Four Equivalent Versions of Non-Abelian Gerbes

F our Equiv alen t V ersions o f Non-Ab elian Gerb es Thomas Nik olaus a and Konrad W aldorf b a F ac hbereich Mathematik, Bereich Algebra und Zahle ntheorie Bundesstraße 55, 2 0146 Hamburg, Ger many b F akult¨ at f¨ ur Mathematik, Universit¨ at Regensburg Univ er s it¨ atsstra ße 31, 93053 Rege ns burg, Germa ny Abstract W e recall and partially improv e four vers ions of smo oth , n on-ab elian gerbes: ˇ Cec h cocycles, classifying maps, bundle gerb es, and principal 2-bun dles. W e pro ve that all these four ver- sions are equiv alent, and so establish new rela tions betw een in teresting recent develo pments. Prominen t partial res ults that w e pro ve are a bijection betw een t he continuous and smo oth non- abelian cohomology , and an explicit equiv alence betw een bund le gerb es and p rin cipal 2-b undles as 2-stacks. Keywords : non-abelian gerb e, principal 2-bundle, 2-group, non-ab elian cohomology , 2-stack MSC 20 10 : Pr imary 55R65, Secondary 53C08,55N05,22A22 Con ten ts 1 In tro duction 2 2 Preliminaries 5 2.1 Lie Group oids and Group o id Actions on Ma nifolds . . . . . . . . . . . . . . . . . . . 5 2.2 Principal Group oid Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Anafunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Lie 2-Groups a nd cr o ssed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 3 V ersion I: Groupoi d-v alued Cohomolo gy 16 4 V ersion I I: Classifying Maps 18 5 V ersion I I I: Group oid Bundle Gerb es 21 5.1 Deﬁnition via the Plus Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 Prop erties of Groupo id Bundle Gerb es . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 Classiﬁcation by ˇ Cech Co homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6 V ersion IV: Principal 2-Bundles 33 6.1 Deﬁnition of P rincipal 2-Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.2 Prop erties of Principal 2 -Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7 Equiv alence b etw een Bundle Gerb es and 2-Bundles 38 7.1 F rom Pr incipal 2-B undles to Bundle Ger be s . . . . . . . . . . . . . . . . . . . . . . . 3 9 7.2 F rom Bundle Gerb es to Pr incipal 2-Bundles . . . . . . . . . . . . . . . . . . . . . . . 48 A Equiv arian t Anafunctors and Group Actions 57 B Cons tructing Equiv al e nces be t w een 2-Stac ks 59 References 62 1 In tro duc tion Let G be a Lie gro up and M be a smo oth manifold. There are (among others) the following four wa ys to say what a smo oth G -bu nd le ov er M is: (1) ˇ Ce ch 1-Co cycles : an op en cover { U i } of M , and for each non-empty in terse ction U i ∩ U j a smo oth ma p g ij : U i ∩ U j / / G satisfying the co cycle condition g ij · g j k = g ik . (2) Cla ssifying maps : a contin uous map f : M / / B G to the classifying space B G of the gr oup G . (3) Bund le 0-gerb es : a surjective submersio n π : Y / / M and a smo o th map g : Y × M Y / / G satisfying π ∗ 12 g · π ∗ 23 g = π ∗ 13 g , – 2 – where π ij : Y × M Y × M Y / / Y × M Y denotes the pr o jection to the i th and the j th factor s. (4) Princi p al bund les : a sur jective submer sion π : P / / M with a s mo oth a ction of G on P that preserves π , such that the ma p P × G / / P × M P : ( p, g ) ✤ / / ( p, p.g ) is a diﬀeomorphism. It is well-known that these f our v ersions of “smo oth G -bundles” are a ll equiv alent. Indeed, (1) forms the smoo th G -v a lued ˇ Cech cohomo lo gy in degree o ne , whereas (2) is known to be equiv alent to co nt inuous G -v alued ˇ Cech co homology , which in turn coincides with the smo oth G -v alued ˇ Cech cohomolog y . F urther, (3) a nd (4) form equiv alent categor ie s ; and is omorphism cla sses of the o b jects (3) are in bijection with eq uiv alence clas ses of the co cycles (1). In this article w e provide an analogo us picture fo r smo oth Γ -gerb es , where Γ is a strict Lie 2-gro up. In particular, Γ can be the automor phism 2-gr oup o f an o rdinary Lie group G , in whic h case the ter m “non-ab elia n G -gerb e” is commonly used. W e compare the following four versions: V ersio n I: Smo oth, non-ab elian ˇ Ce ch Γ -c o cycles (Deﬁnition 3.6). These form the clas sical, smoo th group oid-v alued cohomology ˇ H 1 ( M , Γ) in the s ense of Giraud [Gir71] and Breen [Br e 9 0, Ch. 4], [B r e94]. V ersio n I I: Classifyi ng maps (Deﬁnition 4 .4). These are co nt inuous maps f : M / / B | Γ | to the classifying space of the geometric realization of Γ; s uch maps hav e b een in tro duced and studied by Ba ez a nd Stevenson [BS0 9]. V ersio n I I I: Γ -bund le gerb es (Deﬁnition 5.1.1). These hav e b een developed by Aschieri, Ca nt ini and Jurco [A CJ05] as a generalization of the abelian bundle gerb es of Murray [Mur9 6]. Here we present an equiv alent de ﬁnitio n by applying a higher ca tegorica l v ersion [NS11] o f Grothendieck’s stackiﬁcation construction to the monoidal pre - 2-stack of principal Γ-bundles. V ersio n IV: Princip al Γ -2-bund les (Deﬁnition 6.1.5). These hav e b een intro duced by Ba rtels [Bar04]; their total spaces ar e Lie group oids on which the L ie 2-group Γ a cts in a cer tain wa y . Compared to Ba rtels’ deﬁnition, ours uses a stricter and e a sier no tion of such a n ac tion. W e prove tha t all four v ersio ns are equiv a lent, and follow the sa me line of a rguments as in the ca se of G -bundles outlined ab ov e: • Ba ez and Stev enson have sho wn that homotopy classes of classifying maps of V ersion II are in bijection with the contin uous group oid-v alued ˇ Cech cohomology ˇ H 1 c ( M , Γ). W e pro ve (Prop ositio n 4.1) that the inclusion of smo oth into c ont inuous ˇ Cech Γ-co cy c le s induces a bijection ˇ H 1 c ( M , Γ) ∼ = ˇ H 1 ( M , Γ). These tw o results establish the equiv alence betw een our V ersio ns I and I I (Theore m 4.6). – 3 – • Γ- bundle gerb es and principa l Γ-2-bundles over M form bicategories . W e prov e (Theorem 7.1) that these bicategor ie s are equiv alent, a nd so establis h the eq uiv alence b etw een V ersions II I and IV. Our pro of pr ovides ex plic it 2-functor s in b oth directions. • W e prov e the equiv alence betw een V ersions I and II I by showing that no n-ab elian Γ-bundle gerb es a re classiﬁed by the non-ab elian cohomolo gy group ˇ H 1 ( M , Γ) (Theor e m 5.3 .2). The ﬁrst a im of this paper is to simplify a nd clarify the notion of a non-ab elian ger b e. This concerns the notion of a Γ-bundle ger b e (V ersion II I), for whic h we give a new, co nceptually clear, a nd manifestly 2 -categor ical deﬁnition. It also concerns the no tion of a pr incipal 2-bundle (V ersio n IV), for whic h w e provide a new deﬁnition that is car efully balanced b etw een generality and simplicity . The second aim of this pap er is to mak e it p ossible to co mpa re and transfer av ailable results betw een the v arious versions. Indeed, none of th e thr e e equiv alences ab ove is av ailable in the existing literature. As an example why such equiv alences can be useful, we use Theo rem 7.1 – the equiv alenc e betw een Γ-bundle gerb es a nd principal Γ- 2-bundles – in o rder to carry tw o facts ab out Γ-bundle gerb es over to principal Γ-2-bundles. W e prove: 1. Principal Γ-2-bundles form a 2-stack ov er smo oth manifolds (Theorem 6.2.1). This is a new and evidently impo rtant result, s inc e it expla ins precisely in which wa y one can glu e 2-bundles from lo cal patches. 