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G -gerbes, prin ipal 2-group bundles and hara teristi lasses ∗ †‡ Grégory Ginot Mathieu Stiénon 8 0 Abstra t 0 2 We give an expli it des ription of a 1-1 orresponden e between Morita [G → Aut(G)] n equivalen e lasses of, on the one hand, prin ipal 2-group - G a bundles over Lie groupoids and, on the other hand, -extensions of Lie J [G → Aut(G)] 2 grouGpoids (i.e. between -bundles over di(cid:27)erentiable sta ks and -gerbes over di(cid:27)erentiable sta ks). We also introdu e universal har- 2 G a teristi lasses for 2-group bundles. For groupoid entral -extensions, ] we provethat the universal hara teristi lasses oin ide with the Diximer T Douady lasses that an be omputed from onne tion-type data. A . h t Contents a m [ 1 Introdu tion 2 2 v 2 Generalized morphisms and prin ipal Lie 2-group bundles 4 8 3 2.1 Lie 2-groupoids, Crossed modules and Morita morphisms . . . . . . 4 2 1 2.2 Generalized morphisms of Lie 2-groupoids . . . . . . . . . . . . . . 7 . 1 2.3 2-group bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 0 8 G 0 3 Groupoid -extensions 11 : G [G → Aut(G)] v 3.1 From groupoid -extensions to -bundles . . . . . . . 12 i X [G → Aut(G)] G 3.2 From -bundles to groupoid -extensions . . . . . . . 14 r a 3.3 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Universal hara teristi maps and Dixmier-Douady lasses 16 4.1 Cohomology of Lie 2-groupoids . . . . . . . . . . . . . . . . . . . . 16 4.2 Cohomology hara teristi map for 2-group bundles . . . . . . . . . 18 G 4.3 DD lasses for groupoid entral -extensions . . . . . . . . . . . . 19 4.4 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 S1 4.5 The ase of entral -extensions . . . . . . . . . . . . . . . . . . . 22 4.6 Proof of Theorem 4.14 . . . . . . . . . . . . . . . . . . . . . . . . . 24 ∗ UPMC Paris 6, Institut Mathématique de Jussieu, Équipe Analyse Algébrique, Case 82, 4 p†la e Jussieu, 75252 Paris, Fran e ginotmath.jussieu.fr Pennsylvania State University, Department of Mathemati s, 109 M Allister Building, Uni‡versityPark, PA 16802, U.S.A. stienonmath.psu.edu Resear hsupportedbytheEuropeanUnionthroughtheFP6MarieCurieR.T.N.ENIGMA (Contra t numberMRTN-CT-2004-5652). 1 1 Introdu tion G This paper is devoted to the study of the relation between groupoid -extensions and prin ipal Lie 2-group bundles, and of their hara teristi lasses. Γ ⇉ Γ Γ 2 1 1 A Lie 2-group is a Lie groupoid , whose spa es of obje ts and of mor- Γ 2 phisms are Lie grouρps and all of whose stru ture mapsρare group morphisms. (G −→ H) G −→ H A rossed module is a Lie group morphism together with an H G a tion of on satisfying suitable ompatibility onditions. It is standardρthat [G −→ H] Lie 2-groups are in bije tion with rossed modules [3,12℄. In thρis paper, (G −→ H) denotes the 2-group orresponding to the rossed module . Lie2-groupsarisenaturallyinmathemati alphysi s. Forinstan e,inhighergauge theory [2,4℄, Lie 2-group bundles provide a well suited framework for des ribing theparalleltransportofstrings[1,4,29℄. Severalre entworkshaveapproa hedthe on ept of bundles with a (cid:16)stru ture Lie 2-group(cid:17) over a manifold from various perspe tives [1,4,6,36℄. Here we take