequen e insidethe (Hausdor(cid:27)) manifold G (1) .Let x be a point of M . We know the orbit S = G x is an immersedsubmanifold of M (pre isely, there exists a unique manifold stru ture on S su h that t : G x → S is a prin ipal right G| x -bundle and the in lusion S ֒ → M an immersion). Now, it follows from the properness of G that S isa tually a submanifold of M . To see this, (cid:28)x a point s ∈ S . Sin e thereexists a lo al equivariant hart G ( x , W ) ≈ W × G| x where W is both anopen neighborhood of s in S and a submanifold of M , it will be enoughto prove the existen e of an open ball B ⊂ M at s su h that S ∩ B ⊂ W .To do this, take a sequen e of open balls B i shrinking to s : the de reasingsequen e Σ i = G ( x , B i ) − G ( x , W ) of losed subsets of the manifold G ( x , - ) is ontained in the ompa t subset G ( x , B ) and therefore, sin e T Σ i = ∅ ,there exists some i su h that G ( x , B i ) ⊂ G ( x , W ) .Ÿ2 Classi al RepresentationsIn this se tion we introdu e the ostumary notion of representation of aLie groupoid on a smooth ve tor bundle and we explain, by means of a ounterexample, why this notion is inadequate for the purpose of building apossible Tannaka duality theory for proper Lie groupoids.Let G be a Lie groupoid and let M be its base. We let R ∞ ( G ; C ) denotethe ategory of all C-linear lassi al representations of G . The obje ts of this ategory are the pairs ( E, ̺ ) onsisting of a smooth omplex ve tor bundle E (of lo ally (cid:28)nite rank) over M and a Lie groupoid homomorphism G ( s , t ) (cid:15) (cid:15) ̺ / / GL ( E ) ( s , t ) (cid:15) (cid:15) M × M id × id / / M × M ;(1)2. CLASSICAL REPRESENTATIONS 25the arrows, let us say those a : ( E, ̺ ) → ( F, ς ) , are the morphisms of ve torbundles a : E → F su h that the square E xa x (cid:15) (cid:15) ̺ ( g ) / / E x ′ a x ′ (cid:15) (cid:15) F x ς ( g ) / / F x ′ (2) ommutes for all x, x ′ ∈ M and g ∈ G ( x, x ′ ) . There is an entirely analogousnotion of R-linear lassi al representation of G , where real ve tor bundlesare used instead of omplex ones. One obtains a orresponding ategory R ∞ ( G ; R ) . Insofar as a parti ular hoi e of oe(cid:30) ients is not relevant to thesubje t matter of a dis ussion, we shall write simply R ∞ ( G ) and suppressany further referen e to oe(cid:30) ients.Lie groupoids annot always be distinguished from one another just on thebasis of knowledge of the respe tive ategories of lassi al representations;this onsideration motivates our approa h to Tannaka duality as des ribedin Chapter IV. We are going to substantiate our assertion by means of a ounterexample whi h we dis overed independently in 2005: only re entlyA. Henriques pointed out to us that the same ounterexample was alreadyknown in the ontext of orbispa e theory, see Lü k and Oliver (2001) [23℄.Re all that a Lie bundle (also known as bundle of Lie groups) is a Liegroupoid whose sour e and target map oin ide.Fix a Lie group H and hoose an automorphism χ ∈ Aut( H ) . There isa general pro edure(cid:22) ompletely analogous to the onstru tion of Möbiusbands, Klein bottles et similia(cid:22)by means of whi h one an obtain a lo allytrivial Lie bundle G = G H ; χ → S with (cid:28)bre H over the unit ir le. Put G (1) = ( R × H ) / ∼ where ∼ is the equivalen e relation(3) ( t, h ) ∼ ( t ′ , h ′ ) ⇔ t ′ − t = ℓ ∈ Z and h ′ = χ ℓ ( h ) .The bundle (cid:28)bration G (1) → S (= sour e map of G = target map of G ) isde(cid:28)ned as the unique map that makes the squareR × H / / quot. proj. (cid:15) (cid:15) R t e πit (cid:15) (cid:15) G (1) / / ____ S (4) ommute. In terms of representatives of elements of G (1) , the omposition law c : G (1) × S G (1) → G (1) an be de(cid:28)ned by setting(5) [ t ′ , h ′ ] · [ t, h ] = [ t ′ , h ′ · χ k ( h )] ,where k = t ′ − t ∈ Z and the square bra ket notation indi ates that we aretaking equivalen e lasses. This operation turns
G → S into a bundle ofgroups over the ir le, with (cid:28)bre H .6 CHAPTER I. LIE GROUPOIDS, CLASSICAL REPRESENTATIONSConsider the open over of S determined by the lo al exponentialparametrizations (0 , ∼ → U and ( − , ) ∼ → V . One has two orrespond-ing mutually ompatible trivializing harts for G (1) over S , namely(6) τ U : G (1) | U ∼ → U × H and τ V : G (1) | V ∼ → V × H :the former sends g ∈ G (1) | U to the pair ( e πit , h ) with [ t, h ] = g and < t < ,the latter sends g ∈ G (1) | V to the pair ( e πit , h ) with [ t, h ] = g and − < t < .These harts determine the di(cid:27)erentiable stru ture. Noti e, by the way, thatthe transition map between them, namely(7) τ U ◦ τ V − : ( U ∩ V ) × H ∼ → ( U ∩ V ) × H ,is given by the identity over W × H and by ( w ′ , h ) ( w ′ , χ ( h )) over W ′ × H ,if one lets (0 , ) ∼ → W and ( , ∼ → W ′ denote the two onne ted omponentsof the interse tion U ∩ V .We start by studying the omplex lassi al representations of the Liebundle G H ; χ , whi h are te hni ally easier to handle. The analogous result forreal representations will be dedu ed as a orollary.Fix a lassi al representation ( E, ̺ ) ∈ Ob R ∞ ( G ; C ) on a smooth omplexve tor bundle E of rank ℓ over S . Sin e U and V are ontra tible opensubsets of S , the ve tor bundle E will be trivial over ea h of them i.e. therewill exist smooth ve tor bundle isomorphisms(8) E | U ∼ → U × C ℓ and E | V ∼ → V × C ℓ .These will form a trivializing atlas for E over S , whose unique transitionmapping will be given by, let us say,(9) Q : W → GL ( ℓ ; C ) and Q ′ : W ′ → GL ( ℓ ; C ) .A ordingly, the Lie bundle GL ( E ) over S (that is, by abuse of notation,the restri tion of the linear groupoid GL ( E ) to the diagonal S ֒ → S × S )will be des ribed by trivializing harts of the following form(10) GL ( E ) | U ∼ → U × GL ( ℓ ; C ) and GL ( E ) | V ∼ → V × GL ( ℓ ; C ) ,whose transition map ( U ∩ V ) × GL ( ℓ ; C ) ∼ → ( U ∩ V ) × GL ( ℓ ; C ) will send w ∈ W to A Q ( w ) AQ ( w ) − and w ′ ∈ W ′ to A Q ′ ( w ′ ) AQ ′ ( w ′ ) − .In this situation one an write down orresponding lo al expressions for ̺ , namely ̺ U ( u, h ) = (cid:0) u, A U ( u, h ) (cid:1) over U and ̺ V ( v, h ) = (cid:0) v, A V ( v, h ) (cid:1) over V with A U : U × H → GL ( ℓ ; C ) a smooth family of representations of H et ., whi h make the following squares G (1) | U ̺ | U / / τ U ≈ (cid:15) (cid:15) GL ( E ) | U ≈ U (cid:15) (cid:15) G (1) | V ̺ | V / / τ V ≈ (cid:15) (cid:15) GL ( E ) | V ≈ V (cid:15) (cid:15) U × H ̺ U / / ___ U × GL ( ℓ ; C ) V × H ̺ V / / ___ V × GL ( ℓ ; C ) (11)2. CLASSICAL REPRESENTATIONS 27 ommute. If we take their restri tions to W , W ′ respe tively, we obtain W × H ̺ V / / ___ W × GL ( ℓ ; C ) W ′ × H ̺ V / / ___ W ′ × GL ( ℓ ; C ) G (1) | W ̺ | W / / τ U ≈ (cid:15) (cid:15) τ V ≈ O O GL ( E ) | W ≈ U (cid:15) (cid:15) ≈ V O O G (1) | W ′ ̺ | W ′ / / τ U ≈ (cid:15) (cid:15) τ V ≈ O O GL ( E ) | W ′ ≈ U (cid:15) (cid:15) ≈ V O O W × H ̺ U / / ___ W × GL ( ℓ ; C ) W ′ × H ̺ U / / ___ W ′ × GL ( ℓ ; C ) (12)and hen e, making use of the expli it expression (7) for the transition map τ U ◦ τ V − , we are led to the following relations: for all h ∈ H (13) ( A U ( w, h ) = Q ( w ) A V ( w, h ) Q ( w ) − for all w ∈ WA U (cid:0) w ′ , χ ( h ) (cid:1) = Q ′ ( w ′ ) A V ( w ′ , h ) Q ′ ( w ′ ) − for all w ′ ∈ W ′ .From now on, we assume that H is ompa t. We also (cid:28)x two points w ∈ W and w ′ ∈ W ′ . There is a ontinuous path γ U : [0 , → U from w to w ′ . This gives a ontinuous map(14) [0 , × H γ U × id −−−→ U × H A U −−→ GL ( ℓ ; C ) whi h is learly a homotopy of representations of H onne ting A U ( w , - ) to A U ( w ′ , - ) . Then, as remarked in Note 30, there will be an invertible matrix R ∈ GL ( ℓ ; C ) su h that(15) A U ( w , - ) = RA U ( w ′ , - ) R − .A se ond path γ V : [0 , → V onne ting w to w ′ will analogously yield amatrix S ∈ GL ( ℓ ; C ) su h that(16) A V ( w , - ) = SA V ( w ′ , - ) S − .Making the appropriate substitutions in (13), we (cid:28)nally (cid:28)nd an invertiblematrix P ∈ GL ( ℓ ; C ) su h that(17) A U (cid:0) w , χ ( h ) (cid:1) = P A U ( w , h ) P − for all h ∈ H .Next, we further spe ialize down to the ase where H is abelian and onne ted. Motivated by Eq. (17), we fo us our attention on those matrixrepresentations A : H → GL ( ℓ ; C ) su h that(18) ∃ P ∈ GL ( ℓ ; C ) for whi h A ( χ ( h )) = P A ( h ) P − .By S hur's Lemma, every irredu ible matrix representation of an Abelian Liegroup must be one-dimensional ( f. for instan e Brö ker and tom Die k p. 69)8 CHAPTER I. LIE GROUPOIDS, CLASSICAL REPRESENTATIONSand therefore, be ause of the ompa tness of H , ne essarily a hara ter i.e. aLie group homomorphism of H into the 1-torus T . Sin e every representationof a ompa t Lie group is a dire t sum of irredu ible ones (ibid. p. 68), it isno loss of generality to assume Eq. (18) to be of the following form(19) ( α ◦ χ )( h ) · · · ... . . . ... · · · ( α ℓ ◦ χ )( h ) = P α ( h ) · · · ... . . . ... · · · α ℓ ( h ) P − ,where α , . . . , α ℓ : H → T are hara ters of H .The two omplex diagonal matri es o urring in Eq. (19) must have thesame hara teristi polynomial p ( h, X ) ∈ C [ X ] . Thus, if we put(20) β j = α j ◦ χ : H → T and F ij = (cid:8) h ∈ H : α i ( h ) = β j ( h ) (cid:9) ,we an in parti ular express H as a (cid:28)nite union F ∪ · · · ∪ F ℓ of losedsubsets. Now, it follows by a standard indu tive argument that one of them,let us say F , must have nonempty interior; therefore, the two hara ters α and β oin ide on all of H , be ause a homomorphism of onne ted Liegroups is determined by its di(cid:27)erential at the neutral element (ibid. p. 24).Can elling the two orresponding linear fa tors in p ( h, X ) we obtain(21) (cid:0) X − β ( h ) (cid:1) · · · (cid:0) X − β ℓ ( h ) (cid:1) = (cid:0) X − α ( h ) (cid:1) · · · (cid:0) X − α ℓ ( h ) (cid:1) .Then, arguing by indu tion on the degree of the polynomial, we on ludethat there is a permutation σ on ℓ letters su h that α i = β σ ( i ) = α σ ( i ) ◦ χ forall i = 1 , . . . , ℓ .Now, onsider for instan e α . Write σ as a produ t of disjoint y les and onsider the y le (cid:0) , σ (1) , . . . , σ r (1) (cid:1) where r ≧ and σ r +1 (1) = 1 . Then wehave α = α σ (1) ◦ χ = (cid:0) α σ ( σ (1)) ◦ χ (cid:1) ◦ χ = α σ (1) ◦ χ = · · · = α σ r (1) ◦ χ r = (cid:0) α σ ( σ r (1)) ◦ χ (cid:1) ◦ χ r = α σ r +1 (1) ◦ χ r +1 = α ◦ χ r +1 . Therefore α is an exampleof a hara ter α : H → T with the spe ial property(22) ∃ r ≧ su h that α = α ◦ χ r +1 .Finally, let us take H = T = T × T to be the 2-torus. Fix an arbitrary ℓ ∈ Z, and onsider the map(23) χ ℓ : T → T de(cid:28)ned by the rule ( s, t ) ( s, s ℓ t ) .This is an automorphism of the Lie group T , with inverse χ − ℓ .Any 2- hara ter α : T → T an be written as the produ t α ( s, t ) = µ ( s ) ν ( t ) of the two 1- hara ters µ ( s ) = α ( s, and ν ( t ) = α (1 , t ) . If weassume that α enjoys the property (22) then we get µ ( s ) ν ( t ) = α ( s, t ) = α (cid:0) s, s ℓ ( r +1) t (cid:1) = µ ( s ) ν ( s ) ℓ ( r +1) ν ( t ) and therefore ν ( s ) ℓ ( r +1) = 1 for all s ∈ T .Now, if ℓ = 0 then ν must be trivial, be ause r + 1 > . It follows that(24) α ( s, t ) = µ ( s ) does not depend on t .2. CLASSICAL REPRESENTATIONS 2925 Proposition Fix any integer = ℓ ∈ Z and let G T ; χ ℓ → S bethe lo ally trivial Lie bundle with (cid:28)bre T over the ir le, onstru ted asexplained above by using χ ℓ ∈ Aut(T ) as twisting automorphism.Then there exists an embedding of Lie bundles over the ir le S × T (cid:31) (cid:127) ϕ / / (cid:15) (cid:15) G T ; χ ℓ (cid:15) (cid:15) S × S id × id / / S × S (26)with the property that every lassi al representation ( E, ̺ ) in R ∞ ( G T ; χ ℓ ) pulls ba k to a trivial representation ( E, ̺ ◦ ϕ ) of S × T .Proof De(cid:28)ne the embedding ϕ as follows. Given ( x, z ) ∈ S × T , send itto the equivalen e lass [ t, (1 , z )] , no matter what t you hoose as long as e πit = x . With respe t to either of the two harts τ U and τ V of Eq. (6), thelo al expression of this embedding is simply ( x, z ) ( x ; 1 , z ) .Now, let ( E, ̺ ) be a C-linear representation of G and let w ∈ W bethe point we sele ted in the ourse of the dis ussion above. In the hart τ U the isotropy group G| w and the torus T are identi(cid:28)ed by the indu edLie group isomorphism G| w ≈ T . The subgroups ϕ ( { w } × T ) ⊂ G| w and { } × T ⊂ T orrespond to one another under this isomorphism; moreover,the homomorphism ̺ w : G| w → GL ( E w ) is given the matrix representation A = A U ( w , - ) : T → GL ( ℓ ; C ) of Eq. (18). Therefore, sin e from Eq. (24)we know that { } × T is ontained in Ker A , we on lude that the image ϕ ( { w } × T ) is ontained in Ker ̺ w . By the standard homotopy argumentof Note 30 we (cid:28)nally get ϕ ( { x } × T ) ⊂ Ker ̺ x for all x ∈ S . This ompletesthe proof in the C-linear ase.Finally, let R = ( E, ̺ ) be any R-linear lassi al representation of G . It willbe enough to take the omplexi(cid:28) ation R ⊗ C = ( E ⊗ C , ̺ ⊗ C ) and observethat Ker ̺ x = Ker ̺ x ⊗ C = Ker ( ̺ ⊗ C ) x for all x . q.e.d.Consider the map R × T → S × T given by ( t ; z, z ′ ) ( e πit , z ) . Thisindu es an epimorphism of Lie bundles over S ψ : G T ; χ ℓ −→ T (cid:0) T def = S × T (cid:1) (27)whose kernel is pre isely the image of the embedding ϕ of the pre edingproposition. The latter map yields an identi(cid:28) ation of forgetful fun tors R ∞ (T ) forg. fun . (cid:15) (cid:15) ψ ∗ ≃ / / R ∞ ( G T ; χ ℓ ) forg. fun . (cid:15) (cid:15) V ∞ (S ) V ∞ (S ) (28)0 CHAPTER I. LIE GROUPOIDS, CLASSICAL REPRESENTATIONSde(cid:28)ned as ψ ∗ ( E, ̺ ) def = ( E, ψ ◦ ̺ ) . One easily re ognizes that the fun tor ψ ∗ is an isomorphism of ategories. Indeed, its inverse ψ ∗ an be onstru tedexpli itly by means of the familiar universal property of the quotient (whi hin the present ase follows immediately from Proposition 25), namely G T ; χ ℓ ψ (cid:15) (cid:15) ̺ / / GL ( E )T ψ ∗ ̺ ssssss (29)for every ( E, ̺ ) ∈ Ob R ∞ ( G T ; χ ℓ ) , so that ( E, ψ ∗ ̺ ) is an obje t of R ∞ (T ) (one obviously sets ψ ∗ ( a ) = a for all morphisms a ).The existen e of the identi(cid:28) ation of ategories (28) shows in a very on-vin ing way that, in general, a ategory of lassi al representations does notprovide enough information to re over the Lie groupoid from whi h it origi-nates; this is true independently of the re ipe one might invent for a possiblere onstru tion theory. Note also that this failure already o urs under ir- umstan es where the Lie groupoid is a very reasonable one ( ompa t, abelianand onne ted). Of ourse, what we are saying does not ex lude the possi-bility that in some spe ial ases the re onstru tion may be feasible; we shallgive a few elementary examples of this sort later on in Ÿ28.30 Note (Compare also Brö ker and tom Die k [4℄ p. 84) Let G bea Lie group and let ̺ t : G → GL ( V ) be a family of representations ̺ t depending ontinuously on g ∈ G and t ∈ [0 , , in other words, a homotopyof representations. We laim that if G is ompa t, the representations ̺ and ̺ are isomorphi (cid:22)i.e. there exists some A ∈ GL ( V ) whi h onjugates ̺ into ̺ . To begin with, let G ∨ denote the set of isomorphism lasses ofirredu ible G -modules. For ea h γ ∈ G ∨ , sele t a representative V γ . Then forevery t ∈ [0 , one an de ompose the G -module V t = ( V, ̺ t ) into a dire tsum V t ≈ ⊕ γ ∈ G ∨ n tγ V γ in whi h the integer n tγ = multipli ity of V γ in V t = R χ t χ γ , where χ γ is the hara ter of V γ and χ t = P γ ∈ G ∨ n tγ χ γ is the hara terof V t , depends ontinuously on t and is therefore onstant. This proves the laim.More generally, one has that if f t : G → H is any homotopy of homo-morphisms of a ompa t Lie group G into a Lie group H then f and f are onjugate: see Conner and Floyd (1964) [7℄ Lemma 38.1. Their result isa onsequen e of the following theorem of Montgomery and Zippin (1955)(whi h an be found in [30℄ p. 216):Theorem Let G be a Lie group and F a ompa t subgroup of G . Thenthere exists an open set O in G , F ⊂ O , with the property that if H is a ompa t subgroup of G and H ⊂ O , then there is a g in G su h that g − Hg ⊂ F .3. NORMALIZED HAAR SYSTEMS 31Moreover given any neighborhood W of e , O an be so hosen that forevery H ⊂ O the desired g an be sele ted in W .Compare Bredon (1972) [3℄ II.5.6.Ÿ3 Normalized Haar SystemsNormalized Haar systems on proper Lie groupoids are the analogue of Haarprobability measures on ompa t Lie groups. In the present se tion we reviewthe basi de(cid:28)nitions and give some details about the onstru tion of Haarsystems on proper Lie groupoids; an exposition of this material an also befound in Craini [10℄. Let G be a Lie groupoid over a manifold M .1 De(cid:28)nition A positive Haar system on G is a family of positive Radonmeasures { µ x } ( x ∈ M ), ea h one with support on entrated in the respe tivesour e (cid:28)bre G x = G ( x, - ) = s − ( x ) , satisfying the following onditions:i) R ϕ d µ x > for all nonnegative ϕ ∈ C ∞ c ( G x ) , ϕ = 0 ;ii) for every ϕ ∈ C ∞ c ( G (1) ; C ) , the fun tion Φ : M → C de(cid:28)ned by(2) Φ( x ) def = Z G x ϕ | G x dµ x is of lass C ∞ ;iii) right invarian e: for arbitrary g ∈ G ( x, x ′ ) and ϕ ∈ C ∞ c ( G x ) ,(3) Z G x ′ ϕ ◦ τ g d µ x ′ = Z G x ϕ d µ x where τ g : G ( x ′ , - ) → G ( x, - ) denotes right translation h hg .In this de(cid:28)nition the term `positive' refers to the (cid:28)rst ondition whereas theterm `smooth' is o asionally used to emphasize the se ond ondition.The existen e of positive (smooth) Haar systems on a Lie groupoid G anbe established if G is proper. (Re all that G is proper if it is Hausdor(cid:27) and themap ( s , t ) : G → M × M is proper in the usual sense.) One way to do thisis the following. One starts by (cid:28)xing a Riemann metri on the ve tor bundle g → M , where g is the Lie algebroid of G ( fr. Craini [10℄ or Moerdijk andMr£un [27℄, Chapter 6; note the use of para ompa tness). Right translationsdetermine isomorphisms T G ( x, - ) ≈ t ∗ g | G ( x, - ) for all x ∈ M . These an beused to indu e, on the sour e (cid:28)bres G ( x, - ) , Riemann metri s whose asso iatedvolume forms provide the desired system of measures.Positive Haar systems are not entirely adequate for our purposes. We will(cid:28)nd the following notion more useful:2 CHAPTER I. LIE GROUPOIDS, CLASSICAL REPRESENTATIONS4 De(cid:28)nition A normalized Haar system on G is a family of positive Radonmeasures { µ x } ( x ∈ M ), ea h with support on entrated in the respe tivesour e (cid:28)bre G ( x, - ) , with the following properties: (a) all smooth fun tionson G ( x, - ) are integrable with respe t to µ x , that is to say(5) C ∞ (cid:0) G ( x, - ); C (cid:1) ⊂ L ( µ x ; C ) ;(b) Conditions ii) and iii) of the pre eding de(cid:28)nition hold for an arbitrarysmooth fun tion ϕ on G (1) , respe tively G ( x, - ) ; ( ) the following normality ondition is satis(cid:28)ed:i ′ ) R d µ x = 1 , for all x ∈ M .Every proper Lie groupoid admits normalized (smooth) Haar systems.For su h a groupoid G , one an prove this by using a ut-o(cid:27) fun tion, namelya positive, smooth fun tion c on the base M , su h that the sour e map s restri ts to a proper map on supp ( c ◦ t ) and R ( c ◦ t ) d ν x = 1 for all x ∈ M ,where { ν x } is a (cid:28)xed positive (smooth) Haar system on G . The system ofpositive measures µ x = ( c ◦ t ) ν x has the desired properties.Observe that if E ∈ Ob V ∞ ( M ) is a smooth ve tor bundle of lo ally (cid:28)niterank over the base of G and ψ : G → E is a smooth mapping su h that forea h x ∈ M the (cid:28)bre G ( x, - ) is mapped into the ve tor spa e E x , then theintegral(6) Ψ( x ) def = Z ψ x d µ x makes sense and de(cid:28)nes a smooth se tion of E . This follows easily from theproperties listed in De(cid:28)nition 4, by working in lo al oordinates.Ÿ4 The Lo al Linearizability TheoremLet G be a Lie groupoid and let M be its base manifold. We say that asubmanifold N of M is a sli e at the point z ∈ N if the orbit immersion G z ֒ → M is transversal to N at z . A submanifold S of M will be alled asli e if it is a sli e at all of its points. The following remark will be used veryoften: Let N be a submanifold of M and let g ∈ G N ≡ s − ( N ) ; then N isa sli e at z = s ( g ) if and only if the interse tion G N ∩ t − ( z ′ ) , z ′ = t ( g ) istransversal at g . Indeed, from the equalities T g G N = T z N ⊕ T g G z and T g t − ( z ′ ) = T z G z ′ ⊕ W = T z G z ⊕ W ,where W is a linear subspa e of T g G z , it follows immediately that(1) T g G N + T g t − ( z ′ ) = (cid:0) T z N + T z G z (cid:1) ⊕ T g G z .4. THE LOCAL LINEARIZABILITY THEOREM 33By virtue of this fa t, one obtains that for ea h submanifold N of M , thesubset of all points at whi h N is a sli e is an open subset of N . In order toas ertain it, (cid:28)x a point z belonging to this subset. Sin e the interse tion of G N with the (cid:28)bre t − ( z ) must be transversal at u ( z ) ∈ G ( z, z ) , there will bea neighbourhood Γ N of u ( z ) in G N su h that for all g ∈ Γ N the interse tion G N ∩ t − ( t g ) is transversal at g . Now, if S is an open neighbourhood of z in N su h that u ( S ) ⊂ Γ N , one has that S is a sli e.Let R , S be mutually transversal submanifolds of a manifold N : then R ∩ S is a submanifold of N , of dimension r + s − n .Next, let p : Y → X be a submersion, let S be any submanifold of Y and(cid:28)x s ∈ S . Put x = p ( s ) . Then S interse ts the (cid:28)bre p − ( x ) transversallyat s if and only if the restri tion p | S : S → X is submersive at that point;from this, it immediately follows that when the interse tion S ∩ p − ( x ) istransversal at s , there exists a neighbourhood A of s in S su h that at allpoints a ∈ A the interse tion S ∩ p − ( x ) , x = p ( a ) is also transversal. In orderto he k the previous laim, it is not restri tive to assume that Y = X × Z is a Cartesian produ t and that p = pr is the proje tion onto the (cid:28)rst fa tor.Setting s = ( x , z ) , one obtains for the tangent spa es the pi ture(2) T s S + T z Z ⊂ T s ( X × Z ) = T x X ⊕ T z Z pr ∗ −−→ T x X ,from whi h it is evident that T s S ontains a linear subspa e W su h that pr ∗ ( W ) = T x X if and only if the in lusion (2) is an equality.3 Note If a submanifold S of M is a sli e then the interse tion s − ( S ) ∩ t − ( S ) is transversal and the restri tion G| S is a Lie groupoid over S . Indeed, let us (cid:28)x g ∈ G ( z, z ′ ) with z, z ′ ∈ S . Sin e(4) T g s − ( S ) + T g t − ( z ′ ) ⊂ T g s − ( S ) + T g t − ( S ) ,one immediately obtains the transversality at g of the interse tion writ-ten above. The target map t will indu e a submersion of s − ( S ) onto anopen subset of M and this submersion will in turn indu e a submersion of s − ( S ) ∩ t − ( S ) onto S .5 Note Let S be a sli e; then G · S is an open subset of M . To verifythis it will be enough to show that given any point z ∈ S there exists aneighbourhood U of z in M su h that s − ( S ) ∩ t − ( u ) = ∅ for all u ∈ U . Thisis true be ause the interse tion s − ( S ) ∩ t − ( z ) is nonempty and transversal.Then U ⊂ G · S , from whi h the in lusion G · z ⊂ G · U ⊂ G · S (cid:28)nallyfollows.Theorem (N.T. Zung) Let G be a proper Lie groupoid and let X beits base manifold. Let x ∈ X be a point whi h is not moved by thetautologi al a tion of G on its own base.4 CHAPTER I. LIE GROUPOIDS, CLASSICAL REPRESENTATIONSThen there exists a ontinuous linear representation G → GL ( V ) ofthe isotropy group G ≡ G| x on a (cid:28)nite dimensional ve tor spa e V , su hthat for some open neighbourhood U of x one an (cid:28)nd an isomorphismof Lie groupoids G| U ≈ G ⋉ V whi h makes x orrespond to .Proof See Zung's paper [38℄. q.e.d.We want to give a areful proof of the statement that any proper Liegroupoid is lo ally Morita equivalent to the translation groupoid asso iatedwith a (linear) ompa t Lie group a tion; this will of ourse follow fromZung's theorem. The latter statement is a key ingredient in the proof ofour (cid:19) main re onstru tion theorem (cid:20), Theorem 20.28. Let us begin with ate hni al observation about sli es.Let S , T be two sli es in M . Let g ∈ G ( S, T ) ; put s ≡ s ( g ) ∈ S and t ≡ t ( g ) ∈ T . To (cid:28)x ideas, suppose dim S ≦ dim T . Then we laim thatthere exists a smooth se tion τ : B → G (1) to the target map of G , de(cid:28)nedover some open neighbourhood B of t in T , su h that τ ( t ) = g and the omposite map s ◦ τ indu es a submersion of B onto an open neighbourhoodof s in S . To begin with, let us noti e(cid:22)in general(cid:22)that if one is givena ouple of smooth submersions Y p ←− X q −→ Z with dim Y ≧ dim Z thenfor ea h point x ∈ X there exists a smooth p -se tion π : U → X , de(cid:28)nedover some open neighbourhood U of p ( x ) , su h that π ( p ( x )) = x and the omposite q ◦ π : U → N is a submersion onto an open neighbourhood of q ( x ) in Z ; this is seen by means of an obvious argument based on elementarylinear algebra: there exists a omplementary subspa e F to Ker T x p in T x X su h that F + Ker T x q = T x X . Now, the interse tion(6) X ≡ s − ( S ) ∩ t − ( T ) ⊂ G (1) is transversal, be ause for all g ∈ G ( s, t ) with s ∈ S and t ∈ T , s − ( S ) willinterse t t − ( t ) and hen e a fortiori t − ( T ) transversally at g (sin e S is asli e). Moreover, the sour e map s : G → M restri ts to a submersion of X onto S , for(cid:22)sin e T is a sli e(cid:22)the submanifold t − ( T ) is transversal toevery s -(cid:28)bre it interse ts and therefore the restri tion s : t − ( T ) → M is asubmersion. Symmetri ally, the indu ed mapping t | X : X → T will be sub-mersive. Thus we an apply the foregoing general remark about submersionsto get a smooth target se tion τ with the desired properties.7 Corollary Let G be a proper Lie groupoid over a manifold M .Then for ea h point x ∈ M there exist a (cid:28)nite dimensional linearrepresentation G → GL ( V ) of a ompa t Lie group G , and a G -invariantopen neighbourhood U of x in M along with a Morita equivalen e ι : G ⋉ V ֒ → G| U , su h that ι (0) : V ֒ → U is an embedding of manifoldsmapping to x .4. THE LOCAL LINEARIZABILITY THEOREM 35Proof By properness, we an (cid:28)nd a sli e S ⊂ M su h that S ∩ G · x = { x } . Then G| S is a proper Lie groupoid for whi h the point x is invariant.By Zung's theorem, we an assume that there exists an isomorphism of Liegroupoids G ⋉ V ≈ G| S , x , for some linear ompa t Lie group a tion G → GL ( V ) . We ontend that G ⋉ V ≈ G| S ֒ → G| U , where U is the opensubset G · S ⊂ M , is the Morita equivalen e ι we are looking for. It will besu(cid:30) ient to prove that the surje tive mapping V × U G| U → U , ( v, g ) t ( g ) is a submersion. This learly follows from the pre eding observation aboutsli es when we take T ≡ U . q.e.d.We on lude this se tion with some remarks relating the groupoids G| S and G| T indu ed on two di(cid:27)erent sli es S and T . Suppose dim S ≦ dim T .Let s ∈ S and t ∈ T be two points lying on the same G -orbit. Theni) for some open neighbourhoods B ⊂ T of t and A ⊂ S of s thereexists a Morita equivalen e G| B ։ G| A mapping t to s and indu inga submersion of B onto A ;ii) for some open neighbourhood A ⊂ S of s there exists an embeddingof Lie groupoids G| A ֒ → G| T mapping s to t and indu ing a sli eembedding A ֒ → T (ie an embedding whose image is a sli e);iii) if in parti ular dim S = dim T then the Lie groupoids G| S and G| T arelo ally isomorphi about the points s and t .Let us verify the assertion i). Choose any g ∈ G ( s , t ) . By the te hni alobservations pre eding Corollary 7, we an (cid:28)nd a smooth target se tion τ : B → G (1) so that s ◦ τ is a submersion onto an open neighbourhood A ⊂ S of s . The latter map an be lifted to(8) G| B → G A , h τ ( t h ) − · h · τ ( s h ) ;this formula sets up the required Morita equivalen e. In an entirely analogousmanner assertion ii) an be proved by onsidering a suitable smooth sour ese tion σ : A → G (1) su h that t ◦ σ is a sli e embedding of A into T andthen by lifting this embedding to one of Lie groupoids(9) G| A ֒ → G| T , g σ ( t g ) · g · σ ( s g ) − .10 Note Let σ : U → G (1) be a lo al bise tion. Suppose S ⊂ U is asli e. Then T ≡ t (cid:0) σ ( S ) (cid:1) is also a sli e; moreover, there exists a Lie groupoidisomorphism G| S ≈ −→ G| T whi h lifts the map t ◦ σ .Let us prove that T is a sli e. Put V = t (cid:0) σ ( U ) (cid:1) . Fix a point s ∈ S andlet t ≡ t ( σ ( s )) . Then(11) t (cid:0) σ ( G · s ∩ U ) (cid:1) = G · t ∩ V t ◦ σ is a di(cid:27)eomorphism of U onto V , the orbit G · s interse ts the submanifold S transversally at s if and only if G · t interse ts T transversally at t ; our laim follows. Next, observe that t ◦ σ is ertainlya di(cid:27)eomorphism of S onto T , whi h an be lifted(cid:22)via σ , as in (9)(cid:22)to a Liegroupoid isomorphism with the expe ted properties.Ÿ5 Global QuotientsThe material presented in this se tion is not dire tly relevant to the problemdis ussed in the thesis; if the reader wishes to do so, he may go dire tly tothe next hapter. As before, we lay no laim to originality.1 Lemma Let H be a proper Lie groupoid, a ting without isotropy onits own base F (i.e. all isotropy groups of H are assumed to be trivial).Then the orbit spa e F/ H has a unique manifold stru ture su h thatthe quotient map q : F → F/ H is a submersion.Proof The mapping ( s , t ) : H → F × F is an inje tive immersion. Indeed,for a (cid:28)xed h ∈ H ( f, f ′ ) , f, f ′ ∈ F , the tangent map(2) T h H T h ( s , t ) −−−−−→ T ( f,f ′ ) ( F × F ) ∼ = T f F ⊕ T f ′ F equals the linear map T h s ⊕ T h t ; therefore(3) Ker T h ( s , t ) = Ker T h s ∩ Ker T h t = T h H ( f, f ′ ) = 0 ( fr. for example [27℄, proof of Thm. 5.4, p. 117; by the triviality of theisotropy groups of H , the latter tangent spa e must be zero).Moreover, be ause of properness, ( s , t ) : H → F × F is also a losed map,hen e in fa t an embedding of smooth manifolds.It follows that the equivalen e relation R = Im ( s , t ) = { ( f, f ′ ) |∃ h : f → f ′ in H} is a submanifold of F × F ; the proje tion onto the se ond fa tor learly restri ts to a submersion of R onto F . Therefore, by Godement'sTheorem (see [33℄, p. 92), there is a manifold stru ture on the quotient spa e F/R = F/ H , making the quotient map q : F → F/ H a submersion. q.e.d.This lemma applies when a proper Lie groupoid G with base M a ts freelyfrom the left on a manifold F along some smooth mapping p : F → M . Byde(cid:28)nition, this means that the orresponding a tion groupoid H ≡ G ⋉ F has trivial isotropy groups. In order to on lude that there exists a smoothmanifold stru ture on the quotient spa e F/ G , for whi h the proje tion F → F/ G is submersive, one needs to he k that the groupoid G ⋉ F is also proper.So, let C ⊂ F × F be any ompa t subset and put C = pr ( C ) ⊂ F ; sin e F is a Hausdor(cid:27) manifold, the inverse image ( s H , t H ) − ( C ) will be a losedsubset of the manifold G × F and hen e of the ompa t set(4) ( s G , t G ) − (cid:0) ( p × p )( C ) (cid:1) × C ⊂ G × F ,5. GLOBAL QUOTIENTS 37where p × p denotes the smooth map ( f, f ′ ) ( p ( f ) , p ( f ′ )) .Now, suppose that a Lie group K a ts smoothly on F from the right, insu h a way that p : F → M turns out to be a prin ipal K -bundle. Assumethat this a tion ommutes with the given left a tion of G . Then there is awell-de(cid:28)ned indu ed right a tion of K on the quotient manifold F/ G . Thisis in fa t a smooth a tion be ause of an elementary property of submersions(see e.g. p. 147 below): the a tion map F/ G × K → F/ G has to be smoothbe ause upon omposing it with the submersion F × K → F/ G × K oneobtains a smooth map, namely F × K → F → F/ G .The next result should probably be regarded as folklore. Its statement,along with the key idea for the proof presented here, was suggested to me byI. Moerdijk as early as the beginning of 2006.5 Theorem Suppose a proper Lie groupoid G admits a global faithfulrepresentation on a smooth ve tor bundle.Then G is Morita equivalent to the translation groupoid asso iatedwith a ompa t, onne ted Lie group a tion.Proof Let ̺ : G ֒ → GL ( E ) be a faithful representation on a(cid:22)let us say,real(cid:22)smooth ve tor bundle E over the base M of G . By properness of G ,we an (cid:28)nd a ̺ -invariant metri 1 on E , whi h we (cid:28)x on e and for all. Thenlet F = Fr( E ) p −→ M be the orthonormal frame bundle of E (relative tothe hosen invariant metri ): re all that the (cid:28)bre F x above x is the spa eof all linear isometries f : R d ∼ → E x , where d is the rank of E x . The totalspa e F of this (cid:28)bre bundle is a para ompa t Hausdor(cid:27) manifold; moreover,the (cid:28)bration p is a prin ipal bundle for the anoni al right a tion of theorthogonal group K = O ( d ) on F (de(cid:28)ned by f k = f ◦ k ). Sin e ̺ a ts on E by isometries, the Lie groupoid G will a t on the manifold F from theleft(cid:22)via the representation ̺ , that is by the rule gf = ̺ ( g ) ◦ f (cid:22)along thebundle map p . Clearly, the two a tions ommute.Next, let the (cid:16)double a tion groupoid(cid:17) G ⋉ F ⋊ K be the Lie groupoidover the manifold F that is obtained as follows. Its manifold of arrows is ( G ⋉ F ) × K , viz. the submanifold of the Cartesian produ t ( G × F ) × K onsisting of all triples ( g, f, k ) with s ( g ) = p ( f ) . The sour e map sends thearrow ( g, f, k ) to f and the target map to gf k . The omposition of arrows isde(cid:28)ned to be ( g ′ , f ′ , k ′ ) · ( g, f, k ) = ( g ′ g, f, kk ′ ) . Then the identity arrow at f is ( p ( f ) , f, id ) and the inverse must be given by ( g, f, k ) − = ( g − , gf k, k − ) .All these stru ture maps are obviously smooth.Now, we laim that there are Morita equivalen es(6) G ˜ p ←−−−− G ⋉ F ⋊ K ˜ q −−−−→ F/ G ⋊ K G is Morita equivalentto the a tion groupoid F/ G ⋊ K , as ontended. Perhaps it is good to spenda ouple of words to state the formulas for right a tion groupoids; these areobtained by regarding a given right a tion of a Lie group H on a manifold X as a left a tion of the opposite group. Thus ( x, h ) x , resp. x · h is thesour e, resp. target map, and ( x ′ , h ′ ) · ( x, h ) = ( x, hh ′ ) is the omposition.We start with the onstru tion of the equivalen e to the left(7) ˜ p : G ⋉ F ⋊ K −→ G .As the notation ˜ p suggests, this equivalen e is to be given by the surje tivesubmersion p : F → M on base manifolds; as to arrows, we put ˜ p ( g, f, k ) = g .It is immediate to he k that ˜ p de(cid:28)nes a Lie groupoid homomorphism of G ⋉ F ⋊ K onto G . All one needs to do now in order to show that ˜ p is aMorita equivalen e is to solve, within the ategory of smooth manifolds, theuniversal problem stated in the left-hand diagram below: X ( f,f ′ ) & & GGGGG g X ( f,f ′ ) & & GGGGG ( q ◦ f,k ) % % G ⋉ F ⋊ K ˜ p / / (cid:15) (cid:15) G (cid:15) (cid:15) G ⋉ F ⋊ K ˜ q / / (cid:15) (cid:15) F/ G ⋊ K (cid:15) (cid:15) F × F p × p / / M × M F × F q × q / / F/ G × F/ G .(8)It will be enough to noti e that the map X → K , x κ ( x ) , whi h assignsthe linear isometry κ ( x ) = f ′ ( x ) − ◦ ̺ ( g ( x )) ◦ f ( x ) to ea h x , is of lass C ∞ . Then we an de(cid:28)ne the dashed arrow in the aforesaid diagram to be x ( g ( x ) , f ( x ) , κ ( x )) . This is learly the unique possible solution.We turn our attention now to the other equivalen e(9) ˜ q : G ⋉ F ⋊ K −→ F/ G ⋊ K .This is given by q on obje ts and by ˜ q ( g, f, k ) = ( q ( f ) , k ) on arrows. Clearly,the map ˜ q so de(cid:28)ned is a homomorphism of Lie groupoids. Sin e the basemapping q : F → F/ G is known to be a surje tive submersion by Lemma 1,in order to show that ˜ q is a Morita equivalen e it will be enough to solve theright-hand universal problem of (8). We observe that from the propernessof G and the faithfulness of ̺ it follows(cid:22)see for instan e Corollary 23.10below(cid:22)that the image ̺ ( G ) ⊂ GL ( E ) is a submanifold; moreover, it anbe shown(cid:22) fr. Lemma 26.3, for example(cid:22)that ̺ : G ≈ −→ ̺ ( G ) is a tually adi(cid:27)eomorphism. Now, the map X → GL ( E ) , x γ ( x ) , that sends x to theisometry γ ( x ) = f ′ ( x ) ◦ k ( x ) ◦ f ( x ) − , is learly smooth and fa tors throughthe submanifold ̺ ( G ) . Then we may use the fa t that ̺ is a di(cid:27)eomorphismof G onto ̺ ( G ) and de(cid:28)ne the dashed arrow as x (cid:0) ̺ − ( γ ( x )) , f ( x ) , k ( x ) (cid:1) ;this is of ourse a smooth orresponden e. q.e.d.hapter IIThe Language of TensorCategoriesWith the ex eption of Ÿ10, the present hapter o(cid:27)ers an introdu tion to the ategori al setting of the modern theory of Tannaka duality originating fromthe ideas of A. Grothendie k and N. Saavedra Rivano; fr [32, 12, 11, 18℄.In Se tion 10 we prove a key te hni al lemma whi h will be used in theproof of our re onstru tion theorem in Ÿ20; sin e this lemma deals with afairly abstra t ategori al situation, we thought it was more appropriate toin lude it in this hapter.Ÿ6 Tensor CategoriesA tensor stru ture on a ategory C onsists of the following data:(1) a bifun tor ⊗ : C × C −→ C , a distinguished obje t ∈ Ob( C ) and a list of natural isomorphisms, alled ACU onstraints: α R,S,T : R ⊗ ( S ⊗ T ) ∼ → ( R ⊗ S ) ⊗ T , γ R,S : R ⊗ S ∼ → S ⊗ R , λ R : R ∼ → ⊗ R and ρ R : R ∼ → R ⊗ (2)satisfying Ma Lane's oheren e onditions ( fr for example Ma Lane (1971),pp. 157 (cid:27). and espe ially p. 180 for a detailed exposition). A tensor ategoryis a ategory endowed with a tensor stru ture. In the terminology of [24℄, thepresent notion orresponds to that of (cid:16)symmetri monoidal ategory(cid:17). Thenatural isomorphism α resp. γ is alled the asso iativity resp. ommutativity onstraint; λ and ρ are the tensor unit onstraints.In order to state Ma Lane's Coheren e Theorem for tensor ategories, itwill be onvenient to introdu e the on epts of (cid:19) anoni al multi-fun tor (cid:20) and390 CHAPTER II. THE LANGUAGE OF TENSOR CATEGORIES(cid:19) anoni al transformation (cid:20). These will onstitute respe tively the obje tsand the morphisms of a ategory C an ( C ) .A multi-fun tor on C is a fun tor Φ : C I → C for some (cid:28)nite set I . The ardinality | I | = Card I will be alled the (cid:19) -ariety (cid:20) of Φ .The anoni al multi-fun tors are, roughly speaking, those obtained asformal iterates of ⊗ , possibly involving . The adje tive `formal' here meansthat a ` anoni al multi-fun tor' is not just a ertain type of multi-fun tor,in that one should regard the parti ular indu tive onstru tion, by whi ha anoni al multi-fun tor is obtained, as part of the de(cid:28)ning data; we donot want to go into details here: the interested reader may onsult [24℄. There ursive rules for generating anoni al multi-fun tors are listed below:i) the unique -ary anoni al multi-fun tor is : C ∅ = { ⋆ } → C , ⋆ ;ii) the (cid:16)identity(cid:17): C { ⋆ } → C is anoni al;iii) if Φ : C I → C and Ψ : C J → C are anoni al then so is Φ ⊗ Ψ : C I ⊔ J → C where I ⊔ J indi ates disjoint union;iv) if I σ −→ J is a bije tion of (cid:28)nite sets and Φ : C I → C is anoni al then Φ σ : C J → C I → C is also anoni al.Canoni al multi-fun tors are the obje ts of C an ( C ) . As to anoni al naturaltransformations, they are re ursively generated as follows:a) the identity id : Φ → Φ is anoni al; if η : Φ → Φ ′ , with Φ , Φ ′ : C I → C ,and θ : Ψ → Ψ ′ , with Ψ , Ψ ′ : C J → C , are anoni al transformations of anoni al multi-fun tors, then so is η ⊗ θ : Φ ⊗ Ψ → Φ ′ ⊗ Ψ ′ (naturaltransformations of multi-fun tors C I ⊔ J → C ); if I σ −→ J is a bije tion ofsets then θ σ : Φ σ → Ψ σ is also anoni al;b) α Φ , Ψ , X : (cid:2) Φ ⊗ (Ψ ⊗ X) (cid:3) σ ∼ → (cid:2) (Φ ⊗ Ψ) ⊗ X (cid:3) τ and its inverse α Φ , Ψ , X − are anoni al transformations, where σ , τ are the bije tions I ⊔ ( J ⊔ K ) → I ⊔ J ⊔ K ← ( I ⊔ J ) ⊔ K ; ) γ Φ , Ψ : Φ ⊗ Ψ ∼ → [Ψ ⊗ Φ] σ (along with its inverse) is anoni al, where I ⊔ J σ ←− J ⊔ I is the obvious bije tion;d) λ Φ : Φ ∼ → ( ⊗ Φ) σ and ρ Φ : Φ ∼ → (Φ ⊗ ) τ (along with their inverses)are anoni al, where ∅ ⊔ I σ −→ I τ ←− I ⊔ ∅ are the obvious bije tions.It is lear that all anoni al transformations are isomorphisms.Ma Lane's Coheren e Theorem for (cid:19) symmetri monoidal ategories (cid:20)((cid:19) tensor ategories (cid:20) in our terminology) an now be stated as follows:Theorem The ategory C an ( C ) is a preorder. That is to say, for any anoni al multi-fun tors Φ and Ψ there is at most one anoni al naturaltransformation Φ → Ψ .Proof See [Ma Lane℄, xi.1 p. 253. q.e.d.6. TENSOR CATEGORIES 41This theorem says that any diagram of anoni al multi-fun tors and anoni al natural transformations one an possibly onstru t will ommute.When one is given su h a diagram, let us say of multi-fun tors C I → C , onemay hoose an identi(cid:28) ation { , . . . , i } ∼ → I and denote a generi obje t of C I by ( R , . . . , R i ) , R , . . . , R i ∈ Ob( C ) . Evaluating the given diagram atthis i -tuple of obje ts(cid:22)so that Φ θ −→ Ψ be omes Φ( R , . . . , R i ) θ ( R ,...,R i ) −−−−−−→ Ψ( R , . . . , R i ) , for instan e(cid:22)one obtains a ommutative diagram in C .3 Note (See also Saavedra, 1.3.3.1) Let ( C , ⊗ , ) be a tensor ategory.Then End C ( ) is a ommutative ring. To see this, observe that the tensor unit onstraint ∼ = ⊗ establishes a anoni al isomorphism of rings between End( ) and End( ⊗ ) . Now, if e, e ′ ∈ End( ) then ee ′ ∼ = (1 ⊗ e )( e ′ ⊗
1) = e ′ ⊗ e = ( e ′ ⊗ ⊗ e ) ∼ = e ′ e in this isomorphism and hen e ee ′ = e ′ e . Notethat this proof only uses the oheren e identity λ = ρ for the tensor unit onstraints; the ommutativity onstraint plays no role.Rigid tensor ategoriesA tensor ategory ( C , ⊗ ) is said to be losed, whenever one an exhibit abifun tor hom : C op × C −→ C , alled `internal hom' and denoted by ( X, Y ) Y X ≡ hom( X, Y ) ,along with natural transformations (in the variable Y ) η XY : Y → ( Y ⊗ X ) X and ε XY : Y X ⊗ X → Y ,satisfying the triangular identities for an adjun tion C (cid:0) X ⊗ T , Y (cid:1) ∼ → C (cid:0) X, hom( T, Y ) (cid:1) (in the variables ( X, Y ) ∈ C op × C )between the fun tors (cid:19) - ⊗ T (cid:20) and (cid:19) hom( T, - ) (cid:20) and making Y X ′ ⊗ X id ⊗ a (cid:15) (cid:15) Y a ⊗ id / / Y X ⊗ X ε (cid:15) (cid:15) ( Y ⊗ X ) X ( id ⊗ a ) id / / ( Y ⊗ X ′ ) X Y X ′ ⊗ X ′ ε / / Y Y η O O η / / ( Y ⊗ X ′ ) X ′ id a O O (4) ommute for every arrow a : X → X ′ .Suppose now that an `internal hom' bifun tor and natural transformations η , ε with these properties have been (cid:28)xed. Then there is an obvious arrow(5) δ S,TX,Y : X S ⊗ Y T → ( X ⊗ Y ) S ⊗ T ,namely the unique solution d to the equation ε ◦ ( d ⊗ id ) = ( ε ⊗ ε ) ◦ ∼ = , ∼ = is the unique anoni al isomorphism. Be ause of (4), the arrow δ must be natural in all variables. By the same reason, the solution(6) ι X : X → X ∨∨ (where we put X ∨ ≡ hom( X, ) , to be alled the dual of X ) to the equation ε ◦ ( ι X ⊗ id ) = ε ◦ ∼ = is natural in X .A di(cid:27)erent hoi e of internal hom bifun tor and natural transformations η and ε will yield the same natural arrows δ and ι up to isomorphism: thus itmakes sense to all a losed tensor ategory rigid when these natural arrowsare isomorphisms.One an also formulate this notion in terms of duals, sin e for a rigidtensor ategory one has the identi(cid:28) ation(7) hom( X, Y ) ≈ X ∨ ⊗ Y , f. Deligne (1990), [11℄ 2.1.2.Let ( C , ⊗ ) be a rigid tensor ategory. The ontravariant fun tor X X ∨ , f t f is an equivalen e between C and its opposite ategory C op (be ause it isinvolutive, ie its omposite with itself is naturally isomorphi to the identity,sin e rigidity implies that (6) is a natural isomorphism).This gives in parti ular a bije tion between the hom-sets f t f : Hom C ( X, Y ) ∼ → Hom C ( Y ∨ , X ∨ ) .One also has an (cid:16)internal(cid:17) isomorphism Y X ∼ → X ∨ Y ∨ , namely the omposite Y X ≈ ←− X ∨ ⊗ Y id ⊗ ι Y −−−−→ X ∨ ⊗ Y ∨∨ ≈ −→ Y ∨∨ ⊗ X ∨ ≈ −→ X ∨ Y ∨ . For every obje t of C there is an arrow X X ∼ → X ∨ ⊗ X ε −→ . If we applythe fun tor Hom C ( , · ) to this, we obtain the tra e map(8) Tr X : End C ( X ) → End C ( ) . The rank of X is de(cid:28)ned as Tr X (1 X ) . There are the formulas Tr X ⊗ X ′ ( f ⊗ f ′ ) = Tr X ( f )Tr X ′ ( f ′ ) , Tr ( f ) = f. (9)6. TENSOR CATEGORIES 43A tensor ategory ( C , ⊗ ) is said to be additive if the ategory C is endowedwith an additive stru ture su h that the bifun tor ⊗ is biadditive, that isadditive in ea h variable separately. Moreover, if the hom-sets C ( A, B ) areendowed with a real (or omplex) ve tor spa e stru ture in su h a way that omposition of arrows and the bifun tor ⊗ are bilinear, then we say that ( C , ⊗ ) is a linear tensor ategory.10 Example Let V ec C be the ategory of ve tor spa es over C of (cid:28)nitedimension. Then this is an abelian rigid tensor ategory, and all the pre edingde(cid:28)nitions have their usual meaning.11 Example Let M be a smooth manifold. Let C = V ∞ ( M ; C ) be the ategory of smooth omplex ve tor bundles of lo ally (cid:28)nite rank over M .The dire t sum operation ( E, F ) E ⊕ F makes it into an additive C-linear ategory, although in general not an abelian one, sin e a map of ve torbundles over M need not have a kernel, for instan e. We shall identify the ategory of (cid:28)nite dimensional ve tor spa es over C with V ∞ ( ⋆ ; C ) where ⋆ isthe one-point manifold.The ategory V ∞ ( M ; C ) is endowed with a anoni al rigid tensor stru -ture, obtained from the rigid tensor stru ture of V ec C by means of the generalpro edure des ribed in Lang 2001 [22℄ p. 58, as follows. Re all that a multi-fun tor
Φ : V ec C × · · · × V ec C n times −→ V ec C(where ase n = 0 orresponds to the hoi e of an obje t Φ( · ) ∈ Ob( V ec C ) ,and we allow Φ to be ontravariant in some variables), su h that the indu edmappings L ( V , W ) × · · · × L ( V n , W n ) → L (Φ( V , . . . , V n ) , Φ( W , . . . , W n )) are of lass C ∞ , determines a orresponding multi-fun tor Φ : V ∞ ( M ; C ) × · · · × V ∞ ( M ; C ) −→ V ∞ ( M ; C ) with the same varian e and satisfying the following properties:i) for every x ∈ M , the (cid:28)ber above x is(12) Φ( E , . . . , E n ) x = { x } × Φ( E x , . . . , E nx ) ≈ Φ( E x , . . . , E nx ); ii) for arbitrary morphisms of ve tor bundles a i : E i → F i , i = 1 , . . . , n , Φ( a , . . . , a n ) x orresponds to Φ( a x , . . . , a nx ) up to the anoni al iden-ti(cid:28) ations (12);4 CHAPTER II. THE LANGUAGE OF TENSOR CATEGORIESiii) If the ve tor bundles E i ≈ M × E i are trivial, then these trivializations ≈ i determine a trivialization Φ( E , . . . , E n ) ≈ M × Φ( E , . . . , E n ) in a anoni al way; in parti ular, in the ase n = 0 , Φ( - ) ≈ M × Φ( - ) (the standard notation is then Φ( - ) = Φ( - ) ).A natural transformation λ : Φ → Ψ of multi-fun tors with the same vari-an e indu es a natural transformation λ : Φ → Ψ , su h that λ ( E , . . . , E n ) x orresponds to λ ( E x , . . . , E nx ) up to the identi(cid:28) ations (12). Observe that λ ◦ µ = λ ◦ µ and id = id .We an apply these onstru tions to the multifun tors and natural trans-formations whi h de(cid:28)ne the rigid tensor stru ture of V ec C, in order to obtaina similar stru ture on V ∞ ( M ; C ) .Ÿ7 Tensor Fun torsLet C , D be tensor ategories. A tensor fun tor : C −→ D onsists of thedata ( F, τ, υ ) , where F : C −→ D is a fun tor, τ is a natural isomorphism of bifun tors τ R,S : F ( R ) ⊗ F ( S ) ∼ → F ( R ⊗ S ) su h that the diagrams F R ⊗ ( F S ⊗ F T ) α (cid:15) (cid:15) id ⊗ τ / / F R ⊗ F ( S ⊗ T ) τ / / F ( R ⊗ ( S ⊗ T )) F ( α ) (cid:15) (cid:15) ( F R ⊗ F S ) ⊗ F T τ ⊗ id / / F ( R ⊗ S ) ⊗ F T τ / / F (( R ⊗ S ) ⊗ T ) and F ( R ) ⊗ F ( S ) τ (cid:15) (cid:15) γ / / F ( S ) ⊗ F ( R ) τ (cid:15) (cid:15) F ( R ⊗ S ) F ( γ ) / / F ( S ⊗ R ) ommute, and υ : ∼ → F ( ) is an isomorphism in D su h that F ( R ) F ( λ ) / / λ (cid:15) (cid:15) F ( ⊗ R ) F ( R ) F ( ρ ) / / ρ (cid:15) (cid:15) F ( R ⊗ ) ⊗ F ( R ) υ ⊗ id / / F ⊗ F ( R ) τ O O F ( R ) ⊗ id ⊗ υ / / F ( R ) ⊗ F τ O O
7. TENSOR FUNCTORS 45 ommute. (Commutativity of one square implies ommutativity of the other,be ause of the symmetry of the monoidal stru ture.)Now suppose that C and D are losed tensor ategories. Let F : C −→ D be a tensor fun tor. (We shall usually omit writing down the full triple ofdata.) Then there is a anoni al arrow p RS : F ( S R ) → F S
F R , namely the unique solution p to the problem F ( S R ) ⊗ F R τ (cid:15) (cid:15) p ⊗ id / / F S
F R ⊗ F R ε (cid:15) (cid:15) F ( S R ⊗ R ) F ( ε ) / / F S.
