Groupoid representations and modules over the convolution algebras

GR OUPOID REPRESENT A TI ONS AND MODULES O VER THE CONV OLUTION ALGEBRAS J. KALI ˇ SNIK Abstract. The classical Serre-Swan’s theorem defines a bijective corresp on- dence betw een v ector bundles and finitely generated pro jective mo dules ov er the algebra of con tinuo us functions on some compact H ausdorff topological space. W e extend these results to obtain a correspondence b et we en the cate- gory of represent ations of an ´ etale Li e group oid and the category of mo dules o ve r its con v olution algebra th at are of finite type and of constan t rank. Both of these constructions are functorially defined on the Morita bicat egory of ´ etale Lie group oids and the given corr espondence represents a natural equiv alenc e betw een them. 1. Introduction There a re many phenomena in different are a s of ma thema tics and ph ysics that are most naturally describ ed in the language of smo oth manifolds and s mooth ma ps betw een them. How ev er, some natural constructio ns, coming from the theory of foliations or from Lie g r oup actions, result in slightly more singular spaces and require a different a pproach. The Morita categ ory of Lie g roupo ids and pr incipal bundles [7, 8, 15, 18, 20] provides a natur al framework in which to study ma n y such s ingular spaces like s paces of leaves of foliations, spaces o f orbits of Lie gro up actions o r for example orbifolds [1, 4 , 16, 17]. A Lie gr oupo id can b e cons idered as an atlas for the given s ingular space. It turns out that different Lie group oids represent the same geometric spa ce precis e ly when they are Morita e q uiv alent, i.e. when they are iso morphic in the Mo rita catego ry of Lie gro up oids. F or this reaso n we are primarily interested in those a lgebraic inv aria n ts of Lie g roup oids that are functorially defined on the Mo rita ca tegory of Lie gro upoids. T angent bundles, bundles of higher or der tensors, line bundles and other vector bundles play a cen tral role in the study of smo oth manifolds. The theor y of represen- tations of Lie gro upoids naturally extends all these notions to the catego r y of ´ etale Lie gr oupoids [1, 17] and s hows their close connectio n to the theory of Lie groups representations. It encompas ses many w ell known c o nstructions like equiv ariant vector bundles [2 , 27], or bibundles over orbifolds [1], foliated and tra nsv ersal v ector bundles ov er spaces of leaves of foliations [1 7], as well as ordinar y vector bundles ov er manifolds or r epresentations o f discrete groups. The cons tr uction of the ca te- gory of representations of a Lie g roupo id is inv ar ian t under the Mo rita equiv alence and thus represents one of the basic algebraic in v arian ts of Lie group oids. The Connes conv olution a lgebra of smoo th functions with compact s upport [4, 2 2, 23] o n a n ´ etale Lie group oid is another exa mple of suc h an in v arian t. Smo oth functions with compact suppo rt on a smo oth manifold and group algebras of dis- crete gr oups are all sp ecial examples o f co n volution algebr a s. Finitely generated pro jective mo dules ov er the algebr a of smo oth functions on a compact manifold are closely connected to smo oth vector bundles ov er that manifold by a theorem due to Serre and Sw an [28, 29]. Represen tations of a discrete group Γ on complex vector 2000 Mathematics Subje ct Classific ation. 16D40, 16D90, 22A22. This work was supported b y the Slov enian Ministry of Science. 1 2 J. KALI ˇ SNIK spaces on the other hand corr e spond bijectively to mo dules over the gr oup algebra C [Γ] of the g roup Γ. One is thus lead to be lie ve that b oth of these ex a mples repr e- sent a s imilar type of o b jects, namely a corre spondence b etw een r epresentations of an ´ etale Lie gro upoid and a c e rtain clas s of mo dules over its conv olution algebr a. W e show in this pap er (Theorem 3.2) ho w to ex tend the classica l Serre-Swan’s corres p ondence to the ca tegory of ´ etale Lie gro up oids. The functor of smo oth sec- tions with compact suppo rt defines an equiv alence be t w een the categ ory of represen- tations of an ´ etale Lie gro upoid G and the ca tegory of mo dules over the conv olution algebra of G which are of finite type and of consta n t ra nk . Thes e mo dules generalize the well known finitely ge ner ated pro jective modules ov er the alg ebras of functions and coincide with them if the manifold of ob jects of G is compac t and connected. F or example, such a corresp ondence makes it p ossible to study the ca tegory of orbibundles over a compact orbifold with the to ols o f noncommutativ e geometry , applied to the co n v olution a lgebra of the corres ponding orbifold group oid. The co nstructions of the categories o f re present ations and o f the mo dules of finite t yp e a nd of constant rank extend to morphisms from the Morita bicatego ry of ´ etale Lie g roupo ids to the 2- c ategory of additive ca tegories. In this more g eneral context, the given co rresp ondence ca n b e descr ibed (Theorem 4.3) as a natural equiv a lence betw een these t wo morphisms. 2. Basic definitions and exampl es 2.1. The M orita category of Lie group oids. F or the conv enience of the reader , and to fix the nota tions, we be g in by s ummarizing s ome bas ic definitions and r esults concerning Lie gro up oids tha t will be used througho ut this pa per. W e refer the reader to one of the b o oks [13, 17, 18] for a mor e detailed exp osition and further examples. A Lie gr ou p oid over a s mooth, second countable, Haus dorff manifold M is given by a smo oth manifold G and a structur e of a ca tegory o n G with o b jects M , in which a ll the ar rows are in vertible and all the structure maps G × s,t M G mlt / / G inv / / G s / / t / / M uni / / G are smoo th, with the so ur ce map s a submersion. W e allow the manifold G to be non-Hausdorff, but we as sume that the fib ers of the source map a re Haus dorff. W e write G ( x, y ) = s − 1 ( x ) ∩ t − 1 ( y ) for the set o f arrows from x ∈ M to y ∈ M and G x = G ( x, x ) for the iso tr op y gr oup of the group oid G a t x . The s e t G ( x, y ) is a submanifold of G and the isotropy group G x is a Lie gro up. If g ∈ G is a n y a rrow from x to y , and g ′ ∈ G is an arr o w from y to z , then the pr oduct g ′ g = mlt( g ′ , g ) is an arrow from x to z . The map uni ass igns to each x ∈ M the ident ity arrow 1 x = uni( x ) in G , and we o ften iden tify M with its imag e uni ( M ) in G . The map inv maps each arr ow g ∈ G to its inv erse g − 1 . A Lie group oid is ´ etale if all o f its str ucture ma ps a re lo cal diffeomorphisms. A bise ctio n of a n ´ etale Lie group oid G is an op en subset U of G such that b oth s | U and t | U are injective. T o any suc h bisection U corr esponds a diffeomor phism τ U : s ( U ) → t ( U ) defined by τ U = t | U ◦ ( s | U ) − 1 . Bisectio ns of the gro upoid G for m a basis for the top ology on G , so in par ticular they can b e chosen arbitr arily small. Generalised morphisms [18, 20] turn out to b e the right notion o f a ma p b e- t ween Lie gr oupoids in the repr esen tation theory of Lie group oids. They ar e clo sely connected to group oid actions and principa l bundles, which we briefly descr ibe. A smo oth left action o f a Lie group oid G on a smo oth manifold P alo ng a smo o th map π : P → M is a