2. If Γ and Ω ar e w eak ly equiv alent Lie 2-g roups, the 2-stacks of principal Γ-2-bundles and prin- cipal Ω - 2-bundles a re equiv ale nt (Theor em 6.2 .3). This is another new result that gener alizes the well-known fact that principal G -bundles a nd principa l H -bundles form equiv a lent stacks, whenever G and H a re isomorphic Lie gro ups. The tw o facts ab out Γ-bundle gerb es (Theorems 5.1.5 and 5 .2.2) on which these results are ba sed are prov ed in an outmos t abstr act wa y: the ﬁrst is a mer e consequence of the de ﬁnitio n of Γ-bundle gerb es that w e give, namely via a 2 -stackiﬁcation pro cedure for principal Γ-bundles. The seco nd follows from the fact that principal Γ-bundles and principal Ω-bundles form equiv alent monoidal pre-2-s ta cks, which we deduce as a c o rollar y of their description by ana functors. The present paper is par t of a la rger progr am. In a forthcoming pa p er , we addres s the discussion of non-ab elian lift ing problems, in particular string s tructures. In a second for thcoming paper we will presen t the picture of fo ur equiv alent versions in a setting with c onne ctions , ba s ed on the res ults of the presen t paper. Our motiv ation is to under stand the role of 2-bundles with connections in higher gaug e theories , wher e they serve as “B-ﬁelds”. Her e, t wo (non-ab elian) 2 - groups ar e esp ecially imp or tant, namely the string gr o up [BCSS07] and the Jandl gr oup [NS11]. More pr ecisely , string-2-bundles app ea r in super symmetric sig ma mo dels that describ e fermionic – 4 – string theories [Bun11]; while Jandl-2- bundles app ea r in unoriented sigma mo dels that describ e e.g. bo sonic t yp e-I string theo ries [SSW07]. This pap er is o rganized as follows. I n Section 2 we recall and summar ize the theory o f principa l group oid b undles and their description b y anafunctors. The r e st o f the pap er is bas e d on this theory . In Sections 3, 4, 5 and 6 we int ro duce our four versions of smo oth Γ-gerb es, and establish all but one equiv alence. The r emaining equiv alence, the one betw een bundle gerb es a nd principal 2-bundles, is dis cussed in Section 7. Ac kno wledgem en ts. W e thank Christoph W o ck el for helpful discussions. W e also thank the Erwin Schr¨ odinger Institute in Vienna and the Instituto Sup erior T´ ecnico in Lisb o n for kind invita- tions. TN is s uppo rted by the Collab o r ative Research Cen tre 67 6 “ Particles, Strings and the E arly Univ er s e - the Str uc tur e o f Matter and Sp ace - Time” and the cluster of excellence “C o nnecting particles with the cosmo s”. 2 Preliminaries There is no claim of origina lity in this section. O ur sources ar e Lerma n [Ler], Metzler [Met], Heinloth [Hei04] and Mo erdijk-Mrˇ cun [MM03]. A s lightly diﬀerent but equiv alent approach is developed in [MRS]. 2.1 Lie Gr oup oids and Group oid Actions on Manifolds W e assume that t he reader is familia r with the notions o f Lie group oids, smo oth functors and smo oth na tural transformations. In this pap er, the following examples of Lie group oids a pp e ar: Example 2.1.1. (a) Ev ery smo oth manifold M deﬁnes a Lie group oid deno ted by M dis whose ob jects and morphisms are M , and a ll o f whose structur e maps are ident ities. (b) Every Lie group G deﬁnes a Lie group oid denoted by B G , with one ob ject, with G as its smo oth manifold o f morphisms, a nd with the co mpo sition g 2 ◦ g 1 := g 2 g 1 . (c) Supp o s e X is a s mo oth manifold and ρ : H × X / / X is a smoo th left action of a Lie gr oup H on X . Then, a Lie group oid X/ /H is deﬁned with ob j ects X and morphisms H × X , and with s ( h, x ) := x , t ( h, x ) := ρ ( h, x ) and id x := (1 , x ). The co mp o s ition is ( h 2 , x 2 ) ◦ ( h 1 , x 1 ) := ( h 2 h 1 , x 1 ), – 5 – where x 2 = ρ ( h 1 , x 1 ). The Lie group oid X/ /H is called the action gr oup oid of the actio n of H on X . (d) Let t : H / / G be a homomorphism of Lie groups. Then, ρ : H × G / / G : ( h, g ) ✤ / / ( t ( h ) g ) deﬁnes a smo oth left action o f H on G . Thus, we hav e a Lie group oid G/ /H . (e) T o every Lie gro up oid Γ one can asso ciate an opp osite Lie gr oup oid Γ op which has the source and the targ et map exchanged. W e say that a right action of a Lie group oid Γ on a smo oth manifold M is a pair ( α, ρ ) consisting of smo o th maps α : M / / Γ 0 and ρ : M α × t Γ 1 / / M such that ρ ( ρ ( x, g ) , h ) = ρ ( x, g ◦ h ) , ρ ( x, id α ( x ) ) = x and α ( ρ ( x, g )) = s ( g ) for all po ssible g , h ∈ Γ 1 , p ∈ Γ 0 and x ∈ M . The map α is called anchor . Later o n we will replace the letter ρ for the action by the symbol ◦ that denotes the co mpo sition of the group o id. A left action of Γ on M is a right action o f the opp osite Lie group oid Γ op . A smo oth map f : M / / M ′ betw een Γ-spa c es with actions ( α, ρ ) and ( α ′ , ρ ′ ) is called Γ -e qu ivariant if α ′ ◦ f = α a nd f ( ρ ( x, g )) = ρ ′ ( f ( x ) , g ). Example 2.1.2. (a) Let Γ be a Lie g r oup oid. Then, Γ acts on the right on its mo rphisms Γ 1 by α := s and ρ := ◦ . It acts on the left on its morphisms by α := t and ρ := ◦ . (b) Let G b e a Lie g roup. Then, a rig ht/left action o f the Lie group oid B G (see Example 2 .1.1 (b)) on M is the same as a n or dinary s mo oth r ight/left a ction of G on M . (c) L et X be a smo oth manifold. A right/left action o f X dis (see Example 2.1 .1 (a)) on M is the same as a smo oth map α : M / / X . 2.2 Principal Groupoid Bundles W e giv e the deﬁnition of a pr incipal bundle in exactly the sa me wa y as we are going to deﬁne principal 2- bundles in Section 6. Deﬁnition 2.2.1. L et M b e a s m o oth manifo ld, and let Γ b e a Lie gr ou p oid. – 6 – 1. A princip al Γ -bun d le over M is a smo oth manifold P with a surje ctive submersion π : P / / M and a right Γ -action ( α, ρ ) that r esp e cts the pr oje ction π , such that τ : P α × t Γ 1 / / P × M P : ( p, g ) ✤ / / ( p, ρ ( p, g )) is a diﬀe omorphism. 2. L et P 1 and P 2 b e princip al Γ - bu nd les over M . A morphism ϕ : P 1 / / P 2 is a Γ -e quivariant smo oth map that r esp e cts t he pr oje ctions to M . Principal Γ-bundles over M form a ca tegory B u n Γ ( M ). In fact, this catego r y is a group oid, i.e. all morphisms b etw e e n principal Γ-bundles are in vertible. There is an evident notion of a pullback f ∗ P of a principal Γ-bundle P ov er M along a smo oth map f : X / / M , and similar ly , mor phis ms betw een principal Γ-bundles pull back. These deﬁne a functor f ∗ : B u n Γ ( M ) / / B un Γ ( X ). These functors make principa l Γ-bundles a prestack ov er s mo oth ma nifo lds . O ne can easily show that this pr estack is a stack (for the Grothendieck top ology of sur jective s ubmersions). Example 2.2.2 (Ordinar y principal bundles) . F or G a Lie group, w e hav e an equality of catego ries B un B G ( M ) = B un G ( M ), i.e. Deﬁnition 2.2.1 reduces consistent ly to the deﬁnition of an ordinary principal G -bundle. Example 2.2.3 (T riv ial pr incipal gro upo id bundles) . F or M a smo oth manifold and f : M / / Γ 0 a smo oth map, P := M f × t Γ 1 and π ( m, g ) := m deﬁne a surjective submersion, and α ( m, g ) := s ( g ) and ρ (( m, g ) , h ) := ( m, g ◦ h ) deﬁne a rig ht action of Γ on P that preserves the ﬁb ers . The map τ we have to lo ok at ha s the inv erse τ − 1 : P × M P / / P π × t Γ 1 : (( m, g 1 ) , ( m, g 2 )) ✤ / / (( m, g 1 ) , g − 1 1 ◦ g 2 ), which is smoo th. Thus we have deﬁned a principal Γ-bundle, whic h we deno te by I f and which we call the trivial bund le for the map f . An y bundle that is isomorphic to a trivial bundle is called trivializabl e . Example 2.2.4 (Discrete structure gro upo ids) . F or X a smo oth manifo ld, we have an equiv a lence of categories B un X dis ( M ) ∼ = C ∞ ( M , X ) dis . Indeed, for a giv en principal X dis -bundle P one obse rves that the a nchor α : P / / X desc ends along the bundle pro jection to a smo o th map f : M / / X , and that isomorphic bundles determine the same map. Conv ersely , o ne asso ciates to a smoo th map f : M / / X the trivial principa l X dis - bundle I f ov er M . – 7 – Example 2.2.5 (Exact sequenc e s ) . Let 1 / / H t / / G p / / K / / 1 (2.2.1) be an exa c t sequenc e of Lie g roups, a nd let Γ := G/ /H b e the action gr oup oid as so ciated to the Lie group ho momorphism t : H / / G as explained in Example 2.1.1 (d). In this situation, p : G / / K is a sur jective submer sion, and α : G / / Γ 0 : g ✤ / / g and ρ : G α × t Γ 1 / / G : ( g , ( h, g ′ )) ✤ / / g ′ deﬁne a smo o th rig ht actio n of Γ on G that preser ves p . The inv erse of the map τ is τ − 1 : G × K G / / G α × t Γ 1 : ( g 1 , g 2 ) ✤ / / ( g 1 , ( t − 1 ( g 1 g − 1 2 ) , g 2 )), which is smoo th b ecause t is an embedding. Thus, G is a principal Γ-bundle ov er K . Next we pr ovide so me elementary statemen ts abo ut trivial pr inc ipa l Γ- bundles. Lemma 2.2.6. A princip al Γ -bund le over M is triviali zable if and only if it has a smo oth se ction. Pro of. A trivial bundle I f has the sectio n s f : M / / I f : x ✤ / / ( x, id f ( x ) ); and so an y trivia liz able bundle has a section. Conv ersely , supp ose a principal Γ-bundle P has a smo oth s ection s : M / / P . Then, with f := α ◦ s , ϕ : I f / / P : ( m, g ) ✤ / / ρ ( s ( m ) , g ) is a n isomorphism.  The following consequence shows that pr incipal Γ-bundles of Deﬁnition 2 .2.1 are lo ca lly tr iv ial- izable in the usual sense. Corollary 2.2. 7. L et P b e a princip al Γ -bund le over M . Then, every p oint x ∈ M ha s an op en neighb orho o d U over which P has a trivializa tion: a smo oth map f : U / / Γ 0 and a morphism ϕ : I f / / P | U . Pro of. One can choo se U s uch that the surjective submers io n π : U / / P has a smo oth section. Then, Lemma 2.2 .6 a pplies to the restriction P | U .  