This arrow is natural in the variables
R, S . A rigid fun tor is a tensor fun -tor between losed tensor ategories su h that this natural arrow is an iso-morphism. If C and D are both rigid, then a tensor fun tor F : C −→ D isautomati ally rigid.1 Example Let f : M → N be a C ∞ -mapping of smooth manifolds.This map indu es the base hange or pullba k fun tor f ∗ : V ∞ ( N ) −→ V ∞ ( M ) . Re all that for x ∈ M the (cid:28)ber ( f ∗ F ) x oin ides with { x } × F f ( x ) , sin e f ∗ F is by onstru tion a subset of M × F . For every fun tor of several variables Φ as in the last example of Se tion 6, we have a anoni al natural isomorphism(2) f ∗ Φ( E , . . . , E n ) ≈ Φ( f ∗ E , . . . , f ∗ E n ) . It follows at on e from the existen e of these anoni al natural isomorphismsthat f ∗ an be regarded as a tensor fun tor (with respe t to the standardtensor stru ture des ribed in the last example of the pre eding se tion). Itis also lear from (2) that this tensor fun tor is rigid. (Of ourse, rigidityof the pullba k fun tor follows also indire tly from rigidity of the ategories V ∞ ( M ) , V ∞ ( N ) .)3 De(cid:28)nition Let λ : F → G be a natural transformation of tensor fun tors. λ is said to be tensor-preserving, or a morphism of tensor fun tors, wheneverthe diagrams F R ⊗ F S τ (cid:15) (cid:15) λ ( R ) ⊗ λ ( S ) / / GR ⊗ GS τ (cid:15) (cid:15) υ (cid:15) (cid:15) id / / υ (cid:15) (cid:15) F ( R ⊗ S ) λ ( R ⊗ S ) / / G ( R ⊗ S ) F λ ( ) / / G ommute. The olle tion of all su h λ 's will be denoted by Hom ⊗ ( F, G ) .6 CHAPTER II. THE LANGUAGE OF TENSOR CATEGORIESŸ8 Complex Tensor CategoriesAn anti-involution on a C-linear tensor ategory C = ( C , ⊗ ) is an anti-lineartensor fun tor(1) ∗ : C → C , R R ∗ for whi h there exists a tensor preserving natural isomorphism(2) ι R : R ∗∗ ∼ → R with ι ( R ∗ ) = ι ( R ) ∗ .By (cid:28)xing one su h isomorphism, one obtains a mathemati al stru ture whi hwe all omplex tensor ategory. A morphism of omplex tensor ategories,or omplex tensor fun tor, is obtained by atta hing, to an ordinary C-lineartensor fun tor F , a tensor preserving natural isomorphism(3) ξ R : F ( R ) ∗ ∼ → F ( R ∗ ) su h that the following diagram ommutes F ( R ) ∗∗ ∼ = ∗ / / ∼ = ' ' OOOOOOO F ( R ∗ ) ∗ ∼ = / / F ( R ∗∗ ) F ∼ = w w ooooooo F R .(4) 5 Example: the ategory of ve tor spa es If V is a omplex ve tor spa e,we let V ∗ denote the spa e obtained by retaining the additive stru ture of V but hanging the s alar multipli ation into zv ∗ = ( zv ) ∗ ; the star hereindi ates that a ve tor of V is to be regarded as one of V ∗ . Sin e any linearmap f : V → W will map V ∗ linearly into W ∗ , we an also regard f asa linear map f ∗ : V ∗ → W ∗ . Moreover, the unique linear map of V ∗ ⊗ W ∗ into ( V ⊗ W ) ∗ sending v ∗ ⊗ w ∗ ( v ⊗ w ) ∗ is an isomorphism, and omplex onjugation sets up a linear bije tion between C and C ∗ . This turns ve torspa es into a omplex tensor ategory with V ∗∗ = V .6 Example: the ategory of ve tor bundles over a manifold By using thepro edure des ribed in Example 6.11 one an transport the omplex tensorstru ture of the pre eding example to the ategory V ∞ ( M ; C ) of smooth omplex ve tor bundles (of lo ally (cid:28)nite rank) over a manifold M .Consider a omplex tensor ategory ( C , ⊗ , ∗ ) . By a sesquilinear form onan obje t R ∈ Ob( C ) we mean any arrow b : R ⊗ R ∗ → . A sesquilinearform b on the obje t R will be said to be Hermitian when the sesquilinearform ˜ b on R , de(cid:28)ned as the omposite(7) R ⊗ R ∗ ∼ = R ∗∗ ⊗ R ∗ ∼ = ( R ⊗ R ∗ ) ∗ b ∗ −−→ ∗ ∼ = ,8. COMPLEX TENSOR CATEGORIES 47 oin ides with b itself, i.e. ˜ b = b . Note that one always has the equality ˜˜ b = b .Clearly, in the examples above one re overs the familiar notions.Suppose now that our omplex tensor ategory is rigid. Then for ea hobje t R we an (cid:28)nd another obje t R ′ , along with arrows e R : R ′ ⊗ R → and d R : → R ⊗ R ′ , su h that the following ompositions are identities:(8) R ∼ = ⊗ R d R ⊗ R −−−−→ R ⊗ R ′ ⊗ R R ⊗ e R −−−−→ R ⊗ ∼ = RR ′ ∼ = R ′ ⊗ R ′ ⊗ d R −−−−−→ R ′ ⊗ R ⊗ R ′ e R ⊗ R ′ −−−−−→ ⊗ R ′ ∼ = R ′ .We make the assumption that for ea h obje t R we have sele ted one su htriple ( R ∨ , e R , d R ) . Then for ea h R we obtain a well-de(cid:28)ned isomorphism q R : R ∨∗ ∼ → ( R ∗ ) ∨ , namely the unique arrow q su h that(9) R ∨∗ ⊗ R ∗ q ⊗ R ∗ −−−→ R ∗∨ ⊗ R ∗ e R ∗ −−→ equals R ∨∗ ⊗ R ∗ ∼ = ( R ∨ ⊗ R ) ∗ ( e R ) ∗ −−−→ ∗ ∼ = .We say that a sesquilinear form b on R is nondegenerate, when the arrows b - : R → R ∗∨ and b - : R ∗ → R ∨ , de(cid:28)ned as the unique solutions to(10) R ⊗ R ∗ b - ⊗ R ∗ −−−−→ R ∗∨ ⊗ R ∗ e R ∗ −−→ equals b and b equals R ⊗ R ∗ R ⊗ b - −−−→ R ⊗ R ∨ ∼ = R ∨ ⊗ R e R −−→ ,are isomorphisms. If b is Hermitian then b - is an isomorphism if and only ifso is b -. Indeed, the diagrams R ∗ b - (cid:15) (cid:15) ( ˜ b - ) ∗ / / R ∗∨∗ R ∗∗∼ = (cid:15) (cid:15) ( ˜ b ) - ∗ / / R ∨∗ q R ≈ (cid:15) (cid:15) R ∨ ∼ = / / R ∨∗∗≈ ( q R ) ∗ O O R b - / / R ∗∨ (11) ommute for an arbitrary sesquilinear form b on R .Let ( C , ⊗ , ∗ ) be a omplex tensor ategory. By a des ent datum on anobje t R ∈ Ob( C ) we mean an isomorphism µ : R ∼ → R ∗ su h that the omposition R µ ≈ R ∗ µ ∗ ≈ R ∗∗ ∼ = R equals id R . We let R C denote the ategorywhose obje ts are the pairs ( R, µ ) onsisting of an obje t R of C and a des entdatum µ on R and whose morphisms a : ( R, µ ) → ( R ′ , µ ′ ) are the morphisms a : R → R ′ su h that µ ′ · a = a ∗ · µ . Note that R C is naturally an R-linear ategory; moreover, there is an obvious indu ed tensor stru ture, whi h turnsR C into an R-linear tensor ategory.As an example of this onstru tion, observe that one has an obvious equiv-alen e of real tensor ategories between V ec R and R ( V ec C ) : in one dire tion,8 CHAPTER II. THE LANGUAGE OF TENSOR CATEGORIESto any real ve tor spa e V one an assign the pair ( C ⊗ V , z ⊗ v z ⊗ v ) ; onversely, any des ent datum µ : U ∼ → U ∗ on a omplex ve tor spa e U determines the real subspa e U µ ⊂ U of µ -invariant ve tors. More generally,one has analogous equivalen es of real tensor ategories between V ∞ ( M ; R ) and R (cid:0) V ∞ ( M ; C ) (cid:1) , R ∞ ( M ; R ) and R (cid:0) R ∞ ( M ; C ) (cid:1) and so on.Noti e that any omplex tensor fun tor F : C → D will indu e a lineartensor fun tor R F : R C → R D . By using the fa t that the isomorphism R ⊕ R ∗ ≈ ( R ⊕ R ∗ ) ∗ is a des ent datum on R ⊕ R ∗ for ea h R , one aneasily show that setting ˆ λ ( R, µ ) = λ ( R ) de(cid:28)nes a bije tion(12) Hom ⊗ , ∗ ( F, G ) ∼ → Hom ⊗ ( R F , R G ) , λ ˆ λ between the self- onjugate tensor preserving transformations F → G andthe tensor preserving transformations R F → R G , for any omplex tensorfun tors F, G : C → D .Ÿ9 Review of Groups and Tannaka DualityThroughout the present se tion, k is a (cid:28)xed (cid:28)eld. We let V ec k denote the ategory of (cid:28)nite dimensional ve tor spa es over k ; this is a rigid abelianlinear tensor ategory with End( ) = k . All k -algebras are understood to be ommutative.Let G = Spec A be an a(cid:30)ne group s heme over k , ie a group obje tin the ategory Sch ( k ) of (a(cid:30)ne) s hemes over k (s hemes endowed witha morphism G → Spec k , in other words with A a k -algebra). This meansthat we have morphisms of s hemes: (cid:16)multipli ation(cid:17) G × k G → G , (cid:16)unitelement(cid:17) Spec k → G , (cid:16)inverse(cid:17) G → G (over k ), satisfying the usual grouplaws; equivalently, one is given morphisms of k -algebras ∆ : A → A ⊗ k A , ε : A → k and σ : A → A (the omultipli ation, ounit and oinverse maps)su h that the following axioms hold: oasso iativity, oidentity A ∆ (cid:15) (cid:15) ∆ / / A ⊗ A id ⊗ ∆ (cid:15) (cid:15) A ≈ FFFFFFFFF ∆ / / A ⊗ A ε ⊗ id (cid:15) (cid:15) A ⊗ A ∆ ⊗ id / / A ⊗ A ⊗ A k ⊗ A and oinverse A ε (cid:15) (cid:15) ∆ / / A ⊗ A ( σ, id ) (cid:15) (cid:15) k (cid:31) (cid:127) / / A. If A is a (cid:28)nitely generated k -algebra, we say that G is algebrai or that it isan algebrai group. One de(cid:28)nes a oalgebra over k to be a ve tor spa e C over k endowed with linear maps ∆ : C → C ⊗ k C and ε : C → k satisfying the9. REVIEW OF GROUPS AND TANNAKA DUALITY 49 oasso iativity and oidentity axioms. A (right) omodule over a oalgebra C is a ve tor spa e V over k together with a linear map ρ : V → V ⊗ C su hthat the following diagrams ommute V ρ (cid:15) (cid:15) ρ / / V ⊗ C ρ ⊗ ∆ (cid:15) (cid:15) V ≈ GGGGGGGGG ρ / / V ⊗ C id ⊗ ε (cid:15) (cid:15) V ⊗ C ρ ⊗ id / / V ⊗ C ⊗ C V ⊗ k For example, ∆ de(cid:28)nes a C - omodule stru ture on C itself.An a(cid:30)ne group s heme G = Spec A an be regarded as a fun tor G : k - alg −→ groups of k -algebras with values into groups ( f. also Waterhouse1979 [35℄): G ( R ) = Hom k - alg ( A, R ) , for every k -algebra R, so in parti ular, when R = k , G ( k ) = Hom k - alg ( A, k )= Hom
Sch ( k ) (Spec k, G ) is the set of losed k -rational points of G . The group stru ture on G ( R ) isobtained as follows: for s, t ∈ G ( R ) , the produ t s · t , the neutral elementand the inverse s − are respe tively de(cid:28)ned as A ∆ −→ A ⊗ k A s ⊗ k t −−→ R ⊗ k R mult. −−−→ R,A ε −→ k unit −−→ R,A σ −→ A s −→ R. Let C be a rigid abelian k -linear tensor ategory, and let ω : C −→ V ec k be an exa t faithful k -linear tensor fun tor. Then one an de(cid:28)ne a fun tor Aut ⊗ ( ω ) : k - alg −→ groups , as follows. For R a k -algebra, there is a anoni al tensor fun tor φ R : V ec k −→ Mod R , V V ⊗ k R into the ategory of R -modules (this is an abelian tensor ategory with End( ) = R , but in general it will not be rigid be ause not all R -modules will be re(cid:29)exive). If F, G : C −→ V ec k are tensor fun tors, thenwe an de(cid:28)ne Hom ⊗ ( F, G ) to be the fun tor of k -algebras Hom ⊗ ( F, G )( R ) = Hom ⊗ ( φ R ◦ F, φ R ◦ G ) . Thus
Aut ⊗ ( ω )( R ) onsists of families ( λ X ) , X ∈ Ob( C ) where λ X is an R -linear automorphism of ω ( X ) ⊗ k R su h that λ X ⊗ X = λ X ⊗ λ X , λ is theidentity mapping of R , and ω ( X ) ⊗ R ω ( a ) ⊗ id (cid:15) (cid:15) λ X / / ω ( X ) ⊗ R ω ( a ) ⊗ id (cid:15) (cid:15) ω ( Y ) ⊗ R λ Y / / ω ( Y ) ⊗ R a : X → Y in C . In the spe ial ase where C = R ( G ; k ) for some a(cid:30)ne group s heme G over k , and ω is the forgetful fun tor R ( G ; k ) −→ V ec k , it is lear that every element of G ( R ) de(cid:28)nes an elementof Aut ⊗ ( ω )( R ) . One has the following result1 Proposition The natural transformation G → Aut ⊗ ( ω ) (of fun torsof k -algebras with values into groups) is an isomorphism.2 Theorem Let C be a rigid abelian tensor ategory su h that End( ) = k , and let ω : C −→ V ec k be an exa t faithful k -linear tensor fun tor. Theni) the fun tor Aut ⊗ ( ω ) of k -algebras is representable by an a(cid:30)ne groups heme G ;ii) ω de(cid:28)nes an equivalen e of tensor ategories C −→ R ( G ; k ) . k is a rigid abelian k -linear tensor ategory C for whi h there exists an exa t faithful k -linear tensorfun tor ω : C −→ V ec k . Any su h fun tor is said to be a (cid:28)bre fun tor for C .Ÿ10 A Te hni al Lemma on Compa t GroupsThroughout the present se tion, let V ec denote the omplex tensor ategoryof omplex ve tor spa es of (cid:28)nite dimension (see Note 8.5).Let C be an arbitrary additive omplex tensor ategory. Let F : C → V ec be a omplex tensor fun tor. Moreover, let H be a topologi al group. Supposewe are given a homomorphism of monoids(1) π : H → End ⊗ , ∗ ( F ) .We shall say that π is ontinuous if for every obje t R ∈ Ob( C ) the indu edrepresentation(2) π R : H → End (cid:0) F ( R ) (cid:1) de(cid:28)ned by h π R ( h ) ≡ π ( h )( R ) is ontinuous.3 Proposition (Te hni al Lemma.) Let C , F and H be as above.Suppose in addition that H is a ompa t Lie group. Finally, let π : H → End ⊗ , ∗ ( F ) be a ontinuous homomorphism.Assume the following ondition holds:10. A TECHNICAL LEMMA ON COMPACT GROUPS 51(*) for any ouple of obje ts R, S ∈ Ob( C ) and for ea h homomorphism A : F ( R ) → F ( S ) of the orresponding H -modules(cid:22)in other words,for ea h C-linear map A su h that the diagram F ( R ) A (cid:15) (cid:15) π R ( h ) / / F ( R ) A (cid:15) (cid:15) F ( S ) π S ( h ) / / F ( S ) (4) ommutes ∀ h ∈ H (cid:22)there is an arrow R a −→ S su h that A = F ( a ) .Then π is surje tive; in parti ular, End ⊗ , ∗ ( F ) = Aut ⊗ , ∗ ( F ) is ne essarilya group.Proof Put K def = Ker π ⊂ H . This is a losed normal subgroup, be auseit oin ides with the interse tion T Ker π X over all obje ts X of C . On thequotient G def = H/K there is a unique ( ompa t) Lie group stru ture su h thatthe quotient homomorphism H ։ G is a Lie group homomorphism. Every π X an indi(cid:27)erently be thought of as a ontinuos representation of H or a ontinuous representation of G , and every linear map A : F ( X ) → F ( Y ) is amorphism of G -modules if and only if it is a morphism of H -modules. Being ontinuous, every π X is also smooth.We laim there exists an obje t R of C su h that the orresponding π R isfaithful as a representation of G . This an be seen in a ompletely standardway, f. for instan e Brö ker and tom Die k (1985), pp. 136(cid:21)137; nonetheless,in the present more abstra t situation it will be useful to have a look at theargument in detail anyway. The `Noetherian' property of the ompa t Liegroup G allows us to (cid:28)nd X , . . . , X ℓ ∈ Ob( C ) with the property that(5) Ker π X ∩ · · · ∩ Ker π X ℓ = { e } as representations of G , where e denotes the neutral element. Then, setting R def = X ⊕ · · · ⊕ X ℓ , the representation π R will be faithful be ause of theexisten e of an isomorphism of G -modules(6) F ( X ⊕ · · · ⊕ X ℓ ) ≈ F ( X ) ⊕ · · · ⊕ F ( X ℓ ) .(The existen e of su h isomorphisms follows from the remark that a naturaltransformation of additive fun tors is additive: for instan e, when ℓ = 2 , F X
F i X (cid:15) (cid:15) π ( h )( X ) / / F X
F i X (cid:15) (cid:15) F X ⊕ F Y ≈ (cid:15) (cid:15) ⇒ π X ( h ) ⊕ π Y ( h ) / / F X ⊕ F Y ≈ (cid:15) (cid:15) F ( X ⊕ Y ) π ( h )( X ⊕ Y ) / / F ( X ⊕ Y ) F ( X ⊕ Y ) π X ⊕ Y ( h ) / / F ( X ⊕ Y ) F Y
F i Y O O π ( h )( Y ) / / F Y
F i Y O O F ( X ) ⊕ F ( Y ) ≈ F ( X ⊕ Y ) is also anisomorphism of H -modules or, equivalently, G -modules.)It follows that the G -module F ( R ) is a tensor generator for the omplextensor ategory R ( G ; C ) of ontinuous (cid:28)nite dimensional omplex G -modules.Indeed, every irredu ible su h G -module V embeds as a submodule of sometensor power F ( R ) ⊗ k ⊗ ( F ( R ) ∗ ) ⊗ ℓ (see for instan e Brö ker and tom Die k,1985); sin e by assumption ea h π ( h ) is a self- onjugate tensor preservingnatural transformation, this tensor power will be naturally isomorphi to F (cid:0) R ⊗ k ⊗ ( R ∗ ) ⊗ ℓ (cid:1) as a G -module and hen e, as a onsequen e of the existen eof the G -module isomorphisms (6), for ea h obje t V of R ( G ; C ) there willbe some obje t X of C su h that V embeds into F ( X ) as a submodule.Next, onsider an arbitrary natural transformation λ ∈ End( F ) . Let X bean obje t of C and let V ⊂ F X be a submodule. The hoi e of a omplementto V in F X determines a module endomorphism P : F X → V ֒ → F X whi h,by ondition (*), omes from some endomorphism X p −→ X ∈ C . Therefore F X P (cid:15) (cid:15) λ ( X ) / / F X P (cid:15) (cid:15) F X λ ( X ) / / F X (7) ommutes and, onsequently, λ ( X ) maps V into itself. I will usually omit X from the notation and simply write λ V : V → V for the linear map that λ ( X ) indu es on V by restri tion.Given any other submodule W ⊂ F Y and any module homomorphism B : V → W , the diagram V B (cid:15) (cid:15) λ V / / V B (cid:15) (cid:15) W λ W / / W (8)is ne essarily ommutative. To prove this, extend the given homomorphism B : V → W to a homomorphism B ′ : F X → F Y (for instan e, by hoosinga omplement to V in F X and then by taking the omposite map
F X → V B −→ W ֒ → F Y ) and then argue as before, by invoking the assumption (*).Next, we de(cid:28)ne an isomorphism of omplex algebras(9) θ : End( F ) ∼ → End( ω G ) so that the following diagram ommutes H pr (cid:15) (cid:15) π / / End( F ) θ (cid:15) (cid:15) G π G / / End( ω G ) ,(10)10. A TECHNICAL LEMMA ON COMPACT GROUPS 53where ω G : R ( G ; C ) → V ec is the standard forgetful fun tor (whi h to any G -module asso iates the underlying omplex ve tor spa e) and π G ( g ) is thenatural transformation ̺ π G ( g )( ̺ ) ≡ ̺ ( g ) . Given a module V , there existsan obje t X of C together with an embedding V ֒ → F X , so we may de(cid:28)ne θ ( λ )( V ) to be the restri tion of λ ( X ) to V (this makes sense in view of theabove remarks). Of ourse, it is ne essary to he k that θ is well-de(cid:28)ned.Suppose we are given two obje ts X, Y ∈ Ob( C ) , along with G -moduleembeddings of V into F X , F Y respe tively. Sin e it is always possible toembed everything equivariantly into F ( X ⊕ Y ) and sin e doing this does nota(cid:27)e t the indu ed λ V 's, it will be no loss of generality to assume that X = Y .Let W, W ′ ⊂ F X be the submodules orresponding to the two di(cid:27)erentembeddings of V into F X . Then by our remark (8) there is a ommutativediagram V ≈ / / W λ W / / ≈ (cid:15) (cid:15) W ≈ (cid:15) (cid:15) ≈ − / / VV ≈ / / W ′ λ W ′ / / W ′ ≈ − / / V ,(11)whi h shows that the two di(cid:27)erent embeddings pre isely determine the samelinear endomorphism of V .Clearly, (8) implies that θ ( λ ) ∈ End( ω G ) . For µ ∈ End( ω G ) and X ∈ C ,put µ F ( X ) = µ ( F X ) ; then µ F ∈ End( F ) and θ ( µ F ) = µ , be ause of theexisten e of embeddings V ֒ → F X and be ause of naturality of µ : hen e θ is surje tive. The latter map is also inje tive sin e λ ( X ) = θ ( λ )( F X ) . It isstraightforward to he k that the diagram (10) ommutes.Now, to on lude the proof, it will be enough to show that θ indu es abije tion between End ⊗ , ∗ ( F ) and End ⊗ , ∗ ( ω G ) = T ( G ) , be ause then from(10) we get at on e the following ommutative square H pr (cid:15) (cid:15) π / / End ⊗ , ∗ ( F ) θ ≈ (cid:15) (cid:15) G π G / / T ( G ) ,(12)where the map on the bottom is a bije tion (by the lassi al Tannaka dualitytheorem for ompa t groups), when e surje tivity of π is evident.For instan e, suppose λ ∈ End ⊗ ( F ) and let V and W be G -modulesthat admit equivariant embeddings V ֒ → F X and
W ֒ → F Y for some
X, Y ∈ Ob( C ) . Sin e we are dealing with (cid:28)nite dimensional spa es, V ⊗ W ֒ → F X ⊗ F Y ∼ = F ( X ⊗ Y ) will be also an embedding of G -modules. Then, bythe de(cid:28)nition of θ and the assumption that λ is tensor preserving, we see4 CHAPTER II. THE LANGUAGE OF TENSOR CATEGORIESthat the diagram F ( X ⊗ Y ) λ ( X ⊗ Y ) / / F ( X ⊗ Y ) V ⊗ W ?(cid:31) O O λ V ⊗ λ W / / V ⊗ W ?(cid:31) O O (13)must ommute. This shows that θ ( λ )( V ⊗ W ) = θ ( λ )( V ) ⊗ θ ( λ )( W ) . Thereverse dire tion is straightforward. q.e.d.The argument that we used above in order to (cid:28)nd the tensor generator R admits the following generalization to the non- ompa t ase. Let C and F be as in the statement of the pre eding proposition.14 Proposition Let G be a Lie group. Suppose that(15) π : G −→ Aut( F ) is a faithful ontinuous homomorphism(cid:22)in other words, a ontinuoushomomorphism su h that for ea h g = e ∈ G there exists an obje t X in C with π X ( g ) = id F X .Then there exists an obje t R ∈ Ob( C ) for whi h Ker π R is a dis retesubgroup of G or, equivalently, for whi h the ontinuous representation(16) π R : G → GL ( F R ) is faithful(cid:22)i.e. inje tive(cid:22)on some open neighbourhood of e .Proof Let X be an arbitrary obje t of C . Then K def = Ker π X is a losed Liesubgroup of G . The onne ted omponent K e of e in K is also a losed Liesubgroup of G ; in parti ular, the in lusion map K e ֒ → G is an embedding ofLie groups (that is, a Lie subgroup and an embedding of manifolds). So, if Y is another obje t, the ontinuous representation π Y : G → GL ( F Y ) indu esby restri tion a ontinuous representation of K e .The kernel D def = K e ∩ Ker π Y is a losed Lie subgroup(cid:22)in parti ular, a losed submanifold(cid:22)of K e again. Thus, either dim D < dim K e or D = K e ,be ause K e is onne ted. Sin e π is faithful, when dim K e > we an always(cid:28)nd some obje t Y su h that D $ K e .Then it follows that for ea h X ∈ Ob( C ) one an always (cid:28)nd anotherobje t Y su h that the submanifold Ker π X ⊕ Y has dimension stri tly smallerthan the dimension of Ker π X , unless dim Ker π X = 0 . Hen e an indu tiveargument using additivity of the ategory C will yield an obje t R su h that dim Ker π R = 0 i.e. Ker π R is dis rete, as ontended. q.e.d.hapter IIIRepresentation Theory RevisitedIn the present hapter we introdu e our language of smooth sta ks of (addi-tive, real or omplex) tensor ategories, or brie(cid:29)y smooth (real or omplex)tensor sta ks. We propose this language as the general foundational frame-work for the theory of representations of Lie groupoids.Some general onventions. We use the expressions `smooth' and `of lass C ∞ ' as synonyms. The apital letters X, Y and Z stand for manifolds of lass C ∞ , the orresponding lower- ase letters x, x ′ , . . . , y, et . denote pointson these manifolds. C ∞ X indi ates the sheaf of smooth fun tions on X (weusually omit the subs ript). Sheaves of C ∞ X -modules will also be referred toas sheaves of modules over X . For pra ti al purposes, we need to onsidermanifolds whi h are possibly neither Hausdor(cid:27) nor para ompa t.Ÿ11 The Language of Fibred TensorCategoriesFibred tensor ategories. Fibred tensor ategories will be denoted by meansof apital Gothi type variables. Of ourse, as in Ÿ8, we have to distinguishbetween the notions of real and omplex (cid:28)bred tensor ategory. We do the omplex version; the real ase is entirely analogous.A (cid:28)bred omplex tensor ategory T assigns, to ea h smooth manifold X ,an additive omplex tensor ategory(1) T ( X ) = (cid:0) T ( X ) , ⊗ X , X , ∗ X (cid:1) or (cid:0) T ( X ) , ⊗ , , ∗ (cid:1) for short(cid:22)omitting subs ripts when they are lear fromthe ontext(cid:22)and, to ea h smooth mapping X f −→ Y , a omplex tensor fun tor(2) f ∗ : T ( Y ) −→ T ( X ) alled (cid:19) pull-ba k along f (cid:20). Moreover, for ea h pair of omposable smoothmaps X f −→ Y g −→ Z and for ea h manifold X , any (cid:28)bred omplex tensor556 CHAPTER III. REPRESENTATION THEORY REVISITED ategory provides self- onjugate tensor preserving natural isomorphisms(3) ( δ : f ∗ ◦ g ∗ ∼ → ( g ◦ f ) ∗ ε : Id ∼ → id X ∗ .These are altogether required to make the following diagrams ommute f ∗ g ∗ h ∗ δ · h ∗ (cid:15) (cid:15) f ∗ δ / / f ∗ ( hg ) ∗ δ (cid:15) (cid:15) id X ∗ f ∗ δ (cid:15) (cid:15) f ∗ f ∗ ε (cid:15) (cid:15) vvvvvvvvvvvvvvvvvvvv ε · f ∗ o o ( gf ) ∗ h ∗ δ / / ( hgf ) ∗ f ∗ f ∗ id Y ∗ . δ o o (4)This is all of the mathemati al data we need to introdu e in order to speakabout smooth tensor sta ks and, later on, representations of Lie groupoids.All the required on epts an(cid:22)and will(cid:22)be de(cid:28)ned in terms of the given ategori al stru ture T , i.e. anoni ally. We now explain how.Smooth tensor presta ksThroughout the present subse tion we let P denote a (cid:28)bred omplex tensor ategory, (cid:28)xed on e and for all.Notation. For i U : U ֒ → X the in lusion of an open subset, we shallput E | U = i U ∗ E and a | U = i U ∗ a for any obje t E and morphism a of the ategory P ( X ) . (More generally, we shall adopt this abbreviation for thein lusion i S : S ֒ → X of any submanifold.)For any pair of obje ts E, F ∈ Ob P ( X ) , we let H om P X ( E, F ) denote thepresheaf of omplex ve tor spa es over X de(cid:28)ned by(5) U Hom P ( U ) ( E | U , F | U ) ,with the obvious restri tion maps a j ∗ a orresponding to the in lusions j : V ֒ → U of open subsets. (To be pre ise, restri tion along j sends a to the unique morphism E | V → F | V whi h orresponds to j ∗ a up to the anoni al isomorphisms j ∗ ( E | U ) ∼ = E | V and j ∗ ( F | U ) ∼ = F | V of (3).) Now,the requirement that P be a presta k means exa tly that any su h presheafis in fa t a sheaf; in parti ular, it entails that one an glue any family of ompatible lo al morphisms over X . Two spe ial ases will be of parti ularinterest to us: the sheaf Γ E = H om P X ( , E ) , to be referred to as the sheaf ofsmooth se tions of E ∈ Ob P ( X ) , and the sheaf E ∨ = H om P X ( E, ) , to bereferred to as the sheaf dual of E . For any open subset U , the elements of Γ E ( U ) will be of ourse referred to as the smooth se tions of E over U ; it isperhaps useful to point out that it makes sense, for smooth se tions over U ,to take linear ombinations with omplex oe(cid:30) ients, be ause Γ E ( U ) has a anoni al ve tor spa e stru ture.11. THE LANGUAGE OF FIBRED TENSOR CATEGORIES 57Sin e a morphism a : E → F in P ( X ) yields a morphism Γ a : Γ E → Γ F of sheaves of omplex ve tor spa es over X (by omposing | U → E | U a | U −−→ F | U ), we obtain a anoni al fun tor(6) Γ = Γ X : P ( X ) −→ { sheaves of C X - modules } ,where C X denotes the onstant sheaf over X of value C. (Note that a sheaf of omplex ve tor spa es over a topologi al spa e X is exa tly the same thingas a sheaf of C X -modules.)This fun tor is ertainly linear. Moreover, there is an evident way to makeit a pseudo-tensor fun tor of the tensor ategory (cid:0) P ( X ) , ⊗ X , X (cid:1) into the ategory of sheaves of C X -modules (with the standard tensor stru ture). Indetail, a natural transformation τ E,F : Γ X E ⊗ C X Γ X F → Γ X ( E ⊗ F ) arises,in the obvious manner, from the lo al pairings(7) Γ E ( U ) × Γ F ( U ) −→ Γ ( E ⊗ F )( U )( | U a −→ E | U , | U b −→ F | U ) | U ∼ = | U ⊗ | U a ⊗ b −−→ E | U ⊗ F | U ∼ = ( E ⊗ F ) | U (whi h are bilinear with respe t to lo ally onstant oe(cid:30) ients), and a morph-ism υ : C X → Γ X an be easily de(cid:28)ned as follows(8) locally constant complexvalued functions on U −→ Γ ( U ) t : U → C t · U : | U → | U (where U = id : | U → | U is the (cid:16)unity onstant se tion(cid:17)); the operationof multipli ation by t in (7) and (8) is well-de(cid:28)ned be ause t is a omplex onstant, at least lo ally. It is easy to he k that these morphisms of sheavesmake all the diagrams in the de(cid:28)nition of a tensor fun tor ommute.Note that for X = ⋆ , where ⋆ is the one-point manifold, one has thestandard identi(cid:28) ation { sheaves of C ⋆ - modules } = { complex vector spaces } of omplex tensor ategories. One may therefore regard, for X = ⋆ , thefun tor (6) as a linear pseudo-tensor fun tor(9) P ( ⋆ ) −→ { complex vector spaces } .It will be onvenient to have a short notation for this; making the aboveidenti(cid:28) ation of ategories expli it, we put, for all obje ts E ∈ Ob P ( ⋆ ) ,(10) E ∗ = ( Γ ⋆ E )( ⋆ ) (so this is a omplex ve tor spa e), and do the same for morphisms. Now,as a part of the de(cid:28)nition of the general notion of smooth tensor sta k, weask that the following ondition be satis(cid:28)ed: the morphism of sheaves (8) isan isomorphisms for X = ⋆ . Let us re ord an immediate onsequen e of thisrequirement: there is a anoni al isomorphism of omplex ve tor spa es(11) C ∼ → ∗ .8 CHAPTER III. REPRESENTATION THEORY REVISITED12 Note When dealing with the ase of (cid:28)bred omplex tensor ategories,one also has a natural morphism of sheaves of modules over X (13) ( Γ X E ) ∗ −→ Γ X ( E ∗ ) de(cid:28)ned by means of the anti-involution and the obvious related anoni alisomorphisms. Sin e ζ ∗∗ = ζ (up to anoni al isomorphism), it follows aton e that (13) is a natural isomorphism for an arbitrary omplex tensorpresta k; in fa t, (13) is an isomorphism of pseudotensor fun tors viz. it is ompatible(cid:22)in the sense of Ÿ7(cid:22)with the natural transformations (7) and (8).Be ause of these onsiderations, we will not need to worry about omplexstru ture in our subsequent dis ussion of (cid:16)axioms(cid:17) in Ÿ15.Notation. (Fibres of an obje t) Besides the fundamental notion of (cid:19) sheafof smooth se tions (cid:20) we are now able to introdu e a se ond one, that of (cid:19) (cid:28)breat a point (cid:20). Namely, given an obje t E ∈ Ob P ( X ) , we de(cid:28)ne the (cid:28)bre of E at x to be the (cid:28)nite dimensional omplex ve tor spa e E x = ( x ∗ E ) ∗ ; we usethe same name for the point x and for the (smooth) mapping ⋆ → X, ⋆ x ,so that x ∗ is just the ordinary notation (2) for the pull-ba k, x ∗ E belongsto P ( ⋆ ) and we an apply our notation (10). Similarly, whenever a : E → F is a morphism in P ( X ) , we let a x : E x → F x denote the linear map ( x ∗ a ) ∗ .Sin e - ( - ) x is by onstru tion the omposite of two omplex pseudo-tensorfun tors, it may itself be regarded as a omplex pseudo-tensor fun tor. If inparti ular we apply this to a lo al smooth se tion ζ ∈ Γ E ( U ) and make useof the anoni al identi(cid:28) ation (11), we get, for u in U , a linear map(14) C ∼ → ( ⋆ ) ∗ ∼ = ( u ∗ | U ) ∗ ( u ∗ ζ ) ∗ −−−→ ( u ∗ E | U ) ∗ ∼ = ( u ∗ E ) ∗ = E u ,whi h orresponds to a ve tor ζ ( u ) ∈ E u (the image of the unity ∈ C) tobe alled the value of ζ at u . One has the intuitive formula(15) a u · ζ ( u ) = [ Γ a ( U ) ζ ]( u ) .Noti e also that the ve tors ζ ( u ) ⊗ η ( u ) and ( ζ ⊗ η )( u ) orrespond to oneanother in the anoni al linear map E u ⊗ F u → ( E ⊗ F ) u (we may state thisloosely by saying they are equal).We have not explained yet what we mean when we say that a tensorpresta k is (cid:19) smooth (cid:20). This was not ne essary before be ause all we havesaid so far does not depend on that spe i(cid:28) property. However, from thispre ise moment we begin to develop systemati ally on epts whi h, even inorder to be de(cid:28)ned, presuppose the smoothness of the tensor presta k, so itbe omes ne essary to (cid:28)ll the gap.Consider the tensor unit ∈ Ob P ( X ) and let x be any point. There isa anoni al isomorphism C ∼ = x analogous to (11), namely the omposite11. THE LANGUAGE OF FIBRED TENSOR CATEGORIES 59C ∼ = ( ⋆ ) ∗ ∼ = ( x ∗ ) ∗ = x . This identi(cid:28) ation allows us to de(cid:28)ne a anoni alhomomorphism of omplex algebras(16) End P ( X ) ( ) −→ { functions X → C } , e ˜ e by putting ˜ e ( x ) = the omplex onstant su h that the linear map (cid:16)s alarmultipli ation by ˜ e ( x ) (cid:17) (of C into itself) orresponds to e x : x → x underthe linear isomorphism C ∼ = x . We shall say that the tensor presta k P issmooth if the homomorphism (16) determines a one-to-one orresponden eonto the subalgebra of smooth fun tions on X (17) End P ( X ) ( ) ∼ = C ∞ ( X ) .A (cid:28)rst onsequen e of the smoothness of P is the possibility to endowea h spa e Hom P ( X ) ( E, F ) with a C ∞ ( X ) -module stru ture, anoni al and ompatible with the already de(cid:28)ned operation of multipli ation by lo ally onstant fun tions. Indeed, the natural a tion(18) End P ( X ) ( ) × Hom P ( X ) ( E, F ) −→ Hom P ( X ) ( E, F ) , ( e, a ) E ∼ = ⊗ E e ⊗ a −−→ ⊗ F ∼ = F turns Hom P ( X ) ( E, F ) into a left End P ( X ) ( ) -module, hen e we an use theidenti(cid:28) ation of C-algebras (17) to make Hom P ( X ) ( E, F ) a C ∞ ( X ) -module;in short, the module multipli ation an be written as (˜ e, a ) e ⊗ a .A ordingly, H om P X ( E, F )( U ) = Hom P ( U ) ( E | U , F | U ) inherits a anoni alstru ture of C ∞ ( U ) -module, for ea h open subset, and one veri(cid:28)es at on ethat this makes H om P X ( E, F ) a sheaf of C ∞ X -modules. Of ourse, the remarkapplies in parti ular to any sheaf of `smooth' se tions Γ X E , partly justifyingthe terminology; moreover, one readily sees that any morphism a : E → F in the ategory P ( X ) indu es a morphism Γ X a : Γ X E → Γ X F of sheaves of C ∞ X -modules. So we get a C ∞ ( X ) -linear fun tor(19) P ( X ) −→ { sheaves of C ∞ X - modules } ,still denoted by Γ X . (Noti e that both ategories have Hom -sets enri hed witha C ∞ ( X ) -module stru ture1. The C ∞ ( X ) -linearity of the fun tor amountsby de(cid:28)nition to the C ∞ ( X ) -linearity of all the maps Hom P ( X ) ( E, F ) → Hom C ∞ X ( Γ X E, Γ X F ) , a Γ X a . )If one also takes into a ount the tensor stru ture then the pro ess of(cid:19) upgrading (cid:20) the fun tor (6) an be pursued further by observing that theoperations des ribed in (7), (8) may now be used to de(cid:28)ne morphisms ofsheaves of C ∞ X -modules(20) ( τ : Γ X E ⊗ C ∞ X Γ X F → Γ X ( E ⊗ F ) , υ : C ∞ X → Γ X ;1Su h that the omposition of morphisms is C ∞ ( X ) -bilinear.0 CHAPTER III. REPRESENTATION THEORY REVISITEDthe morphism τ = τ E,F is natural in the variables
E, F and, along with υ ,makes (19) a pseudo-tensor fun tor of the tensor ategory P ( X ) into thetensor ategory of sheaves of C ∞ X -modules. This is loser than (6) to beinga tensor fun tor, in that the morphism υ is evidently an isomorphism ofsheaves of C ∞ X -modules.Consider next a smooth mapping of manifolds f : X → Y . Suppose that U ⊂ X and V ⊂ Y are open subsets with f ( U ) ⊂ V , and let f U denote theindu ed mapping of U into V . For any obje t F of the ategory P ( Y ) , weobtain a orresponden e of lo al smooth se tions(21) ( Γ Y F )( V ) −→ Γ X ( f ∗ F )( U ) , η η ◦ f by putting η ◦ f equal by de(cid:28)nition to the omposite(22) | U ∼ = ( f ∗ ) | U ∼ = f ∗ U ( | V ) f ∗ U ( η ) −−−→ f ∗ U ( F | V ) ∼ = ( f ∗ F ) | U .One easily veri(cid:28)es that for U (cid:28)xed and V variable, the maps (21) form anindu tive system indexed over the in lusions of neighbourhoods V ⊃ V ′ ⊃ f ( U ) , and that eventually they indu e a morphism of sheaves of C ∞ X -modules(23) f ∗ ( Γ Y F ) −→ Γ X ( f ∗ F ) ,where f ∗ ( Γ Y F ) is the ordinary pull-ba k in the sense of sheaves of modulesover smooth manifolds. It is also lear that the morphism (23) is naturalin F , and also a morphism of pseudo-tensor fun tors (in other words, it istensor preserving). To on lude, let us give some motivation for the notation(cid:19) η ◦ f (cid:20). There is an obvious anoni al isomorphism of ve tor spa es(24) ( f ∗ F ) x = ( x ∗ f ∗ F ) ∗ ∼ = ( f ( x ) ∗ F ) ∗ = F f ( x ) .Now, we have the two ve tors η ( f ( x )) ∈ F f ( x ) and ( η ◦ f )( x ) ∈ ( f ∗ F ) x ,and you an easily he k that they orrespond to one another in the aboveisomorphism. We an state this loosely as(25) ( η ◦ f )( x ) = η ( f ( x )) .The last expression evidently justi(cid:28)es our notation.Ÿ12 Smooth Tensor Sta ksIt will be onvenient to regard the open overings of a manifold X as smoothmappings onto X . This an be made pre ise as follows. Borrowing somestandard terminology from algebrai geometers, we shall say that a smoothmapping p : X ′ → X is (cid:29)at, if it is surje tive and it restri ts to an openembedding p U ′ : U ′ ֒ → X on ea h onne ted omponent U ′ of X ′ ; we may12. SMOOTH TENSOR STACKS 61think of p as odifying a ertain open overing of X , indexed by the set of onne ted omponents of X ′ . A re(cid:28)nement of X ′ p −→ X will be obtained by omposing p ba kwards with another (cid:29)at mapping X ′′ p ′ −→ X ′ . The funda-mental property of (cid:29)at mappings is that they an be pulled ba k, preserving(cid:29)atness, along any smooth map: pre isely, for any Y f −→ X the pull-ba k(1) Y × X X ′ = (cid:8) ( y, x ′ ) : f ( y ) = p ( x ′ ) (cid:9) will make sense in the ategory of C ∞ -manifolds and the (cid:28)rst proje tion pr : Y × X X ′ → Y will be a (cid:29)at mapping. Parti ularly relevant is the asewhere f is also a (cid:29)at mapping, leading to the (cid:16)standard(cid:17) ommon re(cid:28)nementfor f and p .Some standard abbreviations. For any (cid:29)at mapping p : X ′ → X , let(2) X ′′ = X ′ × X X ′ = (cid:8) ( x ′ , x ′ ) : p ( x ′ ) = p ( x ′ ) (cid:9) ,with the two proje tions p , p : X ′′ → X ′ ; and the triple (cid:28)bred produ t(3) X ′′′ = X ′ × X X ′ × X X ′ = (cid:8) ( x ′ , x ′ , x ′ ) : p ( x ′ ) = p ( x ′ ) = p ( x ′ ) (cid:9) with its proje tions p , p , p : X ′′′ → X ′′ resp. given by ( x ′ , x ′ , x ′ ) ( x ′ , x ′ ) and so forth.A des ent datum for a smooth omplex tensor presta k P , relative tothe (cid:29)at mapping p : X ′ → X , will be a pair ( E ′ , θ ) onsisting of an obje t E ′ ∈ P ( X ′ ) and an isomorphism θ : p ∗ E ′ ∼ → p ∗ E ′ in P ( X ′′ ) , su h that p ∗ ( θ ) = p ∗ ( θ ) ◦ p ∗ ( θ ) up to the anoni al isos. A morphism of des entdata, let us say of ( E ′ , θ ) into ( F ′ , ξ ) , will be a morphism a ′ : E ′ → F ′ in P ( X ′ ) ompatible with θ and ξ in the sense that p ∗ ( a ′ ) ◦ θ = ξ ◦ p ∗ ( a ′ ) .Des ent data of type P and relative to X ′ p −→ X (and their morphisms) forma ategory D es P ( X ′ /X ) . There is an obvious fun tor(4) P ( X ) −→ D es P ( X ′ /X ) , E ( p ∗ E, φ E ) , a p ∗ a de(cid:28)ned by letting φ E be the anoni al isomorphism p ∗ ( p ∗ E ) ∼ = ( p ◦ p ) ∗ E =( p ◦ p ) ∗ E ∼ = p ∗ ( p ∗ E ) . Whenever the fun tor (4) is an equivalen e of at-egories for every (cid:29)at mapping of manifolds p : X ′ → X , one says that thepresta k P is a sta k.5 Note Depending on one's purposes, the ondition that the fun tors (4)be equivalen es of ategories for all (cid:29)at mappings X ′ → X an be weakenedto some extent. For example, one ould ask it to be satis(cid:28)ed just for all (cid:29)at X ′ → X over a Hausdor(cid:27), para ompa t X . In fa t, the latter ondition willprove to be su(cid:30) ient for all our purposes: no relevant aspe t of the theoryseems to depend on the stronger requirement. We propose to use the term(cid:19) parasta k (cid:20) for the weaker notion; we will often be sloppy and use `sta k'as a synonym to `parasta k'.2 CHAPTER III. REPRESENTATION THEORY REVISITEDLo ally trivial obje tsLet S be any smooth tensor presta k. An obje t E ∈ Ob S ( X ) will be alled trivial if there exists some V ∈ Ob S ( ⋆ ) for whi h one an (cid:28)nd anisomorphism E α ≈ c X ∗ V in S ( X ) , where c X : X → ⋆ denotes the ollapsemap. Any su h pair ( V, α ) will be said to onstitute a trivialization of E .For an arbitrary manifold X , let V S ( X ) denote the full sub ategoryof S ( X ) formed by the lo ally trivial obje ts of lo ally (cid:28)nite rank; moreexpli itly, E ∈ Ob S ( X ) will be an obje t of V S ( X ) provided one an over X with open subsets U su h that E | U trivializes in S ( U ) by means of atrivialization of the form ( ⊕ · · · ⊕ , α ) or, equivalently, su h that in S ( U ) there exists an isomorphism E | U ≈ U ⊕ · · · ⊕ U . It follows at on e from thebilinearity of ⊗ , the triviality of and the linearity of f ∗ that the operation X V S ( X ) determines a (cid:28)bred (additive, omplex) tensor sub ategory of S . Hen e X V S ( X ) inherits a (cid:28)bred tensor stru ture from S . It is easyto see that one gets in fa t a smooth tensor presta k V S ; moreover, it isobvious that V S is a parasta k resp. a sta k if su h is S .The omplex tensor ategory V S ( X ) very losely relates to that of smooth omplex ve tor bundles over X . Let us make this pre ise. Clearly, everyobje t E ∈ V S ( X ) yields a smooth omplex ve tor bundle over X : just put ˜ E = { ( x, e ) : x ∈ X, e ∈ E x } , with the lo al trivializing harts obtainedfrom lo al trivializations E | U α ≈ U ⊕ · · · ⊕ U , α = ( α , . . . , α d ) by setting ( u, e ) = (cid:0) u ; α ,u ( e ) , . . . , α d,u ( e ) (cid:1) ∈ U × C d . Sin e any morphism a : E → E ′ in V S ( X ) an be lo ally des ribed in terms of (cid:16)matrix expressions(cid:17) with smooth oe(cid:30) ients, setting ˜ a · ( x, e ) = ( x, a x · e ) de(cid:28)nes a morphism of smooth ve torbundles ˜ a : ˜ E → ˜ E ′ . It is an exer ise to show that the assignment E ˜ E de(cid:28)nes a faithful omplex tensor fun tor of V S ( X ) into smooth omplexve tor bundles. Under extremely mild hypotheses, this fun tor will a tuallyprove to be an equivalen e of omplex tensor ategories; this will happen, forexample, when S is a parasta k and X is para ompa t, or when S is sta k.In on lusion, we see that for S a smooth tensor (para-)sta k (and X areasonable manifold), the ategory S ( X ) will essentially in lude(cid:22)as a fulltensor sub ategory(cid:22)all smooth ve tor bundles over X . One arrives at thesame results, alternatively, by onsidering the fun tor Γ X and the ategoryof lo ally free sheaves of C ∞ X -modules of lo ally (cid:28)nite rank. This last remark an be summarized in the diagram V S ( X ) (cid:21) u Γ X ( ( PPPPPP - f ( - ) ≃ / / V ∞ ( X ) I i Γ X v v nnnnnn { sheaves of C ∞ X - modules } (6)( ommutative up to anoni al natural isomorphism). The smooth tensorsta k V ∞ is therefore, in a very pre ise sense, the (cid:16)smallest(cid:17) possible.13. FOUNDATIONS OF REPRESENTATION THEORY 63Ÿ13 Foundations of Representation TheoryWe develop our theory of representations relative to a (cid:19) type (cid:20). This an beany smooth omplex tensor parasta k S , in the sense of Note 12.5. On e atype S has been (cid:28)xed, one an asso iate to any Lie groupoid a mathemati alobje t alled (cid:19) (cid:28)bre fun tor (cid:20).This is done as follows. Let G be a Lie groupoid, let us say, with base M .We are going to onstru t a ategory R S ( G ) , along with a fun tor ω S ( G ) of R S ( G ) into S ( M ) that we shall all the (cid:19) standard (cid:28)bre fun tor (cid:20) of G (oftype S ). An obje t of the ategory R ( G ) = R S ( G ) (every time we like we an omit writing the type S , as this is (cid:28)xed) is de(cid:28)ned to be a pair ( E, ̺ ) with E an obje t of S ( M ) and ̺ a morphism in S ( G ) (1) ̺ : s ∗ E → t ∗ E (where s , t : G → M denote the sour e, resp. target map of G ), su h thatthe appropriate onditions for ̺ to be an a tion(cid:22)in other words, for it to be ompatible with the groupoid stru ture(cid:22)are satis(cid:28)ed, namely:i) p u ∗ ̺ q = id E , where u : M → G denotes the unit se tion. (Here and inthe sequel we adopt the devi e of putting orners around a morphismto indi ate the morphism(cid:22)whi h one, will always be lear from the ontext(cid:22)that orresponds to it up to some anoni al identi(cid:28) ations;for instan e, the last equality, spelled out expli itly, means that thediagram u ∗ s ∗ E ∼ = an. " " DDDDDDD u ∗ ̺ / / u ∗ t ∗ E ∼ = an. } } zzzzzzz E (2) ommutes, where we use the identi(cid:28) ations u ∗ s ∗ E ∼ = ( s ◦ u ) ∗ E = id M ∗ E ∼ = E et . provided by the (cid:28)bred tensor stru ture onstraintsasso iated with S );ii) if we let G (2) = G s × t G denote the manifold of omposable arrows of G , c : G (2) → G , ( g ′ , g ) g ′ g the omposition of arrows and p , p : G (2) → G the two proje tions ( g ′ , g ) g ′ , g onto the (cid:28)rst andse ond fa tor respe tively, we have the identity p c ∗ ̺ q = p p ∗ ̺ q · p p ∗ ̺ q ;that is to say, a ording to our onvention, we have the ommutativityof the following diagram in the ategory S ( G (2) ) : c ∗ s ∗ E jjjjjjjj c ∗ ̺ / / c ∗ t ∗ E TTTT TTTT p ∗ s ∗ E p ∗ ̺ ' ' OOOOOOOOOOOOO p ∗ t ∗ E p ∗ ̺ ooooooooooooo p ∗ t ∗ E p ∗ s ∗ E (3)4 CHAPTER III. REPRESENTATION THEORY REVISITED(whi h involves the anoni al identi(cid:28) ations c ∗ s ∗ E ∼ = ( s ◦ c ) ∗ E =( s ◦ p ) ∗ E ∼ = p ∗ s ∗ E et . provided by the stru ture onstraints of S ).We shall also write (cid:19) c ∗ ̺ = p ∗ ̺ · p ∗ ̺ (mod ∼ = ) (cid:20).This on ludes the des ription of the obje ts of R S ( G ) ; we shall all themrepresentations of G , or G -a tions (of type S ). As morphisms of G -a tions a : ( E, ̺ ) → ( E ′ , ̺ ′ ) we take all those morphisms a : E → E ′ in S ( M ) whi hmake the following square ommutative s ∗ E s ∗ a (cid:15) (cid:15) ̺ / / t ∗ E t ∗ a (cid:15) (cid:15) s ∗ E ′ ̺ ′ / / t ∗ E ′ .