smo oth map µ : G × s,π M P → P , ( g , p ) 7→ g · p , which satisfies π ( g · p ) = t ( g ), 1 π ( p ) · p = p and g ′ · ( g · p ) = ( g ′ g ) · p , for all g ′ , g ∈ G and p ∈ P with 3 s ( g ′ ) = t ( g ) and s ( g ) = π ( p ). W e define right actions of Lie gr o upoids o n smo oth manifolds in a simila r wa y . Let G b e a Lie gro upoid over M and let H b e a Lie gro upoid over N . A princip al H -bund le over G is a smo oth ma nifold P , e q uipped with a left action µ of G along a smo oth submersio n π : P → M and a right action η o f H along a smo oth map φ : P → N , such that (i) φ is G -inv ariant, π is H -inv ariant and both actions commute: φ ( g · p ) = φ ( p ), π ( p · h ) = π ( p ) and g · ( p · h ) = ( g · p ) · h for ev ery g ∈ G , p ∈ P and h ∈ H with s ( g ) = π ( p ) and φ ( p ) = t ( h ), (ii) π : P → M is a principal right H -bundle: (pr 1 , η ) : P × φ,t N H → P × π ,π M P is a diffeomorphism. A map f : P → P ′ betw een principal H -bundles P a nd P ′ ov er G is equiv a riant if it satisfies π ′ ( f ( p )) = π ( p ), φ ′ ( f ( p )) = φ ( p ) and f ( g · p · h ) = g · f ( p ) · h , for every g ∈ G , p ∈ P a nd h ∈ H with s ( g ) = π ( p ) and φ ( p ) = t ( h ). An y such map is automatically a diffeomorphism. Pr incipal H - bundles P a nd P ′ ov er G are isomorphic if there exists an equiv ariant diffeomorphism b e tw een them. There is a na tural str ucture of a ca tegory (a ctually a group oid) on the set of principa l H -bundles ov er G for any tw o Lie group oids G and H . It has principal H -bundles ov er G as ob jects and equiv ariant diffeomo rphisms as morphisms b et w een them. T o any smo oth functor ψ : G → H there c orresp onds a pr incipal H -bundle P ( ψ ) = M × ψ ,t N H over G with the actions given by the ma ps g · ( x, h ) = ( t ( g ) , ψ ( g ) h ) for g ∈ G ( x, y ) and ( x, h ) · h ′ = ( x, hh ′ ) for h, h ′ ∈ H suc h that s ( h ) = t ( h ′ ). A principal bundle P is isomorphic to o ne induced by a functor if and only if it is trivial, i.e. if there exists a globa l smo oth sectio n of the bundle P . If P is a principal H -bundle ov er G and if P ′ is a pr incipal K -bundle ov er H , for a nother Lie gro upoid K , one can define the comp osition P ⊗ H P ′ [18, 1 9, 20], which is a principa l K -bundle over G . It is the quo tien t of P × φ,π ′ N P ′ with res pect to the dia gonal action of the gr oupo id H . So defined comp osition is a ssoc iativ e only up to a na tural isomorphis m. The Morita c a te gory GPD of Lie g r oupo ids consists of Lie gro upoids as ob jects and isomor phis m classes of principal bundles a s mo r phisms b et ween them. The morphisms in GPD a re so metimes refer red to as Hilsum-Sk anda lis maps o r gener- alised mor phisms b etw een Lie group oids. Two Lie gr o upoids are Morita equiv alen t if they are is o morphic in the catego ry GPD . T he Morita categ ory EtGPD of ´ etale Lie gro upoids is the full sub category of the categ o ry GPD with ´ etale Lie group oids as ob jects. If G a nd H are ´ etale Lie g roupo ids and if P is a pr incipal H -bundle over G , then the corr esponding map π : P → M is automatically a lo cal diffeomo rphism. W e will b e primarily interested in the Mo rita bicatego ry of Lie group oids which we describ e later on in the pap er. 2.2. Representations of Lie group oids. Lie group oids a dmit a t wofold inter- pretation. They can b e used to descr ibe symmetries of fibr e bundles in a similar wa y as Lie groups a re used to s tudy symmetr ie s of top ological spaces. How ever, the extra transversal part o f the struc tur e, which is enco ded in the ma nifo ld of ob jects, makes them a co n v enient mo del for singular geometric spaces such as o rbifolds, spaces of leav es of foliations or spa ces of o rbits of Lie group a ctions. The theory of repr esen tations of Lie g roupo ids provides a unified framework for the study of vector bundles on such geometric s paces and shows their intimate connectio n to representations of Lie groups. Let G b e a Lie gr oupo id ov er a smo oth manifold M and let E be a s mo oth complex vector bundle of rank k ov er M . A r epr esentation of the group oid G on E is a smo oth left action ρ : G × M E → E , denoted by ρ ( g , v ) = g · v , of G on E along the bundle pro jection p : E → M [17], suc h that for any arr o w g ∈ G ( x, y ) the induced map g ∗ : E x → E y , v 7→ g · v , is a linear isomorphis m. 4 J. KALI ˇ SNIK Example 2.1. (i) Represe ntations of a un it gr oup oid a s socia ted to a smo oth man- ifold M corr espond precisely to smo oth complex vector bundles over M . (ii) Let G b e a p oi nt gr oup oid with only o ne o b ject, i.e. G is a Lie group K . The representation theory of G then coincides with the re present ation theory of the Lie group K on finite dimensional complex v ector spaces. (iii) The p air gr oup oi d G = M × M ov er a smo oth manifo ld M has both pro jec- tions as source and ta r get maps a nd multiplication defined in a natura l wa y . E v ery representation of G on a vector bundle E ov er M a mo un ts to a natural iden tification of a ll the fib ers of E and is thus isomo rphic to a trivial representation. (iv) Let G = K ⋉ M b e the tr anslation gr o up oid of a smo oth left action of a Lie gr oup K on a manifold M . In this ca se the representations o f the gro up oid G corres p ond to K -eq uiv aria n t vector bundles ov er M [2, 2 7]. (v) Any ´ etale Lie gro upoid G over a manifold M has a natural r e pr esen tation on the complex ified tang en t bundle of the manifold M , where the action of any ar r o w is defined b y the differen tial of the lo cal diffeo morphism c o rresp onding to s o me bisection thro ugh that arr o w [1 , 17]. The cota ng en t bundle a nd tenso r bundles all inherit this natural repr esen tation, so it makes s ense to sp eak of vector fields, differential forms or riemannian metrics on ´ etale Lie gro upoids. (vi) Let G be an orbifold gr oup oid (i.e. a prop er ´ etale Lie gro upoid) over M . Such gr oupoids a re used as mo dels for orbifo lds [1 , 16, 26]. Representations of such group oids cor resp ond to orbibundles as defined in [1 , 2 4, 25]. (vii) Let ( M , F ) b e a foliated manifold. Represe n tations of the holonomy gr oup- oid Hol( M , F ) a re so metimes r eferred to as tra nsv ersal vector bundles, while those of the mono dr omy gr oup o id Mon( M , F ) are referr ed to as folia ted vector bundles. A morphism b et ween representations E a nd F of the gr oupoid G ov er M is a G -equiv arian t morphism φ : E → F of vector bundles. More precisely , φ : E → F is a fiber wise linear smooth map which commutes with bundle pro jections and satisfies φ ( g · e ) = g · φ ( e ) for all g ∈ G and all e ∈ E s ( g ) . Representations of a Lie group oid G tog ether with G -e q uiv aria n t mor phis ms b etw een them form a category Rep ( G ) of representations of G . Catego ries of (equiv a riant) vector bundles ov er a manifold, categorie s of orbibundles over a n orbifold or categor ies of repres en tations of Lie gr oups a re so me examples of categor ies o f r epresent ations of Lie gro upoids that aris e naturally in v a rious contexts. Direc t sums, tensor pro ducts, duals and other o pera tions on vector bundles genera lize to repres en tations of Lie gr oupoids and tur n the categ ory Rep ( G ) into an additive categ ory for every gro upoid G . Generalised maps b etw een group oids can