Differentiable Categories, gerbes and G-structures

Tsemo Aristide, Coll` ege Bor´ eal Campus de T oronto 951, av enue Carlaw T oronto O N M4 K 3M2 tsemo58@yaho o.ca Diﬀerent iable Categories, diﬀerentiable gerb es and G -structur es. Abstract. The theories of strings and D -bra nes ha v e motiv ated the developmen t of non Abelia n cohomo logy tec hniques in diﬀerential geometry , on the purp ose to ﬁnd a geometric interpretation of characteristic clas ses. The spaces studied here, like or bifolds a re not often smooth. In classical diﬀeren tial geometry , non smo oth spaces app ear also naturally , for example in the theory of foliations, the s pace of leav es can b e an or bifo ld with singula rities. The scheme to study these structures is identical: classical to ols used in diﬀerential ge ometry , like connections and curv atures are adapted. The purp os e of this pap er is to present the no tio n o f diﬀerentiable catego ry which uniﬁes all these po ints o f view. This enables us to pro vide a geometric in terpretation of 5-ch arac teristic clas ses, and to interpret classical problems which a ppea r in the theory of G -str uctur es by using gerb es. 1 In tro duction. Diﬀerent ial geometry is the study o f the analy tic prop erties of top ologica l spaces. Most of the main to ols developed in this theory are is sued from calculus, and a manifold do es not hav e sing ularities. Mo duli spa ces in diﬀerential geometry are r arely smo oth as sho w the space of orbits o f the action o f a compact Lie group on a ma nifold, the space of leaves of a folia tion, the co mpactiﬁcation of a space of curves,... Since these singular structures arise natura lly in diﬀerential geometry , it is normal to try to study them b y using metho ds created in the smo oth case . An example of such a metho d is the theory of orbifolds created by Sa take [37 ], which enables to study the action o f ﬁnite gro ups on manifolds which may have ﬁxed p oints (see Audin [4 ]), foliations with bundle-like metrics (see Mo lino a nd Pierrot [33]), strings theory (see Chen and Ruan [11]), quotient of compact aﬃne manifolds, in particular quotient of ﬂat aﬃne spaces forms are s tudied by many author s (see Lo ng and Reid [2 0 ], Ratcliﬀe and Tschant z [36]).... W e ca n als o quote other theories such as the theor y of foliage s which 1 studies the diﬀer e n tial ge o metry of the space of leaves of a foliation (see Mo lino [31]), the study of homogeneous ( X, H )-manifolds see (Goldman [15 ]). One of the main goa l of this pap er is to pr op ose the theory o f diﬀeren- tiable categories to unify the gener alizations of cla ssical diﬀeren tial geometry men tioned ab ov e: a diﬀer en tiable catego ry is a category whose ob jects are dif- ferentiable manifolds , and the morphisms b etw een its ob jects are diﬀerentiable maps. This p oint of view enables to handle a lso new situations like gener alized orbifolds s uch as the or bit spa ce of the action of a compact Lie gro up, or the space of leav es of a foliatio n endow ed with a bundle-lik e metr ic. (see Molino and Pierrot [33]). In mathematics, the classiﬁcatio n problem is the cornerstone on which re- lies every theory T ; this is p erfo rmed by assigning to ob jects which occ ur in T simpler inv a riants whic h enable to describ e them co mpletely: fo r exa mple, the genus of a clo sed surfa c e. The scheme usually followed in cla ssical diﬀerential geometry is to deﬁne ob jects and inv ar iants lo cally , and glue them with par- titions of unit y . Lo cal inv ar iants in the theor y of diﬀerentiable catego ries a re more diﬃcult to study , since even when there exists a topo logy , neigh b orho o ds of diﬀerent p oints are not a lways isomor phic, for example the notion of a fr a mes bundle is not straightforw ard deﬁned since the dimension of the ob jects in a diﬀerentiable categor y may v a ry . This situation is analo g to algebra ic geome- try , and we intensiv ely us e the machinery de velop ed by Grothendieck a nd his student s in this context (see Gira ud [13], [14]). In fact sheav es of categ ories a nd gerb es are no wada ys intensiv ely studied by diﬀerential geometers (se e Brylin- ski [8], Brylins k i and McLaug hlin [9], [10]): the functional action in class ical mechanic which describ es the motion of a point is expr essed b y using a con- nection on a principa l bundle. In the purpose to unify all existing fundamental strengths, physicists hav e deﬁned strings and branes theor ies. The functional action which describ es the motio n of a string is deﬁned by a ger b e, and one exp ects that a go o d notion of n -g erb es will ena ble to handle branes theories . In fact sheav es of categor ie s in this context are examples of diﬀerentiable cate- gories. A curr ent ly v e r y active resear ch topic is the adaptation of to o ls deﬁned in classical geometry like connectio ns on principal bundles to these ob jects. W e star t this pap er by studying the diﬀerential geometry of diﬀerentiable categorie s without using Grothendieck top olo gies. W e deﬁne the notions of principal bundles, which ar e to rsors whose ﬁb ers a re L ie g r oups, the tangent space of a diﬀerentiable catego ry and its DeRham cohomolo gy . In this setting we introduce connectio ns for ms and distributions a nd study their holonomy . In mo dern geometry , global ob jects are constructed by g luing loca l ob jects. F or example, a manifold is obtained by g luing op en subs e ts of a vector space, schemes in algebraic g eometry are deﬁned by gluing sp ectrums of comm utative rings. Algebraic geometer s hav e remar ked tha t in many situations, the transi- tion functions that a re us e d to glue ob jects do not verify the Chasles relation. This ha s mo tiv a ted descent theory which is pr esented in the s etting of categor ies theory by Gira ud [13 ]. W e study diﬀer e n tial descen t; or equiv a lent ly descen t 2 in the theory of diﬀere n tiable ca tegories. This is an a daptation of the a nalysis situs of Giraud; we int ro duce diﬀerentiable ﬁbered principal functors and their connective structures. Recall that the notion of connective str ucture has b een int ro duced by Brylinski [8 ] in the context of gerb es o n manifolds to provide a geometric in terpretation of characteristic classes. The lo cal ana ly sis in tensively use d in diﬀerential geometry relies on the ex- istence of neig h b orho o ds of points. This is ac hieved in this context by diﬀer e n- tiable Grothendieck top olog ies: ex amples of Grothendieck top olog ies are deﬁned on orbifolds , gene r alized or bifolds, folia ges,... the CechDeRham complex is then used to study coho mology . Chen and Ruan [11] hav e de ﬁned a new cohomology theory for orbifolds to understand mathematical strings theory . W e adapt to generalized orbifolds this new cohomolog y theory . With the no tio n of Grothendieck top olog ies deﬁned, we can study sheav e s o f categorie s and gerb es in the theory of diﬀerentiable categ ories. The ﬁrst example of such a c o nstruction can be obtained b y gerb es deﬁned on the Grothendieck top ology asso ciated to an orbifold. Luper cio and Urib e [21] hav e provided such a construction by using gr oup o ids. One of the fundamental ex ample o f a dif- ferentiable gerbe is the cano nical gerb e deﬁned o n a compact simple Lie gr o up H (see B rylinski [8 ]). The classifying co cycle of this gerb e is the ca nonical 3- cohomolog y class deﬁned by the Killing for m. Medina and Revo y [26], [27 ] have classiﬁed Lie groups endow ed with a non degenera ted bi-inv ariant scalar pro d- uct which also deﬁnes a canonical 3 -form. Rema r k that these Lie groups are not alwa y s co mpact and are even cont ractible when they are nilpotent and sim- ply connected. The theory o f lattices in Lie groups pres en ted by Ragh unathan [35] and the Leray-Serre sp ectral sequence ena ble us to construct fundamen tal examples of gerbes on compact manifolds whic h are the quotient of a nilp otent Lie group b y a lattice. The notion of Grothendieck topo logy of a diﬀerentiable ca tegory enables us to construct the cur ving, and the curv ature of a connective s tructure on a diﬀerentiable principal gerb e. W e also deﬁne the holono m y form which is used to study functional action on lo op spaces. An approach of the study of the diﬀer e n tial geometry of a ger be can b e done by using r ig ht inv ar iant distributions deﬁned in the thesis of Molino [29]. W e outline how to a gerb e deﬁned on a manifold one ca n asso cia te an in v ariant distribution whic h enables to construct the holo nomy around curves. In the last part of the pap er, we study sequences of ﬁb ered categ ories. An example of s uch a construction has b een done by Brylinski and McLaughlin [9] to pr ovide a geometric repres ent ation o f the Pon tryagin class o f degr ee 4 . T o a pr incipal gerb e , we asso ciate a 2-sequence of ﬁb ered catego ries which must be a n example of a U (1) 3- g erb e (rec all that the notion of 3-gerb e is no t well- understo o d yet) . W e asso ciate to such a 2-sequence of ﬁb e red categor ies a 5-integral coho mo logy class. This new to ols fo r diﬀerential geo meters can b e us ed to tackle well-known problems in diﬀere ntial geo metry . A G -structure is a reduction of the bundle o f 3 jets deﬁned on a ma nifold. This theory has b een intensiv ely studied in the sev- ent ies (see Molino [3 2] and the thesis of Alb ert [1], Medina [25], Nguiﬀo-Bo yom [34]). W e can a sso ciate to a manifold a sheaf of catego r ies which represe nts the geometric obstruction to the existence of a G -structure. Plan. 1. Intro duction. 2. No ta tion. 3. B a sic deﬁnitions and examples. 3.1 Orbifolds ( X , H )-manifolds and diﬀerentiable categor ies. 3.2 Actions of compact Lie groups and diﬀerent iable categ ories. 3.3 F oliages and diﬀerentiable categories. 3.4 Pro jective pr esented manifolds. 4. Diﬀerentiable ﬁbere d categories . 4.1 Connections on diﬀerentiable bundles to r sors. 4.2 Diﬀerentiable tensors o f a diﬀeren tiable category . 4.3 F rames bundle and diﬀerentiable categories. 4.4 Diﬀerentiable descen t and connections in ﬁb ered categ ories. 5. Gr othendieck top olo gies in diﬀeren tiable categories. 5.1 Grothendieck topolog ie s a nd cohomology of diﬀeren tiable catego r ies. 5.2 Chen-Ruan cohomolog y fo r generalized orbifolds. 6. Shea f of categories and gerb es in diﬀeren tia ble categories . 6.1 The classifying co cycle of a gerb e. 7. E xamples of sheav es of categories and gerb es. 7.1 Gerb es and G -structures. 7.2 Gerb es and inv ariant sca lar pro ducts on Lie groups. 7.3 A sheaf of categor ies on a n orbifold with singularities. 8. Diﬀerential geometry of sheav es of categor ies. 8.1 Induced bundles. 8.2 Reduction to the motiv ating example. 8.3 Holonomy and functor on lo op spaces. 8.4 Canonical relations asso cia ted to the connective structure o n a gerbe. 8.5 Uniform distribution and gerb es. 9. Seq uences of ﬁbered ca tegories in diﬀerentiable ca teg ories. 9.1 4-co cycles and sequences of ﬁb ered categorie s References 4 2 Notations. Let C be a category which has a ﬁnal ob ject e , and I a small set relatively to a g iven universe (se e [3 ] SGA 4 p. 4). In fact the cardina lit y used thro ughout this paper are n umerable. Since we ar e studying diﬀerentiable manifolds, we wan t our spaces to b e at least paracompact, to insur e e xistence of partitions o f unit y , one of the main to ols used in diﬀeren tial geometry to sho w the existence of global ob jects. Consider a small family ( X i ) i ∈ I of ob jects of C . W e denote by X i 1 ...i n the ﬁber pr o duct (if it exists) of the ﬁnite subset { X i 1 , ..., X i n } of ( X i ) i ∈ I ov er e . Let P b e a pres heaf o f categories deﬁned ov er C . F or every ob jects e i , e ′ i ∈ P ( X i 1 ), and a map u : e i → e ′ i , we deno te by e i 2 ...i n i and by u i 2 ...i n the resp ective restrictions of e i and u to U i 1 ...i n . 3 Basic Deﬁnitions and examples. The diﬀer en tiable manifolds used in this pa per are C ∞ , a nd ﬁnite dimensio nal. Deﬁnition 3.1. A diﬀerentiable category C is a categor y such that: - every element X o f the class of ob jects of C is a diﬀeren tiable manifold, - every mor phism of C is a diﬀerentiable map. Examples. The category D if f , whose ob jects are ﬁnite dimensional diﬀerentiable man- ifolds, and suc h that the set of morphisms H om Dif f ( M , N ) b etw een t wo diﬀer- ent iable manifo lds M and N is the set of diﬀerentiable maps b etw een M and N , is a diﬀerentiable ca tegory . Remark that this categ ory is not small relatively to an universe U which contains the set o f real n umber s, but is U -small (see [3] SGA4 p. 5 ). Let N be a manifold, and C N the catego ry whose ob jects a r e op en subs e ts of N . The morphisms betw een ob jects o f C N are the ca nonical imbeddings. The category C N is endow ed with the s tr ucture o f a diﬀerent iable catego ry , for whic h each op en subset of N is endow ed with the diﬀerentiable structure inherited from N . 