an alternative point of view and give a de(cid:28)nition of prin ipal Lie 2-group bundles of a global nature (i.e. not resorting to a des ription expli itly involving lo al harts and o y les) and whi h allows for the base spa e to be a Lie groupoid. In other words, we onsider 2-group prin ipal bundles over di(cid:27)erentiable sta ks [7℄. Our approa h immediately leads to a natural onstru tion of (cid:16)universal hara teristi lasses(cid:17) for prin ipal 2-group bundles. G P M Let us start with Lie (1-)groups. A prin ipal -bundle over a manifold M anoni ally determines a homotopy lass of maps from to the lassifying spa e BG G G of the group . In fa t, the set of isomorphism lasses of -prin ipal bundles f M M −→ BG over is in bije tion with the set of homotopy lasses of maps [14, H∗(BG) 38,39℄. Pulling ba k the generators of (the universal lasses) through f P M , one obtains hara teristi lasses of the prin ipal bundle over . These hara teristi lasses oin ide with those obtained from a onne tion by applying the Chern-Weil onstru tion [17,32℄. There is an analogue but mu h less known, di(cid:27)erential geometri rather than G M purely topologi al, point of view: a prin ipal -bundle over a manifold an be thought of as a (cid:16)generalized morphism(cid:17) (in the sense of Hilsum-Skandalis [24℄) M G from the manifold to the Lie group both onsidered as 1-groupoids. To see G this, re all that a prin ipal -bundle an be de(cid:28)ned as a olle tion of transition g : U → G ij ij fun tions on the double interse tions of some open overing, satis- g g = g ij jk ik fying the o y le ondition . These transition fun tions onstitute a U ⇉ U ij i morphism of groupoids from the ƒe h groupoid ` ` asso iated to the {U } G⇉ ∗ i i∈I open overing to the Lie group . Hen e we have a diagram (M ⇉ M)←∼− ( U ⇉ U )→ (G ⇉ ∗) a ij a i in the ategory of Lie groupoids and their morphisms whose leftward arrow is a M Morita equivalen e, in other words a generalized morphism from the manifold G to the Lie group . This se ond point of view, or more pre isely its generalization to the 2-groupoid ontext, onstitutes the foundations on whi h our approa h is built. The general- izationofthe on eptof(cid:16)generalizedmorphism(cid:17) to2-groupoidsisstraightforward: Γ ∆ Γ ←−φ E −→f ∆ a generalized morphism of Lie 2-groupoids is a diagram ∼ 2Gpd φ in the ategory of Lie 2-groupoids and their morphisms, where is a 2 Morita equivalen e (a (cid:16)smooth(cid:17) equivalen e of 2-groupoids). It is sometimes useful to think of two Morita equivalent Lie 2-groupoids as two di(cid:27)erent hoi es of an atlas (or open over) on the same geoρmetri obje t (whi h is a di(cid:27)erentiable [G −→ H] Γ 2-sta k [7,11℄). Wede(cid:28)ne aprin ipal ρ -bundle overaLiegroupoid to be Γ [G −→ H] a generalized morphism from to (up to equivalen e). See Se tion 2.3. The on ept of (geometri ) nerve of Lie groupoids extends to the 2- ategori al ontext as a fun tor from the ategory of Lie 2-groupoids to the ategory of simpli ial manifolds [40℄. By onvention, the ohomology of a 2-groupoid is the ohomology of its nerve, whi h an be omputed via a double omplex (for instan e, see [20℄). Cru ially, Morita equivalen es indu e isomorphisms in