G-gerbes, principal 2-group bundles and characteristic classes

G -GERBES, PRINCIP AL 2-GR OUP BUNDLES AND CHARA CTERISTIC CLASSES GRÉGOR Y GINOT AND MA THIEU STIÉNON Abstra ct. Let G b e a Lie group and G → Aut( G ) b e the canonical group homomorphism induced b y the adjoint action of a group on itself. W e giv e an explicit description of a 1-1 corresp ondence betw een Morita equiv alence classes of, on the one hand, principal 2-group [ G → Aut( G )] - bundles o ver Lie group oids and, on the other hand, G -extensions of Lie group oids (i.e. b et ween principal [ G → Aut( G )] -bundles ov er diﬀeren- tiable stacks and G -gerb es ov er diﬀerentiable stacks). This approach also allows us to iden tify G -b ound gerb es and [ Z ( G ) → 1] -group bundles o ver diﬀerentiable stacks, where Z ( G ) is the cen ter of G . W e also intro- duce univ ersal characteristic classes for 2-group bundles. F or group oid cen tral G -extensions, w e introduce Dixmier–Douady classes that can b e computed from conn ection-t yp e data generalizing the ones for bundle gerb es. W e prov e that these classes coincide with universal character- istic classes. As a corollary , w e obtain further that Dixmier–Douady classes are in tegral. 1. Intr oduction This pap er is devoted to the relation b et ween group oid G -extensions and princip al Lie 2-gr oup bund les and to their characteristic classes. A Lie 2-gr oup is a Lie group oid Γ 2 ⇒ Γ 1 , whose spaces of ob jects Γ 1 and of morphisms Γ 2 are Lie groups and all of whose structure maps are group morphisms. One shall note that in this pap er w e are interested in strict Lie 2-groups only , though we b eliev e all our results can b e extended to w eak ones as well. A cr osse d mo dule ( G ρ − → H ) is a Lie group morphism G ρ − → H together with an action of H on G satisfying suitable compatibility conditions. It is standard that Lie 2-groups are in bijection with crossed mo dules [3, 15]. In this pap er, [ G ρ − → H ] denotes the 2-group corresp onding to the crossed mo dule ( G ρ − → H ) . Lie 2-groups arise naturally in mathematical physics. F or instance, in higher gauge theory [2, 4], Lie 2-group bundles pro vide a well suited frame- w ork for describing the parallel transp ort of strings [1, 4, 37]. Several recent w orks ha v e approached the concept of bundles with a “structure Lie 2-group” o ver a manifold from v arious p ersp ectives [1, 4, 6, 50, 58]. Here we take an Researc h supp orted b y the European Union through the FP6 Marie Curie R.T.N. ENIGMA (Con tract num ber MR TN-CT-2004-5652). 1 2 GRÉGOR Y GINOT AND MA THIEU STIÉNON alternativ e p oint of view and giv e a deﬁnition of principal Lie 2-group bun- dles of a global nature (i.e. not resorting to a description explicitly in volving lo cal c harts and co cycles) and which allo ws for the base space to b e a Lie group oid. In other words, w e consider 2-group principal bundles o ver diﬀer- en tiable stacks [9]. Our approach immediately leads to a natural construction of “univ ersal c haracteristic classes” for principal 2-group bundles. Let us start with Lie (1-)groups. A principal G -bundle P ov er a manifold M canonically determines a homotop y class of maps from M to the classi- fying space B G of the group G . In fact, the set of isomorphism classes of G -principal bundles ov er M is in bijection with the set of homotopy classes of maps M f − → B G [17, 53, 54]. Pulling bac k the generators of H ∗ ( B G ) (the uni- v ersal classes) through f , one obtains characteristic classes of the principal bundle P ov er M . These c haracteristic classes coincide with those obtained from a connection by applying the Chern–W eil construction [21, 42]. There is an analogue but muc h less known, diﬀerential geometric rather than purely top ological, p oin t of view: a principal G -bundle ov er a manifold M can b e thought of as a “generalized morphism” (in the sense of Hilsum & Sk andalis [30]) from the manifold M to the Lie group G b oth considered as 1-group oids. T o see this, recall that a principal G -bundle can b e deﬁned as a collection of transition functions g ij : U ij → G on the double in tersections U ij of some open co v ering { U i } i ∈ I of M , satisfying the co cycle condition g ij g j k = g ik . These transition functions constitute a morphism of group oids from the Čec h group oid ` U ij ⇒ ` U i asso ciated to the op en cov ering { U i } i ∈ I to the Lie group G ⇒ ∗ . Hence we ha ve a diagram ( M ⇒ M ) ∼ ← − ( a U ij ⇒ a U i ) → ( G ⇒ ∗ ) in the category of Lie group oids and their morphisms whose leftw ard arrow is a Morita equiv alence, in other words a generalized morphism from the manifold M to the Lie group G . This second p oin t of view, or more precisely its generalization to the 2- group oid con text, constitutes the foundation on whic h our approach is built. The generalization of the concept of “generalized morphism” to 2-group oids is straigh tforw ard: a generalized morphism of Lie 2-group oids Γ ∆ is a diagram Γ φ ← − ∼ E f − → ∆ in the category 2Gpd of Lie 2-group oids and their morphisms, where φ is a Morita equiv alence (a “smo oth” equiv alence of 2-group oids). It is sometimes useful to think of tw o Morita equiv alen t Lie 2-group oids as tw o diﬀerent choices of an atlas (or op en cov er) on the same geometric ob ject (whic h is a diﬀerentiable 2-stack [9, 14]). W e deﬁne a prin- cipal [ G ρ − → H ] -bundle ov er a Lie group oid Γ to b e a generalized morphism from Γ to [ G ρ − → H ] (up to equiv alence). See Section 2.3. The concept of (geometric) nerv e of Lie group oids extends to the 2- categorical con text as a functor from the category of Lie 2-groupoids to the category of simplicial manifolds [55]. By conv en tion, the cohomology of a 2-group oid is the cohomology of its nerve, which can b e computed via a PRINCIP AL 2-GROUP BUNDLES 3 double complex (for instance, see [25]). Crucially , Morita equiv alences in- duce isomorphisms in cohomology . Therefore, any generalized morphism of 2-group oids Γ F [ G → H ] deﬁning a principal [ G → H ] -bundle B o v er the group oid Γ yields a pullbac k homorphism F ∗ : H • ([ G → H ]) → H • (Γ) in cohomology , which is called the c ohomolo gy char acteristic map (c haracter- istic map for short). The cohomology classes in H • ([ G → H ]) [25] should b e viewed as univ ersal characteristic classes and their images by F ∗ as the c haracteristic classes of B . Lie 2-group principal bundles are closely related to non-abelian gerb es. Geometrically , non-ab elian G -gerb es ov er diﬀerentiable stacks can b e con- sidered as groupoid G -extensions mo dulo Morita equiv alence [32]. By a group oid G -extension, we mean a short exact sequence of group oids 1 → M × G i − → ˜ Γ φ − → Γ → 1 , where M × G is a bundle of groups. One of our main results is an equiv alence b etw een (strict) G -gerb es ov er a diﬀerentiable stack and principal bundles ov er the 2-group [ G → Aut( G )] . More precisely , w e establish an explicit 1-1 correspondence betw een group oid G -extensions up to Morita equiv alence (i.e. strict G -gerb es o v er diﬀeren tiable stac ks) and principal [ G → Aut( G )] -bundles o ver Lie groupoids mo dulo Morita equiv alence (i.e. [ G → Aut( G )] -principal bundles ov er diﬀeren tiable stac ks). This is Theorem 3.4. Note that a restricted version of this corre- sp ondence is highlighted in [33, Theorem 4]. It is kno wn that Giraud’s second non abelian cohomology group H 2 ( X , G ) classiﬁes the G -gerbes o v er a diﬀerentiable stack X [26] while Dedec k er’s H 1 ( X , [ G → Aut( G )]) classiﬁes the principal [ G → Aut( G )] -bundles [18, 19]. In [12, 13], Breen show ed that these tw o cohomology groups are isomorphic. In some sense, our theorem ab ov e can b e considered as an explicit geometric pro of of Breen’s theorem in the smo oth context. Indeed, one of the main motiv ations b ehind the presen t paper is the relation b et w een G -extensions and 2-group principal bundles. W e b eliev e our result thro ws a bridge b et ween the group oid extension approach to the di ﬀerential geometry of G -gerb es dev elop ed in [32] and the one based on higher gauge theory due to Baez & Sc hreib er [4]. This will b e inv estigated somewhere else. An imp ortant class of G -extensions is formed b y the so called c entr al G - extensions [32], those for whic h the structure 2-group [ G → Aut( G )] reduces to the 2-group [ Z ( G ) → 1] (where Z ( G ) stands for the center of G ). They corresp ond to G -gerb es with trivial band or G -b ound gerb es [32]. Eac h such extension determines a principal [ Z ( G ) → 1] -bundle o ver the base group oid Γ . In [9], Behrend & Xu gav e a natural construction asso ciating a class in H 3 (Γ) to a central S 1 -extension of a Lie group oid Γ . When the base Lie group oid is Morita equiv alen t to a smo oth manifold (view ed as a trivial 2- group oid), a cen tral S 1 -extension is what has been studied by Murray and Hitc hin under the name bundle gerb e [31, 45]. The Behrend–Xu class of a bundle gerb e coincides with its Dixmier–Douady class , whic h can b e de- scrib ed b y the 3 -curv ature. In the present pap er, we extend the construction 4 GRÉGOR Y GINOT AND MA THIEU STIÉNON of Behrend & Xu and deﬁne a Dixmier–Douady class D D ( α ) ∈ H 3 (Γ) ⊗ Z ( g ) for any central G -extension, where G is connected with a reductive Lie al- gebra. Since