(4)We endow the ategory R S ( G ) with the linear stru ture of S ( M ) . Then theforgetful fun tor(5) ω S ( G ) : R S ( G ) −→ S ( M ) , ( E, ̺ ) E is linear and faithful. We all it the standard (cid:28)bre fun tor of G (of type S ).Observe that the linear ategory R S ( G ) is additive. Indeed, (cid:28)x any obje ts R , R ∈ R ( G ) , let us say R i = ( E i , ̺ i ) , and hoose a representative E i ֒ → E ⊕ E i ← ֓ E for the dire t sum in S ( M ) . Then, sin e the linear fun tors s ∗ , t ∗ have to preserve dire t sums ( f. Ma Lane (1998), p. 197), there willbe a unique `universal' isomorphism in S ( G ) s ∗ ( E ⊕ E ) = s ∗ E ⊕ s ∗ E ̺ ⊕ ̺ −−−−−→ t ∗ E ⊕ t ∗ E = t ∗ ( E ⊕ E ) .One he ks that the pair R ⊕ R = ( E ⊕ E , ̺ ⊕ ̺ ) is a G -a tion, that R i ֒ → R ⊕ R i ← ֓ R are morphisms of G -a tions, and that they yield adire t sum in R ( G ) . The pro ess to obtain a null representation is entirelyanalogous, starting from a null obje t in S ( M ) .6 Lemma For an arbitrary G -a tion ( E, ̺ ) ∈ R S ( G ) , the morphism ̺ : s ∗ E → t ∗ E is ne essarily an isomorphism in S ( G ) .Proof Let C be any ategory. De(cid:28)ne two arrows a, a ′ to be `equivalent',and write a ∼ a ′ , if they are isomorphi as obje ts of the arrow ategory Ar ( C ) (in other words, if there exist isomorphisms between their domainsand odomains whi h transform the one arrow into the other). Then thefollowing assertions hold: a) for any fun tor F : C → D , a ∼ a ′ implies F a ∼ F a ′ ; b) the existen e of a natural iso F ∼ → G implies F a ∼ Ga forevery a ; ) if a ∼ a ′ and a is left (resp. right) invertible, then the same istrue of a ′ ; d) ba ∼ id implies that a is left invertible and b right invertible.14. HOMOMORPHISMS AND MORITA INVARIANCE 65Let i : G → G , g g − be the inverse, and onsider the two maps ( i , id ) , ( id , i ) : G → G (2) given by g ( g − , g ) , ( g, g − ) respe tively. Thenone has the following equivalen es of arrows in the ategory S ( G ) id s ∗ E = s ∗ id E a) ∼ s ∗ u ∗ ̺ b) ∼ ( u ◦ s ) ∗ ̺ = [ c ◦ ( i , id )] ∗ ̺ b) ∼ ( i , id ) ∗ c ∗ ̺ a) ∼ ( i , id ) ∗ p c ∗ ̺ q (3) = ( i , id ) ∗ ( p p ∗ ̺ q · p p ∗ ̺ q )= ( i , id ) ∗ p p ∗ ̺ q · ( i , id ) ∗ p p ∗ ̺ q ,hen e ( i , id ) ∗ p p ∗ ̺ q is left invertible in S ( G ) , by d). Sin e this is in turnequivalent to ( i , id ) ∗ p ∗ ̺ ∼ [ p ◦ ( i , id )] ∗ ̺ = id G∗ ̺ ∼ ̺ , ̺ itself will be leftinvertible in S ( G ) , by ). An analogous reasoning will establish the rightinvertibility of ̺ . It follows that ̺ is invertible. q.e.d.Next, we dis uss the standard tensor stru ture on the ategory R ( G ) . Thisstru ture makes R ( G ) an additive linear tensor ategory. The standard (cid:28)brefun tor ω = ω ( G ) turns out to be a stri t tensor fun tor of R ( G ) into S ( M ) ,in the sense that the identities ω ( R ⊗ S ) = ω ( R ) ⊗ ω ( S ) and ω ( ) = hold,so that they an be taken respe tively as the natural onstraints τ and υ inthe de(cid:28)nition of tensor fun tor.We start with the onstru tion of the bifun tor ⊗ : R ( G ) × R ( G ) → R ( G ) .For two arbitrary representations R, S ∈ R ( G ) , let us say R = ( E, ̺ ) and S = ( F, σ ) , we put R ⊗ S = ( E ⊗ F , p ̺ ⊗ σ q ) , where(cid:22)following the usual onvention(cid:22) p ̺ ⊗ σ q stands for the omposite morphism(7) s ∗ ( E ⊗ F ) ∼ = s ∗ E ⊗ s ∗ F ̺ ⊗ σ −−−−→ t ∗ E ⊗ t ∗ F ∼ = t ∗ ( E ⊗ F ) .It is easy to re ognize that the pair R ⊗ S is itself a G -a tion, i.e. an obje tof the ategory R ( G ) ; moreover, if ( E, ̺ ) a −→ ( E ′ , ̺ ′ ) and ( F, σ ) b −→ ( F ′ , σ ′ ) aremorphisms in R ( G ) then so is a ⊗ b : R ⊗ S → R ′ ⊗ S ′ .We de(cid:28)ne the tensor unit of R ( G ) to be the pair ( M , p id q ) , where M the tensor unit of S ( M ) and p id q is the omposite anoni al isomorphism(8) s ∗ M ∼ = G ∼ = t ∗ M .The ACU natural onstraints α , γ , λ , ρ for the tensor stru ture of the base ategory S ( M ) will provide analogous onstraints for the tensor produ twe just introdu ed on R ( G ) . (For example, onsider three representations R, S, T ∈ R ( G ) and let E, F, G ∈ S ( M ) be the respe tive supports; then theisomorphism α E,F,G : E ⊗ ( F ⊗ G ) ∼ → ( E ⊗ F ) ⊗ G is also an isomorphism α R,S,T : R ⊗ ( S ⊗ T ) ∼ → ( R ⊗ S ) ⊗ T in R ( G ) .) A fortiori, the oheren ediagrams for su h `inherited' onstraints will ommute.Ÿ14 Homomorphisms and Morita Invarian eWe now pro eed to study the operation of taking the inverse image of arepresentation along a homomorphism of Lie groupoids. Then we on entrate6 CHAPTER III. REPRESENTATION THEORY REVISITEDon the spe ial ase of Morita equivalen es; in order to give a satisfa torytreatment of these, it will be ne essary to analyze natural transformations ofLie groupoid homomorphisms (cid:28)rst.Let ϕ : G → H be a homomorphism of Lie groupoids and let M f −→ N bethe smooth map indu ed by ϕ on the base manifolds.Suppose ( F, σ ) ∈ R S ( H ) . Consider the morphism(cid:22)whi h we also denoteby ϕ ∗ σ , slightly abusing notation(cid:22)de(cid:28)ned as follows:(1) s G∗ ( f ∗ F ) ∼ = ϕ ∗ s H∗ F ϕ ∗ σ −−−→ ϕ ∗ t H∗ F ∼ = t G∗ ( f ∗ F ) ;the equalities f ◦ s G = s H ◦ ϕ et . a ount, of ourse, for the existen e of the anoni al isomorphisms o urring in (1). It is straightforward to he k thatthe pair ( f ∗ F , ϕ ∗ σ ) onstitutes an obje t of the ategory R S ( G ) and thatif ( F, σ ) b −→ ( F ′ , σ ′ ) is a morphism of H -a tions then f ∗ b is a morphism of ( f ∗ F , ϕ ∗ σ ) into ( f ∗ F ′ , ϕ ∗ σ ′ ) in R S ( G ) . Hen e we get a fun tor(2) ϕ ∗ : R S ( H ) −→ R S ( G ) ,whi h we agree to all the inverse image or pull-ba k (of representations)along ϕ .It is fairly easy to he k that the onstraints(3) ( υ : M ∼ → f ∗ N τ F,F ′ : f ∗ F ⊗ f ∗ F ′ ∼ → f ∗ ( F ⊗ F ′ ) ,asso iated with the tensor fun tor f ∗ , an also fun tion as isomorphisms of G -a tions, υ : ∼ → ϕ ∗ ( ) and τ S,S ′ : ϕ ∗ ( S ) ⊗ ϕ ∗ ( S ′ ) ∼ → ϕ ∗ ( S ⊗ S ′ ) , for all S, S ′ ∈ R ( H ) with, let us say, S = ( F, σ ) and S ′ = ( F ′ , σ ′ ) . A fortiori,these isomorphisms are natural and they provide appropriate tensor fun tor onstraints for ϕ ∗ , thus making ϕ ∗ a tensor fun tor of the tensor ategory R ( H ) into the tensor ategory R ( G ) .Let G ϕ −→ H ψ −→ K be two omposable homomorphisms of Lie groupoidsand let X ϕ −→ Y ψ −→ Z denote the respe tive maps on bases. Note that for anarbitrary a tion T = ( G, τ ) ∈ R ( K ) the anoni al isomorphism ϕ ∗ ψ ∗ G ∼ =( ψ ◦ ϕ ) ∗ G = ( ψ ◦ ϕ ) ∗ G is a tually a morphism ϕ ∗ ( ψ ∗ T ) ∼ → ( ψ ◦ ϕ ) ∗ T inthe ategory R ( G ) . Hen e we get an isomorphism of tensor fun tors(4) ϕ ∗ ◦ ψ ∗ ∼ = −→ ( ψ ◦ ϕ ) ∗ .It is worthwhile remarking that ϕ ∗ (cid:28)ts in the following diagram R S ( H ) ω S ( H ) (cid:15) (cid:15) ϕ ∗ / / R S ( G ) ω S ( G ) (cid:15) (cid:15) S ( N ) f ∗ / / S ( M ) ,(5)14. HOMOMORPHISMS AND MORITA INVARIANCE 67whose ommutativity is to be interpreted as an equality of omposite tensorfun tors(cid:22)thus, involving also the onstraints.The notion from Lie groupoid theory we want to dualize next is thatof natural transformation; this omes about espe ially when one onsidersMorita equivalen es, as we shall see soon. Re all that a transformation τ : ϕ → ϕ (between two Lie groupoid homomorphisms ϕ , ϕ : G → H ) is asmooth mapping τ of the base manifold M of G into the manifold of arrowsof H , su h that τ ( x ) : f ( x ) → f ( x ) ∀ x ∈ M and the familiar diagram f ( x ) ϕ ( g ) (cid:15) (cid:15) τ ( x ) / / f ( x ) ϕ ( g ) (cid:15) (cid:15) f ( x ′ ) τ ( x ′ ) / / f ( x ′ ) (6)is ommutative for all g ∈ G (1) , g : x → x ′ . Suppose an a tion S = ( F, σ ) ∈ R S ( H ) is given. Then one an apply τ ∗ to the isomorphism σ : s ∗ F ≈ −→ t ∗ F to obtain an isomorphism f ∗ F ≈ −→ f ∗ F in the ategory S ( M ) (7) f ∗ F ∼ = τ ∗ s ∗ F τ ∗ σ −−→ τ ∗ t ∗ F ∼ = f ∗ F ,whi h may be denoted by the symbol σ ◦ τ . (Here one uses the identities f = s H ◦ τ and f = t H ◦ τ .) By expressing (6) as an identity betweensuitable smooth maps, one an he k that σ ◦ τ is a tually an isomorphismof G -a tions between ϕ ∗ S and ϕ ∗ S : in detail, onsider the maps ( τ ◦ t , ϕ ) and ( ϕ , τ ◦ s ) , of G (1) (manifold of arrows) into H (2) ≡ H s × t H (mani-fold of omposable arrows), respe tively given by g ( τ ( t g ) , ϕ ( g )) and g ( ϕ ( g ) , τ ( s g )) ; the ommutativity of (6) implies that upon ompos-ing these maps with multipli ation c : H (2) → H one gets the same result, c ◦ ( τ ◦ t , ϕ ) = c ◦ ( ϕ , τ ◦ s ) ; from the latter identity it is easy to see that(7) is a morphism in R S ( G ) . Then the rule ( F, σ ) σ ◦ τ de(cid:28)nes a nat-ural isomorphism(cid:22)in fa t, a tensor preserving one(cid:22)between the fun tors ϕ ∗ , ϕ ∗ : R S ( H ) → R S ( G ) ; we will use the notation(8) τ ∗ : ϕ ∗ ∼ −→ ϕ ∗ , τ ∗ ∈ Iso ⊗ ( ϕ ∗ , ϕ ∗ ) .We are now ready to dis uss Morita equivalen es. Re all that a homo-morphism ϕ : G → H is said to be a Morita equivalen e in ase G ( s , t ) (cid:15) (cid:15) ϕ / / H ( s , t ) (cid:15) (cid:15) M × M f × f / / N × N (9)is a pullba k diagram in the ategory of C ∞ manifolds and the mapping(10) t ◦ pr : M f × s H → N ,8 CHAPTER III. REPRESENTATION THEORY REVISITEDwhi h, loosely speaking, sends f ( x ) h −→ y to y , is a surje tive submersion.Our main goal in this se tion is to show that the pull-ba k fun tor ϕ ∗ : R ( H ) → R ( G ) asso iated with a Morita equivalen e ϕ is an equivalen e oftensor ategories.2 Clearly, it will be enough to show that ϕ ∗ is a ategori alequivalen e (in the familiar sense): this means that we have to look for afun tor ϕ ! : R ( G ) → R ( H ) su h that natural isomorphisms ϕ ! ◦ ϕ ∗ ≃ Id R ( H ) and ϕ ∗ ◦ ϕ ! ≃ Id R ( G ) exist.Noti e that the ondition that the map (10) should be a surje tive sub-mersion will of ourse be satis(cid:28)ed when f itself is a surje tive submersion.As a (cid:28)rst step, we show how the task of onstru ting a quasi-inverse for thepullba k fun tor ϕ ∗ asso iated with an arbitrary Morita equivalen e ϕ maybe redu ed to the spe ial ase where f is pre isely a surje tive submersion.To this end, onsider the weak pullba k (see [27℄, pp. 123(cid:21)132) P χ (cid:15) (cid:15) ψ / / G ϕ (cid:15) (cid:15) H Id / / τ & . H .(11)Let P be the base manifold of the Lie groupoid P . It is well-known (ibid.p. 130) that the Lie groupoid homomorphisms ψ and χ are Morita equiva-len es with the property that the respe tive base maps ψ (0) : P → M and χ (0) : P → N are surje tive submersions. Now, if we su eed in proving that ψ ∗ and χ ∗ are ategori al equivalen es then, sin e by (4) and (8) above wehave a natural isomorphism (a tually, a tensor preserving one)(12) χ ∗ ≈ −→ ( ϕ ◦ ψ ) ∗ ∼ = ←− ψ ∗ ◦ ϕ ∗ ,the same will be true of ϕ ∗ .From now on we will work under the hypothesis that the given Moritaequivalen e ϕ (9) determines a surje tive submersion f : M ։ N on basemanifolds. This being the ase, there exists an open over N = ∪ i ∈ I V i of themanifold N by open subsets V i su h that for ea h of them one an (cid:28)nd asmooth se tion s i : V i ֒ → M to f . We (cid:28)x su h a over and su h se tions on eand for all.Let an arbitrary obje t R = ( E, ̺ ) ∈ R S ( G ) be given. For ea h i ∈ I one an take the pull-ba k E i ≡ s i ∗ E ∈ S ( V i ) . Fix a ouple of indi es i, j ∈ I .Then, sin e (9) is a pull-ba k diagram, for ea h y ∈ V i ∩ V j there is exa tlyone arrow g ( y ) : s i ( y ) → s j ( y ) su h that ϕ ( g ( y )) = y . More pre isely, let y g ( y ) = g ij ( y ) be the smooth mapping de(cid:28)ned as the unique solution to2Re all that a tensor fun tor Φ :
C → D is said to be a tensor equivalen e in ase thereexists a tensor fun tor
Ψ :
D → C along with tensor preserving natural isomorphisms Ψ ◦ Φ ≃ Id C and Φ ◦ Ψ ≃ Id D .14. HOMOMORPHISMS AND MORITA INVARIANCE 69the following universal problem (in the C ∞ ategory) V ij ( s i ,s j ) ' ' g ij % % JJJJJJ u | V ij G ( s , t ) (cid:15) (cid:15) ϕ / / H ( s , t ) (cid:15) (cid:15) M × M f × f / / N × N ,(13)where u : N → H denotes the unit se tion and V ij ≡ V i ∩ V j . Then, putting E i | j = E i | V i ∩ V j and E j | i = E j | V i ∩ V j , one may pull the a tion ̺ ba k along themap g ij so as to get an isomorphism θ ij : E i | j ∼ → E j | i in the ategory S ( V ij ) :(14) E i | j ∼ = ( s ◦ g ij ) ∗ E ∼ = g ij ∗ s ∗ E g ij ∗ ̺ −−−−→ g ij ∗ t ∗ E ∼ = ( t ◦ g ij ) ∗ E ∼ = E j | i or, as an identity up to anoni al isomorphisms, θ ij = g ij ∗ ̺ . (cid:0) mod ∼ = (cid:1) (15)(Note that the fa t that ̺ is an isomorphism in the ategory S ( G ) , thatis to say Lemma 13.6, is used in an essential way.) Next, from the obviousremark that for an arbitrary third index k ∈ I one has g ik ( y ) = g jk ( y ) g ij ( y ) ∀ y ∈ V ijk ≡ V i ∩ V j ∩ V k (or better g ik | j = c ◦ ( g jk | i , g ij | k ) , where g ik | j denotesthe restri tion of g ik to V ijk et .), and from the multipli ative axiom (13.3)for ̺ , it follows that the system of isomorphisms { θ ij } onstitutes a (cid:16) o y le(cid:17)or (cid:16)des ent datum(cid:17) for the family { E i } i ∈ I ∈ S (cid:16)` i ∈ I V i (cid:17) , relative to the (cid:29)atmapping ` i ∈ I V i → N . Sin e N is a para ompa t manifold and S is a smoothparasta k, there exists some obje t ϕ ! E of S ( N ) along with isomorphisms θ i : ( ϕ ! E ) | i ≡ ( ϕ ! E ) | V i ≈ −→ E i in S ( V i ) , ompatible with { θ ij } in the sensethat, modulo the identi(cid:28) ation ( ϕ ! E ) i | V ij ∼ = ( ϕ ! E ) j | V ij , one has the identity θ j | i = θ j | V ij = θ ij · θ i | V ij = θ ij · θ i | j . (cid:0) mod ∼ = (cid:1) (16)For simpli ity, let us put F ≡ ϕ ! E . Our next step will be to de(cid:28)ne amorphism σ = ϕ ! ̺ : s H∗ F → t H∗ F , whi h is to provide the H -a tion on F .For ea h pair V i , V i ′ we introdu e the abbreviation H i,i ′ ≡ H ( V i , V i ′ ) ; we alsowrite H ij,i ′ j ′ ≡ H ( V ij , V i ′ j ′ ) . Then the subsets H i,i ′ ⊂ H (1) form an open overof the manifold H (1) . Now, let g i,i ′ : H i,i ′ → G be the smooth map obtainedby solving the following universal problem H i,i ′ ( s , t ) (cid:15) (cid:15) g i,i ′ & & NNNNNNN in lusion V i × V i ′ s i × s i ′ - - G ( s , t ) (cid:15) (cid:15) ϕ / / H ( s , t ) (cid:15) (cid:15) M × M f × f / / N × N .(17)0 CHAPTER III. REPRESENTATION THEORY REVISITEDWe an use this map to de(cid:28)ne a morphism σ i,i ′ : ( s H∗ F ) | i,i ′ → ( t H∗ F ) | i,i ′ inthe ategory S ( H i,i ′ ) , as follows:(18) ( s H∗ F ) | i,i ′ ∼ = ( s H | i,i ′ ) ∗ ( F | i ) ( s H | i,i ′ ) ∗ θ i −−−−−−−→ ( s H | i,i ′ ) ∗ E i ∼ = g i,i ′ ∗ s G∗ E g i,i ′ ∗ ̺ −−−−→ g i,i ′ ∗ t G ∗ E ∼ = ( t H | i,i ′ ) ∗ E i ′ ( t H | i,i ′ ) ∗ θ i − −−−−−−−−→ ( t H | i,i ′ ) ∗ ( F | i ′ ) ∼ = ( t H∗ F ) | i,i ′ or, in the form of an identity modulo anoni al identi(cid:28) ations, σ i,i ′ = ( t H | i,i ′ ) ∗ θ i − · g i,i ′ ∗ ̺ · ( s H | i,i ′ ) ∗ θ i . (cid:0) mod ∼ = (cid:1) (19)Starting from the equality of mappings(20) g i,i ′ | j,j ′ = ( g j ′ i ′ ◦ t H | ij,i ′ j ′ ) g j,j ′ | i,i ′ ( g ji ◦ s H | ij,i ′ j ′ ) (note that g j ′ i ′ = i G ◦ g i ′ j ′ where i G is the inverse map of G ) and the (cid:16)mod ∼ = (cid:17) identities (15), (16) and (19), one an he k that σ i,i ′ | j,j ′ = σ j,j ′ | i,i ′ in S ( H ij,i ′ j ′ ) ; hen e the morphisms σ i,i ′ glue together into a unique morphism σ = ϕ ! ̺ of S ( H (1) ) , with the property that σ | i,i ′ = σ i,i ′ .Next, suppose we are given a morphism a : R → R ′ in R S ( G ) , where R ′ = ( E ′ , ̺ ′ ) , let us say. Then we an obtain a morphism ϕ ! a : ϕ ! R → ϕ ! R ′ ,where ϕ ! R = ( ϕ ! E, ϕ ! ̺ ) et ., by (cid:28)rst letting b i = s i ∗ a and the observing that(21) θ ′ ij · b i | j = b j | i · θ ij in S ( V ij ) (be ause of the de(cid:28)nition of θ ij = θ Rij and θ ′ ij = θ R ′ ij and be ause a is a G -equivariant morphism). In this way we get a fun tor of R S ( G ) into R S ( H ) .The onstru tion of the isomorphisms ϕ ∗ ◦ ϕ ! ≃ Id R ( G ) and ϕ ! ◦ ϕ ∗ ≃ Id R ( H ) is left as an exer ise, to be done along the same lines.hapter IVGeneral Tannaka TheoryIn the pre eding hapter we laid down the foundations of RepresentationTheory in the abstra t setting of smooth tensor sta ks. The assumptions onthe type S were quite mild there, nothing more than just smoothness andthe property of being a sta k. However, in order to get our re onstru tiontheory to work e(cid:27)e tively, we need to impose further restri tions on the type S . We will all a smooth tensor sta k a sta k of smooth (cid:28)elds when it meetssu h additional requirements.The additional properties whi h hara terize sta ks of smooth (cid:28)elds areintrodu ed in Ÿ15. The sta k of smooth ve tor bundles is an example. In thesubsequent se tion we provide another fundamental example, the sta k ofsmooth (Eu lidean) (cid:28)elds, whi h will play a major role in the a hievement ofour Tannaka duality theorem for proper Lie groupoids in Ÿ20. This sta k is anontrivial extension of the sta k of smooth ve tor bundles, but its de(cid:28)nitionis as simple. Ÿ15 Sta ks of Smooth FieldsThe expression (cid:19) sta k of smooth (cid:28)elds (cid:20) will be employed to indi ate asmooth (real or omplex) tensor sta k1 for whi h the axiomati onditionslisted below are satis(cid:28)ed. When dealing spe i(cid:28) ally with sta ks of smooth(cid:28)elds we shall prefer them to be represented by the letter F , whi h is moresuggestive than the usual S . The axiomsOur (cid:28)rst axiom is about the tensor produ t and pull-ba k operations.Roughly speaking, it states that the se tions of a tensor produ t or a pull-ba kare exa tly what one would expe t them to be on the basis of the standard1In a ordan e with the philosophy of Note 12.5, we use the word `sta k' but we reallymean `parasta k'. 712 CHAPTER IV. GENERAL TANNAKA THEORYde(cid:28)nition of tensor produ t and pull-ba k of sheaves of C ∞ -modules; how-ever, for su h se tions the relation of equality may be oarser, in the sensethat more se tions may be regarded as being identi al.1 Axiom I (tensor produ t & pull-ba k) The anoni al natural morph-isms (11.20) and (11.23) ( Γ E ⊗ C ∞ X Γ E ′ → Γ ( E ⊗ E ′ ) f ∗ ( Γ Y F ) → Γ X ( f ∗ F ) are surje tive (= epimorphisms of sheaves).Thus, every lo al smooth se tion of E ⊗ E ′ will possess, in the vi inity of ea hpoint, an expression as a (cid:28)nite linear ombination, with smooth oe(cid:30) ients,of se tions of the form ζ ⊗ ζ ′ . Similarly, given any partial smooth se tion of f ∗ F , it will be possible to express it lo ally as a (cid:28)nite linear ombination,with oe(cid:30) ients in C ∞ X , of se tions of the form η ◦ f .Suppose E ∈ F ( X ) . Let us go ba k for a moment to the map Γ E ( U ) → E x , ζ ζ ( x ) we de(cid:28)ned in Ÿ11 (for ea h open neighbourhood U of the point x ). These maps are evidently ompatible with the restri tion to a smalleropen neighbourhood of x , hen e on passing to the indu tive limit they willdetermine a linear map(2) ( Γ E ) x → E x , ζ ζ ( x ) of the stalk of Γ E at x into the (cid:28)bre of E at the same point. We all thismap the evaluation (of germs) at x . Noti e, by the way, that the identity(3) ( αζ )( x ) = α ( x ) ζ ( x ) holds for all germs of smooth se tions ζ ∈ ( Γ E ) x and of smooth fun tions α ∈ C ∞ X,x . It follows from Axiom i (pull-ba k) that for any sta k of smooth(cid:28)elds, the evaluation of germs at a point is a surje tive linear map. Indeed,the stalk ( Γ E ) x oin ides, as a ve tor spa e, with the spa e of global se tionsof x ∗ ( Γ E ) (re all that ( Γ E ) x = lim −→ U ∋ x Γ E ( U ) = x − ( Γ E )( ⋆ ) , a tually as a C ∞ X,x -module), and the (cid:28)bre E x is de(cid:28)ned as the spa e of global se tions of Γ ( x ∗ E ) ; it is immediate to re ognize that the evaluation of germs is just themap of global se tions indu ed by (11.23).The se ond axiom says that a di(cid:27)eren e between any two morphisms anbe dete ted by looking at the linear maps they indu e on the (cid:28)bres.4 Axiom II (vanishing) Let a : E → E ′ be a morphism in F ( X ) .Suppose that a x : E x → E ′ x is zero ∀ x ∈ X . Then a = 0 .15. STACKS OF SMOOTH FIELDS 73As a (cid:28)rst, immediate onsequen e, an arbitrary se tion ζ ∈ Γ E ( U ) willvanish if and only if all its values ζ ( u ) will be zero as u ranges over U : thus,one sees that smooth se tions are hara terized by their values; intuitively,one an think of the elements of Γ E ( U ) as se tions(cid:22)in the usual sense(cid:22)ofthe `bundle' of (cid:28)bres { E u } .Furthermore, by ombining Axioms ii and i, it follows that the fun tor Γ X : F ( X ) → { sheaves of C ∞ X - modules } is faithful. This is an easy onse-quen e of the surje tivity of the evaluation of germs at a point; the argumentwe propose now will also be preparatory to the next axiom.For ea h morphism a : E → F in F ( X ) , onsider the `bundle' of linearmaps { a x : E x → F x } and the morphism α = Γ a : Γ E → Γ F of sheavesof C ∞ X -modules. We start by asking what relation there is between thesedata. The link between the two is obviously provided by the above anoni alevaluation maps of the stalks onto the (cid:28)bres ( Γ E ) x ։ E x : it is lear that thestalk homomorphism α x and the linear map a x have to be ompatible, in thesense that the following square should ommute ( Γ E ) x eval. (cid:15) (cid:15) (cid:15) (cid:15) α x / / ( Γ F ) x eval. (cid:15) (cid:15) (cid:15) (cid:15) E x a x / / F x .(5)In general, we shall say that a morphism of sheaves of modules α : Γ E → Γ F and a `bundle' of linear maps { a x : E x → F x } are ompatible, whenever thediagram (5) ommutes for all x ∈ X . Noti e that, in view of the pre edingaxioms, ompatibility implies that the morphism of sheaves and the bundle oflinear maps determine ea h other unambiguously. (Indeed, in one dire tion,the morphism α learly determines the maps a x through the ommutativityof (5). Conversely, the ommutativity of (5) for all x entails that for anysmooth se tion ζ ∈ Γ E ( U ) one has the formula [ α ( U ) ζ ]( x ) = a x (cid:0) ζ ( x ) (cid:1) , andtherefore, if α and β are both ompatible with { a x } , it follows by Axiom iithat α ( U ) ζ = β ( U ) ζ for all ζ and hen e that α = β .) In parti ular, from Γ a = Γ b it will follow that a x = b x for all x and therefore that a = b .Let us all a morphism of sheaves of modules α : Γ E → Γ F representable,if it admits a ompatible bundle of linear maps { a x : E x → F x } . Our nextaxiom, whi h omplements the pre eding one by providing a general riterionfor the existen e of morphisms in F ( X ) , states that the olle tion of su hmorphisms is (cid:16)as big as possible(cid:17):6 Axiom III (morphisms) For every representable α : Γ E → Γ F , thereexists a morphism a : E → F in F ( X ) su h that Γ a = α .This axiom will not be used anyhere in the present se tion. It will playa role only in Ÿ17, where it is needed in order to onstru t morphisms ofrepresentations by means of (cid:28)brewise integration.4 CHAPTER IV. GENERAL TANNAKA THEORYWe annot yet dedu e, from the axioms we have introdu ed so far, ertainvery intuitive properties that are surely reasonable for a (cid:16)smooth se tion(cid:17); forinstan e, if a se tion(cid:22)or, more generally, a morphism(cid:22)vanishes over a denseopen subset of its domain of de(cid:28)nition, it would be natural to expe t it to bezero everywhere. Analogously, if the value of a se tion is non zero at a pointthen it should be non zero at all nearby points. The next axiom yields su hproperties, among many other onsequen es.We shall say that a Hermitian(cid:22)or, in the real ase, symmetri (cid:22)form φ : E ⊗ E ∗ → in F ( X ) is a Hilbert metri on E , when for every point x theindu ed form φ x on the (cid:28)bre E x (7) E x ⊗ E x ∗ an. −−→ ( E ⊗ E ∗ ) x φ x −−→ x ∼ = Cis a Hilbert metri (in the familiar sense, viz. positive de(cid:28)nite).8 Axiom IV (metri s) Any obje t E ∈ Ob F ( X ) supports lo al metri s;that is to say, the open subsets U su h that one an (cid:28)nd a Hilbert metri on E | U over X .In general, one an only assume lo al metri s to exist, think e.g. of smoothve tor bundles; however, as for ve tor bundles, global metri s an be on-stru ted from lo al ones as soon as smooth partitions of unity are availableon the manifold X (e.g. when X is para ompa t).Let E ∈ Ob F ( X ) and let φ be a Hilbert metri on E . By a φ -orthonormalframe for E about a point x of X we mean a list of se tions ζ , . . . , ζ d ∈ Γ E ( U ) , de(cid:28)ned over a neighbourhood of x , su h that for all u in U theve tors ζ ( u ) , . . . , ζ d ( u ) are orthonormal in E u (with respe t to φ u ) and(9) Span (cid:8) ζ ( x ) , . . . , ζ d ( x ) (cid:9) = E x .Orthonormal frames for E exist about ea h point x for whi h the (cid:28)bre E x is(cid:28)nite dimensional. Indeed, over some neighbourhood N of x we an (cid:28)rst ofall (cid:28)nd lo al smooth se tions ζ , . . . , ζ d with the property that the ve tors ζ ( x ) , . . . , ζ d ( x ) form a basis of the spa e E x (Axiom i). Sin e for all n ∈ N the ve tors ζ ( n ) , . . . , ζ d ( n ) are linearly dependent if and only if there isa d -tuple of omplex numbers ( z , . . . , z d ) with | z | + · · · + | z d | = 1 and d P i =1 z i ζ i ( n ) = 0 , the ontinuous fun tion N × S d − → R, ( n ; s , t , . . . , s d , t d ) (cid:12)(cid:12)(cid:12)(cid:12) d P ℓ =1 ( s ℓ + it ℓ ) ζ ℓ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) must have a minimum c > at n = x , hen e a lower bound c on a suitableneighbourhood U of x so that the ζ i ( u ) must be linearly independent for all u ∈ U . At this point it is enough to apply the Gram(cid:21)S hmidt pro ess in15. STACKS OF SMOOTH FIELDS 75order to obtain an orthonormal frame about x . This elementary observation(existen e of orthonormal frames) will prove to be very useful. Let us startto illustrate its importan e with some basi appli ations.Consider an embedding e : E ′ ֒ → E in the ategory F ( X ) , that is to say,a morphism su h that the linear map e x : E ′ x ֒ → E x is inje tive for all x .2Suppose there exists a global metri φ on the obje t E ; also assume that E ′ ∈ Ob V F ( X ) is lo ally trivial of (lo ally) (cid:28)nite rank. Then e admits a o-se tion, i.e. there exists a morphism p : E → E ′ with p ◦ e = id (so e isa se tion in the ategori al sense). To prove this, note (cid:28)rst of all that themetri φ will indu e a metri φ ′ on E ′ ֒ → E . Fix any point x ∈ X . Sin e E ′ x is (cid:28)nite dimensional, there exists a φ ′ -orthonormal frame for E ′ about x , letus say ζ ′ , . . . , ζ ′ d ∈ Γ E ′ ( U ) . Put ζ i = Γ e ( U ) ζ ′ i ∈ Γ E ( U ) , let φ U be the metri indu ed on E | U , and onsider(10) ζ i : E | U ∼ = E | U ⊗ U ∼ = E | U ⊗ | U ∗ E | U ⊗ ζ ∗ i −−−−→ E | U ⊗ E | U ∗ φ U −→ U .De(cid:28)ne p U : E | U → E ′ | U as the omposite of E | U ζ ⊕ ··· ⊕ ζ d −−−−−−→ ⊕ · · · ⊕ and ⊕ · · · ⊕ ζ ′ ⊕ ··· ⊕ ζ ′ d −−−−−−→ E ′ | U . Note that ( p U ) u : E u → E ′ u is the orthogonalproje tion, with respe t to φ u , onto E ′ u ֒ → E u : it follows by Axiom ii that p U does not a tually depend on U or the other hoi es involved, so that weget a well-de(cid:28)ned morphism p : E → E ′ , by the presta k property; moreover,we have p ◦ e = id for similar reasons.Another appli ation: let E ∈ Ob F ( X ) , and suppose that the dimensionof the (cid:28)bres is ((cid:28)nite and) lo ally onstant over X ; then E ∈ Ob V F ( X ) i.e. E is lo ally trivial, of lo ally (cid:28)nite rank. Indeed, (cid:28)x an arbitrary point x . By Axiom iv, there exists an open neighbourhood U of x su h that E | U supports a metri φ U . Sin e E x is (cid:28)nite dimensional, it is no loss of generalityto assume that a φ U -orthonormal system ζ , . . . , ζ d ∈ Γ E ( U ) an be found;one an also assume dim E u = d onstant over U . Take e def = ζ ⊕ · · · ⊕ ζ d : E ′ def = ⊕ · · · ⊕ ֒ → E | U and p : E | U → E ′ as above. It is immediate to seethat e and p are (cid:28)brewise inverse to one another.11 Lemma Let X be a para ompa t manifold and let S i S ֒ → X be a losed submanifold. Let F be a sta k of smooth (cid:28)elds.Let E, F ∈ Ob F ( X ) , and suppose that E ′ = E | S belongs to V F ( S ) ,i.e. is lo ally free, of lo ally (cid:28)nite rank.Then every morphism a ′ : E ′ → F ′ in F ( S ) an be extended to amorphism a : E → F in F ( X ) , i.e. a ′ = a | S for su h an a .2It follows immediately from Axiom ii that an embedding is a monomorphism. The onverse need not be true be ause the fun tor E E x doesn't have any exa tness prop-erties. For example, let a be a smooth fun tion on R su h that a ( t ) = 0 if and only if t = 0 . Then a , regarded as an element of End( ) , is both mono and epi in F ( R ) while a = 0 : C → C is neither inje tive nor surje tive.6 CHAPTER IV. GENERAL TANNAKA THEORYProof Fix a point s ∈ S . Then there exists an open neighbourhood A of s in S su h that over A we an (cid:28)nd a trivialization ( d summands)(12) E ′ | A ≈ A ⊕ · · · ⊕ A .Let ζ ′ , . . . , ζ ′ d ∈ Γ E ′ ( A ) be the se tions orresponding to this trivialization(so for instan e ζ ′ is the omposite S | A ∼ = A ֒ → A ⊕ · · · ⊕ A ≈ E ′ | A ).Also, let U be any open subset of X su h that U ∩ S = A .Now, by Axiom i (pull-ba k ase), taking smaller U and A about s ifne essary, it is no loss of generality to assume that there exist lo al se tions ζ , . . . , ζ d ∈ Γ E ( U ) with ζ ′ k = ζ k ◦ i S , k = 1 , . . . , d . To see this, observe thatlo ally about s ea h ζ ′ k is a (cid:28)nite linear ombination P j α j,k ( ζ j,k ◦ i S ) with ζ j,k ∈ Γ E ( U ) and α j,k ∈ C ∞ ( A ) , by the ited axiom; hen e if U is hosen onveniently, let us say say so that there exists a di(cid:27)eomorphism of U ontoa produ t A × R n , the oe(cid:30) ients α j,k will extend to some smooth fun tions ˜ α j,k ∈ C ∞ ( U ) and ζ k = P j ˜ α j,k ζ j,k will meet our requirements.We have already observed (11.24) that there is a anoni al isomorphismof ve tor spa es ( i ∗ S E ) s ∼ = E i ( s ) whi h makes ( ζ k ◦ i S )( s ) orrespond to ζ k ( x ) ,where we put x = i S ( s ) . Hen e the values ζ k ( x ) , k = 1 , . . . , d are linearlyindependent in the (cid:28)bre E x , be ause the same is true of the values ζ ′ k ( s ) , k = 1 , . . . , d in E ′ s (the trivializing isomorphism (12) above yields a linearisomorphism ( E ′ ) s ≈ C d whi h, as one an easily he k, makes ζ ′ k ( s ) or-respond to the k -th standard basis ve tor of C d ). This implies that if U issmall enough then the morphism ζ = ζ ⊕ · · · ⊕ ζ d : U ⊕ · · · ⊕ U → E | U isan embedding and admits a ose tion p : E | U → U ⊕ · · · ⊕ U , by Axiom iv(existen e of lo al metri s).Next, set η ′ k = Γ a ′ ( A ) ζ ′ k ∈ Γ F ′ ( A ) . As remarked earlier in the proof, itis no loss of generality to assume that there exist partial se tions η , . . . , η d in Γ F ( U ) with η ′ k = η k ◦ i S . Again, these se tions an be ombined into amorphism η : U ⊕ · · · ⊕ U → F | U ( d -fold dire t sum).Finally, we an take the omposite E | U p −→ U ⊕ · · · ⊕ U | {z } d summands η −→ F | U .It is immediate to he k that the restri tion of this morphism to the sub-manifold A ֒ → U oin ides with a ′ | A , up to the anoni al identi(cid:28) ations ( E | U ) | A ∼ = E ′ | A and ( F | U ) | A ∼ = F ′ | A . Let us summarize brie(cid:29)y what wehave done so far: starting from an arbitray point s ∈ S , we have found anopen neighbourhood U = U s of x = i S ( s ) in X , along with a morphism a s : E | U → F | U whose restri tion to A = U ∩ S agrees with a ′ | A . This meansthat we have solved our problem lo ally.To on lude the proof, onsider the open over of X formed by the opensubsets { U s : s ∈ S } and the omplement U = ∁ X S . (Here we use, of15. STACKS OF SMOOTH FIELDS 77 ourse, the losedness of S .) Sin e X is a para ompa t manifold, we an (cid:28)nda smooth partition of unity { θ i : i ∈ I } ∪ { θ } subordinated to this open over. Then(cid:22)by the presta k property(cid:22)the sum a def = P i ∈ I θ i a s i orrespondsto a well-de(cid:28)ned morphism E → F in F ( X ) , learly extending a ′ . q.e.d.The last two axioms impose various (cid:28)niteness requirements, both on the(cid:28)bres and on the sheaf of smooth se tions of an obje t.To begin with, there is a sto k of onditions we shall impose on F inorder that the ategory F ( ⋆ ) may be equivalent, as a tensor ategory, to the ategory of ve tor spa es of (cid:28)nite dimension. We gather these onditions intowhat we all the (cid:16)dimension axiom(cid:17):13 Axiom V (dimension) It is required of the anoni al pseudo-tensorfun tor (11.9) : F ( ⋆ ) → { vector spaces } thata) it is fully faithful;b) it fa tors through the sub ategory whose obje ts are the (cid:28)nite dimen-sional ve tor spa es, in other words E ∗ (11.10) is (cid:28)nite dimensionalfor all E ∈ F ( ⋆ ) ; ) it is a genuine tensor fun tor, i.e. (11.7) and (11.8) be ome iso-morphisms of sheaves for X = ⋆ .In parti ular, for ea h obje t V ∈ F ( ⋆ ) there exists a trivialization of V ,i.e. an isomorphism V ≈ ⊕ · · · ⊕ ((cid:28)nite dire t sum). The number of opiesof in any su h de omposition determines the dimension of an obje t.Moreover, it follows from this axiom, and pre isely from ), that thefun tor `(cid:28)bre at x ', E E x is a omplex tensor fun tor. (In general, it isonly a omplex pseudo-tensor fun tor, see Ÿ11.)An obje t E of F ( X ) is lo ally (cid:28)nite, if Γ E is a lo ally (cid:28)nitely generated C ∞ X -module. In other words, E is lo ally (cid:28)nite if the manifold X admitsa over by open subsets U su h that there exist lo al se tions ζ , . . . , ζ d ∈ Γ E ( U ) with the property(14) Γ E | U = C ∞ U { ζ , . . . , ζ d } .(The expression on the right-hand side has a lear meaning as a presheaf ofse tions over U ; sin e it is always possible to assume U para ompa t, thispresheaf is in fa t a sheaf, as one an easily see by means of partitions ofunity.) The ondition on U amounts to the existen e of an epimorphism ofsheaves of modules(15) C ∞ U ⊕ · · · ⊕ C ∞ U | {z } d summands ։ Γ E | U .8 CHAPTER IV. GENERAL TANNAKA THEORY16 Axiom VI (lo al (cid:28)niteness) Let X be a smooth manifold. Everyobje t E ∈ Ob F ( X ) is lo ally (cid:28)nite.The present axiom, like Axiom iii above, will play a role in the proof ofthe `Averaging Lemma' only, in Ÿ17.