oho- Γ F [G → H] mology. Therefore, any generalized morphism of 2-groupoids [G → H] B Γ de(cid:28)ning a prin ipal -bundle over the groupoid yields a pullba k F∗ : H•([G → H]) → H•(Γ) homorphism in ohomology, whi h is alled the ohomology hara teristi map ( hara teristi map for short). The ohomology H•([G → H]) lasses in [20℄ should be viewed as universal hara teristi lasses F∗ B and their images by as the hara teristi lasses of . Lie 2-group prin ipal bundles are losely related to non-abelian gerbes. Ge- G ometri ally, non-abelian -gerbes over di(cid:27)erentiable sta ks an be onsidered G G as groupoid -extensions modulo Morita equivalen e [26℄. By a groupoid - 1 → M × G −→i Γ˜ −→φ extension, we mean a short exa t sequen e of groupoids Γ → 1 M × G , where is a bundle of groups. Here we establish an expli it 1-1 G G orresponden e between groupoid -extensions up to Morita equivalen e (i.e. - [G → Aut(G)] gerbes over di(cid:27)erentiable sta ks) and prin ipal -bundles over Lie [G → Aut(G)] groupoids modulo Morita equivalen e (i.e. -prin ipal bundles over di(cid:27)erentiable sta ks). This is Theorem 3.4. Note that a restri ted version of this orresponden e is highlighted in [27, Theorem 4℄. H2(X,G) It is known that Giraud's se ond non abelian ohomology group lassi- G X H1(X,[G → (cid:28)es the -gerbes over a di(cid:27)erentiable sta k [21℄ while Dede kers' Aut(G)]) [G → Aut(G)] lassi(cid:28)es the prin ipal -bundles [15,16℄. In [9,10℄, Breen showed that thesetwo ohomology groups areisomorphi . Insomesense, our theorem above an be onsidered as an expli it geometri proof of Breen's theorem in the smooth ontext. Indeed, one of the main motivations behind the present G paperistherelation between -extensionsand2-group prin ipal bundles. Webe- lieve our result throws a bridge between the groupoid extension approa h to the G di(cid:27)erential geometry of -gerbes developed in [26℄ and the one based on higher gauge theory due to Baez-S hreiber [4℄. This will be investigated somewhere else. G G Animportant lassof -extensionsisformedbytheso alled entral -extensions [G → Aut(G)] [26℄, those for whi h the stru ture 2-group redu es to the 2-group [Z(G) → 1] Z(G) G G (where stands for the enter of ). They orrespond to - G gerbes with trivial band or -bound gerbes [26℄. Ea h su h extension determines [Z(G) → 1] Γ a prin ipal -bundle over the base groupoid . In [7℄, Behrend-Xu H3(Γ) S1 gave a natural onstru tion asso iating a lass in to a entral -extension Γ of a Lie groupoid . When the base Lie groupoid is Morita equivalent to a S1 smooth manifold (viewed as a trivial 2-groupoid), a entral -extension is what has been studied by Murray and Hit hin under the name bundle gerbe [25,35℄. The Behrend-Xu lass of a bundle gerbe oin ides with its Dixmier-Douady lass, 3 whi h an be des ribed by the - urvature. In the present paper, we extend DD ∈ (α) the onstru tion of Behrend-Xu and de(cid:28)ne a Dixmier-Douady lass H3(Γ)⊗Z(g) G G for any entral -extension, where is a onne ted redu tive Lie G [Z(G) → 1] group. Sin e a entral -extension indu es a -prin ipal bundle over 3 Γ H3([Z(G) → 1]) → H3(Γ) , there is also a harateristi map . Dualizing, one CC ∈ H3(Γ)⊗Z(g) φ obtains a lass . We prove