W e determine the Hom-set H om ( I f 1 , I f 2 ) b etw een trivial pr incipal Γ-bundles deﬁned by smo oth maps f 1 , f 2 : M / / Γ 0 . T o a bundle morphism ϕ : I f 1 / / I f 2 one asso ciates the smo oth function g : M / / Γ 1 which is uniquely deﬁned by the condition ( ϕ ◦ s f 1 )( x ) = s f 2 ( x ) ◦ g ( x ). for all x ∈ M . It is str a ightforw ard to see that – 8 – Lemma 2.2.8. The ab ove c onstruction deﬁnes a bije ction H om ( I f 1 , I f 2 ) / / { g ∈ C ∞ ( M , Γ 1 ) | s ◦ g = f 1 and t ◦ g = f 2 } , under whic h identity morphisms c orr esp ond to c onstant maps and the c omp osition of bund le mor- phisms c orr esp onds to the p oint-wise c omp osition of functions. Finally , we co nsider the ca se o f principa l bundles for action group oids. Lemma 2 . 2.9. Supp ose X/ / H is a smo oth action gr oup oid. The c ate gory B un X/ /H ( M ) is e quivalent to a c ate gory with • O bje cts: princip al H -bund les P H over M to gether with a smo ot h, H -anti-e quivariant map f : P H / / X , i.e. f ( p · h ) = h − 1 f ( p ) . • Morphisms: bund le morphi sms ϕ H : P H / / P ′ H that r esp e ct the maps f and f ′ . Pro of. F or a principal X/ / H -bundle ( P , α, ρ ) w e set P H := P with the giv en pr o jection to M . The action of H on P H is deﬁned by p ⋆ h := ρ ( p, ( h, h − 1 · α ( p ))). This a ction is smo oth, and it follows from the axioms of the pr incipal bundle P that it is principal. The map f : P H / / X is the anc hor α . The r emaining steps are s traightforward and left as an exercise.  2.3 Anafunctors An anafunctor is a ge ner alization of a smo oth functor betw een Lie groupo ids, similar to a Mor ita equiv alenc e , and also k nown as a Hilsum-Sk anda lis morphism. The idea go es back to Bena bo u [B´ e n73], als o see [J oh77]. The re fer ences for the following deﬁnitions a re [Ler, Met]. Deﬁnition 2.3.1. L et X and Y b e Lie gr oup oids. 1. A n anafunctor F : X / / Y is a smo oth manifold F , a left action ( α l , ρ l ) of X on F , and a right action ( α r , ρ r ) of Y on F su ch that the actions c ommu te and α l : F / / X 0 is a princip al Y -bund le over X 0 . 2. A tr ansformation b etwe en anafunctors f : F + 3 F ′ is a smo oth map f : F / / F ′ which is X -e quivariant, Y -e quivariant, and satisﬁes α ′ l ◦ f = α l and α ′ r ◦ f = α r . The smo oth manifold F of an ana functor is called its total sp ac e . Notice that the co ndition that the tw o actions on F c o mmut e implies that each re s pe cts the anchor of the other. F or ﬁxed Lie – 9 – group oids X and Y , anafunctors F : X / / Y and transfor mations form a categor y A na ∞ ( X , Y ). Since tra nsformations are in particular morphisms b etw een pr incipal Y -bundles, every transfo rma- tion is inv ertible so tha t A na ∞ ( X , Y ) is in fact a gr oup oid. Example 2.3.2 (Anafunctors fr om ordinary functors) . Given a smo oth functor φ : X / / Y , we obtain an a nafunctor in the follo wing wa y . W e set F := X 0 φ × t Y 1 with anchors α l : F / / X 0 and α r : F / / Y 0 deﬁned b y α l ( x, g ) := x and α r ( x, g ) := s ( g ), and actions ρ l : X 1 s × α l F / / F and ρ r : F α r × t Y 1 / / F deﬁned b y ρ l ( f , ( x, g )) := ( t ( f ) , φ ( f ) ◦ g ) and ρ r (( x, g ) , f ) := ( x, g ◦ f ). In the same w ay , a smo oth natural transfor mation η : φ + 3 φ ′ deﬁnes a transfor mation f η : F + 3 F ′ by f η ( x, g ) := ( x, η ( x ) ◦ g ). Conv er s ely , one can show that an a nafunctor comes from a smo oth functor, if its pr incipal Γ-bundle has a smo oth section. Example 2.3.3 (Anafunctor s with discrete source) . F o r M a smooth manifold a nd Γ a L ie group oid, we have an equality o f categories B un Γ ( M ) = A na ∞ ( M dis , Γ). F urther, trivial principal Γ-bundles corresp ond to smo o th functors . In particular , with Exa mple 2.2.2 we ha ve, (a) F or G a Lie g r oup and M a smo oth manifold, an anafunctor F : M dis / / B G is the same as an o rdinary principal G -bundle ov er M . (b) F or M and X smo oth manifolds, an anafunctor F : M dis / / X dis is the sa me a s a smo oth map. Example 2.3 . 4 (Anafunctors with discrete targ et) . F o r Γ a Lie g roup oid and M a smo oth ma nifold, we have a n equiv alence of categories C ∞ (Γ 0 , M ) Γ dis ∼ = A na ∞ (Γ , M dis ) where C ∞ (Γ 0 , M ) Γ denotes the set of smoo th maps f : Γ 0 / / M s uch that f ◦ s = f ◦ t as maps Γ 1 / / M . The equiv a le nc e is induced by reg arding a map f ∈ C ∞ (Γ 0 , M ) Γ as a s mo oth functor f : Γ / / M dis , whic h in tur n induces an anafunctor. Conv ersely , an anafunctor F : Γ / / M dis is in particular an M dis -bundle ov er Γ 0 , which is nothing but a smo o th function f : Γ 0 / / M by Example 2.2.4. The additional Γ-action a ssures the Γ-inv aria nce of f . Example 2.3 .5 (Anafunctors b etw een one-o b ject Lie group oids) . Let G a nd H b e Lie groups, and let B G and B H b e the as so ciated one-ob ject Lie gr oup oids (Example 2.1.1 b). Then, there is an eq uiv alence of categories Hom( G, H ) / /H ∼ = A na ∞ ( B G, B H ), – 10 – where the actio n of H on Hom( G, H ) is b y p oint-wise conjugation. The functor whic h establishes this equiv a lence sends a smo oth gro up homo morphism α : G / / H to the evident smo oth functor F α : B G / / B H a nd conv er ts this into an anafunctor (Ex ample 2 .3.2). A mor phis m h : α 1 / / α 2 is se nt to the na tural transforma tion η h : F α 1 / / F α 2 whose comp onent at the single ob ject is the morphism h ∈ H . In order to see that this is essen tially surjective, it suﬃces to notice that the principal H -bundle o f an y smooth anafunctor F : B G / / B H ha s a section. The pro of that the functor is full and faithful is str aightforw ar d. F or the follo wing deﬁnition, w e supp ose X , Y and Z are Lie group oids, and F : X / / Y and G : Y / / Z ar e a nafunctors g iven by F = ( F, α l , ρ l , α r , ρ r ) and G = ( G, β l , τ l , β r , τ r ). Deﬁnition 2.3. 6. The c omp osition G ◦ F : X / / Z is the anafunctor deﬁne d in the fol lowing way: 1. Its total sp ac e is E := ( F α r × β l G ) / ∼ wher e ( f , τ l ( h, g )) ∼ ( ρ r ( f , h ) , g ) fo r al l h ∈ Y 1 with α r ( f ) = t ( h ) and β l ( g ) = s ( h ) . 2. The anchors ar e ( f , g ) ✤ / / α l ( f ) and ( f , g ) ✤ / / β r ( g ) . 3. The actions X 1 s × α E / / E and E β × t Z 1 / / E ar e given, r esp e ct ively, by ( γ , ( f , g )) ✤ / / ( ρ l ( γ , f ) , g ) and (( f , g ) , γ ) ✤ / / ( f , τ r ( g , γ )) . Remark 2. 3.7. Lie group oids, anafunctors a nd transfor ma tions form a bicatego r y . This bicatego ry is equiv alent to the bicategory o f diﬀeren tiable stacks (als o known as geometric s tacks) [Pro9 6]. In this a rticle, anafunctors serve t wo purp o ses. The ﬁr st is that one can use con venien tly the comp osition of anafunctors to deﬁne extensions of principal group oid bundles: Deﬁnition 2.3. 8. If P : M dis / / Γ is a princip al Γ -bund le over M , and Λ : Γ / / Ω is an anafunctor, t hen the princip al Ω -bund le Λ P := Λ ◦ P : M dis / / Ω is c al le d the extension of P along Λ . Un winding this deﬁnition, the principal Ω-bundle Λ P has the total space Λ P = ( P α × α l Λ) / ∼ (2.3.1) where ( p, ρ l ( γ , λ )) ∼ ( ρ ( p, γ ) , λ ) for all p ∈ P , λ ∈ Λ and γ ∈ Γ 1 with α ( p ) = t ( γ ) and α l ( λ ) = s ( γ ). Here α is the anc hor and ρ is the action o f P , a nd Λ = (Λ , α l , α r , ρ l , ρ r ). The bundle pro jection – 11 – is ( p, λ ) ✤ / / π ( p ), wher e π is the bundle pro jection of P , the anchor is ( p, λ ) ✤ / / α r ( λ ), a nd the action is ( p, λ ) ◦ ω = ( p, ρ r ( λ, ω )). Extensions of bundles are acco mpa nied b y extensio ns of bundle morphisms. If ϕ : P 1 / / P 2 is a morphism b etw ee n Γ-bundles, a morphism Λ ϕ : Λ P 1 / / Λ P 2 is deﬁned b y Λ ϕ ( p 1 , λ ) := ( ϕ ( p 1 ) , λ ) in terms of (2.3.1). Summarizing, w e hav e Lemma 2.3. 