3.1 Orbifolds, ( X , H ) -m anifolds and diﬀeren tiable categories. The theo ry o f or bifolds has b een intro duced by Satake (see Satake [37], Chen a nd Ruan[11]). Orbifolds app ear in diﬀerent branches of mathematics, like s trings theory , foliatio ns theory: the singular foliation deﬁned by the adherence of the leav es o f a foliation endow ed with a bundle-like metr ic can deﬁne an or bifold (see Molino and Pierro t [33 ] p. 208 ). Deﬁnition 3.1.1. 5 An n -dimensio nal orbifold N (see als o Chen and Rua n [11] deﬁnition 2.1), is a separated top ological space N , suc h that: - for every element x ∈ N , there exists an open s ubset U x of N , - a n o pen subset V x of R n , a ﬁnite gro up of diﬀeomorphisms Γ x of V x , an element ˆ x ∈ V x such that for every elemen t γ x in Γ x , γ x ( ˆ x ) = ˆ x . -There exis ts a contin uous map φ x : V x → U x , such that φ x ( ˆ x ) = x , fo r each y ∈ V x , and γ x in Γ x , φ x ( γ x ( y )) = φ x ( y ), a nd the induced mor phism V x / Γ x → U x is an homeomorphism. The triple ( V x , φ x , Γ x ) is called an orbifold chart. W e supp o se that the following co ndition is satisﬁed: Let ( V x , φ x , Γ x ) and ( V y , φ y , Γ y ) be tw o o r bifolds c harts. W e denote by p x : V x × N V y → V x the canonical pro jection. W e supp ose that there exis ts a n equiv ariant diﬀeomorphis m in r esp ect of Γ x and Γ y : φ xy : p y ( V x × N V y ) → p x ( V x × N V y ) such that: φ y | p y ( V x × N V y ) = φ x | p x ( V x × N V y ) ◦ φ xy . The fact that the mo r phism φ xy is equiv ar iant is equiv alent to saying that for every element γ y in Γ y , there exists an element Φ xy ( γ y ) in Γ x such that: φ xy ◦ γ y = Φ xy ( γ y ) ◦ φ xy . . Remark tha t the ﬁb er product V x × N V y is not necessar ily a manifold. This can b e illustrated by the following example: consider the quotient N , of the real line R, by the map x → − x , the ﬁb er pro duct R × N R is the union o f t wo no n parallel lines in R 2 ; but p x ( V x × N V y ) is an open subs e t of V x . The maps φ x ◦ φ xy ◦ φ y z | p z ( V x × N V y × N V z ) and φ x ◦ φ xz p z ( | V x × N V y × N V z ) are equal. Since Γ x is ﬁnite, there exists an element c xy z in Γ x such that: φ xy ◦ φ y z | p z ( V x × N V y × N V z ) = c xy z φ xz p z ( V x × N V y × N V z ) . Let γ z be an elemen t of Γ z , w e hav e: φ xy ◦ φ y z ◦ γ z | p z ( V x × N V y × N V z ) = Φ xy (Φ y z ( γ z )) ◦ c xy z φ xz p z ( | V x × N V y × N V z ) . W e also hav e : c xy z φ xz ◦ γ z p z ( V x × N V y × N V z ) = c xy z ◦ Φ xz ( γ z ) ◦ φ xz p z ( V x × N V y × N V z ) This implies that: 6 Φ xy ◦ Φ y z = c xy z ◦ Φ xz ◦ c xy z − 1 . An example of an orbifold is the quotient of a manifold by a ﬁnite s ubgroup o f its g roup of diﬀeomorphisms. W e are g oing to characterize some or bifolds aris ing from the theory of aﬃne manifolds . (See also Long and Reid [20]; Ratcliﬀe, and Tschan tz [36]). Let H be a Lie group which acts diﬀeren tiably and tra nsitively o n the man- ifold X , we ﬁrst r ecall s ome basic facts of the theory of ( X, H ) manifolds (See Goldman [15]). A ( X, H )-manifold, is a diﬀerentiable manifold, endo wed with an o pen cov- ering ( U i ) i ∈ I such that: - F or each i ∈ I , there exis ts a diﬀere ntiable map d i : U i → X , such that d i : U i → d i ( U i ) is a diﬀeomorphism. - The transition function d j ◦ d i − 1 | d i ( U i ∩ U j ) : d i ( U i ∩ U j ) → d j ( U i ∩ U j ) coincides with the restriction of the action of an element of H on d i ( U i ∩ U j ). An ( X , H )-map f : N → N ′ betw een the ( X , H ) -manifolds N and N ′ , is a diﬀerentiable map which preserves their ( X , H ) structures. Examples of ( X , H ) s tructures are n -dimensiona l aﬃne manifolds: here H is the g roup Af f (R n ) of aﬃne trans formations of R n , and X = R n ; n - dimensional pro jective manifolds where X is the real pro jective s pace P n (R), and H is P Gl ( n, R) the group of pro jectiv e transformations. A (R n , Af f (R n ))- automorphism is called an aﬃne transforma tion. W e can show the following result relating o rbifolds to aﬃne manifolds, and to pro jective manifolds : Prop ositio n 3 .1.1. L et N b e an aﬃne manifold, and Γ a ﬁ nite gr oup of aﬃne t r ansformations of N , such that the set C of elements of N , s u ch that for every element u of C , ther e exists a n on trivial element γ of Γ such that γ ( u ) = u , is ﬁnite; mor e over we supp ose that every element of C is ﬁxe d by every element of Γ . L et p : N → N/ Γ b e the c anonic al pr oje ction. The blowing-up (not in the classic al sense) of N / Γ at p ( C ) is a pr oje ct ive manifold. Pro of. First w e are going to blow-up the ac tio n of Γ . Let u b e an element of C , a nd U an aﬃne chart aro und u . Thus U − { u } can be identiﬁed with a ball without the orig in. Conside r the submanifold P ′ n of R n × P n − 1 R deﬁned by the equa tio ns: ( x 1 , ..., x n , [ X 1 , ..., X n ]) ∈ P ′ n ⇐ ⇒ x i X j − x j X i = 0 . There exists a pro jectio n p ′ : P ′ n → R n , the res tr iction of the ca nonical R n × P n − 1 R → R n . The blowing up of N at u is the op eration whic h replace s U by p ′ − 1 ( U ) (see McDuﬀ-Salamon [23] p. 23 3-235 , See also Tsemo [3 9]). W e can cov er P n − 1 R by t wo op en aﬃne subsets U 1 and U 2 ,which ar e the trivializ a tions of the the R- line bundle P ′ n ov er P n − 1 R. The co ordina tes change of these trivializations is the map: 7 u 12 : U 1 ∩ U 2 × R − → U 1 ∩ U 2 × R ( x, y ) − → ( x, − y ) Thu s the imbedding maps: u i : i = 1 , 2 : U i × R → P n +1 R ( x, y ) − → [ x, y ] deﬁnes a pro jective structur e aro und u which can b e g lued with the aﬃne atlas of N − C to obtain a pro jective struc tur e on the blo w ing -up ˆ N , of N . W e can identify the restrictio n of the a ction o f the element γ of Γ o n U , to a linear map A γ , a nd extends it to a map A ′ γ of P ′ n deﬁned by A ′ γ ( x 1 , ..., x n [ X 1 , ..., X n ]) = ( A γ ( x 1 , ..., x n ) , A γ ([ X 1 , ..., X n ]). W e thus obtain a free a ction of Γ on ˆ N by pr o- jective maps. The quotient of ˆ N by this action is a pr o jective manifold N ”, obtained from N / Γ , b y replacing a neighbor ho o d of ev ery element p ( u ), u ∈ C by the quotient o f p ′ − 1 ( U ) by Γ. W e also s ay that N ” is a blowing-up (not in the classical sense) of N / Γ • W e asso cia te to a orbifold N the diﬀerentiable catego ry C N deﬁned as fol- lows: An ob ject of C N is a triple ( M , φ M , Γ M ) where M is a ma nifold, Γ M a ﬁnite group of diﬀeomorphisms of M , and φ M : M → N , a cont inuous map suc h that for every element γ M in Γ M , for every elemen t x ∈ M , φ M ( γ M ( x )) = φ M ( x ), and the induced map M / Γ M → N is a lo cal ho meo morphism. F or ev er y y = φ M ( x ) in N , there exists a c hart of the orbifold ( V x , φ x , Γ x ) around y = φ x ( x ) (see deﬁnition 3 .1.1) such that: if p M : V x × N M → M , and p x : V x × N M → V x are the natural pr o jections, there exists an equiv a riant lo cal diﬀeomo rphism φ M V x : p M ( V x × N M ) → p x ( V x × N M ) such that φ x ◦ φ M V x = φ M | p M ( V x × N M ) . In particular a c hart of M is an o b ject of C N . A morphism b etw ee n the ob jects ( M , φ M , Γ M ), and ( M ′ , φ M ′ , Γ M ′ ), is an equiv ariant diﬀerentiable ma p φ : ( M , Γ M ) → ( M ′ , Γ M ′ ) suc h that φ M = φ M ′ ◦ φ . 3.2 Actions of compact Lie groups and diﬀeren tiable c at- egories. Let M be a ﬁnite dimensional manifold, and G a compact Lie group whic h acts eﬀectively on M . This is equiv a le nt to saying that if a n element o f G ﬁxe s every element of M , it is the iden tit y . W e denote b y N the quotient of M by G . The orbits of G are submanifolds, a nd when the action is free, a well-known elementary res ult implies that N is a manifold (see Audin [4 ] p. 13 - 19). In the general situation N is an orbifold with singular ities. This can be seen b y using the slice theorem of Koszul that w e recall now: 8 Theorem 3.2.1 Ko s zul [18] . L et G b e a c omp act Lie gr oup which acts eﬀe ctively and diﬀer entiably on the manifold M . L et u b e an element of M . Denote by G u the sub gr oup of G which ﬁxes u , ther e exists an invariant neighb orho o d U of u which is isomorphi c to a neighb orho o d of the zer o se ct ion in t he quotient of G × V by G u , wher e V is the quotient of the tangent sp ac e T u M by it s su bsp ac e tangent to t he orbit. Thu s, the slice theorem allows to construct an open co vering of M /G whose elements are quotient of op en subsets of a vector space by the actio n of a c o mpact Lie group (take a transversal to the zero section in theorem 3.2.1). Let N b e the quotient space of M b y G . W e asso c iate to the action o f G on M the following diﬀeren tiable category C N deﬁned as follows: An ob ject of C N is a triple ( P , H , φ P ) where P is a manifold endowed with an eﬀective ac tio n of a compa c t Lie group H , such that there exists a loc al equiv ariant diﬀeomo rphism φ P : ( P, H ) → ( M , G ) such that the induced map P / H → N = M / G is a local homeomo rphism. A mor phism f b et ween the ob jects ( P , H , φ P ) and ( P ′ , H ′ , φ P ′ ) is deﬁned by an equiv aria nt diﬀerentiable map f : P → P ′ such that φ P ′ ◦ f = φ P . The previous construction can b e generalize d in the following setting: Deﬁnition 3.2.1. Let N be a separated top ologic al space , A ge ne r alized o rbifold on N is deﬁned by the following data: F or every elemen t u ∈ N , there exis ts a manifold M u , a compact Lie group H u which a cts diﬀerentiably o n M u , and a contin uous map φ u : M u → N whos e image contains u ; suc h that for every h u in H u , for every x in M u , φ u ( h u ( x )) = φ u ( x ), a nd the induced map M u /H u → N is a lo c a l homeo morphism. The triple ( M u , H u , φ u ) is called a chart of the gener alized orbifold. Let ( M u , H u , φ u ) and ( M v , H v , φ v ) b e tw o charts. Denote by p u : M u × N M v → M u the cano nical pr o jection. There exists a lo cal equiv ariant diﬀeo- morphism φ uv : p v ( M u × N M v ) → p u ( M u × N M v ) such that φ v | p v ( M u × N M u ) = φ u ◦ φ uv . Mo duli spaces app ear in diﬀer e n t domains o f diﬀeren tial geometr y ; many of them can be endow ed with the str ucture of a diﬀerentiable catego ry C . F or example, consider the diﬀerentiable categor y whose class of ob jects is the class of isomor phism classes of hyp erb olic surfa ces of g enus h , h ﬁxed. Let [ X h ] b e the cla ss o f the s urface of genus h , X h . The diﬀeren tiable structure of [ X h ] is the diﬀerentiable structure of one element picked in the cla s s of [ X h ], for example X h itself. The set of morphisms H om ([ X h ] , [ X h ′ ]) is the set o f hyperb olic maps betw een X 1 h and X 2 h ′ , where X 1 h and X 2 h ′ are the re s pec tive repres ent ants pick ed in the classe s of [ X h ] a nd [ X h ′ ] to deﬁne the structure of the diﬀerentiable category . 9 3.3 F oliages and diﬀeren tiable categor ies. Let N be a n - dimens io nal manifold, a foliation F on N of co dimension q is deﬁned by an atlas ( U i , φ i : U i → R p × R q ) i ∈ I , such that φ i ◦ φ j − 1 | φ j ( U i ∩ U j ) ( x, y ) = ( u ij ( x, y ) , v ij ( y )). This is equiv alent to deﬁne a partition of N by immer sed manifolds of dimensio n p called the leaves. In this situation we say that the couple ( N , F ) is a fo lia ted manifold. One of the main impo rtant problem in foliation theory is the study of the to po logy o f the spa ce o f lea ves, which is not alwa y s endowed with the s tr ucture o f a ma nifold. F o r example consider the quotient T 2 of R 2 by the gr o up Γ generated t wo translations t e 1 and t e 2 whose directions e 1 and e 2 are indep endent vectors. Le t θ be an irrational int eger; the foliation of R 2 by aﬃne lines parallel to e 1 + θ e 2 deﬁnes on T 2 a foliation for which every leaf is dense. Thus the spa ce o f leav es o f this foliation is not separated. In [31] Mo lino has int ro duced the notion of foliage to study these situations which can b e interpreted with diﬀeren tiable categories : Deﬁnitions 3.3.1 . Two foliated manifolds ( N 1 , F 1 ) and ( N 2 , F 2 ) are transversally eq uiv a lent if and o nly if there exists a foliated manifold ( ˆ N , ˆ F ), tw o submers ions π i , i = 1 , 2 : ˆ N → N i such that the leaves of ˆ F are the preimag es of the leaves of F i by π i . Let N b e a top ological space a foliage on N is a diﬀerentiable categor y C N whose ob jects are quadruples ( U , V , F , π ), where U is an op en subset of N , ( V , F ) a foliated ma nifold. W e assume that the s pace of leaves of F is U and π : V → U is the natura l pro jection. Let ( U ′ , V ′ , F ′ , π ′ ) another ob ject of C N , w e deno te by p V : V × N V ′ → V the natural pro jection. W e assume the quadruples ( π ( p V ( V × N V ′ )) , p V ( V × N V ′ ) , F | p V ( V × N V ′ ) , π | p V ( V × N V ′ ) ) a nd ( π ( p V ′ ( V × N V ′ )) , p V ′ ( V × N V ′ ) , F ′ | p V ′ ( V × N V ′ ) , π ′ | p V ′ ( V × N V ′ ) ) are o b jects of C N and tr ansversally equiv a lent , where F | p V ( V × N V ′ ) is the res tric- tion of F to p V ( V × N V ′ ). Finally we supp ose that for every element y ∈ N , there ex ists a n ob ject ( U, V , F , π ) of C N , such that y ∈ U . A morphism b etw een the ob jects ( V , U, F , π ) and ( V ′ , U ′ , F ′ , π ′ ) of C N is a diﬀerentiable map φ : V → V ′ such that π = π ′ ◦ φ . 3.4 Pro ject iv e presen ted manifolds. In or der to study a ge ne r alized equiv a lence Carta n problem, Molino (see Molino [32]) has studied pro jective presented manifolds whic h are examples o f diﬀeren- tiable categories: Deﬁnition 3.4.1. A pro jective pre s ent ed manifold is a small diﬀerentiable categ ory whose class of o b jects is a pro jective sys tem of manifolds ( V i , π i j : V j → V i ) i ∈ I . The following conditions need to be satisﬁed: The maps π i j are submersions, 10 Let ˆ V b e the topolo gical pro jective limit of the family ( V i , π i j ) i ∈ I , a nd ( x i ) i ∈ I , x i ∈ V i an element of ˆ V . Let P i : ˆ V → V i which ass o ciates to ( x i ) i ∈ I the element x i in V i . F or each i , there exists an op en neighbor ho o d U i of x i in V i , a map c i : U i → ˆ V , such that P i ◦ c i = I d U i , and P j ◦ c i is diﬀeren tiable. Let ( x i ) i ∈ I be a n element of ˆ V , there exists i 0 , a nd a neig hbo rho o d U i 0 of x i 0 in V i 0 such that for every i > i 0 , for every y ∈ U i 0 , π i 0 i − 1 ( y ) is connected in U i . Deﬁnition 3.4.2. Let C a nd C ′ be tw o diﬀeren tiable catego r ies; a diﬀerentiable morphism betw een C and C ′ is deﬁned by a functor F : C → C ′ , such that for every ob ject X of C , ther e exists a diﬀerent iable map h F X : X → F ( X ), such that fo r every morphis m f : X → X ′ in C , the following square is comm uta tiv e: X f − → X ′ ↓ h F X ↓ h F X ′ F ( X ) F ( f ) − → F ( X ′ ) W e can supp ose that the categor ie s C and C ′ are im be dded in D if f the category of diﬀerentiable manifolds. In this setting, a diﬀerentiable functor is a morphism b etw een the identit y functor o f C , a nd a functor F : C → Di f f whose image is contained in C ′ , mor eov er for every ob ject X , the map h F X : X → F ( X ) which deﬁnes the mor phism of functors is a diﬀeren tiable map. Examples. Let f : M → N b e a diﬀerentiable map, f can b e v ie w ed as a diﬀerentiable functor F : C → C ′ where the unique ob ject of C is M and the uniq ue o b ject of C ′ is N . W e supp ose that the o nly morphisms in C and C ′ are the identities. The functor F assigns N to M , and h F M = f . Suppo se that a Lie g roup G a cts diﬀerent ially o n M , and N , we deﬁne C , to b e the diﬀeren tiable c ategory whic h has M has a unique ob ject, and suc h that H om C ( M , M ) is the image o f the map G → D if f ( M ) which deﬁnes the action. Similarly , w e deﬁne C ′ to be the ca tegory whose unique ob ject is N and such that H om C ′ ( N , N ) is the image of the map G → Dif f ( N ). Let φ b e a n endomorphism o f G , each φ -equiv aria nt map f : M → N ; tha t is a map such that for each g ∈ G , f ◦ g = φ ( g ) ◦ f deﬁnes a diﬀerentiable functor F b etw e e n C and C ′ , suc h that F assig ns N to M , h F M = f , and F ( g ) = φ ( g ). 4 Diﬀeren tiable ﬁb ered categori es. T o study the diﬀerentiable structure o f diﬀerentiable categor ies, we are g oing to use the theory of ﬁb ered categories. On this purpos e, we r ecall the following facts adapted to our setting: Deﬁnition 4.1. 11 Let F : P → C b e a diﬀerentiable functor, a nd f : x → y a map o f C . Let x ′ , z ′ be tw o ob jects of the ﬁb er of x , and y ′ an ob ject of the ﬁb er of y . Denote by H om f ( z ′ , y ′ ) the subset of the set of mor phisms H om P ( z ′ , y ′ ) such that for every element l ∈ H om f ( z ′ , y ′ ), F ( l ) = f . A mor phism f ′ : x ′ → y ′ is Cartesian, if a nd o nly if the map H om I d x ( z ′ , x ′ ) → H om f ( z ′ , y ′ ) which ass igns to h the ma p f ′ ◦ h is bijective for every z ′ in the ﬁber o f x . Deﬁnition 4.2. A diﬀerentiable bundle functor F : P → C is a Cartes ia n functor which satisﬁes the following conditio ns: - The ﬁber of an ob ject x of C has a unique element p x . - F or every ob ject x o f C , there exists a Lie g roup H x such that the canonica l pro jection p x → x deﬁnes on p x the structure of a tota l space of a H x -principal bundle, who se base space is x . Morphisms b etw e en ob jects of P are morphisms betw een principal diﬀerentiable bundles. If the group H x is indepe nden t of x , w e say that F : P → C is a H -principal diﬀerentiable bundle functor. Example. Let H be a compact gr oup which a cts on the manifold N b y diﬀeomor phisms. W e have attached a diﬀeren tiable categ o ry C N to this action (see p. 8). Let ( P, H P , φ P ), an ob ject of C N , we c a n construct the pr incipal H P -bundle P H P which is the quo tien t o f P × H P by the diagona l action of H P . Let f : ( P , H ) → ( P ′ , H ′ ) a morphism in C N which is induced b y a mo rphism l H,H ′ : H → H ′ such that fo r ev er y elements h in H , and p in P , f ( hp ) = l H,H ′ ( h ) f ( p ) since by deﬁnition f is an equiv a riant map. W e deduce a mo rphism ψ H,H ′ ( f ) : P × H → P ′ × H ′ which s ends ( p, h ) to ( f ( p ) , l H,H ′ ( h )). F o r every h 0 ∈ H , we ha ve: ψ H,H ′ ( f )( h 0 p, h 0 h ) = ( f ( h 0 p ) , l H,H ′ ( h 0 h )) = l H,H ′ ( h 0 ) ψ ( p, h ) . Thu s the morphism ψ H,H ′ ( f ) induces a morphism ψ ′ H,H ′ ( f ) : P H → P ′ H ′ . W e deduce the existence of a diﬀere ntiable ca tegory P C N whose class of ob jects are the bundles P H P , and a diﬀerentiable bundle functor F N : P C N → C N which sends the ob ject P H P to P . The Ca rtesian map a bove f is ψ ′ H,H ′ ( f ). Deﬁnition 4.3. Let F : P → C b e a diﬀerentiable bundle functor, and H : C ′ → C , a morphism b et ween diﬀeren tiable categories . The pull-back o f F by H is the diﬀerentiable bundle functor F ′ : P ′ → C ′ deﬁned as follows: Let X ′ be an ob ject of C ′ , h H X ′ : X ′ → F ( X ′ ) the map which deﬁnes H ; let p F ( X ′ ) the ob ject of the ﬁb er o f F ( X ′ ) fo r F , and h p ( F ( X ′ )) : p F ( X ′ ) → F ( X ′ ) the bundle ma p. The ﬁber of X ′ is the ﬁb er pro duct of the maps h H X ′ and h p ( F ( X ′ )) . 12 4.1 Connection on diﬀerentia ble bundle functors. In this part we are going to study connectio ns on diﬀerentiable bundles functors. First we recall the notion of connection on a principal bundle (See Lic hner owicz [19] p. 56 , McDuﬀ and Salamon [23] p. 20 7-209 ). Let H b e a Lie gro up w ho se Lie algebra is denoted by H , a nd p : P → N a H -pr inc ipa l bundle over the n -dimensional manifold N , for every element A ∈ H , we de no te A ∗ the vector ﬁeld deﬁned on P by the formula: A ∗ ( x ) = l im t → 0 d dt xexp ( tA ) , x ∈ P . A connectio n deﬁned on the H -principal bundle p : P → N , is a 1-form θ : P → H which v er iﬁes the following conditions: Let A ∗ be the fundamen tal vector ﬁeld deﬁned b y A ∈ H , i A ∗ θ = A . F or every elemen t h ∈ H , h ∗ θ = Ad ( h − 1 ) θ . A co nnection is also deﬁned by a distribution on P tra nsverse to the ﬁb ers and inv ar iant by H , whose r ank is n the dimension of N . T o a co nnec tio n form θ , the distribution asso ciated is: Θ x = { u ∈ T P x , θ ( x ) = 0 } . The curv ature of θ is the H -v alued 2 -form on P deﬁned by: Ω = dθ + 1 2 [ θ, θ ]. W e adapt now this deﬁnition to diﬀeren tiable categor ie s: Deﬁnition 4.1.1. A connection on the principal bundle functor p : P → C is deﬁned by a connection form θ X on the principal H X -bundle p X → X of the ﬁb er of X , such that for a ma p h : p X → p Y (necessarily Cartesian), the distr ibution deﬁned b y the k ernel o f h ∗ ( θ Y ) is the distribution which deﬁnes the connection form of θ X . Example. Consider the interv al I =] − 1 , 1[ of R, a nd N the orbifo ld which is the quotient of I by the symmetry h : x → − x , we asso ciate to this orbifold the diﬀerentiable categ ory C N whose class o f ob jects contains o nly I , and the s et of morphisms o f I , H om C N ( I , I ) = { I d I , h } , remark that this is not the canonical diﬀerentiable categor y asso cia ted to an orbifold deﬁned a t p. 8. The rea l 1- form α = xdx is in v ariant by h , th us deﬁnes a connection on the trivial bundle functor P → C N in cir cles over C N as follows: let C 1 be the cir cle, P is the category whic h unique ob ject is e I = I × C 1 . The unique non trivia l mo r phism of e I is the map h ′ deﬁned b y h ′ ( x, y ) = ( − x, y ). Let ( u, v ) b e a vector tangent to ( x, y ) ∈ I × C 1 we set θ I ( x,y ) ( u, v ) = α x ( u ) + v = xu + v . Deﬁnition: Holo no m y of a connection of a principal bundle functor 4.1.2. Consider C I , the canonica l diﬀerentiable categor y deﬁned on the int erv al by its str ucture of manifold, and F : P → C a H - pr incipal bundle functor, endow ed with a connection form θ and L : C I → C a diﬀerentiable functor. The pull-back of F and θ by L is a pr inc ipa l bundle over the int erv al endowed with a connection form whose holono my map is the holono my of F : P → C , around L . 13 4.2 Diﬀeren tiable t ensors of a diﬀeren tiable category . In this section, we are go ing to a sso ciate to a diﬀerentiable ca tegory C , principa l bundles functors whic h allow to deﬁne tensor ﬁelds. Such a theory is o bviously known for ma nifo lds . It has als o b een developed in the category of o rbifolds see (Chen and Ruan [11]), and for foliages (see Molino [31]). Deﬁnition 4.2.1. Let C be a diﬀerentiable category , the diﬀerentiable tangent bundle of C is the diﬀeren tiable category T ( C ) deﬁned as fo llows: the elemen ts of the class o f ob jects of T ( C ) are ta ngent space s T ( X ), where X is a n ob ject of C . A map betw een T ( X ) and T ( Y ), is a map T ( h ) : T ( X ) → T ( Y ) induced by a morphism h : X → Y in C . A diﬀerentiable functor F : C → C ′ , induces a tangent functor T ( F ) : T ( C ) → T ( C ′ ) deﬁned as follows: let X b e an o b ject of C , the map h F X : X → F ( X ) induces the tangen t ma p T ( h F X ) : T ( X ) → T ( F ( X )) which deﬁnes the tangent functor. The diﬀeren tiable category of p -forms of C , Λ p ( C ), is the category whose class of ob jects is the class whose elemen ts are Λ p T ( X ), where X is an o b ject in C . A morphis m b etw een the ob jects Λ p T ( X ) and Λ p T ( Y ) is a map of the form Λ p ( h ) where h : X → Y is a morphism in C . A diﬀer e ntiable p - fo rm is a functor α : Λ p ( C ) → C R , where C R is endowed with the structure of a diﬀerentiable ca tegory which as a unique ob ject: the real line R, and such that the endo morphisms of R are diﬀerentiable ma ps o f R. W e deduce from the deﬁnition of a diﬀerent iable functor that the following condition is satisﬁed: let α X be a p -form, for every ma p f : X → Y , there exists a diﬀeomorphism α ( f ) o f R such that the following square is comm uta tiv e: Λ p X d p f − → Λ p Y ↓ α X ↓ α Y R α ( f ) − → R Let f : X → Y b e a morphism in C , w e do n’t ass ume that α ( f ) is the ident ity of R. Thus α X is not necessarily the pull-back of α Y by f . W e deno te by Λ p I d ( C ) the set of p -forms such that for every ma p f in C , α ( f ) is the identit y . Deﬁnition-Prop o sition 4.2.2. L et C b e a diﬀer ent iable c ate gory, and α a p - form deﬁne d on C , ther e exists a fun ctor d : Λ p ( C ) → Λ p +1 ( C ) , the diﬀer ential such that d ◦ d = 0 . Pro of. Let α be a p -form deﬁned on C , fo r each ob ject X of C , α X is a p -form, we can deﬁne dα X the diﬀerential o f α X . Let f : X → Y b e a morphism in Λ p ( C ), we deﬁne d ( α )( f ) = d ( α ( f )) • Examples. Let N be a manifold, a nd H a Lie group which a cts diﬀerentially on N . Consider the diﬀeren tial categ o ry C H N whose unique ob ject is N , and such that the set of endomo r phisms of N in C H N is the image of the map H → Dif f ( N ) 14 which deﬁnes the a ction. Let χ b e a character of H , w e ca n deﬁne Λ p H,χ ( N ) to be the set of p -forms on C H N such that for every element α ∈ Λ p H,χ ( N ), the following squar e is comm uta tiv e: Λ p N d p f − → Λ p N ↓ α X ↓ α Y R χ ( f ) − → R In pa rticular if χ is the trivial c har acter, we obtain the space of H - in v ariant p -forms, and the equiv ariant co homology . Let C N be the diﬀerent iable categ ory asso cia ted to a fo lia ge. The set of Λ p I d C N forms on C N is the set of basic forms (see Molino [31]). 