a central G -extension induces a [ Z ( G ) → 1] -principal bun- dle ov er Γ , there is also a charateristic map H 3 ([ Z ( G ) → 1]) → H 3 (Γ) . Dualizing, one obtains a class C C φ ∈ H 3 (Γ) ⊗ Z ( g ) . W e prov e that the Dixmier–Douady class D D ( α ) coincides with the characteristic class C C φ . In a certain sense, this is the gerb e analo gue of the Chern–W eil isomorphism for principal bundles [21, 42]. The pap er is organized as follows. Section 2 is dev oted to generalized mor- phisms of Lie 2-group oids and to 2-group bundles and recalls some standard material on Lie 2-group oids. The main feature of Section 3 is Theorem 3.4 on the equiv alence of group oids G -extensions and principal [ G Ad − − → Aut( G )] - bundles. In Section 4 we deﬁne the characteristic map/classes of principal Lie 2-group bundles, we presen t the construction of the Dixmier–Douady classes of group oid cen tral G -extensions and we prov e that the Dixmier–Douady class of a central G -extension coincides with the univ ersal c haracteristic class of the induced [ Z ( G ) → 1] -bundle — see Theorem 4.18. Since the universal c haracteristic map can b e deﬁned in cohomology with in teger coeﬃcients, w e obtain that the Dixmier–Douady class of a central G -extension is in tegral as a corollary of our study . This applies, in particular, in the classical case of a bundle gerb e. Note that, when G is discrete, the relation b etw een group oid G -extensions and 2-group principal bundles w as also indep endently studied by Haeﬂi- ger [28]. Some of the results of the present pap er are related to results announced b y Baez & Stevenson [5]. Recently , Sati, Stasheﬀ & Schreiber hav e studied c haracteristic classes for 2-group bundles b y the mean of L ∞ -algebras [50]. It w ould b e very interesting to relate their construction to ours using in tegration of L ∞ -algebras as in [23, 29]. 2. Generalized morphisms and princip al Lie 2-group bundles 2.1. Lie 2-group oids, Crossed mo dules and Morita morphisms. This section is concerned with Lie 2-group oids and Morita equiv alences. The material is rather standard. F or instance, see [39, 44] for the general theory of Lie group oids and [3, 60] for Lie 2-group oids. W e only deal with the case of strict 2-group oids and strict 2-groups. Deﬁnition 2.1. A Lie 2-gr oup oid is a double Lie group oid Γ 2 s / / t / / l   u   Γ 0 id   id   Γ 1 s / / t / / Γ 0 (1) PRINCIP AL 2-GROUP BUNDLES 5 in the sense of [15], where the righ t column Γ 0 id / / id / / Γ 0 denotes the trivial group oid asso ciated to the smo oth manifold Γ 0 . It makes sense to use the sym b ols s and t to denote the source and target maps of the group oid Γ 2 ⇒ Γ 0 since s ◦ l = s ◦ u and t ◦ l = t ◦ u . R emark 2.2 . A Lie 2-group oid is th us a small 2-category in which all ar- ro ws are inv ertible, the sets of ob jects, 1-arrows and 2-arrows are smo oth manifolds, all structure maps are smo oth and the sources and targets are surjectiv e submersions. In the sequel, the 2-group oid (1) will b e denoted Γ 2 l / / u / / Γ 1 s / / t / / Γ 0 or just Γ . The so called vertical (resp. horizontal) m ultiplication in the group oid Γ 2 l / / u / / Γ 1 (resp. Γ 2 s / / t / / Γ 0 ) will b e denoted by  (resp. ∗ ) Clearly , a Lie group oid Γ 1 s / / t / / Γ 0 can b e seen as a Lie 2-group oid Γ 1 id / / id / / Γ 1 s / / t / / Γ 0 . A Lie 2-groupoid where Γ 0 is the one-point space ∗ is known as a Lie 2-gr oup . There is a well-kno wn equiv alence b etw een Lie 2-group oids and crossed mo dules of group oids [15]. Deﬁnition 2.3. A cr osse d mo dule of group oids ( X ρ − → Γ) is a morphism of group oids X 1 ρ / / p   p   Γ 1 s   t   X 0 id / / Γ 0 , from a family of groups X 1 p − → X 0 to a groupoid Γ 1 ⇒ Γ 0 sharing the same unit space X 0 = Γ 0 , together with a right action by automorphisms ( γ , x ) 7→ x γ of Γ 1 ⇒ Γ 0 on X 1 → X 0 satisfying: ρ ( x γ ) = γ − 1 ρ ( x ) γ ∀ ( x, γ ) ∈ X 1 × Γ 0 Γ 1 , (2) x ρ ( y ) = y − 1 xy ∀ ( x, y ) ∈ X 1 × Γ 0 X 1 . (3) Note that the equalities (2) and (3) make sense b ecause X 1 is a family of groups. Example 2.4. Giv en any Lie group G , w e obtain a crossed mo dule by setting X 1 = G , Γ 1 = Aut( G ) , Γ 0 = ∗ and ρ ( g ) = Ad g (the conjugation by g ). 6 GRÉGOR Y GINOT AND MA THIEU STIÉNON Example 2.5. A Lie group oid Γ 1 ⇒ Γ 0 induces a crossed mo dule in the follo wing wa y . Let S Γ = { x ∈ Γ 1 | s ( x ) = t ( x ) } be the set of closed lo ops in Γ 1 . Then S Γ is a family of groups ov er Γ 0 and Γ 1 acts by conjugation on S Γ . Therefore, we obtain a crossed mo dule S Γ i / /     Γ 1     Γ 0 id / / Γ 0 where i is the inclusion map. A 2-group oid Γ 2 l / / u / / Γ 1 s / / t / / Γ 0 determines a crossed mo dule of group- oids ( G ρ − → H ) as follo ws. Here the group oid H is Γ 1 ⇒ Γ 0 , G 1 = { g ∈ Γ 2 | l ( g ) ∈ Γ 0 ⊂ Γ 1 } , ρ is the restriction of u to G 1 and the action of H 1 = Γ 1 on G 1 ⊂ Γ 2 is b y conjugation. More precisely , if 1 h is the unit o v er an ob ject h in the group oid Γ 2 l / / u / / Γ 1 , then g h = 1 h − 1 ∗ g ∗ 1 h . Conv ersely , given a crossed mo dule of group oids ( X ρ − → Γ) , one gets a Lie 2-group oid X 1 n Γ 1 l / / u / / Γ 1 s / / t / / Γ 0 , where X 1 n Γ 1 ⇒ Γ 1 is the transformation group oid and X 1 n Γ 1 ⇒ Γ 0 is the semi-direct pro duct of group oids. More precisely , for all x, x 0 ∈ X 1 and γ , γ 0 ∈ Γ 1 , the structures maps are deﬁned by l ( x, γ ) = γ , ( x 0 , γ 0 ) ∗ ( x, γ ) = ( x 0 x γ 0 − 1 , γ 0 γ ) , u ( x, γ ) = ρ ( x ) γ , ( x 0 , ρ ( x ) γ )  ( x, γ ) = ( x 0 x, γ ) . In the sequel, we will denote the Lie 2-group oid asso ciated to the crossed mo dule ( G ρ − → H ) b y [ G ρ − → H ] . Example 2.6. The crossed mo dule of groups  G Ad − − → Aut( G )  yields the 2-group G n Aut( G ) l / / u / / Aut( G ) / / / / ∗ with structure maps l ( g , ϕ ) = ϕ u ( g , φ ) = Ad g ◦ ϕ ( g 1 , Ad g 2 ◦ ϕ 2 )  ( g 2 , ϕ 2 ) = ( g 1 g 2 , ϕ 2 ) ( g 1 , ϕ 1 ) ∗ ( g 2 , ϕ 2 ) = ( g 1 ϕ 1 ( g 2 ) , ϕ 1 ◦ ϕ 2 ) A (strict) morphism Γ φ − → ∆ of Lie 2-group oids is a triple ( φ 0 , φ 1 , φ 2 ) of smo oth maps φ i : Γ i → ∆ i ( i = 0 , 1 , 2 ) comm uting with all structure maps. Morphisms of crossed mo dules are deﬁned similarly . PRINCIP AL 2-GROUP BUNDLES 7 Let ∆ b e a Lie 2-group oid. Given a surjective submersion f : M → ∆ 0 , w e can form the pul lb ack Lie 2-group oid ∆ [ M ] : ∆ 2 [ M ] l / / u / / ∆ 1 [ M ] s / / t / / M , where ∆ i [ M ] = { ( m, γ , n ) ∈ M × ∆ i × M s.t. s ( γ ) = f ( m ) , t ( γ ) = f ( n ) } , for i ∈ { 1 , 2 } . The maps s, t are the pro jections on the ﬁrst and last factor resp ectiv ely . The maps u, l , the horizon tal and vert ical multiplications are induced by the ones on ∆ as follows: u ( m, γ , n ) = ( m, u ( γ ) , n ) , ( m, γ , n ) ∗ ( n, γ 0 , p ) = ( m, γ ∗ γ 0 , p ) , l ( m, γ , n ) = ( m, l ( γ ) , n ) , ( m, γ , n )  ( m, γ 0 , n ) = ( m, γ  γ 0 , n ) . There is a natural map of group oids ∆ [ M ] → ∆ deﬁned by m 7→ f ( m ) and ( m, γ , n ) 7→ γ . Pullbac k of 2-group oids yield a con ve nient deﬁnition of a higher analogue for Lie 2-group oids of the notion of Morita morphism or w eak equiv alence of Lie 1-group oids. These maps are higher analogues of the notion of a cov er and thus are called hypercov ers to agree with the terminology of [59, 60]. Deﬁnition 2.7. A morphism of Lie 2-group oids Γ φ − → ∆ is a hyp er c over if φ is the comp osition of tw o morphisms Γ 2     / / ∆ 2 [Γ 0 ]     / / ∆ 2     Γ 1     / / ∆ 1 [Γ 0 ]     / / ∆ 1     Γ 0 id / / Γ 0 φ 0 / / ∆ 0 suc h that Γ 0 → ∆ 0 and Γ 1 → ∆ 1 [Γ 0 ] are surjective submersions and Γ 2     / / ∆ 2 [Γ 0 ]     Γ 1 / / ∆ 1 [Γ 0 ] is a Morita morphism 1 of 1-group oids. Deﬁnition 2.8. The (w eak est) equiv alence relation generated b y the h yp er- co vers is called Morita e quivalenc e . 2 More precisely , tw o Lie 2-group oids Γ 1 Here we follow the terminology of [9, 32]. Mo erdijk calls suc h maps weak equiv a- lences [44]. 2 By analogy with [9], w e will sometimes refer to a hypercov er as a Morita morphism since it induces a Morita equiv alence. 