b e used to pull back repr esen tations in the sa me sense as vector bundles can be pulled back a long smo oth maps . L e t G and H b e Lie gr oupoids over M resp ectively N and let P b e a principal H -bundle ov er G . F or any representation E o f the gr oupoid H we get the pull back represe ntation P ⊗ H E as follows (see [9] for details). The pull back bundle φ ∗ E = P × N E has a natur a l s tructure of a vector bundle ov er P with pro jection onto the fir st factor as the pro jection map. Groupoid H acts diago nally fro m the right on the s pace φ ∗ E along the fibe r s of the pr o jection onto M a nd it is eas y to see that the natural map P ⊗ H E = φ ∗ E /H → M is well defined, smo oth and makes P ⊗ H E a vector bundle ov er M . Finally , the action o f the group oid G on the spa ce P induces a representation of the group oid G on the bundle P ⊗ H E by acting on the first factor. The c o nstruction o f pulling back representations alo ng a principal bundle P extends to a functor from the categ ory o f repre sen tations of H to the catego ry of repres en tations of G . Define a re pr esen tation R ep ( P )( E ) = P ⊗ H E of G for any repres en tation E of H and a morphism Rep ( P )( φ ) : P ⊗ H E → P ⊗ H F of representations of G by Rep ( P )( φ )( p ⊗ v ) = p ⊗ φ ( v ) for any morphism φ : E → F 5 of r epresentations o f the gr o upoid H . W e thus obtain a cov ar ian t functor Rep ( P ) : Rep ( H ) → Rep ( G ) . One can use a n alternative des cription of the ab ov e ope ration in the cas e o f triv- ial bundles, i.e. when the pr incipal bundle comes from a smo oth functor. Suppos e that ψ : G → H is a smo oth functor b etw een Lie g roup oids and let E b e a r ep- resentation of the gr oupo id H . One defines a representation ψ ∗ E of the group oid G on the vector bundle ψ ∗ 0 E ov er M with the action g · ( x, v ) = ( t ( g ) , ψ ( g ) v ) for g ∈ G ( x, y ) and v ∈ E ψ 0 ( x ) . So defined r epresentation is na turally isomo rphic to the representation P ( ψ ) ⊗ H E of G via the isomorphism f : P ( ψ ) ⊗ H E → ψ ∗ E , which s e nds the element ( x, h ) ⊗ v to the element ( x, hv ). 2.3. Con v olution algebras and principal bim o dules. A smo oth manifold M is closely co nnected with the commut ative algebr a C ∞ c ( M ) of smo oth functions with compact supp ort on M . By r eplacing a manifold M with an ´ etale Lie gr o upoid G ov er M a nd b y defining a pr oper notion of the conv olution pro duct on the space of functions o ne obtains the conv olution algebr a C ∞ c ( G ) of the gr oupoid G . It is in gener a l nonco mm utativ e but it contains the algebra C ∞ c ( M ) as a commut ative subalgebra . The above construction natur ally extends to a cov ariant functor fro m the Mo rita category of ´ etale Lie group oids to the Morita catego ry of alg ebras, i.e. for each principal bundle b et w een gro upoids one naturally co nstructs a bimo dule betw een cor respo nding algebra s. In this subsection we briefly recall the definition of the conv olution alg ebra [4, 5, 2 0, 22] assigned to an ´ etale, not necessarily Haus dorff, Lie g roupo id and of the principal bimo dule [10, 20] a ssigned to a principa l bundle. W e first recall the definition of the co nvolution pro duct on the vector s pace C ∞ c ( G ) of smo oth functions with compact supp ort on a Hausdorff ´ etale Lie gro upoid G . Define a bilinear op eration on the spa ce C ∞ c ( G ) by the formula (1) ( ab )( g ) = X g = g ′ g ′′ a ( g ′ ) b ( g ′′ ) , for a ny a, b ∈ C ∞ c ( G ). Equipp ed with this pro duct the space C ∞ c ( G ) beco mes an asso ciative a lgebra called the c onvolution algebr a [4 ] of the ´ etale Lie group oid G . In the case of a ge ner al ´ etale Lie g r oupo id a suita ble notio n of a smo oth function with compact supp ort on a non-Hausdo rff manifold, a s given in [5], is needed. Considering that smo oth functions on a Hausdorff ma nifo ld M corres pond precisely to the contin uous sectio ns of the sheaf of germs of smoo th complex v a lued functions on M it makes sense to use this alternative approa c h to define smo oth functions with compact suppor t on an arbitrar y manifold P . One first considers the vector space o f all (not-necessar ily contin uous) sections of the sheaf of ger ms of s mo oth functions o n P . The tr ivial extension of a n y smo oth function with a compact suppo rt in a Hausdo rff op en subset of P naturally repre sen ts a section of that sheaf. The vector space C ∞ c ( P ) of s mo oth functions with compact supp ort on P is then defined to b e the subspace of the space of a ll sections, genera ted by such sections. This definition of the vector spa c e C ∞ c ( P ) agr ees with the cla ssical one if P is Hausdor ff, but in genera l there exists no natural multiplicativ e str ucture on the spa ce C ∞ c ( P ). The supp ort, i.e. the set where the v a lues o f the section are non trivial, of any function in C ∞ c ( P ) is always a compact subset of P but not necessarily closed if P is a non-Hausdor ff manifold. The stalk of the sheaf of g erms of smo oth functions on P at any p oin t of P is a co mm utativ e algebra with identit y , which e na bles us to per fo rm p oint wise op erations such as addition, multiplication or pullbacks alo ng smo oth maps. In particular, for any non-Hausdo rff ´ etale Lie group oid G a formula analo gous to the 6 J. KALI ˇ SNIK formula (1) c a n be used to define the conv olution algebra C ∞ c ( G ) of the ´ etale Lie group oid G . W e refer the r eader to [22] for details. Example 2.2. (i) The co n v olution pr o duct on the space C ∞ c ( G ) coinc ide s with the or dinary po in t wise pro duct of functions in C ∞ c ( M ) if G is the unit gr oupoid asso ciated to a smo oth manifold M . O n the other hand, if G is the p oin t group oid of a discrete g r oup Γ, it follows C ∞ c ( G ) = C [Γ], where C [Γ] is the g r oup alg ebra o f the gro up Γ. The suba lgebra C ∞ c ( M ) = { λ · 1 Γ | λ ∈ C } is is omorphic to co mplex nu mbers and is cent ral in C [Γ ]. (ii) The conv olution algebra C ∞ c ( G ) of the pair group oid on n p oin ts coincides with the algebr a of n × n complex matrices and c o n tains the subalgebra C ∞ c ( M ) of diagonal matrices. This example shows that C ∞ c ( M ) need not lie in the center of the algebra C ∞ c ( G ). (iii) Let G b e the translation group oid of a smo oth action of a discrete gro up Γ on a ma nifold M . The conv olution alg ebra C ∞ c ( G ) of G is known in the literature [4] a s the cro ssed pro duct algebra Γ ⋉ C ∞ c ( M ). One c an na turally extend the conv olution alge br a cons tr uction to a functor fro m the Morita ca tegory of ´ etale Lie gr oupoids to the Morita c a tegory of algebr as. Let G and H b e ´ etale Lie group oids and let P b e a pr incipal H -bundle over G . One can define conv olution actions of the algebra s C ∞ c ( G ) and C ∞ c ( H ) on the vector space C ∞ c ( P ) to turn it in to a C ∞ c ( G )- C ∞ c ( H )-bimo dule, whic h is calle d the princip al bimo dule a sso ciated to the principal H -bundle P ov er G . The functor C ∞ c from the Morita categor y of ´ etale Lie g roupo ids to the Mor ita category of algebras assigns to every ´ etale Lie group oid its c o n volution alg ebra a nd to an iso mo rphism clas s of a principal bundle the iso morphism cla ss of the asso ciated principal bimo dule [20] (see a lso [10] for a treatment of the non-Hausdo rff c ase). Throughout