Ÿ16 Smooth Eu lidean FieldsOur next goal in this se tion is to elaborate a on rete model for the axiomswe just proposed. Of ourse, in order to be useful, su h a model ought to ontain mu h more than just ve tor bundles: in fa t, we intend to exploitit later on, in Ÿ20, to prove a general re onstru tion theorem for proper Liegroupoids. We (cid:28)rst introdu e a somewhat weaker notion whi h, however, isof some interest on its own.1 De(cid:28)nition By a smooth Hilbert (cid:28)eld we mean an obje t H onsistingof (a) a family { H x } of Hilbert spa es, indexed over the set of points of amanifold X , and (b) a sheaf Γ H of C ∞ X -modules of lo al se tions of { H x } ,subje t to the following onditions:i) (cid:8) ζ ( x ) : ζ ∈ ( Γ H ) x (cid:9) , where ( Γ H ) x indi ates the stalk at x , is a denselinear subspa e of H x ;ii) for ea h open subset U , and for all se tions ζ , ζ ′ ∈ Γ H ( U ) , the fun tion h ζ , ζ ′ i on U de(cid:28)ned by u (cid:10) ζ ( u ) , ζ ′ ( u ) (cid:11) turns out to be smooth.We refer to the manifold X as the base of H ; we an also say that H is asmooth Hilbert (cid:28)eld over X .Some explanations are perhaps in order. By a (cid:19) lo al se tion of { H x } (cid:20) wemean here an element of the produ t Q x ∈ U H x of all the spa es over some opensubset U of X . The de(cid:28)nition establishes in parti ular that for ea h opensubset U the set of se tions Γ H ( U ) is a submodule of the C ∞ ( U ) -moduleof all the se tions of { H x } over U . Γ H will be alled the sheaf of smoothse tions of H and the elements of Γ H ( U ) will be a ordingly referred to asthe smooth se tions of H over U . This terminology, overlapping with thatof Ÿ11, has been introdu ed intentionally and will be justi(cid:28)ed soon.Next, we need a suitable notion of morphism. There are various possibil-ities here. We hoose the notion whi h seems to (cid:28)t our purposes better: abundle of bounded linear maps indu ing a morphism of sheaves of modules.Pre isely, let E and F be smooth Hilbert (cid:28)elds over X . A morphism of E into F is a family of bounded linear maps { a x : E x → F x } , indexed overthe set of points of X , su h that for ea h open subset U ⊂ X and for all ζ ∈ Γ E ( U ) the se tion over U given by u a u · ζ ( u ) belongs to Γ F ( U ) .Smooth Hilbert (cid:28)elds over X and their morphisms form a ategory whi hwill be denoted by H ∞ ( X ) . We want to turn the operation X H ∞ ( X ) into16. SMOOTH EUCLIDEAN FIELDS 79a (cid:28)bred ( omplex) tensor ategory H ∞ , in the sense of Ÿ11. This (cid:28)bred tensor ategory will prove to be a smooth tensor parasta k (but not a sta k: thisis the reason why we work with the weaker notion of parasta k) satisfyingsome of the axioms, although(cid:22)of ourse(cid:22)not all of them: for this reason, H ∞ onstitutes a sour e of interesting examples.Let us start with the de(cid:28)nition of the tensor stru ture on the ategory H ∞ ( X ) of smooth Hilbert (cid:28)elds.We shall on ern ourselves with the tensor produ t of Hilbert (cid:28)elds in amoment; before doing that however we review the tensor produ t of Hilbertspa es. Let V be a omplex ve tor spa e. We denote by V ∗ the spa e ob-tained by retaining the additive stru ture of V while hanging the s alarmultipli ation into zv ∗ = ( zv ) ∗ ; the star here indi ates that a ve tor of V isto be regarded as one of V ∗ . If φ : E ⊗ E ∗ → C and ψ : F ⊗ F ∗ → C aresesquilinear forms then we an ombine them into a sesquilinear form on thetensor produ t E ⊗ F (2) ( E ⊗ F ) ⊗ ( E ⊗ F ) ∗ ∼ = ( E ⊗ E ∗ ) ⊗ ( F ⊗ F ∗ ) φ ⊗ ψ −−−→ C ⊗ C ∼ = C.If we ompute this form on the generators of E ⊗ F we get(3) h e ⊗ f, e ′ ⊗ f ′ i = h e, e ′ i h f, f ′ i .Suppose now that both φ and ψ are Hilbert spa e inner produ ts. Thenthis formula shows that the form (2) is Hermitian. Moreover, if we expressan arbitrary element w of E ⊗ F as a linear ombination k P i =1 ℓ P j =1 a i,j e i ⊗ f j with e , . . . , e k , resp. f , . . . , f ℓ orthonormal in E , resp. F , we see from (3)that a i,j = h w, e i ⊗ f j i = 0 for all i, j implies w = 0 . Hen e the form isnon degenerate. The same expression an be used to show that the form ispositive de(cid:28)nite: h w, w i = P i,i ′ P j,j ′ a i,j a i ′ ,j ′ δ j,j ′ i,i ′ = P i,j | a i,j | ≧ .The spa e E ⊗ F an be ompleted with respe t to the pre-Hilbert innerprodu t (2) to a Hilbert spa e alled the (cid:19) Hilbert tensor produ t (cid:20) of E and F . We agree that from now on, when E and F are Hilbert spa es, thesymbol E ⊗ F will denote the Hilbert tensor produ t of E and F . It is equallyeasy to see that if a : E → E ′ and b : F → F ′ are bounded linear maps ofHilbert spa es then their tensor produ t extends by ontinuity to a boundedlinear map of E ⊗ F into E ′ ⊗ F ′ that we still denote by a ⊗ b . Moreover,the anoni al isomorphisms of ve tor spa es u ⊗ ( v ⊗ w ) ( u ⊗ v ) ⊗ w et .extend by ontinuity to unitary isomorphisms E ⊗ ( F ⊗ G ) ∼ → ( E ⊗ F ) ⊗ G et . of Hilbert spa es.Suppose now that E and F are Hilbert (cid:28)elds over X . Consider the bundleof tensor produ ts { E x ⊗ F x } . For arbitrary lo al se tions ζ ∈ Γ E ( U ) and0 CHAPTER IV. GENERAL TANNAKA THEORY η ∈ Γ F ( U ) , we let ζ ⊗ η denote the se tion of { E x ⊗ F x } given by u ζ ( u ) ⊗ η ( u ) . The law(4) U C ∞ ( U ) (cid:8) ζ ⊗ η : ζ ∈ Γ E ( U ) , η ∈ Γ F ( U ) (cid:9) de(cid:28)nes a sub-presheaf of the sheaf of lo al se tions of { E x ⊗ F x } . (We useexpressions of the form C ∞ ( U ) {· · · } to indi ate the C ∞ ( U ) -module spannedby a olle tion of se tions over U .) Let E ⊗ F denote the Hilbert (cid:28)eld over X onsisting of the bundle { E x ⊗ F x } and the sheaf (of se tions of this bundle)generated by the presheaf (4), in other words, the smallest subsheaf of thesheaf of lo al se tions of { E x ⊗ F x } ontaining (4). We all E ⊗ F the tensorprodu t of E and F . Observe that for all morphisms E α −→ E ′ and F β −→ F ′ of Hilbert (cid:28)elds over X , the bundle of bounded linear maps { a x ⊗ b x } yieldsa morphism α ⊗ β of E ⊗ F into E ′ ⊗ F ′ .Another operation whi h applies to Hilbert spa es is onjugation. Thisoperation sends a Hilbert spa e E to the onjugate ve tor spa e E ∗ endowedwith the Hermitian produ t h v ∗ , w ∗ i = h w, v i . We now arry onjugation ofHilbert spa es over to a fun torial onstru tion on Hilbert (cid:28)elds. Let E bea Hilbert (cid:28)eld over X . We get the onjugate (cid:28)eld E ∗ by taking the bundle { E x ∗ } of onjugate spa es, along with the lo al smooth se tions of E regardedas lo al se tions of { E x ∗ } . If α = { a x } : E → F is a morphism of Hilbert(cid:28)elds over X then, sin e a linear map a x : E x → F x also maps E x ∗ linearlyinto F x ∗ , we get a morphism α ∗ = { a x ∗ } : E ∗ → F ∗ . Observe that the orresponden e α α ∗ is anti-linear. Note also that E ∗∗ = E .The rest of the onstru tion (tensor unit, the various onstraints . . . ) is ompletely obvious. One obtains a omplex tensor ategory, that is easilyre ognized to be additive as a C-linear ategory. It remains to onstru t the omplex tensor fun tor f ∗ : H ∞ ( Y ) → H ∞ ( X ) asso iated with a smooth map f : X → Y , and to de(cid:28)ne the onstraints (11.3).Let H be a Hilbert (cid:28)eld over Y . The pull-ba k of H along f , denotedby f ∗ H , is the Hilbert (cid:28)eld over X whose des ription is as follows: theunderlying bundle of Hilbert spa es, indexed by the points of X , is (cid:8) H f ( x ) (cid:9) ;the sheaf of smooth se tions is generated(cid:22)as a subsheaf of the sheaf of alllo al se tions of the bundle (cid:8) H f ( x ) (cid:9) (cid:22)by the presheaf(5) U C ∞ X ( U ) (cid:8) η ◦ f : η ∈ Γ H ( V ) , V ⊃ f ( U ) (cid:9) .Sin e this is a presheaf of C ∞ X -modules (of se tions), it follows that Γ ( f ∗ H ) is a sheaf of C ∞ X -modules (of se tions). Moreover, it is lear that for anymorphism β : H → H ′ of Hilbert (cid:28)elds over Y , the family of boundedlinear maps { b f ( x ) } de(cid:28)nes a morphism f ∗ β : f ∗ H → f ∗ H ′ of Hilbert (cid:28)eldsover X .Observe that f ∗ H ⊗ f ∗ H ′ and f ∗ ( H ⊗ H ′ ) are exa tly the samesmooth Hilbert (cid:28)eld over X , essentially be ause ( η ⊗ η ′ ) ◦ f = ( η ◦ f ) ⊗ ( η ′ ◦ f ) ; also C ∞ X = f ∗ C ∞ Y . These identities an fun tion as tensor fun tor17. CONSTRUCTION OF EQUIVARIANT MAPS 81 onstraints. Similarly f ∗ ( H ∗ ) = ( f ∗ H ) ∗ an be taken as a onstraint, so weget a omplex tensor fun tor f ∗ : H ∞ ( Y ) → H ∞ ( X ) .Sin e the identities f ∗ ( g ∗ H ) = ( g ◦ f ) ∗ H and id X ∗ H = H hold, theoperation X H ∞ ( X ) is a (cid:16)stri t(cid:17) (cid:28)bred omplex tensor ategory.Note that the `sheaf of se tions'(cid:22)de(cid:28)ned abstra tly only in terms of thepresta k stru ture of H ∞ , as explained in Ÿ11(cid:22)turns out to be pre isely the`sheaf of smooth se tions' whi h we introdu ed in the above de(cid:28)nition as oneof the two onstituent data of a smooth Hilbert (cid:28)eld. However, note that the(cid:28)bre H x (in the sense of Ÿ11) will be in general only a dense subspa e of theHilbert spa e H x (this is the reason why we use two distin t notations); of ourse, H x = H x whenever H x is (cid:28)nite dimensional.Let E ∞ ( X ) be the full sub ategory of H ∞ ( X ) onsisting of all obje ts E whose sheaf of se tions is lo ally (cid:28)nitely generated over X , in the sense ofAxiom vi. E ∞ ( X ) is a omplex tensor sub ategory i.e. it is losed under ⊗ , ∗ and it ontains the tensor unit: indeed, Γ E ⊗ C ∞ Γ E ′ , whi h is a lo ally(cid:28)nitely generated sheaf of modules over X be ause su h are Γ E and Γ E ′ ,surje ts (as a sheaf) onto Γ ( E ⊗ E ′ ) , by Axiom i, so the latter will be lo ally(cid:28)nite too, as ontended. Moreover, the pull-ba k fun tor f ∗ : H ∞ ( Y ) → H ∞ ( X ) arries E ∞ ( Y ) into E ∞ ( X ) . We obtain a smooth substa k E ∞ ⊂ H ∞ of additive omplex tensor ategories; it is lear that E ∞ satis(cid:28)es Axiomsi(cid:21)vi.The obje ts of the sub ategory E ∞ ( X ) ⊂ H ∞ ( X ) will be referred to assmooth Eu lidean (cid:28)elds over X .Ÿ17 Constru tion of Equivariant MapsLet F denote an arbitrary sta k of smooth (cid:28)elds, to be regarded as (cid:28)xedthroughout the present se tion.The next lemma is to be used in ombination with Lemma 15.11.1 Lemma Let G be a (lo ally) transitive Lie groupoid, and let X beits base manifold. Consider any representation ( E, ρ ) ∈ R F ( G ) . Then E ∈ V F ( X ) i.e. E is a lo ally trivial obje t of F ( X ) .Proof Lo al transitivity means that the mapping ( s , t ) : G → X × X is asubmersion. Fix a point x ∈ X . Sin e ( x, x ) lies in the image of the map ( s , t ) , the latter admits a lo al smooth se tion U × U → G over some openneighbourhood of ( x, x ) . Let us onsider the `restri tion' g : U → G of thisse tion to U ≡ U × { x } : g will be a smooth map for whi h the identities s ( g ( u )) = u and t ( g ( u )) = x hold for all u ∈ U .Let ⋆ x −→ X denote the map ⋆ x . We have already noti ed that, bythe `dimension' Axiom (15.13), there is an isomorphism x ∗ E ≈ ⊕ · · · ⊕ (a trivialization) in F ( ⋆ ) . Now, it will be enough to pull ρ ba k to U alongthe smooth map g and observe that there is a fa torization of the map t ◦ g c : U → ⋆ followed by x : ⋆ → X in order to on lude thatthere is also a trivialization E | U ≈ U ⊕ · · · ⊕ U in F ( U ) . Indeed, sin e ρ isan isomorphism, one an form the following long invertible hain E | U = i ∗ U E = ( s ◦ g ) ∗ E ∼ = g ∗ s ∗ E g ∗ ρ −−→ g ∗ t ∗ E ∼ = ( t ◦ g ) ∗ E == ( x ◦ c ) ∗ E ∼ = c ∗ ( x ∗ E ) ≈ c ∗ ( ⊕ · · · ⊕ ) = U ⊕ · · · ⊕ U (re all that the pull-ba k c ∗ preserves dire t sums). q.e.d.Let i : S ֒ → X be an invariant immersed submanifold, viz. one whoseimage i ( S ) is an invariant subset under the `tautologi al' a tion of G on itsown base. The pull-ba k of G along i makes sense and proves to be a Liesubgroupoid3 ι : G| S ֒ → G of G . (Observe that G| S = G S = s − G ( S ) .) In thespe ial ase of an orbit immersion, G| S will be a transitive Lie groupoid over S . Then the lemma says that for any ( E, ρ ) ∈ Ob R ( G ) the pull-ba k i ∗ S E isa lo ally trivial obje t of F ( S ) , be ause the transitive Lie groupoid R ( G| S ) a ts on i ∗ S E via ι ∗ S ρ . In parti ular, when the orbit S ֒ → X is a submanifold,we an also write E | S = i ∗ S E ∈ V F ( S ) .2 Note The notion of Lie groupoid representation we have been workingwith so far is ompletely intrinsi . We were able to prove all results by meansof purely formal arguments, involving only manipulations of ommutativediagrams. For the purposes of the present se tion, however, we have to hangeour point of view.Let G be a Lie groupoid. Consider a representation ( E, ρ ) ∈ Ob R ( G ) , s ∗ E ρ −→ t ∗ E . Ea h arrow g determines a linear map ρ ( g ) : E s ( g ) → E t ( g ) de(cid:28)ned via the ommutativity of the diagram [ g ∗ s ∗ E ] ∗ [ g ∗ ρ ] ∗ (cid:15) (cid:15) [ ∼ =] ∗ / / [ s ( g ) ∗ E ] ∗ def. E s ( g ) ρ ( g ) (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) [ g ∗ t ∗ E ] ∗ [ ∼ =] ∗ / / [ t ( g ) ∗ E ] ∗ def. E t ( g ) (3)where the notation (11.10) is used. It is routine to he k that the o y le onditions (13.2) and (13.3) in the de(cid:28)nition of representation imply thatthe orresponden e g ρ ( g ) is multipli ative i.e. that ρ ( g ′ g ) = ρ ( g ′ ) ◦ ρ ( g ) and ρ ( x ) = id for ea h point of the base manifold X .Next, onsider any arrow g . Also, let ζ ∈ Γ E ( U ) be a se tion de(cid:28)nedover a neighbourhood of s ( g ) in X . Re all that a ording to (11.21) ζ willdetermine the se tion ζ ◦ s ∈ Γ G ( s ∗ E )( G U ) , de(cid:28)ned over the open subset G U = s − ( U ) of the manifold of arrows G (1) ; the morphism of sheaves ofmodules Γ ρ an be evaluated at ζ ◦ s : [ Γ ρ ( G U )]( ζ ◦ s ) ∈ Γ ( t ∗ E )( G U ) . Axiom3In general, a (cid:19) Lie subgroupoid (cid:20) is a Lie groupoid homomorphism ( ϕ, f ) su h thatboth ϕ and f are inje tive immersions.17. CONSTRUCTION OF EQUIVARIANT MAPS 83(15.1) implies that there exists an open neighbourhood Γ ⊂ G U of g overwhi h [ Γ ρ ( G U )]( ζ ◦ s ) an be expressed as a (cid:28)nite linear ombination, with oe(cid:30) ients in C ∞ (Γ) , of se tions of the form ζ ′ i ◦ t with ζ ′ i , i = 1 , · · · , d de(cid:28)ned over t (Γ) . Expli itly,(4) (cid:2) Γ ρ (Γ) (cid:3) ( ζ ◦ s | Γ ) = d P i =1 r i ( ζ ′ i ◦ t ) | Γ with r , . . . , r d ∈ C ∞ (Γ) and ζ ′ , . . . , ζ ′ d ∈ ( Γ E )( t (Γ)) . This equality an beevaluated at g ∈ Γ in the abstra t sense of (11.14), also taking (3) intoa ount, to get a more intuitive expression(5) ρ ( g ) · ζ ( s g ) = d P i =1 r i ( g ) ζ ′ i ( t g ) .To summarize: any G -a tion ( E, ρ ) determines an operation g ρ ( g ) whi h assigns a linear isomorphism E x ρ ( g ) −−→ E x ′ to ea h arrow x g −→ x ′ in su ha way that the omposition of arrows is respe ted; moreover, the operationenjoys a `smoothness property' whose te hni al formulation is synthesized inEquation (5). Conversely, it is yet another exer ise to re ognize that su hdata determine an a tion of G on E , by Axiom (15.6). Therefore we see thatfor the representations whose type is a sta k of smooth (cid:28)elds the intrinsi de(cid:28)nition of Ÿ13 is equivalent to a more on rete de(cid:28)nition involving anoperation g ρ ( g ) and a `smoothness ondition' expressed pointwise.Let G be a Lie groupoid over a manifold X . Consider any representation ( E, ρ ) ∈ Ob R ( G ) . Fix an arbitrary point x ∈ X . Using the remarks of thepre eding note, the fa t that the (cid:28)bre E def = E x is a (cid:28)nite dimensional ve torspa e, by Axiom (15.13), and the fa t that the evaluation map (15.2) ( Γ E ) → E , ζ ζ ( x ) is surje tive, one sees at on e that the operation(6) ρ : G → GL ( E ) , g ρ ( g ) is a smooth representation of the Lie group G = G (= the isotropy group at x ) on the (cid:28)nite dimensional ve tor spa e E .Now, suppose we are given a G -equivariant linear map A : E → F ,for some other G -a tion ( F, σ ) . Let S ֒ → X be the orbit through x ; just to(cid:28)x ideas, assume it is a submanifold. The theory of Morita equivalen es ofŸ14 says that there exists a unique morphism A ′ : ( E | S , ρ | S ) → ( F | S , σ | S ) in R ( G| S ) su h that ( A ′ ) = A , up to the standard anoni al identi(cid:28) ations.A tually, for any point z ∈ S and any arrow g ∈ G ( x , z ) one has(7) ( A ′ ) z = σ ( g ) · A · ρ ( g ) − : E z → F z .4 CHAPTER IV. GENERAL TANNAKA THEORYSet E ′ = E | S . As remarked earlier, sin e the groupoid G| S is transitive itfollows that the obje t E ′ is lo ally trivial, by Lemma 1. If the submanifold S ֒ → X is in addition losed then, sin e base manifolds of Lie groupoidsare always para ompa t, Lemma 15.11 will yield a morphism a : E → F extending A ′ and hen e, a fortiori, A .The averaging operatorWe are now ready to des ribe an (cid:19) averaging te hnique (cid:20) whi h is of entralimportan e in our work(cid:22)as the reader will see. We explain in detail how,starting from any (right-invariant) Haar system µ = { µ x } on a proper Liegroupoid G over a manifold M , one an onstru t, for ea h pair of represen-tations R = ( E, ρ ) , S = ( F, σ ) ∈ R ( G ) (of type F ), a linear operator(8) Av µ : Hom F ( M ) ( E, F ) → Hom R ( G ) ( R, S ) alled the (cid:19) averaging operator (of type F ) (cid:20) asso iated with µ , with theproperty that Av µ ( a ) = a whenever a already belongs to the subspa e Hom R ( G ) ( R, S ) ⊂ Hom F ( M ) ( E, F ) . This onstru tion will be ompatible withthe restri tion to an invariant submanifold of the base: namely, if N ⊂ M isany su h submanifold then, letting ν denote the Haar system indu ed by µ on the subgroupoid G| N = G N ι N ֒ → G (what we are saying makes sense be ause N is invariant), the following diagram will ommute Hom F ( M ) ( E, F ) i ∗ N (cid:15) (cid:15) Av µ / / Hom R ( G ) ( R, S ) ι ∗ N (cid:15) (cid:15) Hom F ( N ) ( E | N , F | N ) Av ν / / Hom R ( G| N ) ( ι ∗ N R, ι ∗ N S ) .(9)Thus, in parti ular, if a restri ts to an invariant morphism over N then Av µ ( a ) | N = a | N . Sin e µ will be (cid:28)xed throughout the present dis ussion, weabbreviate Av µ ( a ) into ˜ a from now on.We start from a very simple remark, valid even without assuming G to beproper. Suppose that ζ ∈ Γ E ( U ) and η , . . . , η n ∈ Γ F ( U ) are se tions oversome open subset of M , and moreover that η , . . . , η n are lo al generators for Γ F over U ; then for ea h g ∈ G U = s − ( U ) there exists an open neighbour-hood g ∈ Γ ⊂ G U , along with smooth fun tions φ , . . . , φ n ∈ C ∞ (Γ) , su hthat the identity(10) σ ( g ) − · a t ( g ) · ρ ( g ) · ζ ( s g ) = n P j =1 φ j ( g ) η j ( s g ) holds in the (cid:28)bre F s ( g ) for all g ∈ Γ . To see this, re all that, a ording toNote 2, there are an open neighbourhood Γ of g in G U and lo al smooth17. CONSTRUCTION OF EQUIVARIANT MAPS 85se tions ζ ′ , . . . , ζ ′ m of E over U ′ = t (Γ) , su h that ρ ( g ) ζ ( s g ) = m P i =1 r i ( g ) ζ ′ i ( t g ) for some smooth fun tions r , . . . , r m ∈ C ∞ (Γ) . For i = 1 , . . . , m , put η ′ i = Γ a ( U ′ )( ζ ′ i ) ∈ Γ F ( U ′ ) . Sin e Γ − is a neighbourhood of g − we anassume(cid:22)again by Note 2, using the hypothesis that the η j 's are genera-tors(cid:22) Γ to be so small that for ea h i = 1 , . . . , m there exist smooth fun -tions s ,i , . . . , s n,i ∈ C ∞ (Γ − ) with σ ( g − ) η ′ i ( t g ) = n P j =1 s j,i ( g − ) η j ( s g ) ∀ g ∈ Γ .Hen e for all g ∈ Γ we get σ ( g ) − · a t ( g ) · ρ ( g ) · ζ ( s g ) = σ ( g − ) · a t ( g ) · m P i =1 r i ( g ) ζ ′ i ( t g ) == m P i =1 r i ( g ) σ ( g − ) η ′ i ( t g ) = n P j =1 (cid:20) m P i =1 r i ( g ) s j,i ( g − ) (cid:21) η j ( s g ) ,whi h is (10) with φ j ( g ) = m P i =1 r i ( g ) s j,i ( g − ) , j = 1 , . . . , n .Let α = Γ a ∈ Hom C ∞ ( Γ E, Γ F ) . We an use the last remark to obtaina morphism ˜ α : Γ E → Γ F of sheaves of modules over M , in the followingway. Let ζ be a lo al smooth se tion of E , de(cid:28)ned over an open subset U ⊂ M so small that there exists a system η , . . . , η n of lo al generators for F over U (su h a system an always be found lo ally, be ause F satis(cid:28)esAxiom (15.16)). For ea h g ∈ G U = s − ( U ) , sele t an open neighbourhood Γ( g ) , along with smooth fun tions φ g , . . . , φ g n ∈ C ∞ (cid:0) Γ( g ) (cid:1) , as in (10).Sin e the manifold of arrows of G , and(cid:22) onsequently(cid:22)its open submanifold G U , is para ompa t (we are assuming G proper now; f. Ÿ1), there will be asmooth partition of unity { θ i } , i ∈ I on G U subordinated to the open over { Γ( g ) } , g ∈ G U . Then we put(11) ˜ α ( U ) ζ = n P j =1 Φ j η j , where Φ j ( u ) = Z G u P i ∈ I θ i ( g ) φ ij ( g ) d µ u ( g ) (note that the integrand P i ∈ I θ i φ ij is a smooth fun tion on G U and hen e Φ j ∈ C ∞ ( U ) , j = 1 , . . . , n ). Of ourse, many arbitrary hoi es are involved here,so one has to make sure that this de(cid:28)nition is not ambiguous (however, assoon as (11) is known to be independent of all these hoi es, it will ertainlyde(cid:28)ne a morphism of sheaves of modules over M ). One an do this, in twosteps, by introdu ing independently a ertain bundle of linear maps { λ x : E x → F x } over M (cid:28)rst and then he king that [ ˜ α ( U ) ζ ]( u ) = λ u (cid:0) ζ ( u ) (cid:1) for all u ∈ U . Sin e the right-hand term in the last equality will not depend on any hoi e, Axiom (15.4) will imply at on e that ˜ α ( U ) ζ is a well-de(cid:28)ned se tionof F over U . The same equality will furthermore yield the on lusion that ˜ α ∈ Hom C ∞ M ( Γ E, Γ F ) is equal to Γ ˜ a for a unique ˜ a ∈ Hom F ( M ) ( E, F ) , byAxiom (15.6). It should be lear how to pro eed now, but let us arry out6 CHAPTER IV. GENERAL TANNAKA THEORYthe details anyway, for ompleteness. If we look at (10) with s ( g ) = x (cid:28)xed,we immediately re ognize that the map(12) G x → F x , g σ ( g ) − · a t ( g ) · ρ ( g ) · ζ ( x ) ,of the manifold G x = s − ( x ) into the (cid:28)nite dimensional ve tor spa e F x , isof lass C ∞ and hen e ontinuous. Sin e for ea h v ∈ E x there is some lo alse tion ζ of E about x su h that v = ζ ( x ) , by Axiom (15.1), we an writedown the integral(13) a µ ( x ) · v def = Z G x σ ( g ) − · a t ( g ) · ρ ( g ) · v d µ x ( g ) for ea h v ∈ E x . Clearly v a µ ( x ) · v de(cid:28)nes a linear map of E x into F x , sowe get our bundle of linear maps (cid:8) a µ ( x ) : E x → F x (cid:9) . It remains to he k,for an arbitrary u ∈ U , the equality [ ˜ α ( U ) ζ ]( u ) = a µ ( u ) · ζ ( u ) with ˜ α ( U ) ζ given by (11). The omputation is straightforward: [ ˜ 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 ij ( g ) d µ u ( g ) η j ( u )= Z G u P i ∈ I θ i ( g ) n P j =1 φ ij ( g ) η j ( s g ) d µ u ( g )= Z G u P i ∈ I θ i ( g ) (cid:2) σ ( g ) − · a t ( g ) · ρ ( g ) · ζ ( s g ) (cid:3) d µ u ( g )= a µ ( u ) · ζ ( u ) .In on lusion, we de(cid:28)ne Av µ ( a ) as the unique morphism ˜ a : E → F ∈ F ( M ) su h that Γ ˜ a = g ( Γ a ) . The linearity of a Av µ ( a ) follows now from(13), the relation [ ˜ α ( U ) ζ ]( u ) = a µ ( u ) · ζ ( u ) and the faithfulness of a Γ a .It remains to show that Av µ ( a ) belongs to Hom R ( G ) ( R, S ) and that Av µ ( a ) equals a when a already belongs to Hom R ( G ) ( R, S ) ; although the al ulationis ompletely standard, we review it be ause of its importan e. In order toprove that ˜ a ≡ Av µ ( a ) is a morphism of G -a tions, it will be enough (byAxiom 15.4) to he k the identity ˜ a t ( g ) ◦ ̺ ( g ) = σ ( g ) ◦ ˜ a s ( g ) or equivalently,letting x = s ( g ) and x ′ = t ( g ) , the identity a µ ( x ′ ) ◦ ̺ ( g ) = σ ( g ) ◦ a µ ( x ) forea h arrow g ; the orresponding omputation reads as follows: a µ ( x ′ ) ◦ ̺ ( g ) = Z G ( x ′ , - ) σ ( g ′ ) − a t ( g ′ ) ̺ ( g ′ ) ̺ ( g ) d µ x ′ ( g ′ ) by (13) = Z G ( x, - ) σ ( g ) σ ( h ) − a t ( h ) ̺ ( h ) d µ x ( h ) by right-invarian e = σ ( g ) ◦ a µ ( x ) by (13) again.17. CONSTRUCTION OF EQUIVARIANT MAPS 87Next, whenever a is an element of Hom R ( G ) ( R, S ) , the omputation a µ ( x ) = Z G ( x, - ) σ ( g ) − a t ( g ) ̺ ( g ) d µ x ( g ) by (13) = Z G ( x, - ) a x d µ x ( g ) be ause a ∈ Hom R ( G ) ( R, S )= a x be ause µ is normalizedproves the identity ˜ a = a . Appli ationsFor the reader's onvenien e and for future referen e, it will be useful to olle t the on lusions of the previous subse tion into a single statement. Asever, F will denote an arbitrary sta k of smooth (cid:28)elds, for example the sta kof smooth ve tor bundles or the sta k of smooth Eu lidean (cid:28)elds.14 Proposition (Averaging Lemma) Let G be a proper Lie groupoidover a manifold M , and let µ be a right-invariant Haar system on G .Then for any given G -a tions R = ( E, ̺ ) and S = ( F, σ ) of type F ,ea h morphism a : E → F in the ategory F ( M ) determines a (unique)morphism ˜ a = Av µ ( a ) : R → S ∈ R F ( G ) through the requirement thatfor ea h x ∈ M the map ˜ a x : E x → F x should be given by the formula ˜ a x ( v ) = Z G x σ ( g ) − · a t ( g ) · ̺ ( g ) · v d µ x ( g ) . (cid:0) ∀ v ∈ E x (cid:1) (15)In parti ular, ˜ a = a for all G -equivariant a .We will now derive a series of useful orollaries, whi h enter as key ingredientsin many proofs throughout Ÿ20.16 Corollary (Isotropy Extension Lemma) Let G be a proper Liegroupoid over a manifold M , and let x ∈ M be any point.Let R = ( E, ̺ ) and S = ( F, σ ) be G -a tions of type F and put E ≡ E x and F ≡ F x . Moreover, let A : E → F be a G -equivariant linearmap, where G ≡ G denotes the isotropy group of G at x .Then there exists a morphism a : R → S in R F ( G ) su h that a ≡ a x = A .Proof Apply Lemma 15.11 and then the Averaging Lemma to the morphism A S : ( E | S , ̺ | S ) → ( F | S , σ | S ) ∈ R F ( G| S ) (7), where S = G · x . The orollarywill follow from the formula (15) written at x = x . q.e.d.8 CHAPTER IV. GENERAL TANNAKA THEORY17 Corollary (Existen e of Invariant Metri s) Let G be a proper Liegroupoid over a manifold M . Let R = ( E, ̺ ) ∈ R ( G ) be a representation.Then there exists a metri m : R ⊗ R ∗ → in R ( G ) .Proof Choose any metri φ : E ⊗ E ∗ → in F ( M ) (su h metri s existbe ause F satis(cid:28)es Axiom 15.8 and M is para ompa t); also (cid:28)x any right-invariant Haar system µ on G . By applying the averaging operator we obtaina morphism ˜ φ = Av µ ( φ ) : R ⊗ R ∗ → in R ( G ) . We ontend that ˜ φ is aninvariant metri on R . It su(cid:30) es to prove that for ea h x ∈ M the indu edform ˜ φ x : E x ⊗ E x ∗ → C is a Hilbert metri (i.e. Hermitian and positivede(cid:28)nite). Formula (15) reads ˜ φ x ( v, w ) = Z G x (cid:10) ̺ ( g ) v, ̺ ( g ) w (cid:11) φ d µ x ( g ) , (cid:0) ∀ v, w ∈ E x (cid:1) (18)when e our laim is evident. q.e.d.Let R = ( E, ̺ ) be any G -a tion. By a G -invariant se tion of E , de(cid:28)nedover an invariant submanifold N of the base M of G , we mean any se tion ζ ∈ Γ( N ; E | N ) whi h is at the same time a morphism → R | N in R ( G| N ) .19 Corollary (Invariant Se tions) Let S be a losed invariant submani-fold of the base M of a proper Lie groupoid G . Let R = ( E, ̺ ) ∈ R ( G ) bea representation.Then ea h G -invariant se tion ξ of E over S an be extended to aglobal G -invariant se tion; in other words, there exists some G -invariant Ξ ∈ Γ( M ; E ) su h that Ξ | S = ξ .Proof Apply Lemma 15.11 and the Averaging Lemma. q.e.d.In general, we shall say that a partial fun tion ϕ : S → C, de(cid:28)ned onan arbitrary subset S ⊂ M , is smooth when for ea h x ∈ M one an (cid:28)ndan open neighbourhood B of x in M and a smooth fun tion B → C thatrestri ts to ϕ over B ∩ S .20 Corollary (Invariant Fun tions) Let S be any invariant subset ofthe base manifold M of a proper Lie groupoid G . Suppose ϕ : S → Ris a smooth invariant fun tion (i.e. ϕ ( g · s ) = ϕ ( s ) for all g ). Then thereexists a smooth invariant fun tion Φ : M → R extending ϕ outside S .Proof Apply the Averaging Lemma to any smooth fun tion extending ϕ outside S (su h an extension an be obtained by means of a partition ofunity over M , be ause of the smoothness of ϕ ). q.e.d.18. FIBRE FUNCTORS 89Ÿ18 Fibre Fun torsLet F be a sta k of omplex smooth (cid:28)elds, to be regarded as (cid:28)xed on e andfor all. Let M be a para ompa t smooth manifold.1 De(cid:28)nition By a (cid:28)bre fun tor (of type F ) over M , or with base M , wemean a faithful omplex tensor fun tor(2) ω : C −→ F ( M ) ,of some additive omplex tensor ategory C , with values into F ( M ) . We donot assume C to be rigid.When a (cid:28)bre fun tor ω is assigned over M , one an onstru t a groupoid T ( ω ) having the points of M as obje ts. Under reasonable assumptions, itis possible to make T ( ω ) a topologi al groupoid over the (topologi al) spa e M ; the hoi e of a topology is di tated by the idea that the obje ts of C should give rise to (cid:19) ontinuous representations (cid:20) of T ( ω ) and that, vi eversa, ontinuity of these representations should be enough to hara terizethe topologi al stru ture. An improvement of the same idea leads one tostudy a ertain fun tional stru ture on T ( ω ) , in the sense of Bredon (1972),p. 297, and the important related problem of determining su(cid:30) ient onditionsfor this fun tional stru ture to be ompatible with the groupoid operations.Another fundamental issue here is to understand whether one gets in fa t amanifold stru ture4 making T ( ω ) a Lie groupoid over M ; if this proves to bethe ase, we say that the (cid:28)bre fun tor ω is smooth.Some notation is needed (cid:28)rst of all. Let x be a point of M . If x alsodenotes the (smooth) map ⋆ → M , ⋆ x , one an onsider the omplextensor fun tor `(cid:28)bre at x ' whi h was introdu ed in Ÿ11(3) F ( M ) → { vector spaces } , E E x def = ( x ∗ E ) ∗ .Let ω x be the omposite omplex tensor fun tor(4) C ω −→ F ( M ) ( - ) x −−→ { vector spaces } , R ω x ( R ) def = (cid:0) ω ( R ) (cid:1) x .De(cid:28)ne the omplex, resp. real, Tannakian groupoid of ω in the following way:for x, x ′ ∈ M , put(5) ( T ( ω ; C )( x, x ′ ) def = Iso ⊗ ( ω x , ω x ′ ) T ( ω ; R )( x, x ′ ) def = Iso ⊗ , ∗ ( ω x , ω x ′ ) .(Re all that the right-hand side of the se ond equal sign denotes the set ofall the self- onjugate tensor preserving natural isomorphisms ω x ∼ → ω x ′ , that4A manifold an be de(cid:28)ned as a topologi al spa e endowed with a fun tional stru turelo ally looking like the stru ture of smooth real valued fun tions on some R d .0 CHAPTER IV. GENERAL TANNAKA THEORYis to say, the subset of Iso ⊗ ( ω x , ω x ′ ) onsisting of those λ whi h make thefollowing square ommutative for ea h obje t R ∈ Ob( C ) : ω x ( R ) ∗ an. ∼ = (cid:15) (cid:15) λ ( R ) ∗ / / ω x ′ ( R ) ∗ an. ∼ = (cid:15) (cid:15) ω x ( R ∗ ) λ ( R ∗ ) / / ω x ′ ( R ∗ ) .)(6)Setting ( λ ′ λ )( R ) = λ ′ ( R ) ◦ λ ( R ) and x ( R ) = id , one obtains two groupoidsover the set of points of M , with inverse given by λ − ( R ) = λ ( R ) − . Wemay also express (5) in short by writing T ( ω ; C ) = Aut ⊗ ( ω ) and T ( ω ; R ) =Aut ⊗ , ∗ ( ω ) .Let us investigate the relationship between the omplex tannakian group-oid T ( ω ; C ) and its subgroupoid T ( ω ; R ) (cid:28)rst. As a onvenient notationaldevi e, we omit writing ω when we simply refer to the set of arrows of thetannakian groupoid; thus for instan e T ( C ) is the set of arrows of the group-oid T ( ω ; C ) . We de(cid:28)ne a map T ( C ) → T ( C ) , λ λ , whi h we all omplex onjugation, by setting λ ( R ) = λ ( R ∗ ) ∗ ; more pre isely, λ ( R ) is de(cid:28)ned byimposing the ommutativity of ω x ( R ∗ ) ∗ λ ( R ∗ ) ∗ (cid:15) (cid:15) ∼ = / / ω x ( R ∗∗ ) ω x ( ∼ =) / / ω x ( R ) λ ( R ) (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) ω x ′ ( R ∗ ) ∗ ∼ = / / ω x ′ ( R ∗∗ ) ω x ′ ( ∼ =) / / ω x ′ ( R ) .(7)It is straightforward to he k that λ ∈ Hom ⊗ ( ω x , ω x ′ ) implies λ ∈ Hom ⊗ ( ω x , ω x ′ ) and that λ λ is a groupoid homomorphism of T ( ω ; C ) into itself, identi al on obje ts; this endomorphism is moreover involutiveviz. λ = λ . Then we an hara terize the arrows belonging to the subgroup-oid T ( ω ; R ) as the (cid:28)xed points of the involution λ λ :(8) T ( R ) = { λ ∈ T ( C ) : λ = λ } .Next, we endow the set T = T ( C ) or T ( R ) with a topology. In order todo this, we need to introdu e the notion of (cid:19) metri (cid:20) in F ( M ) . Let E bean obje t of F ( M ) . A metri on E , or supported by E , is a Hermitian form φ : E ⊗ E ∗ → in F ( M ) su h that for all x ∈ M the indu ed Hermitianform φ x on the (cid:28)bre E x (9) E x ⊗ E x ∗ ∼ = ( E ⊗ E ∗ ) x φ x −→ x ∼ = Cis positive de(cid:28)nite (and hen e turns E x into a omplex Hilbert spa e of (cid:28)nitedimension).We start by de(cid:28)ning a olle tion R of omplex valued fun tions on T ,whi h we may all the (cid:19) representative fun tions (cid:20). (Whenever we need to18. FIBRE FUNCTORS 91distinguish between T ( C ) and T ( R ) , we an write R ( C ) or R ( R ) as the asemay be.)Choose an obje t R ∈ Ob( C ) , and let φ be a metri on the obje t ω ( R ) of F ( M ) . Also (cid:28)x a pair of global smooth se tions ζ , ζ ′ ∈ Γ ( ω R )( M ) . Youget a omplex fun tion(10) r R,φ,ζ,ζ ′ : T → C, λ (cid:10) λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ ) (cid:11) φ ≡ φ t ( λ ) (cid:0) λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ ) (cid:1) .Then put(11) R = (cid:8) r R,φ,ζ,ζ ′ : R ∈ Ob( C ) , φ metri on ω ( R ) in F ( M ) , ζ , ζ ′ ∈ Γ ( ω R )( M ) (cid:9) .We endow T with the oarsest topology making all the fun tions in R ontin-uous. From now on in our dis ussion T ( C ) and T ( R ) will always be regardedas topologi al spa es, with this topology. Observe that T ( R ) turns out tobe a subspa e of T ( C ) ; more expli itly, the topology on T ( R ) indu ed by R ( R ) oin ides with the topology indu ed from T ( C ) along the in lusion T ( R ) ⊂ T ( C ) .We now want to establish a few fundamental algebrai properties of the olle tion R of omplex valued fun tions on T . We are going to show that R is a omplex algebra of fun tions, and moreover that R ( R ) is losed undertaking the omplex onjugate. Both assertions are immediate onsequen esof the following identities:i) For all smooth fun tions a ∈ C ∞ ( M ) ,(12) ( a ◦ s ) r R,φ,ζ,ζ ′ = r R,φ,aζ,ζ ′ and ( a ◦ t ) r R,φ,ζ,ζ ′ = r R,φ,ζ,aζ ′ ;in parti ular, if c ∈ C is onstant, r R,φ,cζ,ζ ′ = c r R,φ,ζ,ζ ′ = r R,φ,ζ,cζ ′ .ii) If we let τ denote the metri on ω ( ) orresponding to the trivial metri ⊗ ∗ ∼ = ⊗ ∼ = on the obje t of F ( M ) , and ∈ Γ ( ω )( M ) orrespond to the (cid:16)unity se tion(cid:17) of ∈ F ( M ) under the iso υ : ∼ → ω ( ) , then(13) `unity onstant fun tion' = r ,τ, , .iii) For any hoi e of a dire t sum R ֒ → R ⊕ S ← ֓ S in C ,(14) r R,φ,ζ,ζ ′ + r S,ψ,η,η ′ = r R ⊕ S,φ ⊕ ψ,ζ ⊕ η,ζ ′ ⊕ η ′ ,where ζ ⊕ η ∈ Γ ( ω ( R ⊕ S ))( M ) et . are obtained by setting ω ( R ) ⊕ ω ( S ) = ω ( R ⊕ S ) .2 CHAPTER IV. GENERAL TANNAKA THEORYiv) Allowing the obvious ( anoni al) identi(cid:28) ations,(15) r R,φ,ζ,ζ ′ r S,ψ,η,η ′ = r R ⊗ S,φ ⊗ ψ,ζ ⊗ η,ζ ′ ⊗ η ′ .(For instan e, ζ ⊗ η here denotes really the global se tion of ω ( R ⊗ S ) orresponding to the (cid:16)true(cid:17) ζ ⊗ η in the iso τ R,S : ω ( R ) ⊗ ω ( S ) ∼ → ω ( R ⊗ S ) .)v) Allowing again some loose notation,(16) r R,φ,ζ,ζ ′ = r R ∗ ,φ ∗ ,ζ ∗ ,ζ ′∗ ◦ (cid:16) λ λ (cid:17).In parti ular, sin e the omplex onjugation (cid:16) λ λ (cid:17) restri ts to theidentity on T ( R ) , it follows that r R,φ,ζ,ζ ′ = r R ∗ ,φ ∗ ,ζ ∗ ,ζ ′∗ in R ( R ) .Noti e that from the fa t that R ( R ) is losed under omplex onjugationit follows immediately that the real and imaginary parts of any fun tion in R ( R ) will belong to R ( R ) as well. Thus, if we let R [ R ] ⊂ R ( R ) denote thesubset of all the real valued fun tions, we an express R ( R ) = C ⊗ R [ R ] asthe omplexi(cid:28) ation of a real fun tional algebra.For the rest of the se tion(cid:22)and for the purposes of the present thesis(cid:22)wewill only be interested in studying the real tannakian groupoid T ( ω ; R ) . Sofrom now on we forget about T ( ω ; C ) and simply write T ( ω ) for T ( ω ; R ) .There is one further pie e of stru ture we want to onsider on T ( ω ) , besidesthe topology.Let the sheaf of ontinuous (real valued) fun tions on an arbitrary to-pologi al spa e T be denoted by C T . Then re all that a ording to Bredon(1972), a (cid:16)fun tionally stru tured spa e(cid:17) is a topologi al spa e T , endowedwith a sheaf of real algebras of ontinuous fun tions on T (cid:22)in other words,a subsheaf of algebras of C T . A morphism of su h (cid:16)fun tionally stru turedspa es(cid:17) is then de(cid:28)ned as a ontinuous mapping su h that the pullba k of ontinuous fun tions along the mapping is ompatible with the fun tionalstru tures. For more details, we refer the reader to lo . it., p. 297. We adoptthis point of view in order to obtain a natural surrogate on T ( R ) of the no-tion of (cid:19) smooth fun tion (cid:20), drawing on the intuition that the representativefun tions should be regarded as the prototype (cid:19) smooth fun tions (cid:20).It is obvious that if we start from the idea that the (real) representativefun tions are (cid:16)smooth(cid:17) then so we have to regard any fun tion obtained by omposing them with a smooth fun tion f : R d → R. De(cid:28)ne R ∞ to be thesheaf, of ontinuous real valued fun tions on the spa e T = T ( R ) , generatedby the presheaf(17) Ω (cid:8) f ( r | Ω , . . . , r d | Ω ) : f : R d → R of lass C ∞ , r , . . . , r d ∈ R [ R ] (cid:9) .In other words, R ∞ is the smallest subsheaf of C T ontaining (17) as a sub-presheaf. The expression f ( r | Ω , . . . , r d | Ω ) denotes of ourse the fun tion λ
18. FIBRE FUNCTORS 93 f (cid:0) r ( λ ) , . . . , r d ( λ ) (cid:1) , λ ∈ Ω . Sin e (17) is evidently a presheaf of R-algebrasof ontinuous fun tions on T , R ∞ will be a sheaf of su h algebras and hen ethe pair ( T , R ∞ ) will onstitute a fun tionally stru tured spa e.Of ourse, we would like to say that the fun tional stru ture R ∞ on T is ompatible with the groupoid stru ture of T ( ω ) . This means that thestru ture maps of T ( ω ) should be all morphisms of fun tionally stru turedspa es, the base M being regarded as su h a spa e by means of its own sheafof smooth real valued fun tions; in parti ular, the stru ture maps shouldbe all ontinuous. What we are saying is not very pre ise, of ourse, unlesswe turn the spa e of omposable arrows itself into a fun tionally stru turedspa e. Let us begin by observing that if ( X, F ) and ( Y, G ) are any fun tion-ally stru tured spa es then so is their Cartesian produ t endowed with thesheaf F ⊗ G lo ally generated by the fun tions ( ϕ ⊗ ψ )( x, y ) = ϕ ( x ) ψ ( y ) .Then one an repeat the foregoing pro edure to obtain, on X × Y , a sheaf ( F ⊗ G ) ∞ of lass C ∞ , i.e. losed under omposition with arbitrary smoothfun tions as in (17). Any subspa e S ⊂ X × Y may be (cid:28)nally regardedas a fun tionally stru tured spa e by endowing it with the indu ed sheaf ( F ⊗ G ) ∞ | S def = i S ∗ [( F ⊗ G ) ∞ ] , where i S denotes the in lusion mapping of S into X × Y . (Re all that if f : S → T is any ontinuous mapping into afun tionally stru tured spa e ( T, T ) then f ∗ T is the fun tional sheaf on S asso iated with the presheaf U lim −→ V ⊃ f ( U ) T ( V ) . (cid:1) Noti e that in ase X and Y are smooth manifolds and S ⊂ X × Y is asubmanifold, one re overs the orre t fun tional stru tures: ( C ∞ X ⊗ C ∞ Y ) ∞ = C ∞ X × Y and C ∞ X × Y | S = C ∞ S . It is therefore perfe tly reasonable to endow thespa e of omposable arrows T (2) = T s × t T with the fun tional stru ture R (2) , ∞ def = ( R ∞ ⊗ R ∞ ) ∞ | T (2) and to all the omposition map c : T (2) → T (cid:16)smooth(cid:17) whenever it is a morphism of su h fun tionally stru tured spa einto ( T , R ∞ ) .Later on we will show that T ( ω ) is a tually a fun tionally stru turedgroupoid in the two ases of major interest for us, namely when ω is thestandard (cid:28)bre fun tor ω ( G ) asso iated with a proper Lie groupoid (Ÿ20) orwhen ω is a (cid:19) lassi al (cid:20) (cid:28)bre fun tor (Ÿ21). However, we an already veryeasily he k the (cid:16)smoothness(cid:17) (in parti ular, the ontinuity) of some of thestru ture maps:(a) The sour e map s : T → M . First of all observe that for an arbitrary a ∈ C ∞ ( M ) we have a ◦ s ∈ R , by (12) and (13). Let U ⊂ M be open.For ea h u ∈ U there exists f u ∈ C ∞ ( M ) with supp f u ⊂ U and f u ( u ) =1 . Sin e f u ◦ s ∈ R , the subset ( f u ◦ s ) − ( C =0 ) ⊂ T must be open. Now ( f u ◦ s ) − ( C =0 ) = s − (cid:0) f u − ( C =0 ) (cid:1) ⊂ s − ( U ) , so s − ( U ) an be expressed asa union of open subsets of T and therefore it is open. This shows that s is ontinuous; sin e a ◦ s ∈ R [ R ] whenever a is real valued, it also follows that4 CHAPTER IV. GENERAL TANNAKA THEORY s is a morphism of fun tionally stru tured spa es.(b) The target map t : T → M . The dis ussion here is entirely analogous,starting from the other identity a ◦ t = r ,τ, ,a ∈ R .( ) The unit se tion u : M → T . This time let r = r R,φ,ζ,ζ ′ ∈ R be given; wemust show that r ◦ u ∈ C ∞ ( M ) . This is trivial be ause ( r ◦ u )( x ) = (cid:10) x ( R ) · ζ ( x ) , ζ ′ ( x ) (cid:11) φ = (cid:10) ζ ( x ) , ζ ′ ( x ) (cid:11) φ = h ζ , ζ ′ i φ ( x ) .Finally, let us remark that, as a onsequen e of the existen e of metri son any obje t of F ( M ) (be ause F is a sta k of smooth (cid:28)elds and M admitspartitions of unity), the spa e T of arrows of T ( ω ) is always Hausdor(cid:27).Indeed, let µ = λ ∈ T . We an assume s ( µ ) = x = s ( λ ) and t ( µ ) = x ′ = t ( λ ) otherwise we are immediately done by using the Hausdor(cid:27)ness of M and the ontinuity of either the sour e or the target map. Then there exists R ∈ Ob( C ) with µ ( R ) = λ ( R ) . Choose any metri φ on ω ( R ) (there is at least one): sin e φ x ′ is in parti ular non-degenerate on E x ′ , there will be global(cid:22)again, be auseof the existen e of partitions of unity(cid:22)se tions ζ , ζ ′ ∈ Γ ( ω R )( M ) with z µ = (cid:10) µ ( R ) · ζ ( x ) , ζ ′ ( x ′ ) (cid:11) φ = (cid:10) λ ( R ) · ζ ( x ) , ζ ′ ( x ′ ) (cid:11) φ = z λ .Let D µ , D λ ⊂ C be disjoint open disks about z µ , z λ respe tively. Then, setting r = r R,φ,ζ,ζ ′ , the inverse images r − ( D µ ) and r − ( D λ ) will be disjoint openneighbourhoods of µ and λ in T .Ÿ19 PropernessWe shall say that a metri φ on the obje t ω ( R ) , R ∈ Ob( C ) of F ( M ) is ω -invariant, when there exists a Hermitian form m : R ⊗ R ∗ → in C su hthat φ oin ides with the indu ed Hermitian form(1) ω ( R ) ⊗ ω ( R ) ∗ ∼ = ω ( R ⊗ R ∗ ) ω ( m ) −−−→ ω ( ) ∼ = .We express this in short by writing φ = ω ∗ m . Note that by the faithfulnessof ω there is at most one su h m .2 De(cid:28)nition A (cid:28)bre fun tor ω : C −→ F ( M ) will be alled proper ifi) the ontinuous mapping ( s , t ) : T → M × M is proper, andii) for every obje t R ∈ Ob( C ) , the obje t ω ( R ) of F ( M ) supports an ω -invariant metri .We an express the se ond ondition more su in tly by saying that (cid:19) thereare enough ω -invariant metri s (cid:20).3 Example Let ω be the standard fun tor ω ( G ) : R ( G ) −→ F ( M ) , oftype F , asso iated with a proper Lie groupoid G over M . Then ω is a proper(cid:28)bre fun tor.19. PROPERNESS 95In order to see this, observe ( fr. also Ÿ20) that there is an obvious homo-morphism of groupoids(4) π : G −→ T ( G ) def = T ( ω ( G )) ,identi al on the base, alled the (cid:19) F -envelope homomorphism (cid:20) of G and de-(cid:28)ned by setting π ( g )( R ) = ̺ ( g ) for ea h obje t R = ( E, ̺ ) of R ( G ) ; thenotation ̺ ( g ) was introdu ed in Ÿ17. The mapping π (1) : G (1) → T (1) is on-tinuous. Indeed, if we (cid:28)x any representative fun tion r = r R,φ,ζ,ζ ′ ∈ R , let ussay with R = ( E, ̺ ) , and a small open subset Γ ⊂ G on whi h we have, for ̺ a ting on ζ , the sort of expression ̺ ( g ) · ζ ( s g ) = ℓ P i =1 r i ( g ) ζ ′ i ( t g ) , r i ∈ C ∞ (Γ) we derived in Ÿ17, then for all g ∈ Γ we obtain ( r ◦ π )( g ) = (cid:10) π ( g )( R ) · ζ ( s g ) , ζ ′ ( t g ) (cid:11) φ = ℓ P i =1 r i ( g ) (cid:0) h ζ ′ i , ζ i φ ◦ t (cid:1) ( g ) .Therefore, we on lude that r ◦ π ∈ C ∞ ( G ) and hen e, in parti ular, that r ◦ π is ontinuous. Note that in fa t this argument shows that the map π is a morphism of fun tionally stru tured spa es, of ( G , C ∞G ) into ( T , R ∞ ) .We will prove in Ÿ20 that the envelope mapping π is also surje tive; theproperness of ( s , t ) : T → M × M is now a trivial onsequen e of this fa tand the properness of ( s , t ) : G → M × M . The existen e of enough invariantmetri s was established in Ÿ17 as a orollary to the Averaging Lemma.Ba k to general notions, it turns out that in order to hara terize thetopology of T the ω -invariant metri s are (for ω proper) as good as thegeneri , `not ne essarily invariant' ones. More exa tly, let R ′ ⊂ R be theset of all the representative fun tions r R,φ,ζ,ζ ′ with φ = ω ∗ m an ω -invariantmetri on ω ( R ) . Note that R ′ is a subalgebra of R , losed under omplex onjugation; this follows from the identities proved above, by observing that ω ∗ m ⊗ ω ∗ n = ω ∗ ( m ⊗ n ) and so on. Then we laim that5 Lemma The topology on T is also the oarsest making all the fun -tions in R ′ ontinuous.Proof Re all that the topology on T was de(cid:28)ned as the oarsest making allthe fun tions belonging to R ontinuous. We have already observed that R ′ is an algebra of ontinuous omplex fun tions on T , losed under onjugation.Moreover, it separates points, be ause of the existen e of enough ω -invariantmetri s, f. the argument used to prove Hausdor(cid:27)ness of T . Hen eforth, forevery open subset Ω ⊂ T with ompa t losure Ω , the involutive subalgebra6 CHAPTER IV. GENERAL TANNAKA THEORY R ′ Ω ⊂ C (Ω; C ) , formed by the restri tions to Ω of elements of R ′ , is sup-norm dense in C (Ω) and a fortiori in R Ω = { r | Ω : r ∈ R } , as a onsequen eof the Stone(cid:21)Weierstrass theorem.This remark applies in parti ular to Ω =