that the Dixmier-Douady lass DD CC (α) φ oin ides with the hara teristi lass . In a ertain sense, this is the gerbe analogue of the Chern-Weil isomorphism for prin ipal bundles [17,32℄. The paper is organized as follows. Se tion 2 is on erned with generalized morphisms of Lie 2-groupoids and with 2-group bundles and re alls some standard material on Lie 2-groupoids. The main feature of Se tion 3 is Theorem 3.4 on the Ad G [G −−→ Aut(G)] equivalen e of groupoids -extensions and prin ipal -bundles. In Se tion4wede(cid:28)nethe hara teristi map/ lassesofprin ipalLie2-groupbundles, G we present the onstru tion of the Dixmier-Douady lasses of groupoid entral - G extensions and we prove that the Dixmier-Douady lass of a entral -extension [Z(G) → 1] oin ides with the universal hara teristi lassof theindu ed -bundle (cid:22) see Theorem 4.14. G G Note that, when is dis rete, the relation between groupoid -extensions and 2-group prin ipal bundles was also independently studied by Hae(cid:29)iger [22℄. Some of the results of the present paper are related to results announ ed by Baez Stevenson [5℄. Re ently, Sati, Stashe(cid:27) and S hreiber have studied hara teristi L ∞ lasses for 2-group bundles by the mean of -algebras [36℄. It would be very L ∞ interesting to relate their onstru tion to ours using integration of -algebras as in [19,23℄. A knowledgments The authors are grateful to André Hae(cid:29)iger, Jim Stashe(cid:27), Urs S hreiber and Ping Xu for many useful dis ussions and suggestions. 2 Generalized morphisms and prin ipal Lie 2-group bundles 2.1 Lie 2-groupoids, Crossed modules and Morita morphisms This se tion is on erned with Lie 2-groupoids and Morita equivalen es. The material is rather standard. For instan e, see [31,34℄ for the general theory of Lie groupoids and [3,42℄ for Lie 2-groupoids. A Lie 2-groupoid is a double Lie groupoid s Γ2 //// Γ0 t u l id id (1) (cid:15)(cid:15) (cid:15)(cid:15) s (cid:15)(cid:15) (cid:15)(cid:15) Γ1 //// Γ0 t id Γ0 ////Γ0 in the sense of [12℄, where the right olumn id denotes the trivial Γ 0 groupoid asso iated to the smooth manifold . It makes sense to use the sym- s t Γ ⇉ Γ 2 0 bols and to denote the sour e and target maps of the groupoid sin e s◦l = s◦u t◦l = t◦u and . Remark 2.1. A Lie 2-groupoid isthus a small 2- ategory in whi h all arrows are invertible, the sets of obje ts, 1-arrows and 2-arrows are smooth manifolds, all stru turemapsaresmoothandthesour esandtargetsaresurje tivesubmersions. 4 l s Γ2 ////Γ1 ////Γ0 In the sequel, the 2-groupoid (1) will be denoted u t . The so l Γ2 ////Γ1 alledverti al(resp. horizontal)multipli ationinthegroupoid u (resp. s Γ2 //// Γ0 ⋆ ∗ t ) will be denoted by (resp. ) id s Γ1 //// Γ1 //// Γ0 Clearly, a Lie groupoid an be seen as a Lie 2-groupoid id t . Γ ∗ 0 A Lie 2-groupoid where is the one-point spa e is known as a Lie 2-group. There is a well-known equivalen e between Lie 2-groupoids and rossed modules of groupoids [12℄. A rossed module of groupoids is a morphism of groupoids ρ X //Γ 1 1 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) X //Γ 0 = 0 X = Γ X ⇉ X 0 0 1 0 whi h is the identity on the base spa es (i.e. ) and where is a family of groups (i.e. sour e and target maps oin ide), together with a right (γ,x) 7→ xγ Γ X a tion by automorphisms of on satisfying: ρ(xγ) = γ−1ρ(x)γ ∀(x,γ)∈ X × Γ , 1 Γ0 1 (2) xρ(y) = y−1xy ∀(x,y)∈ X × X . 