the r est of the pap er we will restr ict ourselves to Hausdor ff gr oupoids for simplicity , although essentially the same for m ulas apply in the non-Hausdor ff case as w ell. 3. Groupoid represent a tio ns and modules over co nv olution algebras The vector space of sections of a smo oth vector bundle E ov er a manifold M admits a natural actio n of the alge bra of smo oth functions o n M . Additional struc- ture o f a representation of an ´ etale L ie gro upoid G on the bundle E allows a natural extension of that action to the action o f the convolution algebra o f the gro upoid G . In this s e ction we characterize the mo dules over the conv olution alg ebra of the group oid G that aris e in this fashion fr om s ections of repres e n tations of G . Results o f this type were first considered by Serre [28] in the the categ ory of algebraic v arieties and Swan in the catego ry of compact Hausdorff top ological spaces [29]. All o ur vector bundles will b e a ssumed to be of globally constant rank, a condition which is automatically sa tisfied if the manifold of ob jects of the gr oupo id is co nnected. Similar results hold howev er in the ca se of vector bundles of g lobally bo unded r a nk as well. 3.1. Mo dule of sections of a representation. Let M be a smo oth, Hausdorff and s econd countable ma nifo ld and denote by C ∞ c ( M ) the algebr a of smo oth func- tions with compa c t suppo rt o n M . W e will denote by Rep ( M ) the ca tegory of smo oth vector bundles ov er M since it coincides with the c a tegory of r epresentations of the unit g roupo id asso ciated to the manifold M . F or any v ector bundle E ov er the manifold M , the vector spa ce Γ ∞ c ( E ) of smo oth sections of E with c ompact supp ort admits a natural struc tur e of a left C ∞ c ( M )-mo dule given b y ( f u )( x ) = f ( x ) u ( x ) for any f ∈ C ∞ c ( M ) and any u ∈ Γ ∞ c ( E ). E v ery mor phism φ : E → F of vec- tor bundles ov er M induces a ho momorphism Γ ∞ c ( φ ) : Γ ∞ c ( E ) → Γ ∞ c ( F ) of left 7 C ∞ c ( M )-mo dules by comp osing with φ , i.e. Γ ∞ c ( φ )( u ) = φ ◦ u . As a res ult we obtain the c o v a riant functor Γ ∞ c = (Γ ∞ c ) M : Rep ( M ) → M Mo d from the category of smo oth vector bundles over the manifold M to the ca tegory of left modules ov er the commutative a lgebra C ∞ c ( M ). Now let G be an ´ etale Lie g roup o id over M . W e will ass o ciate a n a ction of the conv olution alge br a C ∞ c ( G ) on the space of sections Γ ∞ c ( E ) to any representation E o f the group oid G . Define a bilinear map C ∞ c ( G ) × Γ ∞ c ( E ) → Γ ∞ c ( E ) by the for m ula ( au )( x ) = X t ( g )= x a ( g )( g · u ( s ( g ))) , for a ∈ C ∞ c ( G ) and u ∈ Γ ∞ c ( E ). Since the function a ∈ C ∞ c ( G ) has a compact suppo rt, there ar e only finitely many g ∈ t − 1 ( x ) with a ( g ) 6 = 0 for each x ∈ M , hence au is a well defined section o f the v ector bundle E . It r emains to be pr o ven that au b elongs to the space Γ ∞ c ( E ) and tha t the ab ov e map r eally defines an action of the a lg ebra C ∞ c ( G ) on the space Γ ∞ c ( E ). Strictly sp eaking, the ab o ve formula only holds for Hausdo rff Lie gr oupoids . How ev er, by ev aluating the sections of the s heaf o f germs of smo oth functions, one can use virtually the sa me for m ula for non-Hausdor ff group oids a s well. Prop osition 3 .1. The ve ctor sp ac e of se ctions Γ ∞ c ( E ) has a natur al st ructur e of a left mo dule over the c onvolution algebr a C ∞ c ( G ) . Pr o of. First w e show that au represents a smo oth section of the vector bundle E for any a ∈ C ∞ c ( G ) and any u ∈ Γ ∞ c ( E ). W e can decompo se any function a ∈ C ∞ c ( G ) as a sum a = P a j of functions, each of which has supp ort contained in some bisection, so we c an assume right fro m the s ta rt that the supp ort o f a is contained in some bisec tio n U . Let us denote W = t ( U ) and V = s ( U ). W e then hav e the following co mm uting diagram E | W µ ← − − − − U × V E pr E − − − − → E | V p   y pr U   y p   y W t | U ← − − − − U s | U − − − − → V of maps of vector bundles. The maps µ and pr E are isomorphisms of vector bundles cov ering the diffeomo rphisms t | U : U → W res pectively s | U : U → V . Let us denote by σ U = ( t | U ) ◦ ( s | U ) − 1 : V → W the diffeomorphism cor respo nding to the bisection U , and by τ U = µ ◦ (pr E ) − 1 : E | V → E | W the corr esponding is omorphism of vector bundles. The smo oth section u | V of the bundle E | V gets ma pped by the a bov e isomor phis m to the smo oth s ection u ′ = τ U ◦ u ◦ ( σ U ) − 1 of the bundle E | W . F urthermore, since a has compact supp ort in U , the function a ◦ ( t | U ) − 1 has compact suppor t in W . One ca n now expr ess the section au of the bundle E as ( au )( x ) = ( a ◦ ( t | U ) − 1 )( x ) u ′ ( x ) to prove that it is a smo oth s ection of E with compact suppo rt in W . T o see that the space o f sections Γ ∞ c ( E ) is a mo dule over the a lgebra C ∞ c ( G ), the eq ualit y a ( bu ) = ( ab ) u must hold for all a, b ∈ C ∞ c ( G ) and a ll u ∈ Γ ∞ c ( E ). T o 8 J. KALI ˇ SNIK this effect w e compute (( ab ) u )( x ) = X t ( g )= x ( ab )( g )( g · u ( s ( g ))) = X t ( g )= x  X g = g ′ g ′′ a ( g ′ ) b ( g ′′ )  ( g · u ( s ( g ))) = X t ( g ′ )= x s ( g ′ )= t ( g ′′ ) a ( g ′ ) b ( g ′′ )  ( g ′ g ′′ ) · u ( s ( g ′′ ))  . On the o ther hand we hav e ( a ( bu ))( x ) = X t ( g )= x a ( g )  g ·  ( bu )( s ( g ))  = X t ( g )= x a ( g )( g ·  X t ( g ′ )= s ( g ) b ( g ′ )  g ′ · u ( s ( g ′ ))   = X t ( g )= x s ( g )= t ( g ′ ) a ( g ) b ( g ′ )  g · ( g ′ · u ( s ( g ′ )))  . In the last line we have used the linearity of the map g ∗ : E x → E y for ea c h g ∈ G ( x, y ). Since E is a representation o f G , the equality g · ( g ′ · e ) = g g ′ · e holds for all pairs of co mp osable arr o ws g , g ′ ∈ G a nd all e ∈ E s ( g ′ ) , th us ( ab ) u = a ( bu ).  By the ab ov e pro cedure we obtain a left mo dule of s ections Γ ∞ c ( E ) over the conv olution algebr a C ∞ c ( G ) for a ny repres en tation E of the gro up oid G . Any mor- phism φ : E → F of representations of G is in particular a morphism of vector bundles and therefo r e pro duces a homomo rphism Γ ∞ c ( φ ) : Γ ∞ c ( E ) → Γ ∞ c ( F ) of C ∞ c ( M )-mo dules. Considering that the map φ is fib erwise linear and G -equiv arian t we g et the equalities (Γ ∞ c ( φ )( au ))( x ) = φ  ( au )( x )  = φ  X t ( g )= x a ( g )( g · u ( s ( g )))  = X t ( g )= x a ( g )( g · φ ( u ( s ( g )))) = ( a Γ ∞ c ( φ )( u ))( x ) , for a ∈ C ∞ c ( G ) and u ∈ Γ ∞ c ( E ). The ho momorphism Γ ∞ c ( φ ) is therefo r e a ho mo- morphism of C ∞ c ( G )-modules so we have the cov a riant functor Γ ∞ c = (Γ ∞ c ) G : Rep ( G ) → G Mo d from the category of r e presen tations of the gro upoid G to the ca tegory o f left mo dules over the conv olution a lg ebra C ∞ c ( G ) of G . 