T | U × U ′ where U and U ′ are opensubsets of M with ompa t losure. (Here is where we use the properness of T ( s , t ) −−→ M × M .) Note that the subset T | U × U ′ is also open in the spa e T ′ = (cid:19) T ( R ) with the topology generated by R ′ (cid:20) be ause T ′ ( s , t ) −−→ M × M is learly still ontinuous. Sin e the subsets T | U × U ′ over T , we are now redu edto showing that the identity mappings T | U × U ′ = −→ T ′ | U × U ′ are homeomorphisms.To simplify the notation, we reformulate our laim as follows: given asubset Ω ⊂ T ( R ) , open in both T and T ′ and with ompa t losure in T ,show that the identity mapping Ω ′ = −→ Ω is ontinuous (here Ω ′ denotes of ourse the open subset, viewed as a subspa e of T ′ ). Noti e that the topologyon Ω generated by the olle tion of fun tions R Ω = { r | Ω : r ∈ R } oin ideswith the subspa e topology indu ed from T . Then, let r ∈ R be (cid:28)xed; sin e Ω is ompa t in T , the restri tion r | Ω will be, as remarked at the beginning,a uniform limit of ontinuous fun tions on Ω ′ and hen e itself a ontinuousfun tion on Ω ′ . q.e.d.We shall make impli it use of the lemma throughout the rest of the presentsubse tion.Another easy, although important observation is that all λ ∈ T ( R ) willa t unitarily under any ω -invariant metri . More pre isely, for any obje t R ∈ Ob( C ) and any ω -invariant metri φ on ω ( R ) , the linear isomorphism λ ( R ) will preserve the inner produ t h , i φ :(6) (cid:10) λ ( R ) · v, λ ( R ) · v ′ (cid:11) φ = h v, v ′ i φ .We use this observation to prove the following7 Proposition Let ω be a proper (cid:28)bre fun tor. Then T ( ω ) is a(Hausdor(cid:27), proper) topologi al groupoid.Proof We must show that the inverse and omposition maps of T ( ω ) are ontinuous.a) Continuity of the inverse map i : T → T . It must be proved thatthe omposite r ◦ i is ontinuous on T , for any r = r R,φ,ζ,ζ ′ ∈ R with φ an ω -invariant metri on ω ( R ) . This is immediate, be ause ( r R,φ,ζ,ζ ′ ◦ i )( λ ) = (cid:10) λ ( R ) − · ζ ( t λ ) , ζ ′ ( s λ ) (cid:11) φ = (cid:10) ζ ( t λ ) , λ ( R ) · ζ ′ ( s λ ) (cid:11) φ = (cid:10) λ ( R ) · ζ ′ ( s λ ) , ζ ( t λ ) (cid:11) φ = r R,φ,ζ ′ ,ζ ( λ ) ,19. PROPERNESS 97in view of (6).b) Continuity of omposition c : T s × t T → T (the domain of the mapbeing topologized as a subspa e of the artesian produ t
T × T ). We makea te hni al observation (cid:28)rst.Fix λ ∈ T , let us say λ : x → x ′ . Let R ∈ Ob C and let φ beany ω -invariant metri on E = ω ( R ) . Fix a lo al φ -orthonormal system ζ ′ , . . . , ζ ′ d ∈ Γ ( ω R )( U ′ ) for E about x ′ as in (15.9); hen e, in parti ular,(8) E x ′ = Span { ζ ′ ( x ′ ) , . . . , ζ ′ d ( x ′ ) } .Sin e M is para ompa t, it is no loss of generality to assume that for every i = 1 , . . . , d ζ ′ i is the restri tion to U ′ of a global se tion ζ i of ω ( R ) . Let ζ ∈ Γ ( ω R )( M ) be another global se tion. Consider an open neighbourhood Ω of λ in T su h that t (Ω) ⊂ U ′ . Also let Φ i ∈ C (Ω; C ) ( i = 1 , . . . , d ) be alist of ontinuous omplex fun tions on Ω . Then the norm fun tion(9) µ (cid:12)(cid:12)(cid:12)(cid:12) µ ( R ) · ζ ( s µ ) − d P i =1 Φ i ( µ ) ζ ′ i ( t µ ) (cid:12)(cid:12)(cid:12)(cid:12) is ertainly ontinuous on Ω : indeed, its square is (cid:12)(cid:12) µ ( R ) ζ ( s µ ) (cid:12)(cid:12) − X i ℜ e h Φ i ( µ ) (cid:10) µ ( R ) ζ ( s µ ) , ζ ′ i ( t µ ) (cid:11)i + (cid:12)(cid:12)(cid:12)(cid:12) d P i =1 Φ i ( µ ) ζ ′ i ( t µ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) ζ ( s µ ) (cid:12)(cid:12) − X i ℜ e h Φ i ( µ ) (cid:10) µ ( R ) ζ ( s µ ) , ζ i ( t µ ) (cid:11)i + d P i =1 (cid:12)(cid:12) Φ i ( µ ) (cid:12)(cid:12) (be ause µ ( R ) is unitary (6) and the ve tors ζ ′ i ( t µ ) , i = 1 , . . . , d form anorthonormal system in E t ( µ ) ). Now, make Φ i ( µ ) = (cid:10) µ ( R ) ζ ( s µ ) , ζ i ( t µ ) (cid:11) in(9) and evaluate the fun tion you get at µ = λ : the result will be zero,be ause the ve tors ζ i ( x ′ ) , i = 1 , . . . , d onstitute an orthonormal basis of E x ′ . Hen e, by the just observed ontinuity, for ea h ε > there will be anopen neighbourhood of λ in T , let us all it Ω ε ( λ ) , over whi h the followingestimate holds(10) (cid:12)(cid:12)(cid:12)(cid:12) µ ( R ) · ζ ( s µ ) − d P i =1 r R,φ,ζ,ζ i ( µ ) ζ i ( t µ ) (cid:12)(cid:12)(cid:12)(cid:12) < ε .With this preliminary observation at hand it is easy to show ontinuityof the omposition of arrows. Indeed, onsider an arbitrary obje t R ∈ Ob C ,an arbitrary ω -invariant metri φ on ω ( R ) , and arbitrary global se tions ζ , η ∈ Γ ( ω R )( M ) . We have to he k the ontinuity of the fun tion(11) ( µ ′ , µ ) ( r R,φ,ζ,η ◦ c )( µ ′ , µ ) = (cid:10) µ ′ ( R ) · µ ( R ) · ζ ( s µ ) , η ( t µ ′ ) (cid:11) φ on the spa e of omposable arrows T (2) . Let x λ −→ x ′ λ ′ −→ x ′′ be an arbitrary pairof omposable arrows, whi h we regard as (cid:28)xed. Choose a lo al φ -orthonormal8 CHAPTER IV. GENERAL TANNAKA THEORYsystem about x ′ as before. Then, by the estimate (10) and our remark (6)that µ ′ ( R ) is unitary, for all ( µ ′ , µ ) lose enough to ( λ ′ , λ ) , let us say for µ ∈ Ω ε ( λ ) , the fun tion (11) will di(cid:27)er from the fun tion d P i =1 r R,φ,ζ,ζ i ( µ ) (cid:10) µ ′ ( R ) · ζ i ( s µ ′ ) , η ( t µ ′ ) (cid:11) φ = d P i =1 r R,φ,ζ,ζ i ( µ ) r R,φ,ζ i ,η ( µ ′ ) up to ε k η k , where k η k is a positive bound for the norm of η in a neighbour-hood of x ′′ . This proves the desired ontinuity, be ause the last fun tion is ertainly ontinuous on T × T and hen e on T (2) . q.e.d.Ÿ20 Re onstru tion TheoremsWhen applying the formal apparatus of Ÿ18 to the standard (cid:28)bre fun tor ω F ( G ) asso iated with a Lie groupoid G , we prefer to use the alternativenotation T F ( G ) for the real Tannakian groupoid T (cid:0) ω F ( G ); R (cid:1) and refer tothe latter as the (real) F -envelope of G . If expli it mention of type is notne essary, we normally just write T ( G ) .The F -envelope homomorphism asso iated with a Lie groupoid G is thegroupoid homomorphism π : G → T ( G ) , or, more pedanti ally,(1) π F ( G ) : G −→ T F ( G ) de(cid:28)ned by the formula π ( g )( E, ̺ ) def = ̺ ( g ) . (Having a look at Note 17.2 onemore time might be useful at this point.) The study of properties of theenvelope homomorphism π ( G ) for proper G will onstitute our main on ernin this se tion.Let M/ G be the topologi al spa e obtained by endowing the set of orbits {G · x | x ∈ M } with the quotient topology indu ed by the orbit map(2) o : M → M/ G (the map sending a point x to the respe tive G -orbit o ( x ) = G · x ). Notethat the map o is open: indeed, if U ⊂ M is an open subset then so is o − ( o ( U )) = t ( s − ( U )) be ause t is an open map(cid:22)a tually, a submersion.Furthermore, M/ G is a lo ally ompa t Hausdor(cid:27) spa e. Indeed, suppose G ( x, x ′ ) empty. Properness of G , applied to some sequen e of balls B i × B i ′ shrinking to the point ( x, x ′ ) , will yield open balls B, B ′ ⊂ M at x, x ′ su hthat ( s , t ) − ( B × B ′ ) is empty, in other words, su h that o ( B ) ∩ o ( B ′ ) = ∅ ,as ontended. In parti ular, every orbit G · x = o − { o ( x ) } is a losed subsetof M .3 Theorem Let F be any sta k of smooth (cid:28)elds. Let G be a proper Liegroupoid. Then the F -envelope homomorphism π F ( G ) : G → T F ( G ) is full(i.e. surje tive, as a mapping of the spa es of arrows).20. RECONSTRUCTION THEOREMS 99Proof To begin with, let us prove that G ( x, x ′ ) empty implies T ( G )( x, x ′ ) empty. Put S = G x ∪ G x ′ and let ϕ : S → C be the fun tion whi h takes thevalue over the orbit G x and the value over the orbit G x ′ ; ϕ is well-de(cid:28)nedbe ause G x ∩ G x ′ = ∅ . S is an invariant submanifold of M . Sin e S is theunion of two disjoint losed subsets of M , it is also a losed submanifold.Moreover, ϕ is equivariant with respe t to the trivial representation of G ,i.e. ϕ ( g · s ) = ϕ ( s ) . Corollary 17.20 says that there is some smooth invariantfun tion Φ : M → C, extending ϕ , equivalently, some smooth fun tion Φ on M , onstant along the G -orbits and with Φ( x ) = 1 , Φ( x ′ ) = 0 . By setting b z def = Φ( z ) id , one gets an endomorphism b of the trivial representation with b x = id and b x ′ = 0 . Now, suppose there exists some λ ∈ T ( G )( x, x ′ ) : then,by the naturality of λ , one gets a ommutative squareC id (cid:15) (cid:15) λ / / C (cid:15) (cid:15) C λ / / Cwhi h ontradi ts the invertibility of λ ( ) .In order to (cid:28)nish the proof of the theorem, it will be su(cid:30) ient to provesurje tivity of all isotropy homomorphisms indu ed by π , be ause G| xg - ≈ (cid:15) (cid:15) π x / / T ( G ) | xπ ( g ) - ≈ (cid:15) (cid:15) G ( x, x ′ ) π x,x ′ / / T ( G )( x, x ′ ) ommutes for all g ∈ G ( x, x ′ ) . More expli itly, it will be su(cid:30) ient to provethat π x : G| x → T ( G ) | x is an epimorphism of groups, for every x ∈ M . Thisfollows immediately from Proposition 10.3 and Corollary 17.16. q.e.d.We ontinue to work with an arbitrary sta k of smooth (cid:28)elds.4 De(cid:28)nition A Lie groupoid G will be said to be F -re(cid:29)exive, or self-dualrelative to F , if its F -envelope homomorphism π F ( G ) : G → T F ( G ) is anisomorphism of topologi al groupoids.It turns out, for proper Lie groupoids, that the requirement that the ontinuous mapping π (1) : G (1) → T ( G ) (1) should be open is super(cid:29)uous; morepre isely, one has the following statement:5 Theorem Let G be a proper Lie groupoid. Let F be any sta k ofsmooth (cid:28)elds. Then G is F -re(cid:29)exive if and only if the homomorphism π F ( G ) is faithful (i.e. inje tive, as a mapping of the spa es of arrows).00 CHAPTER IV. GENERAL TANNAKA THEORYProof The assertion that inje tivity implies bije tivity, or, to say the samething di(cid:27)erently, that faithfulness implies full faithfulness, is an immediate onsequen e of Theorem 3 above.As to the statement that the mapping π is open, we have to show thatwhenever Γ is an open subset of G (1) and g a point of Γ , the image π (Γ) isa neighbourhood of π ( g ) in T ( G ) .To (cid:28)x ideas, suppose g ∈ G ( x , x ′ ) . Let us start by observing that,as in the proof of Proposition 10.3, it is possible to (cid:28)nd a representation R = ( E, ̺ ) ∈ Ob R ( G ) whose asso iated x -th isotropy homomorphism ̺ : G → GL ( E ) is inje tive (same notation as in Eq. (17.6)); for su h an R ,the map G ( x , x ′ ) → Lis( E x , E x ′ ) , g ̺ ( g ) is also inje tive. We regard R as (cid:28)xed on e and for all. Moreover, we hoose an arbitrary Hilbert metri φ on E . As we know from Ÿ15, there are lo al φ -orthonormal frames for E (6) ( ζ , . . . , ζ d ∈ Γ E ( U ) about x and ζ ′ , . . . , ζ ′ d ∈ Γ E ( U ′ ) about x ′ ;their ardinality turns out to be the same be ause E x ≈ E x ′ . Sin e M ispara ompa t, it is no loss of generality to assume that the ζ i and the ζ ′ i ′ are(restri tions of) global se tions. Finally, we sele t any ompa tly supportedsmooth fun tions a, a ′ : M → C with supp a ⊂ U and supp a ′ ⊂ U ′ , su hthat a ( x ) = 1 ⇔ x = x and a ′ ( x ′ ) = 1 ⇔ x ′ = x ′ .Let us put, for all ≦ i, i ′ ≦ d , ̺ i,i ′ = r i,i ′ ◦ π def = r R,φ,ζ i ,ζ ′ i ′ ◦ π : G →
C, [using notation (18.10)℄(7)and for i = 0 and ≦ i ′ ≦ d , resp. ≦ i ≦ d and i ′ = 0 ,5(8) ( ̺ ,i ′ = r ,i ′ ◦ π def = a ◦ s G = (cid:0) a ◦ s T ( G ) (cid:1) ◦ π : G →
C, resp. ̺ i, = r i, ◦ π def = a ′ ◦ t G = (cid:0) a ′ ◦ t T ( G ) (cid:1) ◦ π : G →
C.Also, put z i,i ′ = ̺ i,i ′ ( g ) ∈ C. We laim that, as a onsequen e of properness,there exist open disks D i,i ′ ⊂ C entred at z i,i ′ su h that(9) \ ≦ i,i ′ ≦ d ̺ i,i ′ − ( D i,i ′ ) ⊂ Γ .Before we go into the proof of this laim, let us show how the statement that π (Γ) is a neighbourhood of π ( g ) follows from (9). Sin e, by Theorem 3, π is5For i = i ′ = 0 either hoi e will do; for d = 0 there are obvious modi(cid:28) ations whi h weleave to the reader. The only thing that really matters is that both a ◦ s and a ′ ◦ t shouldo ur in the interse tion (9) at least on e.20. RECONSTRUCTION THEOREMS 101surje tive as a mapping of G (1) into T ( G ) (1) , we have \ r i,i ′ − (cid:0) D i,i ′ (cid:1) = ππ − (cid:18)\ r i,i ′ − (cid:0) D i,i ′ (cid:1)(cid:19) = π (cid:18)\ π − r i,i ′ − (cid:0) D i,i ′ (cid:1)(cid:19) = π (cid:18)\ ̺ i,i ′ − (cid:0) D i,i ′ (cid:1)(cid:19) ⊂ π (Γ) . (by the in lusion (9))Now we are done, be ause g ∈ r i,i ′ − (cid:0) D i,i ′ (cid:1) and r i,i ′ ∈ C ( T ( G ) (1) ; C ) for all ≦ i, i ′ ≦ d .In order to prove our laim (9), let us onsider, for ea h ≦ i, i ′ ≦ d , ade reasing sequen e of open disks(10) · · · ⊂ D ℓ +1 i,i ′ ⊂ D ℓi,i ′ ⊂ · · · ⊂ D i,i ′ ⊂ C entred at z i,i ′ and whose radius δ ℓi,i ′ tends to zero. If we make the inno uousassumption δ i,i ′ = 1 then it will follow from our hypotheses on the fun tions a, a ′ that the sets Σ ℓ def = \ ≦ i,i ′ ≦ d r i,i ′ − (cid:16) D ℓi,i ′ (cid:17) − Γ (cid:0) ℓ = 1 , , . . . (cid:1) (11)are losed subsets of the ompa t spa e G ( K, K ′ ) , where K = supp a and K ′ = supp a ′ . The sets Σ ℓ form a de reasing sequen e. Their interse tion ∞ ∩ ℓ =1 Σ ℓ has to be empty be ause of the faithfulness of g ̺ ( g ) on G ( x , x ′ ) and our hypotheses on a , a ′ . Hen e, by ompa tness, there will be some ℓ su h that Σ ℓ = ∅ . This proves the laim, and therefore, the theorem. q.e.d.12 Note (The present remark will be used nowhere else and therefore itmay be skipped without onsequen es. You should read ŸŸ24(cid:21)25 (cid:28)rst, anyway.)Observe that whenever G and H are Morita equivalent Lie groupoids, oneof them is F -re(cid:29)exive if and only if the other is. Indeed, by naturality of theenvelope transformation π F ( - ) : Id → T F ( - ) , one gets a ommutative squareof topologi al groupoid homomorphisms G ϕ Morita eq. (cid:15) (cid:15) π ( G ) / / T ( G ) T ( ϕ ) (cid:15) (cid:15) H π ( H ) / / T ( H ) (13)in whi h both ϕ and T ( ϕ ) are fully faithful. It follows immediately that π ( G ) is fully faithful if and only if the same is true of π ( H ) . With a bit more work,it an be shown that π ( G ) is an open map if and only if π ( H ) is so (use thesimplifying assumption that ϕ (0) : G (0) → H (0) is a surje tive submersion).02 CHAPTER IV. GENERAL TANNAKA THEORYBy de(cid:28)nition, a Lie groupoid G is F -re(cid:29)exive if and only if one an solvetopologi ally the problem of re onstru ting G from its representations of type F (that is to say one an re over G up to isomorphism of topologi al groupoidsfrom su h representations). In the ase of Lie groups, a topologi al solutionprovides a ompletely satisfa tory answer be ause the smooth stru ture ofany Lie group is uniquely determined by the topology of the group itself.However, in the present more general ontext it is not evident a priori thatthe notion of re(cid:29)exivity we introdu ed above is as strong as to settle thesmoothness problem mentioned at the beginning of Ÿ18, think e.g. of G = M a smooth manifold. More pre isely, we onsider the following question: doesre(cid:29)exivity of G , in the foregoing purely topologi al sense, a tually imply thatthe fun tionally stru tured spa e ( T ( G ) (1) , R ∞ ) de(cid:28)ned in Ÿ18 is a smoothmanifold and the envelope map π (1) : G (1) → T ( G ) (1) a di(cid:27)eomorphism? Theanswer proves to be a(cid:30)rmative, as we shall now see.Let G be an arbitrary Lie groupoid. Choose an arrow g ∈ G ( x , x ′ ) and a representation R = ( E, ̺ ) of G (cid:28)rst of all. Then hoose an arbitrarymetri φ on E and global se tions ζ , . . . , ζ d , resp. ζ ′ , . . . , ζ ′ d , forming a lo al φ -orthonormal frame for E about x , resp. x ′ , as in the proof of Theorem 5.These data determine a smooth mapping(14) ̺ ζ ...,ζ d ζ ′ ,...,ζ ′ d : G (1) −→ M × M × M ( d ; C ) ,as follows: g (cid:0) s ( g ); t ( g ); ̺ , ( g ) , . . . , ̺ i,i ′ ( g ) , . . . , ̺ d,d ( g ) (cid:1) (the fun tions ̺ i,i ′ are those de(cid:28)ned in (7); M ( d ; C ) = End( C d ) is the spa eof d × d omplex matri es).If the envelope homomorphism π ( G ) : G → T ( G ) of the Lie groupoid G is faithful, it follows from Lemma 10.14 that for every point x of the basemanifold M of G there exists a representation ( E, ̺ ) ∈ Ob R ( G ) su h that Ker ̺ x is a dis rete subgroup of the isotropy group G x = G| x . Consequently,for an arbitrary arrow g ∈ G ( x , x ′ ) there will exist ( E, ̺ ) ∈ Ob R ( G ) su hthat the map G ( x , x ′ ) → Lis( E x , E x ′ ) , g ̺ ( g ) is inje tive on some openneighbourhood of g in G ( x , x ′ ) . Then the following lemma applies:15 Lemma Let G be a Lie groupoid. Fix an arrow g ∈ G ( x , x ′ ) andlet ( E, ̺ ) ∈ Ob R ( G ) be a representation. Suppose the map g ̺ ( g ) : G ( x , x ′ ) → Lis( E x , E x ′ ) is inje tive on some open neighbourhood of g in G ( x , x ′ ) .Then the smooth mapping ̺ ζζ ′ : G (1) → M × M × M ( d ; C ) (14) is animmersion at g , for any hoi e of a metri and of related orthonormalframes ζ = { ζ , . . . , ζ d } , ζ ′ = { ζ ′ , . . . , ζ ′ d } .Proof Let M be the base manifold of G . Fix open balls U, U ′ ⊂ M , entredat x , x ′ respe tively and so small that the se tions ζ , . . . , ζ d , resp. ζ ′ , . . . , ζ ′ d form a lo al orthonormal frame for E over U , resp. U ′ . Sin e the sour e map20. RECONSTRUCTION THEOREMS 103 s of G is a submersion, one an always hoose U also so small that there existsa lo al trivialization Γ ≈ U × B pr −→ U for s in a neighbourhood Γ of g in G (1) , where B is an open eu lidean ball. It is no loss of generality to assume t (Γ) ⊂ U ′ . Then we obtain, for the restri tion of the mapping ̺ ζζ ′ = ̺ ζ ...,ζ d ζ ′ ,...,ζ ′ d to Γ , a (cid:16) oordinate expression(cid:17) of the following form(16) U × B → U × U ′ × M ( d ; C ) , ( u, b ) (cid:0) u, u ′ ( u, b ) , ̺ ( u, b ) (cid:1) where ̺ ( g ) ∈ M ( d ; C ) denotes the matrix { ̺ i,i ′ ( g ) } ≦ i,i ′ ≦ d . The di(cid:27)erential ofthe mapping (16) at, let us say, g = ( x , reads Id ∗ D u ′ ( x , ∗ D ̺ ( x , (17)and it is therefore inje tive if and only if su h is the di(cid:27)erential of the partialmap b (cid:0) u ′ ( x , b ) , ̺ ( x , b ) (cid:1) : B → U ′ × M ( d ; C ) at the origin of B .We are now redu ed to showing that the restri tion ̺ ζζ ′ : G ( x , - ) −→ M × GL ( d ) = { x } × M × GL ( d ; C ) is an immersion at g . Let G = G| x be the isotropy group at x and hoose,in the vi inity of g , a lo al (equivariant) trivialization G ( x , S ) ≈ S × G for the prin ipal G -bundle t x : G ( x , - ) → G x ; we an assume that S is asubmanifold of U ′ and that in this lo al hart g = ( x ′ , e ) , where e standsfor the neutral element of G . We then obtain a new oordinate expressionfor the restri tion of ̺ ζζ ′ to G ( x , - ) , namely(18) S × G → U ′ × GL ( d ; C ) , ( s, g ) (cid:0) s, ̺ ( s, g ) (cid:1) .Sin e its (cid:28)rst omponent is the in lusion of a submanifold, this map will bean immersion at g = ( x ′ , e ) provided the partial map g ̺ ( x ′ , g ) is animmersion at e . The latter orresponds to the diagonal of the square G g - ≈ (cid:15) (cid:15) ̺ / / Aut( E x ) ρ ( g ) - ≈ (cid:15) (cid:15) G ( x , x ′ ) ̺ / / Lis( E x , E x ′ ) ,so our problem redu es to proving that the homomorphism ̺ : G → GL ( E x ) is immersive. By hypothesis, this is inje tive in an open neighbourhood of e and hen e our laim follows at on e. q.e.d.We are now ready to establish our previous laims about the fun tionalstru ture R ∞ on the Tannakian groupoid T ( G ) . Let G be any F -re(cid:29)exive Liegroupoid ( F an arbitrary sta k of smooth (cid:28)elds, as ever).04 CHAPTER IV. GENERAL TANNAKA THEORYFix an arrow λ ∈ T ( G ) (1) . Our (cid:28)rst task will be to (cid:28)nd some openneighbourhood Ω of λ su h that (Ω , R ∞ Ω ) turns out to be isomorphi , as afun tionally stru tured spa e, to a smooth manifold ( X, C ∞ X ) . Sin e we areworking under the hypothesis that G is re(cid:29)exive, there is a unique g ∈ G (1) su h that λ = π ( g ) . By Lemma 15 and the omments pre eding it, we an(cid:28)nd, for a onveniently hosen ( E, ̺ ) ∈ Ob R ( G ) , an open neighbourhood Γ of g in G (1) su h that the smooth map ̺ ζζ ′ : G (1) → M × M × M ( d ; C ) (14)indu es a di(cid:27)eomorphism of Γ onto a submanifold X ⊂ M × M × M ( d ; C ) .Noti e that the same data whi h determine the map (14) also determine amap of fun tionally stru tured spa es(19) r ζζ ′ = r ζ ...,ζ d ζ ′ ,...,ζ ′ d : T ( G ) (1) −→ M × M × M ( d ; C ) , λ (cid:0) s ( λ ); t ( λ ); { r i,i ′ ( λ ) } ≦ i,i ′ ≦ d (cid:1) ,where we put r i,i ′ = r R,φ,ζ i ,ζ ′ i ′ ∈ R (18.11). From the re(cid:29)exivity of G again, itfollows that the envelope map π indu es a homeomorphism between Γ andthe open subset Ω def = π (Γ) of T ( G ) (1) . The following diagram Γ π | Γ ≈ homeo & & NNNNNNNNNNNNNN ̺ ζζ ′ | Γ ≈ di(cid:27)eo / / X ⊂ M × M × M ( d ; C )Ω r ζζ ′ | Ω nnnnnnnnnnnnnn (20)is learly ommutative. We ontend that the map r ζζ ′ | Ω provides the desiredisomorphism of fun tionally stru tured spa es. Expli itly, this means that anarbitrary fun tion f : X ′ → C belongs to C ∞ ( X ′ ) if and only if its pullba k h = f ◦ r ζζ ′ belongs to R ∞ (Ω ′ ) , for ea h (cid:28)xed pair of orresponding opensubsets Ω ′ ⊂ Ω , X ′ ⊂ X . Note that sin e the problem is lo al, we an makethe simplifying assumption Ω ′ = Ω , X ′ = X . Thus, suppose f ∈ C ∞ ( X ) (cid:28)rst;be ause of the lo al hara ter of the problem again, it is not restri tive toassume that f admits a smooth extension ˜ f ∈ C ∞ (cid:0) M × M × M ( d ) (cid:1) . Then h oin ides with the restri tion to Ω of a global fun tion ˜ h = ˜ f ◦ r ζζ ′ : T (1) = T ( G ) (1) → C belonging to R ∞ ( T (1) ) be ause (19) is a map of fun tionallystru tured spa es. Conversely, suppose h = f ◦ r ζζ ′ ∈ R ∞ (Ω) . We know,from Example 19.3, that the envelope map π is a morphism of fun tionallystru tured spa es. Hen e the omposite h ◦ π will belong to C ∞ (Γ) . Sin e h ◦ π = f ◦ r ζζ ′ ◦ π = f ◦ ̺ ζζ ′ and ̺ ζζ ′ | Γ is a di(cid:27)eomorphism of Γ onto X , itfollows that f ∈ C ∞ ( X ) , as ontended.We have therefore proved that if a Lie groupoid G is F -re(cid:29)exive then thespa e ( T F ( G ) (1) , R ∞ ) is a tually a (Hausdor(cid:27)) smooth manifold. There islittle work left to be done by now:21 Proposition Let F be an arbitrary sta k of smooth (cid:28)elds and let G be a Lie groupoid. Suppose G is F -re(cid:29)exive.20. RECONSTRUCTION THEOREMS 105Then the Tannakian groupoid T F ( G ) , endowed with its anoni alfun tional stru ture R ∞ , turns out to be a Lie groupoid; moreover, the F -envelope homomorphism(22) π F ( G ) : G ≈ −−→ T F ( G ) turns out to be an isomorphism of Lie groupoids.Proof We know from the foregoing dis ussion that ( T (1) , R ∞ ) is a smoothmanifold. Then all we have to show now, learly, is that the envelope map π : G (1) → T (1) is a di(cid:27)eomorphism. Equivalently, we have to show that π is an isomorphism of fun tionally stru tured spa es between ( G (1) , C ∞G (1) ) and ( T (1) , R ∞ ) . This follows immediately, lo ally, from the ommutativity of thetriangles (20) and the previously established fa t that both ̺ ζζ ′ | Γ and r ζζ ′ | Ω are fun tionally stru tured spa e isomorphisms onto ( X, C ∞ X ) . q.e.d.Let us pause for a moment to summarize our urrent knowledge of the F -envelope π F ( G ) : G → T F ( G ) of an arbitrary proper Lie groupoid G . Firstof all, we know that π ( G ) is faithful (Thm. 3). We have also as ertainedthat T ( G ) is a topologi al groupoid (Ex. 19.3 and Prop. 19.7). Moreover, ithas been established that π ( G ) is ne essarily an isomorphism of topologi algroupoids in ase π ( G ) is faithful (Thm. 5); whenever this happens to betrue, one an ompletely solve the re onstru tion problem for G (Prop. 21).Now observe that faithfulness of π ( G ) is equivalent to the following property:if g = u ( x ) in the isotropy group G| x then there exists a representation ( E, ̺ ) ∈ Ob R ( G ) su h that ̺ ( g ) = id ∈ Aut( E x ) . We an therefore on ludeby saying that an arbitrary proper Lie groupoid an be re overed from itsrepresentations of type F if and only if su h representations are (cid:19) enough (cid:20)in the sense of the above-mentioned property.The (cid:28)nal part of the present se tion will be devoted to showing that anyproper Lie groupoid admits enough representations of type E ∞ (= smoothEu lidean (cid:28)elds, fr. Ÿ16). By the foregoing remarks, this will immediatelyimply the general re onstru tion theorem we were striving for. Re all thatour approa h via smooth Eu lidean (cid:28)elds is motivated by the impossibilityto obtain that result by using representations of type V ∞ (smooth ve torbundles), as illustrated by the examples dis ussed in Ÿ2.We begin with some preliminary remarks of a purely topologi al nature.Let G be a proper Lie groupoid and let M denote the base manifold of G .Re all that a subset S ⊂ M is said to be invariant when s ∈ S implies g · s ∈ S for all arrows g ∈ G (1) . If S is an arbitrary(cid:22)viz., not ne essarilyinvariant(cid:22)subset of M , we let G · S denote the saturation of S , that is tosay the smallest invariant subset of M ontaining S , so that S is invariantif and only if G · S = S ; note that the saturation of an open subset is also06 CHAPTER IV. GENERAL TANNAKA THEORYopen. Now let V be any open subset with ompa t losure: we ontend that G · V = G · V . The dire tion ` ⊂ ' of this equality is valid even for a non-properLie groupoid; it follows for instan e from the existen e of lo al bise tions.To he k the opposite in lusion, one an resort to the well-known fa t thatthe orbit spa e6 of a proper Lie groupoid is Hausdor(cid:27) and then use the ompa tness of V ; in detail: sin e the image of the ompa t set V under the ontinuous mapping o : M → M/ G is a ompa t and hen e losed subsetof the Hausdor(cid:27) spa e M/ G , the inverse image G · V = o − (cid:0) o (cid:0) V (cid:1)(cid:1) mustbe losed as well. Next, let U be an invariant open subset of M . From theequality we have just proved, it follows immediately that U oin ides withthe union of all its open invariant subsets V , V ⊂ U . Indeed, sin e anygiven point u ∈ U admits an open neighbourhood W with ompa t losure ontained in U , one has u ∈ G · W = V ⊂ V = G · W = G · W ⊂ G · U = U .The latter remark applies to the onstru tion of G -invariant partitions ofunity on M ; for our purposes it will be enough to illustrate a spe ial aseof this onstru tion. Consider an arbitrary point x ∈ M and let U be anopen invariant neighbourhood of x . Choose another open neighbourhood V of x , invariant and with losure ontained in U . The orbit G · x and theset-theoreti omplement ∁ V are invariant disjoint losed subsets of M , soCorollary 17.20 provides us with an invariant fun tion Φ ∈ C ∞ ( M ; R ) su hthat Φ( x ) = 1 and Φ = 0 outside V .We are now ready to establish a basi extension property enjoyed by therepresentations of type E ∞ of proper Lie groupoids; our (cid:19) main theorem (cid:20)below will be essentially a onsequen e of this property and of Zung's resultson lo al linearizability. Our goal will be a hieved by means of an obvious ut-o(cid:27) te hnique whi h is of ourse not available when one limits oneself torepresentations on ve tor bundles.Sin e throughout the subsequent dis ussion the type F = E ∞ is (cid:28)xed, weagree to systemati ally suppress any referen e to type. Let G be an arbitraryproper Lie groupoid and let M denote its base as usual. Let U ⊂ M be a G -invariant open neighbourhood of a point x ∈ M , and suppose we aregiven a partial representation ( E U , ̺ U ) ∈ R ( G| U ) . We know from Ÿ17 thatthere is an indu ed Lie group representation(23) ̺ U, : G −→ GL ( E U, ) of the isotropy Lie group G = G| x on the ve tor spa e E U, = ( E U ) x . We ontend that one an onstru t a global representation ( E , ̺ ) ∈ R ( G ) forwhi h it is possible to exhibit an isomorphism of G -spa es E def = E x ≈ E U, .6The quotient of M asso iated with the equivalen e x ∼ g · x . We will indi ate by o the map (of M into this quotient) whi h sends x to its equivalen e lass.20. RECONSTRUCTION THEOREMS 107(The G -spa e stru ture on E omes from the indu ed representation(24) ̺ : G −→ GL ( E ) ,that on E U, from (23).)To begin with, let us (cid:28)x any invariant smooth fun tion a ∈ C ∞ ( M ) with a ( x ) = 1 and supp a ⊂ U ; su h fun tions always exist(cid:22)as we saw before(cid:22)inview of the properness of G . Let V ⊂ M denote the open subset onsistingof all x su h that a ( x ) = 0 . One an de(cid:28)ne the following bundle { E x } ofEu lidean spa es over M :(25) E x = ( E U,x if x ∈ V { } otherwise.Let Γ E be the smallest sheaf of se tions of the bundle { E x } whi h ontainsthe following presheaf(26) W (cid:8) aζ : ζ ∈ Γ ( E U )( U ∩ W ) (cid:9) .(Here of ourse aζ is to be understood as the appropriate (cid:16)prolongation byzero(cid:17) of the indi ated se tion; note that sin e M admits partitions of unity(25) a tually equals Γ E .) One veri(cid:28)es immediately that these data de(cid:28)ne asmooth Eu lidean (cid:28)eld E over M . Next, introdu e ̺ by putting(27) ̺ ( g ) = ( ̺ U ( g ) for g ∈ G| V otherwise.This law must be understood as des ribing a bundle (cid:8) ̺ ( g ) : ( s ∗ E ) g ∼ → ( t ∗ E ) g (cid:9) of linear isomorphisms indexed over the manifold G . The ompati-bility of this family of maps with the omposition of arrows, amounting tothe equalities ̺ ( g ′ g ) = ̺ ( g ′ ) ̺ ( g ) and ̺ ( x ) = id , is lear. Now, ̺ will be ana tion of G on E provided it is a morphism s ∗ E → t ∗ E of Eu lidean (cid:28)eldsover G : this is obvious, be ause for suitable fun tions r i ∈ C ∞ one has ̺ ( g ) aζ ( s g ) = a ( s g ) ̺ ( g ) ζ ( s g ) = a ( t g ) ℓ P i =1 r i ( g ) ζ ′ i ( t g ) = ℓ P i =1 r i ( g ) aζ ′ i ( t g ) ,in view of the G -invarian e of a . Hen e ( E , ̺ ) ∈ R ( G ) . Finally, the identity E = E x def = E U,x = E U, provides the required G -equivariant isomorphism.28 Theorem (General Re onstru tion Theorem, Main Theorem) Ea hproper Lie groupoid is E ∞ -re(cid:29)exive.Proof Let G be any su h groupoid and (cid:28)x a point x of its base manifold M . We need to show the existen e of a Eu lidean representation ( E , ̺ ) ∈ Ob R ( G ) indu ing a faithful isotropy representation ̺ : G ֒ → GL ( E ) (24)08 CHAPTER IV. GENERAL TANNAKA THEORY(we freely use the notation above). By the previously established extensionproperty of Eu lidean representations, it will be enough to (cid:28)nd a partial rep-resentation ( E U , ̺ U ) ∈ Ob R ( G| U ) de(cid:28)ned over some invariant open neigh-bourhood U of x and with ̺ U, : G ֒ → GL ( E U, ) (23) inje tive.It was observed in Ÿ4 that Zung's Lo al Linearizability Theorem yieldsthe existen e of (a) a smooth representation G → GL ( V ) on some (real)(cid:28)nite dimensional ve tor spa e (b) an embedding of manifolds V i ֒ → M su hthat x and su h that U def = G · i ( V ) is an open subset of M ( ) a Moritaequivalen e G ⋉ V ι −→ G| U indu ing V i ֒ → U at the level of base manifolds.Note that the isotropy of G ⋉ V at equals G and that the equivalen e ι indu es an automorphism ι ∈ Aut( G ) (whi h an be assumed to be theidentity, just to (cid:28)x ideas).Now let Φ : G ֒ → GL ( E ) be any faithful representation on a (cid:28)nitedimensional omplex ve tor spa e. One has an indu ed faithful representation e Φ of G ⋉ V on V × E ( fr. the end of Ÿ28). By the theory of Ÿ14, thereexists some representation ( E U , ̺ U ) ∈ Ob R ( G| U ) su h that ι ∗ ( E U , ̺ U ) ≈ ( V × E , e Φ) ; this is pre isely the one we are looking for, be ause ̺ U, : G ֒ → GL ( E U, ) ≈ GL ( E ) must oin ide with Φ . q.e.d.hapter VClassi al Fibre Fun torsIn the present hapter we will again o upy ourselves with the study of theabstra t notion of (cid:28)bre fun tor. However, we shall be ex lusively interestedin (cid:28)bre fun tors whi h take values in the ategory of smooth ve tor bundlesover a manifold, in other words (cid:28)bre fun tors of the form ω : C → V ∞ ( M ) or,equivalently, of type V ∞ . Moreover, sin e in all examples of su h fun tors wehave in mind the tensor ategory C invariably turns out to be rigid, we shallmake the assumption that C is rigid even though this is not indispensable;note that in this ase End ⊗ ( ω ) = Aut ⊗ ( ω ) ie λ tensor preserving implies λ invertible, see, for instan e, [12℄ Prop. 1.13. We shall use the adje tive` lassi al' to refer to (cid:28)bre fun tors of this sort.Se tion 21 is devoted to the study of some general properties of lassi al(cid:28)bre fun tors. To start with, the Tannakian groupoid T ( ω ) asso iated witha lassi al (cid:28)bre fun tor ω proves to be a C ∞ -stru tured groupoid, that isto say all the stru ture maps of T ( ω ) turn out to be morphisms of fun -tionally stru tured spa es; ompare Ÿ18. This allows us to introdu e the ategory R ∞ ( T ( ω )) of C ∞ -representations of the C ∞ -stru tured groupoid T ( ω ) , along with an (cid:16)evaluation(cid:17) fun tor ev : C −→ R ∞ ( T ( ω )) .The latter is in fa t a tensor fun tor, by whi h the ategory C is put inrelation to R ∞ ( T ( ω )) ; we shall say more about this fun tor in Ÿ26. Finally,we observe that a lassi al (cid:28)bre fun tor ω whi h admits enough ω -invariantmetri s (in the sense of De(cid:28)nition 19.2) is proper(cid:22)in other words, so is the orresponding map ( s , t ) : T ( ω ) → M × M .Se tion 22 deals with the te hni al notion of tame submanifold, and ispreliminary to ŸŸ23(cid:21)25. However, in order to read the latter se tions a thor-ough understanding of Ÿ22 is not really ne essary: it is a tually enough toknow what tame submanifolds are and the statements of Propositions 22.5,22.11; one may skip what remains of Ÿ22 at (cid:28)rst reading.Se tion 23 provides, for the Tannakian groupoid T ( ω ) asso iated with a lassi al (cid:28)bre fun tor ω : C → V ∞ ( M ) , an alternative hara terization of the10910 CHAPTER V. CLASSICAL FIBRE FUNCTORSproperty of smoothness in terms of what we all representative harts. Su h harts arise from the obje ts of the ategory C , and their de(cid:28)nition involvestame submanifolds of linear groupoids GL ( E ) over the manifold M .Se tions 24(cid:21)25 are devoted to morphisms of (cid:28)bre fun tors. For ea hmorphism between two lassi al (cid:28)bre fun tors there exists a orrespondinghomomorphism between the asso iated Tannakian groupoids, whi h turnsout to be (cid:16)smooth(cid:17) ie a homomorphism of C ∞ -stru tured groupoids. In Ÿ25we introdu e, as a spe ial ase, the notion of weak equivalen e; the alterna-tive hara terization of smoothness provided in Ÿ23 is here put to work toshow that the property of smoothness is, for lassi al (cid:28)bre fun tors, invari-ant under weak equivalen e. Finally, the homomorphism asso iated with aweak equivalen e of smooth lassi al (cid:28)bre fun tors is proved to be a Moritaequivalen e.Ÿ21 Basi De(cid:28)nitions and PropertiesIn this se tion we study general properties of lassi al (cid:28)bre fun tors. Let usbegin by giving a pre ise de(cid:28)nition:1 De(cid:28)nition We shall all a (cid:28)bre fun tor ω : C → F ( M ) lassi al if itmeets the following requirements:i) the domain tensor ategory C is rigid;ii) for every R ∈ Ob( C ) , ω ( R ) is a lo ally trivial obje t of F ( M ) .Observe that sin e the type F is a sta k of smooth (cid:28)elds, ω ( R ) in ii) willa tually belong to Ob V F ( M ) ie it will be a lo ally trivial obje t of F ( M ) of lo ally (cid:28)nite rank ( fr Ÿ11). Sin e V F ( M ) is equivalent to the ategory V ∞ ( M ) of smooth ve tor bundles of lo ally (cid:28)nite rank over M (re all thatthe base M is always para ompa t), it follows that the theory of lassi al (cid:28)brefun tors essentially redu es to just one type F = V ∞ . Be ause of this, for therest of the present hapter(cid:22)a tually, for the rest of the present work(cid:22)weshall omit any referen e to type. So, for instan e, we will write V ∞ ( M ) or V ∞ ( M ) at all pla es where we would otherwise write F ( M ) .The pivotal fa t of lassi al (cid:28)bre fun tor theory is that for su h (cid:28)brefun tors one has lo al formulas analogous to (17.5). Namely, let ω : C → V ∞ ( M ) be a lassi al (cid:28)bre fun tor. Let an obje t R ∈ Ob( C ) and an arrow λ ∈ T ≡ T ( ω ) (1) be given. Choose, on E ≡ ω ( R ) , an arbitrary Hilbertmetri φ , whose existen e is guaranteed by the para ompa tness of M . Bythe lo al triviality assumption on E , it will be possible to (cid:28)nd a lo al φ -orthonormal frame ζ ′ , . . . , ζ d ′ ∈ Γ E ( U ′ ) about x ′ ≡ t ( λ ) su h that E u ′ =Span (cid:8) ζ ′ ( u ′ ) , . . . , ζ d ′ ( u ′ ) (cid:9) for all u ′ ∈ U ′ . (Note that here one really needslo al triviality of E within F , in the sense of Ÿ11, and not just the hypothesisthat Γ E is lo ally free as a sheaf of modules over M .) Then for any given21. BASIC DEFINITIONS AND PROPERTIES 111lo al se tion ζ ∈ Γ E ( U ) , de(cid:28)ned in a neighbourhood U of x ≡ s ( λ ) , onegets, by letting Ω ≡ s − ( U ) ∩ t − ( U ′ ) ⊂ T , λ ( R ) · ζ ( s λ ) = d P i ′ =1 r R,φ,ζ,ζ i ′′ ( λ ) ζ i ′ ′ ( t λ ) , ( ∀ λ ∈ Ω) (2)where r R,φ,ζ,ζ i ′′ ∈ R ∞ (Ω) denotes(cid:22)as in Ÿ18(cid:22)the representative fun tion λ (cid:10) λ ( R ) · ζ ( s λ ) , ζ i ′ ′ ( t λ ) (cid:11) φ .We shall immediately put this basi remark to work in the proof of thefollowing3 Proposition For every lassi al (cid:28)bre fun tor ω : C → V ∞ ( M ) , theTannakian groupoid T ( ω ) is a C ∞ -stru tured groupoid (with respe t tothe standard C ∞ -stru ture R ∞ de(cid:28)ned in Ÿ18). T ( ω ) is, in parti ular, atopologi al groupoid for every lassi al ω .Proof Let us take an arbitrary representative fun tion r = r R,φ,ζ,ζ ′ : T →
Con the spa e
T ≡ T ( ω ) (1) , as in (18.10). We shall regard r as (cid:28)xed throughoutthe entire proof.To begin with, we onsider the omposition map T (2) = T s × t T c −→ T . Ourgoal is to show that the fun tion r ◦ c is a global se tion of the sheaf R (2) , ∞ ≡ ( R ∞ ⊗ R ∞ ) ∞ | T (2) . (Review, if ne essary, the dis ussion about fun tionallystru tured groupoids in Ÿ18.) Fix any pair of omposable arrows ( λ ′ , λ ) ∈T (2) . There will be some φ -orthonormal frame ζ ′ , . . . , ζ d ′ ∈ Γ ( ω R )( U ′ ) about x ′ ≡ t ( λ ) , su h that Eq. (2) above holds for all λ ∈ Ω ′ ≡ t − ( U ′ ) . Then,for every pair ( λ ′ , λ ) belonging to the open subset Ω ′′ ≡ s − ( U ′ ) s × t t − ( U ′ ) ⊂T (2) , one gets the identity ( r ◦ c )( λ ′ , λ ) = r ( λ ′ ◦ λ ) = (cid:10) λ ′ ( R ) · λ ( R ) · ζ ( s λ ) , ζ ′ ( t λ ′ ) (cid:11) φ = d P i ′ =1 r R,φ,ζ,ζ i ′′ ( λ ) r R,φ,ζ i ′ ′ ,ζ ′ ( λ ′ ) , by (2)whi h expresses ( r ◦ c ) | Ω ′′ in the desired form, namely as an element of R (2) , ∞ (Ω ′′ ) .Next, onsider the inverse map T i −→ T . Fix any λ ∈ T . In a neighbour-hood U of x = s ( λ ) it will be possible to (cid:28)nd a trivializing φ -orthonormalframe ζ , . . . , ζ d ∈ Γ ( ω R )( U ) . One an write down (2) for ea h ζ i ( i =1 , . . . , d ): λ ( R ) · ζ i ( s λ ) = d P i ′ =1 r R,φ,ζ i ,ζ ′ i ′ ( λ ) ζ ′ i ′ ( t λ ) . ( λ ∈ Ω = s − ( U ) ∩ t − ( U ′ ) . ) (4)Letting { r ′ i ′ ,i ( λ ) : 1 ≦ i ′ , i ≦ d } denote the inverse of the matrix { r R,φ,ζ i ,ζ ′ i ′ ( λ ) :1 ≦ i ′ , i ≦ d } for ea h λ (this makes sense be ause λ ( R ) is a linear iso), wesee from the standard formula involving the inverse of the determinant that12 CHAPTER V. CLASSICAL FIBRE FUNCTORS r ′ i ′ ,i ∈ R ∞ (Ω) for all ≦ i ′ , i ≦ d . If we now put a ′ i ′ = h ζ , ζ ′ i ′ i φ ∈ C ∞ ( U ′ ) forall i ′ = 1 , . . . , d and a i = h ζ i , ζ ′ i φ ∈ C ∞ ( U ) for all i = 1 , . . . , d , we obtain thefollowing expression for ( r ◦ i ) | Ω ( r ◦ i )( λ ) = r ( λ − ) = (cid:10) λ ( R ) − · ζ ( t λ ) , ζ ′ ( s λ ) (cid:11) φ == d P i ′ =1 a ′ i ′ ( t λ ) (cid:10) λ ( R ) − · ζ ′ i ′ ( t λ ) , ζ ′ ( s λ ) (cid:11) φ == d P i ′ =1 d P i =1 a ′ i ′ ( t λ ) r ′ i ′ ,i ( λ ) a i ( s λ ) ,whi h learly shows membership of ( r ◦ i ) | Ω in R ∞ (Ω) .The (cid:16)smoothness(cid:17) of the remaining stru ture maps was already proved inŸ18 for an arbitrary (cid:28)bre fun tor. q.e.d.By exploiting the ategori al equivalen e V ( M ) ≈ −→ V ∞ ( M ) , E ˜ E (12.6), one an make sense of the expression GL ( E ) for every E ∈ Ob V ( M ) simply by regarding GL ( E ) as short for GL ( ˜ E ) . If ω : C → V ( M ) is a lassi al(cid:28)bre fun tor, ea h obje t R ∈ Ob( C ) will determine a homomorphism offun tionally stru tured groupoids(5) ev R : T ( ω ) −→ GL ( ω R ) , λ λ ( R ) (note that if φ is any Hilbert metri on E = ω ( R ) , the fun tions q φ,ζ,ζ ′ : GL ( E ) (1) → C, µ (cid:10) µ · ζ ( s µ ) , ζ ′ ( t µ ) (cid:11) φ will provide suitable lo al oordinatesystems for the manifold GL ( E ) (1) ), whi h may be thought of as a (cid:16)smooth(cid:17)representation of T ( ω ) .It is worthwhile mentioning the following universal property, whi h har-a terizes the fun tional stru ture (and topology) we endowed the Tannakiangroupoid with. Let ω be a lassi al (cid:28)bre fun tor. Then for any fun tionallystru tured spa e ( Z, F ) , a mapping f : Z → T = T ( ω ) (1) is a morph-ism of ( Z, F ) into ( T , R ∞ ) (or simply, a ontinuous mapping of Z into T )if and only if su h is ev R ◦ f for every R ∈ Ob C . The `only if' dire tionis lear be ause of the foregoing remarks about the (cid:16)smoothness(cid:17) of ev R .Conversely, onsider any representative fun tion r = r R,φ,ζ,ζ ′ : T →