8 GRÉGOR Y GINOT AND MA THIEU STIÉNON and ∆ are Morita equiv alent if there exists a ﬁnite collection E 0 , E 1 , . . . , E n of Lie 2-group oids with E 0 = Γ and E n = ∆ , and, for each i ∈ { 1 , . . . , n } , either a hypercov er E i − 1 ∼ − → E i or a hypercov er E i ∼ − → E i − 1 . In fact, by Lemma 2.17 4), one has the following w ell-known lemma. Lemma 2.9. If Γ and ∆ ar e Morita e quivalent, ther e exits a chain of hy- p er c overs Γ ∼ ← − E ∼ − → ∆ of length 2 in b etwe en Γ and ∆ . R emark 2.10 . F rom the categorical point of view, a hyperco ver φ : Γ ∼ − → ∆ is in particular a 2-equiv alence of 2-categories preserving the smo oth structures. W e exp ect that the notion of Morita equiv alence in tro duced here will help shed light on the integration problem for Couran t algebroids [35] and more sp eciﬁcally on the relation b etw een the diﬀerent prop osed approac hes [34, 40, 52]. It is exp ected that the ob jects integrating Courant algebroids are symplectic 2-stacks [41]. R emark 2.11 . Similar to [9], one can deﬁne diﬀerentiable 2-stacks. T wo Lie 2-group oids deﬁne the same diﬀerentiable 2-stack if, and only if, they are Morita equiv alen t. In fact a Lie 2-group oid can b e though t of as a c hoice of a diﬀeren tiable atlas on a diﬀeren tiable 2-stac k. 2.2. Generalized morphisms of Lie 2-group oids. Generalized morph- isms of Lie 2-group oids are a straightforw ard generalization of generalized morphisms of Lie (1-)groupoids [30, 44]. They also ha v e b een considered in [60]. Let 2Gp d denote the category of Lie 2-group oids and morphisms of Lie 2-group oids. Deﬁnition 2.12. A gener alize d morphism F is a zigzag Γ ∼ ← − E 1 → . . . ∼ ← − E n → ∆ , where all leftw ard arrows are hypercov ers. W e use a squig arro w F : Γ ∆ to denote a generalized morphism. The comp osition of tw o generalized morphisms is deﬁned by the concatenation of tw o zigzags. In fact we are interested in equiv alence classes of generalized morphisms: In the sequel, w e will consider t wo morphisms of 2-group oids f : Γ → ∆ and g : Γ → ∆ to b e e quivalent if there exists t wo smo oth applications ϕ : Γ 0 → ∆ 1 and ψ : Γ 1 → ∆ 2 suc h that, for any x ∈ Γ 2 and an y pair of comp osable arrows i, j ∈ Γ 1 , the following relations are satisﬁed:  g 2 ( x ) ∗ 1 ϕ ( s ( x ))   ψ ( l ( x )) = ψ ( u ( x ))   1 ϕ ( t ( x )) ∗ f 2 ( x )  , (4) ψ ( j ∗ i ) =  1 g 1 ( j ) ∗ ψ ( i )    ψ ( j ) ∗ 1 f 1 ( i )  . (5) In other words, f and g are “conjugate” b y a (inv ertible) map ψ compatible with the horizontal multiplication. It is easy to c hec k that the conditions (4) and (5) are equiv alent to the data of a natural 2-transformation from f to g [11, 38]. Recall that a natural 2-transformation is giv en by the following data: an arro w ϕ ( m ) ∈ ∆ 1 for PRINCIP AL 2-GROUP BUNDLES 9 eac h ob ject m ∈ Γ 0 , and a 2-arrow ψ ( γ ) ∈ ∆ 2 for each arrow γ ∈ Γ 1 as in the diagram f ( s ( j )) ϕ ( s ( j )) / / f ( j )   g ( s ( j )) g ( j )   f ( t ( j )) ϕ ( t ( j )) / / ψ ( j ) 5 = g ( t ( j )) and satisfying obvious compatibility conditions with resp ect to the comp o- sitions of arrows and 2-arrows. W e no w introduce the notion of equiv alence of generalized morphisms; it is the natural equiv alence relation on generalized morphism extending the equiv alence of group oids morphisms. Deﬁnition 2.13. Equivalenc e of gener alize d morphisms is the weak est equiv- alence relation satisfying the following three prop erties: (1) If there exists a natural transformation betw een a pair f , g of ho- momorphisms of 2-group oids, f and g are equiv alent as generalized morphisms. (2) If Γ φ − → ∆ is a h yp erco ver of 2-group oids, the generalized morphisms ∆ φ ← − Γ φ − → ∆ and Γ φ − → ∆ φ ← − Γ are equiv alen t to ∆ id − → ∆ and Γ id − → Γ , resp ectively . (3) Pre- and p ost-comp osition with a third generalized morphism pre- serv es the equiv alence. Generalized morphisms can b e seen as the 1 -morphisms in a bicategory of fractions 2Gp d [ M − 1 ] , where w e ha v e “ formally in verted” the collection M of hypercov ers, and equiv alence b etw een generalized morphisms as b eing (in vertib le) 2-morphisms. W e refer to [36, Chapter 7] for details on lo caliza- tion of categories with resp ect to a multiplicativ e system (see Lemma 2.17 b elo w) and to [48, Chapter 3] and [49, Section 2] for details on the construc- tion of the tw o cells of the asso ciated bicategory of fractions. In particular, the 2 -morphisms in the bicategory of fractions 2Gp d [ M − 1 ] are represented b y diagrams F α  & φ 1 u u f 1 ) ) β x  Γ E ε 2   ε 1 O O ∆ G φ 2 i i f 2 5 5 in whic h φ i ◦ ε i are Morita morphisms (i.e. hypercov ers) and α , β are 2- transformations as ab ov e. See [48, Section 3.2.3] or [49, Section 2]. Ho wev er, we will most y only need the 1-category underlying 2Gp d [ M − 1 ] . 10 GRÉGOR Y GINOT AND MA THIEU STIÉNON Example 2.14. Let F 1 : Γ φ 1 ← − ∼ E 1 f 1 − → ∆ and F 2 : Γ φ 2 ← − ∼ E 2 f 2 − → ∆ b e t wo generalized morphisms. Supp ose that there exists a morphism E 1 ε − → E 2 suc h that the diagram E 1 φ 1 x x ε   f 1 & & Γ ∆ E 2 φ 2 f f f 2 8 8 comm utes up to 2-transformations (in particular, φ 2 ◦ ε is an hypercov er). Then F 1 and F 2 are equiv alen t generalized morphisms. Example 2.15. By its v ery deﬁnition, a Morita equiv alence of group oids Γ ∼ ← − E 1 ∼ − → . . . ∼ ← − E n ∼ − → ∆ deﬁnes tw o generalized morphisms F : Γ ∆ and G : ∆ Γ . The comp ositions F ◦ G and G ◦ F are b oth equiv alen t to the identit y . R emark 2.16 . As previously mentionned, generalized morphisms are ob- tained b y formally inv erting the Morita morphisms (i.e. h yp ercov ers). In fact, the following Lemma can b e chec k ed. L emma 2.17 . The c ol le ction M of al l hyp er c overs of Lie 2-gr oup oids is a left multiplic ative system [36, Deﬁnition 7.1.5; 56, Deﬁnition 10.3.4] in 2Gp d . Inde e d, the fol lowing pr op erties hold: (1) ( Γ id − → Γ ) ∈ M , ∀ Γ ∈ 2Gp d ; (2) M is close d under c omp osition; (3) given Γ f − → ∆ φ ← − ∼ E in 2Gp d with φ ∈ M , ther e exists Γ ψ ← − ∼ Z g − → E in 2Gp d with ψ ∈ M such that Z ψ ∼ w w g ' ' Γ f ' ' E φ ∼ w w ∆ c ommutes; (4) given Γ φ ∼ / / ∆ f / / g / / E in 2Gp d with φ ∈ M , f ◦ φ = g ◦ φ implies f = g . Pr o of. Prop erties 2) and 3) follow from [59, Theorem 2.12] or [60, Section 2]. Prop ert y 4) follows from the fact that the map φ 0 : Γ 0 → ∆ 0 is a surjective submersion and that φ is an equiv alence of 2 -categories.  Since M is a left multiplicativ e system in the category 2Gp d , we can consider the lo calization 2Gp d M of 2Gp d with resp ect to M [36, Chap- ter 7; 56, Section 10.3]. This new category 2Gp d M has the same ob jects as 2Gp d but its arrows are equiv alence classes of generalized morphisms. An PRINCIP AL 2-GROUP BUNDLES 11 isomorphism in 2Gp d M corresp onds to (the equiv alence class of ) a Morita equiv alence in 2Gp d . In particular, the category 2Gp d M is the 1 -category obtained from the bicategory 2Gp d [ M − 1 ] (by identifying all 1-morphisms that are connected b y 2-morphisms). Lemma 2.17 3) implies that any generalized morphism can b e represented b y a c hain of length 2: Lemma 2.18. Any gener alize d morphism b etwe en two Lie 2-gr oup oids Γ and ∆ is e quivalent to a diagr am Γ φ ← − ∼ E f − → ∆ in the c ate gory 2Gp d in which φ is a hyp er c over (i.e. φ ∈ M ). R emark 2.19 . There is a bijection b etw een maps of (representable) diﬀer- en tiable 2-st acks and equiv alence classes of generalized morphisms of Lie 2-group oids up to Morita equiv alences. 2.3. Lie 2-group bundles. In this section, w e give a deﬁnition of Lie 2- group bundles of a global nature and form ulated in terms of generalized morphisms of Lie 2-group oids. Deﬁnition 2.20. A princip al (Lie 2-gr oup) [ G → H ] -bund le over a Lie gr oup oid Γ 1 ⇒ Γ 0 is a generalized morphism B from Γ 1 ⇒ Γ 0 (seen as a Lie 2-group oid) to the Lie 2-group [ G → H ] asso ciated to the crossed mo dule ( G → H ) . In particular, a principal [ G → Aut( G )] -bundle o v er a group oid Γ 1 ⇒ Γ 0 is a generalized morphism from Γ 1 ⇒ Γ 0 (seen as a Lie 2-group oid) to the Lie 2-group [ G → Aut( G )] . T wo principal [ G → H ] -bundles B and B 0 o ver the Lie group oid Γ 1 ⇒ Γ 0 are said to b e isomorphic if, and only if, these tw o generalized morphisms are equiv alen t. A [ G → H ] -bundle ov er a manifold M is a (2-group) [ G → H ] -bundle o ver the Lie group oid M ⇒ M . Let B b e a [ G → H ] -bundle ov er a Lie group oid Γ 1 ⇒ Γ 0 . If Γ 0 1 ⇒ Γ 0 0 and [ G 0 → H 0 ] are Morita equiv alent to Γ 1 ⇒ Γ 0 and [ G → H ] resp ectively , then the comp osition  Γ 0 1 ⇒ Γ 0 0  !  Γ 1 ⇒ Γ 0  B [ G → H ] ! [ G 0 → H 0 ] . deﬁnes a principal [ G 0 → H 0 ] -bundle ov er Γ 0 1 ⇒ Γ 0 0 denoted B b y abuse of notation. Here the left and right squig arro ws are the Morita equiv alences seen as inv ertible generalized morphisms as in Example 2.15. Deﬁnition 2.21. A principal (Lie 2-group) [ G → H ] -bundle B ov er a Lie group oid Γ 1 ⇒ Γ 0 and a principal (Lie 2-group) [ G 0 → H 0 ] -bundle B 0 o ver a Lie group oid Γ 0 1 ⇒ Γ 0 0 are said to b e Morita e quivalent if, and only if, 12 GRÉGOR Y GINOT AND MA THIEU STIÉNON Γ 0 1 ⇒ Γ 0 0 is Morita equiv alent to Γ 1 ⇒ Γ 0 , [ G 0 → H 0 ] is Morita equiv alen t to [ G → H ] , and B (viewed as a generalized morphism  Γ 0 1 ⇒ Γ 0 0  [ G 0 → H 0 ] ) and B 0 are equiv alen t generalized morphisms. R emark 2.22 . When the groupoid is just a manifold, our deﬁnition is equiv a- len t to the usual deﬁnition of Lie 2-group bundles in [4, 6, 50, 58] as suggested b y Examples 2.23 and 2.24 b elo w. F urthermore our notion of isomorphism of principal 2-group bundles o ver a ﬁxed manifold agrees with the one of [58] and is in fact a particular case of the one describ ed in [59], see Remark 2.26. Example 2.23. Let P π − → M b e a principal H -bundle. Then the diagram M     P × M P o o     / / H     M     P × M P φ o o s   t   f / / H     M P π o o / / ∗ where s ( x, y ) = x , t ( x, y ) = y , π ( x ) = φ ( x, y ) = π ( y ) and x · f ( x, y ) = y , deﬁnes a generalized morphism from the manifold M to the 2 -group [1 → H ] . Hence, it is a 2 -group bundle o ver M . Note that a principal H -bundle P o ver M is Morita equiv alen t (as a 2-group bundle) to a principal H 0 -bundle P 0 o ver M 0 if, and only if, H and M are isomorphic to H 0 and M 0 resp ectiv ely and P and P 0 are isomorphic principal bundles. Example 2.24. Let M b e a smo oth manifold and G b e a (non-ab elian) Lie group. A non ab elian 2-co cycle [18, 19, 26, 43] on M with v alues in G relativ e to an op en cov ering { U i } i ∈ I of M is a collection of smo oth maps λ ij : U ij → Aut( G ) and g ij k : U ij k → G satisfying the following relations: λ ij ◦ λ j k = Ad g ij k ◦ λ ik g ij l g j k l = g ikl λ − 1 kl ( g ij k ) . Suc h a non-ab elian 2-co cycle deﬁnes a [ G → Aut( G )] -bundle o v er the mani- fold M ; for it can b e seen as the generalized morphism M     ` i,j U ij × G × G o o l   u   f / / G n Aut( G )     M     ` i,j U ij × G φ o o     / / Aut( G )     M ` i U i o o / / ∗ PRINCIP AL 2-GROUP BUNDLES 13 b et w een the manifold M and the 2-group [ G → Aut( G )] . Here l ( x ij , g 1 , g 2 ) = ( x ij , g 1 ) φ ( x ij , g ) = x u ( x ij , g 1 , g 2 ) = ( x ij , g 2 ) f ( x ij , g 1 , g 2 ) =  g 2 g − 1 1 , Ad g 1 ◦ λ ij ( x )  where x ij denotes a p oint x ∈ M seen as a p oint of the op en subset U ij = U i ∩ U j , x i the p oint x ∈ M seen as a p oin t of the op en subset U i , and g , g 1 , g 2 arbitrary elements of G . The horizontal and vertical multiplication are given by ( x ij , g 1 ) ∗ ( x j k , g 2 ) =  x ik , g ij k λ − 1 j k ( g 1 ) g 2  , ( x ij , g 1 , g 2 )  ( x ij , g 2 , g 3 ) = ( x ij , g 1 , g 3 ) . Example 2.25. Let { U i } i ∈ I b e an op en cov ering of a smo oth manifold M . A family of smo oth maps g ij k : U ij k → S 1 deﬁnes a Lie group oid structure on ` i,j U ij × S 1 ⇒ ` i U i with multiplication ( x ij , e iϕ ) · ( x j k , e iψ ) = ( x ik , g ij k e i ( ϕ + ψ ) ) if, and only if, g ij k is a Čech 2-co cycle. In that case, w e get the generalized morphism of 2-group oids M     ` i,j U ij × S 1 × S 1 o o l   u   f / / S 1     M     ` i,j U ij × S 1 o o     / / ∗     M ` i U i o o / / ∗ with f ( x ij , e iϕ , e iψ ) = e i ( ψ − ϕ ) . It deﬁnes an [ S 1 → ∗ ] -bundle ov er M . R emark 2.26 . Our deﬁnition of principal 2-groups bundles ov er a stack also agrees with one whic h was in tro duced more recently b y W olfson in [59]; more precisely our deﬁnition agrees with a sp ecial case of principal bundle o v er a simplicial Lie group in lo c. cit. when the simplicial Lie group is a strict Lie 2- group (viewed as a sp ecial kind of simplicial Lie groups as in [59, §6]). Indeed, giv en a generalized morphism  Γ 1 ⇒ Γ 0  B [ G → H ] , we get a strict Lie 2-group oid map U → [ G → H ] , for some hypercov er U φ − → Γ , which in turns yields a map b etw een the asso ciated simplicial manifolds. Pulling bac k the univ ersal bundle W ([ G → H ]) of [59, Deﬁnition 6.2] along this map yields a t wisted Cartesian pro duct U × [ G → H ] W ([ G → H ]) → U ov er (the simplicial manifold asso ciated to) U (whic h is a stac k by construction). Since U φ − → Γ is a hypercov er, this is precisely the data of a lo cal 2 -bundle which is principal with resp ect to (the simplicial Lie group asso ciated to) [ G → H ] in the sense of [59, §5]. 14 GRÉGOR Y GINOT AND MA THIEU STIÉNON F urthermore, by choosing a common reﬁnement of t w o hypercov ers, an isomorphism of principal 2-group bundle s in the sense of Deﬁnition 2.21 yields an equiv alence of lo cal 2-bundles (as t wisted cartesian pro duct and as stac ks), since in both cases it b oils do wn to b eing a collection of equiv alences of lo cal ob jects of the form E ∼ = U × [ G → H ] . Example 2.27. Principal 2 -group bundles arise whenever one studies group actions on stac ks, whic h, in general are only we ak actions. F or instance, there is a canonical (but subtle) w eak action of S 1 on the inertia stack of an y diﬀeren tiable stack giving rise to a canonical principal bundle, which shall b e detailed elsewhere. F or the moment we just explain brieﬂy how it can be deﬁned in terms of Lie 2 -group(oid)s. Let Γ : Γ 1 ⇒ Γ 0 b e a Lie group oid. Its inertia group oid 3 Λ Γ : S Γ × Γ 0 Γ 1 ⇒ S Γ (where S Γ = { γ ∈ Γ 1 s.t. d ( γ ) = s ( γ ) } is the space of lo ops) has a c anonic al action of the group stac k asso ciated to the 2-group [ Z → 1] . This action is given, for ( γ , g ) ∈ S Γ × Γ 1 and n ∈ Z b y ( γ , g ) · n := ( γ , γ n · g ) . Hence it also inherits an action of the group stack S 1 ∼ = [ Z → R ] (induced by the canonical map R → 0 ). One shall note that this action is almost never represen ted by a strict action of the group S 1 on the inertia group oid but really by an action of the 2 -group [ Z → R ] . Assume the inertia group oid is a Lie group oid — which is true if Γ is étale and prop er. It follo ws from [24, Theorem 0.2] that the quotient of the stac k represented by Λ Γ by the (group stack represented by) S 1 ∼ = [ Z → R ] is a diﬀeren tiable stac k and further a [ Z → R ] -principal bundle. Indeed this quotient stack can b e presen ted b y a Lie group oid ˜ Γ which is Morita equiv alent to the Lie 2-group oid ˜ ˜ Γ : S Γ × Γ 0 Γ 1 × Z × R ⇒ S Γ × Γ 0 Γ 1 × R ⇒ S Γ . The horizontal multiplications are giv en b y the pro duct of the Lie (2-)group oids structures of Γ and [ Z → R ] while v ertical mutiplication is induced by the action of Z on the inertia group oid describ ed abov e. : The canonical pro jection ˜ ˜ Γ → [ Z → R ] = Z × R ⇒ R ⇒ 1 giv es righ t to the generalized morphism B : ˜ Γ ∼ ← − ˜ ˜ Γ → [ Z → R ] deﬁning the bundle structure. R emark 2.28 . There is a nerv e functor from Lie 2-group oids to simplicial spaces generalizing the nerve for Lie 1-group oids. F or instance, see [16, 51, 55] and Section 4.1 b elo w. Comp osing it with the (fat) realization functor, w e obtain the classifying space functor Γ 7→ B Γ from Lie 2-group oids to top o- logical spaces. Since the realization of a Morita morphism (i.e. hypercov er) is a homotopy equiv alence, a generalized morphism Γ F ∆ induces a map B Γ B F − − → B ∆ in the homotopy category of top ological spaces. In particular, a [ G → H ] -group bundle ov er a manifold induces a map M → B [ G → H ] in the homotopy category . This is the top ological side of generalized morphisms and 2-group bundles. In fact, using standard arguments on homotopy for manifolds, it should be p ossible to pro v e that [ G → H ] -group bundles o v er Γ 3 The inertia group oid of a group oid Γ 1 ⇒ Γ 0 is a group oid representing the inertia stac k of the quotient stac k [Γ 0 / Γ 1 ] . See [8] for details on inertia group oids and stacks. PRINCIP AL 2-GROUP BUNDLES 15 (up to Morita equiv alences) are in bijection with homotopy classes of maps B Γ → B [ G → H ] . 3. Gr oupoid G -extensions W e ﬁx a Lie group G . W e recall the following deﬁnition (see [32]) Deﬁnition 3.1. A Lie gr oup oid G -extension is a short exact sequence of Lie group oids ov er the identit y map on the unit space M 1 → M × G i − → ˜ Γ φ − → Γ → 1 (6) Here b oth Γ and ˜ Γ are Lie group oids ov er M and M × G ⇒ M is a (trivial) bundle of groups. The map φ being a map o v er the iden tit y map on the unit space M means that b oth ˜ Γ and Γ ha ve M for unit space and that the restriction of φ to M is the identit y on M : ˜ Γ     φ / / Γ     M id / / M . In the sequel, an extension lik e (6) will b e denoted ˜ Γ φ − → Γ ⇒ M and we will write g m instead of i ( m, g ) . Lie group oid G -extensions can b e in terpreted in terms of crossed mo dules as follows. Prop osition 3.2. The morphism of gr oup oids ˜ Γ φ − → Γ ⇒ M is a gr oup oid G -extension if, and only if, ( M × G i  − → ˜ Γ) is a cr osse d mo dule of gr oup oids with quotient gr oup oid ˜ Γ /i ( M × G ) isomorphic to Γ . Prop osition 3.2 follows easily from Remark 3.7 and Lemma 3.9 b elow. Deﬁnition 3.3 ([32]) . A Morita morphism b etwe en Lie gr oup oid G -exten- sions is a homomorphism of Lie group oid G -extensions ˜ Γ f   / / Γ / / / / f   M f   ˜ ∆ / / ∆ / / / / N suc h that M f − → N is a surjective submersion and Γ     f / / ∆     M f / / N and ˜ Γ     f / / ˜ ∆     M f / / N 16 GRÉGOR Y GINOT AND MA THIEU STIÉNON are Morita morphisms 4 of 1-group oids. As in the Lie 2-groupoid case, the Morita morphisms of Lie groupoid extensions form a left m ultiplicativ e system in the category of Lie group oid extensions and homomorphisms of Lie group oid extensions. Hence, one can lo calize this category b y its Morita morphisms. T wo Lie group oid extensions are Morita e quivalent if they are isomorphic in the lo calized category . (As in the Lie 2-group oids case, there is a notion of gener alize d morphisms for Lie group oid extensions. In that language, a Morita equiv alence is an inv ertible generalized morphism.) Here is our ﬁrst main theorem. Theorem 3.4. Ther e exists a bije ction b etwe en the Morita e quivalenc e class- es of Lie gr oup oid G -extensions and the Morita e quivalenc e classes of [ G → Aut( G )] -bund les over Lie gr oup oids. R emark 3.5 . The abov e theorem can b e regarded as a geometric v ersion of a theorem of Breen [12], which states that H 2 ( X , G ) is isomorphic to H 1 ( X , ( G → Aut( G ))) . The pro of of Theorem 3.4 is the ob ject of the next tw o sections. 