4.3 F rames bundle and diﬀeren tiable categories. Let C b e a diﬀerentiable ca tegory , we ca nnot always deﬁne the bundle of linear frames, since tw o ob jects of C do not hav e necessar ily the s ame dimension. Suppo se tha t ev e ry ob jects C has dimension n . W e can deﬁne the set of vector frames V ( C ) as follows: Let X be an ob ject of C , a nd x b e an element of X , we denote by V ( C ) x the set of linear ma ps u : R n → T x X , wher e T x X is the tangent space of x . W e c a n thus deﬁne the vector bundle V ( C )( X ) over X whose ﬁb er at x is V ( C )( X ) x . Let f : X → Y be a diﬀerentiable map, and u ∈ V ( C )( X ) x , the linea r map d f x ◦ u is a vector frame of V ( C )( Y ) f ( x ) . W e hav e th us deﬁne the catego ry of vector frames of C . Since the morphisms in C are not neces sarily lo cal diﬀeomorphisms, we cannot a ssume that the elements of V ( C ) x are isomorphis ms . Let C H N be the diﬀerentiable ca tegory as so ciated to the action of a co mpact Lie gro up H on N (see p. 9 ). Let ( P, H P , φ P ) b e an ob ject of C N . W e can deﬁne the vector s pa ce T C N x , the quotient of the tang ent space at x , T P x of P , by the image of the inﬁnitesimal action at x of H P . This space is called the tangent space a t x of the action. Remark that the dimension di m ( T C N x ) of T C N x depe nds only of φ P ( x ). But this dimensio n can v ar y if x v ar ie s in P . W e c an deﬁne the diﬀerentiable categor y of linear frames L ( C N ) o f C . F or each ob ject X o f C N , ther e exists ﬁbration L ( C N )( X ) → X such that for every element x ∈ X , L ( C N )( X ) x is the set of linear isomorphisms R dim ( T C N x ) → T C N x . Prop ositio n 4 .3.1. Supp ose that the dimension of T C N ( P ) x do es not dep end of P , then ther e exists a c onn e ction on the fr ames bund le L ( C N ) , of t he diﬀer en t iable c ate gory C N deﬁne d by t he diﬀer entiable c ate gory deﬁne d ab ove. Pro of. Let θ b e a c onnection on the frames bundle L ( C N )( N ) of T C N ( N ) inv ariant by the action o f H . Let ( P , H P , φ P ) b e an o b ject of C N . Since the dimension o f T C N ( P ) x do es not dep end neither of P , nor of x in P , the pull- back of θ by φ P deﬁnes a connection form on L ( C N )( P ) • 15 Deﬁnition 4.3.1. Let C N be the diﬀerent iable categor y C N deﬁned b y the actio n of the c om- pact Lie g r oup H on N . W e supp ose that the dimension of T C N x do es no t depe nd of x . Let ( X , H X , φ X ) b e an element of C N , and α X the fundamental 1-form of the bundle L ( C N )( X ). It is the form R dim ( T C N ) -v alued form deﬁned by: α X u ( v ) = u − 1 ( dp X ( v )) where u is an element of L ( C N )( X ) x , v element of the tangent space of L ( C N )( X ) a t u , and p X : L ( C N )( X ) → X the canonical pro jectio n. The family of 1-forms ( α X ) X ∈ C N deﬁnes an inv ariant form on C N . 4.4 Diﬀeren tiable descen t and connection in ﬁb ered cate- gories. In this pa rt we are going to study the notions of connection and holono m y on diﬀerentiable ﬁb ered ca tegories. Let us recall so me facts on the analysis situs in diﬀeren tiable categories (see Giraud [13]): Let F : P → C b e a Car tesian functor, a cliv age is a family L of morphisms of P such that: every elemen t in L is cartes ia n, for every mor phism f : x → y in C , and y ′ ∈ P y , there exists a unique morphism f ′ ∈ L whos e target is y ′ and such that F ( f ′ ) = f . A cliv a ge is a scindage if a nd only if it is stable by comp osition o f maps. A cliv ag e is the analog of a reduction in diﬀeren tial geometry . Let l : x → y b e a map in C , and L a cliv age. The cliv age L and l induce a functor l ∗ : P y → P x deﬁned as follows: The image of the ob ject z ∈ P y is the source of the unique Cartesian map c l ( z ) : l ∗ ( z ) → z in L ov er l . Consider tw o maps l and m , such that the target of l is the source of m , there exists a natural transformation: c l,m : ( m ◦ l ) ∗ → l ∗ ◦ m ∗ which s atisﬁes the relation: c ml ◦ c l,m = c l ◦ c m (See also Giraud [13] p.3). Let p : P → C b e a Car tesian functor betw een diﬀerentiable categories. W e assume that there ex ists a L ie gr o up H such tha t for every o b ject X o f C , every ob ject e X in the ﬁber of X is endo wed with the structure of an H -space; i.e the group H acts on the righ t and freely on e X . 16 There exists a pro jectio n p : e X → X , suc h that for e very h ∈ H , and x ∈ e X , p ( xh ) = p ( x ). A mor phism f : e X → e X ′ in C is a diﬀerentiable map f such that for e very element h ∈ H , we have f ◦ h = h ◦ f . Let f be an e ndo morphism of e X , and x ∈ e X . W e denote by u ( x ) the element of H suc h that f ( x ) = xu ( x ). F or every element h ∈ H , we hav e f ( xh ) = ( xh ) u ( xh ) = f ( x ) h = xu ( x ) h . This implies that: u ( xh ) = h − 1 u ( x ) h. W e supp ose that there exists a principal H -bundle functor Aut ( P ) → C , such that for every o b ject e X in the ﬁber of X ∈ C , there exists a c a nonical isomorphism Aut ( P )( X ) → E nd ( e X ), whic h is natura l in r e spe c t of morphisms betw een ob jects. Let A b e an elemen t o f H the Lie algebra o f H , for every o b ject e X , we c a n deﬁne the vector ﬁeld: d dt t =0 xexp ( tA ) . which is a fundamen tal vector ﬁeld. This allows to identify H with a subbundle of the tangent space T e X x of e X . Let us start by a motiv ating exa mple. Let: 1 − → H → L ′ → L → 1 be an exact sequence o f Lie gr oups. Cons ider a principa l L -bundle p : P → N ov er the manifold N . The obstruction to extend the structural gr oup L , to L ′ , is deﬁned by a sheaf of categories C H deﬁned as follows: for every open subse t U of N , C H ( U ) is the categor y whos e o b jects are L ′ -principal bundles over U whose quotient by H is the restriction o f p to U . Morphisms b etw e e n ob jects of C H ( U ) are mor phis ms o f L ′ -bundles which induce the identit y on the restr iction of p to U . Let L and L ′ be the resp ective Lie algebra s of L and L ′ . W e know the deﬁnition of a c onnection form θ on p , and we wan t to g eneralize this deﬁnition. A natura l wa y is to take for each ob ject e U ∈ C H ( U ) a co nnection α U , suc h that the comp osition of α U with the natural pro jection L ′ → L desc e nds to the r e s triction θ U of θ to U . The choice o f α U is no t ca no nical since it is not necessarily preserved b y every a utomorphism h of e U . Remark that the form: α h = h ∗ ( α U ) − α U = Ad ( h − 1 )( α U ) − α U + h − 1 dh is a H -v alued form. This motiv ates the following deﬁnition (compar e with Br ylinski [8] p. 2 06, and with Breen and Messing [7]): Deﬁnition 4.4.1. 17 Let p : P → C b e a H -principal ﬁbe red ca tegory , a nd H the Lie algebra o f H . A co nnective structure on C is a map which assigns to every ob ject e U of P U , an aﬃne space C o ( e U ) such that: The vector spa ce of C o ( e U ) is the set of H -forms Ω 1 ( U, H ). F or e very mo rphisms h ′ : U ′ → U ”, h : U → U ′ , and for every ob ject e U in the ﬁb er o f U , e U ′ in the ﬁb er of U ′ and e U ” in the ﬁbe r of U ”, ther e exists a morphism: h ∗ : C o ( e U ) → C o ( e U ′ ) which is compa tible with comp osition: ( h ′ h ) ∗ = h ′ ∗ h ∗ . There exists a morphism: u h : h ∗ ( C o ( e U ′ )) − → C o ( h ∗ ( e U ′ )) such that the following square is commutativ e: h ∗ ( h ′ ∗ C o ( e U ” )) u h ′ → h ∗ C o ( h ′ ∗ e U ” ) u h → C o ( h ∗ h ′ ∗ e U ” ) ↓ α ∗ h ′ ,h ↓ c h,h ′ − 1 ∗ ( h ′ h ) ∗ C o ( e U ” ) u h ′ h − → C o (( h ′ h ) ∗ e U ” ) where the mo r phisms c h,h ′ is the mor phism deﬁned by a morphism in the analysis situs (see p. 16 ), and α h ′ ,h the canonical isomorphism of torsor s . Let u : e U → e ′ U ′ be a Ca rtesian mor phism ab ov e h : U → U ′ , w e have the compatibility diag ram: h ∗ C o ( e U ) u ∗ − → h ∗ C o ( e ′ U ′ ) ↓ u h ↓ u h C o ( h ∗ e U ) u ∗ − → C o ( h ∗ ( e ′ U ′ )) . There exists an action of Aut U ( e U ) o n C o ( e U ) such that for every element h of Aut ( e U ), and every element θ in C o ( e U ). W e have the relation: h ∗ ( θ ) = h. ( θ ) + h − 1 dh. And for every ele ment α ∈ Ω 1 ( U, H ), w e have: h ∗ ( θ + α ) = h ∗ ( θ ) + Ad ( h − 1 )( α ) . An alter native deﬁnition of connective structure can be done by considering torsor s C o ( e U ) whose vector space is the space of closed H -v alued 1-forms if H is comm utative. A fundamental relatio n. Suppo se now that F : P → C is a diﬀerentiable ﬁb ered catego ry , cons ide r a cliv ag e L . F o r each ob jects X of C , and X ′ in the ﬁber of X , consider a 18 morphism u X Y : Y → X , and its lift to a Car tesian morphism u X ′ Y ′ : Y ′ → X ′ of L . Let α b e a connective structure deﬁned on this diﬀerentiable ﬁb ered bundle, we denote by α Y ′ an element of C o ( Y ′ ) , a nd by α X ′ Y ′ the 1-form such that α X ′ = α X ′ Y ′ + u X ′ Y ′ ∗ ( α Y ′ ) we have: u X ′ Y ′ ∗ ( α Y ′ Z ′ ) − α X ′ Z ′ + α X ′ Y ′ = = u X ′ Y ′ ∗ ( α Y ′ − u Y ′ Z ′ ∗ ( α Z ′ )) − ( α X ′ − u X ′ Z ′ ∗ ( α Z ′ )) + ( α X ′ − u X ′ Y ′ ∗ ( α Y ′ )) = u X ′ Z ′ ∗ ( α Z ′ ) − u X ′ Y ′ ∗ u Y ′ Z ′ ∗ ( α Z ′ ) Since F : P → C is a ﬁbered categ ory , there ex is ts a mor phism c X ′ ,Y ′ ,Z ′ such that u X ′ Y ′ u Y ′ Z ′ = u X ′ Z ′ c X ′ ,Y ′ ,Z ′ , we deduce that: u X ′ Y ′ ∗ ( α Y ′ Z ′ ) − α X ′ Z ′ + ( α X ′ Y ′ ) = u X ′ Z ′ ∗ ( α Z ′ − c X ′ ,Y ′ ,Z ′ ∗ ( α Z ′ )) = u X ′ ,Z ′ ∗ (( α Z ′ ) − c X ′ ,Y ′ ,Z ′ . ( α Z ′ ) − c X ′ ,Y ′ ,Z ′ − 1 dc X ′ ,Y ′ ,Z ′ ) 5 Grothendiec k top ologies in d iﬀeren tiable cat- egories. W e hav e studied diﬀerentiable categor ies witho ut emphasizing on the global top ology . This can b e achiev ed by using the notion of diﬀer e ntiable Gro thendiec k top ology (see [3] S.G.A 4-1; p. 21 9; or Giraud [14]). Deﬁnitions 5.1. Let C b e a diﬀeren tiable ca tegory , a sieve R in C is a subclass R of the class of ob jects of C such that if U is an ob ject of R , and V → U is a morphism in C , then V is in R . A Gro thendieck top ology o n C is deﬁned by assig ning to each o b ject U of C a non empty fa mily o f s ieves J ( U ) o f the ca teg ory over U , C ↑ U such that the following conditions ar e satisﬁed: F or every mor phism h : V → U , a nd every sieve R ∈ J ( U ), the pull-back sieve R h is in J ( V ). A sieve R o f C ↑ U is in J ( U ) if and only if for every ma p h : V → U , R h ∈ J ( V ). Examples. An exa mple of a Grothendieck top olog y ca n b e deﬁned as follows: Let N be a top olo gical space, and C N the categ ory w ho se o b jects are o p en subsets, 19 and whos e maps a re canonica l imbeddings b et ween op en subsets. F or an op en subset U , an element of J ( U ) is a family of op en subsets ( U i ) i ∈ I of U such that S i ∈ I U i = U . This topo logy is often called the small site. Let N b e a gene r alized o rbifold (see deﬁnition 3.2 .1). W e c an deﬁne on N the following Grothendieck topolo gy: A covering o f an op en s ubset U of N , is a family of ob jects ( P i , H i , φ i ) i ∈ I which is U -joint ly sur jective. This e q uiv a lent to saying that S i ∈ I φ i ( P i ) = U . In particular if for every ob ject ( P , H P , φ P ) in C N , the gr oups H P is discrete, we obtain a Grothendieck topolog y on orbifolds. Consider the spa ce of hyperb olic surfaces of a given genus h . Eac h of this surface can b e cut in pa n ts. The hyperb olic length of the bo undaries cycle s of these pa n ts a re the F enschel-Nielsen co ordinates which iden tify the set of isomorphic clas ses o f hyperb olic s urfaces of ge n us h to a cell. (See [6] X. Buﬀ and al p.13-15). Deﬁnition 5.2. Let ( C, J ) b e a category endow e d with a Grothendieck top olo gy , w e s upp os e that C has a ﬁnal ob ject e . A g lobal cov ering of C is a cover of e , that is an element of J ( e ). Deﬁnition 5.3. A presheaf deﬁned on the diﬀerentiable categ o ry C , is a contra v ariant funct or F , from C to the category of sets. A shea f is a pres heaf which satisﬁes 1-descent in res pect to any s ieve R in J ( U ). This is equiv ale nt to s aying that fo r ev er y ob ject U of C , and every sieve R in J ( U ), the natural map: F ( U ) → l i m V → U ∈ R F ( V ) is bijectiv e. 5.1 Grothend iec k top ologies and cohomology of diﬀeren- tiable c at egor ies. The cohomolo gy of or bifo lds is studied in alg ebraic geometr y a nd symplectic ge- ometry , sinc e orbifolds a rise a s pha se spaces in theor etical physics. Gr othendieck and his colla b or ators (se e S.G.A. 4 I I, p.16) have deﬁned Cech c ohomology in Grothendieck sites. W e shall apply this p oint of view to ge neralized orbifolds. W e shall also genera lize Chen and Ruan cohomology of or bifolds (see [1 1]) to generalized orbifolds. Let J N be the Grothendiec k topolog y assoc ia ted to the gener alized orbifold N . W e ca n deﬁne the preshea f Ω p N , such that for each ob ject e = ( P, H P , φ P ) of C N , Ω p N is the vector space of p -diﬀerentiable forms inv aria nt by H P deﬁned on P (see also p. 14 ). If h : e → e ′ is a morphism in C N , the restriction is deﬁned by the pull-back of diﬀeren tiable forms. 