C; if q φ,ζ,ζ ′ : GL ( ω R ) (1) → C is the smooth fun tion de(cid:28)ned above then one has r ◦ f = q φ,ζ,ζ ′ ◦ ev R ◦ f ∈ F ( Z ) , be ause by assumption ev R ◦ f is a morph-ism of ( Z, F ) into the smooth manifold GL ( ω R ) (1) . The equivalen e is nowproven.In a manner entirely analogous to Ÿ2, one an de(cid:28)ne the omplex tensor ategory R ∞ ( T ( ω ); C ) of all (cid:16)smooth(cid:17) representations of the fun tionallystru tured groupoid T ( ω ) on smooth omplex ve tor bundles over the basemanifold M of ω . Pre isely, any su h representation will onsist of a omplexve tor bundle E ∈ Ob V ∞ ( M ) and a homomorphism ̺ : T ( ω ) → GL ( E ) of21. BASIC DEFINITIONS AND PROPERTIES 113fun tionally stru tured groupoids over M ( ̺ identi al on M ). Then one hasthe omplex tensor fun tor(6) ev : C −→ R ∞ ( T ( ω ); C ) , R ( ] ω ( R ) , ev R ) (the so- alled (cid:16)evaluation fun tor(cid:17)). The parallel with the situation depi tedin Ÿ9 leads us to formulate the problem of determining whether or not thefun tor (6) is in general(cid:22)for an arbitrary lassi al (cid:28)bre fun tor(cid:22)a ategori alequivalen e. The answer is known to be yes, a tually in the strong form of anisomorphism of ategories, for a large lass of examples: see Ÿ26, Proposition(26.21) and related omments.We on lude this introdu tory se tion with an observation about proper lassi al (cid:28)bre fun tors ( fr. Ÿ19). We intend to show that, in the lassi al ase, existen e of enough invariant metri s is su(cid:30) ient to ensure propernessand hen e that the (cid:28)rst ondition of De(cid:28)nition 19.2 is a tually redundant forany lassi al (cid:28)bre fun tor.Noti e (cid:28)rst of all that ea h Hilbert metri φ on a omplex ve tor bundle E ∈ Ob V ∞ ( M ) determines a subgroupoid U ( E, φ ) ⊂ GL ( E ) , onsisting ofall φ -unitary linear isomorphisms between the (cid:28)bres of E ; more expli itly,the arrows x → x ′ in U ( E, φ ) are the unitary isomorphisms of ( E x , φ x ) onto ( E x ′ , φ x ′ ) . Clearly, U ( E, φ ) is a proper Lie groupoid over the manifold M ,embedded into GL ( E ) . When there is no danger of ambiguity about themetri , we will just suppress φ from the notation.From our elementary remark (19.6) it follows that for any ω -invariantHilbert metri φ on ω ( R ) ( R ∈ Ob C ) the evaluation homomorphism ev R (5)must fa tor through the subgroupoid U ( ω R ) ֒ → GL ( ω R ) . Hen e one mayview ev R as a (cid:16)smooth(cid:17) homomorphism(7) ev R : T ( ω ) −→ U ( ω R ) , λ λ ( R ) .8 Proposition Let ω : C → V ∞ ( M ) be a lassi al (cid:28)bre fun tor.Suppose there are enough ω -invariant metri s ( fr Ÿ19, De(cid:28)nition 2). Then ω is proper; in parti ular, T ( ω ) is a proper groupoid.Proof Let us assign, to ea h obje t R ∈ Ob C , an ω -invariant metri φ R on ω ( R ) on e and for all. We shall simply write U ( ω R ) in pla e of U ( ω ( R ) , φ R ) .Let K be an arbitrary ompa t subset of the base manifold M . We haveto show that T | K = ( s , t ) − ( K × K ) is a ompa t subset of the topologi alspa e T = T ( ω ) (1) . Consider the auxiliary spa e(9) Z K def = Y R ∈ Ob C U ( ω R ) | K (produ t of topologi al spa es) and observe that Z K is ompa t be ause thesame is true of ea h fa tor U ( ω R ) | K . There is an obvious ontinuous inje tive14 CHAPTER V. CLASSICAL FIBRE FUNCTORSmap e : T | K ֒ → Z K given by λ
7→ { λ ( R ) } R ∈ Ob C . We laim that this map isa tually a topologi al embedding of T | K onto a losed subset of Z K : this willentail the required ompa tness of T | K .The map e is an embedding. This will be implied at on e by the followingextension property of representative fun tions: for every r = r R,φ,ζ,ζ ′ ∈ R (18.11), there exists a ontinuous fun tion h : Z K → C su h that r = h ◦ e on T | K . In order to obtain su h an extension of r , note simply that on T | K one has r R,φ,ζ,ζ ′ = ( q φ,ζ,ζ ′ ◦ π R ) ◦ e , where π R : Z K → U ( ω R ) | K is the R -thproje tion and q φ,ζ,ζ ′ is the (restri tion to U ( ω R ) | K of) the smooth fun tion GL ( ω R ) → C, µ (cid:10) µ · ζ ( s µ ) , ζ ′ ( t µ ) (cid:11) φ .The image of e is a losed subset of Z K . It is su(cid:30) ient to observe thatthe onditions expressing membership of µ = { µ R } R ∈ Ob C ∈ Q U ( ω R ) | K inthe image of e (cid:22)namely that s ( µ R ) = s ( µ S ) and t ( µ R ) = t ( µ S ) ∀ R, S ∈ Ob C , naturality of µ and its being tensor preserving and self- onjugate(cid:22)areea h stated in terms of a huge number of identities whi h involve only the oordinates µ R = π R ( µ ) in a ontinuous way. q.e.d.10 Note A very marginal omment about proper lassi al (cid:28)bre fun tors,improving, in the lassi al ase, Lemma 19.5: for any proper lassi al ω , theequality R = R ′ holds. In order to see this, noti e (cid:28)rst of all that if U isany open subset of M on whi h E | U ( E = ω ( R ) ) trivializes then we an(cid:28)nd a ∈ Aut( E | U ) su h that φ u ( v, v ′ ) = φ R,u ( v, a u · v ′ ) for all u ∈ U ( φ an arbitrary metri on E , φ R as in the proof of the pre eding proposition, v, v ′ ∈ E u ). Now, if we put ξ ′ U = a ( U ) ζ ′ U where ζ ′ U is the restri tion to U of ζ ′ , we get r R,φ,ζ,ζ ′ = r R,φ R ,ζ,ξ ′ U on t − ( U ) ⊂ T . We an use a partition ofunity over all su h U 's to obtain a global se tion ξ ′ with the property that r R,φ,ζ,ζ ′ = r R,φ R ,ζ,ξ ′ ∈ R ′ .Ÿ22 Tame Submanifolds of a Lie GroupoidLet G be a Lie groupoid over a manifold M .1 De(cid:28)nition A submanifold Σ of the manifold of arrows G (1) will be saidto be prin ipal if it an be overed with lo al parametrizations (viz inversesof lo al harts or, equivalently, open embeddings) of the form(2) ( Z × A ֒ → Σ( z, a ) τ ( z ) · η ( a ) ,where Z is a submanifold of M , τ : Z → G ( x, - ) is, for some point x ∈ M ,a smooth se tion to the target map of the groupoid, η : H ֒ → G x is a Liesubgroup of the x -th isotropy group G x of G and A is an open subset of H su h that η restri ts to an embedding of A into G x .22. TAME SUBMANIFOLDS OF A LIE GROUPOID 115Note that the image Σ = τ ( Z ) · η ( A ) of a map of the form (2) is alwaysa submanifold of G (1) and that the same map indu es a smooth isomorphismof Z × A onto Σ . So, in parti ular, it makes sense to use su h maps as lo alparametrizations. (Details an be found in Note 6 below.)Note also that any prin ipal submanifold of G (1) admits an open overby lo al parametrizations of type (2) with the additional property that theLie group H is onne ted and A is an open neighbourhood in H of theneutral element e . (Indeed, let σ ∈ Σ be a given point and hoose a lo alparametrization τ · η of the form (2). Suppose σ = ( z, a ) ∈ Z × A in thislo al hart. Repla ing A with a − A and τ with τ · η ( a ) a omplishes theredu tion to the situation where A is a neighbourhood of e and σ = ( z, e ) ;interse ting with the onne ted omponent of e in H (cid:28)nishes the job.)3 Lemma Let ϕ : G → G ′ be a Lie groupoid homomorphism, indu ingan immersion f : M → M ′ at the level of manifolds of obje ts. Assumethat Σ and Σ ′ are prin ipal submanifolds of G and G ′ respe tively, withthe property that ϕ maps Σ inje tively into Σ ′ .Then ϕ restri ts to an immersion of Σ into Σ ′ .Proof Fix any point σ ∈ Σ and let x ≡ s ( σ ) , z ≡ t ( σ ) . Choose lo alparametrizations τ · η : Z × A ֒ → Σ and τ ′ · η ′ : Z ′ × A ′ ֒ → Σ ′ of type (2)with, let us say, σ = ( z , e ) ∈ Z × A and ϕ ( σ ) = ( f ( z ) , e ′ ) ∈ Z ′ × A ′ ,where e , resp. e ′ is the neutral element of the Lie subgroup η : H ֒ → G x ,resp. η ′ : H ′ ֒ → G ′ f ( x ) . As remarked above, the Lie groups H and H ′ an beassumed to be onne ted. Let the domain of the (cid:28)rst parametrization shrinkaround the point ( z , e ) until the smooth inje tion ϕ : Σ ֒ → Σ ′ admits a lo alrepresentation relative to the hosen parametrizations, namely Σ ϕ / / Σ ′ Z × A (cid:31) ? τ · η O O ˜ ϕ / / ____ Z ′ × A ′ . (cid:31) ? τ ′ · η ′ O O ˜ ϕ will be a smooth inje tive map, of the form ( z, a ) (cid:0) z ′ ( z, a ) , a ′ ( z, a ) (cid:1) . Notethat z ′ ( z, a ) = f ( z ) so that, in parti ular, f maps Z into Z ′ ; this follows by omparing the target of the two sides of the equality τ ′ ( z ′ ) · η ′ ( a ′ ) = ϕ ( τ ( z )) · ϕ ( η ( a )) .Sin e the restri tion of f to Z is an immersion of Z into Z ′ , the mapping ˜ ϕ is immersive at ( z , e ) if and only if the orresponding partial map a a ′ ( z , a ) is immersive at e ∈ A . Now, onsider the following huge ommutative16 CHAPTER V. CLASSICAL FIBRE FUNCTORSdiagram, where we put x ′ ≡ f ( x ) and z ′ ≡ f ( z ) : G x G x ϕ / / G ′ f ( x ) G ′ f ( x ) G ( x , z ) τ ( z ) − · O O ϕ / / G ′ ( x ′ , z ′ ) ϕ ( τ ( z )) − · O O A ?(cid:31) η O O { z } × A (cid:31) ? τ · η O O / / { z ′ } × A ′ (cid:31) ? τ ′ · η ′ O O A ′ ?(cid:31) η ′ O O [the re tangle on the right ommutes be ause ϕ ( τ ( z )) = ϕ ( σ ) = τ ′ ( f ( z )) = τ ′ ( z ′ ) ℄. The ommutativity of the outer re tangle entails that the bottommap in this diagram, namely a a ′ ( z , a ) , oin ides with the restri tion to A of a (ne essarily unique) Lie group homomorphism ζ : H → H ′ ; the samemap is therefore an immersion, be ause a Lie group homomorphism whi his inje tive in a neighbourhood of e must be immersive, see eg Brö ker andtom Die k [4℄, p. 27. The proof of the existen e of the homomorphism of Liegroups ζ is deferred to Note 9 below. q.e.d.4 De(cid:28)nition A submanifold Σ of the arrow manifold of a Lie groupoid G will be said to be tame if the following onditions are satis(cid:28)ed:i) the sour e map of G restri ts to a submersion of Σ onto an open subsetof the base manifold M of G ;ii) for ea h point x ∈ M , the orresponding sour e (cid:28)bre Σ( x, - ) ≡ Σ ∩ G ( x, - ) is a prin ipal submanifold.Note that from the (cid:28)rst ondition it already follows that the sour e (cid:28)bre Σ( x, - ) is a submanifold (of Σ and hen e) of G (1) .5 Proposition Let ϕ : G → G ′ be a Lie groupoid homomorphism,indu ing an immersion f : M → M ′ at the level of base manifolds.Suppose that Σ , resp. Σ ′ is a tame submanifold of G , resp. G ′ and that ϕ maps Σ inje tively into Σ ′ .Then ϕ restri ts to an immersion of Σ into Σ ′ .Proof Fix σ ∈ Σ , and put x = s ( σ ) . Choose lo al parametrizations U × B ֒ → Σ at σ ≈ ( x , ∈ U × B , and U ′ × B ′ ֒ → Σ ′ at ϕ ( σ ) ≈ ( f ( x ) , ∈ U ′ × B ′ , lo ally trivializing the respe tive sour e map(cid:22)whi his a submersion be ause of Condition i) of De(cid:28)nition 4(cid:22)over the open sub-sets U ⊂ M , U ′ ⊂ M ′ . (Here B and B ′ are open balls.) This means, forinstan e, that the (cid:28)rst parametrization makes the diagram U × B pr GGGGGGGGG (cid:31) (cid:127) / / Σ s (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) U
22. TAME SUBMANIFOLDS OF A LIE GROUPOID 117 ommute. If the domain of the (cid:28)rst parametrization is made to be onve-niently small around the enter ( x , , the mapping ϕ : Σ ֒ → Σ ′ will indu ea smooth and inje tive lo al expression Σ ϕ / / Σ ′ U × B (cid:31) ? O O / / ___ U ′ × B ′ (cid:31) ? O O of the form ( x, b ) ( x ′ ( x, b ) , b ′ ( x, b )) = ( f ( x ) , b ′ ( x, b )) , so that, in parti ular, f will map U into U ′ . Sin e f : U → U ′ is then an immersion by assumption,the above lo al expression is an immersive map at ( x , if and only if thepartial map b b ′ ( x , b ) is immersive at ∈ B . At this point we an useLemma 3 to on lude the proof. q.e.d.In parti ular, it follows that when a homomorphism ϕ of Lie groupoids(let us say over the same manifold M and with f = id ) indu es a homeo-morphism between two tame submanifolds Σ and Σ ′ , then it restri ts in fa tto a di(cid:27)eomorphism of Σ onto Σ ′ . This will be for us the most useful propertyof tame submanifolds, and we shall make repeated appli ation of it in thesubsequent se tions. A tually, the motivation for introdu ing the on ept oftame submanifold was pre isely to ensure this kind of automati (cid:16)di(cid:27)erentia-bility out of ontinuity(cid:17).6 Note Let S = G m be the m -th orbit. As a notational onvention, weshall use the letter S when we think of this orbit as a manifold, endowedwith the unique di(cid:27)erentiable stru ture that turns the target map(7) t : G ( m, - ) → S into a prin ipal bundle with (cid:28)bre the Lie group G m (a ting on the manifold G ( m, - ) from the right, in the obvious way); (7) is in parti ular a (cid:28)bre bundle,whi h is in fa t equivariantly lo ally trivial. The in lusion S ֒ → M is aninje tive immersion, although not in general an embedding of manifolds. Seealso Moerdijk and Mr£un (2003), [27℄ pp. 115(cid:21)117.To begin with, we show that the in lusion map is an embedding of themanifold Z into S . Of ourse, Z is a submanifold of M and we have thein lusion Z ⊂ G m , but from this fa t we annot a priori on lude that Z embeds into S , not even that the in lusion map Z ֒ → S is ontinuous;the reason why we an do away with this di(cid:30) ulty is that over Z thereexists, by assumption, a smooth se tion τ to the target map G ( m, - ) → M .(In identally, observe that any su h τ : Z → G ( m, - ) is an embedding ofmanifolds. Clearly, it will be enough to see that τ is an embedding of Z into G . Sin e τ is a smooth se tion over Z to t : G → M , it is an inje tiveimmersion; moreover, for any open subset U of M we have τ ( Z ∩ U ) = τ ( Z ) ∩ t − ( U ) .)18 CHAPTER V. CLASSICAL FIBRE FUNCTORSNow, from the existen e of τ it follows immediately that the in lusion t ◦ τ of Z into S is a smooth mapping; moreover, we have that this is a tually aninje tive immersion, be ause on omposing it with S ֒ → M one obtains theembedding Z ֒ → M . It only remains to noti e that if U is open in M then Z ∩ U oin ides with Z ∩ W where W = t G ( m, U ) is open in S .Next, we show that8 Lemma For every z ∈ Z , there is a lo al trivialization of the prin ipalbundle (7), of the form G ( m, W ) ≈ W × G m over an open neighbourhood W of z in S , su h that its unit se tion agreeswith τ on Z ∩ W . (Re all that the unit se tion of su h a lo al trivializationis the mapping that orresponds to W ֒ → W × G m , w ( w, m ) .)Proof Sin e Z embeds as a submanifold of S , it is possible to (cid:28)nd an openneighbourhood W of z in S di(cid:27)eomorphi to a produ t of manifolds W ≈ ( W ∩ Z ) × B , z ≈ ( z , ,where B is an open eu lidean ball. Moreover, it is learly not restri tive toassume that the prin ipal bundle (7) an be trivialized over W . Then, afterhaving (cid:28)xed one su h trivialization, we an take the omposite mapping W ≈ ( W ∩ Z ) × B pr −−→ W ∩ Z τ −→ G ( m, W ) ≈ W × G m pr −−→ G m ,whi h we denote by θ : W → G m , and use it to produ e an equivariant hange of harts and hen e a new lo al trivialization for (7), namely W × G m ∼ → W × G m ≈ G ( m, W ) , ( w, g ) ( w, θ ( w ) g ) ,whose unit se tion is immediately seen to agree with τ on Z ∩ W . q.e.d.Our aim was to prove that Σ = τ ( Z ) · η ( A ) is a submanifold of G andthat τ · η is a smooth isomorphism between Z × A and Σ . Thus, (cid:28)x σ ∈ Σ ,an let z = t ( σ ) ; the latter is a point of Z . Fix also a trivializing hart forthe prin ipal bundle (7) as in the statement of Lemma 8; then W × G m ≈ di(cid:27)eo. / / G ( m, W )( Z ∩ W ) × A (cid:31) ? embed. O O bije t. / / Σ ∩ G ( m, W ) (cid:31) ? set-th. in l. O O ommutes, where on the left we have the obvious embedding of manifolds,and the bottom map is ( z, a ) τ ( z ) · η ( a ) , the restri tion of τ · η . (Thediagram ommutes pre isely be ause the unit se tion of the hart agreeswith τ over Z ∩ W .) It is then lear that Σ ∩ G ( m, W ) is a submanifold of22. TAME SUBMANIFOLDS OF A LIE GROUPOID 119the open neighbourhood G ( m, W ) of σ in G ( m, - ) , and that τ · η restri ts toa di(cid:27)eomorphism of ( Z ∩ W ) × A onto this submanifold.Hen eforth, Σ is a submanifold of G ( m, - ) and τ · η is a bije tive lo aldi(cid:27)eomorphism between Z × A and Σ . (Note that the statement that Z ֒ → S is an embedding is really used here.)9 Note Assume that a ommutative re tangle A (cid:15) (cid:15) (cid:31) (cid:127) / / H ∃ ! ζ (cid:15) (cid:15) (cid:31) (cid:127) η / / G ϕ (cid:15) (cid:15) A ′ (cid:31) (cid:127) / / H ′ (cid:31) (cid:127) η ′ / / G ′ is given, where G , G ′ are Lie groups, ϕ is a Lie group homomorphism, η : H ֒ → G and η ′ : H ′ ֒ → G ′ are Lie subgroups with H onne ted, A ⊂ H , A ′ ⊂ H ′ are open neighbourhoods of the unit elements e , e ′ of H , H ′ respe tively,and A → A ′ is a smooth mapping. Then there exists a unique Lie grouphomomorphism ζ : H → H ′ whi h (cid:28)ts in the diagram as indi ated.Indeed, sin e A is an open neighbourhood of e in H and H is onne ted, A generates H as a group, see Brö ker and tom Die k (1995), [4℄ p. 10. So ϕη ( A ) generates ϕη ( H ) , and therefore ϕη ( H ) ⊂ η ′ ( H ′ ) be ause ϕη ( A ) ⊂ η ′ ( A ′ ) ⊂ η ′ ( H ′ ) . Sin e η ′ : H ′ → η ′ ( H ′ ) is a bije tive homomorphism ofgroups, there exists a unique group-theoreti solution ζ : H → H ′ to theproblem η ′ ◦ ζ = ϕ ◦ η . The restri tion of ζ to A oin ides with the givensmooth map A → A ′ , thus ζ is smooth in a neighbourhood of e ; sin e lefttranslations are Lie group automorphisms, the ommutativity of H ≈ h · (cid:15) (cid:15) ζ / / H ′≈ ζ ( h ) · (cid:15) (cid:15) H ζ / / H ′ shows that ζ is smooth in the neighbourhood of any h ∈ H , and hen eglobally smooth, in other words a Lie group homomorphism.Tameness and Morita equivalen eThere is still one fundamental point we need to dis uss, for the treatmentof weak equivalen es of lassi al (cid:28)bre fun tors in Se tion 25 below. Namely,suppose one is given a Morita equivalen e of Lie groupoids ϕ : G → G ′ su h that at the level of manifolds of obje ts it is given by a submersion ϕ : M → M ′ . Let Σ be a subset of the manifold of arrows of G , and assumethat every point of Σ has an open neighbourhood Γ in G with(10) ϕ − (Σ ′ ) ∩ Γ ⊂ Σ ,20 CHAPTER V. CLASSICAL FIBRE FUNCTORSwhere we put Σ ′ = ϕ (Σ) ; note that this is equivalent to saying that ∀ γ ∈ Γ , γ ∈ Σ ⇔ ϕ ( γ ) ∈ Σ ′ .Then one has what follows1. Σ is a submanifold of G if and only if Σ ′ is a submanifold of G ′ ;2. Σ is a submanifold of G verifying Condition i) of De(cid:28)nition 4 if andonly if the same is true of Σ ′ in G ′ ;3. for every m ∈ M , the restri tion ϕ : Σ( m, - ) → Σ ′ ( ϕ ( m ) , - ) is an openmapping between topologi al subspa es of the manifolds G and G ′ ;4. for every m ∈ M , the (cid:28)bre Σ( m, - ) is a prin ipal submanifold of G ifand only if its image ϕ (Σ( m, - )) is a prin ipal submanifold of G ′ .Before we start with the proofs, let us show how these statements 1-4 maybe used to derive the following main result11 Proposition Let ϕ : G −→ G ′ be a Morita equivalen e of Liegroupoids indu ing a submersion at the level of base manifolds. Let Σ be a subset of the manifold of arrows of G whi h satis(cid:28)es ondition (10)above, and put Σ ′ = ϕ (Σ) . Then Σ is a tame submanifold of G if and onlyif Σ ′ is a tame submanifold of G ′ .Proof ( ⇐ ) Suppose m ∈ M is given: we must show that Σ( m, - ) is a prin i-pal submanifold of G . Be ause of Statement 3, ϕ (Σ( m, - )) is an open subset ofthe subspa e Σ ′ ( ϕ ( m ) , - ) ⊂ G ′ . Sin e the latter is by assumption a prin ipalsubmanifold of G ′ , it follows that the open subset ϕ (Σ( m, - )) is a prin i-pal submanifold of G ′ as well, and hen e, by Statement 4, that Σ( m, - ) is aprin ipal submanifold of G .( ⇒ ) Fix m ′ ∈ M ′ . A ording to Statement 3, we have the open overing Σ ′ ( m ′ , - ) = [ m ∈ ϕ − ( m ′ ) ϕ (Σ( m, - )) ,and every open set belonging to this overing is a prin ipal submanifoldof G ′ , by Statement 4 and the assumption. Hen e the whole submanifold Σ ′ ( m ′ , - ) ⊂ G ′ is a prin ipal submanifold of G ′ . q.e.d.Now we ome to the proofs of Statements 1 to 4:Proof of Statement 1. Re all from Note 15, (16) below that, up todi(cid:27)eomorphism, one has for the morphism ϕ a anoni al de omposition Γ ( s , t ) (cid:15) (cid:15) ≈ / / Γ ′ × B × C (cid:15) (cid:15) pr / / Γ ′ ( s ′ , t ′ ) (cid:15) (cid:15) U × V ≈×≈ / / U ′ × B × V ′ × C pr × pr / / U ′ × V ′
22. TAME SUBMANIFOLDS OF A LIE GROUPOID 121in a neighbourhood Γ of every point of Σ , with Γ verifying ondition (10).We have that Σ ′ ∩ Γ ′ is a submanifold of Γ ′ if and only if (Σ ′ ∩ Γ ′ ) × A isa submanifold of Γ ′ × A , where A = B × C . Thus, sin e (Σ ′ ∩ Γ ′ ) × A = pr − (Σ ′ ∩ Γ ′ ) orresponds to ϕ − (Σ ′ ∩ Γ ′ ) ∩ Γ = ϕ − (Σ ′ ) ∩ Γ = Σ ∩ Γ in the di(cid:27)eomorphism Γ ≈ Γ ′ × B × C , this is in turn equivalent to sayingthat Σ ∩ Γ is a submanifold of Γ . Thus we see that Σ is a submanifold of G if and only if Σ ′ is a submanifold of G ′ .Proof of Statement 2. From the previous diagram, we get that, up todi(cid:27)eomorphism, s : Γ → U orresponds to s ′ × pr : Γ ′ × B × C → U ′ × B ,so it restri ts to a submersion Σ ∩ Γ → U if and only if s ′ × pr restri ts to asubmersion (Σ ′ ∩ Γ ′ ) × B × C → U ′ × B ; and this is in turn true if and onlyif s ′ : Σ ′ ∩ Γ ′ → U ′ is a submersion.Proof of Statement 3. Fix a point σ ∈ Σ( m, - ) and an open neighbour-hood of that point in G . Then from Note 15 below, we have for the restri tionof ϕ to Σ a anoni al lo al de omposition Σ ∩ Γ s (cid:15) (cid:15) ≈ / / (Σ ′ ∩ Γ ′ ) × B × C s ′ × id (cid:15) (cid:15) pr / / Σ ′ ∩ Γ ′ s ′ (cid:15) (cid:15) U ≈ / / U ′ × B pr / / U ′ at σ = ( σ ′ , , , where Γ an be hoosen as small as one likes around σ ,simply by taking a smaller Γ ′ = ϕ (Γ) at σ ′ = ϕ ( σ ) and redu ing the radiusof the open balls B , C ; in parti ular, Γ an be hosen so small that it (cid:28)ts inthe previously assigned open neighbourhood of σ in G .It is immediate to re ognize that ϕ (Σ( m, - ) ∩ Γ) = Σ ′ ( ϕ ( m ) , - ) ∩ Γ ′ , wherethe latter is learly an open subset of the subspa e Σ ′ ( ϕ ( m ) , - ) of G ′ . Indeed,in the left-hand square of the pre eding diagram, the s -(cid:28)bre above m ∈ U ,namely (Σ ∩ Γ)( m, - ) = Σ( m, - ) ∩ Γ , orresponds to the s ′ × pr -(cid:28)bre above ( ϕ ( m ) , , namely (Σ ′ ∩ Γ ′ )( ϕ ( m ) , - ) × × C .The latter is mapped by the proje tion pr onto (Σ ′ ∩ Γ ′ )( ϕ ( m ) , - ) = Σ ′ ( ϕ ( m ) , - ) ∩ Γ ′ ,hen e ϕ maps Σ( m, - ) ∩ Γ onto Σ ′ ( ϕ ( m ) , - ) ∩ Γ ′ , as ontended.Proof of Statement 4. This will be based on the following lemma:22 CHAPTER V. CLASSICAL FIBRE FUNCTORS12 Lemma Let ϕ : G → G ′ be a fully faithful homomorphism of Liegroupoids and let ϕ : M → M ′ be the map indu ed on base manifolds.Suppose that Σ ⊂ G and Σ ′ = ϕ (Σ) ⊂ G ′ are submanifolds. Suppose alsothat a ommutative diagram(13) Σ t (cid:15) (cid:15) ≈ / / Σ ′ × C t ′ × id (cid:15) (cid:15) pr / / Σ ′ t ′ (cid:15) (cid:15) V ≈ / / V ′ × C pr / / V ′ is given, where V ⊂ M and V ′ ⊂ M ′ are open subsets, C is an open balland the ≈ 's are di(cid:27)eomorphisms su h that the top row oin ides with ϕ (arrows) and the bottom one with ϕ (obje ts). Let σ ∈ Σ be a pointwith σ ≈ ( σ ′ , ∈ Σ ′ × C .Then Σ admits a lo al parametrization of type (2) at σ if and onlyif Σ ′ admits su h a parametrization at σ ′ .Proof Notation: let z = t ( σ ) ∈ V and z ′ = t ′ ( σ ′ ) = ϕ ( z ) ∈ V ′ . Observethat from (13) it follows that z orresponds to ( z ′ , in the di(cid:27)eomorphism V ≈ V ′ × C , be ause σ orresponds to ( σ ′ , in Σ ≈ Σ ′ × C .( ⇐ ) Suppose that Σ ′ admits a type (2) lo al parametrization σ ′ · η ′ : Z ′ × A ′ ֒ → Σ ′ at σ ′ ≈ ( z ′ , e ′ ) ∈ Z ′ × A ′ . It is learly no loss of generality toassume that the whole Σ ′ is the image of this lo al parametrization. Z ′ = t ′ ( σ ′ ( Z ′ )) ⊂ t ′ (Σ ′ ) ⊂ V ′ is a submanifold, be ause so is Z ′ ⊂ M ′ . Writethe di(cid:27)eomorphism V ≈ V ′ × C as v ( ϕ ( v ) , c ( v )) and let Z ⊂ V bethe submanifold orresponding to Z ′ × C . De(cid:28)ne σ : Z → Σ as σ ( z ) =( σ ′ ( ϕ ( z )) , c ( z )) ∈ Σ ′ × C ≈ Σ , and η by(14) G ( m, m ) ϕ ≈ / / G ′ ( m ′ , m ′ ) H ′ R2 η d d I I I I I (cid:12), η ′ tttttttttt so that σ is learly a smooth t -se tion t ( σ ( z )) ≈ (cid:0) t ′ × id (cid:1)(cid:0) σ ′ ( ϕ ( z )) , c ( z ) (cid:1) = (cid:0) t ′ ( σ ′ ( ϕ ( z ))) , c ( z ) (cid:1) = ( ϕ ( z ) , c ( z )) ≈ z with σ ( z ) ≈ (cid:0) σ ′ ( ϕ ( z )) , c ( z ) (cid:1) = ( σ ′ , ≈ σ , and η : H ֒ → G m is a Liesubgroup, where we put H = H ′ . Let A = A ′ . It is immediate to al ulatethat the image of σ · η : Z × A ֒ → G is the whole Σ : thus we have onstru teda global parametrization of Σ at σ .( ⇒ ) In the other dire tion, suppose we are given a lo al parametrization σ · η : Z × A ֒ → Σ of type (2) su h that σ ∈ Σ orresponds to ( z , e ) =( t ( σ ) , e ) ∈ Z × A . Clearly, Z = t ( σ ( Z )) ⊂ t (Σ) ⊂ V is a submanifold sin eso is Z ⊂ M .22. TAME SUBMANIFOLDS OF A LIE GROUPOID 123To begin with, observe that it is not restri tive to assume that the sub-manifold Z ⊂ V orresponds to Z ′ × C under the di(cid:27)eomorphism V ≈ V ′ × C , where of ourse Z ′ = ϕ ( Z ) . Pre isely, the di(cid:27)eomorphism Σ ≈ Σ ′ × C , that identi(cid:28)es σ with ( σ ′ , , allows one to hoose a smaller openneighbourhood ( σ ′ , ∈ Σ ′ × C ⊂ Σ ′ × C su h that Σ ≈ Σ ′ × C is on-tained in the domain of the lo al hart ( σ · η ) − . From the ommutativity ofthe diagram Z × A pr (cid:15) (cid:15) Σ ? _ ( σ · η ) − open emb. o o t (cid:15) (cid:15) ≈ / / Σ ′ × C t ′ × id (cid:15) (cid:15) Z t (Σ ) ? _ in lusion o o ≈ / / t ′ (Σ ′ ) × C it follows at on e that Z = t (Σ ) ⊂ Z is an open subset su h that V ≈ V ′ × C indu es a bije tion Z ≈ Z ′ × C , where Z ′ = t ′ (Σ ′ ) . Sin e it is ompatible with the aims of the present proof to repla e C with a smaller C entered at , we an work with the smaller lo al parametrization obtainedby restri ting σ to the open subset Z of Z .Se ondly, the t -se tion σ : Z → Σ indu es, by means of the di(cid:27)eomorph-isms Z ≈ Z ′ × C and Σ ≈ Σ ′ × C , a smooth mapping Z ′ × C → Σ ′ × C ofthe form ( z ′ , c ) ( σ ′ ( z ′ , c ) , c ) ; indeed ( z ′ , c ) ≈ z = t ( σ ( z )) ≈ ( t ′ × id ) (cid:0) σ ′ ( z ′ , c ) , c ( z ′ , c ) (cid:1) = (cid:0) t ′ ( σ ′ ( z ′ , c )) , c ( z ′ , c ) (cid:1) ,hen e it follows t ′ ( σ ′ ( z ′ , c )) = z ′ and c ( z ′ , c ) = c . We laim that it is no lossof generality to assume that it a tually is of the form ( z ′ , c ) ( σ ′ ( z ′ ) , c ) , iethat σ ′ does not really depend on the variable c . Indeed, de(cid:28)ne τ : Z → Σ as τ ( z ) = (cid:0) σ ′ ( ϕ ( z ) , , c ( z ) (cid:1) ∈ Σ ′ × C = Σ ; su h a map is also a smooth t -se tion t ( τ ( z )) ≈ ( t ′ × id ) (cid:0) σ ′ ( ϕ ( z ) , , c (cid:1) = (cid:0) t ′ ( σ ′ ( z ′ , c )) , c (cid:1) = ( ϕ ( z ) , c ) ≈ z with τ ( z ) = (cid:0) σ ′ ( z ′ , , (cid:1) = σ ( z ) = σ . Then we an apply Lemma 20 below,the `Reparametrization Lemma', to obtain a new type (2) lo al parametriza-tion of Σ at σ , for whi h su h an assumption holds as well. Then we an introdu e a smooth t ′ -se tion σ ′ : Z ′ → Σ ′ su h that σ ′ ( z ′ ) = σ ′ ,by setting σ ′ ( z ′ ) = σ ′ ( z ′ , ; also, we de(cid:28)ne η ′ by means of (14) and put H ′ = H and A ′ = A . Thus, from the simplifying assumption above, it followsthat σ ′ ( ϕ ( z )) = ϕ ( σ ( z )) for every z ∈ Z , and therefore that the image of σ ′ · η ′ : Z ′ × A ′ ֒ → G ′ oin ides with ϕ (Im σ · η ) . But Im σ · η ⊂ Σ is an opensubset, and ϕ : Σ → Σ ′ = ϕ (Σ) is an open mapping, when e Im σ ′ · η ′ is anopen subset of Σ ′ . This on ludes the proof. q.e.d.24 CHAPTER V. CLASSICAL FIBRE FUNCTORS15 Note Fix a point σ ∈ Σ . Sin e f is a submersion, one an hoose openneighbourhoods U and V of s ( σ ) and t ( σ ) in M respe tively, so small that,up to di(cid:27)eomorphism, f | U be omes an open proje tion U ≈ U ′ × B pr −→ U ′ ( U ′ is an open subset of M ′ and B is an open ball; moreover, we shall assumethat s ( σ ) orresponds to ( f ( s ( σ )) , in the di(cid:27)eomorphism U ≈ U ′ × B ),and f | V be omes an open proje tion V ≈ V ′ × C pr −→ V ′ ( V ′ is an open subsetof M ′ , and C is an open ball; also, t ( σ ) orresponds to ( f ( t ( σ )) , in thedi(cid:27)eomorphism V ≈ V ′ × B ). Sin e ϕ is a Morita equivalen e, we have thefollowing pullba k in the ategory of di(cid:27)erentiable manifolds of lass C ∞ G ( U, V ) ( s , t ) (cid:15) (cid:15) ϕ / / G ′ ( U ′ , V ′ ) ( s ′ , t ′ ) (cid:15) (cid:15) U × V f × f / / U ′ × V ′ whi h has therefore, up to di(cid:27)eomorphism, the following aspe t G ( U, V ) ≈ di(cid:27)eo. / / ( s , t ) (cid:15) (cid:15) G ′ ( U ′ , V ′ ) × B × C ( s ′ , t ′ ) × id × id (cid:15) (cid:15) pr / / G ′ ( U ′ , V ′ ) ( s ′ , t ′ ) (cid:15) (cid:15) U × V ≈ × ≈ / / U ′ × B × V ′ × C pr × pr / / U ′ × V ′ ,where the top omposite arrow oin ides with ϕ and the bottom one with f × f . Next, take an open neighbourhood Γ of σ in G su h that the relation(10) holds. Then the same relation is learly also satis(cid:28)ed by any smaller openneighbourhood of σ in G , hen e it is no loss of generality to assume that Γ is ontained in G ( U, V ) and that it orresponds to a produ t Γ ′ × B × C (with Γ ′ = ϕ (Γ) ne essarily open in G ′ ( U ′ , V ′ ) , be ause ϕ : G ( U, V ) → G ′ ( U ′ , V ′ ) isopen as it is lear from the latter diagram, and with B ⊂ B , C ⊂ C openballs entered at of smaller radius) in the di(cid:27)eomorphism G ( U, V ) ≈ G ′ ( U ′ , V ′ ) × B × C .Then, by our hoi e of Γ we obtain a ommutative diagram Γ ≈ di(cid:27)eo. / / ( s , t ) (cid:15) (cid:15) Γ ′ × B × C s ′ , t ′ ) × id × id (cid:15) (cid:15) pr / / Γ ′ ( s ′ , t ′ ) (cid:15) (cid:15) U × V ≈ × ≈ / / U ′ × B × V ′ × C pr × pr / / U ′ × V ′ (16)where the top omposite arrow oin ides with ϕ and the bottom one with f × f . Finally, by pasting the following ommutative diagram U × V pr (cid:15) (cid:15) ≈ × ≈ / / U ′ × B × V ′ × C pr (cid:15) (cid:15) pr × pr / / U ′ × V ′ pr (cid:15) (cid:15) V ≈ / / V ′ × C pr / / V ′
22. TAME SUBMANIFOLDS OF A LIE GROUPOID 125to the former one along the ommon edge, we obtain Γ ≈ / / t (cid:15) (cid:15) Γ ′ × B × C t ′ × pr (cid:15) (cid:15) pr / / Γ ′ t ′ (cid:15) (cid:15) V ≈ / / V ′ × C pr / / V ′ (17)and then, sin e property (10) holds for Γ , Σ ∩ Γ ≈ / / t (cid:15) (cid:15) (Σ ′ ∩ Γ ′ ) × B × C t ′ × pr (cid:15) (cid:15) pr / / Σ ′ ∩ Γ ′ t ′ (cid:15) (cid:15) V ≈ / / V ′ × C pr / / V ′ .(18)Both in (17) and in (18), the top omposite arrow oin ides with the restri -tion of ϕ and the bottom one with the restri tion of f . Of ourse, one hasanalogous diagrams with sour e maps repla ing target maps.19 Note Here we shall state and prove the Lo al ReparametrizationLemma, whi h was needed in the proof of Lemma 12.20 Lemma (Lo al Reparametrization) Let G ⇒ M be a Lie groupoid.Suppose we are given: a point m ∈ M , a smooth t -se tion τ : Z → G ( m, - ) de(cid:28)ned over a submanifold Z ⊂ M , a Lie subgroup η : H ֒ → G m and anopen neighbourhood A of the unit e in H su h that the restri tion of η isan embedding. Let Σ = τ ( Z ) · η ( A ) be the image of the mapping of type(2) obtained from these data.Let σ ≈ ( z , e ) ∈ Z × A be a given point in Σ , and suppose that σ : Z → Σ is any other smooth t -se tion su h that σ ( z ) = σ = τ ( z ) .Then there exists a smaller open neighbourhood Z × A of the point ( z , e ) in Z × A su h that σ · η : Z × A ֒ → Σ is still a lo al parametrization for Σ at σ .Proof If we onsider the omposite ( τ · η ) − ◦ σ : Z → Σ → Z × A , weget smooth oordinate maps z ( ζ ( z ) , α ( z )) , hara terized by the equation σ ( z ) = τ ( ζ ( z )) · η ( α ( z )) . Comparing the target of the sides of this equationwe get ζ ( z ) = z . Thus σ is ompletely determined by the smooth mapping α : Z → A via the relation σ ( z ) = τ ( z ) · η ( α ( z )) .Now, we hoose a smaller open neighbourhood A ⊂ A of the unit e su hthat A · A ⊂ A , whi h exists by ontinuity of the multipli ation of H , andnext an open neighbourhood Z of z in Z su h that α ( Z ) ⊂ A ; this ispossible be ause α ( z ) = e , whi h follows from σ ( z ) = τ ( z ) = τ ( z ) · η ( e ) .It is then lear that σ · η maps Z × A into Σ : indeed, ∀ ( z, a ) ∈ Z × A ,26 CHAPTER V. CLASSICAL FIBRE FUNCTORS σ ( z ) · η ( a ) = ( τ ( z ) · η ( α ( z ))) · η ( a ) = τ ( z ) · η ( α ( z ) · a ) , and this is learly anelement of τ ( Z ) · η ( A · A ) ⊂ τ ( Z ) · η ( A ) = Σ .If again we ompose ( τ · η ) − ◦ ( σ · η ) : Z × A → Σ → Z × A , we getsmooth oordinate maps ( z, a ) (cid:0) ζ ( z, a ) , α ( z, a ) (cid:1) , hara terized by the rela-tion σ ( z ) · η ( a ) = τ ( ζ ( z, a )) · η ( α ( z, a )) . Taking the target yields ζ ( z, a ) = z ,thus we have a smooth mapping Z × A → Z × A of the form ( z, a ) ( z, α ( z, a )) hara terized by the equation σ ( z ) · η ( a ) = τ ( z ) · η ( α ( z, a )) . (So,in parti ular, α ( z, e ) = α ( z ) and α ( z , e ) = e .)To on lude, it will be enough to observe that this mapping has invertibledi(cid:27)erential at ( z , e ) ∈ Z × A , be ause if that is the ase then the mappingindu es a lo al di(cid:27)eomorphism of an open neighbourhood of ( z , e ) in Z × A (whi h an be assumed to be Z × A itself, up to shrinking) onto an openneighbourhood of ( z , e ) ∈ Z × A , so that if we then ompose ba k with τ · η we see that σ · η is a di(cid:27)eomorphism of Z × A onto an open subset of Σ .To see the invertibility of the di(cid:27)erential, it will be su(cid:30) ient to prove thatthe partial map a α ( z , a ) has invertible di(cid:27)erential at e ∈ A . But fromthe hara terizing equation (setting z = z ) α ( z , a ) = η − ( τ − ( z ) σ ( z )) · a = η − (1 m ) · a = a we see at on e that this di(cid:27)erential is in fa t the identity. q.e.d.21 Note We in lude here a dis ussion of tame submanifolds in onne tionwith embeddings of Lie groupoids, parallel to the one on erning Moritaequivalen es. Suppose one is given su h an embedding, ie a Lie groupoidhomomorphism ι : G ֒ → G ′ su h that the mapping ι itself and the mapping i : M ֒ → M ′ indu ed on bases are embeddings of manifolds. Let Σ be asubset of G , and put Σ ′ = ι (Σ) ⊂ G ′ . The following statements holdi) Σ is a submanifold of G if and only if Σ ′ is a submanifold of G ′ , in whi h ase the restri tion ι : Σ → Σ ′ is a di(cid:27)eomorphism;ii) Σ is a prin ipal submanifold of G if and only if Σ ′ is a prin ipal sub-manifold of G ′ ;iii) in ase i : M ֒ → M ′ is an open embedding, Σ is a tame submanifold of G if and only if Σ ′ is a tame submanifold of G ′ .Note that, as a spe ial ase, we get invarian e of tame submanifolds underisomorphisms of Lie groupoids.Ÿ23 Smoothness and Representative ChartsIn Ÿ21 we dis ussed some general properties of lassi al (cid:28)bre fun tors, whi hhold quite apart from the eventuality that the anoni al C ∞ -stru ture onthe spa e of arrows of the Tannakian groupoid might prove not to be asmooth manifold stru ture. On the ontrary, in the present se tion we turn23. SMOOTHNESS, REPRESENTATIVE CHARTS 127our attention spe i(cid:28) ally to the problem of (cid:28)nding e(cid:27)e tive riteria to de idewhether a given lassi al (cid:28)bre fun tor is (cid:16)smooth(cid:17) in the sense illustrated atthe beginning of Ÿ18. Su h riteria will be employed in Ÿ26; they involve thete hni al notion of tame submanifold introdu ed in the pre eding se tion.To motivate our de(cid:28)nitions (whi h may appear rather arti(cid:28) ial at (cid:28)rstglan e) let us onsider a smooth lassi al (cid:28)bre fun tor ω over a manifold M . Re all that ω being smooth means by de(cid:28)nition that the standard C ∞ -stru ture R ∞ on the spa e T ( ω ) (1) turns T ( ω ) into a Lie groupoid over M ; ompare Ÿ18. Consider any lassi al representation ̺ : T ( ω ) → GL ( E ) on a smooth ve tor bundle E ; we know from Lemma 20.15 that if the map λ ̺ ( λ ) is inje tive in the vi inity of λ within the subspa e T ( ω )( x , x ′ ) [ x ≡ s ( λ ) , x ′ ≡ t ( λ ) ℄ of T ( ω ) (1) , the same map must be an immersion,into the manifold of arrows of GL ( E ) , of some open neighbourhood Ω ⊂ T of λ and therefore it must indu e, provided Ω is hosen small enough, adi(cid:27)eomorphism of Ω onto a submanifold ̺ (Ω) of GL ( E ) . When, in parti ular, ̺ = ev R for some R ∈ Ob( C ) , we agree to write R (Ω) for the submanifold [ofthe manifold of arrows of GL ( ω R ) ℄ that orresponds to Ω , namely we put(1) R (Ω) def = ev R (Ω) .It is not ex eedingly di(cid:30) ult to see that the submanifolds of GL ( E ) of theform ̺ (Ω) , for all ̺ and Ω su h that ̺ indu es a di(cid:27)eomorphism of Ω onto ̺ (Ω) , are ne essarily tame submanifolds of GL ( E ) , fr Lemma 26.3 below.It will be onvenient to have a name for the lo al di(cid:27)eomorphisms of theabove-mentioned type:2 De(cid:28)nition We shall all representative hart any pair (Ω , R ) onsistingof an open subset Ω of the spa e of arrows of T ( ω ) and an obje t R ∈ Ob( C ) ,su h that ev R : T ( ω ) → GL ( ω R ) restri ts to a homeomorphism of Ω onto atame submanifold R (Ω) of the linear groupoid GL ( ω R ) .Note that this de(cid:28)nition has been formulated so that it makes sense foran arbitrary lassi al (cid:28)bre fun tor ω ; when ω is smooth and (Ω , R ) is arepresentative hart, the map λ λ ( R ) indu es a di(cid:27)eomorphism of Ω ontothe submanifold R (Ω) of GL ( ω R ) : this justi(cid:28)es our de(cid:28)nition.Observe that if R and S are two isomorphi obje ts of C then (Ω , R ) isa representative hart of T ( ω ) if and only if the same is true of (Ω , S ) (seeNote 11 below). Moreover, if (Ω , R ) is a representative hart of T ( ω ) , thesame is obviously true of (Ω ′ , R ) for ea h open subset Ω ′ ⊂ Ω .We know from Lemma 10.14 that if a lassi al (cid:28)bre fun tor ω is smooththen for ea h λ there exists some R ∈ Ob( C ) su h that the map λ λ ( R ) is inje tive in a neighbourhood of λ within the subspa e T ( ω )( s λ , t λ ) of T ( ω ) (1) . Now, as remarked before, this implies that λ lies in the domain Ω of a representative hart (Ω , R ) : thus we see that for any smooth lassi al(cid:28)bre fun tor, the domains of representative harts form an open overing ofthe spa e of arrows of the orresponding Tannakian groupoid.28 CHAPTER V. CLASSICAL FIBRE FUNCTORSNext, let us onsider an arbitrary representative hart (Ω , R ) of T ( ω ) ,for a smooth ω . Let S be an arbitrary obje t of C . By hoosing dire t sumrepresentatives onveniently, we may suppose that ω ( R ⊕ S ) = ω R ⊕ ω S .The evaluation map ev R ⊕ S will yield a one-to-one orresponden e between Ω and the subspa e ( R ⊕ S )(Ω) of GL ( ω R ⊕ ω S ) : indeed, sin e λ ( R ⊕ S ) = λ ( R ) ⊕ λ ( S ) for all λ ∈ T ( ω ) , it is lear that the map λ λ ( R ⊕ S ) fa torsthrough the submanifold GL ( ω R ) × M GL ( ω S ) ֒ → GL ( ω R ⊕ ω S ) ( fr Note16 below) as the map λ (cid:0) λ ( R ) , λ ( S ) (cid:1) (the latter is evidently inje tive,be ause so is λ λ ( R ) , by hypothesis). We ontend that ev R ⊕ S a tuallyindu es a homeomorphism of Ω onto the respe tive image; sin e ev R ⊕ S is im-mersive (by Lemma 20.15), our ontention will imply at on e that ( R ⊕ S )(Ω) is a submanifold of GL ( ω R ⊕ ω S ) and that ev R ⊕ S yields a di(cid:27)eomorphismbetween Ω and this submanifold. Now, let Ω ′ ⊂ Ω be a given open subset; (cid:28)xany open subset Λ ′ ⊂ GL ( ω R ) su h that R (Ω) ∩ Λ ′ = R (Ω ′ ) (su h Λ ′ existbe ause Ω and R (Ω) are homeomorphi via ev R ): then(3) ( R ⊕ S )(Ω) ∩ (cid:0) Λ ′ × M GL ( ω S ) (cid:1) = ( R ⊕ S )(Ω ′ ) ,whi h proves our ontention. From the remarks that pre ede De(cid:28)nition 2 weimmediately on lude that the following property is satis(cid:28)ed by any smooth lassi al (cid:28)bre fun tor ω : when (Ω , R ) is a representative hart of T ( ω ) , somust be (Ω , R ⊕ S ) for ea h obje t S ∈ Ob( C ) .The onverse holds:4 Proposition Let ω be a lassi al (cid:28)bre fun tor. Then ω is smooth ifand only if the following two onditions are satis(cid:28)ed:i) the domains of representative harts over the spa e of arrows ofthe Tannakian groupoid T ( ω ) , ie for ea h λ ∈ T ( ω ) there exists arepresentative hart (Ω , R ) with λ ∈ Ω ;ii) if (Ω , R ) is a representative hart of T ( ω ) then the same is true of (Ω , R ⊕ S ) for every obje t S ∈ Ob( C ) .Proof We have already proved that a smooth lassi al (cid:28)bre fun tor satis(cid:28)es onditions i) and ii). Vi e versa, suppose these onditions are satis(cid:28)ed: the ru ial point now is to show that any representative hart (Ω , R ) establishesan isomorphism of fun tionally stru tured spa es between (Ω , R ∞ Ω ) and thesubmanifold X def = R (Ω) ⊂ GL ( ω R ) (endowed with the stru ture C ∞ X ).Sin e ev R : T → GL ( ω R ) is a morphism of fun tionally stru turedspa es, it is lear that f ∈ C ∞ ( X ) implies f ◦ ev R ∈ R ∞ (Ω) ( fr. theproof of Proposition 20.21). The onverse impli ation is less obvious: wewill make use of the spe ial properties of tame submanifolds we derived inthe pre eding se tion. Suppose r = r S,ψ,η,η ′ ∈ R ∞ (Ω) and let f be the fun -tion on X su h that f ◦ ev R = r ; we must show that f ∈ C ∞ ( X ) . Sin e23. SMOOTHNESS, REPRESENTATIVE CHARTS 129 f = q ψ,η,η ′ ◦ ev S ◦ ev R − where q ψ,η,η ′ is the smooth fun tion on GL ( ω S ) given by ν (cid:10) ν · η ( s ν ) , η ′ ( t ν ) (cid:11) ψ and ev R − : X ≈ −→ Ω is the inverse map,it will be enough to show that ev S ◦ ev R − is a smooth mapping of X into GL ( ω S ) . Put E = ω ( R ) , F = ω ( S ) . Re all that GL ( E ) × M GL ( F ) is theprodu t of GL ( E ) and GL ( F ) in the ategory of Lie groupoids over M (seeNote 16 below) and that therefore it omes equipped with two proje tions pr E , pr F that are morphisms of Lie groupoids over M . One an build thefollowing ommutative diagram ( R ⊕ S )(Ω) ≈ homeo (cid:15) (cid:15) (cid:31) (cid:127) e R,S / / GL ( E ) × M GL ( F ) pr E (cid:15) (cid:15) Ω ev R ⊕ S qqqqqqqqqqqqq ev R / / X = R (Ω) (cid:31) (cid:127) submanifold / / GL ( E ) ,(5)where e R,S is the smooth embedding whose omposition with(6) GL ( E ) × M GL ( F ) ֒ → GL ( E ⊕ F ) = GL (cid:0) ω ( R ⊕ S ) (cid:1) , ( µ, ν ) µ ⊕ ν equals the in lusion of ( R ⊕ S )(Ω) into GL ( ω R ⊕ S ) . Now, (Ω , R ⊕ S ) is arepresentative hart of T ( ω ) and hen e ( R ⊕ S )(Ω) is a tame submanifoldof GL ( ω R ⊕ S ) , so we an apply Proposition 22.5 to on lude that the tran-sition homeomorphism in (5) is in fa t a di(cid:27)eomorphism. This immediatelyimplies the desired smoothness of the transition mapping ev S ◦ ev R − : X → GL ( F ) , be ause of the ommutativity of the following diagram: ( R ⊕ S )(Ω) (cid:31) (cid:127) e R,S smooth / / GL ( E ) × M GL ( F ) pr F (cid:15) (cid:15) X trans. di(cid:27)eo ≈ ppppppppppppp ev R − / / Ω (cid:31) (cid:127) ev S / / ev R ⊕ S O O GL ( F ) .