3.1. F rom groupoid G -extensions to [ G → Aut( G )] -bundles. Giv en a Lie group oid G-extension ˜ Γ φ / / Γ a / / b / / M , one can deﬁne a Lie 2- group oid ˜ Γ × Γ ˜ Γ l / / u / / ˜ Γ s / / t / / M , where ˜ Γ × Γ ˜ Γ = n ( ˜ γ 1 , ˜ γ 2 ) ∈ ˜ Γ × ˜ Γ | φ ( ˜ γ 1 ) = φ ( ˜ γ 2 ) o l ( ˜ γ 1 , ˜ γ 2 ) = ˜ γ 2 u ( ˜ γ 1 , ˜ γ 2 ) = ˜ γ 1 s ( ˜ γ ) = a  φ ( ˜ γ )  t ( ˜ γ ) = b  φ ( ˜ γ )  ( ˜ γ 1 , ˜ γ 2 )  ( ˜ γ 2 , ˜ γ 3 ) = ( ˜ γ 1 , ˜ γ 3 ) ( ˜ γ 1 , ˜ γ 2 ) ∗ ( ˜ δ 1 , ˜ δ 2 ) = ( ˜ γ 1 · ˜ δ 1 , ˜ γ 2 · ˜ δ 2 ) . Here · stands for the multiplication in ˜ Γ ⇒ M . The group oid homomorphism φ naturally induces a Morita morphism (i.e. h yp erco ver) of 2-group oids: ˜ Γ × Γ ˜ Γ l   u   / / Γ id   id   ˜ Γ s   t   φ / / Γ a   b   M id / / M (7) 4 W eak equiv alences in [44] and h ypercov ers in [59, 60]. PRINCIP AL 2-GROUP BUNDLES 17 where the 2-group oid Γ id / / id / / Γ a / / b / / M is simply the 1-group oid Γ a / / b / / M seen as a 2-group oid in the trivial wa y . Consider the map ˜ Γ → Aut( G ) : ˜ γ 7→ Ad ˜ γ deﬁned b y  Ad ˜ γ g  t ( ˜ γ ) = ˜ γ · g s ( ˜ γ ) · ˜ γ − 1 . It gives a morphism of Lie group oids ˜ Γ Ad / /     Aut( G )     M / / ∗ (8) whic h, together with the map ˜ Γ × Γ ˜ Γ → G n Aut( G ) : ( ˜ γ 1 , ˜ γ 2 ) 7→ ( g , Ad ˜ γ 2 ) , where ˜ γ 1 ˜ γ − 1 2 = g t ( ˜ γ 1 ) , deﬁnes a homomorphism of Lie 2-group oids ˜ Γ × Γ ˜ Γ     / / G n Aut( G )     ˜ Γ     Ad / / Aut( G )     M / / ∗ (9) R emark 3.6 . Note that the induced map ˜ Γ × Γ ˜ Γ     / / G n Aut( G )     ˜ Γ / / Aut( G ) is a fully faithful functor. R emark 3.7 . In terms of crossed mo dules, the ab o ve discussion go es as follo ws. The extension 1 → M × G i − → ˜ Γ φ − → Γ → 1 leads to an ac- tion of ˜ Γ on the groupoid M × G ⇒ M by conjugation, i.e. via the map ˜ γ 7→ Ad ˜ γ . Then ˜ Γ × Γ ˜ Γ l / / u / / ˜ Γ s / / t / / M is the Lie 2-group oid corresp ond- ing to the crossed mo dule ( M × G i − → ˜ Γ) . The pro jection on to the ﬁrst factor M × G → M and the morphism φ : ˜ Γ → Γ induce the Morita equiv alence of crossed mo dules ( M × G → ˜ Γ) → ( M → Γ) corresp onding to the map (7). Moreo ver , the map Ad : ˜ Γ → Aut( G ) yields the map of crossed mo dules 18 GRÉGOR Y GINOT AND MA THIEU STIÉNON  M × G i − → ˜ Γ  (pr 2 , Ad ) − − − − − →  G → Aut( G )  corresp onding to the morphism of Lie 2-group oids (9). Prop osition 3.8. (1) A Lie gr oup oid G -extension ˜ Γ → Γ ⇒ M induc es a princip al [ G → Aut( G )] -bund le over Γ ⇒ M , which c an b e de- scrib e d explicitely by the fol lowing gener alize d morphism: Γ     ˜ Γ × Γ ˜ Γ o o     / / G n Aut( G )     Γ     ˜ Γ o o     / / Aut( G )     M M o o / / ∗ (2) If ˜ Γ → Γ ⇒ M and ˜ ∆ → ∆ ⇒ N ar e Morita e quivalent G - extensions, then the c orr esp onding 2 -gr oup bund les ar e Morita e quiv- alent. Pr o of. Claim 1) follo ws from the ab ov e discussion. Supp ose giv en a Morita morphism of G -extensions ˜ Γ f   / / Γ / / / / f   M f   ˜ ∆ / / ∆ / / / / N . Since f commutes with the ˜ Γ and ˜ ∆ -actions on G , there is a comm utativ e diagram [ M → Γ] ( f ,f ) ∼   [ M × G → ˜ Γ] ( f × id ,f ) ∼   ∼ o o (pr 2 , Ad ) / / [ G → Aut( G )] id   [ N → ∆] [ N × G → ˜ ∆] ∼ o o (pr 2 , Ad ) / / [ G → Aut( G )] , where p 2 denotes the canonical pro jection on the second comp onen t. Now, Claim 2) follows from Example 2.14.  3.2. F rom [ G → Aut( G )] -bundles to group oid G -extensions. In this section, w e show ho w to reverse the pro cedure. Starting from a [ G → Aut( G )] -bundle, we recov er a group oid G -extension. F or future reference, we state the following tec hnical result without pro of. PRINCIP AL 2-GROUP BUNDLES 19 Lemma 3.9. L et ∆ 2     φ 2 / / Γ 2     ∆ 1     φ 1 / / Γ 1     ∆ 0 φ 0 / / Γ 0 b e a hyp er c over of Lie 2-gr oup oids. And let L j   φ / / K i   ∆ 1 φ 1 / / Γ 1 b e the induc e d map of cr osse d mo dules. Then φ maps j − 1 (1 m ) onto i − 1 (1 φ ( m ) ) bije ctively (for every m ∈ ∆ 0 ) and induc es a functor fr om the gr oup oid ∆ 1 j ( L ) to the gr oup oid Γ 1 i ( K ) , which is ful ly faithful and surje ctive on the obje cts. 5 No w, given a [ G → Aut( G )] -bundle o ver a Lie group oid Γ ⇒ Γ 0 , we pro ceed with the construction of a Lie group oid G -extension. Supp ose the [ G → Aut( G )] -bundle is given by the generalized morphism of 2-group oids Γ     ∆ 2 φ 2 o o     f 2 / / G n Aut( G )     Γ     ∆ 1 φ 1 o o     f 1 / / Aut( G )     Γ 0 ∆ 0 φ 0 o o f 0 / / ∗ and let Γ 0 i   L φ o o j   f / / G Ad   Γ ∆ 1 φ 1 o o f 1 / / Aut( G ) b e the induced generalized morphism of crossed mo dules. Hence L = { α ∈ ∆ 2 | u ( α ) = 1 x for some x ∈ ∆ 0 } 5 The crossed mo dules [1 → ∆ 1 j ( L ) ] and [1 → Γ 1 i ( K ) ] are the ‘cok ernels’ of the crossed mo dules [ L j − → ∆ 1 ] and [ K i − → Γ 1 ] resp ectiv ely . 20 GRÉGOR Y GINOT AND MA THIEU STIÉNON and j : L → ∆ 1 is the restriction of the structure map l : ∆ 2 → ∆ 1 to L . Since φ is a hypercov er and Γ 0 i − → Γ is an injection, b y Lemma 3.9, L j − → ∆ 1 is also injective and ∆ 1 j ( L )     φ / / Γ i (Γ 0 )     ∆ 0 φ / / Γ 0 is a fully faithful functor. Since Γ i (Γ 0 ) is diﬀeomorphic to Γ , the group oid ∆ 1 j ( L ) ⇒ ∆ 0 is the pullbac k of Γ ⇒ Γ 0 through the surjectiv e submersion ∆ 0 φ − → Γ 0 . Therefore ∆ 1 j ( L ) = ∆ 0 × φ, Γ 0 ,s Γ × t, Γ 0 ,φ ∆ 0 is a smo oth manifold. Consider the group oid structure on ∆ 1 × G ⇒ ∆ 0 with source s ( δ, g ) = s ( δ ) , target t ( δ, g ) = t ( δ ) , m utliplication ( δ 1 , g 1 ) · ( δ 2 , g 2 ) =  δ 1 δ 2 , f 1 ( δ − 1 2 )[ g 1 ] · g 2  , and in verse ( δ, g ) − 1 =  δ − 1 , f 1 ( δ )[ g − 1 ]  for all δ, δ 1 , δ 2 ∈ ∆ 1 and g , g 1 , g 2 ∈ G . The map H : L → ∆ 1 × G deﬁned by H ( α ) =  j ( α ) , f ( α − 1 )  is a morphism of group oids from L ⇒ ∆ 0 to ∆ 1 × G ⇒ ∆ 0 . One chec ks that ( δ, g ) ·  j ( α ) , f ( α − 1 )  =  j ( δ ∗ α ∗ δ − 1 ) , f ( δ ∗ α ∗ δ − 1 ) − 1  · ( δ, g ) =  δ · j ( α ) , f ( α − 1 ) · g  , for all δ ∈ ∆ 1 , g ∈ G , and α ∈ L . Th us the image of L under H is a normal subgroup oid of ∆ 1 × G ⇒ ∆ 0 . Since j : L → ∆ 1 is injective, the action of L on ∆ 1 × G by m ultiplication from the righ t ( δ, g ) • α = ( δ, g ) ·  j ( α ) , f ( α − 1 )  =  δ · j ( α ) , f ( α − 1 ) · g  is free and its orbit space (∆ 1 × G ) / H ( L ) is a smo oth manifold. No w consider the group oid G -extension ∆ 1 × G → ∆ 1 ⇒ ∆ 0 . The mor- phism of group oids ∆ 1 × G 3 ( δ, g ) 7→ δ ∈ ∆ 1 in tertwin es the right action of L on ∆ 1 × G with the right action δ • α = δ · j ( α ) of L on ∆ 1 , whose orbit space is the smo oth manifold ∆ 1 j ( L ) . Therefore, passing to quotients, we obtain the G -extension of group oids (∆ 1 × G ) / H ( L ) → ∆ 1 /j ( L ) ⇒ ∆ 0 . Note that the corresp onding crossed mo dule is  ∆ 0 × G → (∆ 1 × G ) / H ( L )  . Prop osition 3.10. (1) Every [ G → Aut( G )] -bund le over a Lie gr oup oid Γ ⇒ Γ 0 induc es a Lie gr oup oid G -extension. (2) Morita e quivalent [ G → Aut( G )] -bund les induc e Morita e quivalent extensions. PRINCIP AL 2-GROUP BUNDLES 21 Pr o of. 1) As was outlined ab o v e, a bundle [Γ 0 → Γ] φ ← − ∆ f − → [ G → Aut( G )] determines a G -extension (∆ 1 × G ) / H ( L ) → ∆ 1 /j ( L ) ⇒ ∆ 0 , where j : L → ∆ 1 is the restriction of the structure map l : ∆ 2 → ∆ 1 to L = { α ∈ ∆ 2 | u ( α ) = 1 x for some x ∈ ∆ 0 } and H : L → ∆ 1 × G is the morphism of group oids from L ⇒ ∆ 0 to ∆ 1 × G ⇒ ∆ 0 deﬁned by H ( α ) =  j ( α ) , f ( α − 1 )  . 2) It is suﬃcient to chec k that for any diagram E φ 1 ∼ w w ε   f 1 ) ) [Γ 0 → Γ] [ G Ad − − → Aut( G )] F φ 2 ∼ g g f 2 5 5 comm uting up to natural 2-equiv alences, the G -extensions corresp onding to the lo wer and upp er generalized morphisms are Morita equiv alen t. Since φ 1 , φ 2 are Morita equiv alences, ε is also a Morita equiv alence. Therefore, b y Lemma 2.18, w e can assume that ε is a hypercov er. Then, denoting by ( K → E 1 ) and ( L → F 1 ) the crossed mo dules corresp onding to E and F resp ectiv ely , the map ε induces a commutativ e diagram ( E 1 × G ) / H ( K ) ( ε, id)   / / E 1 /j ( K ) / / / / ε   E 0 ε   ( F 1 × G ) / H ( L ) / / F 1 /j ( L ) / / / / F 0 whic h is a Morita equiv alence of extensions by Lemma 3.9.  3.3. Pro of of Theorem 3.4. It remains to prov e that the constructions of Section 3.1 and Section 3.2 are inv erse of each other. Supp ose that a [ G → Aut( G )] -principal bundle B o ver Γ is giv en b y the generalized morphism Γ φ ← − ∆ f − → [ G → Aut( G )] . Let ∆ 1 × G H ( L ) → ∆ 1 j ( L ) ⇒ ∆ 0 b e the induced G -principal extension as in Prop osition 3.10 1). The corresp onding crossed mo dule is (∆ 0 × G → ∆ 1 × G H ( L ) ) . W e hav e the following comm utativ e diagram of crossed mo dules: (Γ 0 → Γ) ( L → ∆ 1 ) ∼ o o   / / ( G → Aut( G )) (∆ 0 → ∆ 1 j ( L ) ) ∼ O O (∆ 0 × G → ∆ 1 × G H ( L ) ) ∼ o o 5 5 , 22 GRÉGOR Y GINOT AND MA THIEU STIÉNON where ( L → ∆ 1 ) is the crossed mo dule corresp onding to ∆ . It follo ws that the generalized morphism [Γ 0 → Γ] ∼ ← − [ L → ∆ 1 ] → [ G → Aut( G )] w e started from is equiv alent to the generalized morphism [Γ 0 → Γ] ∼ ← − [∆ 0 × G → ∆ 1 × G H ( L ) ] → [ G → Aut( G )] asso ciated to the G -extension ∆ 1 × G H ( L ) → ∆ 1 j ( L ) ⇒ ∆ 0 . Hence they represen t the same ( G → Aut( G )) -bundle ov er Γ ⇒ Γ 0 . Recipro cally , if ˜ Γ → Γ ⇒ M is a G -extension, then the asso ciated principal [ G → Aut( G )] -bundle is given by the generalized morphism [ M → Γ] φ ← − [ M × G → ˜ Γ] (pr 2 , Ad ) − − − − − → [ G → Aut( G )] (10) according to Remark 3.7. Direct insp ection of the pro of of Prop osition 3.10 sho ws that the G -extension induced by the generalized morphism (10) is exactly ˜ Γ → Γ ⇒ M . 