20 Consider a covering ( U i , H i , φ i ) i ∈ I of N . W e cannot deﬁned the c lassical Cech reso lutio n, since the diﬀere ntiable categor y C N asso ciated to N is not necessarily sta ble ﬁb er pro ducts. Let ( U i 1 , H i 1 , φ i 1 ) , ..., ( U i n , H i n , φ i n ), b e o b- jects o f C N , φ i 1 ( U i 1 ...i n ) is a manifold, W e can deﬁned the bi-graded complex Ω l N ( φ i 1 ( U i 1 ...i p )) e ndowed with tw o deriv a tions: the Cech-deriv ation a nd the canonical deriv ation of diﬀerent iable forms. W e denote by H ∗ , ∗ ( U i ,H i ,φ i ) i ∈ I ( N ) the induced bigraded cohomolog y groups . W e say that the cov er ing ( U ′ i ′ , H ′ i ′ , φ i ′ ) i ′ ∈ I ′ is ﬁner than the c overing ( U i , H i , φ i ) i ∈ I , if and only if for e very i ′ ∈ I ′ , there exists i ∈ I such that φ i ′ ( U ′ i ′ ) ⊂ φ i ( U i ). This relation deﬁnes an inductive system on the set o f cov ering s, the inductiv e limit of H ∗ , ∗ ( U i ,H i ,φ i ) i ∈ I ( N ) is the Cech-DeRham co homology of the genera lized orbifold. 5.2 Chen-Ruan c ohomology for generalized orbifolds. Suppo se that the gener alized orbifold N is co mpact. W e are going to adapt the cohomolog y theory deﬁned by Chen and Ruan [11 ] for or bifolds. Firstly we rec a ll the following construction in C he n and Ruan (page 6-7 ): let N be an orbifold, ( U x , H x , φ x ) a lo ca l chart at x , deﬁne ˆ N to be the set whose elements a re ( x, ( h x )), where ( h x ) is the conjugacy class of the element h x of H x . Remark that ˆ N is well-deﬁned despite the use of lo cal charts. The or bifold ˆ N is not necessar ily connected. Its connected comp onents are called twisted sector s (Chen and Ruan p.8). Ther e exists a na tural surjection p : ˆ N → N , the connected co mpo nent s of elements of p − 1 ( U x ) can b e par ameterized by the set of conjugacy cla sses ( h x ), h x in H x . Suppose that the or bifold is endow ed with a pseudo- complex structure, which deﬁnes a repre s ent ation ρ H x : H x → Gl ( n, C) ( n = dim C N ). F or every elemen t h x in H x , ρ H x ( h x ) dep ends only of the conjuga cy class ( h x ) of h x in H x , they deﬁne i x, ( h x ) = − i 2 π Log ( det ( ρ H x ( h x ))). This enables Chen and Ruan to deﬁne the orbifold d -cohomolog y gr o up: H d ( X ) = ⊕ H d − 2 i ( h ) ( X ( h ) ) . Let N b e a generalized co mpact or bifo ld, we can ﬁnd a ﬁnite cov er ( U i , H i , φ i ) for the Grothendieck to po logy , such that ea ch open s ubset U i is deﬁned by the slice theorem (see theo rem 3 .2.1), this is equiv alent to s aying that U i is the quo- tien t H i × H ′ i V i by H i where H ′ i is the s tabilizer of an e le men t x i of U i , V i is the quotient o f the ta ngent space T U i at x i , by the image of the inﬁnitesimal action of H i at x i (see Audin [4] p. 15). Let C i be an op en subset of V i inv ariant by H i . Then ( C i , H i , φ ′ i ) is a chart of the generalize d orbifold, wher e φ ′ i : C i → C i /H i is the cano nical pro jection. Th us fo r ev er y elemen t x in N , there exists a chart ( U x , H x , φ x ), x ′ in U x such tha t φ x ( x ′ ) = x , and H x ( x ′ ) = x ′ . W e are g oing to co ns ider only this type of charts in the sequel. The exis tence of such c har ts is rela ted to the deﬁnition of holo no m y of singular foliations. See Molino and Pierro t [33] p. 208, for the deﬁnition of the holonomy a foliation deﬁned b y the action of a compact Lie group, or Debord [1 2]. 21 Let H be a clo s ed subg roup of H x , w e deno te by ( H ) the conjuga c y cla ss of H in H x . Let y b e an element of U x , and H y x the subgroup of H x which ﬁxes y . W e say that y and y ′ hav e the sa me type if and only if ( H y x ) = ( H y ′ x ). Let ( U y , H y , φ y ) b e a chart such that H y ( y ) = y , denote by λ y : H y → H x the mor phis m induced by the tr a nsition function φ xy . W e supp ose that the stabilizer of φ xy ( y ) in U x is λ y ( H y ). W e can deﬁne: ˆ N = { ( x, ( H )) , H ⊂ H x , ( H ) = ( H y x ) } where x ∈ N , ( U x , H x , φ x ) is a lo ca l chart at x . W e denote by H ” x the set of subgroups of H x which a re t yp e of an orbit, and by H ′ x the set of conjugacy classes of these subgroups. Remark tha t the argument in Audin [4] p. 17 prop osition 2.2 .3 implies that we ca n assume that the n umber of types of orbits contained in every chart is ﬁnite. The reunio n D H of the o rbits whose t yp e is ( H ) is a submanifold. The follo wing prop ositio n is sho wn for orbifolds by Chen and Ruan [11] p.7. Prop ositio n 5 .2.1. Ther e exists a gener alize d orbifold structu re on ˆ N . L et ( U x , H x , p H x ) b e a chart of N , and ( H ) ∈ H ′ x . W e denote by U H the ﬁxe d p oint subset of U by the action of H , and by C ( H ) the normalizer of H in H x , then (( U H , C ( H )) , C ( H ) , φ H ) is a chart of the gener alize d orbifo ld ˆ N , wher e φ H : U H → U H /C ( H ) is the nat- ur al pr oje ction. Pro of. Co nsider ( U x , H x , φ x ) a chart at x . Let y b e an element of φ x ( U x ). Consider a chart ( U y , H y , φ y ), such that U y contains a n element y ′ such tha t φ y ( y ′ ) = y and H y ( y ′ ) = y ′ . The equiv ar iant transitio n function φ xy : p y ( U x × N U y ) → p x ( U x × N U y ) where p x : U x × N U y → U x is the canonical pro jection induces a morphism λ y ′ : H y → H x . Le t H = H z y ′ and h ∈ H , the element λ y ′ ( h ) ﬁxes φ xy ( z ). W e deduce a map Φ which a sso ciates to ( y , ( H )) the pr o- jection of φ xy ( y ′ ) in S H = H y x ∈ H ” x U H x /H x , wher e an element h of H x acts on S H = H y x ∈ H ” x U H x by sending the element c ∈ U H x to h ( c ) ∈ U hH h − 1 x . If instead o f tak ing H , we take the element H ′ in ( H ), H ′ = aH a − 1 , φ xy ( az ) ∈ U λ y ′ ( H ′ ) , and Φ( y , ( aH a − 1 )) is the pro jection to S H = H y x ∈ H ” x U H x /H x of φ xy ( y ′ ) in U λ y ′ ( hH h − 1 ) x . If we take y ” such that φ x ( y ′ ) = φ x ( y ”), y ” = b y ′ , b ∈ H x , and φ xy ( bz ) ∈ U bλ y ′ ( H ) b − 1 , and Φ( y , ( H )) is the pro jection of y ” ∈ U bλ y ′ ( H ) b − 1 to S H = H y x ∈ H ” x U H x /H x . Thu s the map Φ is well deﬁned. This map is s urjective; this can b e shown by the fact that we can linea r ize the action of compact Lie gr oup. It is injective: If φ ( y , ( H )) = φ ( y 1 , ( H 1 )), and Φ( y , ( H )) and Φ( y 1 , ( H 1 )) ar e the pro jections of y ′ and y ′ 1 in S H = H y x ∈ H ” x U H x /H x , there exists a ∈ H x such that y ′ 1 = ay ′ . This implies that y = y 1 . The deﬁnition of Φ implies then that ( H ) = ( H ′ ). Remar k that the image of the previous map is in bijection with S ( H ) ∈ H ′ x U H / ( C ( H )). W e endow ˆ N with the topo logy the topolog y genera ted by the imag e of the maps U H → ˆ N . The tr iples ( U H , C ( H ) , φ H ) deﬁnes a covering atlas of the generalized orbifold where φ H : U H → U H /C ( H ) is the pro jection map • 22 Let H = H y x , U H /H is an ope n subset of a sub orbifold of ˆ N completely determined by ( H ) if N is connected that we denote N H . Consider a pseudo-complex structure deﬁned on C N , this is equiv alent to suppo se that each chart is endo wed with a pseudo- complex structure, and mor- phisms in C N preserve pseudo-complex structures. Consider a chart ( U x , H x , φ x ). F or every and ( H ) in H ′ x , we deﬁne 2 i ( H ) = di m C ( U x ) − di m C ( N H ). W e can deﬁne: H d ( N ) = ⊕ H d − 2 i ( H ) ( N H ) . 6 Sheaf of c ategories and gerb es in diﬀeren- tiable categ ories. Recall that if C is a diﬀerentiable category endow ed with a top ology , U an ob ject of C and R a sieve in J ( U ). The forg etful functor from R to C which sends a map V → U to V is Cartesian. Deﬁnition 6.1. Let F : P → C be a diﬀerentiable ﬁb ered functor, where the catego ry C is equipp e d with a Gro thendieck top olog y , w e say that F is a shea f of categor ies, if for ev ery ob ject U of C , and for ev ery s ie ve R ∈ J ( U ), the natura l restriction map: C ar t C ( C ↑ U, F ) → C ar t C ( R, F ) is a 2 -descent map, otherwise s aid, an equiv alence of categor ies. (See Giraud [14]) The s heaf of c ategories is called a gerb e bounded by the she a f H if the following conditions ar e satisﬁed: F is lo ca lly connected: this is equiv alent to saying that for every ob ject U of C , there exis ts a sieve R ∈ J ( U ) such that for every ma p V → U ∈ R , the ob jects of the ﬁber P V of V are is o morphic each o ther. There exists a shea f in gro ups H deﬁned o n ( C, J ) suc h that for every o b ject U ∈ C , and e U ∈ P U the group Aut U ( e U ) of automorphisms of e U ov er the ident ity of U is isomorphic to H ( U ), and these family of iso mo rphisms commute with mor phisms b etw een ob jects and restr ic tions. The s hea f H is called the band of the gerb e. Let ( C, J ) b e a site, tw o ﬁb ered categories F i , i = 1 , 2 : C i → C are equiv a - lent , if there exists a Cartesian isomor phism b etw een C 1 and C 2 . An equiv alence b etw ee n the ger be s F i , i = 1 , 2 : P i → C is a Ca rtesian isomorphism which commutes with their bands. Let H b e a sheaf deﬁned on the diﬀerentiable site ( C, J ), w e denote by H 2 ( C, H ) the set of e q uiv a lences cla sses of H -gerb es. This s et is often ca lled the non- ab elian 2-coho mology gro up of the sheaf H . 23 6.1 The classifying co cycle of a gerb e. Suppo se that the diﬀerentiable categor y C ha s inductive limits, ﬁnite pr o jective limits, a ﬁnal and initial ob ject. Let R b e a cov ering of the ﬁnal ob ject e . W e s uppos e that R is a g o o d cov er ing, that is e very gerb e deﬁned on an ob ject X i of C s uc h that there exists a map X i → e in R is trivial and connected. Let F : P → C b e a g erb e, and e i an ob ject o f the ﬁb er P X i . There exists an isomorphism: u ij : e i j → e j i W e denote by c ij l the isomorphism u j li ◦ u l ij ◦ u i j l . W e hav e the relatio n: c i 2 i 1 i 3 i 4 u i 1 i 2 i 4 i 3 c i 4 i 1 i 2 i 3 u i 1 i 2 i 3 i 4 = c i 3 i 1 i 2 i 4 c i 1 i 2 i 3 i 4 The family of 2-chains c i 1 i 2 i 3 which sa tisﬁes the relation above is called a non-ab elian 2-co c y cle. Gira ud [14] has shown that there exists a 1 to 1 c o rre- sp ondence b e tween the se t of ger be s b ounded by H and non ab elian H 2-co c y cles (see also the pro of in Brylinski [8] p. 2 00-203 for comm utative gerb es). 7 Examples of sheaf of categori es and gerb es. The diﬀerentiable catego ry C H which repres ent s the g eometric obstruction to extend the structural group of a principal bundle is a gerb e (see page 17). Recently , Lupercio a nd Urib e [21] hav e intro duced Ab elian gerb es o n o rb- ifolds. F or an orbifold N , we can deﬁne a gerb e on N to be a gerb e deﬁned on the Grothendieck site J N ( see p. 20). 7.1 Gerb es and G -structures. W e are going to deﬁne a fundamen tal example of a gerb e, that we are go ing to a pply to the study of G -s tructures. Let G b e a Lie group, and H a clos ed subgroup of G . Consider a principal G -bundle p : P → N ov er the manifold N . A natural question is to ask wether the bundle has an H -reduction, that is wether ther e ex is ts co o rdinates change whic h take their v a lues in H . This problem is eq uiv a lent to the following question: Consider the bundle p ′ : P ′ → N whose typical ﬁb er is the ho mogeneous spa ce G/H o bta ined by making the quotient of ea ch ﬁb er of p b y H . Is there exists a global section of p ′ ?