9. L et M b e a smo oth manifold and Λ : Γ / / Ω b e an anafunct or. Then, exten sion along Λ is a functor Λ : B un Γ ( M ) / / B un Ω ( M ) . Mor e over, it c ommutes with pul lb acks and so extends t o a m orphism b etwe en stacks. Next we suppo se tha t t : H / / G is a Lie group homomorphis m, and G/ /H is the asso cia ted action gr oup oid of Ex ample 2.1.1 (d). W e loo k at the functor Θ : G/ /H / / B H which is deﬁned by Θ ( h, g ) := h . Co mbining L e mma 2 .2 .9 with the extension a long Θ, we obtain Lemma 2.3.10 . The c ate gory B un G/ /H ( M ) of princip al G/ /H -bun d les over a smo oth manifold M is e quivalent to a c ate gory with • O bje cts: princip al H - bund le s P H over M to gether with a se ction of Θ ( P H ) . • Morphisms: morphisms ϕ of H -bun d les so that Θ( ϕ ) pr eserves the se ctions. The second motiv ation for intro ducing anafunctor s is that they pro vide the inverses to certain smo oth functor s which ar e no t necess arily eq uiv alences of categor ies. Deﬁnition 2 .3.11. A smo oth functor or anafunctor F : X / / Y is c al le d a we ak e quivalenc e , if ther e ex ists an anafunctor G : Y / / X to gether with tr ansformations G ◦ F ∼ = id X and F ◦ G ∼ = id Y . W e hav e the following immedia te cons equence fo r the stack mor phisms o f Lemma 2.3.9. Corollary 2.3.12 . L et Λ : Γ / / Ω b e a we ak e quivalenc e b etwe en Lie gr oup oids. Then, extension of princip al bund les along Λ is an e qu ivalenc e Λ : B u n Γ ( M ) / / B un Ω ( M ) of c ate gories. Mor e over, these deﬁne an e quivalenc e b etwe en the stacks B un Γ and B un Ω . Concerning the claimed generaliza tion of in vertibilit y , w e ha ve the following w ell-known theor em, see [Le r, L e mma 3 .3 4], [Met, Prop o sition 60]. Theorem 2 .3.13. A smo oth functor F : X / / Y is a we ak e quivalenc e if and only if the fol lowi ng two c onditions ar e satisﬁe d: (a) it is smo othly essential ly surje ctive: the m ap s ◦ pr 2 : X 0 F 0 × t Y 1 / / Y 0 is a s u rje ctive submersion. – 12 – (b) it is smo othly ful ly faithful: the diagr am X 1 F / / s × t   Y 1 s × t   X 0 × X 0 F × F / / Y 0 × Y 0 is a pul lb ack dia gr am. Remark 2.3.14. One can show that any smoo th functor F : X / / Y that is a weak equiv alence actually has a c anonic al inv erse a nafunctor. 2.4 Lie 2 -Groups and crossed Mo dules A (strict) Lie 2-gr oup is a Lie group o id Γ whose ob jects and morphisms are L ie groups , and all of whose str ucture ma ps are Lie gro up homo mo rphisms. One can co nv enient ly bundle the m ultiplications and the in versions into smo oth functors m : Γ × Γ / / Γ and i : Γ / / Γ. Example 2.4 .1. F o r A an ab elian Lie g roup, the Lie group o id B A from E xample 2.1.1 (b) is a Lie 2-gro up. The c ondition tha t A is ab elian is nece s sary . Example 2.4 .2. Let t : H / / G b e a ho momorphism of Lie groups , and let G/ /H b e the cor- resp onding Lie group oid from Example 2.1.1 (d). This Lie gro upo id beco mes a Lie 2-gr oup if the following structur e is given: a smo oth left a ction of G on H by L ie group homo mo rphisms, denoted ( g , h ) ✤ / / g h , satisfying t ( g h ) = g t ( h ) g − 1 and t ( h ) x = hxh − 1 for all g ∈ G and h, x ∈ H . Indeed, the ob jects G of G/ / H alr eady form a Lie gro up, a nd the m ultiplication on the morphisms H × G of G/ /H is the semi-direct pro duct ( h 2 , g 2 ) · ( h 1 , g 1 ) = ( h 2 g 2 h 1 , g 2 g 1 ). (2.4.1) The homomor phism t : H / / G together with the action of G on H is called a smo oth cr osse d mo dule . Summar iz ing, every smo o th cr o ssed module deﬁnes a Lie 2-g roup. Remark 2.4.3. Every Lie 2-gr o up Γ can b e obtained from a smo oth crossed module. Indee d, one puts G := Γ 0 and H := ker( s ), equipp e d with the Lie group structure s deﬁned by the multiplication functor m of Γ. The homomor phism t : H / / G is the ta rget map t : Γ 1 / / Γ 0 , and the action of G o n H is given b y the formu la g γ := id g · γ · id g − 1 for g ∈ Γ 0 and γ ∈ k er( s ). Thes e tw o constructions ar e inv erse to each other (up to canonica l Lie gr oup isomorphisms and strict Lie 2-gro up isomor phis ms , resp ectively). – 13 – Example 2.4.4 . Consider a co nnected Lie gr o up H , so that its automor phism gro up Aut( H ) is a gain a Lie group [OV91]. Then, we have a smo oth crossed module (Aut( H ) , H , i , ev), wher e i : H / / Aut( H ) is the assignment of inner automor phis ms to gr oup elements, and ev : Aut( H ) × H / / H is the ev a luation action. T he a sso ciated Lie 2-gro up is denoted AUT( H ) and is called the automorphi sm 2-gr oup of H . Example 2.4.5. Let 1 / / H t / / G p / / K / / 1 be an exa ct sequence of Lie gr o ups, i.e. a n exact sequence in whic h p is a submersion and t is an embedding. The homomorphisms t : H / / G and p : G / / K deﬁne action g roup oids G/ /H and K/ / G as explained in Example 2.1.1. The ﬁrst one is even a Lie 2 -group: the actio n of G on H is deﬁned b y g h := t − 1 ( g t ( h ) g − 1 ). This is well-deﬁned: since p ( g t ( h ) g − 1 ) = p ( g ) p ( t ( h )) p ( g − 1 ) = p ( g ) p ( g ) − 1 = 1, the element g t ( h ) g − 1 lies in the imag e of t , and has a unique preimage. The action is smo oth bec ause t is an embedding. The axioms of a cr o ssed mo dule are ob vio us ly satisﬁed. If a L ie group o id Γ is a Lie 2-gro up in virtue o f a m ultiplication functor m : Γ × Γ / / Γ, then the category B un Γ ( M ) of principal Γ-bundles over a smo oth manifold M is monoidal: Deﬁnition 2.4. 6. L et P : M dis / / Γ and Q : M dis / / Γ b e princip al Γ -bu nd les. The tensor pr o duct P ⊗ Q is the anafunctor M dis diag / / M dis × M dis P × Q / / Γ × Γ m / / Γ . Example 2.4.7. (a) Since trivial principal Γ-bundles I f corres p o nd to smo oth functors f : M dis / / Γ (Example 2.3.3), it is clear that I f ⊗ I g = I f g . (b) Un winding Deﬁnition 2.4.6 in the g eneral ca se, the tensor product of tw o principal Γ-bundles P 1 and P 2 with anc hors α 1 and α 2 , resp ectively , and ac tio ns ρ 1 and ρ 2 , resp ectively , is given by P 1 ⊗ P 2 = (( P 1 × M P 2 ) m ◦ ( α 1 × α 2 ) × t Γ 1 ) / ∼ , (2.4.2) where ( p 1 , p 2 , m ( γ 1 , γ 2 ) ◦ γ ) ∼ ( ρ 1 ( p 1 , γ 1 ) , ρ 2 ( p 2 , γ 2 ) , γ ) (2.4.3) for all p 1 ∈ P 1 , p 2 ∈ P 2 and morphisms γ , γ 1 , γ 2 ∈ Γ 1 satisfying t ( γ i ) = α i ( p i ) for i = 1 , 2 and s ( γ 1 ) s ( γ 2 ) = t ( γ ). The bundle pr o jection is ˜ π ( p 1 , p 2 , γ ) := π 1 ( p 1 ) = π 2 ( p 2 ), the anchor is ˜ α ( p 1 , p 2 , γ ) := s ( γ ), a nd the Γ-action is g iven by ˜ ρ (( p 1 , p 2 , γ ) , γ ′ ) := ( p 1 , p 2 , γ ◦ γ ′ ). – 14 – As a conse quence of Lemma 2 .3.9 and the fact that the co mpo sition o f anafunctors is ass o ciative up to coher e nt trans formations, we have Prop ositi on 2. 4.8. F or M a smo oth m anifold and Γ a Lie 2-gr oup, the tensor pr o duct ⊗ : B un Γ ( M ) × B u n Γ ( M ) / / B un Γ ( M ) e quips the gr oup oid of princip al Γ -bund les over M with a monoidal structur e. Mor e over, it turn s the stack B un Γ into a monoi dal stack. Notice that the tenso r unit of the mo no idal gr oup oid B un Γ ( M ) is the trivia l pr incipal Γ - bundle I 1 asso ciated to the constant map 1 : M / / Γ 0 , or , in ter ms of ana functors, the one asso ciated to the constant functor 1 : M / / Γ. A (weak) Lie 2-gr oup homomorphi sm b etw een Lie 2-gr oups (Γ , m Γ ) and (Ω , m Ω ) is an anafunctor Λ : Γ / / Ω tog ether with a transformatio n Γ × Γ m Γ / / Λ × Λ   Γ Λ   η ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ x  ③ ③ ③ ③ ③ ③ ③ ③ Ω × Ω m Ω / / Ω (2.4.4) satisfying the eviden t coherence condition. Under the equiv alence with smo oth crossed mo dules (Remark 2.4.3), Lie 2-gro up homomorphisms co rresp ond to so- c a lled butterﬂies [AN09]. A Lie 2 - group homomor phism is called we ak e quivalenc e , if the a nafunctor Λ is a weak equiv alence. Since extensions and tensor pro ducts a re bo th deﬁned via compos ition of anafunctors, w e immediately obtain Prop ositi on 2. 4.9. Extension along a Lie 2-gr oup homomorphism Λ : Γ / / Ω b etwe en Lie 2- gr oups is a monoidal functor Λ : B un Γ ( M ) / / B un Ω ( M ) b etwe en monoidal c ate gories. Mor e over, these form a monoidal morphism b etwe en monoidal st acks. Since a monoida l functor is an equiv ale nc e of monoidal catego ries if it is an equiv a lence of the underlying categories, Corollary 2.3 .12 implies: Corollary 2.4.10. F or Λ : Γ / / Ω a we ak e quivalenc e b etwe en Lie 2-gr oups, the monoidal functor of Pr op osition 2.4.9 is an e quivalenc e of monoidal c ate gories. Mor e over, these form a monoidal e quivalenc e b etwe en monoidal st acks. If we repr esent the Lie 2-g roup Γ by a smo oth crossed mo dule t : H / / G as describ ed in Example 2.4.2, we wan t to deter mine explicitly what the tensor pro duct lo oks like under the corres - – 15 – po ndence of G/ /H - bundles a nd principal H -bundles with anti-equiv ar ia nt maps to G , see Lemma 2.2.9. Lemma 2.4. 