4. Universal characteristic maps and Dixmier–Douad y classes 4.1. Cohomology of Lie 2-group oids. T o eac h Lie 2-group oid Γ 2 ⇒ Γ 1 ⇒ Γ 0 is asso ciated a simplicial manifold: its (ge ometric) nerve N • Γ . It is the nerv e of the underlying 2-category as deﬁned by Street [55]. In par- ticular, N 0 Γ = Γ 0 , N 1 Γ = Γ 1 , N 2 Γ is a submanifold of Γ 2 × Γ 1 × Γ 1 × Γ 1 parameterizing the 2-arrows of Γ 2 ﬁtting in a commutativ e triangle A 1 f 0 α   A 0 f 1 / / f 2 > > A 2 (11) and N 3 Γ is a submanifold of (Γ 2 ) 4 × (Γ 1 ) 6 parameterizing the comm utative tetrahedra like A 3 A 1 f 02 O O α 2 W _ α 3   f 03 & & α 0 W _ A 0 f 13 / / f 23 8 8 f 12 E E α 1 R Z A 2 f 01 Y Y (12) with faces giv en by elements of N 2 Γ . By the commutativit y of the tetrahe- dron (12), we mean that ( α 3 ∗ f 01 )  α 1 = ( f 23 ∗ α 0 )  α 2 . F or p ≥ 3 , N p Γ is the manifold of all p -simplices suc h that each subsimplex of dimension 3 is a tetrahedron of the form (12) ab ov e [22, 55]. The nerv e of a Lie group oid considered as a Lie 2-group oid is isomorphic to its usual (1-)nerve [51]. The PRINCIP AL 2-GROUP BUNDLES 23 nerv e N • deﬁnes a functor from the category of Lie 2 -group oids to the cate- gory of simplicial manifolds. T aking the fat realization of the nerve deﬁnes a functor from Lie 2 -group oids to top ological spaces. The de Rham cohomology groups of a Lie 2-groupoid Γ are deﬁned to b e the total cohomology groups of the bicomplex (Ω • ( N • Γ ) , d DR , ∂ ) , where d DR : Ω p ( N q Γ ) → Ω p +1 ( N q Γ ) stands for the de Rham diﬀeren tial and ∂ : Ω p ( N q Γ ) → Ω p ( N q +1 Γ ) is deﬁned by ∂ = ( − 1) p P q +1 i =0 ( − 1) i d ∗ i , where d i : N q +1 Γ → N q Γ denotes the i th face map. W e use the shorter notation Ω • tot ( Γ ) for the asso ciated total complex. Hence Ω n tot ( Γ ) = L p + q = n Ω p ( N q Γ ) with (total) diﬀerential d DR + ∂ . W e denote the subspaces of co cycles and cob oundaries b y Z • DR ( Γ ) and B • DR ( Γ ) resp ectively , and the cohomology of Γ b y H • ( Γ ) . The following Lemma is folklore (see [20] for a more general statement with resp ect to hypercov ers). Lemma 4.1. L et F : Γ → ∆ b e a hyp er c over of Lie 2-gr oup oids. Then F ∗ : H • ( ∆ ) → H • ( Γ ) is an isomorphism. Pr o of. It is w ell-kno wn that a natural transformation betw een t wo 2-functors f and g from Γ to ∆ induces a simplicial homotopy b etw een f ∗ : N • ( Γ ) → N • ( ∆ ) and g ∗ : N • ( Γ ) → N • ( ∆ ) , for instance see [16, Prop osition 4]. In particular equiv alent (top ological) 2-categories hav e homotopic nerves. The result follows for a hypercov er with a section. Since lo cal sections alwa ys exist, the general case reduces to a h ypercov er Γ [ ` U i ] → Γ induced b y pullbac k along the canonical map ` U i → Γ where ( U i ) is a cov er of Γ 0 . The result follows from a classical May er-Vietoris argument as in [7].  By Lemma 4.1 ab ov e, a generalized morphism F : Γ φ 1 ← − ∼ E 1 f 1 − → . . . φ n ← − ∼ E n f n − → ∆ induces a pullback map in cohomology F ∗ : H • ( ∆ ) f ∗ n − → H • ( E n ) ( φ ∗ n ) − 1 − − − − → . . . f ∗ 1 − → H • ( E 1 ) ( φ ∗ 1 ) − 1 − − − − → H • ( Γ ) Clearly , ( F ◦ G ) ∗ = G ∗ ◦ F ∗ and, if F is a Morita equiv alence, then F ∗ is an isomorphism. Lemma 4.2. If F and G ar e e quivalent gener alize d morphisms fr om Γ to ∆ , the maps F ∗ and G ∗ , which they induc e at the c ohomolo gy level, ar e e qual. Pr o of. A natural transformation b etw een t wo 2-functors f and g from Γ to ∆ induces a simplicial homotopy b etw een f ∗ : N • ( Γ ) → N • ( ∆ ) and g ∗ : N • ( Γ ) → N • ( ∆ ) (see [16]). Therefore the lemma follows from the deﬁnition of equiv alence of generalized morphisms and Lemma 4.1.  24 GRÉGOR Y GINOT AND MA THIEU STIÉNON R emark 4.3 . Note that, for a Lie 2-group oid Γ : Γ 2 l / / u / / Γ 1 s / / t / / Γ 0 , N 2 Γ ma y b e iden tiﬁed to Γ 2 × s, Γ 0 ,t Γ 1 so that the face maps take the form d 0 : N 2 Γ → N 1 Γ : ( α, c ) 7→ u ( α ) d 1 : N 2 Γ → N 1 Γ : ( α, c ) 7→ l ( α ) · c d 2 : N 2 Γ → N 1 Γ : ( α, c ) 7→ c More precisely , A 2 A 1 u ( α )   l ( α ) X X A 0 c o o α   ∈ Γ 2 × s, Γ 0 ,t Γ 1 is iden tiﬁed to A 1 u ( α ) ~ ~ α ∗ c   A 2 A 0 l ( α ) ∗ c o o c ` ` ∈ N 2 Γ . R emark 4.4 . The singular cohomology H • sing ( Γ , R ) of Γ with co eﬃcient in a ring R is deﬁned similarly to the de Rham cohomology . More precisely it is the cohomology of ( C • sing ( N • Γ ) , d sing + ∂ ) where ( C • sing ( X , R ) , d sing ) denotes the singular cochain complex of a space X with co eﬃcients in R . As for manifolds, for R = R , one has a natural isomorphism H • sing ( Γ , R ) ∼ = H • ( Γ ) . 4.2. Cohomological c haracteristic map for 2-group bundles. Fix a crossed mo dule ( G → H ) and let B be a principal [ G → H ] -bundle ov er Γ . In this section w e construct a universal characteristic homomorphism C C B : H • ([ G → H ]) → H • ( Γ ) generalizing the usual characteristic classes of a principal bundle. By deﬁnition, B is a generalized morphism Γ [ G → H ] . Therefore, passing to cohomology , we obtain the homomorphism C C B : H • ([ G → H ]) B ∗ − − → H • ( Γ ) (13) whic h we call the char acteristic homomorphism of the [ G → H ] -bundle B . It dep ends only on the isomorphism class of the 2 -group bundle. Prop osition 4.5. If B and B 0 ar e isomorphic [ G → H ] -bund les over Γ , then C C B = C C B 0 : H • ([ G → H ]) → H • ( Γ ) . Pr o of. It is an immediate consequence of Lemma 4.1 since isomorphic prin- cipal 2-group bundles are equiv alen t as generalized morphisms.  PRINCIP AL 2-GROUP BUNDLES 25 R emark 4.6 . By analogy with the case of principal bundles, one can think of the elements of H • ([ G → H ]) as univ ersal c haracteristic classes and their images in H • ( Γ ) b y C C B as c haracteristic classes of the [ G → H ] -bundle o ver Γ . F or instance, it is pro ved [25, Proposition 6.3] that the characteristic classes asso ciated to the string 2 -group asso ciated to a compact simple Lie group coincide with the usual ones mo dulo the Pon try agin class. Example 4.7. Let P π − → M b e a principal H -bundle. Then, by Exam- ple 2.23, P induces a structure of [1 → H ] -bundle o v er M . Since H • ([1 → H ]) ∼ = H • ( B H ) , the characteristic map C C P of this bundle coincides with the classical map H • ( B H ) → H • ( M ) induced b y the principal H -bundle structure on P . In particular, for a compact Lie group H , the c haracteristic map coincides with the Chern-W eil map S ( h ∗ ) h → H • ( M ) induced by the c hoice of a connection on P . Example 4.8. F rom example 2.27, we kno w that the inertia group oid of a Lie group oid Γ gives rise to a principal [ Z → R ] -bundle. In that case H • ([ Z → R ]) ∼ = H • ( B S 1 ) ∼ = R [ x ] where x is a generator of degree 2 . In particular, w e get a c haracteristic class C C Γ ( x ) ∈ H 2 ([Λ Γ /S 1 ]) = H 2 S 1 (Λ Γ ) (see [24] for equiv ariant cohomology of stac ks). F or instance if Γ is the group oid G ⇒ 1 with G a simply connected compact Lie group, then its inertia group oid is the transformation Lie groupoid G × G ⇒ G with G acting on itself b y the adjoint action. F rom the Gysin sequence in equiv arian t homology of stacks ([24, §8]) and the fact that the homology H • ( G × G ⇒ G ) is trivial in degrees 1 and 2 , we see that C C G × G ⇒ G ( x ) is an in tegral 6 generator of H 2 ([Λ Γ /S 1 ]) . Let Γ F ∆ b e a generalized morphism of Lie (1-)groupoids and let B : ∆ φ ← − ∼ E f − → [ G → H ] be a 2 -group bundle with base ∆ . The pullbac k F ∗ ( B ) of the [ G → H ] -bundle B from ∆ to Γ b y F is the comp osition B ◦ F of the t wo generalized morphisms. It is a principal [ G → H ] -bundle ov er Γ . The Whitney sum of tw o 2-group bundles is deﬁned as follows. Let B : Γ φ ← − ∼ E f − → [ G → H ] and B 0 : Γ φ 0 ← − ∼ E 0 f 0 − → [ G 0 → H 0 ] be tw o 2 -group bundles ov er the same base Γ . Let F b e the “ﬁber product” 2 -group oid E 2 × Γ 2 E 0 2 ⇒ E 1 × Γ 1 E 0 1 ⇒ E 0 × Γ 0 E 0 0 with the obvious structure maps: s ( e, e 0 ) = ( s ( e ) , s ( e 0 )) , ( x, x 0 ) ∗ ( y , y 0 ) = ( x ∗ x 0 , y ∗ y 0 ) , etc. The Whitney sum B ⊕ B 0 is the [ G × G 0 → H × H 0 ] -bundle o v er Γ giv en b y the generalized morphism Γ φ = φ 0 ← − − − ∼ F f × f 0 − − − → [ G × G 0 → H × H 0 ] . By Prop osition 4.5, we obtain Corollary 4.9. (1) C C F ∗ ( B ) = F ∗ ◦ C C B . 6 More precisely , it is the image of a generator of the cohomology with integer coeﬃ- cien ts. See Remark 4.10. 26 GRÉGOR Y GINOT AND MA THIEU STIÉNON (2) C C B ⊕ B 0 = ∆ ∗ ◦  C C B × C C B 0  , wher e ∆ : Γ → Γ × Γ is the diagonal map and × is the cr oss-pr o duct H • ([ G → H ]) ⊗ H • ([ G 0 → H 0 ]) ∼ = H • ([ G × G 0 → H × H 0 ]) . R emark 4.10 . The result of this section easily extends to singular cohomology with any co eﬃcient (see Remark 4.4). In particular the characteristic map C C B : H • sing ([ G → H ] , Z ) B ∗ − − → H • ( Γ , Z ) is deﬁned in cohomology with integer co eﬃcient. R emark 4.11 . By Proposition 3.8, a Lie group oid G -extension ˜ Γ φ − → Γ ⇒ M induces a principal [ G → Aut( G )] -bundle B φ o ver the group oid Γ ⇒ M . Hence w e obtain the universal characteristic map C C B φ : H • ([ G Ad − − → Aut( G )]) → H • ( Γ ) . Unfortunately , the cohomology H • ([ G Ad − − → Aut( G )]) is not known when the center of G is large and it is trivial when the center of G is of dimension less than three [25]. Therefore one cannot hav e muc h hop e of getting interesting characteristic classes except for extensions whose structure 2-group can b e reduced. Indeed, this is the ob ject of the next section. 