(see Alber t and Molino [2] p. 64 ). W e hav e the following: Prop ositio n 7 .1.1. The c orr esp ondenc e deﬁne d on the c ate gory of op en subsets of N , which as- signs to every op en subset U the c ate gory C H ( U ) , whose obje cts ar e H -r e ductions of t he r estriction of p to U , and whose morphi sms, ar e morphisms of H -bund les is a she af of c ate gories. 24 Pro of. Gluing condition for ob jects. Let ( U i ) i ∈ I be an op en cov ering o f U , e i , an ob ject o f C H ( U i ) such that there exists a morphism u ij : e i j → e j i such that u l ij u i j l = u j il . Then ther e ex ists a n ob ject e in C H ( U ) whose restriction to U i is e i , since w e can glue H -bundles. Gluing conditions for arrows: Let e a nd e ′ be tw o ob jects of C H ( U ), the cor resp ondence whic h assigns to every op en subset V of U , H om C H ( U ) ( e | V , e ′ | V ) is a she a f, since it is the sheaf of morphisms betw een tw o bundles • This s heaf of catego r ies ca n b e a pplied to the following situation: supp ose that N is a n -dimensiona l manifold. Let R p ( N ) b e the bundle o f p -linear frames of N , a nd G a subgroup of Gl p ( n, R) the gr oup of invertible p -jets of R n . The geometric o bstruction o f the existence of a G -str uc tur e on N is deﬁned by the sheaf of categor ie s that we ha ve just deﬁned. Let U b e a co ntractible op en subset o f N , C G ( U ) is not empty , s ince the restriction o f P to U is a trivia l bundle. But the ob jects of C G ( U ) ar e not alwa y s isomor phic: suppo se that N is a n - dimensional manifold, and take G = O ( n, R); the G -r e ductions of the bundle of linear fr ames R ( N ) of N deﬁne the diﬀerentiable metrics. It is well-known that tw o diﬀerentiable metrics ar e not lo cally isomorphic if their curv atures are distinct. A particular situation is the e xample of ﬂat G -structures like s y mplectic structures (see Alb ert and Molino [2 ] p. 177). F or every elements x and y in N , there exists neighbor ho o ds U x and U y of x and y in N , and a diﬀeomorphism h : U x → U y which preserves the G -structures induced by N on U x and U y . If U is con tractible open subset of N , t wo elements o f C G ( U ) are connected. The theory of gerb es and G -structures, will b e int ensively studied in [42]. 7.2 Gerb es and in v arian t scalar pro duct on Lie gr oups. Another example of gerbes can b e describ ed as follows: co nsider a Lie gr oup H which is not commutativ e, and L a lattice in H . Consider the manifold H /L , and let suppos e that H is endo wed with an orthogonal bi-inv ar iant metric: this is eq uiv a lent to the existence of a scala r product <, > (i.e a non-degener a ted real v alue d bilinear form not necessarily p ositive deﬁnite) o n the Lie algebra H of H s uc h that for every elements x, y , z ∈ H : < [ x, y ] , z > + < y , [ x, z ] > = 0 . The 3-inv ar iant form ν deﬁned on the Lie algebr a H of H by: ν ( x, y , z ) = < [ x, y ] , z > deﬁnes on H /L a closed 3-form ν L . The space of bilinear symmetric forms on H cor resp onds to rea l 3-co cycles as s hows Koszul [17] p. 9 5. Medina [26] has shown that the dimension of this space is either 1 or 2. 25 Let H b e a n -dimensio nal nilpotent Lie g roup, and L a la ttice of H . Reca ll that there exists a basis e 1 , ..., e n of the Lie algebra H of H , such [ e i , e j ] = P ij l c ij l e l , c ij l ∈ Q (see Raghunathan [35] p. 34 ). W e say in this situation that the cons tant s of s tructure ar e rationa l. A lattice L is the image of Z e 1 ⊕ ... ⊕ Z e n by the exp onential map. Prop ositio n 7 .2.1. Under the notations ab ove, if ther e exists an invariant sc alar pr o duct <, > , such that < e i , e j > ∈ Q , t hen the 3 -form ν on H /L induc es c anonic al ly a r ational 3 -form ν L on H /L . Pro of. The proo f uses a theo r em of Nomizu q uoted in [35] Raghunathan p. 123 in the rea l case. Let H 0 be the c e nter of H , the intersection L 0 = L ∩ H 0 is a lattice in H 0 . (If H is comm utative, we tak e H 0 to b e a non trivial subgroup diﬀerent o f H . See Raghunathan [35] p. 4 0). Thus the foliation o f H /L whos e leav es a re orbits of H 0 , ha s compact lea ves. The space of leaves of this foliation is the quotien t M of H /H 0 by L/L 0 . The natura l pro jection H /L → M is a ﬁbration whose ﬁber s are n - dimensional torus T n , where n is the dimension o f H 0 . W e ca n apply the Leray-Serr e sp ectral sequence to this ﬁbra tion for the rational cohomolog y we obtain: E p,q 2 = H p ( M , H q ( T n , Q)) ≃ H p ( M , ΛQ q ) , E p,q ∞ = ⇒ H p + q ( N , Q) Consider ˆ E p,q ∗ the Leray-Serre s pectr al sequence asso cia ted to the space of H -inv ariant for ms on N and M , w e hav e: ˆ E p,q 2 = H p ( H / H 0 , ΛQ q ) , ˆ E p,q ∞ = ⇒ H p + q ( H , Q ) . The recur sive h yp othesis implies that H ∗ ( M , Q) = H ∗ ( H / H 0 , Q). T his implies that H ∗ ( N , Q) = H ∗ ( H , Q ). The imag e of ν by the isomor phism H 3 ( H , Q ) → H 3 ( N , Q) is the fo rm ν L . The r e sult of Koszul [17] p. 95 s hows that we can rea liz e this fo r m by using an inv a riant bilinear form • The classiﬁca tion theor em of Giraud [14] implies the ex is tence of a ger be ov er H /L whose classifying class is the co homology class of pν L , where p is an int eger. W e call suc h a gerb e, a Medina-Revo y gerb e. Examples o f Med i na-Rev o y gerb es. Lie gro ups endow ed with bi-inv ar iant sca lar pro duct have b een intensively studied b y Aub ert, Dardie, Diatta, Medina and Revo y . Medina and Revo y [27] hav e shown that they can be constructed from simple Lie gr oups and the 1- dimensional Lie g roup by the pro cessus of do uble extensio n. Here is an example 26 of a Medina Revo y gerb e constructed from the double extension of the tw o dimensional Euclidea n space, endow ed with its co mm utative structure of a Lie algebra. Consider the nilp o tent Lie a lgebra cons tr ucted as follo ws: Let ( e 1 , e 2 ) b e an orthogo nal ba sis of the 2-dimensional Euclidean space ( U, <, > ), and h : U → U the linea r endomor phism such that h ( e 1 ) = e 2 , h ( e 2 ) = 0 co nsidered a lso as a deriv ation of the tr ivial underlying Lie a lg ebra o f U . Let V b e the 1-dimensional commutativ e Lie algebra , and V ∗ its dua l. F o r every elements u 1 , u 2 ∈ U , we denote by w ( u 1 , u 2 ) : V → V ∗ the ma p w hich ass igns to v ∈ V = R the scala r < v h ( u 1 ) , u 2 > . The double extension o f ( U, <, > ) b y V and h is the nilp otent Lie algebra L = V ∗ ⊕ U ⊕ V whose brac ket is deﬁned by the form ula : [( v ′ 1 , u 1 , v 1 ); ( v ′ 2 , u 2 , v 2 )] = ( w ( u 1 , u 2 ) , v 1 h ( u 2 ) − v 2 h ( u 1 ) , 0) The Lie algebra V ∗ ⊕ U ⊕ V is endo wed with the scala r product: < ( v ′ 1 , u 1 , v 1 ); ( v ′ 2 , u 2 , v 2 ) > ′ = < u 1 , u 2 > + v 1 v 2 + v ′ 1 ( v 2 ) + v ′ 2 ( v 1 ) The co nstant of structures of L are integral in its canonical bas is. The 3-form ν L deﬁned on L by ( u, v, w ) → < [ u, v ] , w > ′ is rationa l. Let Γ b e the lattice of the 1-connected Lie gr oup L asso ciated to L which is ge ner ated by the image of an integral basis of L . The classiﬁcation theo rem of Giraud implies the ex is tence of a g erb e on L/ Γ w ho se clas s ifying co cycle is pν L , where p ∈ N is such tha t pν L is in tegral. 7.3 A sheaf of categories on an orbifold with singularities. Let H be a compact Lie gr oup which a c ts on a manifold, the quotien t space N /H is an example of a gener alized orbifold C N (see deﬁnition 3.2.1 ). Prop ositio n 7 .3.1. L et N b e a gener alize d c omp act orbifold . The c orr esp ondenc e deﬁne d on the c ate gory of op en subsets of N which assigns t o U the c ate gory C N ( U ) , whose obje cts ar e elements ( P, H P , φ P ) of C N , s u ch that the image of φ P is U is a she af of c ate gories. Pro of. Gluing conditions of ob jects. Let U , b e an o pen subse t of N , and ( U i ) i ∈ I an o pe n covering of U . Consider for each i ∈ I , a n ob ject e i = ( P i , H i , φ i ) in C N ( U i ), and a mor pism u ij : e i j → e j i such that u l ij u i j l = u j il . Since the morphisms u ij are lo cal diﬀeomorphis ms , there exists a manifold P obtained b y gluing the family o f manifolds P i with u ij . W e can glue the Lie g roups H i and their actions to deﬁne a Lie g roup H which acts on P , and such that the map P /H → N is a lo cal homeomorphism: Let l i be the restriction of the action of H i to p i ( P i × N P j ). W e c a n identify p i ( P i × N P j ) with p j ( P i × N P j ) with u ij . W e denote H ij the limit of the maps l i and l j . The Lie group H ij acts on the gluing of P i and P j by u ij . Without restricting the g enerality , we can supp ose tha t I is a numerable set, co nstruct 27 H 01 , H 01 ..n obtained by gluing recursively the action of H 0 , ..., H n . The Lie group H is the limit o f the groups H 01 ..n . Gluing condition of arr ows. Let U b e an op en subset of N , a nd P , and P ′ t wo ob jects of C N ( U ). The corres p ondenc e deﬁned on the category of op en subsets of U , which assigns to V the set H om C N ( V ) ( P | V , P ′ | V ), where P | V is φ − 1 P ( V ) is a sheaf since we can glue diﬀerent iable maps • Another example of sheaf of catego ries is deﬁned by the theo ry o f foliag es (see Molino [31], or deﬁnition 3 .3.1). Let N b e a top olo gical manifold endo wed with a structure of a foliag e. A natural problem is to determine wether this foliage is induced b y a foliation on a manifold. The follo wing prop osition provides the obstruction which so lves this problem. Prop ositio n 7 .3.2. L et N , b e a top olo gic al sp ac e endowe d with the st ructur e of a foliage, for every op en subset U of N , denote by C N ( U ) the class of obje cts ( U, V , F , π ) of C N . The c orr esp ondenc e which assigns C N ( U ) to U is a she af of c ate gories which is the ge ometric obstruction of the ex istenc e of a manifold ˆ N , endowe d with a foliation F N , su ch that N is the sp ac e of le aves of F N . Pro of. Gluing conditions of ob jects. Let U b e a n op en subset o f N , ( U i ) i ∈ I an op en cov er ing of U suc h that for each element i of I , ther e exists an ob ject e i = ( U i , V i , F i , π i ) in C N ( U i ), morphisms u ij : e i j → e j i such that u l ij u i j l = u j il . The mo rphisms u ij allow to glue the family of manifolds V i to o btain a manifold V , o n which is deﬁned a foliation F whose restriction to U i is F i . Gluing condition for arrows. Let e = ( U, V , F , π ) and e ′ = ( U, V ′ , F ′ , π ′ ) t wo ob jects of C N ( U ). The corres p ondenc e deﬁned on the category of op en subsets of U , which assigns to U ′ the set H om C N ( e | U , e ′ | U ), is a sheaf, since we can g lue diﬀer en tiable foliated maps • There exis ts a sheaf L on N which a ssigns to ev er y op en s ubset U of N , the set of isomor phis ms of an ob ject e U of C N ( U ) (the foliated isomorphisms), a nd the sheaf of categor ie s that we ha ve just deﬁned is a gerbe b ounded by L . 