11. L et t : H / / G b e a cr osse d mo dule and let P and Q b e G/ /H -bund les over M . L et ( P H , f ) and ( Q H , g ) b e the princip al H -bu n d les to gether with their H - anti-e quivariant maps that b elong to P and Q , r esp e ctively, under the e quivalenc e of L emma 2.2.9 . Then, the princip al H -bund le that c orr esp onds to the tensor pr o duct P ⊗ Q is given by ( P ⊗ Q ) H =  P × M Q  / ∼ wher e ( p ⋆ h, q ) ∼ ( p, q ⋆ ( f ( p ) − 1 h )) . The action of H on ( P ⊗ Q ) H is [( p, q )] ⋆ h = [( p ⋆ h, q )] , and the H - anti-e qu ivariant map of ( P ⊗ Q ) H is [( p, q )] ✤ / / f ( p ) · g ( q ) . Similar to the tensor pro duct of principal Γ-bundles, the dual P ∨ of a principal Γ-bundle P ov er M is the ex tens io n of P along the inv ersion i : Γ / / Γ of the 2-group, P ∨ := i ( P ). The equality m ◦ (id , i ) = 1 o f functors M / / Γ induces a “death map” d : P ⊗ P ∨ / / I 1 . W e ar e go ing to use this bundle mor phism in Section 5 .2, but omit a further systematica l treatment of duals for the sake o f brevity . 3 V ersion I: Group oid-v alued Cohomology W e ha ve alr eady mentioned group v alued ˇ Cech 1-co cycles in the in tro duction. They consist of an op en cover U = { U i } i ∈ I of M and smo oth functions g ij : U i ∩ U j / / G satisfying the coc ycle condition g ij · g j k = g ik . Sega l r ealized [Seg68] that this is the same as a smo oth functor g : ˇ C ( U ) / / B G where B G denotes the one-ob ject group oid intro duce d in E xample 2.1.1 (b ) and ˇ C ( U ) deno tes the ˇ Ce ch gr oup oid cor resp onding to the cover U . It has ob jects F i ∈ I U i and morphisms F i,j ∈ I U i ∩ U j , and its structure maps are s ( x, i, j ) = ( x, i ) , t ( x, i, j ) = ( x, j ) , id ( x,i ) = ( x, i, i ) and ( x, j, k ) ◦ ( x, i, j ) = ( x, i, k ). Analogously , smo oth natura l transfor mations b etw een smo oth functors ˇ C ( U ) / / B G give rise to ˇ Cech co bo undaries. Th us the set  ˇ C ( U ) , B G  of equiv alence classes of smooth functors equals the usual ﬁrst ˇ Cech co homology with r esp ect to the cover U . The classical ﬁr st ˇ Cech-cohomology ˇ H 1 ( M , G ) of M is hence given by the colimit ov er all op en co vers U of M ˇ H 1 ( M , G ) = lim − → U  ˇ C ( U ) , B G  . W e use this c o incidence in order to deﬁne the 0- th ˇ Cech c o homolog y with co eﬃcients in a g e neral Lie group oid Γ: – 16 – Deﬁnition 3.1. If Γ is a Lie gr oup oid we set ˇ H 0 ( M , Γ) := lim − → U  ˇ C ( U ) , Γ  wher e the c olimit is t aken over al l c overs U of M and  ˇ C ( U ) , Γ  denotes the set of e quivalenc e classes of smo oth functors ˇ C ( U ) / / Γ . Remark 3. 2. The choice of the degree is such that ˇ H 0 ( M , Γ) ag rees in the case Γ = G dis (Example 2.1.1 (a )) with the classical 0-th ˇ Cech-cohomology ˇ H 0 ( M , G ) of M with v alues in G . The geometrica l meaning o f the set is given in the following well-kno wn theor em, which c a n b e prov ed e.g. using Lemma 2 .2.8. Theorem 3.3. Ther e is a bij e ction ˇ H 0 ( M , Γ) ∼ =  Isomorphism classes of princip al Γ -bund les over M  . If Γ is not only a Lie 2- group oid but a Lie 2-g roup one can also deﬁne a ﬁrst cohomo lo gy group ˇ H 1 ( M , Γ). Indeed, in this case one can co nsider the Lie 2- group oid B Γ with one o b ject, morphisms Γ 0 and 2-mor phisms Γ 1 . Multiplication in Γ g ives the comp ositio n of morphisms in B Γ. Let  ˇ C ( U ) , B Γ] denote the set of equiv alence class es of smo oth, weak 2-functors from the ˇ Cech-groupo id ˇ C ( U ) to the Lie 2-group oid B Γ. F or the deﬁnition of weak functors see [B´ en6 7] – below we will determine this set ex plic itly . Deﬁnition 3.4. F or a 2-gr oup Γ we set ˇ H 1 ( M , Γ) := lim − → U  ˇ C ( U ) , B Γ  . Remark 3.5. This agrees for Γ = G dis with the classical ˇ H 1 ( M , G ). F ur thermore, for an ab elian Lie gro up A the Lie gr oup oid B A is ev en a 2-group and ˇ H 1 ( M , B A ) ag rees with the classical ˇ Cech- cohomolog y ˇ H 2 ( M , A ). Un winding the above deﬁnition, we get V ersion I of smo o th Γ-ge rb es: Deﬁnition 3.6. L et Γ b e a Lie 2-gr oup, and let U = { U α } α ∈ A b e an op en c over of M . 1. A Γ -1-c o cycle with r esp e ct to U is a p air ( f αβ , g αβ γ ) of smo oth m aps f αβ : U α ∩ U β / / Γ 0 and g αβ γ : U α ∩ U β ∩ U β / / Γ 1 satisfying s ( g αβ γ ) = f β γ · f αβ and t ( g αβ γ ) = f αγ , and g αβ δ ◦ ( g β γ δ · id f αβ ) = g αγ δ ◦ (id f γ δ · g αβ γ ) . (3.1) Her e, the symb ols ◦ and · s tand for t he c omp osition and multiplic ation of Γ , r esp e ctively. – 17 – 2. Two Γ -1-c o cycles ( f αβ , g αβ γ ) and ( f ′ αβ , g ′ αβ γ ) ar e e qu ivalent, if ther e exist smo oth maps h α : U α / / Γ 0 and s αβ : U α ∩ U β / / Γ 1 with s ( s αβ ) = g ′ αβ · h α , t ( s αβ ) = h β · g αβ and (id h γ · g αβ γ ) ◦ ( s β γ · id f αβ ) ◦ (id f β γ · s αβ ) = s αγ ◦ ( g ′ αβ γ · id h α ) . Remark 3.7. F or a crossed mo dule t : H / / G and Γ := G/ /H the asso ciated Lie 2-g roup (Example 2.4.2) o ne ca n r e duce Γ-1-co cycles to pairs ˜ f αβ : U α ∩ U β / / G and ˜ g αβ γ : U α ∩ U β ∩ U β / / H which satisﬁes then a co cyc le condition similar to (3 .1). Analogo usly , cob oundarie s ca n b e r educed to pa ir s ˜ h α : U α / / G and ˜ s αβ : U α ∩ U β / / H . This yie lds the common deﬁnitio n of non-ab elia n co cycles, whic h ca n for example b e fo und in [Bre90] or [BS0 9]. Example 3. 8. In ca s e of the cro ssed mo dule i : H / / Aut( H ) with Γ = A UT( H ) (see Example 2.4.4) Γ-1-c o cycles consist of pairs ˜ f αβ : U α ∩ U β / / Aut( H ) and ˜ g αβ : U α ∩ U β ∩ U γ / / H . Co cycles of this kind c lassify so-ca lle d Lie group oid H - extensions [CLGX09, Pr op osition 3.14 ], which can hence b e seen as another equiv alent version for A UT( H )-g e rb es. 4 V ersion I I: Classifying Maps It is well known that for a Lie g roup G the smo oth ˇ Cech-cohomology ˇ H 1 ( M , G ) a nd the c ontinuous ˇ Cech-cohomology ˇ H 1 c ( M , G ) a gree if M is a smo oth manifold (in pa r ticular paracompa ct). This can e .g . b e shown by lo cally approximating contin uous co cycles by smo oth ones without changing the cohomolo gy class – see [MW09] (even fo r G inﬁnite-dimensiona l). Below we generalize this fact to non-a be lia n cohomology for cer tain Lie 2-gr oups Γ. Here the c o ntin uo us ˇ Cech-cohomology ˇ H 1 c ( M , Γ) is deﬁned in the s a me w ay as the smoo th one (Deﬁnition 3.4) but with all maps contin uo us instead of smo oth. A Lie group oid Γ is ca lled smo othly sep ar able , if the s et π 0 Γ of isomor phis m classes o f ob jects is a smo o th manifold for which the pro jection Γ 0 / / π 0 Γ is a submersio n. Prop ositi on 4. 1. L et M b e a smo oth m anifold and let Γ b e a smo othly sep ar able Lie 2-gr oup. Then, the inclusion ˇ H 1 ( M , Γ) / / ˇ H 1 c ( M , Γ) of smo oth int o c ontinuous ˇ Ce ch c ohomolo gy is a bije ction. – 18 – Remark 4.2. It is p ossible that the as s umption of being s mo othly separable is no t necess ary , but a proo f not ass uming this w ould certainly b e mor e inv olved than ours. An ywa y , all Lie 2 -gro ups we a re in terested in a re smoo thly sepa rable. Pro of o f Pr op osition 4.1. W e denote by π 1 Γ the Lie subgr oup of Γ 1 consisting o f automo r phisms of 1 ∈ Γ 0 . Since it has tw o commuting group structures – comp ositio n and m ultiplication – it is ab elian. The idea of the pro of is to reduce the statement via long exact s e quences to s tatements prov ed in [MW09]. The exac t sequence w e need c an b e found in [Bre90]: ˇ H 0 ( M , ( π 0 Γ) dis ) / / ˇ H 1 ( M , B π 1 Γ) / / ˇ H 1 ( M , Γ) / / ˇ H 1 ( M , ( π 0 Γ) dis ) / / ˇ H 2 ( M , B π 1 Γ) . Note that ˇ H 1 ( M , Γ) and ˇ H 1 ( M , ( π 0 Γ) dis ) do not hav e gr o up structures: hence, exa ctness is only meant as exactnes s of p ointed sets. But we actually hav e mo re structur e, na mely an action o f ˇ H 1 ( M , B π 1 Γ) on ˇ H 1 ( M , Γ). This action factors to an action of C := coker  ˇ H 0 ( M , ( π 0 Γ) dis ) / / ˇ H 1 ( M , B π 1 Γ)  . In fact o n the non-empty ﬁbres of the morphism ˇ H 1 ( M , Γ) / / ˇ H 1 ( M , ( π 0 Γ) dis ) this a ction is simply transitive. In o ther w ords: ˇ H 1 ( M , Γ) is a C -T orso r ov er K := ker  ˇ H 1 ( M , ( π 0 Γ) dis ) / / ˇ H 2 ( M , B π 1 Γ)  . The same type of sequence a lso exists in contin uous co homology , namely ˇ H 0 c ( M , ( π 0 Γ) dis ) / / ˇ H 1 c ( M , B π 1 Γ) / / ˇ H 1 c ( M , Γ) / / ˇ H 1 c ( M , ( π 0 Γ) dis ) / / ˇ H 2 c ( M , B π 1 Γ) . With C ′ := coker  ˇ H 0 c ( M , ( π 0 Γ) dis ) / / ˇ H 1 c ( M , B π 1 Γ)  K ′ := ker  ˇ H 1 c ( M , ( π 0 Γ) dis ) / / ˇ H 2 c ( M , B π 1 Γ)  , we exhibit ˇ H 1 c ( M , Γ) as a C ′ -T or sor o ver K ′ . The natural inclusio ns of smo oth in to contin uous cohomology form a c hain map betw een the t wo sequences. F rom [MW09] we know that they are isomorphisms on the seco nd, four th and ﬁfth factor. In particula r we hav e an induce d isomorphis m K ∼ / / K ′ . Lemma 4.3 below additionally shows that the induced morphism C / / C ′ is an iso mo rphism. Using these iso morphisms w e see that ˇ H 1 ( M , Γ) and ˇ H 1 c ( M , Γ) ar e b oth C -torso rs ov er K and that the natura l map ˇ H 1 ( M , Γ) / / ˇ H 1 c ( M , Γ) is a mo rphism of to rsors . But each morphism of gr oup torsors is bijective, which concludes the pro of.  – 19 – Lemma 4.3. The ima ges of f : ˇ H 0 ( M , ( π 0 Γ) dis ) / / ˇ H 1 ( M , B π 1 Γ) and f ′ : ˇ H 0 c ( M , ( π 0 Γ) dis ) / / ˇ H 1 c ( M , B π 1 Γ) ar e isomorp hic. Pro of. ˇ H 0 ( M , ( π 0 Γ) dis ) is the g roup of smo oth maps s : M / / π 0 Γ and ˇ H 0 c ( M , ( π 0 Γ) dis ) is the group of contin uous ma ps t : M / / π 0 Γ. The g roups ˇ H 1 ( M , B π 1 Γ) = ˇ H 2 ( M , π 1 Γ) and ˇ H 1 c ( M , B π 1 Γ) = ˇ H 2 c ( M , π 1 Γ) are isomor phic by the result of [MW09]. Under the connecting homo- morphism ˇ H 0 ( π 0 Γ , ( π 0 Γ) dis ) / / ˇ H 1 ( π 0 Γ , B π 1 Γ) the ide ntit y id π 0 Γ is sent to a class ξ Γ with the pro pe rty that f ( s ) = s ∗ ξ Γ and f ′ ( t ) = t ∗ ξ Γ . Hence it suﬃces to show that for each contin uous map t : M / / π 0 Γ there is a smo oth map s : M / / π 0 Γ with s ∗ ξ Γ = t ∗ ξ Γ . It is well known that for e ach con tinuous map t betw een smo oth manifolds a homo- topic s mo oth map s exists. It r emains to show that the pullback ˇ H 1 ( π 0 Γ , B π 1 Γ) / / ˇ H 1 ( M , B π 1 Γ) along s mo oth maps is homotopy in v ariant. This can e.g. b e seen by cho osing s mo oth (ab elian) B π 1 Γ-bundle ge r b es a s r epresentativ es, in which cas e the homotopy in v a riance ca n be deduced from the existence of connections .  It is a standard result in to p o logy tha t the con tinuous G -v alued ˇ Cech cohomology of paracompact spaces is in bijection with homotopy classes of maps to the classifying space B G o f the group G . A mo del for the classifying s pace B G is for exa mple the geometr ic realization of the nerve of the group oid B G , or Milnor’s join co nstruction [Mil56]. Now let Γ b e a Lie 2- g roup, and let | Γ | denote the geometric realiza tion of the nerve of Γ. Since the ne r ve is a simplicial top olo gical gr oup, | Γ | is a to p o logical gr oup. V ersion I I for smo oth Γ-gerb es is: Deﬁnition 4.4 ([BS09]) . A classif ying map for a smo oth Γ -gerb e is a c ontinu ous map f : M / / B | Γ | . W e denote b y  M , B | Γ |  the set o f homotopy classe s of classifying maps. Prop ositi on 4. 5 ([BS09, Theo r em 1 ]) . L et Γ b e a Lie 2-gr oup. Then ther e is a bije ction ˇ H 1 c ( M , Γ) ∼ =  M , B | Γ |  wher e the top olo gic al gr oup | Γ | is the ge ometric re alization of the nerve of Γ . Note that the assumption of [BS09, Theo rem 1] tha t Γ is w ell-p ointed is a utomatically s a tisﬁed bec ause Lie gro ups are well-po int ed. Pr op ositions 4.1 and 4.5 imply the fo llowing equiv alence theorem betw een V ersio n I and V ersion I I. – 20 – Theorem 4.6 . F or M a smo oth manif old and Γ a smo othly s ep ar able Lie 2-gr oup, ther e is a bije ction ˇ H 1 ( M , Γ) ∼ =  M , B | Γ |  . Remark 4. 7. Baez and Stevenson arg ue in [BS09, Section 5.2 .] that the space B | Γ | is homotopy equiv alent to a certain g eometric rea lization of the Lie 2- group oid |B Γ | from Section 3. Baas, B¨ ostedt and Kro hav e shown [B BK12] that |B Γ | class iﬁes c o ncordance classes of charted Γ-2-bundles. In particular, c harted Γ-2-bundles are a further equiv alent version of smo oth Γ-gerb es. 5 V ersion I I I: Group oid Bundle Gerb es Several deﬁnitions of non-ab elian bundle gerbes have app ear e d in liter ature so far [ACJ05, Jur 11, MRS]. The approach w e give her e no t only shows a conceptually clea r way to deﬁne non-ab elia n bundle gerb es, but also produces systematically a whole bica tegory . Mor eov er, these bica tegories form a 2-sta ck ov er smo oth manifolds (with the Grothendieck top olog y of surjective s ubmersions). 5.1 Deﬁnition via the Plus Constr uction Recall that the stack B un Γ of principal Γ-bundles is monoidal if Γ is a L ie 2-gro up (Prop o s ition 2.4.8). Asso ciated to the monoida l sta ck B un Γ we have a pre-2 -stack T r iv G r b Γ := B ( B un Γ ) of trivia l Γ -gerb es . E xplicitly , there is one trivia l Γ- g erb e I ov er ev ery smoo th manifold M . The 1-morphisms fro m I to I are pr incipal Γ -bundles ov er M , and the 2- mo rphisms b etw een those are mo r phisms of principal Γ-bundles. Horizontal compo sition is given by the tens o r pro duct of principal Γ-bundles, and vertical co mpo sition is the ordinary compo sition of Γ-bundle morphisms. Now w e apply the plus c onst ruction of [NS11] in order to stackify this pre-2 -stack. The resulting 2-stack is by deﬁnition the 2-stack of Γ - bund le gerb es , i.e. G r b Γ := ( T r iv G rb Γ ) + . Un winding the details o f the plus construction, we o bta in the following deﬁnitions: Deﬁnition 5. 1.1. L et M b e a smo oth manifold. A Γ -bund le gerb e over M is a surje ctive su bmersion π : Y / / M , a princip al Γ -bund le P over Y [2] and an asso ciative morphism µ : π ∗ 23 P ⊗ π ∗ 12 P / / π ∗ 13 P of Γ -bu nd les over Y [3] . – 21 – The morphism µ is called the bun d le gerb e pr o duct . Its asso ciativit y is the evident condition for bundle morphisms over Y [4] . In order to pro ceed with the 1-morphisms, we say tha t a c ommon r eﬁnement of t wo surjective submersions π 1 : Y 1 / / M a nd π 2 : Y 2 / / M is a s mo oth manifo ld Z together with sur jective submersions Z / / Y 1 and Z / / Y 2 such that the diagram Z ❇ ❇ ❇ ❇ ❇ ❇ ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ Y 1 π 1 ❇ ❇ ❇ ❇ ❇ Y 2 π 2 ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ M is co mm utative. W e ﬁx the following conv ention: s uppo se P 1 and P 2 are Γ-bundles over surjective submersions U 1 and U 2 , respectively , and V is a common r eﬁnement of U 1 and U 2 . Then, a bundle morphism ϕ : P 1 / / P 2 is unders to o d to b e a bundle morphis m betw een the pullbac ks of P 1 and P 2 to the common reﬁnement V . F or exa mple, in the following deﬁnition this conv ention applies to U 1 = Y [2] 1 , U 2 = Y [2] 2 and V = Z [2] . Deﬁnition 5.1. 2. L et G 1 and G 2 b e Γ -bund le gerb es over M . A 1-morphism A : G 1 / / G 2 is a c ommon re ﬁnemen t Z of the surje ctive submersions of G 1 and G 2 to gether with a princip al Γ -bund le Q over Z and a morphism β : P 2 ⊗ ζ ∗ 1 Q / / ζ ∗ 2 Q ⊗ P 1 of Γ -bu nd les over Z [2] , wher e ζ 1 , ζ 2 : Z [2] / / Z ar e the two pr oje ct ions, such that α is c omp atible with the bu nd le gerb e pr o duct s µ 1 and µ 2 . The co mpa tibilit y of α with µ 1 and µ 2 means tha t the diag ram π ∗ 23 P 2 ⊗ π ∗ 12 P 2 ⊗ ζ ∗ 1 Q id ⊗ ζ ∗ 12 β   µ 2 ⊗ id / / π ∗ 13 P 2 ⊗ ζ ∗ 1 Q ζ ∗ 13 β   π ∗ 23 P 2 ⊗ ζ ∗ 2 Q ⊗ π ∗ 12 P 1 ζ ∗ 23 β ⊗ id   ζ ∗ 3 Q ⊗ π ∗ 23 P 1 ⊗ π ∗ 12 P 1 id ⊗ µ 1 / / ζ ∗ 3 Q ⊗ π ∗ 13 P 1 (5.1.1) of morphisms of Γ-bundles ov e r Z [3] is commu tative. – 22 – If A 12 : G 1 / / G 2 and A 23 : G 2 / / G 3 are 1-morphisms betw een bundle gerb es ov er M , the comp osition A 23 ◦ A 12 : G 1 / / G 3 is giv en b y the ﬁbre pro duct Z := Z 23 × Y 2 Z 12 , the principal Γ-bundle Q := Q 23 ⊗ Q 12 ov er Z , and the mo r phism P 3 ⊗ ζ ∗ 1 Q β 23 ⊗ id / / ζ ∗ 2 Q 23 ⊗ P 2 ⊗ ζ ∗ 1 Q 12 id ⊗ β 12 / / ζ ∗ 2 Q ⊗ P 1 . The identit y 1-mor phism id G asso ciated to a Γ-bundle g erb e G is given by Y regar ded as a co mmon reﬁnement of π : Y / / M with itself, the trivial Γ-bundle I 1 (the tensor unit of B un Γ ( Y )), and the eviden t morphism I 1 ⊗ P / / P ⊗ I 1 . In order to deﬁne 2-morphis ms , supp os e that π 1 : Y 1 / / M a nd π 2 : Y 2 / / M a re surjective submersions, and that Z and Z ′ are co mmon reﬁnements of π 1 and π 2 . Let W b e a co mmon reﬁnement of Z and Z ′ with surjective submersions r : W / / Z a nd r ′ : W / / Z ′ . W e o btain t wo maps s 1 : W r / / Z / / Y 1 and t 1 : W r ′ / / Z ′ / / Y 1 , and a nalogous ly , tw o maps s 2 , t 2 : W / / Y 2 . These patch together to maps x W := ( s 1 , t 1 ) : W / / Y 1 × M Y 1 and y W := ( s 2 , t 2 ) : W / / Y 2 × M Y 2 . Deﬁnition 5.1. 