4.3. DD classes for group oid cen tral G -extensions. Let ˜ Γ φ − → Γ ⇒ M b e a G -extension of Lie groupoids. Let φ 0 denote the factorization of the morphism φ through the pro jection q : ˜ Γ → ˜ Γ / Z ( G ) : ˜ Γ φ / / q   Γ ˜ Γ / Z ( G ) φ 0 < < The extension φ is said to b e c entr al [32] if there exists a section σ : Γ → ˜ Γ / Z ( G ) of φ 0 suc h that xg = g x ∀ x ∈ q − 1  σ (Γ)  , ∀ g ∈ G (14) In this case, the subspace ˜ Γ 0 = q − 1  σ (Γ)  of ˜ Γ is a cen tral Z ( G ) -extension of Γ ⇒ M . R emark 4.12 . The deﬁnition of a cen tral G -gerb e here is taken from [32]. A ccording to lo c. cit , it agrees with the one of a G -gerb e with trivial b and . Giv en γ ∈ ˜ Γ , there exists x ∈ ˜ Γ 0 suc h that φ ( x ) = φ ( γ ) . Thus there exists k ∈ G such that γ = x · k . Given γ , both x and k are uniquely determined up to an elemen t of Z ( G ) . Deﬁning a homomorphism of Lie group oids r : ˜ Γ → G/ Z ( G ) by the relation q ( γ ) = σ ( φ ( γ )) r ( γ ) , we obtain that, for any g ∈ G , g γ = g xk = xg k = xk · k − 1 g k = γ g r ( γ ) PRINCIP AL 2-GROUP BUNDLES 27 where g r ( γ ) denotes the conjugate k − 1 g k of g by any elemen t k ∈ G suc h that k Z ( G ) = r ( γ ) . Prop osition 4.13. L et ˜ Γ φ − → Γ ⇒ M b e a G -extension of a Lie gr oup oid Γ and let B denote the c orr esp onding [ G → Aut( G )] -bund le. The extension is c entr al if, and only if, the [ G → Aut( G )] -bund le B r e duc es to a princip al [ Z ( G ) → 1] -bund le, i.e. ther e exists a gener alize d morphism Z B : [ M → Γ] → [ Z ( G ) → 1] such that [ M → Γ] B / / Z B ' ' [ G → Aut( G )] [ Z ( G ) → 1] ?  O O is c ommutative up to e quivalenc e. In particular, b eing central is in v ariant under Morita equiv alences of Lie group oids extension. Pr o of. Let ˜ Γ φ − → Γ ⇒ M b e a cen tral G -extension. The corresp onding 2- group bundle B is the generalized morphism [ M → Γ] ← − [ M × G i − → ˜ Γ] − → [ G Ad − − → Aut( G )] , see Prop osition 3.8 and Remark 3.7. Let τ : ˜ Γ 0 → ˜ Γ b e the inclusion map. The Z ( G ) -extension deﬁnes the crossed mo dule [ M × Z ( G ) i 0 − → ˜ Γ 0 ] and we hav e a commutativ e diagram [ M → Γ] [ M × G i − → ˜ Γ] o o / / [ G Ad − − → Aut( G )] [ M × Z ( G ) i 0 − → ˜ Γ 0 ] g g τ O O / / [ Z ( G ) → 1] ?  O O (15) Note that the righ t square in (15) is commutativ e b ecause the extension is cen tral. Diagram (15) implies that the 2-group bundle B reduces. Recipro cally , assume B reduces. By Prop osition 3.10 2), passing to a Morita equiv alent group oid, we can assume that the G -extension is the ex- tension corresp onding to the generalized morphism [ M → Γ] Z B [ Z ( G ) → 1]  → [ G → Aut( G )] . If Z B is the generalized morphism [ M → Γ] ← − [ M × L → ∆] − → [ Z ( G ) → 1] , the asso ciated extension is, according to Section 3.2, ˜ Γ → Γ ⇒ M , where ˜ Γ =  ∆ × L Z ( G )  × Z ( G ) G . Since the com- p osition ∆ → 1 → Aut( G ) is trivial, Ad e γ is trivial for all e γ ∈ ˜ Γ . Therefore, the extension is central.  R emark 4.14 . If G is a Lie group whose Lie algebra g is reductive, its Lie algebra decomposes as a direct sum g ∼ = Z ( g ) ⊕ m of ideals, where Z ( g ) is the cen ter of g . In the sequel, the sym b ol pr will denote the induced pro jection g → Z ( g ) , which is a homomorphism of Lie algebras and maps [ g , g ] onto 0. Moreov er, if G is connected, this direct sum decomp osition is not only 28 GRÉGOR Y GINOT AND MA THIEU STIÉNON ad Z ( g ) -in v ariant but also Ad G -in v ariant and, consequently , pr ◦ Ad g = pr for all g ∈ G . Moreo v er, for any g ∈ G and an y smo oth path t 7→ f t in G with f 0 = 1 and d dt f t   0 = ξ ∈ g , one has pr  d dt f − 1 t g f t g − 1   0  = pr(Ad g ξ − ξ ) = pr( ξ ) − pr( ξ ) = 0 (16) Prop osition 4.15. L et ˜ Γ φ − → Γ ⇒ M b e a c entr al G -extension with G c on- ne cte d and whose Lie algebr a is r e ductive. 7 L et α ∈ Ω 1 ( ˜ Γ; g ) b e a c onne ction 1-form for the right princip al G -bund le ˜ Γ φ − → Γ . (1) Then ther e exists Ω α ∈ Z 3 DR (Γ • ; Z ( g )) such that pr  dα + ∂ α  = φ ∗ (Ω α ) . (2) Mor e over, if α 1 and α 2 ar e two diﬀer ent c onne ction 1-forms, then Ω α 1 − Ω α 2 ∈ B 3 DR (Γ • ; Z ( g )) . W e c al l D D ( α ) := [Ω α ] ∈ H 3 (Γ) ⊗ Z ( g ) the Dixmier–Douady class of the G -c entr al extension. Pr o of. 1) Being a connection 1-form, α ∈ Ω 1 ( ˜ Γ; g ) enjo ys the following tw o prop erties: R ∗ g α = Ad g − 1 ◦ α, ∀ g ∈ G α ( ˆ ξ x ) = ξ , ∀ x ∈ M , ∀ ξ ∈ g Giv en an y ξ ∈ g and any G -inv arian t v ector ﬁeld v ∈ X ( ˜ Γ) , w e get dα ( ˆ ξ , v ) = ˆ ξ  α ( v )  − v  α ( ˆ ξ )  − α  [ ˆ ξ , v ]  = ˆ ξ  α ( v )  − v ( ξ ) − α ( L ˆ ξ v ) = L ˆ ξ  α ( v )  = − ad ξ  α ( v )  since the v ector ﬁeld v is G -inv arian t and the function α ( v ) is G -equiv arian t. It follows that pr ◦ dα ( ˆ ξ , v ) = pr[ α ( v ) , ξ ] = 0 since pr[ g , g ] = 0 . Moreo v er, w e ha ve R ∗ g ( dα ) = d ( R ∗ g α ) = d (Ad g − 1 ◦ α ) = Ad g − 1 ◦ dα for all g ∈ G . Therefore, by Remark 4.14, the 2-form pr ◦ dα ∈ Ω 2 ( ˜ Γ , Z ( g )) is basic; there exists ω ∈ Ω 2 (Γ , Z ( g )) such that pr ◦ dα = φ ∗ ω . Consider ˜ Γ 2 = ˜ Γ × s, Γ ,t ˜ Γ = n ( x, y ) ∈ ˜ Γ × ˜ Γ | s ( x ) = t ( y ) o , the three face maps p 1 ( x, y ) = x m ( x, y ) = x · y p 2 ( x, y ) = y from ˜ Γ 2 to ˜ Γ and the action of G × G on ˜ Γ 2 giv en b y ( x, y ) ( g ,h ) = ( xg s ( x ) , y h s ( y ) ) . 7 Suc h Lie groups are called reductive, though this terminology sometimes applies only to algebraic groups. Examples of Lie groups with reductive Lie algebras include GL n ( R ) and all compact Lie groups. PRINCIP AL 2-GROUP BUNDLES 29 Then we hav e pr ◦ ∂ α = ∂ (pr ◦ α ) = p ∗ 2 (pr ◦ α ) − m ∗ (pr ◦ α ) + p ∗ 1 (pr ◦ α ) . F rom pr ◦ Ad g = pr and R ∗ g α = Ad g − 1 α , it follows that R ∗ g (pr ◦ α ) = pr ◦ α . This, together with the relations p 2 ◦ R ( g ,h ) = R h ◦ p 2 and p 1 ◦ R ( g ,h ) = R g ◦ p 1 implies that p ∗ 2 (pr ◦ α ) and p ∗ 1 (pr ◦ α ) are inv ariant under the G × G -action ˜ Γ 2 . Giv en a smo oth path t 7→  x t , y t  in ˜ Γ 2 , one also gets R ∗ ( g ,h ) m ∗ (pr ◦ α )  d dt  x t , y t    0  =(pr ◦ α )  d dt x t g y t h   0  =(pr ◦ α )  d dt x t y t g r ( y t ) h   0  =(pr ◦ α )  d dt x t y t g r ( y 0 ) h   0  + (pr ◦ α )  d dt x 0 y 0 g r ( y t ) h   0  While the ﬁrst term of the R.H.S. is equal to m ∗ (pr ◦ α )  d dt  x t , y t    0  since (pr ◦ α ) is G -in v ariant, the second term v anishes. Indeed, using α ( ˆ ξ ) = ξ , R ∗ h α = Ad h − 1 ◦ α and pr ◦ Ad g = pr , we obtain that (pr ◦ α )  d dt x 0 y 0 g r ( y t ) h   0  = pr  d dt g r ( y t ) ( g r ( y 0 ) ) − 1   0  = pr  d dt  g r ( y 0 )  r ( y 0 ) − 1 r ( y t ) ( g r ( y 0 ) ) − 1   0  and the claim follows from (16). Hence R ∗ ( g ,h ) m ∗ (pr ◦ α ) = m ∗ (pr ◦ α ) . There- fore, pr ◦ ∂ α is ( G × G ) -inv ariant. One also has pr ◦ ∂ α  d dt ( xe tξ , y e tη )   0  = pr  α  d dt y e tη   0  − α  d dt xe tξ y e tη   0  + α  d dt xe tξ   0   = pr  η − α  d dt xy e t Ad − 1 r ( y ) ξ e tη   0  + ξ  = pr( η − Ad − 1 r ( y ) ξ − η + ξ ) = pr( ξ ) − pr(Ad − 1 r ( y ) ξ ) =0 . Hence the 1-form pr ◦ ∂ α ∈ Ω 1  ˜ Γ 2 , Z ( g )  is basic with resp ect to the principal ( G × G ) -bundle ˜ Γ 2 → Γ 2 . 2) Clearly , one has i ˆ ξ ( α 1 − α 2 ) = 0 and R ∗ g  pr ◦ ( α 1 − α 2 )  = pr ◦ ( α 1 − α 2 ) . Th us pr ◦ ( α 1 − α 2 ) = φ ∗ A , where A ∈ Ω 1  Γ; Z ( g )  . It follows that φ ∗ (Ω α 1 − Ω α 2 ) = pr  d ( α 1 − α 2 ) + ∂ ( α 1 − α 2 )  = d ( φ ∗ A ) + ∂ ( φ ∗ A ) = φ ∗ ( dA + ∂ A ) 30 GRÉGOR Y GINOT AND MA THIEU STIÉNON and Ω α 1 − Ω α 2 = dA + ∂ A ∈ B 3  Γ • ; Z ( g )  .  R emark 4.16 . The Dixmier–Douady class D D ( α ) of a central G -extension iden tiﬁes with a linear map Z ( g ) ∗ → H 3 ( Γ ) by comp osition with the canon- ical biduality homomorphism Z ( g ) → Z ( g ) ∗ ∗ . R emark 4.17 . When the group G is abelian, then ˜ Γ / Z ( G ) = Γ , ˜ Γ 0 = ˜ Γ and the pro jection map g pr − → Z ( g ) = g is the identit y . In particular, when G = S 1 , the Dixmier–Douady class giv en b y Prop osition 4.15 coincides with the D ixmier–Douady class deﬁned in [9]. 4.4. Main theorem. Let ˜ Γ → Γ ⇒ M b e a cen tral G -extension of Lie group oids. A ccording to Prop osition 4.13, we obtain a universal c haracteristic map C C Φ : H 3 ([ Z ( G ) → 1]) → H 3 ( Γ ) . According to [25], H 3 ([ Z ( G ) → 1]) is isomorphic to Z ( g ) ∗ if G is compact. Th us w e obtain a map C C Φ : Z ( g ) ∗ → H 3 ( Γ ) which, by duality , deﬁnes the universal char acteristic class C C Φ ∈ H 3 ( Γ ) ⊗ Z ( g ) . Our main theorem is Theorem 4.18. L et G b e a c omp act c onne cte d Lie gr oup. F or any c entr al G -extension of Lie gr oup oids ˜ Γ → Γ ⇒ M , the universal char acteristic class c oincides with the Dixmier–Douady class. R emark 4.19 . The theorem ab ov e may b e considered as a higher analogue of Chern–W eil theory where the c haracteristic classes of a principle bundle can b e expressed by geometric data such as connection and curv ature. In particular when G is S 1 , Theorem 4.18 is a higher analogue of the follo wing w ell kno wn fact: the Chern class of an S 1 -bundle can b e computed from its curv ature. The latter play ed an imp ortant role in geometric quantization of symplectic manifolds. W e refer the in terested reader to [57] for prequantiza- tion of symplectic group oids, whic h ma y b e interpreted as a construction of a central S 1 -extension whose Dixmier–Douady class is the prescrib ed sym- plectic form. As a corollary we obtain Corollary 4.20. L et G b e a c omp act c onne cte d Lie gr oup. The Dixmier– Douady class of any c entr al G -extension of Lie gr oup oids is an inte gr al class. Pr o of. The singular cohomology with in teger co eﬃcien ts H • ( Γ ; Z ) of a 2- group oid Γ is deﬁned as for de Rham cohomology , substituting the singular co c hain complex to the de Rham forms in the constructions of Section 4.1, see Remark 4.4. Therefore, given any principal [ Z ( G ) → 1] -bundle B ov er Γ , w e can construct an integer v alued universal characteristic homomorphism C C B : H • ([ Z ( G ) → 1] , Z ) → H • ( Γ , Z ) as in Section 4.2, see Remark 4.10. A ccording to the computations in [25], the image of H 3 ([ Z ( G ) → 1] , Z ) under the canonical morphism H 3 ([ Z ( G ) → 1] , Z ) → H 3 ([ Z ( G ) → 1]) is the lattice in Z ( g ) ∗ generated by the fundamental classes of each circle comp onen t of Z ( G ) ∼ = S 1 × · · · × S 1 . No w the result follows from Theorem 4.18.  