8 Diﬀeren tial geometry of shea v es of categories. In this part, we a re go ing to a nalyze the to ols deﬁned in the gener al context of diﬀerentiable ca tegories to study their geo metry b y using the underlying top ology . Let C b e a diﬀerentiable category endowed with a top olo gy . W e supp ose that C has a ﬁnal ob ject and a g o o d g lo bal cov er ing ( U i ) i ∈ I (see p. 24). Let P → C be a g erb e, we supp ose that ther e exists a Lie gr oup H , a principal H -torsor A : Aut ( P ) → C , such that every ob ject e U ∈ P U , U ∈ C is a bundle p e U : e U → U endow ed with a free rig ht action of H . The set of morphisms 28 betw een tw o o b jects of P U are morphisms b etw een bundles which pr o ject to the identit y of U , and the set of automor phisms o f e U can b e iden tiﬁed with gauge trans formations of Au t ( P )( U ) by a map which co mmutes with mo rphisms betw een o b jects and with restrictions . W e denote by aut ( P ) the vector H -bundle on C ass o ciated to Aut ( P ): If the co o rdinate changes of Aut ( P ) a re deﬁned by the maps ( u ij ) i,j ∈ I , the co or dinate changes of aut ( P ) are deﬁned by the map ( Ad ( u ij )) i,j ∈ I . Such a gerb e is called a H -g erb e. 8.1 Induced gerb es. Let p : P → C b e a n H -principal ge rb e, that is: for every o b ject U of C , the map p e U : e U → U , endo ws e U with the s tructure o f a H -principal bundle. Consider a mor phis m of Lie gr oups h : H → H ′ , we can construc t a principal H ′ -gerb e p ′ : P ′ → C as follows: Let a : A → U, U ∈ C be an ob ject o f P U , it is a H -principal torsor deﬁned by the trivializ a tion ( U i , u ij ∈ H ) i,j ∈ I . W e can deﬁne the image of a by h . It is the torsor whose co ordinates change are deﬁned by: ( U i , h ( u ij )) i,j ∈ I . The family of these images is the induced gerb e. An ex ample is the situation when H is S U ( n ) or O ( n ), and h is the deter - minant morphism. Prop ositio n 8 .1. Ther e exists a c onne ctive structu r e on e ach H -gerb e p : P → C N , wher e N is a manifold, and C N the diﬀer entiable c ate gory asso ciate d to N deﬁne d at p.5. Pro of. L e t ( U i ) i ∈ I be an o pen cov er of N , and e i an o b ject of P U i , we denote b y e ′ i the quotient of e i by the action of H . Consider an isomo rphism u ij : e i j → e j i . W e c a n construct from u ij a mor- phism u ′ ij : e ′ i j → e ′ j i which is its quotient. With these morphisms, we can glue the family of quotients ( e ′ i ) i ∈ I to deﬁne a ﬁb er bundle p ′ : P ′ → N . Consider a c o nnection on p ′ : this is a distribution D ′ of P ′ whose rank is the dimension of N , and w hich is transverse to the ﬁber of p ′ (compare with McDuﬀ and Salamon [23] p. 210). The dis tr ibution D ′ can b e also deﬁned b y a 1-form θ on P ′ which takes its v alues in the canonical bundle o ver P ′ , s uc h that for each x ∈ P ′ , the ﬁb er of this bundle is the tangent spa ce o f P ′ x , the ﬁb er at p ′ ( x ). W e suppo se that if v ∈ T x P ′ , x ∈ U , θ ( v ) = v . Such a dis tribution can be constructed b y using a diﬀeren tiable metric on P ′ , and by taking the orthogo nal of the ﬁb er. Let U b e an op en subs et of N , and e U an ob ject of P U . W e deno te by C o ( e U ) the set of 1- forms deﬁned o n the bundle e U → U whic h take their v a lues in the canonical v ector bundle ov er U whose ﬁb er at x is the ta ngent s pace o f the ﬁb er of e U → U at x . W e suppo se that for every α ∈ C o ( e U ), and every element A ∈ H which generates the fundamen ta l v ector ﬁeld A ∗ , we hav e: α ( A ∗ ) = A. 29 W e supp ose also that the ea ch ele men t of C o ( e U ) descends to the res tr iction of θ to U . Such a co nnection can b e constructed a s follows: Let ( U i ) i ∈ I be a triv ia l- ization of e U → e U /H . W e can deﬁne on U i × H a distribution inv ariant b y H whose pro jectio n to U i is the r estriction of D ′ . Such a distribution is deﬁned by a 1-form α i whose pr o jects to the restriction of θ to U i . By using a partition of unit y , we deduce that C o ( e U ) is not empt y • 8.2 Reduction to a situation similar to the motiv ating example. Now we reduce the study of the diﬀerential geometry o f a H -gerb e to a situa tio n similar to the motiv ating exa mple (see p. 17) by using the following construc- tion. Consider a H -gerb e p : P → N . W e have s een that there exists a ﬁb er bundle p ′ : P ′ → N s uch that for a go o d cov ering ( U i ) i ∈ I of N , the re s triction of P ′ to U i is the quotient o f e very ob ject e i of P U i by H . Let F b e the ﬁb er of this bundle. W e can deﬁne the bundle L ( p ′ ) : L ( F ) → N such that for every element u ∈ N the ﬁber of L ( F ) at u is the set of linear frames o f its ﬁb er, F u . Let U b e a n open subse t o f U , and e U an ob ject of P U . W e can deﬁne the pullback of the maps L ( F ) | U → e U /H and e U → e U /H which is an H -principa l bundle e ′ U ov er the restriction L ( F ) | U . The class P ′ U whose elemen ts a re the e ′ U just constructed deﬁnes a ger be L ( P ) → N . This a llows to dea l only with to a situation simila r to the motiv a ting example in supp osing that the bundle P ′ → N is principa l. In fact in the sequel, w e deal o nly with the motiv ating example. Moreov er we supp ose that connective structures are co nstructed by using connections of P ′ like at the pr op osition 8.2.1. Another v ar iant of the prev ious construction is the following: let p : P → N be a H -pr incipa l bundle deﬁned o n the ma nifold N , to each co cyc le c ∈ H 2 C ech ( N , H ) we can a sso ciate a ge rb e C whose classifying co cy cle is c (see Giraud [14 ]). The pr evious co nstruction allows to construc t a gerb e R ( C ) ab ov e the frames bundle of N , such that an ob ject of R ( C )( U ) is the pull back of an ob ject of C ( U ) b y the pro jectio n map R ( U ) → U , where R ( U ) is the bundle of linear frames o f U . Thus we can deﬁne connectiv e structures ab ove connections of R ( N ). Deﬁnition 8.2.1. A cur ving (see also B rylinski [8 ] p. 211) deﬁned on the connective structure C o of the H -g e rb e p : P → C , is a map which assigns to ev ery o b ject e U of P U , and e very element θ ∈ C o ( e U ), a 2-form L ( e U , θ ) whic h tak es its v alues in H such that the following proper ties are satisﬁed: Let h : e U → e U ′ be a morphism, for ev ery θ ∈ C o ( e U ′ ), we hav e: L (( e U ) , h ∗ ( θ )) = h ∗ ( L ( e U ′ , θ )) . If h is an automorphism of e U , and θ a n elemen t of C o ( e U ), we hav e: L ( e U , h ∗ ( θ )) = Ad ( h − 1 )( L ( e U , θ )) 30 Let α b e an element o f Ω 1 ( U, aut ( U )), we have: L ( e U , θ + α ) = L ( e U , θ ) + d ( α ) + 1 2 ([ α, α ] + [ θ , α ] + [ α, θ ]) . The co rresp ondence ( e U , θ ) → L ( e U , θ ) is natura l in re spe c t to res trictions and morphisms betw een ob jects. Remark tha t since the gerb e considered here is asso cia ted to the motiv ating example, the 2-form L ( e U , θ + α ) − L ( e U , θ ) is H - v alued. Prop ositio n 8 .2.1. L et C o b e a c onne ctive stru ctur e deﬁne d on the H -princip al gerb e p : P → C , ther e exists a curving. Pro of. Compar e the following pro of with Brylyinsk i [8 ] p. 212). W e are going to ass ume that there exists a L ′ -bundle p ′ : P ′ → N , an exa ct sequence of Lie groups 1 → H → L → L ′ → 1 such that the gerb e is the geo metric obstruction to lift the s tructural gr oup L ′ of p ′ to L . W e supp ose a lso that the connective structure de ﬁned on the pr incipal gerb e is constructed as in the prop osition 8.1. Th us there exists a c o nnection α deﬁned on the principal bundle p ′ : P ′ = P /H → N suc h that for every op en subset U of N , e U an ob ject of P U , the elemen ts of C o ( e U ) are connections which pr o jects to α . Let H , L , and L ′ be the resp ective Lie a lgebras of H , L a nd L ′ . Let u b e a linear section o f the ca nonical map L → L ′ . W e can deﬁne the form α 0 = u ◦ α on P ′ . Le t θ b e an element of C o ( e U ), c o nsider the form α e U , the pull-back of the restriction of α 0 by the cano nic a l pro jection e U → e U /H = P ′ | U . W e set: L ( e U , θ ) = dθ + 1 2 [ θ, θ ] − ( dα e U + 1 2 [ α e U , α e U ]) . • Deﬁnition 8.2.2. Let L b e a curv ing of the connective structure C o deﬁned on the H - gerb e P → C N , where N is a manifold, and C N the canonic a l diﬀeren tiable catego ry asso ciated to N (see page. 5). Let ( U i ) i ∈ I be a go o d c ov ering of N , and e i and ob ject of P U i , and α i an element of C o ( e i ) and u ij : e i j → e j i a morphism. Let Ω i be the curv ature o f α i . On U i ∩ U j , the diﬀerence Ω i j − u ∗ ij Ω j i deﬁnes a 1-Cech aut ( P )-co cycle. The c o homology class do e s not dep end o f the e le ments e i in P U i and of the e le ments α i used to deﬁne it, since C o ( e i ) is a torsor whose vector space is a spa ce o f H -v alued 1-for ms. The DeRham-Cech isomor phism a llows to identify this co cycle to a 3 − aut ( P ) fo r m D ca lled a curv ature of the connective str ucture. Deﬁnition 8.2.3: Hol onomy . W e ar e going to reduce this deﬁnition to the commutativ e case, a ll the gr oups are compact and complex , a s well a s the vector bundles. Let p : P → C be a diﬀerentiable H -gerb e, e ndowed with a connectiv e structure and a cur ving L . 31 Suppo se that P is the geometric obstruction to lift a G -bundle P ′ ov er N to a G ′ -bundle g iven the exa ct sequence o f Lie g roups 1 → H → G ′ → G → 1. W e a ssume a lso that in fact, the G - bundle P ′ is the fr ames bundle of a vector bundle V over N , and the ob jects of P ar e also asso c ia ted to v ecto r bundles. Let N b e a sur face, Consider a morphism h : N → C . W e can pull-back p by h a nd obta in a g e rb e p N : P N → N , endowed with a co nnective structure which has a curving. The quotient of p N by H is the pull-back p ′ G : P G N → N of P ′ by h , which is a reduction of the frames bundle of the pull-back V N of V b y h . A w ell-known result implies that V N is isomor phic to the summand of complex line bundles (see McDuﬀ and Salamon [23] p. 80). Thus w e ca n assume that the s tr uctural group of P N is a commutativ e subgr oup G N of G . The ger be p N is als o the geometric obstruction to extend the structural g roup G N to G ′ N given an exa ct sequence of Lie groups 1 → H N → G ′ N → G N → 1 . This implies that we can assume also that H N and G ′ N are comm utative. The pull-back of the connective s tr ucture a nd the curving of p , induces a connective structure C o N of p N and a curving L N , more ov er the connectio n α used to constr uct the connective structure of p (s e e pr op osition 8.2.1 ) is supp os ed to b e Hermitian, as well as the elements of C o ( e U ), whe r e e U an ob ject of the gerb e. Th us the co nnection α N on P G N N , which induces the connective structur e C o N preserves every Hermitian reduction. W e just hav e to r ecall the deﬁnition of the holonomy in the commutativ e case. Let ( U i ) i ∈ I be a g o o d cov er of N . Let e i be an o b ject of P N U i , and let θ i be an element o f C o N ( e i ), we ca n suppo s e that this co nnection takes its v alues in the Lie algebra of G ′ N . W e denote by L ( e i , θ i ) the c ur ving asso ciated to the elemen t θ i ∈ C o ( e i ). Denote b y θ ij the for m θ i j − u ∗ ij ( θ j i ). Since N is 2-dimensional, there exists a 1-form h i such that dh i = L ( e i , θ i ). W e have: θ ij = h j − h i + da ij W e can set d ij l = c − 1 ij l a − 1 j l a il a − 1 ij where c ij l is the classifying co cy cle o f p N . The chain d ij l is the holo nomy co cycle of the gerb e p N endow ed with its connective str ucture. The Ce ch -DeRham iso morphism a llows to iden tiﬁes this form with a 2-fo rm Ω on N . (See also Mack aay and P ick en [22] p. 27). 8.3 Holonom y and functor on lo ops space. Consider the c a tegory C 2 whose ob jects ar e maps: h : C 1 + .. + C 1 → N , where C 1 + ... + C 1 is a ﬁnite disjoin t unio n of circles. A morphism b etw een the ob jects h a nd h ′ , is a map from a sur face l : N → C such that the restrictio n of l to the b oundary of N is the sum of the maps h and h ′ . The holonomy deﬁnes a functor D on C 2 which asso ciates to h the complex line C. Let l : N → C b e a morphism betw een h and h ′ . The real holonomy ar ound N is the image D ( l ) of l by D . 32 W e supp ose here that the struc tur al g roup H of the H - gerb e P → N deﬁned ov er the manifold N and endow e d with a connective structure is co n tained in Gl ( n, C). W e can r elate this gerb e which is asso ciated determinant g erb e as follows: Prop ositio n 8 .3.1. Supp ose that H is include d in Gl ( n, C) , then the tr ac e of t he curvatu re of a princip al H -gerb e P → N is the curvatur e of the asso ciate d determinant gerb e det ( P ) → N ; that is the gerb e induc e d by t he determinant morphism H → C . (Se e 8.1). Pro of. L e t p : P ′ → N b e the quotient of the gerb e b y H . F or every op en subse t U of N , the ob jects of P U are H -pr incipal bundles ov er P ′ | U the restriction of P ′ to U . Let det e U : e U → det ( e U ) the determinant morphism which asso cia tes to the ob ject e U of P U , the cor resp onding ob ject det ( e U ) in det ( P ), and α ∈ C o ( e U ) whose kernel deﬁnes the distribution C α on e U . Since de t e U is a n equiv aria nt morphism b etw een the H -bundle e U → P ′ | U and det ( e U ), the image o f C α is a distribution det ( C α ) inv ariant by the action of U (1 ) o n de t ( e U ) a nd tr ansverse to the ﬁb er. Such a co nstruction thus deﬁnes a co nnective s tructure on the determinant ger be . Suppo se that the form α is L ⊕ g l ( n, C)-v alued, where L is the Lie algebra of L the structural gr oup o f P ′ . Then the c onnection form which deﬁnes the distribution det ( C α ) is the comp osition of α and ( I L , tr ace gl ( n, C) ). This implies that the curving of ( det ( e U ) , C e U ) is the trace of the c ur ving L . This implies the result • 8.4 Canonical relations asso ciated to a connective struc- ture on a gerb e. In this pa rt we are going to determine canonical r elations asso ciated to a con- nective structur e. (compare with Breen and Messing [7] p. 58) The morphisms u ∗ are pull-back, and the morphisms the u ∗ are inv erse of pull-ba ck. Let e i be an ob ject of P U i , and α i ∈ C o ( e i ). Co nsider the r estriction e i j of e j to U ij and u ij : e i j → e j i an arr ow. The 1-for m u ij ∗ ( α j ) is a n element of C o ( e j i ), since C o ( e j i ) is a torsor , ther e exists a 1-for m α ij such that: α j i = u ij ∗ ( α i j ) + α ij W e hav e seen that the family of forms α ij veriﬁes the equations: u ij ∗ α j l − α il + α ij = u il ∗ ( α ij l − Ad ( c − 1 ij l )( α ij l )) − c − 1 ij l dc ij l where c ij l is the map u j li u l ij u i j l . Let L b e a curving of the connective str ucture. Denote by L ij the 2-form L j ( e i j , α j ) − L j ( e i j , u ∗ ij ( α i )). W e have: 33 L j l − L il + u ∗ j l L j l = = L l ( e ij l , α ij l ) − L l ( e ij l , u i j l ∗ ( α il j )) − ( L l ( e ij l , α ij l ) − L l ( e ij l , u j il ∗ ( α j l i )))+ u i j l ∗ ( L j ( e il j , α il j ) − L j ( e il j , u j ij ∗ ( α j l i ))) = L l ( e ij l , u j il ∗ ( α j l i )) − L l ( e ij l , u j j l ∗ u l ij ∗ ( α j l i )) = u j il ∗ ( L i ( e j l i , α j l i ) − Ad ( c ′ ij l − 1 ) L ( e j l i , α j l i )) where c ′ ij l = u l ij u i j l u j li . 