(7) From ondition i) and what we have just proved, we see that ( T , R ∞ ) is a smooth manifold and that ea h representative hart (Ω , R ) indu es adi(cid:27)eomorphism ev R | Ω of Ω onto R (Ω) . Moreover, sin e on the domain of anyrepresentative hart (Ω , R ) the sour e map of T ( ω ) is the omposition of ev R | Ω with the restri tion to R (Ω) of the sour e map of GL ( ω R ) , we alsosee that the sour e map of T ( ω ) is a submersion(cid:22)be ause su h remains thesour e map of GL ( ω R ) when restri ted to the tame submanifold R (Ω) ⊂ GL ( ω R ) . Proposition 21.3 allows us to (cid:28)nish the proof. q.e.d.There is yet one useful remark on erning Condition ii): under the hy-pothesis that (Ω , R ) is a representative hart, the evaluation map ev R ⊕ S es-tablishes, as in (3), a homeomorphism between Ω and the subset ( R ⊕ S )(Ω) of the manifold GL ( ω R ⊕ S ) , wherefore the pair (Ω , R ⊕ S ) is a representa-tive hart if and only if ( R ⊕ S )(Ω) is a tame submanifold of GL ( ω R ⊕ S ) .The usefulness of the last proposition will be ome evident in the study ofweak equivalen es of lassi al (cid:28)bre fun tors ( fr Se tion 25) and in the studyof lassi al (cid:28)bre fun tors asso iated with proper Lie groupoids (Chapter VI).30 CHAPTER V. CLASSICAL FIBRE FUNCTORS8 Corollary Let ω : C → V ∞ ( M ) be a lassi al (cid:28)bre fun tor satisfying onditions i) and ii) of the pre eding proposition.Then there exists a unique manifold stru ture on the spa e of arrowsof the groupoid T ( ω ) , that renders T ( ω ) a Lie groupoid and ev R : T ( ω ) −→ GL ( ω R ) a smooth representation for ea h obje t R . Equivalently, the same mani-fold stru ture an be hara terized as the unique manifold stru ture forwhi h an arbitrary mapping f : X → T is smooth if and only if so is ev R ◦ f for all R . The orresponden e R (cid:0) ω ( R ) , ev R (cid:1) , a ω ( a ) de-termines a faithful tensor fun tor ev of C into R ∞ ( T ( ω )) , whi h makes C ω $ $ IIIIIIIII ev / / R ∞ ( T ( ω )) w w nnnnnnnnn V ∞ ( M ) (9) ommute as a diagram of tensor fun tors (where the unlabelled arrow isthe standard forgetful fun tor of Ÿ13).Proof We only need to he k the assertions on erning the uniqueness ofthe smooth stru ture. Thus, suppose ev R smooth ∀ R . For onvenien e, let T ( ω ) ∗ denote the (cid:16)unknown(cid:17) manifold stru ture on the set T ( ω ) . Sin e thetopology of T ( ω ) ∗ is ne essarily (cid:28)ner than that of T ( ω ) , an open subsetof T ( ω ) must be in parti ular a tame submanifold of T ( ω ) ∗ . Therefore if (Ω , R ) is a representative hart, the homomorphism of Lie groupoids ev R : T ( ω ) ∗ → GL ( ωR ) restri ts to a smooth isomorphism of the open subset Ω ⊂ T ( ω ) ∗ onto the (tame) submanifold R (Ω) of GL ( ωR ) . Thus, we seethat the identity map is, lo ally in the domains of representative harts, adi(cid:27)eomorphism between T ( ω ) and T ( ω ) ∗ ; sin e representative harts over T ( ω ) , we get T ( ω ) ∗ = T ( ω ) , as was to be proved. q.e.d.For the sake of ompleteness, we also re ord the following re(cid:28)nement ofLemma 20.15, whi h may be regarded as a statement about the existen e ofrepresentative harts of a spe ial type:10 Corollary Let G be a proper Lie groupoid over a manifold M .Assume that ( E, ̺ ) is a lassi al representation of G , mapping a subset G ( x, x ′ ) inje tively into Lis( E x , E x ′ ) .Then there exist open balls B and B ′ in M , entred at x and x ′ respe tively, su h that the restri tion ̺ : G ( B, B ′ ) → GL ( E ) is an embedding of manifolds.23. SMOOTHNESS, REPRESENTATIVE CHARTS 131Proof To begin with, observe that for any given arrow g ∈ G ( x, x ′ ) and openneighbourhood Γ of g in G there is an open ball P inside GL ( E ) , entred at ̺ ( g ) , su h that ̺ − ( P ) ⊂ Γ . To see this, we (cid:28)x a sequen e · · · ⊂ P i +1 ⊂ P i ⊂ · · · ⊂ P of open balls inside GL ( E ) , entred at ̺ ( g ) and with lim i radius( P i ) = 0 , andthen we argue as in the proof of Theorem 20.5.By Lemma 20.15, every g ∈ G ( x, x ′ ) admits an open neighbourhood Γ g in G su h that ̺ indu es a smooth isomorphism between Γ g and a submanifoldof GL ( E ) . As observed above, one an then hoose an open ball P g ⊂ GL ( E ) at ̺ ( g ) su h that ̺ − ( P g ) ⊂ Γ g . Now, let Γ = S ̺ − ( P g ) . We laim that ̺ indu es a smooth isomorphism between Γ and a submanifold of GL ( E ) . By onstru tion, ̺ restri ts to an immersion of Γ into GL ( E ) . If g ∈ G ( x, x ′ ) then ̺ (Γ) ∩ P g = ̺ (cid:0) ̺ − ( P g ) (cid:1) is an open subset of the submanifold ̺ (Γ g ) ⊂ GL ( E ) , be ause ̺ is a smoothisomorphism of Γ g onto ̺ (Γ g ) . Sin e the open balls P g over ̺ (Γ) as g rangesover G ( x, x ′ ) , ̺ (Γ) is a submanifold of GL ( E ) . Moreover, sin e ̺ is a lo alsmooth isomorphism of Γ onto ̺ (Γ) , it will be also a global di(cid:27)eomorphismprovided it is globally inje tive over Γ : now, if ̺ ( γ ′ ) = ̺ ( γ ) then γ ′ , γ ∈ ̺ − ( P g ) ⊂ Γ g for some g and therefore γ ′ = γ be ause ̺ is inje tive over Γ g .Finally, one further appli ation of the usual properness argument willyield open balls B, B ′ ⊂ M at x, x ′ su h that G ( B, B ′ ) is ontained in Γ (thisis an open neighbourhood of G ( x, x ′ ) in G ). q.e.d.Note that the pre eding orollary entails in parti ular that the image ̺ ( G ) is a submanifold of GL ( E ) for every proper Lie groupoid G and faithful lassi al representation ( E, ̺ ) of G .Te hni al notes11 Note Suppose one is given an isomorphism E ≈ F of ve tor bundlesover a manifold M . Then there is an indu ed isomorphism of Lie groupoidsover M (ie one that restri ts to the identity mapping on M )(12) GL ( E ) ≈ −→ GL ( F ) ,given, for ea h ( x, x ′ ) ∈ M × M , by the bije tion that makes the linearisomorphisms α and β orrespond to ea h other when they (cid:28)t in the diagram E x ≈ x (cid:15) (cid:15) α / / E x ′ ≈ x ′ (cid:15) (cid:15) F x β / / F x ′ .(13)32 CHAPTER V. CLASSICAL FIBRE FUNCTORSIn parti ular, if two obje ts R, S ∈ Ob( C ) are isomorphi , any indu edisomorphism ω ( ≈ ) : ω ( R ) ≈ ω ( S ) will in turn yield an isomorphism of the orresponding linear groupoids GL ( ω R ) ≈ GL ( ω S ) (identi al on M ), su hthat for ea h λ ∈ T ( ω ) the linear mappings λ ( R ) and λ ( S ) orrespond toone another(cid:22)be ause of naturality of λ : ( ω R ) x ω ( ≈ ) x (cid:15) (cid:15) ω x ( R ) ω x ( ≈ ) (cid:15) (cid:15) λ ( R ) / / ω x ′ ( R ) ω x ′ ( ≈ ) (cid:15) (cid:15) ( ω R ) x ′ ω ( ≈ ) x ′ (cid:15) (cid:15) ( ω S ) x ω x ( S ) λ ( S ) / / ω x ′ ( S ) ( ω S ) x ′ .(14)Thus, the latter isomorphism will transform ev R into ev S : GL ( ω R ) O O ≈ (cid:15) (cid:15) T ( ω ) ev S , , YYYYYYYYYYYYYY ev R eeeeeeeeeeeeee GL ( ω S ) .(15)It follows that if Ω ⊂ T is any open subset then R (Ω) is a tame submanifoldof GL ( ωR ) if and only if S (Ω) is a tame submanifold of GL ( ωS ) (see, forinstan e, Note 22.21) and that R (Ω) and S (Ω) are homeomorphi subsets;hen e ev R will indu e a homeomorphism between Ω and R (Ω) if and only if ev S indu es one between Ω and S (Ω) .16 Note Let G and H be two Lie groupoids over the manifold M . Wewant to onstru t, provided this is possible, their produ t in the ategory ofLie groupoids over M . It ought to be a Lie groupoid over M endowed with anoni al proje tions, satisfying the usual universal property GK ψ / / ϕ / / ( ϕ,ψ ) / / ______ G × M H pr nnnnnnnnnnnnnn pr ( ( QQQQQQQQQQQQQQ H .(17)It must be kept in mind that all the arrows in this diagram are morphismsof Lie groupoids over M , ie they all indu e the identity map id : M → M atthe base level.The onstru tion of the produ t over M an be obtained as a spe ial aseof the so- alled (cid:16)strong (cid:28)bred produ t onstru tion(cid:17) for Lie groupoids, fr.for example Moerdijk and Mr£un (2003), [27℄ p. 123.Namely, we regard the maps / / (cid:15) (cid:15) G ( s , t ) (cid:15) (cid:15) (viz. G ( s , t ) (cid:15) (cid:15) ( s , t ) / / M × M ( pr , pr ) = id (cid:15) (cid:15) H ( s , t ) / / M × M M × M id × id / / M × M et .)23. SMOOTHNESS, REPRESENTATIVE CHARTS 133as morphisms of lie groupoids over M , where M × M is the pair groupoid,and apply the strong (cid:28)bred produ t onstru tion to them: ( set of arrows = (cid:8) ( g, h ) ∈ G × H : ( s , t )( g ) = ( s , t )( h ) (cid:9) ,set of obje ts = (cid:8) ( m, m ′ ) ∈ M × M : m = m ′ (cid:9) ∼ = M .Transversality riteria imply that this de(cid:28)nes a Lie groupoid G × M H over ∆( M ) ∼ = M whenever, for instan e, one of the two maps is a submersion.(Terminology: we say that a Lie groupoid G ⇒ M is lo ally transitive if themap ( s , t ) : G → M × M is a submersion. This appears to be reasonable,sin e G is said to be transitive if that map is a surje tive submersion.) More-over, if the trasversality ondition is satis(cid:28)ed, this onstru tion gives a (cid:28)bredprodu t with the familiar universal property.Suppose that G × M H makes sense, ie that the transversality ondition issatis(cid:28)ed. We remark that the universal property (17) is a onsequen e of theuniversal property of the pullba k. Indeed, (cid:28)rst of all, the two proje tionsof the (cid:28)bred produ t to its own fa tors are morphisms over M , as one seesdire tly at on e. Se ondly, if ϕ : K → G and ψ : K → H are morphismsover M , then the following diagram ommutes (pre isely by de(cid:28)nition ofmorphism over M ) K ψ (cid:15) (cid:15) ( s , t ) & & MMMMMM ϕ / / G ( s , t ) (cid:15) (cid:15) H ( s , t ) / / M × M and therefore there exists a unique morphism of Lie groupoids ( ϕ, ψ ) : K →G × M H su h that diagram (17) ommutes, so we need only verify that ( ϕ, ψ ) is in fa t a morphism over M . This follows at on e from the ommutativityof the diagram K ( s , t ) (cid:15) (cid:15) ( ϕ,ψ ) / / ϕ GGGGGGGGGG
G × M H ( s , t ) (cid:15) (cid:15) pr z z uuuuuuuuuu G ( s , t ) $ $ IIIIIIIIII M × M id × id / / M × M .Observation. By onstru tion, the manifold of arrows of G × M H is asubmanifold of the Cartesian produ t G × H ; it follows that the subsets ofthe form Γ × Λ , for Γ ⊂ G and Λ ⊂ H open, form a basis for the topologyof G × M H . (Of ourse, we write Γ × Λ but we mean (Γ × Λ) ∩ ( G × M H ) .)Thus, one sees immediately that, when the di(cid:27)erentiable stru ture is dis- arded, the same onstru tion yields the produ t in the ategory of topo-logi al groupoids over M .Now, we apply this general onstru tion to the lo ally transitive Liegroupoids GL ( E ) asso iated to ve tor bundles E ∈ Ob V ∞ ( M ) . (These are34 CHAPTER V. CLASSICAL FIBRE FUNCTORSlo ally transitive sin e if E U ≈ U × E and E V ≈ V × F are lo al trivializa-tions of E , then up to di(cid:27)eomorphism the map ( s , t ) oin ides lo ally witha proje tion GL ( E )( U, V ) ≈ U × V × Lis( E , F ) pr −→ U × V and is in parti ular a submersion; note that this makes sense even when Lis( E , F ) = ∅ .)Ÿ24 Morphisms of Fibre Fun torsA morphism of (cid:28)bre fun tors, let us say one ( C , ω ) → ( C ′ , ω ′ ) , onsists of asmooth map f : M → M ′ of the respe tive base manifolds together with alinear tensor fun tor Φ ∗ : C ′ −→ C and a tensor preserving isomorphism α C ′ ω ′ (cid:15) (cid:15) Φ ∗ / / C ω (cid:15) (cid:15) V ∞ ( M ′ ) f ∗ / / α ) V ∞ ( M ) ,(1)where f ∗ = pullba k along f . In pla e of the orre t ( f, Φ ∗ , α ) , our preferrednotation for morphisms of (cid:28)bre fun tors will be the in orre t ( f ∗ , Φ ∗ ) , inorder to emphasize the algebrai symmetry.Composition of morphisms is de(cid:28)ned as(2) ( g ∗ , Ψ ∗ ) · ( f ∗ , Φ ∗ ) = (cid:0) ( g ◦ f ) ∗ , Φ ∗ ◦ Ψ ∗ (cid:1) .Note that if in our de(cid:28)nition we required (1) to ommute in the stri t sense wewould get into trouble be ause ( g ◦ f ) ∗ ∼ = f ∗ ◦ g ∗ are anoni ally isomorphi but not really identi al tensor fun tors.Lemmas 9 and 11 below apply dire tly to (1) to yield maps(3) Hom ⊗ ( ω x , ω y ) Lem. 9 −−−−→
Hom ⊗ (cid:0) ω x ◦ Φ ∗ , ω y ◦ Φ ∗ (cid:1) = Hom ⊗ (cid:0) x ∗ ◦ ω ◦ Φ ∗ , y ∗ ◦ ω ◦ Φ ∗ (cid:1) ≈ (1) + Lem. 11 −−−−−−−−−→ Hom ⊗ (cid:0) x ∗ ◦ f ∗ ◦ ω ′ , y ∗ ◦ f ∗ ◦ ω ′ (cid:1) ∼ = Lem. 11 −−−−−−→
Hom ⊗ (cid:0) f ( x ) ∗ ◦ ω ′ , f ( y ) ∗ ◦ ω ′ (cid:1) = Hom ⊗ (cid:0) ω ′ f ( x ) , ω ′ f ( y ) (cid:1) .Moreover, sin e ( λ ◦ µ ) · Φ ∗ = ( λ · Φ ∗ ) ◦ ( µ · Φ ∗ ) and id · Φ ∗ = id , these anbe pie ed together in a fun torial way, so that they form a homomorphismof groupoids T ( ω ) (cid:15) (cid:15) Φ / / T ( ω ′ ) (cid:15) (cid:15) M × M f × f / / M ′ × M ′ ,(4)24. MORPHISMS OF FIBRE FUNCTORS 135whi h an be hara terized as the unique map making T ( ω ) ev Φ ∗ R ′ (cid:15) (cid:15) Φ / / T ( ω ′ ) ev R ′ (cid:15) (cid:15) GL ( ω Φ ∗ R ′ ) γ ◦ α − ∗ / / GL ( ω ′ R ′ ) (5) ommute for all R ′ ∈ Ob( C ′ ) , where the morphism γ is the (cid:16)proje tion(cid:17) GL ( f ∗ ( ω ′ R ′ )) ∼ = ( f × f ) ∗ ( GL ( ω ′ R ′ )) → GL ( ω ′ R ′ ) and the isomorphism(6) α ∗ : GL ( f ∗ ω ′ R ′ ) ∼ → GL ( ω Φ ∗ R ′ ) omes from α R ′ : f ∗ ω ′ R ′ ∼ → ω Φ ∗ R ′ a ording to Note 23.11. It is alsoimmediate from (5) that su h a solution Φ is ne essarily a morphism of C ∞ -fun tionally stru tured spa es, so (4) proves to be a homomorphism of C ∞ -fun tionally stru tured groupoids.We shall refer to Φ as the realization of the morphism ( f ∗ , Φ ∗ ) . This onstru tion is fun torial with respe t to omposition of morphisms of (cid:28)brefun tors, and therefore de(cid:28)nes a fun tor into the ategory of C ∞ -stru turedgroupoids, alled the realization fun tor.7 Proposition Let ( C , ω ) , ( C ′ , ω ′ ) be smooth lassi al (cid:28)bre fun torsand ( f ∗ , Φ ∗ ) : ( C , ω ) → ( C ′ , ω ′ ) a morphism of (cid:28)bre fun tors. Then the orresponding realization is ahomomorphism of Lie groupoids.Proof It follows from (5) that the omposite ev R ′ ◦ Φ is smooth for everyobje t R ′ of C ′ . The map Φ is then smooth by the (cid:16)universal property(cid:17) of theLie groupoid T ( ω ′ ) . q.e.d.Notes8 Note In this note we re all a ouple of elementary properties of tensorfun tors and tensor preserving natural transformations.9 Lemma Let F , G , S , T be tensor fun tors relating suitable tensor ategories. Then1. the rule λ λ · S maps Hom ⊗ ( F, G ) into Hom ⊗ ( F ◦ S, G ◦ S ) ;2. the rule λ T · λ maps Hom ⊗ ( F, G ) into Hom ⊗ ( T ◦ F , T ◦ G ) .36 CHAPTER V. CLASSICAL FIBRE FUNCTORSProof (1) The natural transformation ( λ · S )( X ) = λ ( SX ) is a morphism oftensor fun tors if su h is λ , be ause F SX ⊗ F SY ∼ = (cid:15) (cid:15) λ ( SX ) ⊗ λ ( SY ) / / GSX ⊗ GSY ∼ = (cid:15) (cid:15) ∼ = (cid:15) (cid:15) id / / ∼ = (cid:15) (cid:15) F ( SX ⊗ SY ) F ∼ = (cid:15) (cid:15) λ ( SX ⊗ SY ) / / G ( SX ⊗ SY ) G ∼ = (cid:15) (cid:15) F F ∼ = (cid:15) (cid:15) λ ( ) / / G G ∼ = (cid:15) (cid:15) F S ( X ⊗ Y ) λ ( S ( X ⊗ Y )) / / GS ( X ⊗ Y ) F S λ ( S ) / / GS .(2) The same an be said of ( T · λ )( X ) = T ( λ ( X )) , sin e T F X ⊗ T F Y ∼ = (cid:15) (cid:15) T λ ( X ) ⊗ T λ ( Y ) / / T GX ⊗ T GY ∼ = (cid:15) (cid:15) ∼ = (cid:15) (cid:15) id / / ∼ = (cid:15) (cid:15) T ( F X ⊗ F Y ) T ∼ = (cid:15) (cid:15) T ( λ ( X ) ⊗ λ ( Y )) / / T ( GX ⊗ GY ) T ∼ = (cid:15) (cid:15) T T ∼ = (cid:15) (cid:15) T ( id ) / / T T ∼ = (cid:15) (cid:15) T F ( X ⊗ Y ) T λ ( X ⊗ Y ) / / T G ( X ⊗ Y ) T F T λ ( ) / / T G .q.e.d.Let ( C , ⊗ ) and ( V , ⊗ ) be tensor ategories. Suppose that F, F ′ , G, G ′ : C −→ V are tensor fun tors, and that F ≈ F ′ , G ≈ G ′ are tensor preserving naturalisomorphisms. For every X ∈ Ob( C ) , there is an obvious bije tive map a a ′ determined by the ommutativity of F X ≈ (cid:15) (cid:15) a / / GX ≈ (cid:15) (cid:15) F ′ X a ′ / / G ′ X .(10)Given a natural transformation λ ∈ Hom(
F, G ) , we put λ ′ ( X ) = λ ( X ) ′ .11 Lemma The rule whi h to λ asso iates λ ′ determines a bije tive orresponden e(12) Hom ⊗ ( F, G ) ∼ → Hom ⊗ ( F ′ , G ′ ) .Proof Obvious. q.e.d.25. WEAK EQUIVALENCES 137Ÿ25 Weak Equivalen es1 De(cid:28)nition A weak equivalen e1 of (cid:28)bre fun tors, symboli ally ( C , ω ) ≈ −→ ( C ′ , ω ′ ) , is a morphism of (cid:28)bre fun tors ( f ∗ , Φ ∗ ) : ( C , ω ) → ( C ′ , ω ′ ) satisfying the following two onditions1. the base mapping f : M → M ′ is a surje tive submersion;2. the fun tor Φ ∗ is a tensor equivalen e, ie there exist a tensor fun tor Φ ∗ : C −→ C ′ and tensor preserving natural isomorphisms (cid:26) Φ ∗ ◦ Φ ∗ ≈ Id C Φ ∗ ◦ Φ ∗ ≈ Id C ′ .In order to on lude that Φ ∗ is a tensor equivalen e, it su(cid:30) es to know it tobe an ordinary ategori al equivalen e. Every quasi-inverse equivalen e Φ ∗ isthen ne essarily a linear fun tor. (Details may be found in Note 10.) Weakequivalen es of (cid:28)bre fun tors are stable under omposition of morphisms of(cid:28)bre fun tors, as de(cid:28)ned in Se tion 24.2 Proposition Let ( f ∗ , Φ ∗ ) : ( C , ω ) ≈ −→ ( C ′ , ω ′ ) be a weak equivalen e of (cid:28)bre fun tors. Then its realization diagram T ( ω ) (cid:15) (cid:15) Φ / / T ( ω ′ ) (cid:15) (cid:15) M × M f × f / / M ′ × M ′ (3)is a topologi al pullba k, ie a pullba k in the ategory of topologi alspa es, and Φ : T ( ω ) ։ T ( ω ′ ) is a surje tive open mapping.Proof Let T be a topologi al spa e, and suppose given a problem T " " a % % & & T ( ω ) ev Φ ∗ R ′ (cid:15) (cid:15) Φ / / T ( ω ′ ) ev R ′ (cid:15) (cid:15) GL ( ω Φ ∗ R ′ ) (cid:15) (cid:15) γ ◦ α − ∗ / / GL ( ω ′ R ′ ) (cid:15) (cid:15) M × M f × f / / M ′ × M ′ (4)1Note on terminology: We shall reserve the term `weak equivalen e' for the ontextof (cid:28)bre fun tors. When dealing with Lie groupoids, we prefer to use the term `Moritaequivalen e'.38 CHAPTER V. CLASSICAL FIBRE FUNCTORSstated in the ategory of topologi al spa es and ontinuous mappings. Thereexists a unique set-theoreti solution a , be ause (3) is already known to bea set-theoreti pullba k (by Note 10 again). Thus, we must he k that a is ontinuous. Note that ∀ R in C , ev R ◦ a is ontinuous if and only if ev Φ ∗ Φ ∗ R ◦ a is ontinuous, be ause of the isomorphism Φ ∗ Φ ∗ R ≈ R , see also the ommentsin Note 23.11. Therefore, if we put R ′ = Φ ∗ R in (4), we on lude at on ethat ev Φ ∗ R ′ ◦ a is ontinuous from the fa t that the lower square of (4) is, byde(cid:28)nition, a topologi al pullba k.Next, observe that if one has a topologi al pullba k X p (cid:15) (cid:15) f / / Y q (cid:15) (cid:15) M g / / N (5)along a submersive morphism g of smooth manifolds, there is the followinglo al de omposition up to di(cid:27)eomorphism X Up (cid:15) (cid:15) f / / Y Vq (cid:15) (cid:15) U g / / V X Up (cid:15) (cid:15) ≈ Y V × P q × id (cid:15) (cid:15) pr / / Y Vq (cid:15) (cid:15) U ≈ V × P pr / / V ,(6)where U ⊂ M is open and so small that, up to di(cid:27)eomorphism, g | U is aproje tion V × P → V = g ( U ) for some open ball P ; of ourse, X U = p − ( U ) et . (Note that in (6), U ≈ V × P is a di(cid:27)eomorphism whereas X U ≈ Y V × P is a homeomorphism.) It follows that f is a `topologi al submersion',in parti ular an open mapping; in addition, if g is surje tive then it is learthat f must be also surje tive. This shows that the statement that Φ isan open mapping follows from the statement that (25.3) is a topologi alpullba k. q.e.d.Suppose a topologi al pullba k (5) along a smooth submersion is given,and let U ⊂ M be an open subset su h that g | U is, up to di(cid:27)eomorphism,a proje tion U ≈ V × P pr −→ V onto an open subset V ⊂ N . Let A ⊂ X bean open subset, and put B = f ( A ) ; B ⊂ Y is open be ause f is an openmapping. We shall be interested in the subspa es p ( A ) ⊂ M and q ( B ) ⊂ N ;note that g restri ts to a ontinuous mapping of p ( A ) onto q ( B ) . Assumethat A has the following property: the ommutative square A ∩ p − ( U ) p (cid:15) (cid:15) f / / B ∩ q − ( V ) q (cid:15) (cid:15) U g / / V (7)25. WEAK EQUIVALENCES 139is a topologi al pullba k. Then there is a trivialization, analogous to (6), whi hshows that the smooth iso U ≈ V × P indu es a orresponden e between p ( A ) ∩ U = p (cid:0) A ∩ p − ( U ) (cid:1) and (cid:0) q ( B ) ∩ V (cid:1) × P = q (cid:0) B ∩ q − ( V ) (cid:1) × P .Thus, ∀ u ∈ U one has u ∈ p ( A ) ⇔ g ( u ) ∈ q ( B ) . Note also that p restri tsto a homeomorphism of A ∩ p − ( U ) onto p ( A ) ∩ U if and only if q restri tsto a homeomorphism of B ∩ q − ( V ) onto q ( B ) ∩ V . The two relevant asesfor the present dis ussion o ur, in the (cid:28)rst pla e, when A = f − ( f ( A )) , andse ondly, when A ⊂ p − ( U ) oin ides with B × P in the trivialization (6).Fix an obje t R ′ ∈ Ob( C ′ ) . Then the outer re tangle of (4) is a topo-logi al pullba k(cid:22)note that it oin ides with (3); the lower square enjoysthe same property. Consequently, the upper square, viz (24.5), must be atopologi al pullba k as well; moreover, sin e the smooth mapping γ ◦ α − ∗ : GL ( ω Φ ∗ R ′ ) → GL ( ω ′ R ′ ) is a (surje tive) submersion, it is a pullba k of theform (6). Hen e the pre eding remarks apply, and we get:1. If (Ω ′ , R ′ ) is a representative hart of ( C ′ , ω ′ ) then (Φ − (Ω ′ ) , Φ ∗ R ′ ) is arepresentative hart of ( C , ω ) . Sin e diagram (24.5) is a topologi al pullba k, Φ − (Ω ′ ) ev Φ ∗ R ′ (cid:15) (cid:15) Φ / / Ω ′ ev R ′ (cid:15) (cid:15) Φ ∗ R ′ (Φ − (Ω ′ )) γ ◦ α − ∗ / / R ′ (Ω ′ ) is also a topologi al pullba k and therefore ev Φ ∗ R ′ indu es a homeomorph-ism between Φ − (Ω ′ ) and its image Φ ∗ R ′ (Φ − (Ω ′ )) , be ause ev R ′ , on theright, does the same. Proposition 22.11 implies that Φ ∗ R ′ (Φ − (Ω ′ )) is a tamesubmanifold of GL ( ω Φ ∗ R ′ ) if and only if R ′ (Ω ′ ) is a tame submanifold of GL ( ω ′ R ′ ) , be ause γ ◦ α − ∗ is a Morita equivalen e and Ω ′ = Φ(Φ − (Ω ′ )) .2. Let Ω ⊂ T ( ω ) be an open subset and λ ∈ Ω . For any given obje t R ′ ∈ Ob( C ′ ) , there is a smaller open neighbourhood λ ∈ Ω ⊂ Ω su h that (Ω , Φ ∗ R ′ ) is a representative hart of ( C , ω ) if and only if (Φ(Ω ) , R ′ ) is arepresentative hart of ( C ′ , ω ′ ) . Let Λ be an open neighbourhood of λ (Φ ∗ R ′ ) in GL ( ω Φ ∗ R ′ ) su h that γ ◦ α − ∗ | Λ is, up to di(cid:27)eomorphism, a proje tion Λ ′ × P → Λ ′ = γ ◦ α − ∗ (Λ) . Making the open ball P , and thus Λ , smallerif ne essary, we (cid:28)nd an open neighbourhood Ω ⊂ ev − ∗ R ′ (Λ) ∩ Ω of λ su hthat the homeomorphism ev − ∗ R ′ (Λ) ≈ ev − R ′ (Λ ′ ) × P of (6) produ es a de- omposition Ω (cid:15) (cid:15) Φ / / Φ(Ω ) (cid:15) (cid:15) Λ γ ◦ α − ∗ / / Λ ′ Ω (cid:15) (cid:15) ≈ Φ(Ω ) × P × id (cid:15) (cid:15) pr / / Φ(Ω ) (cid:15) (cid:15) Λ ≈ Λ ′ × P pr / / Λ ′ (8)40 CHAPTER V. CLASSICAL FIBRE FUNCTORSTherefore, if we put Σ = Φ ∗ R ′ (Ω ) ⊂ Λ and Σ ′ = R ′ (Φ(Ω )) ⊂ Λ ′ we have λ ∈ Σ ⇔ γ ◦ α − ∗ λ ∈ Σ ′ for all λ ∈ Λ , and Proposition 22.11 implies that Σ is a tame submanifold of GL ( ω Φ ∗ R ′ ) if and only if Σ ′ is a tame submanifoldof GL ( ω ′ R ′ ) , sin e γ ◦ α − ∗ is a Morita equivalen e.Clearly, these statements imply that whenever a weak equivalen e of (cid:28)brefun tors ( C , ω ) ≈ −→ ( C ′ , ω ′ ) is given, Condition i) of Proposition 23.4 holds for ( C , ω ) if and only if it holds for ( C ′ , ω ′ ) . (As a onsequen e of the fa t that Φ is surje tive and open: Fix λ ′ = Φ( λ ) . If (Ω , R ) is a hart at λ , then (Φ(Ω ) , Φ ∗ R ) is a hart at λ ′ for some open λ ∈ Ω ⊂ Ω ; onversely, if (Ω ′ , R ′ ) is a hart at λ ′ then (Φ − (Ω ′ ) , Φ ∗ R ′ ) is a hart at λ .)On the other hand, they also imply invarian e of Condition ii) of the sameproposition, p. 128, as follows. Assume the ondition holds for ( C ′ , ω ′ ) : Let (Ω , R ) be a hart of ( C , ω ) and S ∈ Ob( C ) an obje t. Choose a point λ ∈ Ω .There exists a neighbourhood Ω ⊂ Ω of λ su h that (Φ(Ω ) , Φ ∗ R ) , and on-sequently (Φ(Ω ) , Φ ∗ R ⊕ Φ ∗ S ) , is a hart of ( C ′ , ω ′ ) . Sin e Ω ⊂ Φ − Φ(Ω ) and Φ ∗ (Φ ∗ R ⊕ Φ ∗ S ) ≈ R ⊕ S , it follows that (Ω , R ⊕ S ) is a hart of ( C , ω ) .Sin e λ was arbitrary, we on lude that Ω an be overed with open sub-sets Ω su h that (Ω , R ⊕ S ) is a hart, and therefore that (Ω , R ⊕ S ) is a hart as well. Conversely, assume Condition 2 holds for ( C , ω ) : Let (Ω ′ , R ′ ) be a hart of ( C ′ , ω ′ ) and S ′ ∈ Ob( C ′ ) an obje t. Fix a point λ ′ ∈ Ω ′ ;sin e Φ is surje tive, ∃ λ with λ ′ = Φ( λ ) . Sin e (Φ − (Ω ′ ) , Φ ∗ R ′ ) is a hart, (Φ − (Ω ′ ) , Φ ∗ R ′ ⊕ Φ ∗ S ′ ) and, onsequently, (Φ − (Ω ′ ) , Φ ∗ ( R ′ ⊕ S ′ )) are hartsof ( C , ω ) as well. Hen e there exists a neighbourhood Ω ⊂ Φ − (Ω ′ ) of λ su hthat (Φ(Ω ) , R ′ ⊕ S ′ ) is a hart of ( C ′ , ω ′ ) . As before, sin e λ ′ was arbitraryit follows that (Ω ′ , R ′ ⊕ S ′ ) is a hart of ( C ′ , ω ′ ) .We an olle t our on lusions in the following9 Proposition Let ( f ∗ , Φ ∗ ) : ( C , ω ) −→ ( C ′ , ω ′ ) be a weak equivalen e of (cid:28)bre fun tors. Then ( C , ω ) is a smooth lassi al(cid:28)bre fun tor if and only if so is ( C ′ , ω ′ ) . In this ase, ( f, Φ) : T ( ω ) −→ T ( ω ′ ) is a Morita equivalen e of Lie groupoids.Proof That (24.4) is a pullba k in the ategory of manifolds of lass C ∞ follows by the same argument used in the proof of Proposition 2, be ause ofthe universal property of the Tannakian groupoid. q.e.d.25. WEAK EQUIVALENCES 141Notes10 Note List of elementary fa ts.1. Any quasi-inverse equivalen e Φ ∗ is automati ally a linear fun tor.Indeed, the map Hom C ( R, S ) → Hom C (Φ ∗ Φ ∗ R, Φ ∗ Φ ∗ S ) , a Φ ∗ Φ ∗ a is a linear bije tion, as it is lear from the ommutativity of Φ ∗ Φ ∗ R Φ ∗ Φ ∗ a (cid:15) (cid:15) ≈ R / / R a (cid:15) (cid:15) Φ ∗ Φ ∗ S ≈ S / / S ,and the fun tor Φ ∗ is linear and, being a ategori al equivalen e, faithful,hen e the equality Φ ∗ Φ ∗ ( αa + βb ) = α Φ ∗ Φ ∗ a + β Φ ∗ Φ ∗ b = Φ ∗ ( α Φ ∗ a + β Φ ∗ b ) implies the desired linearity Φ ∗ ( αa + βb ) = α Φ ∗ a + β Φ ∗ b .2. The realization Φ : T ( ω ) −→ T ( ω ′ ) of a weak equivalen e is a fullyfaithful morphism of groupoids, in other words (3) is a set-theoreti pullba k.This an be seen as follows.The tensor preserving isomorphism Φ ∗ ◦ Φ ∗ ≈ Id C gives, a ording toLemma 24.9 p. 135, a tensor preserving isomorphism(11) ω x ≈ ω x ◦ Φ ∗ ◦ Φ ∗ ≈ ω ′ f ( x ) ◦ Φ ∗ ;similarly, Φ ∗ ◦ Φ ∗ ≈ Id C ′ yields another su h isomorphism(12) ω ′ f ( x ) ≈ ω ′ f ( x ) ◦ Φ ∗ ◦ Φ ∗ .If now we apply Lemma 24.11 p. 136 to these, we on lude at on e from the ommutativity of the diagram Hom ⊗ ( ω x , ω y ) (11) ≈ (cid:15) (cid:15) Φ x,y / / Hom ⊗ (cid:0) ω ′ f ( x ) , ω ′ f ( y ) (cid:1) (12) ≈ (cid:15) (cid:15) s s hhhhhhhhhhhhhhhhhhh Hom ⊗ (cid:0) ω ′ f ( x ) Φ ∗ , ω ′ f ( y ) Φ ∗ (cid:1) / / Hom ⊗ (cid:0) ω ′ f ( x ) Φ ∗ Φ ∗ , ω ′ f ( y ) Φ ∗ Φ ∗ (cid:1) that the diagonal arrow is a surje tive and inje tive map, and hen e that Φ x,y is bije tive. (The ommutativity of the two triangles follows from the ommutativity of the two squares ω x (Φ ∗ Φ ∗ R ) λ Φ ∗ Φ ∗ R / / ω y (Φ ∗ Φ ∗ R ) ω ′ f ( x ) ( R ′ ) ω ′ f ( x ) ≈ (cid:15) (cid:15) λ R ′ / / ω ′ f ( y ) ( R ′ ) ω ′ f ( y ) ≈ (cid:15) (cid:15) ω x ( R ) ω x ≈ O O λ R / / ω y ( R ) ω y ≈ O O ω ′ f ( x ) (Φ ∗ Φ ∗ R ′ ) λ Φ ∗ Φ ∗ R ′ / / ω ′ f ( y ) (Φ ∗ Φ ∗ R ′ ) expressing naturality of λ, λ ′ respe tively.)42 CHAPTER V. CLASSICAL FIBRE FUNCTORS13 Note Let X and Y be topologi al spa es, and let M and N be smoothmanifolds. Suppose X p (cid:15) (cid:15) f / / Y q (cid:15) (cid:15) M g / / N (14)is a pullba k diagram in the ategory of topologi al spa es, where g is asmooth mapping.1. Given an open subset B ⊂ Y , put A = f − ( B ) . Then the ontinuousmaps in (14) restri t to a ommutative diagram of topologi al spa es A p (cid:15) (cid:15) f / / B q (cid:15) (cid:15) p ( A ) g / / q ( B ) ,(15)whi h is again a topologi al pullba k. Observe that if the restri tion q | B indu es a homeomorphism of B onto q ( B ) , then p | A indu es one between A and p ( A ) . (This is a general property of pullba ks. Indeed, from C g / / p ′ id " " D q − (cid:24) (cid:24) A p (cid:15) (cid:15) f / / B q (cid:15) (cid:15) C g / / D and from the equalities f p ′ p = f and p p ′ p = p , it follows that p ′ p = id , thus p is invertible.)2. Given an open subset U ⊂ M su h that V = g ( U ) is open, p − ( U ) f / / p (cid:15) (cid:15) q − ( V ) q (cid:15) (cid:15) U g / / V (16)makes sense and is learly also a topologi al pullba k.hapter VIStudy of Classi al TannakaTheory of Lie GroupoidsIn this on lusive hapter we are ideally going ba k to the point where westarted from, namely the theory of lassi al representations of Lie groupoidsexpounded in Ÿ2. We will try to see what an be said about su h theoryby the light of the general results of Chapters IV(cid:21)V. In parti ular, we willstudy in detail the standard lassi al (cid:28)bre fun tor asso iated with a Liegroupoid. Re all that in Ÿ2 we introdu ed the ategory R ∞ ( G ) of lassi alrepresentations R = ( E, ̺ ) of a Lie groupoid G , along with the standard lassi al (cid:28)bre fun tor ω ∞ ( G ) de(cid:28)ned as the forgetful fun tor ( E, ̺ ) E of R ∞ ( G ) into the ategory V ∞ ( M ) of smooth ve tor bundles of lo ally (cid:28)niterank over the base M of G . Let us give a brief review of the items we willbe interested in, so as to (cid:28)x the ta it notational onventions to be followedthroughout the hapter.Let T ∞ ( G ) denote the Tannakian groupoid T ( ω ∞ ( G ); R ) asso iated withthe (cid:28)bre fun tor ω ∞ ( G ) . Note that it does not make any di(cid:27)eren e whetherwe use real or omplex oe(cid:30) ients in our theory, be ause eventually thegroupoid T ∞ ( G ) and the other related items dis ussed below will be exa tlythe same; in fa t, all what we are going to say holds for real as well asfor omplex oe(cid:30) ients: for simpli ity, we assume real oe(cid:30) ients wheneverwe need to write them down expli itly. Re all from Ÿ21 that the Tannakian onstru tion de(cid:28)nes an operation G 7→ T ∞ ( G ) , (cid:8) Lie groupoids (cid:9) −→ (cid:8) C ∞ -fun . stru tured groupoids (cid:9) ;also note that the sour e and target map of T ∞ ( G ) are submersions, in thesense that they admit lo al se tions whi h are morphisms of fun tionallystru tured spa es: this follows from the existen e of su h se tions for G andthe fa t that the envelope homomorphism π ∞ (see below) is a morphism offun tionally stru tured spa es.Next, observe that for ea h Lie groupoid homomorphism ϕ : G → H the onstru tions of Ÿ24 may be applied to the equation ω ∞ ( G ) ◦ ϕ ∗ = f ∗ ◦ ω ∞ ( H ) (identity of tensor fun tors), so as to yield a homomorphismof C ∞ -fun tionally stru tured groupoids T ∞ ( ϕ ) : T ∞ ( G ) → T ∞ ( H ) .In spite of the la k of fun toriality of the operation ϕ ϕ ∗ , in other wordsin spite of ( ψ ◦ ϕ ) ∗ ∼ = ϕ ∗ ◦ ψ ∗ being anoni ally isomorphi but not equal,the orresponden e ϕ
7→ T ∞ ( ϕ ) a tually turns out to be a fun tor, i.e. theidentities T ∞ ( ψ ◦ ϕ ) = T ∞ ( ψ ) ◦ T ∞ ( ϕ ) and T ∞ ( id ) = id hold.We let π ∞ ( G ) or, when there is no ambiguity, π ∞ denote the envelopehomomorphism G → T ∞ ( G ) de(cid:28)ned by π ∞ ( g )( E, ̺ ) = ̺ ( g ) . The resultsof Ÿ20 on erning envelope homomorphisms an be applied. In parti ular, π ∞ ( G ) will be a morphism of C ∞ -fun tionally stru tured groupoids. The orresponden e G 7→ π ∞ ( G ) determines, in fa t, a natural transformation π ∞ ( - ) : ( - )
7→ T ∞ ( - ) , that is to say the diagram below ommutes for ea hLie groupoid homomorphism ϕ : G → HG ϕ (cid:15) (cid:15) π ∞ ( G ) / / T ∞ ( G ) T ∞ ( ϕ ) (cid:15) (cid:15) H π ∞ ( H ) / / T ∞ ( H ) .The main result of the present hapter, to be proved in Ÿ27, is: for G proper and regular, the standard lassi al (cid:28)bre fun tor ω ∞ ( G ) is smooth; infa t, T ∞ ( G ) is a proper regular Lie groupoid although, in general, not oneequivalent to G . Furthermore, in Ÿ26 we prove some partial results about thesmoothness of the standard lassi al (cid:28)bre fun tor, that are valid for arbitraryproper Lie groupoids; we also remark that the evaluation fun tor ev : R ∞ ( G ) −→ R ∞ ( T ∞ ( G )) , R = ( E, ̺ ) ( E, ev R ) is an isomorphism of tensor ategories for ea h proper G (re all the de(cid:28)nitionof the ategory R ∞ ( T ∞ ( G )) in Ÿ21). Finally, in Ÿ28 we give a few examplesof lassi ally re(cid:29)exive (proper) Lie groupoids.Ÿ26 On the Classi al Envelope of a Proper LieGroupoidLet G be a Lie groupoid. Re all from Ÿ21 that to ea h lassi al representation R = ( E, ̺ ) of G one an asso iate a representation ev R : T ∞ ( G ) → GL ( E ) ,given by evaluation at the obje t R ∈ Ob R ∞ ( G ) :(1) T ∞ ( G )( x, x ′ ) ∋ λ λ ( R ) ∈ Lis( E x , E x ′ ) ,26. THE CLASSICAL ENVELOPE OF A PROPER GROUPOID 145whi h makes the following triangle ommute G ̺ % % KKKKKKKKKKK π ∞ ( G ) / / T ∞ ( G ) ev R w w ooooooooooo GL ( E ) ,(2)where π ∞ ( G ) denotes the envelope homomorphism π ∞ ( g )( E, ̺ ) = ̺ ( g ) .Throughout the present se tion we shall be interested mainly in properLie groupoids. Therefore, from now on we assume that G is a proper Liegroupoid and we regard this assumption as made on e and for all. As ever, M will denote the base manifold of G . When we want to state a result thatis true under less restri tive assumptions on G , we shall expli itly point itout. We are going to apply the general theory of representative harts (Ÿ23)to the standard lassi al (cid:28)bre fun tor ω ∞ ( G ) .3 Lemma Let ( E, ̺ ) be a lassi al representation of a (not ne essarilyproper) Lie groupoid G . Suppose we are given an open subset Γ of themanifold of arrows of G , su h that the image Σ = ̺ (Γ) is a submanifoldof GL ( E ) and su h that ̺ restri ts to an open mapping of Γ onto Σ .Then Σ is a tame submanifold of GL ( E ) , and the restri tion of ̺ to Γ is a submersion of Γ onto Σ .Moreover, when G is proper then the assumption that ̺ should restri tto an open mapping of Γ onto Σ is super(cid:29)uous.Proof We prove the statement in the proper ase (cid:28)rst, so without makingthe assumption that ̺ is an open map of Γ onto Σ .We start by observing that for ea h x ∈ M the image ̺ (cid:0) G ( x , - ) (cid:1) is aprin ipal submanifold of GL ( E ) and the mapping(4) G ( x , - ) ̺ −→ ̺ (cid:0) G ( x , - ) (cid:1) is a submersion. In parti ular, the latter will be an open mapping and thisfor es the open subset(5) Σ( x , - ) = ̺ (cid:0) G ( x , - ) ∩ Γ (cid:1) ⊂ ̺ G ( x , - ) to be a prin ipal submanifold of GL ( E ) as well.Our argument is as follows. Fix g in G ( x , - ) and let λ = ̺ ( g ) . Choosean open subset V ⊂ M ontaining x ′ = t ( g ) , small enough to ensure thatthe prin ipal bundle G ( x , - ) is trivial over Z = G x ∩ V , ie that a lo alequivariant hart G ( x , Z ) ≈ Z × G an be found, where G denotes theisotropy group at x ; it is no loss of generality to assume g ≈ ( x ′ , e ) insu h a hart whi h we now use, along with the representation ̺ , to obtaina smooth se tion z ( z, e ) ≈ g ̺ ( g ) to the target map of GL ( E ) over46 CHAPTER VI. CLASSICAL TANNAKA THEORY Z . Next, the isotropy homomorphism G → GL ( E ) determined by ̺ at x anoni ally fa tors through the quotient Lie group obtained by dividingout the kernel, thus yielding a losed Lie subgroup H ֒ → GL ( E ) . As usual,this Lie subgroup and the target se tion above an be ombined into anembedding of manifolds of type (22.2), whi h (cid:28)ts in the following square Z × G id × pr (cid:15) (cid:15) ≈ / / G ( x , Z ) ̺ (cid:15) (cid:15) Z × H (cid:31) (cid:127) (22.2) / / GL ( E ) (6)and hen e simultaneously displays ̺ G ( x , Z ) as a prin ipal submanifold of GL ( E ) and, a ording to the initial remarks of Se tion 22, the mapping ̺ : G ( x , Z ) → ̺ G ( x , Z ) as a submersion; sin e the subset(7) ̺ G ( x , Z ) = ̺ G ( x , - ) ∩ t − ( V ) ⊂ ̺ G ( x , - ) is an open neighborhood of λ in ̺ G ( x , - ) , we an on lude.At this point, in order to prove that Σ is a tame submanifold of GL ( E ) we need only verify that the restri tion Σ → M of the sour e map of GL ( E ) is a submersion. So, (cid:28)x σ ∈ Σ , say σ = ̺ ( g ) with g ∈ Γ . There exists alo al smooth sour e se tion γ : U → Γ through g = γ ( s g ) , hen e we analso (cid:28)nd a lo al smooth sour e se tion σ = ̺ ◦ γ : U → Σ through σ .Finally, we ome to the statement that ̺ : Γ → Σ is a submersion. Fix g ∈ Γ and let σ = ̺ ( g ) . Sin e both Γ and Σ are tame submanifolds, thereexist lo al trivializations of the respe tive sour e maps around the points g ≈ ( x , and σ ≈ ( x , , whi h yield a lo al expression for ̺ | Γ Γ ≈ (cid:15) (cid:15) ̺ / / Σ ≈ (cid:15) (cid:15) U × B / / ___ V × C (8)of the form ( u, b ) ( u, c ( u, b )) , where U ⊂ V are open subsets of M and B, C are Eu lidean balls. The partial map b c ( x , b ) is submersive at theorigin be ause it is the lo al expression of (4).Now we turn to the general ase where G is not ne essarily proper. Thus,assume that ̺ restri ts to an open mapping of Γ onto Σ .As explained above, for any given g ∈ G ( x , - ) there is a submanifold Z ⊂ M ontained in G x (cid:22)although, in general, this is no longer of the form Z = G x ∩ V (cid:22)su h that the subset G ( x , Z ) ⊂ G ( x , - ) is open, the image ̺ G ( x , Z ) is a prin ipal submanifold of GL ( E ) and the indu ed mapping ̺ : G ( x , Z ) → ̺ G ( x , Z ) is submersive. On the other hand, from the assumptionthat ̺ : Γ → Σ is open it follows that the restri tion ̺ : Γ( x , - ) → Σ( x , - ) must be open as well, be ause one has(9) ̺ (cid:0) X( x , - ) (cid:1) = ̺ (X)( x , - )