3. L et G 1 and G 2 b e Γ - bund le gerb es over M , and let A , A ′ : G 1 / / G 2 b e 1- morphisms. A 2-morphism ϕ : A + 3 A ′ is a c ommon r eﬁnement W of the c ommon re ﬁn emen t s Z and Z ′ , t o gether with a morphism ϕ : y ∗ W P 2 ⊗ r ∗ Q / / r ′ ∗ Q ′ ⊗ x ∗ W P 1 of Γ -bu nd les over W t hat is c omp atible with the morph isms β and β ′ . The compatibilit y means that a certain diagram o ver W [2] commutes. Fibr ewise ov er a p o int ( w, w ′ ) ∈ W × M W this diag ram lo o ks as follows: P 2 | s 2 ( w ′ ) ,t 2 ( w ′ ) ⊗ P 2 | s 2 ( w ) ,s 2 ( w ′ ) ⊗ Q | r ( w ) id ⊗ β / / µ 2 ⊗ id   P 2 | s 2 ( w ′ ) ,t 2 ( w ′ ) ⊗ Q | r ( w ′ ) ⊗ P 1 | s 1 ( w ) ,s 1 ( w ′ ) ϕ ⊗ id   P 2 | s 2 ( w ) ,t 2 ( w ′ ) ⊗ Q | r ( w ) µ − 1 2 ⊗ id   Q ′ | r ′ ( w ′ ) ⊗ P 1 | s 1 ( w ′ ) ,t 1 ( w ′ ) ⊗ P 1 | s 1 ( w ) ,s 1 ( w ′ ) id ⊗ µ 1   P 2 | t 2 ( w ) ,t 2 ( w ′ ) ⊗ P 2 | s 2 ( w ) ,t 2 ( w ) ⊗ Q | r ( w ) id ⊗ ϕ   Q ′ | r ′ ( w ′ ) ⊗ P 1 | s 1 ( w ) ,t 1 ( w ′ ) id ⊗ µ − 1 1   P 2 | t 2 ( w ) ,t 2 ( w ′ ) ⊗ Q ′ | r ′ ( w ) ⊗ P 1 | s 1 ( w ) ,t 1 ( w ) β ′ ⊗ id / / Q ′ | r ′ ( w ′ ) ⊗ P 1 | t 1 ( w ) ,t 1 ( w ′ ) ⊗ P 1 | s 1 ( w ) ,t 1 ( w ) (5.1.2) – 23 – Finally we identify t wo 2-mor phisms ( W 1 , r 1 , r ′ 1 , ϕ 1 ) and ( W 2 , r 2 , r ′ 2 , ϕ 2 ) if the pullbacks o f ϕ 1 and ϕ 2 to W × Z × × Z ′ W ′ agree. Explicitly , this condition mea ns tha t for all w 1 ∈ W 1 and w 2 ∈ W 2 with r 1 ( w 1 ) = r 2 ( w 2 ) and r ′ 1 ( w 1 ) = r ′ 2 ( w 2 ), and for all p 2 ∈ y ∗ W 1 P 2 = y ∗ W 2 P 2 and q ∈ r ∗ 1 Q = r ∗ 2 Q we have ϕ 1 ( p 2 , q ) = ϕ 2 ( p 2 , q ). Remark 5.1.4. • In the a b ove situation o f a common reﬁnemen t W of tw o common r eﬁnements Z, Z ′ of sur- jective submer sions Y 1 , Y 2 , the diag ram Z ❆ ❆ ❆ ❆ ❆ ❆ ❆ ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ Y 1 W r O O r ′   Y 2 Z ′ ` ` ❆ ❆ ❆ ❆ ❆ ❆ ❆ > > ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ (5.1.3) is not necessa r ily commutativ e. In fact, diagra m (5.1.3) commutes if and o nly if the tw o maps x W : W / / Y 1 × M Y 1 and y W : W / / Y 2 × M Y 2 factor thro ugh the diago nal ma ps Y 1 / / Y 1 × M Y 1 and Y 2 / / Y 2 × M Y 2 , resp ectively . • In the case that a 2 -morphism ϕ is deﬁned on a common reﬁnement Z for which diagram (5.1.3) do es commute, Deﬁnition 5.1 .3 can b e simpliﬁed. As rema rked b efore , the tw o maps x W and y W factor thro ugh the diagonals, over which the bundles P 1 and P 2 hav e canonical trivializations (see Coro llary 5.2.6). Under these trivia lizations, ϕ can b e iden tiﬁed with a bundle morphism ϕ : Q / / Q ′ . F urthermo re, the compatibility diagra m (5.1 .2) simpliﬁes to the diag r am P 2 ⊗ η ∗ 1 Q β / / id ⊗ η ∗ 1 ϕ   η ∗ 2 Q ⊗ P 1 η ∗ 2 ϕ ⊗ id   P 2 ⊗ η ∗ 1 Q ′ β ′ / / η ∗ 2 Q ′ ⊗ P 1 . (5.1.4) Next we deﬁne the vertical co mp o s ition ϕ 23 • ϕ 12 : A 1 + 3 A 3 of 2-mor phisms ϕ 12 : A 1 + 3 A 2 and ϕ 23 : A 2 + 3 A 3 . The reﬁnement is the ﬁbre pro duct W := W 12 × Z 2 W 23 of the cov ers of ϕ 12 and ϕ 23 . The bundle gerb e pro ducts induce isomo rphisms x ∗ W P 1 ∼ = x ∗ W 23 P 1 ⊗ x ∗ W 12 P 1 and y ∗ W P 2 ∼ = y ∗ W 23 P 2 ⊗ y ∗ W 12 P 2 – 24 – ov er W . Under these identiﬁcations, the morphism y ∗ W P 2 ⊗ Q 1 / / Q 3 ⊗ x ∗ W P 1 for the 2-morphism ϕ 23 • ϕ 12 is deﬁned a s y ∗ W 23 P 2 ⊗ y ∗ W 12 P 2 ⊗ Q 1 id ⊗ ϕ 12 / / y ∗ W 23 P 2 ⊗ Q 2 ⊗ x ∗ W 12 P 1 ϕ 23 ⊗ id / / Q 3 ⊗ x ∗ W 23 P 1 ⊗ x ∗ W 12 P 1 . The identit y for vertical co mp o sition is just the identit y r eﬁnement a nd the identit y morphism. Finally we co me to the horizontal co mp o s ition ϕ 23 ◦ ϕ 12 : A 23 ◦ A 12 + 3 A ′ 23 ◦ A ′ 12 of 2-mor phisms ϕ 12 : A 12 + 3 A ′ 12 and ϕ 23 : A 23 + 3 A ′ 23 : its reﬁnement W is giv en by W 12 × ( Y 2 × Y 2 ) W 23 . W e lo ok at the three relev ant maps x W : W / / Y 1 × M Y 1 , y W : W / / Y 2 × M Y 2 and z W : W / / Y 3 × M Y 3 . The morphism ϕ of the 2-morphis m ϕ 23 ◦ ϕ 12 is deﬁned a s the com- po sition z ∗ W P 3 ⊗ Q 23 ⊗ Q 12 ϕ 23 ⊗ id / / Q ′ 23 ⊗ y ∗ W P 2 ⊗ Q 12 id ⊗ ϕ 12 / / Q ′ 23 ⊗ Q ′ 12 ⊗ x ∗ W P 1 . It follows from the prop erties o f the plus construction [NS11] that (a) these deﬁnitions ﬁt tog ether int o a bicategor y G r b Γ ( M ), a nd that (b) these for m a pre- 2-stack G r b Γ ov er smo o th manifolds. That means, ther e are pul lb ack 2-functors f ∗ : G r b Γ ( N ) / / G r b Γ ( M ) asso ciated to smo oth maps f : M / / N , and that these are compatible with the comp osition of smo oth maps. Pullba cks of Γ-bundle gerb es, 1-mo rphisms, and 2 -morphisms ar e obtained by just taking the pullbacks o f all inv olved data. Finally , the plus co nstruction implies (c): Theorem 5.1.5 ([NS11, Theor em 3 .3]) . The pr e-2-stack G r b Γ of Γ -bund le gerb es is a 2-stack. Remark 5. 1.6. Every 2-stack ov er smo oth ma nifo lds deﬁnes a 2- stack over Lie group oids [NS11, Prop os itio n 2.8 ]. This way , our approach pro duces automa tically bicateg ories G rb Γ ( X ) of Γ-bundle gerb es ov er a Lie gr oup oid X . In particular , for a n action group oid X = M / /G we hav e a bicatego ry G r b Γ ( M / /G ) of G -e quivariant Γ -bund le gerb es ov er M . In the remainder of this sectio n we give some examples a nd desc r ib e relatio ns b etw een the deﬁnitions given he r e a nd exis ting o ne s . Example 5. 1 .7. Let A be an ab elian Lie group, for insta nce U(1). Then, B A -bundle gerb e s are the same as the w ell-known A -bundle gerb es [Mur96]. F or more details see Remark 5.1.10 b elow. Example 5.1.8 . Let ( G, H , t, α ) b e a smo oth cross ed mo dule, and let G/ / H b e the as s o ciated action group oid. Then, a ( G/ /H )- bundle ger b e is the same as a cr osse d mo dule bund le gerb e in – 25 – the sense of Jurco [Jur11]. The equiv a lence relatio n “ s tably isomo rphic” o f [Jur11] is given b y “1-iso mo rphic” in terms of the bicatego ry constructed here. These coincidences come from the equiv alenc e b etw een ( G/ /H )-bundles and so-called G - H -bundles used in [Jur 11, ACJ05 ] ex pr essed by Lemma 2.3.1 0. In particula r, in case of the automo r phism 2-gro up AUT( H ) of a co nnected Lie gro up H , a AUT( H )-bundle gerb e is the same as a H - bibundle gerb e in the sense of Aschieri, Cantini and Jurco [ACJ05]. Example 5.1.9. Let G b e a L ie group, so that G dis is a Lie 2- group. Then, there is an equiv alence of 2-categor ies G r b G dis ( M ) ∼ = B un B G ( M ) dis . Indeed, if G is a G dis -bundle g e r b e ov er M , its principa l G dis -bundle over Y [2] is by Example 2.2.4 just a smo oth ma p α : Y [2] / / G , and its bundle gerbe product degenerates to a n equa lity π ∗ 23 α · π ∗ 12 α = π ∗ 13 α for functions on Y [3] . In other words, a G dis -bundle gerb e is the same as a so- called “ G -bundle 0-ger b e”. These form a category that is equiv alent to the one of ordinary principal G -bundles, a s p o int ed out in Section 1. Remark 5. 1.10. There are t wo diﬀerences b etw een the deﬁnitions given here (for Γ = B A ) and the ones of Murray et al. [Mur96, MS00, Ste00]. Firstly , we have a slightly diﬀeren t order ing of tensor pro ducts o f bundles. Thes e order ings ar e not esse nt ial in the ca se of a b elia n gro ups b ecause the tensor ca tegory of ordina ry A -bundles is symmetric. In the non-a b elia n case, a cons is tent theory re quires the co nv ent ions we hav e chosen here. Secondly , the deﬁnitions of 1-mo r phisms and 2-morphisms ha ve be e n generaliz e d step by step: 1. In [Mur96], 1-morphisms did no t include a c ommon reﬁnement , but rather re quired that the surjective submersion of one bundle ge rb e reﬁnes the other. This deﬁnition is to o r e strictive in the sense that e .g . U(1)-bundle ger b es a re not classiﬁe d by H 3 ( M , Z ), as in tended. 2. In [MS00], 1 -morphisms were deﬁned on the c a nonical reﬁnement Z := Y 1 × M Y 2 of the surjective submersions of the bundle gerb es. This deﬁnition solves the prev ious pro blems concerning the cla s siﬁcation o f bundle gerb es, but makes the comp osition of 1-morphis ms quite inv olved [Ste00]. 3. In [W al0 7], 1 -morphisms were deﬁned on reﬁnement s ζ : Z / / Y 1 × M Y 2 . This gener alization allows the same elegant deﬁnition of comp ositio n w e hav e given here, and res ults in the same isomorphism classes of bundle gerb es. Mor eov er, 2 -morphisms are deﬁned with c ommut ative diagrams (5.1 .3) – this makes the structure of the bicategor y outmost simple (see Rema rk 5.1.4). 4. In the present article w e have allow ed for a yet more gener al reﬁnement in the deﬁnition of 1-morphisms. Its achiev ement is that bundle gerb es come out a s an exa mple of a mor e genera l concept – the plus construction – and w e get e.g. Theorem 5.1.5 fo r fr ee. – 26 – Despite these diﬀeren t deﬁnitions of 1-morphisms and 2 - morphisms, the resulting bicategories of B A -bundle g e rb es in 2., 3. and 4. are all equiv alent (see [W al07, Theorem 1 ], [NS11, Remar k 4.5] and Lemma 5.2 .8 b elow). 5.2 Prop erties of Group oid B undle Gerb es W e recall that a homomor phis m Λ : Γ / / Ω b etw een Lie 2 -groups is an anafunctor together with a transformatio n (2.4.4) de s cribing its compatibility with the multiplications. W e recall further from Prop os itio n 2.4.9 that extens ion a long Λ is a 1-morphism Λ : B un Γ / / B un Ω betw een mo no idal stacks over smo oth ma nifolds. Tha t is, extension along Λ is co mpatible with pull- backs, tensor pro ducts, and morphis ms betw een pr incipal Γ-bundles. Applying it to the principal Γ-bundle P of a Γ-bundle g erb e G , and also to the bundle gerb e pro duct µ , we obtain immediately an Ω- bundle gerb e Λ G . The s a me is evidently true for mor phisms a nd 2- morphisms. Summarizing, we g et: Prop ositi on 5.2. 