3.2. Mo dules o f finite t yp e and of constan t rank. According to the prev io us subsection we c an consider a mo dule ov er the conv olution algebra of an ´ etale Lie group oid G as a mo dule of sections of so me re presen tation of G . Ho w ever, not every C ∞ c ( G )-module is o f this kind and it is not to o hard to find counterexamples. In the following subsection we define and expla in the conditions that characterize the mo dules o f sections of repres en tations of the group oid G . Let M be a smoo th manifold and denote b y C ∞ c ( M ) the alge bra of smoo th functions with compact s upport on M . There is a standar d bijective co rresp on- dence b et w een nontrivial homomorphisms η : C ∞ c ( M ) → C of complex algebras and the p oin ts of the manifo ld M . T o any x ∈ M o ne ass ocia tes the ev aluation 9 ǫ x : C ∞ c ( M ) → C at the p oint x given by ǫ x ( f ) = f ( x ) fo r f ∈ C ∞ c ( M ). Conv ersely , the kernel of any nontrivial homo morphism η : C ∞ c ( M ) → C is a maximal idea l of the form k er( η ) = { f ∈ C ∞ c ( M ) | f ( x ) = 0 } for a unique p oin t x ∈ M , thus η = ǫ x . W e will use the nota tion I x C ∞ c ( M ) = { f ∈ C ∞ c ( M ) | f ( x ) = 0 } for the maximal ideal of functions that v anish at x and C ∞ c ( M )( x ) = C ∞ c ( M ) /I x C ∞ c ( M ) for the quotient algebra. Ev aluatio n a t the p oint x induces a ca no nical isomorphism b etw een the algebra C ∞ c ( M )( x ) a nd the field of complex num ber s. Now let G b e an ´ etale Lie group oid ov er M a nd let M b e a left C ∞ c ( G )-module. It follows that M is a C ∞ c ( M )-mo dule as well since C ∞ c ( M ) is a subalgebra o f the c o n volution algebra C ∞ c ( G ). The C ∞ c ( G )-module M is of finite typ e if it is isomorphic, as a C ∞ c ( M )-mo dule, to some submo dule of the mo dule C ∞ c ( M ) k for some natural num ber k . The C ∞ c ( M )-mo dules of the fo r m C ∞ c ( M ) k corres p ond precisely to the mo dules of sections of triv ial v ector bundles M × C k , s o one can roughly think of mo dules of finite type as co rresp onding to s ubfamilies of tr ivial vector bundles. Now cho ose a n arbitra ry p oin t x ∈ M . The C ∞ c ( M )-mo dule I x M = I x C ∞ c ( M ) · M is then a C ∞ c ( M )-submo dule of M and we denote by M ( x ) = M /I x M the quotient C ∞ c ( M )( x )-module a nd consider it as a complex vector space. Supp ose now that the C ∞ c ( G )-module M is of finite type and let Φ : M → C ∞ c ( M ) k be an injective homomo rphism o f C ∞ c ( M )-mo dules. F or e a c h x ∈ M we o btain a n injectiv e co mplex linea r map Φ( x ) : M ( x ) → C ∞ c ( M )( x ) k ∼ = C k , which shows that M ( x ) is a finite dimensional complex vector s pace for each x ∈ M . W e denote by rank x M = dim C M ( x ) the rank of the mo dule M a t the p oint x ∈ M . The C ∞ c ( G )-module M of finite type is of c onstant r ank if the function x 7→ rank x M is a constant function. One ca n s imilarly de fine the notions of mo dules of lo cally constant rank and of mo dules of glo bally b ounded r ank. Suppo se now that M is a smo oth, Hausdo rff and pa racompact manifold and let E b e a v ector bundle ov er M . The mo dule Γ ∞ c ( E ) of s e ctions of the bundle E is a basic example o f a mo dule of finite type and of constant rank. One can see that a s follows. Since M is finite dimensiona l and pa racompact, there exists a vector bundle F over M such that the bundle E ⊕ F is is o morphic to some trivial vector bundle M × C k ov er M ; vector bundles with this prop erty are said to b e of finite type. This prop erty basically follows from the pro of o f Lemma 5 . 9 in [14 ]. As a result we obtain the iso morphism Γ ∞ c ( E ) ⊕ Γ ∞ c ( F ) ∼ = C ∞ c ( M ) k , i.e. the mo dule Γ ∞ c ( E ) is of finite type. F urthermore, there is a na tural isomor phism Γ ∞ c ( E )( x ) → E x of complex vector space s for every x ∈ M , induced by the ev aluation at the po in t x , which s hows that the module Γ ∞ c ( E ) is of constant rank . Our no tio ns of mo dules o f finite type a nd of constant ra nk are closely c o nnected with the clas sical no tions in the Serre-Swan’s theor e m. The algebra C ∞ c ( M ) is unital pr ecisely when the manifold M is c o mpact and in this ca se it ma kes sens e to sp eak of free and pro jective mo dules ov er the algebr a C ∞ c ( M ). Finitely generated, pro jective C ∞ c ( M )-mo dules cor resp ond in this case to the mo dules of finite type and of constant rank if the ma nifold M is connected and to the mo dules of finite t yp e and of lo cally constant rank in general. 3.3. Equiv al e nce b etw een the categories of representa tions and of mo d- ules of finite ty p e and of constant rank. W e will denote by M od ( G ) the full sub c ategory of the catego ry of left mo dules over the conv olution a lgebra of the group oid G consisting of modules of finite type and of constant ra nk. Since every mo dule o f sections of a representation is such a mo dule, we have the functor (Γ ∞ c ) G : Rep ( G ) → Mo d ( G ) which r epresents a na tural equiv alence betw een the g iv en ca teg ories. 10 J. KALI ˇ SNIK Theorem 3.2. L et G b e an ´ etale Lie gr oup oi d over a smo oth manifold M . The functor (Γ ∞ c ) G : Rep ( G ) → Mo d ( G ) is an e quivalenc e b etwe en the c ate gory Rep ( G ) of r epr esent ations of G and the c ate gory Mo d ( G ) of mo dules over the c onvolution algebr a C ∞ c ( G ) of the gr oup oid G which ar e of fi nite typ e and of c onstant r ank. Before we b egin with the pro of of Theorem 3.2 we br ie fly r ecall the class ic a l version of the Ser re-Swan’s theorem in the setting of s mooth ma nifolds and mo dules ov er the algebra s of smo oth functions with compact suppo rt. Theorem 3 . 3. The functor (Γ ∞ c ) M : R ep ( M ) → Mo d ( M ) is an e quivalenc e of c ate gories for any smo oth, Hausdorff and p ar ac omp act manifold M . Pr o of. The cr ucial point in the pro of of the theorem is the observ ation tha t every vector bundle ov er a para compact ma nifold is of finite type, i.e. a subbundle of some triv ial bundle. T aking this into account, basically the sa me pro of as in the Swan’s origina l pa per [29] go es throug h.  Let G b e a n ´ etale Lie gro upoid ov er M . W e will prov e Theor em 3 .2 by con- structing a q uasi-inv erse R G : Mo d ( G ) → Rep ( G ) to the functor Γ ∞ c to show that it is a n equiv alence o f ca tegories. F or any C ∞ c ( G )- mo dule M of finite type a nd o f constant r ank w e define a vector bundle R G ( M ) ov er M a s follows. As a set, the bundle R G ( M ) is defined as a disjoint union of the spaces M ( x ) for x ∈ M R G ( M ) = a x ∈ M M ( x ) , together with a natura l pro jection onto the manifold M . T o define a top ology and a smo oth structure o n the space R G ( M ) we first choos e a vector bundle E ov er M and an isomor phism Φ : Γ ∞ c ( E ) → M of left C ∞ c ( M )-mo dules. Such an isomorphism exists due to the clas sical version of Serre-Swan’s Theorem 3.3. The induced map Φ( x ) : E x → M ( x ) is then an isomorphism of co mplex vector spaces for each x , so we ca n use the fib erwise linear bijection φ = a Φ( x ) : E → R G ( M ) to define a structure of a smo oth vector bundle ov er M on the space R G ( M ). So defined vector bundle str ucture on the space R G ( M ) is well defined. Namely , if E ′ is ano ther vector bundle ov er M and if Φ ′ : Γ ∞ c ( E ′ ) → M is an isomo rphism of C ∞ c ( M )-mo dules, we obtain the isomorphis m (Φ ′ ) − 1 ◦ Φ : Γ ∞ c ( E ) → Γ ∞ c ( E ′ ) of C ∞ c ( M )-mo dules. Bundles E and E ′ are ther e fore iso morphic by Theor em 3 .3 and in turn they define the sa me vector bundle str ucture on the space R G ( M ). W e will next use the extra s tr ucture of a C ∞ c ( G )-module on the space M to de fine a repr e sen tation of the g roup o id G on the vector bundle R G ( M ). Cho ose an y ar row g ∈ G ( x, y ) a nd any v ector v ∈ M ( x ). W e ca n find an element m ∈ M such that v = m ( x ) and a function a ∈ C ∞ c ( G ) with compact supp ort in so me bisection U such tha t a ( g ) = 1. Since M is a left C ∞ c ( G )-module, the element am ∈ M is well defined and we define g · m ( x ) = am ( y ) ∈ M ( y ) . W e will de no te by µ M : G × M R G ( M ) → R G ( M ) the map defined by the ab ove for m ula. Prop osition 3. 