1 Γ0 1 (3) X 1 Note that the equalities (2) and (3) make sense be ause is a family of groups. G Example 2.2. Given any Lie group , we obtain a rossed module by setting N = G Γ = Aut(G) Γ = ∗ ρ(g) = Ad g 1 1 0 g , , and (the onjugation by ). Γ ⇉ Γ 1 0 Example 2.3. ALiegroupoid indu esa rossedmoduleinthefollowing S = {x ∈ Γ /s(x) = t(x)} Γ S Γ 1 1 Γ way. Let be the set of losed loops in . Then Γ Γ S 0 1 Γ is a family of groups over and a ts by onjugation on . Therefore, we obtain a rossed module i S //Γ Γ 1 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) Γ //Γ 0 = 0 i where is the in lusion map. l s Γ2 ////Γ1 ////Γ0 A 2-groupoid u t determines a rossed module of groupoids (G −→ρ H) H = Γ ⇉ Γ G = {g ∈ Γ |l(g) ∈ Γ ⊂ Γ } ρ(g) = 1 0 1 2 0 1 as follows. Here , , u(g) H = Γ G ⊂ Γ 1 1 1 2 and the a tion of on is by onjugation. More pre isely, if l 1h h Γ2 //// Γ1 gh = 1h−1∗g∗1h istheunitoveranobje t inthegroupoid u ,then . ρ X −→ Γ Conversely, givena rossedmoduleofgroupoids ,onegetsaLie2-groupoid X1⋉Γ1 l //// Γ1 s //// Γ0 X1 ⋉Γ1 ⇉ Γ1 u t , where is the transformation groupoid X ⋉Γ ⇉ Γ 1 1 0 and is the semi-dire t produ t of groupoids. More pre isely, for all x,x′ ∈ X γ,γ′ ∈ Γ 1 1 and , the stru tures maps are de(cid:28)ned by l(x,γ) = γ, (x′,γ′)∗(x,γ) = (x′xγ′−1,γ′γ), u(x,γ) = ρ(x)γ, (x′,ρ(x)γ)⋆(x,γ) = (x′x,γ). 5 In thρe sequel, weρwill denote the Lie 2-groupoid asso iated to the rossed module (G −→ H) [G −→ H] by . Ad G −−→ Aut(G) Example 2.4. The rossedmoduleofgroups(cid:0) (cid:1)yieldsthe2-group G⋉Aut(G) l ////Aut(G) ////∗ u with stru ture maps l(g,ϕ) = ϕ u(g,φ) = Ad ◦ϕ g (g ,Ad ◦ϕ )⋆(g ,ϕ ) = (g g ,ϕ ) 1 g2 2 2 2 1 2 2 (g ,ϕ )∗(g ,ϕ )= (g ϕ (g ),ϕ ◦ϕ ) 1 1 2 2 1 1 2 1 2 φ Γ −→ ∆ (φ ,φ ,φ ) 0 1 2 A (stri t) morphism of Lie 2-groupoids is a triple of smooth φ : Γ → ∆ i = 0,1,2 i i i maps ( ) ommuting with all stru ture maps. Morphisms of rossed modules are de(cid:28)ned similarly. ∆ f : M → ∆ 0 Let be a Lie 2-groupoid. Given a surje tive submersion , we an l s ∆[M] : ∆2[M] ////∆1[M] ////M form the pullba k Lie 2-groupoid u t , where ∆ [M] = {(m,γ,n) ∈ M ×∆ ×M s(γ) = f(m), t(γ)= f(n)}, i= 1,2. i i s.t. s,t The maps are the proje tions on the (cid:28)rst and last fa tor respe tively. The u,l maps , the horizontal and verti al multipli ations are indu ed by the ones on ∆ as follows: u(m,γ,n) = (m,u(γ),n), (m,γ,n)∗(n,γ′,p)= (m,γ ∗γ′,p), l(m,γ,n) = (m,l(γ),n), (m,γ,n)⋆(m,γ′,n)= (m,γ ⋆γ′,n). ∆[M] → ∆ m 7→ f(m) There is a natural map of groupoids de(cid:28)ned by and (m,γ,n) 7→ γ . Pullba k of 2-groupoids yield a onvenient de(cid:28)nition of Morita morphism of Lie 2-groupoids. φ Γ −→ ∆ De(cid:28)nition 2.5. A morphism of Lie 2-groupoids is a Morita morphism φ if is the omposition of two morphisms Γ2 //∆2[Γ0] //∆2 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) Γ1 //∆1[Γ0] //∆1 Γ(cid:15)(cid:15) (cid:15)(cid:15) id // Γ(cid:15)(cid:15) (cid:15)(cid:15) φ0 //∆(cid:15)(cid:15) (cid:15)(cid:15) 0 0 0 Γ → ∆ Γ → ∆ [Γ ] 0 0 1 1 0 su h that and are surje tive submersions and Γ //∆ [Γ ] 2 2 0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) Γ // ∆ [Γ ] 1 1 0 is a Morita morphism of 1-groupoids. 