8.5 Uniform distributions and gerb es. Another treatment of the diﬀerent iable structure on gerb es can b e done a s fol- lows: Let p : P → N b e a H -g erb e deﬁned on a manifold. W e reduce the study to the motiv ating example. The natura l way to study the diﬀerential geo me- try of a principal bundle is to use the theory of connectio ns. Unfortunately , connections deﬁned on a princ ipa l bundle ar e not necessar ily in v ariant by the gauge gro up. This motiv ates the deﬁnition of a torso r of co nnections, which is inv ariant by the automorphisms group. There exists a nother p o int of view used b y Molino in his thesis (see [29]). Molino has studied the notion of inv a riant distributions o n principal bundles. An inv aria n t distribution on a principa l bundle is a right in v aria nt distribution. W e do not r equest here that the dimension of the distribution is the dimension of the basis space o f the bundle. The inv aria nt dis tribution is transitive, if its summand with the tangent space of the ﬁber , genera tes the tangent space o f the bundle, (see Molino [29] p. 180), tr ansitive distributions a re no thing but equiv alence class es o f connections. W e fo cus on the mo tiv a ting example of a H - gerb e P → N ; there exists an e x act sequence of Lie groups 1 → H → L ′ → L → 1, a L -principa l bundle P ′ → N , such that the gerb e P → N is the geometr ic obstruction to lift the str uctural group of the previo us principal bundle to L ′ . The o b jects of P U are pr incipal H -bundles ov er the restrictio n of P ′ to U . F or a connection θ deﬁned on P ′ , we ca n deﬁne o n ea ch o b ject e U of P U , the tr ansitive distribution which is the kernel of the pull-back of θ | P ′ | U to e U . This transitive distribution is inv ariant by the automorphisms of e U . Suppo se now that the extensio n which deﬁnes the lifting problem is cen tral. Let H , L ′ and L b e the resp ective Lie algebra s of H , L ′ and L . The co o rdinate changes ( u ij ) i,j ∈ I of the principal L -bundle P ′ → N , deﬁne a L -bundle P L ov er N who se co ordinates changes are ( Ad ( u ij )) i,j ∈ I . W e can lift u ij to an e lemen t u ′ ij of L ′ , since the extension is central, we can deﬁne a L ′ -bundle P L ′ ov er N whose co ordina tes change are ( Ad ( u ′ ij )) i,j ∈ I . Ther e exists a canonical pro jection p 0 : P L ′ → P L . A connection structure deﬁned on the ger be P , is deﬁned as follows: 34 Let U b e an op en s ubset o f N , and e U an ob ject o f P U , there ex is ts a transitive distribution D e U of e U which is rig ht inv aria n t. Let h : e U → e U ′ be a morphism in P , we ass ume that the pull-bac k of D e U ′ by h is D e U . The distribution D e U is not assume to b e uniform, see (Molino [2 9] p. 184) when it is uniform, the connection structure can b e deﬁned by a family of 1- forms θ e U : e U → P L ′ which verify the follo wing conditions: if x is an element of e U , and v a n element of T e U x , the tangent space of e U at x , θ e U ( v ) is an element o f the ﬁb er of p e U ( x ), where p e U : e U → U is the canonical pro jection. Let A be an element of L ′ , and ¯ A the pr o jection of A in L b y the canonical map ¯ p : L ′ → L ′ / H = L . Denote by A ∗ the fundamen tal vector ﬁeld generated by A on e U . W e as sume tha t ¯ p ( θ e U ( A ∗ )) = ¯ A . Let H ∗ e U be the vector space of fundamental vectors generated by elements of H . W e assume that θ e U preserves H ∗ e U , and its r estriction to it is a pr o jection. Let h : e U → e U ′ be a morphism in P , we assume that h ∗ ( θ e U ′ ) = θ e U . An horizontal path in e U is a diﬀeren tia ble path c : I → e U such that for each t in I , the tangent vector to the curve c ′ ( t ) at t is an element of D e U . The holonomy o f a trans itiv e distribution can b e deﬁned as is deﬁned the holonomy of a connection (see Lichnerowicz [1 9] p. 6 2, Molino [29] p. 181 ): Let x be an elemen t of e U , the ho lonomy group H x at x , is the set of ele ments l ′ in L ′ such that there exists an horizontal path b etw een x a nd xl ′ − 1 , of co ur se if we repla c e x b y hx , H hx = Ad ( h − 1 ) H x . Let x b e an elemen t of U , and c : I → U a diﬀerentiable path such that c (0) = c (1) = x . Consider y an elemen t o f the ﬁb er o f x . Since the distribution is transitive, there exists an horizo n tal path ov er c in e U , d : I → e U such that d (0) = y . T his is implied by the fact that a tra nsitive inv ariant distribution contains always a connectio n. (See Mo lino [29] p. 181). The element y d (1 ) − 1 do es not dep ends of y (compar e w ith Lichnero wic z p. 94). It is called the holonomy aro und c . The holo nomy group H e U x at x is the set who se element s are holonomy aro und lo ops at x . The ho lonomy gro up dep ends o f the ob ject since t wo ob jects of P U are not alwa ys isomorphic. Suppo se that U is contractible, then the holono my group do es not dep ends of the o b ject since it is inv ariant by the g auge trans formations which preserve the co nnection since the extension is central, and all the ob jects of P U are isomorphic. T his last gro up can be co mputed b y the Ambrose-Singer theorem. (See Molino [29] p. [1 83]). 9 Sequences of ﬁb ered categories in diﬀeren tiable categories. One of the main motiv ation of the int ro duction of ger be s theory in diﬀerential geometry is the g eometric interpretation of characteristic classes . Let N b e a manifold. A w ell-known r esult iden tiﬁes the 2-dimensional integral cohomolo gy 35 space H 2 ( M , Z) of N , with the space of iso morphic cla sses of U (1)-bundles deﬁned on N . This identiﬁcation is one o f the main to o l used in quantization in ph ys ics. String theory has created the need o f ﬁnding such an in terpretation for higher cohomo lo gy classe s . The space of 3-dimensional in tegr al co homology classes is the classifying space of U (1)-g erb es (See Brylinski [8] p. 20 0). The second Pon tryagin class which is an element of H 4 ( N , Z) ha s been interpreted with 2-gerb es b y Br ylinski and McLaughlin (see [9 ] p. 625). In this pa rt, we are going deﬁne and apply the theory and sequences of ﬁb ered categories analog to the theor y deﬁned in Tsemo [40]) to study characteristic classes in diﬀerentiable categorie s. Deﬁnition 9.1. A 2-sequence of ﬁbe red catego r ies is deﬁned b y the following data: A ﬁber ed categor y p : P → C ov er the Grothendieck site C , such that: Let U be a n ob ject of C , and e U an ob ject of P U . Recall that e U is a diﬀerentiable manifold. There exis ts a corresp ondence whic h ass igns to e U a 2-catego ry Q e U (see Benab ou for the deﬁnition of a 2- category ) whose ob jects are gerb es deﬁned on e U . Let c : U → U ′ a morphism of C , the restriction functor is the pull-back of gerb es. There exists a covering ( U i ) i ∈ I , such that for every ob jets e i and e ′ i of P U i , there exists an isomorphism be tw een the 2-ca tegories Q e i and Q e ′ i . The se t of automor phisms of a n ob ject of Q e U can b e iden tiﬁed with sec- tions o f a sheaf L deﬁned on C , and this identiﬁcation is natural in resp ect to morphisms b et ween ob jects and restrictions. Let P ” b e the category whose o b jects are ob jects of Q e U , e U in P U , and P ′ the ca tegory whose ob jects are o pen subsets o f the ma nifo lds e U . If e in Q e U and e ′ in Q e U ′ are o b jects o f P ”, ther e exists a n o pen subset V of e U , (res p. V ′ of e U ′ ) a Lie gr oup H such that e → V is a H - bundle (resp. e ′ → V ′ is a H -bundle). A mor phis m h ′ : e → e ′ in P ” is a morphism o f H -bundles suc h that there exists h : e U → e U ′ in P such that h ( V ) ⊂ V ′ , and the following square is commutativ e: e h ′ − → e ′ ↓ ↓ V h − → V ′ Our descent co ndition is expr essed by the fact that we a ssume that the corres p ondenc e P ” → P ′ which as signs to e , the o pen subset V of e U is a ﬁber ed category . 9.1 Classiﬁcation 4 -co cycles and sequences of 2 -ﬁbered categories. Before to attach to a 2- sequence of ﬁbered categories a co cycle, we descr ibe an automorphism h ab ove the iden tity of a H - gerb e p : P → N ov er a manifold N . 36 The automorphism h is deﬁned by a family of functors h U of P U ab ov e the ident ity , wher e U is an op en subset in N , such tha t if V is a subset of U the following squar e comm utes: P U h U − → P U ↓ r V ,U ↓ r V ,U P V h V − → P V where r V ,U : P U → P V is the restriction map. Let ( U i ) i ∈ I be a go o d cov ering of N , we a ssume that a n ob ject e i of P U i is a trivial H -bundle. Since the s q uare: P U i h U i − → P U i ↓ r U i ∩ U j ,U i ↓ r U i ∩ U j ,U i P U i ∩ U j h U i ∩ U j − → P U i ∩ U j The automo rphism h is descr ibe d b y a family of morphisms u ij : U i ∩ U j × H → U i ∩ U j × H such tha t u l ij u i j l = u j il . Thus b y a H -bundle. Now can des crib e the classifying 4-co cycle: W e assume that L is commutativ e. Let ( U i ) i ∈ I be a g o o d cover of the s ite ( C, J ), and e i and o b ject of P U i , we cho ose a gerb e d i in Q e i . Since P U i × C U j is co nnec ted, there exists a morphism: u ij : e i j → e j i , and a ma p u ∗ ij : d i j → d j i . The automorphism c ∗ ij l = u ∗ ij u ∗ j l u ∗ li : d j l i → d j l i is not ab ov e the iden tity . B ut c ij lm = u ∗ ij ml c ∗ ij l u ∗ ij lm ◦ c ∗ ij m − 1 ◦ c ∗ ilm ◦ c ∗ j lm − 1 is a morphism ab ov e the identit y that we ide ntiﬁes with a 1-form deﬁned on U ij lm which takes its v alues in L ( U ij lm ). The Cech-DeRham isomorphism ident iﬁes this with a 4-c o cycle which takes its v alues in L if C is a manifold, since L is assumed to be co mm utative. Before to give examples, we are go ing to descr ibe a weak version of a 2- sequence of ﬁbere d categories . Deﬁnition 9.1.2. A 2-sequence of to rsor/ ﬁber ed categories , is a 2-s equence of ﬁbe red categor ie s where the sheaf of categor ie s P → C is in fact a torsor. W e can asso cia te to a 2-sequence of tor sor/ﬁb ere d categories a 3-co cyc le as follows: Let ( U i ) i ∈ I be a g o o d cov er of C , a nd e i the ob ject of P U i , and d i an ob ject of Q e i , and u ij : e i j → e j i . Ther e exists a mo rphism u ∗ ij : d i j → d j i . The mor phism c ∗ ij l = u ∗ j li u ∗ l ij u ∗ l j l is ab ov e the iden tity . It is a 1-form deﬁned o n U ij l which is L ( U ij l ) v alued. The Cec h Der ham isomorphism identiﬁes this cocycle with a 3-form deﬁned on C which is L -v alued. 37 Examples. Let H be a compact simple Lie group. Consider a H - principal bundle p : P → C ov er a diﬀerentiable category C . W e can deﬁne the following 2- sequence of torsor/ﬁb ere d ca tegories: Let U b e an o b ject of C , the ob ject of Q P U are U (1)-gerb es which induces on each ﬁb er of e U → U a gerb e isomor phic to the canonical U (1)-gerb e on H . This example is deﬁned when N is a manifold by Br ylinski a nd McLa ughlin [9 ] p. 625). In this situation the cla ssifying co cycle is an element of H 3 ( N , U (1)) = H 4 ( N , Z). This construction can a lso be a pplied to a sub categ o ry C ′ N of diﬀerentiable category C N asso ciated to a gene r alized or bifold (see deﬁnition 3.2.1) where there exists a simple and compact Lie group H such that ev ery ob ject of C ′ N is of the for m ( P , H , φ P ). 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Differentiable Categories, gerbes and G-structures

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