4. The map µ M defines a r epr esentation of t he Lie gr oup o id G on the ve ctor bu nd le R G ( M ) over M . 11 Pr o of. Before we star t with the pro of of the prop osition, we list some prop erties of the C ∞ c ( M )-mo dules M ( x ) fo r x ∈ M a nd of the conv olution alg ebra C ∞ c ( G ). (i) F or a n y m ( x ) ∈ M ( x ) and a ny function f ∈ C ∞ c ( M ) the action of f on m ( x ) is just the multiplication by f ( x ) in the v ector space M ( x ). In par ticular, f m ( x ) = m ( x ) if f ( x ) = 1 a nd f m ( x ) = 0 ∈ M ( x ) if f ( x ) = 0. (ii) Let a ∈ C ∞ c ( G ) b e a smo oth function with compact supp ort in the bisection U of the gro upoid G and denote W = t ( U ) resp ectiv ely V = s ( U ). F or a ny function f ∈ C ∞ c ( M ) with compact supp ort in V we have the formula af = ( f ◦ ( σ U ) − 1 ) a , where σ U : V → W is the diffeomor phism corr espo nding to the bisection U and f ◦ ( σ U ) − 1 ∈ C ∞ c ( M ) is a smo oth function with co mpact s upport in W . Moreov er, af = a for every function f which is identically equal to 1 on the set s (supp( a )) ⊂ V . Suppo se now that g ∈ G ( x, y ) is an arrow fro m x to y a nd let v ∈ M ( x ) b e an arbitrar y vector. W e will first show that the elemen t µ M ( g , v ) is w ell defined and independent of v a rious choices of repres en tatives for the element s g and v . T o this effect cho ose an ar bitrary function a ∈ C ∞ c ( G ) with supp ort in a bisection U such that a ( g ) = 1 and let m, m ′ ∈ M b oth satisfy m ( x ) = m ′ ( x ) = v . W e then hav e ( m − m ′ )( x ) = 0, so we ca n find a function f ∈ I x C ∞ c ( M ) a nd a n element m ′′ ∈ M with m − m ′ = f m ′′ . No w c ho ose a smo oth function f ′ ∈ C ∞ c ( M ) w ith compact suppo rt in V = s ( U ), such that af ′ = a a s in (ii). The s upp ort o f the function f ′ f is then contained in V , so by (i) it follows a ( m − m ′ )( y ) = af m ′′ ( y ) = af ′ f m ′′ ( y ) = (( f ′ f ) ◦ ( σ U ) − 1 ) am ′′ ( y ) = 0 since (( f ′ f ) ◦ ( σ U ) − 1 )( y ) = ( f ′ f )( x ) = 0. W e obtain the equality am ( y ) = am ′ ( y ) which prov es the indep endence of µ M ( g , v ) of the choice of the r epresentativ e for the element v . Next we prove a similar statement for the choice of the representativ e for the arrow g . Suppo se a ′ ∈ C ∞ c ( G ) is ano ther function with a ′ ( g ) = 1 and with supp ort in a bisection U ′ . One can then find functions f , f ′ ∈ C ∞ c ( M ) with f ( y ) = f ′ ( y ) = 1, such that f a = f ′ a ′ is a function with compact supp ort in the bisection U ∩ U ′ . The equalities am ( y ) = f am ( y ) = f ′ a ′ m ( y ) = a ′ m ( y ) then show that the element µ M ( g , v ) ∈ R G ( M ) is well defined. W e next show that µ M defines an actio n of the gr oupoid G on the vector bundle R G ( M ) ov er M . This will b e tr ue if we show that the equa lities ( g ′ g ) · v = g ′ · ( g · v ) and 1 x · v = v hold for a ll arrows g ∈ G ( x, y ), g ′ ∈ G ( y , z ) and all v ∈ M ( x ). If a ∈ C ∞ c ( G ) with a ( g ) = 1 has suppo r t in a bisection U and if a ′ ∈ C ∞ c ( G ) with a ′ ( g ′ ) = 1 has s upport in a bisec tion U ′ , then a ′ a has suppo rt in the bisection U ′ × s,t M U and ( a ′ a )( g ′ g ) = 1 . W e then have ( g ′ g ) · v = a ′ am ( z ) = g ′ · am ( y ) = g ′ · ( g · v ) . where m ∈ M is an arbitr ary elemen t s uc h that m ( x ) = v . T o prov e the second claim, it is enough to observe that any function f ∈ C ∞ c ( M ) ⊂ C ∞ c ( G ) with f ( x ) = 1 represents the iden tity arrow 1 x at the p oint x ∈ M . The result then follows directly from the definitio n of the ac tio n of the algebra C ∞ c ( M ) on M ( x ). It r emains to be prov en that µ M : G × M R G ( M ) → R G ( M ) is a smo oth map. Cho ose any element ( g , v ) ∈ G × M R G ( M ) and a function a ∈ C ∞ c ( G ) with compact suppo rt in a bisection U ′ such tha t a | U ≡ 1 for some small neighbourho o d U ⊂ U ′ of the arr o w g . There exist elements m 1 , . . . , m k ∈ M with the prop erty that the vectors { m 1 ( x ) , . . . , m k ( x ) } for m a ba sis of the vector space M ( x ) for all x ∈ s ( U ). As a re s ult we o btain smo oth functions λ 1 , . . . , λ k : R G ( M ) | s ( U ) → C , implicitely defined by the formula w = P k i =1 λ i ( w ) m i ( x ) for a n y w ∈ M ( x ) where x ∈ s ( U ). 12 J. KALI ˇ SNIK Lo cally , on a neighbourho o d U × M R G ( M ) of the po in t ( g , v ), we have µ M ( h, w ) = k X i =1 λ i ( w ) am i ( t ( h )) , where am i are smo oth sections o f the v ector bundle R G ( M ). This concludes the pro of of the prop osition.  Now choose left C ∞ c ( G )-modules of finite type a nd o f constant rank M and N and let Φ : M → N b e a homomor phism b et ween them. F or each x ∈ M we have the induced linear maps Φ ( x ) : M ( x ) → N ( x ) that define a fib erwise linear map R G (Φ) = a x ∈ M Φ( x ) : R G ( M ) → R G ( N ) . The map R G (Φ) is a smo oth mo rphism of vector bundles, since it transfor ms smoo th sections o f the bundle R G ( M ) to smo oth sections of the bundle R G ( N ). W e claim that R G (Φ) defines a G -eq uiv ariant morphism of representations of the g roup o id G on R G ( M ) resp ectively R G ( N ) as describ ed ab ov e. T o see this, choos e an arrow g ∈ G ( x, y ) and a vector v ∈ R G ( M ) x = M ( x ). Let a ∈ C ∞ c ( G ) b e a function with suppo rt in a bis e c tion U such that a ( g ) = 1 and suppo se m ∈ M satisfies m ( x ) = v . The statement then follows from the equalities R G (Φ)( g · v ) = Φ( y )( am ( y )) = Φ( am )( y ) = a Φ( m )( y ) = g · R G (Φ)( v ) . The functoriality of the as signmen t Φ 7→ Φ( x ) for ea ch x ∈ M extends to the functoriality of the map R G , so we hav e the functor R G : Mo d ( G ) → Rep ( G ) . Pr o of of The or em 3.2. W e will prov e the theorem by showing that the functors R G : Mo d ( G ) → Rep ( G ) and Γ ∞ c : Rep ( G ) → Mo d ( G ) repr esen t mutual inv erses. W e can natur ally iden tify the C ∞ c ( G )-module M with the space of s e ctions Γ ∞ c (R G ( M )) of the r epresent ation R G ( M ) o f G by assig ning a section x 7→ m ( x ) of the bundle R G ( M ) to the elemen t m ∈ M . Denote by ǫ M : M → Γ ∞ c (R G ( M )) the corresp onding is omorphism of C ∞ c ( G )-modules and let ǫ : Id Mo d ( G ) ⇒ Γ ∞ c ◦ R G be the co rresp onding natural equiv alence o f functors. Let E b e a r epresent ation of the gro upoid G . There is a natur al isomorphism Γ ∞ c ( E )( x ) → E x of complex vector spaces for every x ∈ M , induced by the ev al- uation at the po in t x , whic h induces an iso morphism η E : R G (Γ ∞ c ( E )) → E of representations of the group oid G . The natural equiv a lence of functors η : R G ◦ Γ ∞ c ⇒ Id Rep ( G ) together with the equiv a lence ǫ shows that Γ ∞ c is a n equiv alence of categ ories.  