6 The (weakest) equivalen e relation generated by the Morita morphism is alled Γ ∆ Morita equivalen e. More pre isely, two Lie 2-groupoids and are Morita E ,E ,...,E 0 1 n equivalent if there exists a (cid:28)nite olle tion of Lie 2-groupoids with E = Γ E = ∆ i ∈ {1,...,n} 0 n and , and, for ea h , either a Morita morphism E → E E → E i−1 i i i−1 or a Morita morphism . In fa t, by Lemma 2.12(d), one has the following well-known lemma. Γ ∆ Lemma 2.6. If and are Morita equivalent, there exits a hain of Morita Γ ← E → ∆ Γ ∆ morphisms of length 2 in between and . φ :Γ → ∆ Remark 2.7. In the ategori al point of view, a Morita morphism is in parti ular a 2-equivalen e of 2- ategories preserving the smooth stru tures. Remark 2.8. Similar to [7℄, one an de(cid:28)ne di(cid:27)erentiable 2-sta ks. Two Lie 2- groupoids de(cid:28)ne the same di(cid:27)erentiable 2-sta k if, and only if, they are Morita equivalent. Infa taLie2-groupoid anbethoughtofasa hoi eofadi(cid:27)erentiable atlas on a di(cid:27)erentiable 2-sta k. 2.2 Generalized morphisms of Lie 2-groupoids Generalized morphisms of Lie 2-groupoids are a straightforward generalization of generalized morphisms of Lie (1-)groupoids [24,34℄. They also have been onsid- 2Gpd ered in [42℄. Let denote the ategory of Lie 2-groupoids and morphisms of F Lie 2-groupoids. A generalized morphism is a zigzag ∼ ∼ Γ ←− E → ... ←− E → ∆, 1 n F : where all leftward arrows are Morita morphisms. We use a squig arrow Γ ∆ to denote a generalized morphism. The omposition of two generalized morphismsisde(cid:28)nedbythe on atenationoftwozigzags. Infa tweareinterested in equivalen e lasses of generalized morphisms: f : Γ → ∆ In the sequel, we will onsider two morphisms of 2-groupoids and g : Γ → ∆ ϕ : Γ → ∆ 0 1 to be equivalent if there exists two smooth appli ations ψ : Γ → ∆ x ∈ Γ 1 2 2 and su h that, for any and any pair of omposable arrows i,j ∈ Γ 1 , the following relations are satis(cid:28)ed: g (x)∗1 ⋆ψ(l(x)) = ψ(u(x))⋆ 1 ∗f (x) , 2 ϕ(s(x)) ϕ(t(x)) 2 (cid:0) (cid:1) (cid:0) (cid:1) (4) ψ(j ∗i) = 1 ∗ψ(i) ⋆ ψ(j)∗1 . (cid:0) g1(j) (cid:1) (cid:0) f1(i)(cid:1) (5) f g ψ In other words, and are (cid:16) onjugate(cid:17) by a (invertible) map ompatible with the horizontal multipli ation. It is easy to he k that the onditions (4) and (5) are equivalent to the data f g of a natural 2-transformation from to [8,30℄. Re all that a natural 2- ϕ(m) ∈ ∆ 1 transformation is given by the following data: an arrow for ea h m ∈ Γ ψ(γ) ∈ ∆ γ ∈ Γ 0 2 1 obje t , and a 2-arrow for ea h arrow as in the diagram ϕ(s(j)) f(s(j)) //g(s(j)) tttttt5= f(j) ψ(j) g(j) (cid:15)(cid:15) tttttt (cid:15)(cid:15) f(t(j)) // g(t(j)) ϕ(t(j)) and satisfying obvious ompatibility onditions with respe t to the ompositions of arrows and 2-arrows. 