PRINCIP AL 2-GROUP BUNDLES 31 4.5. The case of cen tral S 1 -extensions. In this section, we establish The- orem 4.18 in the case G = S 1 . Assume ˜ Γ φ − → Γ ⇒ M is a central S 1 -extension. W e consider the following four 2-group oids: A : Γ ⇒ Γ ⇒ M B : ˜ Γ × Γ ˜ Γ ⇒ ˜ Γ ⇒ M C : Z ( G ) ⇒ ∗ ⇒ ∗ D : ˜ Γ ⇒ ˜ Γ ⇒ M The central extension φ determines the (generalized) morphisms D φ / / A and A B φ ∼ o o f / / C A t the nerv e level, w e get N 2 ( D ) d D 2   d D 1   d D 0   φ / / N 2 ( A ) d A 2   d A 1   d A 0   N 2 ( B ) d B 2   d B 1   d B 0   φ o o f / / N 2 ( C ) d C 2   d C 1   d C 0   ˜ Γ φ / / Γ ˜ Γ φ o o f / / ∗ where, according to Remark 4.3, N 2 ( A ) = Γ × t, Γ 0 ,s Γ , N 2 ( C ) = Z ( G ) , N 2 ( D ) = ˜ Γ × t, Γ 0 ,s ˜ Γ , N 2 ( B ) = n ( a, b, c ) ∈ ˜ Γ 3 | φ ( a ) = φ ( b ) and s ( a ) = s ( b ) = t ( c ) o φ : N 2 ( B ) → N 2 ( A ) : ( a, b, c ) 7→  φ ( a ) , φ ( c )  f : N 2 ( B ) → N 2 ( C ) : ( a, b, c ) 7→ ab − 1 and the face maps are given by d A 0 ( a, c ) = a d A 1 ( a, c ) = ac d A 2 ( a, c ) = c d B 0 ( a, b, c ) = a d B 1 ( a, b, c ) = bc d B 2 ( a, b, c ) = c d D 0 ( a, c ) = a d D 1 ( a, c ) = ac d B 2 ( a, c ) = c W e will need one more map: p 13 : N 2 ( B ) → N 2 ( D ) : ( a, b, c ) 7→ ( a, c ) Lemma 4.21. One has d B 0 = d D 0 ◦ p 13 , d B 2 = d D 2 ◦ p 13 , (17) and d D 1 ◦ p 13 ( a, b, c ) = f ( a, b, c ) · d B 1 ( a, b, c ) , ∀ ( a, b, c ) ∈ N 2 ( B ) . (18) Lemma 4.22. F or any pseudo-c onne ction θ ∈ Ω( ˜ Γ) on the c entr al S 1 - extension ˜ Γ φ − → Γ ⇒ Γ 0 , one has ∂ B θ + f ∗ ( dt ) = p ∗ 13 ( ∂ D θ ) Her e dt denotes the Maur er-Cartan (or angular) form on S 1 . 32 GRÉGOR Y GINOT AND MA THIEU STIÉNON Pr o of. Since θ  d dt ˜ γ · e it   0  = 1 and θ is S 1 -in v ariant, it follows from (18) that ( d D 1 ◦ p 13 ) ∗ θ = ( d B 1 ) ∗ θ + f ∗ dt (19) Therefore, ∂ B θ − p ∗ 13 ( ∂ D θ ) =( d B 1 ) ∗ θ − p ∗ 13 ( d D 1 ) ∗ θ b y (17), = − f ∗ dt b y (19).  A ccording to Prop osition 4.15, the connection θ induces a co cycle Ω θ ∈ Z 3 DR ( A ) . Theorem 4.23. φ ∗ [Ω θ ] = f ∗ [ dt ] in H 3 ( B ) Pr o of. By construction, the co cycle Ω θ is the sum Ω θ = η + ω , where φ ∗ ( η ) = ∂ D θ and φ ∗ ( ω ) = d DR θ . ∂ B θ + dθ = p ∗ 13 ( ∂ D θ ) − f ∗ ( dt ) + dθ b y Lemma 4.22 = p ∗ 13 ( φ ∗ η ) − f ∗ ( dt ) + φ ∗ ω = φ ∗ ( η + ω ) − f ∗ ( dt )  No w, Theorem 4.18 in the case G = S 1 follo ws from Theorem 4.23 since C C φ =  φ ∗  − 1 ( f ∗ ( dt )) and D D ( φ ) = [Ω θ ] (by Prop osition 4.15). 4.6. Pro of of Theorem 4.18. By [32] (see also Section 4.3), the central G -extension of Lie group oids ˜ Γ → Γ ⇒ M induces a centr al Z ( G ) -extension ˜ Γ 0 → Γ ⇒ M , where ˜ Γ 0 = q − 1 ( σ (Γ)) . W e recov er ˜ Γ from ˜ Γ 0 b y the formula ˜ Γ ∼ = ˜ Γ 0 × G Z ( G ) , where the Z ( G ) -action on ˜ Γ 0 × G is giv en b y ( x, g ) · z = ( x · z − 1 , z · g ) for x ∈ ˜ Γ 0 , g ∈ G and z ∈ Z ( G ) . The natural inclusion τ : ˜ Γ 0 → ˜ Γ coincides with the map x 7→ [ x, 1 G ] ∈ ˜ Γ 0 × G Z ( G ) . Since G is compact, Z ( g ) is reductive and w e ha ve the Lie algebra morphism pr : g → Z ( g ) . Lemma 4.24. L et α ∈ Ω 1 ( ˜ Γ , g ) b e a c onne ction 1-form on the right princip al G -bund le ˜ Γ → Γ . Then α 0 := pr( τ ∗ ( α )) ∈ Ω 1 ( ˜ Γ 0 , Z ( g )) is a c onne ction 1- form for the right princip al Z ( G ) -bund le ˜ Γ 0 → Γ . Pr o of. Since the inclusion τ : ˜ Γ 0  → ˜ Γ is Z ( G ) -equiv arian t, we hav e α 0 ( ˆ η x ) = pr ◦ α ◦ τ ∗ ( ˆ η x ) = pr ◦ α ◦ ( ˆ η τ ( x ) ) = pr( η ) = η for all η ∈ Z ( g ) and x ∈ ˜ Γ 0 . Similarly , for an y h ∈ Z ( G ) , we hav e R ∗ h ( α 0 ) = R ∗ h (pr( τ ∗ ( α ))) = pr τ ∗ R ∗ h α = pr τ ∗ Ad h − 1 α = α 0 .  Since ˜ Γ 0 → Γ ⇒ M is a Z ( G ) -cen tral extension, by Lemma 4.24 and Prop osition 4.15, w e hav e the Dixmier–Douady class D D ( α 0 ) ∈ H 3 (Γ) ⊗ Z ( g ) . Prop osition 4.25. W e have D D ( α 0 ) = D D ( α ) . PRINCIP AL 2-GROUP BUNDLES 33 Pr o of. By Lemma 4.24 and Prop osition 4.15.(b), w e can use the 1-form α 0 to calculate the Dixmier–Douady class of ˜ Γ 0 → Γ ⇒ M . By construction we ha ve a commutativ e diagram of group oid morphisms ˜ Γ 0 φ 0   τ   ˜ Γ φ / / Γ . A ccording to Prop osition 4.15.(a), the Dixmier–Douady class D D ( α ) is the cohomology class of the co cycle [Ω α ] deﬁned by the identit y pr  dα + ∂ α  = φ ∗ (Ω α ) . (20) Applying τ ∗ to Equation (20), we get τ ∗ pr  dα + ∂ α  = τ ∗ φ ∗ (Ω α ) pr  d + ∂  τ ∗ α = φ 0 ∗ (Ω α )  d + ∂  pr τ ∗ α = φ 0 ∗ (Ω α ) . Therefore, by Prop osition 4.15, D D ( α 0 ) = [Ω α ] = D D ( α ) .  Since G is compact its cen ter is the quotient ( Z 0 ( G ) × C ) / N , where Z 0 ( G ) is the connected comp onent of 1 G in Z ( G ) and C , N are ﬁnite. W e ﬁx an isomorphism of Lie groups Z 0 ( G ) ∼ = S 1 × · · · × S 1 (with n -factors). W e th us obtain isomorphisms Z ( g ) ∼ = R e 1 ⊕ · · · ⊕ R e n and H 3 ([ Z ( G ) → 1]) ∼ = R dt 1 ⊕· · · ⊕ R dt n . Let pr i : Z ( g ) → R e i ( i = 1 . . . n ) b e the natural pro jection. Lemma 4.26. W e have pr i  C C Φ  = C C Φ ( dt i ) in H 3 (Γ) . Pr o of. Let ( ξ 1 , . . . , ξ n ) b e the dual basis of ( e 1 , . . . , e n ) in Z ( g ) ∗ . According to [25], the generator dt i is the left inv arian t vector ﬁeld ξ L i ∈ Ω 1 ( Z ( G )) ⊂ Ω 3 ([ Z ( G ) → 1]) asso ciated to ξ i . The Lemma follows.  Prop osition 4.27. W e have C C Φ = D D ( α 0 ) Pr o of. By linearit y and Lemma 4.24, it is suﬃcient to pro v e that for all i = 1 . . . n , one has pr i  D D ( θ )  = pr i  C C Φ  = C C Φ ( dt i ) (b y Lemma 4.26) . (21) The pro of of Equation (21) is similar to that of Theorem 4.23.  Pr o of of The or em 4.18. By Prop osition 4.25 and Prop osition 4.27 w e obtain C C Φ = D D ( α 0 ) = D D ( α ) and Theorem 4.18 follows.  Example 4.28. Let G b e a simple compact Lie group and let LG denote its lo op group. Supp ose that the Lie algebra g of G is endo wed with m inv arian t 34 GRÉGOR Y GINOT AND MA THIEU STIÉNON non-degenerate bilinear symmetric forms h− , −i i ( i = 1 , . . . , m ) and assume that the Lie algebra 2-co cycle β ∈ Λ 2 ( L g ∗ ) ⊗ R m deﬁned by β ( X , Y ) =  1 2 π Z 2 π 0 h X ( s ) , Y 0 ( s ) i 1 ds, · · · , 1 2 π Z 2 π 0 h X ( s ) , Y 0 ( s ) i m ds  is in tegral on ev ery factor (i.e. the asso ciated closed 2-form is). It thus gives rise to a central extension T m → g LG → LG of the lo op group b y a torus T m of dimension m . The extension b eing central, the adjoin t action of g LG on its Lie algebra f L g = L g ⊕ R m descends to an action on L g (which is compatible with the adjoin t action of LG on L g ). Hence we hav e a central T m -extension  g LG × L g ⇒ L g  − →  LG × L g ⇒ L g  (22) of the asso ciated transformation Lie group oids. Since  LG × L g ⇒ L g  is Morita equiv alent 8 to the transformation group oid G × G ⇒ G (where G acts on itself by conjugation), the extension (22) deﬁnes a central T m -gerb e on G × G ⇒ G . W e compute the universal characteristic class of this cen tral gerb e us- ing the Dixmier–Douady class as follows. Note that the coadjoin t action of g LG on f L g ∗ also restricts to L g ∗ (iden tiﬁed with the aﬃne hyperplane { ( x, 1 , . . . , 1); x ∈ L g } ). Let α = α 1 ⊕ · · · ⊕ α m ∈ Ω 1 ( g LG × L g ) ⊗ R m b e the direct sum of the dual (using on each co ordinate of R m the identi- ﬁcation of L g with L g ∗ giv en b y the form h , i i ) of the restriction of the Liouville 1 -form on g LG × f L g ∗ to Ω 1 ( g LG × L g ∗ ) . By Prop osition 4.15 and Lemma 4.26, we obtain that the Dixmier–Douady class of the gerb e is a sum D D ( α ) = [Ω α 1 ] ⊕ · · · ⊕ [Ω α m ] . By computation (see [10, Proposition 3.2]), we see that each class [Ω α i ] is equal in H 3 ( G × G ⇒ G ) to the class of [ b i + λ i ] where λ i is the bi-inv ariant 3 -form coresp onding to 1 12 h− , [ − , − ] i i ∈ Λ 3 g ∗ and b i = − 1 2 [ h Ad x g ∗ ( θ M C ) , g ∗ ( θ M C ) i i + h g ∗ ( θ M C , x ∗ ( θ M C + θ r M C ) i i ] ∈ Ω 2 ( G × G ) ⊂ Ω 3 ( G × G ⇒ G ) , where θ M C and θ r M C are respectively the left and righ t Maurer-Cartan forms and ( g , x ) denotes the co ordinates on G × G . Since G is a simple compact group, each h− , −i i is an integer multiple h− , −i i = a i h− , −i bas of the basic form of G . Hence [ b i + λ i ] = a i [ b bas + λ bas ] (where b bas , λ bas are deﬁned as ab ov e). F urther, [ b bas + λ bas ] is precisely an integral generator of H 3 ( G × G ⇒ G ) . 8 The equiv alence is induced by the pro jection (ev 0 , Hol) , where ev 0 is the ev aluation map whic h takes a loop in G to its v alue at 0 and Hol : L g → G is the holonomy . 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MR2549951 (2011c:18006) 38 GRÉGOR Y GINOT AND MA THIEU STIÉNON UPMC – Sorbonne Universités – P aris 6, Institut de Ma théma tiques de Jussieu–P aris Rive Ga uche E-mail addr ess : gregory.ginot@imj-prg.fr Penn St a te University, Dep ar tment of Ma thema tics E-mail addr ess : stienon@psu.edu

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