4. Equiv alence of morphisms between bica tegories According to the previous sections we ca n asso ciate to any ´ etale Lie g roup oid the additive categor ies of repr esen tations a nd of mo dules of finite t yp e and of co nstan t rank. W e would like to extend those constr uctions to define functors Rep and Mo d from the categor y of ´ etale Lie group oids a nd pr incipal bundles to the category of additive categor ies and functors such that their v alues a t the gro upoid G ar e Rep ( G ) resp ectively Mo d ( G ). F urthermo re, the family o f functors (Γ ∞ c ) G from T he o rem 3.2 should represent a natural equiv alence b et ween these tw o functors. If o ne wan ts to work in the fra mework of categories, it is necessa ry to work with isomorphis m clas ses o f repres en tations and of mo dules and forget the extra categoric al structure o n these ob jects. A more co n venien t wa y of descr ibing our 13 statement uses the languag e o f bica teg ories a nd mo r phisms be t w een them w hich we now br iefly summar iz e. 4.1. Bicategories and Mo rphisms. W e first recall the definition and some ex- amples of bicatego ries and morphisms b etw een them as given in [3, 11] (se e a lso [12] for a very co nc is e treatment). A bic ate gory C consists of (1) A collection C 0 of ob jects of C. (2) Categor ies C( A, B ) for each pair A, B of ob jects of C. The ob jects a nd arrows of the categor ies C( A, B ) are called 1 -cells resp ectiv ely 2-cells o f the bicategory C. (3) F or a ll A, B , C ∈ C 0 there exist bifunctor s C( A, B ) × C( B , C ) → C( A, C ) which define comp osition of 1-cells of C a nd horizontal comp osition of 2 -cells of C. F urthermore, for ea c h A ∈ C 0 there is the ide ntit y 1-c e ll 1 A ∈ C( A, A ). Comp osition o f 1-cells in a bicategor y C res em bles the usual co mposition of arrows in o rdinary categ ories, altho ugh it is in gener a l asso ciative o nly up to natura l ass o - ciativity coherence isomorphisms, whic h are given as a part of the structure of C. Similarly , the ident ity 1-c ells 1 A for A ∈ C 0 act as units only up to predefined unit coherence isomorphisms. All these coherence is omorphisms have to satisfy so me natural p entagon and triang le axioms (se e [12] for details) as to insure tha t all the diagrams constr ucted out of co herence isomorphis ms commute. Apart from the horizontal co mposition of 2- cells in C there exists a vertical comp osition of 2- c ells, defined by the co mposition in the categor ies C( A, B ). Strict bicategories or 2- categorie s a re bicategor ies in which the comp osition of 1 -cells is s trictly ass ocia tiv e with strict identities. Example 4.1. (i) The bicategory Cat consists of c a tegories as ob jects, functors betw een them as 1-cells and natural transfor mations b etw een functors as 2- cells. F or any categ o ry A ∈ Cat 0 the 1-cell 1 A is represented b y the identit y functor on A . All the coherence is o morphisms ar e ident ities so Cat is in fac t a 2-catego ry . W e will denote by AdCat the full sub 2-ca tegory consisting of additive categ o ries. (ii) Next we des c ribe the Mor ita bicateg o ry GPD of Lie group oids. F or an y t wo Lie g roupo ids G and H there exists a category GPD( G, H ) with principal H - bundles ov er G as ob jects and equiv ariant diffeomorphisms as morphisms b et ween them. Comp osition of 1 -cells P a nd Q is defined by the tensor pr oduct construction P ⊗ H Q to gether with the natura l asso citativity coherence isomor phisms a P,Q, R : ( P ⊗ H Q ) ⊗ K R → P ⊗ H ( Q ⊗ K R ) for P ∈ GPD( G, H ), Q ∈ GPD( H, K ) and R ∈ GP D( K, L ). F or any G ∈ GPD 0 the identit y 1 -cell 1 G is simply the group oid G viewed as a principal G -bundle ov er G . The unit coherence isomo rphisms l P : G ⊗ G P → P and r P : P ⊗ H H → P are induced b y the ac tio n ma ps of the gro up oids on the principal bundles. If f : P → P ′ and g : Q → Q ′ are equiv ariant diffeomorphisms, their hor iz on tal comp osition is naturally defined as f ⊗ g : P ⊗ H Q → P ′ ⊗ H Q ′ . W e will denote by E tGPD the full subbicateg ory of GPD consisting of ´ etale Lie gr oupoids . Now le t C and D b e tw o bicategories . A morphism F : C → D consists of (1) A function F : C 0 → D 0 . (2) F uncto r s F AB : C( A, B ) → D( F ( A ) , F ( B )) for each pair of ob jects o f C. A mor phism of bicategor ies is functor ia l only up to a family of natura l isomorphisms φ f ,g : F ( f ) ◦ F ( g ) → F ( f ◦ g ) and φ A : 1 F ( A ) → F (1 A ) 14 J. KALI ˇ SNIK for ea c h pair of comp osable 1- cells of C and for each o b ject of C. These natural isomorphisms have to satisfy some further na tural coherence ax ioms. Mor phisms of bicategories ar e also r eferred to as Lax functors in the litera ture. Conta v a riant versions of morphisms betw een bicategor ies ca n b e defined analo gously . Our main exa mples of morphisms b etw een bica teg ories come from the corres pon- dence b et w een r epresentations o f a Lie gr oupoid and mo dules of finite type and of constant rank ov er its conv olution algebr a. W e will first descr ib e the contra v ar ian t mo rphism Rep : EtGPD → AdCat from the Morita bicategor y o f ´ etale Lie g r oupo ids to the 2-catego ry o f additive categorie s. T o any ´ etale Lie g roup o id G we a s sign the additive categ ory Rep ( G ) of representations of G . If H is another ´ etale Lie g roup oid a nd if P is a principal H -bundle ov er G , then the functor Rep ( P ) : Rep ( H ) → Rep ( G ) is defined by Rep ( P )( E ) = P ⊗ H E and Rep ( P )( φ ) : P ⊗ H E → P ⊗ H F for any E ∈ R ep ( H ) resp ectively a n y φ ∈ Rep ( H )( E , F ). The morphism R ep ( P )( φ ) of representations of G is explicitely defined by Rep ( P )( φ )( p ⊗ v ) = p ⊗ φ ( v ). F urther more, if f : P → P ′ is an isomorphis m of principal H - bundles over G , we define a na tural tra nsformation R ep ( f ) : R ep ( P ) ⇒ Rep ( P ′ ) by as signing the morphism Rep ( f ) E = f ⊗ id : P ⊗ H E → P ′ ⊗ H E of r epresentations of the group oid G to the r epresentation E ∈ Rep ( H ). So defined morphism of bicategorie s is functor ial up to the family of natural iso morphisms φ P,Q : Rep ( P ) ◦ Rep ( Q ) → Rep ( P ⊗ H Q ) defined by the natural maps φ P,Q ( E ) : P ⊗ H ( Q ⊗ K E ) → ( P ⊗ H Q ) ⊗ K E for an y E ∈ Rep ( K ). Natural transfor mations for the identit y 1 -cells can be defined in a s imilar fashio n by applying the representation maps. Our sec o nd example is the contrav a riant morphism Mo d : EtGPD → AdCat from the Morita bica tegory of ´ etale Lie gro up oids to the 2 - category of additive cat- egories. F or each ´ etale Lie group oid G let Mo d ( G ) denote the categor y o f mo dules ov er the co n volution a lg ebra C ∞ c ( G ) that are of finite type and of constant rank. If P is a principal H - bundle ov er G the functor Mo d ( P ) : M od ( H ) → Mo d ( G ) is simply the restriction of the functor o f tensor ing by C ∞ c ( P ) C ∞ c ( P ) ⊗ C ∞ c ( H ) − : H Mo d → G Mo d T enso ring b y the rig h t C ∞ c ( H )-mo dule C ∞ c ( P ) maps the catego ry of left C ∞ c ( H )- mo dules to the category of left C ∞ c ( G )-modules since C ∞ c ( P ) is a C ∞ c ( G )- C ∞ c ( H )- bimo dule. If M ∈ Mo d ( H ) is a C ∞ c ( H )-mo dule of finite type and of co nstan t r a nk, then C ∞ c ( P ) ⊗ C ∞ c ( H ) M is a C ∞ c ( G )-module of finite type a nd of constant r ank by Prop osition 4.2 which shows that M od ( P ) is a well defined functor. Natural isomorphisms φ P,Q : Mo d ( P ) ◦ M od ( Q ) → Mo d ( P ⊗ H Q ) a re defined by the maps Ω P,Q ⊗ id : C ∞ c ( P ) ⊗ ( C ∞ c ( Q ) ⊗ Γ ∞ c ( E )) → C ∞ c ( P ⊗ H Q ) ⊗ Γ ∞ c ( E ) , where Ω P,Q : C ∞ c ( P ) ⊗ C ∞ c ( H ) C ∞ c ( Q ) → C ∞ c ( P ⊗ H Q ) is the isomorphism of C ∞ c ( G )- C ∞ c ( K )-bimo dules as defined in [20], see