7 We now introdu e the notion of equivalen e of generalized morphisms; it is the natural equivalen e relation on generalized morphism extending the equivalen e of groupoids morphisms. Namely, we onsider the weakest equivalen e relation satisfying the following three properties: f,g (a) If there exists a natural transformation between a pair of homomor- f g phisms of 2-groupoids, and are equivalent as generalized morphisms. φ Γ −→ ∆ (b) If is a Morita morphism of 2-groupoids, the generalized morphisms φ φ φ φ id id ∆ ←− Γ −→ ∆ Γ −→ ∆ ←− Γ ∆ −→ ∆ Γ −→ Γ and are equivalent to and , respe tively. ( ) Pre- and post- omposition with a third generalized morphism preserves the equivalen e. F1 : Γ ←φ−1 E1 −f→1 ∆ F2 : Γ ←φ−2 E2 −f→2 ∆ Example 2.9. Let and ε be two E1 −→ E2 generalized morphisms. Suppose that there exists a morphism su h that the diagram E1 xxqqφq1qqq MMMfM1MM&& Γ ε ∆ ffMMφM2MMM E(cid:15)(cid:15)2 qqqfq2qq88 F F 1 2 ommutes up to 2-transformations. Then and are equivalent generalized morphisms. ∼ Γ ←− EExa−→m∼ p.l.e.2←.∼−10E. By−→∼its∆very de(cid:28)nition, a Morita equivalen e oFf g:roΓup oid∆s 1 n de(cid:28)nes two generalized morphisms and G : ∆ Γ F ◦ G G ◦ F . The ompositions and are both equivalent to the F G identity. Furthermore, the equivalen e lasses of and are independent of the hoi e of the Morita equivalen e. Remark 2.11. Roughly speaking, generalized morphisms are obtained by for- mally inverting the Morita morphisms. In fa t, the following Lemma is easy to he k. M Lemma 2.12. The olle tion of all Morita morphisms of 2-groupoids is a left 2Gpd multipli ative system [28,De(cid:28)nition 7.1.5;41,De(cid:28)nition 10.3.4℄ in . Indeed, the following properties hold: id (Γ −→ Γ) ∈ M ∀Γ∈ 2Gpd (a) , ; M (b) is losed under omposition; f φ ψ g Γ −→ ∆ ←− E 2Gpd φ ∈ M Γ ←− Z −→ E ( ) given ∼ in with , there exists ∼ in 2Gpd ψ ∈ M with su h that Z wwppψpp∼pp NNNNgNN'' Γ E NNfNNNN'' wwpp∼ppφpp ∆ ommutes; φ f (d) given Γ ∼ //∆ g //// E in 2Gpd with φ ∈ M, f◦φ= g◦φ implies f = g. 8 M 2Gpd Sin e isaleftmultipli ative systeminthe ategory ,we an onsiderthe 2Gpd 2Gpd M M lo alization of withrespe tto [28,Chapter7;41,Se tion10.3℄. 2Gpd 2Gpd M This new ategory has the same obje ts as but its arrows are 2Gpd M equivalen e lasses of generalized morphisms. An isomorphism in orre- 2Gpd sponds to (the equivalen e lass of) a Morita equivalen e in . Lemma 2.12(C) implies that any generalized morphism an be represented by a hain of length 2: Γ ∆ Lemma 2.13. Any generalized morphism between two Lie 2-groupoids and is equivalent to a diagram φ f Γ ←− E −→ ∆ 2Gpd φ φ∈ M in the ategory su h that is a Morita morphism (i.e. ). Remark 2.14. There is a bije tion between maps of di(cid:27)erentiable 2-sta ks and equivalen e lasses of generalized morphisms of Lie 2-groupoids up to Morita equivalen es. 2.3 2-group bundles In this se tion, we give a de(cid:28)nition of 2-group bundles of a global nature and formulated in terms of generalized morphisms of Lie 2-groupoids. [G → H] De(cid:28)nition 2.15. A prin ipal (2-group) -bundle over a Lie groupoid Γ ⇉ Γ B Γ ⇉ Γ 1 0 1 0 is a generalized morphism from (seen as a Lie 2-groupoid) [G → H] (G → H) to the 2-group asso iated to the rossed module . [G → Aut(G)] Γ ⇉ Γ 1 0 In parti ular, a prin ipal -bundle over a groupoid is Γ ⇉ Γ 1 0 a generalized morphism from (seen as a 2-groupoid) to the 2-group [G→ Aut(G)] . [G → H] B B′ Γ ⇉ Γ 1 0 Two prin ipal -bundles and over the groupoid are said to be isomorphi if, and only if, these two generalized morphisms are equivalent. [G → H] M [G → H] A -bundle over a manifold is a (2-group) -bundle over the M ⇉ M groupoid . B [G → H] Γ ⇉ Γ Γ′ ⇉ Γ′ Let be a -bundle over a Lie groupoid 1 0. If 1 0 and [G′ → H′] Γ ⇉ Γ [G → H] 1 0 are Morita equivalent to and respe tively, then the omposition Γ′ ⇉ Γ′ ! Γ ⇉ Γ B [G → H]! [G′ → H′]. (cid:0) 1 0(cid:1) (cid:0) 1 0(cid:1) [G′ → H′] Γ′ ⇉ Γ′ B de(cid:28)nes a prin ipal -bundle over 1 0 denoted by abuse of no- tation. Here the left and right squig arrows are the Morita equivalen es seens as invertible generalized morphisms as in Example 2.10. [G → H] B De(cid:28)nition 2.16. A prin ipal (2-group) -bundle over a Lie groupoid Γ ⇉ Γ [G′ → H′] B′ 1 0 and a prin ipal (2-group) -bundle over a Lie groupoid Γ′ ⇉ Γ′ B 1 0 aresaidtobe Moritaequivalent if, and onlyif, (viewed as ageneralized Γ′ ⇉ Γ′ [G′ → H′] B′ morphism(cid:0) 1 0(cid:1) ) and are equivalent generalized morphisms. B B′ Γ ⇉ Γ [G → H] 1 0 In parti ular, if and are Morita equivalent, and are Γ′ ⇉ Γ′ [G′ → H′] 1 0 Morita equivalent to and respe tively. 9 Remark 2.17. When thegroupoid isjustamanifold, our de(cid:28)nition isequivalent totheusualde(cid:28)nitionof2-groupbundlesin[4,6,36℄assuggestedbyExamples2.18 and 2.19 below. π P −→ M H Example 2.18. Let be a prin ipal -bundle. Then the diagram M oo P × P //H M (cid:15)(cid:15) (cid:15)(cid:15) φ (cid:15)(cid:15) (cid:15)(cid:15) f (cid:15)(cid:15) (cid:15)(cid:15) M oo P × P //H M t s (cid:15)(cid:15) (cid:15)(cid:15) π (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) M oo P //∗ s(x,y) = x t(x,y) = y π(x) = φ(x,y) = π(y) x·f(x,y) = y where , , and , de(cid:28)nes a M 2 [1 → H] generalized morphism from the manifold and to the -group . Hen e, 2 M H P M it is a -group bundle over . Note that a prin ipal -bundle over is H′ P′ M′ Morita equivalent (as a 2-group bundle) to a prin ipal -bundle over if, H M H′ M′ P P′ and only if, and are isomorphi to and respe tively and and are isomorphi prin ipal bundles. M G Example 2.19. Let be a smooth manifold and be a (non abelian) Lie M G group. A non abelian 2- o y le [15,16,21,33℄ on with values in relative to {U } M an open overing i i∈I of is a olle tion of smooth maps λ : U → Aut(G) g :U → G ij ij ijk ijk and satisfying the following relations: λ ◦λ = Ad ◦λ ij jk gijk ik g g = g λ−1(g ). ijl jkl ikl kl ijk [G → Aut(G)] Su h a non abelian 2- o y le de(cid:28)nes a -bundle over the manifold M ; for it an be seen as the generalized morphism M oo ì,jUij ×G×G f //G⋉Aut(G) u l (cid:15)(cid:15) (cid:15)(cid:15) φ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) M oo ì,jUij ×G //Aut(G) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) M oo ìUi //∗ M [G → Aut(G)] between the manifold and the 2-group . Here l(x ,g ,g ) = (x ,g ) φ(x ,g) = x ij 1 2 ij 1 ij u(x ,g ,g ) = (x ,g ) f(x ,g ,g )= g g−1,Ad ◦λ (x) ij 1 2 ij 2 ij 1 2 (cid:0) 2 1 g1 ij (cid:1) x x ∈ M U = U ∩U ij ij i j where denotes a point seen asa point of the open subset , x x ∈ M U g,g ,g i i 1 2 the point seen as a point of the open subset , and arbitrary G elements of . The horizontal and verti al multipli ation are given by (x ,α)∗(x ,β) = x ,g λ−1(α)β , ij jk (cid:0) ik ijk jk (cid:1) (x ,g ,g )⋆(x ,g ,g )= (x ,g ,g ). ij 1 2 ij 2 3 ij 1 3 10