26. THE CLASSICAL ENVELOPE OF A PROPER GROUPOID 147for any subset X ⊂ G (1) . Then, sin e Γ ∩ G ( x , Z ) = Γ( x , - ) ∩ G ( x , Z ) is anopen subset of Γ( x , - ) , it is evident that(10) Σ( x , Z ) = ̺ (cid:0) Γ ∩ G ( x , Z ) (cid:1) ⊂ ̺ G ( x , Z ) is both an open neighbourhood of λ in Σ( x , - ) and an open subset of theprin ipal submanifold ̺ G ( x , Z ) of GL ( E ) . This means that Σ( x , - ) is aprin ipal submanifold of GL ( E ) . Moreover, from what we said it is evidentthat ̺ indu es a submersion of Γ( x , - ) onto Σ( x , - ) .The rest of the proof holds without modi(cid:28) ations. q.e.d.Note that the pre eding lemma holds for real as well as for omplex oe(cid:30) ients(cid:22)that is, for ( E, ̺ ) in R ∞ ( G , R ) or in R ∞ ( G , C ) .Our main goal in the present se tion is to show that the standard lassi al(cid:28)bre fun tor ω ∞ ( G ) asso iated with a proper Lie groupoid G always satis(cid:28)es ondition ii) of Proposition 23.4.First of all, note that in order that (Ω , R ) may be a representative hartof T ∞ ( G ) , where Ω is an open subset of the spa e of arrows of T ∞ ( G ) and R = ( E, ̺ ) ∈ Ob R ∞ ( G ) , it is su(cid:30) ient that ev R establishes a one-to-one orresponden e between Ω and a submanifold of GL ( E ) . For if we set Γ =( π ∞ ) − (Ω) , we have ̺ (Γ) = R (Ω) be ause of (2) and the surje tivity of π ∞ ;then Lemma 3 implies that R (Ω) is a tame submanifold of GL ( E ) and that ̺ : Γ → R (Ω) is a submersion(cid:22)so, in parti ular, that the map ev R : Ω → R (Ω) is open and hen e a homeomorphism.Our laim about the ondition ii) of Proposition 23.4 essentially followsfrom a simple general remark about submersions. Namely, suppose that a ommutative triangle of the form X g (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) Y f ′ + + XXXXXXXXXXXXXXX f fffffffffffffff X ′ (11)is given, where X , X ′ and Y are smooth manifolds, f is a submersion onto X , f ′ is a smooth mapping and all we know about g is that it is a set-theoreti solution whi h (cid:28)ts in the triangle. Then the map g is ne essarily smooth; inparti ular, in ase f ′ is also a surje tive submersion, g is a di(cid:27)eomorphism ifand only if it is a set-theoreti bije tion.To see how this may be used to prove ompatibility of harts, suppose weare given an arbitrary representative hart (Ω , R ) of T ∞ ( G ) to start with,where let us say R = ( E, ̺ ) , and an arbitrary lassi al representation S =( F, σ ) . Let Γ = ( π ∞ ) − (Ω) , so that Γ is an open submanifold of G . We havealready observed that ̺ indu es a submersion of Γ onto the submanifold R (Ω) of GL ( E ) ; also, the homomorphism of Lie groupoids(12) ( ̺, σ ) : G −→ GL ( E ) × M GL ( F )
48 CHAPTER VI. CLASSICAL TANNAKA THEORY an be restri ted to Γ to yield a smooth mapping into GL ( E ) × M GL ( F ) .We get an instan e of (11) by introdu ing the following map(13) s = ( ev R , ev S ) ◦ ev R − : R (Ω) → GL ( E ) × M GL ( F ) (note that ev R : Ω → R (Ω) is invertible be ause we assume (Ω , R ) to be arepresentative hart), whi h is then a smooth se tion to the proje tion(14) GL ( E ) × M GL ( F ) → GL ( E ) and thus, in parti ular, an immersion. Now, if s is indeed the embedding of asubmanifold(cid:22)ie if it is an open map onto its image(cid:22)then we are done, sin ein that ase ( R, S )(Ω) = s ( R (Ω)) is a submanifold of GL ( E ) × M GL ( F ) and ( ev R , ev S ) a bije tive map onto it; equivalently, ( R ⊕ S )(Ω) is a submanifoldof GL ( E ⊕ F ) and ev R ⊕ S is a bije tion of Ω onto it. (Cf. Se tion Ÿ23. Asobserved above, this is enough to on lude that (Ω , R ⊕ S ) is a representative hart.) For ea h open subset Λ of GL ( E ) ,(15) s (cid:0) R (Ω) ∩ Λ (cid:1) = s ( R (Ω)) ∩ (cid:0) Λ × GL ( F ) (cid:1) is in fa t an open subset of the subspa e s ( R (Ω)) .We an summarize what we have on luded so far as follows:16 Proposition Let G be a proper Lie groupoid.Then the standard lassi al (cid:28)bre fun tor ω ∞ ( G ) is smooth if and onlyif the spa e of arrows of the lassi al Tannakian groupoid T ∞ ( G ) anbe overed with open subsets Ω su h that for ea h of them one an (cid:28)ndsome R = ( E, ̺ ) ∈ Ob R ∞ ( G ) with the property that ev R establishes abije tion between Ω and a submanifold R (Ω) of GL ( E ) .Moreover, in ase the latter ondition is satis(cid:28)ed then the envelopehomomorphism π ∞ ( G ) : G −→ T ∞ ( G ) will be a surje tive submersion ofLie groupoids.Proof The (cid:28)rst assertion is already proven.The se ond assertion follows from the (previously noti ed) fa t that forea h representative hart (Ω , R ) the mapping ̺ : Γ → R (Ω) is a submersion,where as usual R = ( E, ̺ ) and we put Γ = ( π ∞ ) − (Ω) . (Remember from theproof of Prop. 23.4 that ev R establishes a di(cid:27)eomorphism between Ω and thesubmanifold R (Ω) of GL ( E ) .) q.e.d.Note that, for any proper Lie groupoid G whose asso iated standard las-si al (cid:28)bre fun tor ω ∞ ( G ) is smooth, the pre eding proposition allows us to hara terize the familiar Lie groupoid stru ture on the Tannakian groupoid T ∞ ( G ) as the unique su h stru ture for whi h the envelope homomorphism π ∞ ( G ) be omes a submersion. Indeed, assume that an unknown Lie group-oid stru ture, making π ∞ ( G ) a submersion, is assigned on the Tannakian26. THE CLASSICAL ENVELOPE OF A PROPER GROUPOID 149groupoid of G . Let T ∗ ( G ) indi ate the Tannakian groupoid of G endowedwith the unknown smooth stru ture. Now, the identity homomorphism ofthe Tannakian groupoid into itself (cid:28)ts in the following triangle T ∞ ( G ) id (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) G π ∗ , , YYYYYYYYYYYYYYYYYYY π ∞ eeeeeeeeeeeeeeeeeee T ∗ ( G ) (17)where π ∞ = π ∞ ( G ) = π ∗ are surje tive submersions. It follows that theidentity id : T ∞ ( G ) = T ∗ ( G ) is a di(cid:27)eomorphism.Under the assumption of properness, we an also say something usefulabout ondition i) of Proposition 23.4:18 Note Let G be a proper Lie groupoid. Suppose that for ea h identityarrow x of the Tannakian groupoid T ∞ ( G ) one an (cid:28)nd a representative hart for T ∞ ( G ) about x . Then we ontend that the ondition i) of Propo-sition 23.4 is satis(cid:28)ed by the lassi al (cid:28)bre fun tor ω ∞ ( G ) .Let an arbitrary arrow λ : x → x ′ of T ∞ ( G ) be given. Be ause ofproperness, we have λ = π ∞ ( g ) for some arrow g : x → x ′ of G . Sele tany smooth lo al bise tion σ : U → G (1) , de(cid:28)ned over a neighbourhood U of x and with σ ( x ) = g , and let U ′ = t ( σ ( U )) . Now, let (Ω , R ) be arepresentative hart about x , let us say with Ω ⊂ T ∞ ( G ) | U and R = ( E, ̺ ) .Noti e that one has the following ommutative square G| U ≈ σ - (cid:15) (cid:15) ̺ / / GL ( E ) | U ≈ ( ̺ ◦ σ ) - (cid:15) (cid:15) G ( U, U ′ ) ̺ / / GL ( E )( U, U ′ ) ,(19)where σ - denotes the left translation di(cid:27)eomorphism g σ ( t ( g )) · g and,similarly, ( ̺ ◦ σ ) - denotes the di(cid:27)eomorphism µ ̺ ( σ ( t µ )) · µ . Let Γ =( π ∞ ) − (Ω) , so Γ ⊂ G| U is an open subset. Then Γ σ = σ - (Γ) is an openneighbourhood of g , Ω σ = ( π ∞ ◦ σ ) - (Ω) is an open neighbourhood of λ and Γ σ = ( π ∞ ) − (Ω σ ) . It follows that the subset(20) R (Ω σ ) = ̺ (Γ σ ) = ( ̺ ◦ σ ) - ( ̺ (Γ)) = ( ̺ ◦ σ ) - ( R (Ω)) is a submanifold of GL ( E )( U, U ′ ) . Similarly, one sees that Ω σ bije ts onto R (Ω σ ) via ev R . So (Ω σ , R ) is a representative hart about λ .The next, on lusive result provides, in the spe ial ase under exam, apositive answer to the question raised in Ÿ21 about the evaluation fun torbeing an equivalen e of ategories.50 CHAPTER VI. CLASSICAL TANNAKA THEORY21 Proposition Let G be any proper Lie groupoid.Then the evaluation fun tor ev : R ∞ ( G ) −→ R ∞ ( T ∞ ( G )) , R = ( E, ̺ ) ( E, ev R ) is an isomorphism of ategories, having the pullba k along the envelopehomomorphism of G as inverse.Proof This an be veri(cid:28)ed dire tly, sin e the envelope homomorphism of aproper Lie groupoid is already known to be surje tive. q.e.d.Ÿ27 Smoothness of the Classi al Envelope of aProper Regular GroupoidWe start by re alling a few basi de(cid:28)nitions and properties. For additionalinformation, see Moerdijk (2003) [26℄.Re all that a Lie groupoid G over a manifold M is said to be regular whenthe rank of the di(cid:27)erentiable map t x : G ( x, - ) → M lo ally keeps onstantas the variable x ranges over M ; an equivalent ondition is that the an hormap of the Lie algebroid of G , let us all it ρ : g → T M , should have lo ally onstant rank (as a morphism of ve tor bundles over M ). If G is regularthen the image of the an hor map ρ is a subbundle F of the tangent bundle T M ; in fa t, F turns out to be an integrable subbundle of T M and hen edetermines a foliation F of the base manifold M , alled the orbit foliationasso iated with the regular groupoid G .Re all that a leaf of a foliation F asso iated with an integrable subbundle F of T M is a maximal onne ted immersed submanifold L of M with theproperty of being everywhere tangent to F . The odimension of L in M oin ides with the odimension of F in T M . Also re all that a transversalfor F is a submanifold T of M , everywhere transversal to F and of dimensionequal to the odimension of F . There always exist omplete transversals, i.e.transversals that meet every leaf of the foliation.Bundles of Lie groups, that is to say Lie groupoids whose sour e andtarget map oin ide, form a very spe ial lass of regular Lie groupoids. Properbundles of Lie groups are also alled bundles of ompa t Lie groups.1 Lemma Let G be a bundle of ompa t Lie groups over a manifold M . Let R = ( E, ̺ ) be a lassi al representation of G .Then the image ̺ ( G ) is a submanifold of GL ( E ) .Proof By a result of Weinstein [37℄, every bundle of ompa t Lie groupsis lo ally trivial. This means that for ea h x ∈ M one an (cid:28)nd an openneighborhood U of x in M and a ompa t Lie group G su h that there existsan isomorphism of Lie groupoids over U (viz. a lo al trivialization)(2) G| U ≈ U × G .27. PROPER REGULAR GROUPOIDS 151At the expense of repla ing U with a smaller open neighborhood, one analso assume that there is a lo al trivialization E | U ≈ U × V , where V issome ve tor spa e of (cid:28)nite dimension; as explained in Note 23.11, su h atrivialization will determine an isomorphism GL ( E | U ) ≈ U × GL ( V ) of Liegroupoids over U . Then one an take the following omposite homomorphism U × G ≈ / / (cid:15) (cid:15) G| U ̺ | U / / (cid:15) (cid:15) GL ( E | U ) ≈ / / (cid:15) (cid:15) U × GL ( V ) pr / / (cid:15) (cid:15) GL ( V ) (cid:15) (cid:15) U × U id / / U × U id / / U × U id / / U × U / / ⋆ × ⋆ .(3)This yields a smooth family of representations of the ompa t Lie group G on the ve tor spa e V , parametrized by the onne ted open set U . We willdenote su h family by ̺ U : U × G → GL ( V ) .Now, it follows from the so- alled `homotopy property of representationsof ompa t Lie groups' (Note 2.30) that all the representations of the smoothfamily ̺ U are equivalent to ea h other; in parti ular, they all have the samekernel K ⊂ G . Hen e there exists a unique map f ̺ U making the diagram U × G id × pr (cid:15) (cid:15) id × ̺ U / / U × GL ( V ) U × ( G/K ) f ̺ U nnnnnn (4) ommute. Note that the map f ̺ U must be smooth, be ause id × pr is a sur-je tive submersion; of ourse, the same map is also a faithful representationof the bundle of ompa t Lie groups U × ( G/K ) on the trivial ve tor bundle U × V . Then Corollary 23.10 implies that the image of f ̺ U is a submanifoldof U × GL ( V ) . The latter submanifold oin ides, via the di(cid:27)eomorphism GL ( E ) | ∆ U ≈ U × GL ( V ) , with the interse tion ̺ ( G ) ∩ GL ( E ) | U . q.e.d.It is evident from the above proof that the kernel of the envelope homo-morphism π ∞ : G → T ∞ ( G ) must be a (lo ally trivial) bundle of ompa tLie groups K , embedded into G . Thus, if U is a onne ted open subset of M and R = ( E, ̺ ) is a lassi al representation su h that Ker ̺ u = K| u atsome point u ∈ U , it follows from the aforesaid homotopy property that Ker ̺ | U = K| U and therefore(cid:22)from the ommutativity of (26.1)(cid:22)that theevaluation representation ev R is faithful on T ∞ ( G ) | U .From the latter remark, the dis ussion about smoothness in the pre edingse tion and Lemma 1 it follows immediately that the standard lassi al (cid:28)brefun tor ω ∞ ( G ) asso iated with a bundle of ompa t Lie groups G is smooth.Indeed, let an arbitrary arrow λ ∈ T ∞ ( G ) be (cid:28)xed, let us say λ ∈ T ∞ ( G ) | x with x ∈ M . Take an obje t R ∈ Ob R ∞ ( G ) with the property that therestri tion of the evaluation representation ev R to T ∞ ( G ) | x is faithful (thisexists by Prop. 10.14) and then hoose any onne ted open neighbourhood52 CHAPTER VI. CLASSICAL TANNAKA THEORY U of x in M . Then the pair (cid:0) T ∞ ( G ) | U , R (cid:1) onstitutes a representative hartfor ω ∞ ( G ) about λ .More generally, let G be a proper Lie groupoid with the property thatfor ea h x ∈ M there exists an open neighbourhood U of x in M su hthat G| U is a bundle of ompa t Lie groups. By adapting the above re ipefor the onstru tion of representative harts about the arrows belonging tothe isotropy of T ∞ ( G ) and by taking into a ount Note 26.18, we see that ω ∞ ( G ) is smooth also in the present ase.We are going to generalize the latter remark to arbitrary proper regularLie groupoids. The shortest way to do this is to apply the theory of weakequivalen es of Ÿ25.5 Proposition Let G be a proper regular Lie groupoid.Then the standard lassi al (cid:28)bre fun tor ω ∞ ( G ) asso iated with G issmooth.Re all that in view of Proposition 26.16 this an also be expressed bysaying that there exists a (ne essarily unique) Lie groupoid stru ture onthe Tannakian groupoid T ∞ ( G ) su h that the envelope homomorphism π ∞ ( G ) be omes a smooth submersion.Proof Let M be the base of G . Sele t a omplete transversal T for thefoliation of the manifold M determined by the orbits of G . Note that T is inparti ular a sli e, so the restri tion G| T is a proper Lie groupoid embeddedinto G (by Note 4.3). If i : T ֒ → M denotes the in lusion map then, by theremarks at the end of Ÿ4, the embedding of Lie groupoids G| T (cid:15) (cid:15) (cid:31) (cid:127) in lusion / / G (cid:15) (cid:15) T × T (cid:31) (cid:127) i × i / / M × M (6)is a Morita equivalen e. One may therefore (cid:28)nd another (proper) Lie groupoid K , along with Morita equivalen es G| T M.e. ←−−− K
M.e. −−−→ G indu ing surje tivesubmersions at the level of base manifolds. The orresponding morphisms ofstandard lassi al (cid:28)bre fun tors(7) (cid:0) R ∞ ( G| T ) , ω ∞ ( G| T ) (cid:1) w.e. ←−− (cid:0) R ∞ ( K ) , ω ∞ ( K ) (cid:1) w.e. −−→ (cid:0) R ∞ ( G ) , ω ∞ ( G ) (cid:1) are weak equivalen es. Hen e, by Proposition 25.9, one is redu ed to showingthat ω ∞ ( G| T ) is a smooth (cid:28)bre fun tor.Now, G| T is a proper Lie groupoid over T with the above-mentionedproperty of being, lo ally, just a bundle of ompa t Lie groups. q.e.d.Let ProReg denote the ategory of proper regular Lie groupoids. Onemay summarize the on lusions of the present se tion as follows:28. CLASSICAL REFLEXIVITY: EXAMPLES 1538 Theorem The lassi al Tannakian orresponden e
G 7→ T ∞ ( G ) in-du es an idempotent fun tor(9) T ∞ ( - ) : ProReg −→ ProReg ;moreover, envelope homomorphisms (cid:28)t together into a natural transfor-mation(10) π ∞ ( - ) : Id −→ T ∞ ( - ) .Open Question. It is natural to ask whether this result an be generalized tothe whole ategory of proper Lie groupoids.Ÿ28 A few Examples of Classi ally Re(cid:29)exiveLie GroupoidsRe all that a Lie groupoid G ⇒ X is said to be étale if the sour e and targetmaps s , t : G → X are étale maps, that is to say lo al isomorphisms ofsmooth manifolds. An open subset Γ ⊂ G will be said to be (cid:29)at if the sour eand target map restri t to open embeddings of Γ into X . A Lie groupoid G will be said to be sour e-proper or, for short, s -proper when the sour e mapof G is a proper map.1 Proposition Let G be a sour e-proper étale Lie groupoid.Then G admits globally faithful lassi al representations.Proof The regular representation ( R, ̺ ) of G exists and has lo ally (cid:28)niterank. A ouple of remarks before starting. Let X be the base of G .For every point x of X , the s -(cid:28)ber s − ( x ) is a (cid:28)nite set. Indeed, it isdis rete, be ause if g ∈ s − ( x ) then sin e s is étale there exists a (cid:29)at openneighborhood Γ ⊂ G and therefore { g } = Γ ∩ s − ( x ) is a neighborhood of g in the s -(cid:28)ber. It is also ompa t, be ause of s -properness.Put ℓ ( x ) = k s − ( x ) k , the ardinality of this (cid:28)nite set. Then the (cid:28)ber R x of the ve tor bundle R → X is by de(cid:28)nition the ve tor spa e(2) C ( s − ( x ); R ) ∼ = R ℓ ( x ) of R-valued maps. This makes sense be ause3 Lemma The assignment x ℓ ( x ) de(cid:28)nes a lo ally onstant fun tionon X , with values into positive integers.Proof of the lemma. Fix x ∈ X , and say s − ( x ) = { g , . . . , g ℓ } . For every i = 1 , . . . , ℓ , there exists a (cid:29)at open neighborhood Γ i ⊂ G of g i . Choosing anopen ball B ⊂ T s (Γ i ) at x , we an assume s : Γ i ∼ → B to be an isomorphism ∀ i . Moreover, it is no loss of generality to assume the open subsets Γ , . . . , Γ ℓ
54 CHAPTER VI. CLASSICAL TANNAKA THEORYto be pairwise disjoint. (As a onsequen e of the fa t that a (cid:28)nite union ofopen balls in any manifold(cid:22)not ne essarily Hausdor(cid:27)(cid:22)is a Hausdor(cid:27) opensubmanifold.) Then, ∀ i = 1 , . . . , ℓ and ∀ z ∈ B , the interse tion s − ( z ) ∩ Γ i onsists of a single point g i ( z ) , and these points g ( z ) , . . . , g ℓ ( z ) ∈ G arepairwise distin t, be ause the Γ i are pairwise disjoint. This shows ℓ ( z ) ≧ ℓ ( x ) ∀ z ∈ B . To prove the onverse inequality, it will su(cid:30) e to prove that ∃ N ⊂ B , a smaller ball at x , su h that s − ( N ) ⊂ Γ = Γ ∪ · · · ∪ Γ ℓ . Considera de reasing sequen e of losed balls C n +1 ⊂ C n ⊂ B shrinking to x , andthe orresponding de reasing sequen e Σ n = s − ( C n ) − Γ of losed subsetsof the ompa t subspa e s − ( C ) ⊂ G ; there ∃ n su h that Σ n = ∅ , in otherwords s − ( C n ) ⊂ Γ . This on ludes the proof of the lemma.Thus, it makes sense to regard R → X as the set-theoreti support of aR-linear ve tor bundle of lo ally (cid:28)nite rank. The proof of the lemma ontainsalso a re ipe for the onstru tion of lo al trivializations. Namely, let x ∈ X be (cid:28)xed, and hoose an ordering s − ( x ) = { g , . . . , g ℓ } of the orresponding(cid:28)ber; there exist an open ball B ⊂ X entered at x and disjoint (cid:29)at openneighborhoods Γ , . . . , Γ ℓ ⊂ G of g , . . . , g ℓ su h that s − ( B ) = Γ ∪ · · · ∪ Γ ℓ .Then one gets a bije tion R | B ≈ B × R ℓ by setting, for z ∈ B and f ∈ C ( s − ( z ); R ) , ( z, f ) (cid:0) z, f ( g ( z )) , . . . , f ( g ℓ ( z )) (cid:1) .(Cf. the notation used in the proof of the lemma.) The transition map-pings are smooth, be ause lo ally they are given by onstant permutations ( a , . . . , a ℓ ) (cid:0) a τ (1) , . . . , a τ ( ℓ ) (cid:1) .The R-linear isomorphism ̺ ( g ) ∈ Lis( R x , R y ) ,asso iated with g ∈ G ( x, y ) , is de(cid:28)ned by `translation' f ̺ ( g )( f ) ≡ f ( - g ) .The resulting fun torial map ̺ : G −→ GL ( R ) is learly faithful; it is alsosmooth, be ause in any trivializing lo al harts it looks like a lo ally onstantpermutation. q.e.d.If G is any étale Lie groupoid with base manifold X , there is a morphismof Lie groupoids Ef :
G −→ Γ X , where Γ X is the étale Lie groupoid (withbase X ) of germs of smooth isomorphisms U ∼ → V between open subsets of X . It sends g ∈ G to the germ of the lo al smooth isomorphism asso iatedwith a (cid:29)at open neighborhood of g . An e(cid:27)e tive Lie groupoid is an étale Liegroupoid su h that Ef is faithful, in other words su h that every g ∈ G isuniquely determined by its `lo al a tion' on the base manifold X . (Some ofthe simplest étale groupoids, su h as for instan e the trivial ones X × K , K a dis rete group, are not e(cid:27)e tive at all!)28. CLASSICAL REFLEXIVITY: EXAMPLES 155The lass of e(cid:27)e tive Lie groupoids is stable under weak equivalen eamong étale Lie groupoids. (Cf. Moerdijk and Mr£un (2003), [27℄ p. 137.)The following onditions on a Lie groupoid G are equivalent:1. G is weakly equivalent to a proper e(cid:27)e tive groupoid;2. G is weakly equivalent to the Lie groupoid asso iated with an orbifold.(Cf. ibid. p. 143.) The relevan e of this theorem in the present ontext isthat it tells that if one wants to study orbifolds through their asso iated Liegroupoid and Tannakian duality, it is su(cid:30) ient to prove the duality resultfor proper e(cid:27)e tive groupoids.Any étale Lie groupoid G ⇒ X has a anoni al representation on thetangent bundle T X → X , whi h asso iates to g ∈ G ( x, y ) the invertible R-linear map T x X → T y X of tangent spa es given by the tangent map at x ofthe germ of lo al smooth isomorphisms Ef( g ) . In general, this representationneed not be faithful. However4 Proposition If G is a proper e(cid:27)e tive Lie groupoid with base X , the anoni al representation on the tangent bundle T X is faithful.1Proof If G ⇒ X is a proper étale Lie groupoid and x ∈ X , there exista neighborhood U ⊂ X of x and a smooth a tion of the isotropy group G x = G| x on U , su h that the Lie groupoid G| U ⇒ U is isomorphi to thea tion groupoid G x ⋉ U . I need to re all part of the proof. (Cf. Moerdijkand Mr£un (2003), [27℄ p. 142.) Let G x = { , . . . , ℓ } . There are a onne tedopen neighborhood W ⊂ X of x and s -se tions σ , . . . , σ ℓ : W → G with σ i ( x ) = i ∈ G x ∀ i , su h that the maps f i = t ◦ σ i send W di(cid:27)eomorphi allyonto itself and satisfy f i ◦ f j = f ij for all i, j ∈ G x .Sin e G is also e(cid:27)e tive, the group homomorphism i f i , of G x intothe group Aut( W ; x ) of smooth automorphisms of W that (cid:28)x the point x ,is inje tive. Now, if M is a onne ted manifold and H ⊂ Aut( M ) is a (cid:28)nitegroup of smooth automorphisms of M , the group homomorphism whi h maps f ∈ H x = { f ∈ H | f ( x ) = x } to the tangent map T x f ∈ Aut( T x M ) isinje tive ∀ x ∈ M . (Ibid. p. 36.) In the ase M = W and H = { f i | i ∈ G x } = H x , this says pre isely that the anoni al representation of G on the tangentbundle T X restri ts to a faithful representation G x ֒ → Aut( T x X ) . q.e.d.Another simple example is o(cid:27)ered by a tion groupoids asso iated with om-pa t Lie group a tions.Pre isely, let K be a ompa t Lie group a ting smoothly on a manifold X , say from the left. We denote by K ⋉ X the Lie groupoid over X whose1This was pointed out to me by I. Moerdijk.56 CHAPTER VI. CLASSICAL TANNAKA THEORYmanifold of arrows is the Cartesian produ t K × X , with the se ond pro-je tion ( k, x ) x as sour e map, the a tion K × X → X as range mapand ( k ′ , k · x ) · ( k, x ) = ( k ′ k, x ) as omposition of arrows.If V is a faithful K -module (in other words a faithful representation ̺ of the ompa t Lie group K on a ve tor spa e V ), then we get a faithfulrepresentation of the groupoid K ⋉ X on the trivial ve tor bundle X × V ,de(cid:28)ned by ( k, x ) (cid:0) x, k · x, ̺ ( k ) (cid:1) .ibliography[1℄ I. Androulidakis and G. Skandalis. The holonomy groupoid of a singularfoliation. Preprint arXiv math.DG/0612370.[2℄ R. Bos. Continuous representations of groupoids. Preprint arXivmath.RT/0612639.[3℄ G. E. Bredon. Introdu tion to Compa t Transformation Groups. A a-demi Press, New York, 1972.[4℄ T. Brö ker and T. tom Die k. Representations of Compa t Lie Groups.Graduate Texts in Mathemati s 98. Springer-Verlag, New York, 1985.[5℄ A. Cannas da Silva and A. Weinstein. 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(First Series), 45:1(cid:21)12, 1939.[35℄ W. C. Waterhouse. Introdu tion to A(cid:30)ne Group S hemes. Springer-Verlag, 1979.[36℄ A. Weinstein. Symple ti groupoids and Poisson manifolds. Bull. Amer.Math. So . (New Series), 16:101(cid:21)104, 1987.[37℄ A. Weinstein. Linearization of regular proper groupoids. J. Inst. Math.Jussieu, 1(3):493(cid:21)511, 2002.[38℄ N. T. Zung. Proper groupoids and momentum maps: linearization, a(cid:30)n-ity, and onvexity. Preprint arXiv math.SG/0407208 v4, July 24, 2006.ndexa tion, see representationa tion groupoid G ⋉ M , 23, 155ACU onstraints, 39, 65additive tensor ategory, 43algebroid, 31, 150anti-involution, 46arrows, manifold of -, 21asso iativity onstraint, 39averaging te hnique, 84base, 22, 89bundle of Lie groups, see Lie bundle C ∞ -representation, 24, 112, 143 C ∞ -stru tured groupoid, 93, 111 C an ( C ) ( ategory of anoni al multi-fun tors), 40 anoni al fun tional stru ture, seestandard C ∞ -stru ture anoni al isomorphism, see on-straint anoni al multi-fun tor, 40 anoni al transformation, 40 lassi al (cid:28)bre fun tor, 110 lassi al representation, see C ∞ -representation losed tensor ategory, 41 oheren e onditions, 39, 44, 46, 56 oheren e theorem, 40 ommutativity onstraint, 39 omplete transversal, 150 omplex tensor ategory, 46, 65 omplex tensor fun tor, 46, 65 omposition, 21 onstraint, 39, 44, 46, 56, 65, 66 ut-o(cid:27) fun tion, 32, 106 D es C ( X ′ /X ) ( ategory of des entdata), 61des ent datum, 47, 61dimension axiom, 77dual, 42, 47embedding, 75envelope, see Tannakian groupoidenvelope homomorphism π T ( G ) , π ∞ ( G ) , 95, 98, 144, 152equivalen e of tensor ategories, 68 ev (evaluation fun tor), 113, 150 ev R (evaluation representation), 112,144evaluation fun tor ev , 113, 150evaluation of germs, 72evaluation representation ev R , 112,144(cid:28)bre, 58(cid:28)bre fun tor, 89 lassi al, 110over a manifold, see baseproper, 94, 113smooth, 89, 127, 140, 148, 152(cid:28)bred tensor ategory, 55(cid:28)bred tensor ategory onstraints, 56(cid:29)at map, 60fun tional stru ture, 92fun tionally stru tured groupoid, 93fun tionally stru tured spa e, 92 G ⋉ M (a tion groupoid), 23, 155 Γ E , Γ H (sheaf of se tions), 56, 78 GL ( E ) (linear groupoid), 23, 112groupoid, 21Hausdor(cid:27), 22160NDEX 161lo ally transitive, 81, 133over a manifold, see baseproper, 24re(cid:29)exive, 99, 153regular, 150self-dual, see re(cid:29)exivetransitive, 81, 133Haar systemnormalized, 32positive, 31Hausdor(cid:27) groupoid, 22Hermitian form, 46 H om C X ( E, F ) (sheaf hom), 56homomorphism of groupoids, 22, 65,135, 144internal hom (bifun tor), 41invariant submanifold, subset, 82, 105inverse, 21inverse image, 66isotropy group, 22leaf, 150Lie algebroid, 31, 150Lie bundle, 25Lie groupoid, see groupoidlinear groupoid GL ( E ) , 23, 112linear tensor ategory, 43lo al metri , 74lo ally (cid:28)nite obje t, sheaf, 77lo ally transitive groupoid, 81, 133lo ally trivial obje t, 62, 110main theorem, 107manifold of arrows, obje ts, 21metri , 74, 90 ω -invariant, 94, 113Morita equivalen e, 34, 37, 67, 101,119, 140Morita equivalent, see Morita equiv-alen emorphism of (cid:28)bre fun tors, 134multi-fun tor, 40, 43nondegenerate form, 47 normalized Haar system, 32obje ts, manifold of -, 21 ω T ( G ) , ω ∞ ( G ) (forgetful fun tor), 64,94, 143 ω -invariant metri , 94, 113orbit, 23, 98orbit foliation, 150orbit map, spa e, 98orthonormal frame, 74, 110para ompa t, 22parasta k, 61 π T ( G ) , π ∞ ( G ) (envelope homomorph-ism), 95, 98, 144positive Haar system, 31presta k, 56prin ipal submanifold, 114proper (cid:28)bre fun tor, 94, 113proper groupoid, 24pullba kalong a smooth map, 45, 55, 80of representations, 66of smooth Hilbert (cid:28)elds, 80 R ( olle tion of representative fun -tions), 90 R ∞ ( anoni al fun tional stru tureon the Tannakian groupoid),92 R T ( G ) ( ategory of type T represen-tations), 63 R ∞ ( T ; k ) ( ategory of smooth repre-sentations on ve tor bundles),24, 112, 143rank, 42re(cid:28)nement, 61re(cid:29)exive, 99, 153regular groupoid, 150representation C ∞ - or smooth, 24, 112, 143 lassi al, see C ∞ - or smoothof type T , 63representative hart, 127, 147representative fun tion, 9062 INDEXrigid tensor ategory, 42, 47, 110saturation, 105se tion, 56, 78self- onjugate, 48, 89self-dual, see re(cid:29)exivesesquilinear form, 46sheaf hom H om C X ( E, F ) , 56sheaf of se tions Γ E , Γ H , 56, 78sli e, 32smooth Eu lidean (cid:28)eld, 81smooth (cid:28)bre fun tor, 89, 127, 140,148, 152smooth Hilbert (cid:28)eld, 78smooth representation, see C ∞ -representationsmooth se tion, see se tionsmooth tensor parasta k, 61smooth tensor presta k, 59smooth tensor sta k, 61sour e, 21sta k, 61sta k of smooth (cid:28)elds, 71standard C ∞ -stru ture R ∞ , 92, 111standard (cid:28)bre fun tor( lassi al) ω ∞ ( G ) , 143(of type T ) ω T ( G ) , 64, 94stru ture maps, 21 T T ( G ) , T ∞ ( G ) (Tannakian groupoidasso iated with a Lie group-oid), 98, 143 T ( ω ) (Tannakian groupoid asso i-ated with a (cid:28)bre fun tor), 89,111tame submanifold, 116, 145Tannakian groupoid( lassi al) T ∞ ( G ) , 143(of type T ) T T ( G ) , 98 T ( ω ) , 89, 111target, 21tensor ategory, 39, 65additive, 43 losed, 41 linear, 43rigid, 42, 47, 110tensor equivalen e, 68, 137tensor fun tor, 44, 65, 66tensor fun tor onstraints, 44, 66tensor parasta k, 61tensor preserving, 45tensor presta k, 56smooth, 59tensor produ tof Hilbert spa es, 79of smooth Hilbert (cid:28)elds, 80tensor sta k, 61tensor stru ture, see tensor ategorytensor unit , 39tensor unit onstraints, 39topologi al groupoid, 23tra e, 42transformation, 67transitive groupoid, 81, 133translation groupoid, see a tiongroupoidtransversal ( omplete), 150trivial obje t, 62trivialization, 62type, 63unit map, se tion, 21 V C ( X ) (sub ategory of lo ally trivialobje ts), 62 V ∞ ( X ; k ))
G −→ Γ X , where Γ X is the étale Lie groupoid (withbase X ) of germs of smooth isomorphisms U ∼ → V between open subsets of X . It sends g ∈ G to the germ of the lo al smooth isomorphism asso iatedwith a (cid:29)at open neighborhood of g . An e(cid:27)e tive Lie groupoid is an étale Liegroupoid su h that Ef is faithful, in other words su h that every g ∈ G isuniquely determined by its `lo al a tion' on the base manifold X . (Some ofthe simplest étale groupoids, su h as for instan e the trivial ones X × K , K a dis rete group, are not e(cid:27)e tive at all!)28. CLASSICAL REFLEXIVITY: EXAMPLES 155The lass of e(cid:27)e tive Lie groupoids is stable under weak equivalen eamong étale Lie groupoids. (Cf. Moerdijk and Mr£un (2003), [27℄ p. 137.)The following onditions on a Lie groupoid G are equivalent:1. G is weakly equivalent to a proper e(cid:27)e tive groupoid;2. G is weakly equivalent to the Lie groupoid asso iated with an orbifold.(Cf. ibid. p. 143.) The relevan e of this theorem in the present ontext isthat it tells that if one wants to study orbifolds through their asso iated Liegroupoid and Tannakian duality, it is su(cid:30) ient to prove the duality resultfor proper e(cid:27)e tive groupoids.Any étale Lie groupoid G ⇒ X has a anoni al representation on thetangent bundle T X → X , whi h asso iates to g ∈ G ( x, y ) the invertible R-linear map T x X → T y X of tangent spa es given by the tangent map at x ofthe germ of lo al smooth isomorphisms Ef( g ) . In general, this representationneed not be faithful. However4 Proposition If G is a proper e(cid:27)e tive Lie groupoid with base X , the anoni al representation on the tangent bundle T X is faithful.1Proof If G ⇒ X is a proper étale Lie groupoid and x ∈ X , there exista neighborhood U ⊂ X of x and a smooth a tion of the isotropy group G x = G| x on U , su h that the Lie groupoid G| U ⇒ U is isomorphi to thea tion groupoid G x ⋉ U . I need to re all part of the proof. (Cf. Moerdijkand Mr£un (2003), [27℄ p. 142.) Let G x = { , . . . , ℓ } . There are a onne tedopen neighborhood W ⊂ X of x and s -se tions σ , . . . , σ ℓ : W → G with σ i ( x ) = i ∈ G x ∀ i , su h that the maps f i = t ◦ σ i send W di(cid:27)eomorphi allyonto itself and satisfy f i ◦ f j = f ij for all i, j ∈ G x .Sin e G is also e(cid:27)e tive, the group homomorphism i f i , of G x intothe group Aut( W ; x ) of smooth automorphisms of W that (cid:28)x the point x ,is inje tive. Now, if M is a onne ted manifold and H ⊂ Aut( M ) is a (cid:28)nitegroup of smooth automorphisms of M , the group homomorphism whi h maps f ∈ H x = { f ∈ H | f ( x ) = x } to the tangent map T x f ∈ Aut( T x M ) isinje tive ∀ x ∈ M . (Ibid. p. 36.) In the ase M = W and H = { f i | i ∈ G x } = H x , this says pre isely that the anoni al representation of G on the tangentbundle T X restri ts to a faithful representation G x ֒ → Aut( T x X ) . q.e.d.Another simple example is o(cid:27)ered by a tion groupoids asso iated with om-pa t Lie group a tions.Pre isely, let K be a ompa t Lie group a ting smoothly on a manifold X , say from the left. We denote by K ⋉ X the Lie groupoid over X whose1This was pointed out to me by I. Moerdijk.56 CHAPTER VI. CLASSICAL TANNAKA THEORYmanifold of arrows is the Cartesian produ t K × X , with the se ond pro-je tion ( k, x ) x as sour e map, the a tion K × X → X as range mapand ( k ′ , k · x ) · ( k, x ) = ( k ′ k, x ) as omposition of arrows.If V is a faithful K -module (in other words a faithful representation ̺ of the ompa t Lie group K on a ve tor spa e V ), then we get a faithfulrepresentation of the groupoid K ⋉ X on the trivial ve tor bundle X × V ,de(cid:28)ned by ( k, x ) (cid:0) x, k · x, ̺ ( k ) (cid:1) .ibliography[1℄ I. Androulidakis and G. Skandalis. The holonomy groupoid of a singularfoliation. Preprint arXiv math.DG/0612370.[2℄ R. Bos. Continuous representations of groupoids. Preprint arXivmath.RT/0612639.[3℄ G. E. Bredon. Introdu tion to Compa t Transformation Groups. A a-demi Press, New York, 1972.[4℄ T. Brö ker and T. tom Die k. Representations of Compa t Lie Groups.Graduate Texts in Mathemati s 98. Springer-Verlag, New York, 1985.[5℄ A. Cannas da Silva and A. Weinstein. Geometri Models for Non ommu-tative Algebras. Number 10 in Berkeley Mathemati al Le tures. Ameri- an Mathemati al So iety, Providen e, 1999.[6℄ C. Chevalley. Theory of Lie Groups. Prin eton University Press, 1946.[7℄ P. E. Conner and E. E. Floyd. Di(cid:27)erentiable Periodi Maps. Springer-Verlag, 1964.[8℄ A. Connes. Non ommutative Geometry. A ademi Press, New York,1994.[9℄ A. Coste, P. Dazord, and A. Weinstein. Groupoïdes symple tiques.Publi ations du Département de Mathématiques, Nouvelle Série (A) 87,Univ. Claude-Bernarde, Lyon, 1987. Pages i(cid:21)ii, 1(cid:21)62.[10℄ M. Craini . Di(cid:27)erentiable and algebroid ohomology, Van Est isomorph-ism, and hara teristi lasses. Comm. Math. Helveti i, 78:681(cid:21)721,2003.[11℄ P. Deligne. Catégories tannakiennes. In P. Cartier, L. Illusie, N. M. Katz,et al., editors, The Grothendie k Fests hrift, volume II, pages 111(cid:21)194.Birkhäuser, Basel, 1991.[12℄ P. Deligne and J. S. Milne. Tannakian ategories. In Hodge Cy les, Mo-tives and Shimura Varieties, Le ture Notes in Mathemati s 900, pages101(cid:21)228. Springer-Verlag, 1982.15758 BIBLIOGRAPHY[13℄ J. Dixmier. Les C ∗ -Algébres et leurs Représentations. Gauthier-Villars,Paris, 1969.[14℄ J. Dixmier and A. Douady. Champs ontinues d'espa es Hilbertiens etde C ∗ -algébres. Bull. de la SMF, 19:227(cid:21)284, 1963.[15℄ J.-P. Dufour and N. T. Zung. Poisson stru tures and their normal forms.Number 242 in Progress in Mathemati s. Birkhäuser, Basel, 2005.[16℄ S. Evens, J.-H. Lu, and A. Weinstein. Transverse measures, the modular lass and a ohomology pairing for Lie algebroids. Quart. J. Math.Oxford, 50(2):417(cid:21)436, 1999.[17℄ A. Henriques and D. S. Metzler. Presentations of none(cid:27)e tive orbifolds.Trans. Am. Math. So iety, 356(6):2481(cid:21)2499, 2004.[18℄ A. Joyal and R. Street. An introdu tion to Tannaka duality and quan-tum groups. In A. Carboni, M. C. Pedi hio, and G. 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(First Series), 45:1(cid:21)12, 1939.[35℄ W. C. Waterhouse. Introdu tion to A(cid:30)ne Group S hemes. Springer-Verlag, 1979.[36℄ A. Weinstein. Symple ti groupoids and Poisson manifolds. Bull. Amer.Math. So . (New Series), 16:101(cid:21)104, 1987.[37℄ A. Weinstein. Linearization of regular proper groupoids. J. Inst. Math.Jussieu, 1(3):493(cid:21)511, 2002.[38℄ N. T. Zung. Proper groupoids and momentum maps: linearization, a(cid:30)n-ity, and onvexity. Preprint arXiv math.SG/0407208 v4, July 24, 2006.ndexa tion, see representationa tion groupoid G ⋉ M , 23, 155ACU onstraints, 39, 65additive tensor ategory, 43algebroid, 31, 150anti-involution, 46arrows, manifold of -, 21asso iativity onstraint, 39averaging te hnique, 84base, 22, 89bundle of Lie groups, see Lie bundle C ∞ -representation, 24, 112, 143 C ∞ -stru tured groupoid, 93, 111 C an ( C ) ( ategory of anoni al multi-fun tors), 40 anoni al fun tional stru ture, seestandard C ∞ -stru ture anoni al isomorphism, see on-straint anoni al multi-fun tor, 40 anoni al transformation, 40 lassi al (cid:28)bre fun tor, 110 lassi al representation, see C ∞ -representation losed tensor ategory, 41 oheren e onditions, 39, 44, 46, 56 oheren e theorem, 40 ommutativity onstraint, 39 omplete transversal, 150 omplex tensor ategory, 46, 65 omplex tensor fun tor, 46, 65 omposition, 21 onstraint, 39, 44, 46, 56, 65, 66 ut-o(cid:27) fun tion, 32, 106 D es C ( X ′ /X ) ( ategory of des entdata), 61des ent datum, 47, 61dimension axiom, 77dual, 42, 47embedding, 75envelope, see Tannakian groupoidenvelope homomorphism π T ( G ) , π ∞ ( G ) , 95, 98, 144, 152equivalen e of tensor ategories, 68 ev (evaluation fun tor), 113, 150 ev R (evaluation representation), 112,144evaluation fun tor ev , 113, 150evaluation of germs, 72evaluation representation ev R , 112,144(cid:28)bre, 58(cid:28)bre fun tor, 89 lassi al, 110over a manifold, see baseproper, 94, 113smooth, 89, 127, 140, 148, 152(cid:28)bred tensor ategory, 55(cid:28)bred tensor ategory onstraints, 56(cid:29)at map, 60fun tional stru ture, 92fun tionally stru tured groupoid, 93fun tionally stru tured spa e, 92 G ⋉ M (a tion groupoid), 23, 155 Γ E , Γ H (sheaf of se tions), 56, 78 GL ( E ) (linear groupoid), 23, 112groupoid, 21Hausdor(cid:27), 22160NDEX 161lo ally transitive, 81, 133over a manifold, see baseproper, 24re(cid:29)exive, 99, 153regular, 150self-dual, see re(cid:29)exivetransitive, 81, 133Haar systemnormalized, 32positive, 31Hausdor(cid:27) groupoid, 22Hermitian form, 46 H om C X ( E, F ) (sheaf hom), 56homomorphism of groupoids, 22, 65,135, 144internal hom (bifun tor), 41invariant submanifold, subset, 82, 105inverse, 21inverse image, 66isotropy group, 22leaf, 150Lie algebroid, 31, 150Lie bundle, 25Lie groupoid, see groupoidlinear groupoid GL ( E ) , 23, 112linear tensor ategory, 43lo al metri , 74lo ally (cid:28)nite obje t, sheaf, 77lo ally transitive groupoid, 81, 133lo ally trivial obje t, 62, 110main theorem, 107manifold of arrows, obje ts, 21metri , 74, 90 ω -invariant, 94, 113Morita equivalen e, 34, 37, 67, 101,119, 140Morita equivalent, see Morita equiv-alen emorphism of (cid:28)bre fun tors, 134multi-fun tor, 40, 43nondegenerate form, 47 normalized Haar system, 32obje ts, manifold of -, 21 ω T ( G ) , ω ∞ ( G ) (forgetful fun tor), 64,94, 143 ω -invariant metri , 94, 113orbit, 23, 98orbit foliation, 150orbit map, spa e, 98orthonormal frame, 74, 110para ompa t, 22parasta k, 61 π T ( G ) , π ∞ ( G ) (envelope homomorph-ism), 95, 98, 144positive Haar system, 31presta k, 56prin ipal submanifold, 114proper (cid:28)bre fun tor, 94, 113proper groupoid, 24pullba kalong a smooth map, 45, 55, 80of representations, 66of smooth Hilbert (cid:28)elds, 80 R ( olle tion of representative fun -tions), 90 R ∞ ( anoni al fun tional stru tureon the Tannakian groupoid),92 R T ( G ) ( ategory of type T represen-tations), 63 R ∞ ( T ; k ) ( ategory of smooth repre-sentations on ve tor bundles),24, 112, 143rank, 42re(cid:28)nement, 61re(cid:29)exive, 99, 153regular groupoid, 150representation C ∞ - or smooth, 24, 112, 143 lassi al, see C ∞ - or smoothof type T , 63representative hart, 127, 147representative fun tion, 9062 INDEXrigid tensor ategory, 42, 47, 110saturation, 105se tion, 56, 78self- onjugate, 48, 89self-dual, see re(cid:29)exivesesquilinear form, 46sheaf hom H om C X ( E, F ) , 56sheaf of se tions Γ E , Γ H , 56, 78sli e, 32smooth Eu lidean (cid:28)eld, 81smooth (cid:28)bre fun tor, 89, 127, 140,148, 152smooth Hilbert (cid:28)eld, 78smooth representation, see C ∞ -representationsmooth se tion, see se tionsmooth tensor parasta k, 61smooth tensor presta k, 59smooth tensor sta k, 61sour e, 21sta k, 61sta k of smooth (cid:28)elds, 71standard C ∞ -stru ture R ∞ , 92, 111standard (cid:28)bre fun tor( lassi al) ω ∞ ( G ) , 143(of type T ) ω T ( G ) , 64, 94stru ture maps, 21 T T ( G ) , T ∞ ( G ) (Tannakian groupoidasso iated with a Lie group-oid), 98, 143 T ( ω ) (Tannakian groupoid asso i-ated with a (cid:28)bre fun tor), 89,111tame submanifold, 116, 145Tannakian groupoid( lassi al) T ∞ ( G ) , 143(of type T ) T T ( G ) , 98 T ( ω ) , 89, 111target, 21tensor ategory, 39, 65additive, 43 losed, 41 linear, 43rigid, 42, 47, 110tensor equivalen e, 68, 137tensor fun tor, 44, 65, 66tensor fun tor onstraints, 44, 66tensor parasta k, 61tensor preserving, 45tensor presta k, 56smooth, 59tensor produ tof Hilbert spa es, 79of smooth Hilbert (cid:28)elds, 80tensor sta k, 61tensor stru ture, see tensor ategorytensor unit , 39tensor unit onstraints, 39topologi al groupoid, 23tra e, 42transformation, 67transitive groupoid, 81, 133translation groupoid, see a tiongroupoidtransversal ( omplete), 150trivial obje t, 62trivialization, 62type, 63unit map, se tion, 21 V C ( X ) (sub ategory of lo ally trivialobje ts), 62 V ∞ ( X ; k )) , V ∞ ( X ))