a lso [10]. The identit y co herence isomo r- phisms are pr o vided b y the a ctions of conv olution alg e br as on the mo dules. It is str aight forward to chec k that so defined families o f functors and natural isomorphisms satisfy the axioms for the mo rphisms b etw een bicatego r ies. 15 4.2. Natural equi v alence of the morphisms Rep and M o d. W e will show in this section how one c an int erpret the Serre - Sw an’s cor resp o ndence as a natura l transformatio n betw een the contra v ar ian t morphisms Re p and Mo d fr om the Morita bicategory of ´ etale Lie gro upoids to the 2-catego ry o f additive categor ies. T o this effect we first review the definition of a na tural tra nsformation betw een t wo mo rphisms of bica tegories [11, 12]. Let C and D b e t wo bicategorie s and let F, G : C → D be mor phisms b etw een them. A natur al tr ansformation σ : F → G consists of (1) F o r each A ∈ C 0 a 1-cell σ A : F ( A ) → G ( A ). (2) F o r each 1-cell f : A → B in C a 2 -cell σ f : G ( f ) ◦ σ A ⇒ σ B ◦ F ( f ) F ( A ) F ( f ) − − − − → F ( B ) σ A   y   y σ B G ( A ) G ( f ) − − − − → G ( B ) such that σ f is natural in f and satisfies coherence ax io ms as in [12]. W e b egin with a pr opos ition that is of some int erest indep enden tly of our discussion. Prop osition 4. 2. L et G and H b e ´ etale Lie gr oup oi ds and let P b e a princip al H - bund le over G . F or any r epr esentation E of H ther e exists a natu r al isomorp hism σ P ( E ) : C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) → Γ ∞ c ( P ⊗ H E ) of C ∞ c ( G ) -mo dules. Pr o of. W e fir st define a bilinear map σ P ( E ) : C ∞ c ( P ) × Γ ∞ c ( E ) → Γ ∞ c ( P ⊗ H E ) by the for m ula ( σ P ( E )( f , u ))( x ) = X π ( p )= x f ( p )( p ⊗ u ( φ ( p ))) for f ∈ C ∞ c ( P ) and u ∈ Γ ∞ c ( E ). It is not to o hard to chec k that the map σ P ( E ) is well defined a nd that it induces a homomorphism σ P ( E ) : C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) → Γ ∞ c ( P ⊗ H E ) of C ∞ c ( G )-modules , which we cla im to be an iso morphism. It suffices to show tha t σ P ( E ) is a bijective map. W e firs t cons ider the case when the bundle P is trivia l, i.e . P = P ( ψ ) for s o me smo oth functor ψ : G → H . The represe ntation P ⊗ H E of G is then isomo rphic to the repr esen tation ψ ∗ E of G via the isomorphism f : P ⊗ H E → ψ ∗ E . In particula r, f is an isomorphism of vector bundles ov er M , so we o btain the is o morphism Γ ∞ c ( f ) : Γ ∞ c ( P ⊗ H E ) → Γ ∞ c ( ψ ∗ 0 E ) of C ∞ c ( M )-mo dules. Now co nsider the repr esen tation E of H as a vector bundle ov er N . It is a classica l result (see [6]) that the map σ ψ 0 ( E ) : C ∞ c ( M ) ⊗ C ∞ c ( N ) Γ ∞ c ( E ) → Γ ∞ c ( ψ ∗ 0 E ) , defined analogously as the map σ P ( E ), is an isomor phism of C ∞ c ( M )-mo dules. Finally , since P = M × ψ ,t N H is a tr ivial bundle, we hav e by [20] the isomorphism Ω M ,H : C ∞ c ( P ) ∼ = C ∞ c ( M ) ⊗ C ∞ c ( N ) C ∞ c ( H ) of C ∞ c ( M )- C ∞ c ( H )-bimo dules which g iv es us a n iso morphism ι : C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) → C ∞ c ( M ) ⊗ C ∞ c ( N ) Γ ∞ c ( E ) 16 J. KALI ˇ SNIK of C ∞ c ( M )-mo dules. W e can collect all these is omorphisms into the following com- m utative diagra m o f homomo rphisms of C ∞ c ( M )-mo dules C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) σ P ( E ) − − − − → Γ ∞ c ( P ⊗ H E ) ι   y ∼ = ∼ =   y Γ ∞ c ( f ) C ∞ c ( M ) ⊗ C ∞ c ( N ) Γ ∞ c ( E ) σ ψ 0 ( E ) − − − − − → ∼ = Γ ∞ c ( ψ ∗ 0 E ) Since the remaining thr e e maps ar e bijective, the map σ P ( E ) is bijective as well. A pr incipal H -bundle P ov er G is in genera l only lo cally trivial [18]. Let U b e an o pen subset o f M such that P | U is a trivial H -bundle. W e then have a natural injectiv e homomor phis m i : C ∞ c ( P | U ) → C ∞ c ( P ) of right C ∞ c ( H )-mo dules which induces an injectiv e homomorphis m i ⊗ id : C ∞ c ( P | U ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) → C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) of a belian groups. Injectivity of i ⊗ id follows fr om the fact that C ∞ c ( P ) is a lo cally unital C ∞ c ( M )-mo dule. Namely , for any w = P f i ⊗ u i ∈ C ∞ c ( P | U ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) there exis ts a function f ∈ C ∞ c ( M ) with supp ort in U such that f f i = f i for ea c h i . Left action µ f : C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) → C ∞ c ( P | U ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) b y f is then a homomorphism of ab elian groups suc h that µ f (( i ⊗ id)( w )) = w . W e now have the following co mm uting square o f homo mo rphisms of ab e lian g roups C ∞ c ( P | U ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) σ P | U ( E ) − − − − − → ∼ = Γ ∞ c ( P | U ⊗ H E ) i ⊗ id   y   y C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) σ P ( E ) − − − − → Γ ∞ c ( P ⊗ H E ) where the right vertical map is defined by tr iv ially ex tending the sections outside of the set U . Note that b oth the vertical maps are injective. The injectivit y of the map σ P ( E ) w ill now follow from the injectivity of the map σ P | U ( E ). Indeed, suppo se that η ∈ C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ) is such that σ P ( E )( η ) = 0. Since the mo dule C ∞ c ( P ) is lo cally unital, there exists a function f ∈ C ∞ c ( M ) suc h that f η = η a nd a deco mpositio n f = P f i int o functions with suppo rts contained in op en sets U i such that the bundle P | U i is tr ivial for each i . It now follows fro m the C ∞ c ( M )-linearity of the map σ P ( E ) and from the preceding diagra m tha t σ P | U i ( E )( f i η ) = σ P ( E )( f i η ) = f i σ P ( E )( η ) = 0 . The injectivity of the maps σ P | U i ( E ) no w implies that f i η = 0 for each i and th us η = 0, which pr oves that σ P ( E ) is injective. Surjectivity c a n be prov en by using simila r arguments a nd by noting that the sets C ∞ c ( P | U ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ), a s U v aries over so me trivializing cover o f the bundle P , ge nerate the C ∞ c ( G )-module C ∞ c ( P ) ⊗ C ∞ c ( H ) Γ ∞ c ( E ).  Let us now return to o ur morphisms Rep and M od of bica tegories. Define a natural transforma tion σ : Rep ⇒ Mo d by defining a functor σ G = (Γ ∞ c ) G for each ´ etale Lie gr oupoid G and a na tural transfor mation σ P : Mo d ( P ) ◦ σ H ⇒ σ G ◦ Rep ( P ) for each 1-cell P ∈ EtGPD( G, H ) of the bicategory EtGPD. T ak ing into account Prop osition 4 .2 it is now straightforward to verify the following theor em. 17 Theorem 4. 3. The morphisms Rep and M od fr om the Morita bic ate gory of Lie gr oup oids to the bic ate gory of additive c ate gories ar e natur al ly e quivalent. N atu r al e quiva lenc e is given by the the family of fun ctors (Γ ∞ c ) G : R ep ( G ) → Mo d ( G ) for any gr oup oid G and the family of tr ansformations σ P : Mo d ( P ) ◦ σ H ⇒ σ G ◦ Rep ( P ) for any princip al H -bund le P over G . Ac kno wledgemen ts. I would like to thank J. Mrˇ cun fo r ma n y helpful discussio ns and a dvice rela ted to the pap er. References [1] A . Adem, J. Leida, Y. Ruan, O rbifolds and Stringy T op olo gy . 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Institute of Ma thema tics , Physics and Mechanics, University of Ljubljana, Jadran - ska 19 , 1000 Ljubljana, Slovenia E-mail addr ess : jure.kalis nik@fmf.un i-lj.si