1. Extension of bund le gerb es along a homomorphism Λ : Γ / / Ω b etwe en Lie 2-gr oups deﬁnes a 1-morphism Λ : G rb Γ / / G r b Ω of 2-stacks over smo oth manif olds. W e rec all that a weak equiv alence b etw een Lie 2 -gro ups is a homomorphism Λ : Γ / / Ω that is a weak equiv alence (see Deﬁnition 2.3.1 1). W e have: Theorem 5.2.2. Supp ose Λ : Γ / / Ω is a we ak e quivalenc e b etwe en Lie 2-gr oups. Then, the 1-morphism Λ : G r b Γ / / G r b Ω of Pr op osition 5.2.1 is an e quivalenc e of 2-stacks. Pro of. The monoidal equiv a lence Λ : B u n Γ / / B un Ω betw een the monoidal stacks (Coro llary 2.4.10) induces an e q uiv alence T ri v G r b Γ ( M ) / / T r iv G r b Λ ( M ) b etw een pr e-2-sta cks. Since the plus construction is functoria l, this induces in turn the cla imed equiv alence of 2-stacks.  Next w e gener a lize a c ouple of well-kno wn results from ab elian to non-ab elian bundle ger b es . W e deﬁne a r eﬁn emen t of a surjective submersion π : Y / / M to b e a no ther s urjective submersion ω : W / / M to g ether with a smoo th map f : W / / Y such that ζ = π ◦ f . Notice that suc h a r eﬁnement induces smo oth maps f k : W [ k ] / / Y [ k ] that comm ute with the v arious pr o jections ω i 1 ...i k and π i 1 ...i k . Lemma 5 .2.3. Su pp ose G 1 = ( Y 1 , P 1 , µ 1 ) and G 2 = ( Y 2 , P 2 , µ 2 ) ar e Γ -bund le gerb es over M , f : Y 1 / / Y 2 is a r eﬁnement of surje ctive submersions, and ϕ : f ∗ 2 P 2 / / P 1 is an isomorphism – 27 – of Γ -bu nd les over Y [2] 1 that is c omp atible with the bund le gerb e pr o ducts µ 1 and µ 2 in the sense that the diagr am π ∗ 23 f ∗ 2 P 2 ⊗ π ∗ 12 f ∗ 2 P 2 π ∗ 23 ϕ ⊗ π ∗ 12 ϕ   f ∗ 3 µ / / π ∗ 13 f ∗ 2 P 2 π ∗ 13 ϕ   π ∗ 23 P 1 ⊗ π ∗ 12 P 1 µ / / π ∗ 13 P 1 is c ommutative. Then, G 1 and G 2 ar e isomorp hic. The pro of works just the same w ay as in the ab elian ca s e: one constr ucts the 1-isomo rphism ov er the co mmon r eﬁnement Z := Y 1 × M Y 2 in a straightforward wa y . As a co nsequence of Lemma 5.2.3 we ha ve Prop ositi on 5.2. 4. L et G = ( Y , P, µ ) b e a Γ -bund le gerb e over M , and let f : W / / Y b e a r eﬁn ement of its sur je ctive submersion π : Y / / M . Then, ( W, f ∗ 2 P, f ∗ 3 µ ) is a Γ -bund le gerb e over M , and is iso morphic to G . Lemma 5.2. 5. L et G = ( Y , P , µ ) b e a Γ -bun d le gerb e over M . Then, ther e exist unique smo oth maps i : P / / P and t : Y / / P such that (i) the diagr ams P i / / χ   P χ   Y [2] ﬂip / / Y [2] and P χ   Y diag / / t > > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ Y [2] ar e c ommutative. (ii) the map t is neutr al with r esp e ct to the bund le gerb e pr o duct µ , i.e . µ ( t ( y 2 ) , p ) = p = µ ( p, t ( y 1 )) . for al l p ∈ P with χ ( p ) = ( y 1 , y 2 ) . (iii) the m ap i pr ovides inverses with re sp e ct to the bund le gerb e pr o duct µ , i.e. µ ( i ( p ) , p ) = t ( y 1 ) and µ ( p, i ( p )) = t ( y 2 ) for al l p ∈ P with χ ( p ) = ( y 1 , y 2 ) . Mor e over, α ( t ( y )) = 1 and α ( i ( p )) = α ( p ) − 1 for al l p ∈ P and y ∈ Y . – 28 – Pro of. Concerning uniqueness, supp ose ( t, i ) and ( t ′ , i ′ ) are pairs of maps sa tisfying (i), (ii) and (iii). Firstly , we hav e t ′ ( y ) = µ ( t ( y ) , t ′ ( y )) = t ( y ) and so t = t ′ . Then, µ ( i ( p ) , p ) = t ( y 1 ) = t ′ ( y 1 ) = µ ( i ′ ( p ) , p ) implies i ( p ) = i ′ ( p ), and so i = i ′ . In order to see the existence o f t and i , deno te b y Q := diag ∗ P the pullbac k o f P to Y , denote by Q ∨ the dual bundle and b y d : Q ⊗ Q ∨ / / I 1 the death map. Consider the smoo th map Y s / / I 1 d − 1 / / Q ⊗ Q ∨ µ − 1 ⊗ id Q ∨ / / Q ⊗ Q ⊗ Q ∨ id ⊗ d / / I 1 ⊗ Q ∼ = Q diag / / P where s : Y / / I 1 is the canonical sectio n (see the proo f of Lemma 2.2 .6). It is s traightforward to see that this s atisﬁes the prop er ties of the map t . Since all maps in the a b ov e sequence are (anchor-preserving) bundle morphisms, it is clear that t ◦ α = 1.  Corollary 5.2. 6. L et G = ( Y , P, µ ) b e a Γ -bund le gerb e over M , and let t and i b e t he unique maps of L emma 5.2.5. Then, (i) t is a se ct ion of diag ∗ P , and deﬁnes a t rivializa tion diag ∗ P ∼ = I 1 . (ii) i is a bu nd le isomorp hism i : P ∨ / / ﬂip ∗ P . (iii) C 0 := Y and C 1 := P d eﬁne a Lie gr oup oid with sour c e and tar get maps π 1 ◦ χ and π 2 ◦ χ , r esp e ctively, c omp osition µ , identity t and inversion i . The following statement is w ell-known for abelia n g e rb es; the genera l v ers io n ca n b e proved b y a s traightforw ar d g e neralization of the constructions given in the pro of of [W al07, Prop osition 3 ]. Lemma 5.2.7. Every 1-morp hism A : G / / H b etwe en Γ -bund le gerb es over M is invertible. The last sta tement of this sectio n shows a wa y to bring 1-morphisms and 2-mor phis ms into a s impler form (see Remark 5.1 .10). F or bundle ge rb es G 1 and G 2 with surjective submersions π 1 : Y 1 / / M a nd π 2 : Y 2 / / M we denote by H om ( G 1 , G 2 ) the Hom-catego ry in the bicategory G r b Γ ( M ), and by H om ( G 1 , G 2 ) F P the categor y whose ob jects ar e those 1 -morphisms whose common reﬁnement is Z := Y 1 × M Y 2 , and who se 2-morphisms ar e those 2-mor phisms whose reﬁnement is W := Y 1 × M Y 2 with the maps r , r ′ : W / / Z the identit y maps. Lemma 5.2.8. The inclu s ion H om ( G 1 , G 2 ) F P / / H om ( G 1 , G 2 ) is an e quivalenc e of c ate gories. Pro of. First we show tha t it is essentially surjective. W e ass ume A : G 1 / / G 2 is a g eneral 1-morphism with a principal Γ-bundle Q over a co mmon reﬁnement Z of the surjective submersions π 1 : Y 1 / / M and π 2 : Y 2 / / M of the t wo bundle gerb es. W e lo ok at the principal Γ-bundle ˜ Q := κ ∗ 2 P 2 ⊗ pr ∗ 2 Q ⊗ κ ∗ 1 P 1 – 29 – ov er ˜ Z := Y 1 × M Z × M Y 2 , where κ 1 : ˜ Z / / Y [2] 1 : ( y 1 , z , y 2 ) ✤ / / ( y 1 , y 1 ( z )) and κ 2 : ˜ Z / / Y [2] 2 : ( y 1 , z , y 2 ) ✤ / / ( y 2 ( z ) , y 2 ). The pro jection pr 13 : ˜ Z / / Y 1 × M Y 2 is a surjective submersio n, and ov er ˜ Z × Y 1 × M Y 2 ˜ Z we hav e a bundle morphism α : pr ∗ 1 ˜ Q / / pr ∗ 2 ˜ Q deﬁned ov er a p o int ( ˜ z , ˜ z ′ ) with ˜ z = ( y 1 , z , y 2 ) and ˜ z ′ = ( y 1 , z ′ , y 2 ) b y ˜ Q ˜ z P 2 | y 2 ( z ) ,y 2 ⊗ Q z ⊗ P 1 | y 1 ,y 1 ( z ) µ − 1 2 ⊗ id ⊗ id   P 2 | y 2 ( z ′ ) ,y 2 ⊗ P 2 | y 2 ( z ) ,y 2 ( z ′ ) ⊗ Q z ⊗ P 1 | y 1 ,y 1 ( z ) id ⊗ β ⊗ id   P 2 | y 2 ( z ′ ) ,y 2 ⊗ Q z ′ ⊗ P 1 | y 1 ( z ) ,y 1 ( z ′ ) ⊗ P 1 | y 1 ,y 1 ( z ) id ⊗ id ⊗ µ 1   P 2 | y 2 ( z ′ ) ,y 2 ⊗ Q z ′ ⊗ P 1 | y 1 ,y 1 ( z ′ ) ˜ Q ˜ z ′ . The co mpatibility condition (5.1.1) implies a co cycle condition for α ov er the three-fold ﬁbr e pr o duct of ˜ Z ov er Y 1 × M Y 2 , and since principal Γ-bundles form a stack, the pair ( ˜ Q, α ) deﬁnes a pr incipal Γ- bundle Q F P ov er Z F P := Y 1 × M Y 2 . It is now straig htf or ward to show that the bundle iso morphism β itself descends to a bundle isomor phism β F P ov er Z F P × M Z F P in suc h a w ay that the triple ( Z F P , Q F P , β F P ) forms a 1-morphism A F P : G 1 / / G 2 . In order to show that A F P is an ess ent ial preimage of A , it rema ins to construct a 2-morphism ϕ F P A : A + 3 A F P . In the ter minology of Deﬁnition 5.1 .3, we choose W = ˜ Z with r := pr 2 : W / / Z and r ′ := pr 13 : W / / Z F P . Note that diagram (5.1.3) do es not co mm ute. Th e maps x W : W / / Y [2] 1 and y W : W / / Y [2] 2 are given b y x W = s ◦ κ 1 and y W = κ 2 , wher e s : Y [2] 1 / / Y [2] 1 switches the factors. No w, the bundle isomorphism of the 2-mor phism ϕ F P A we wan t to construct is a bundle isomorphism ϕ : y ∗ W P 2 ⊗ r ∗ Q / / ˜ Q ⊗ x ∗ W P 1 ov er W , and is ﬁbr e w is e over a p oint w = ( y 1 , z , y 2 ) giv en b y P 2 | y 2 ( z ) ,y 2 ⊗ Q z id ⊗ id ⊗ t − 1 / / P 2 | y 2 ( z ) ,y 2 ⊗ Q z ⊗ P y 1 ( z ) ,y 1 ( z ) id ⊗ id ⊗ µ − 1 1   P 2 | y 2 ( z ) ,y 2 ⊗ Q z ⊗ P 1 | y 1 ,y 1 ( z ) ⊗ P 1 | y 1 ( z ) ,y 1 ˜ Q w ⊗ P 1 | s ( y 1 ,y 1 ( z )) , where t is the trivialization of diag ∗ P of Coro llary 5.2.6. The co mpa tibilit y condition (5.1.2) is straightforward to c heck. Now we show that the inclusion H om ( G 1 , G 2 ) F P / / H om ( G 1 , G 2 ) is full a nd faithful. Since it is clearly faithful, it only re ma ins to show that it is full. Given a mor phism A / / A ′ in H om ( G 1 , G 2 ), – 30 – i.e. a common r eﬁnement W of Y 1 × M Y 2 with itself and a bundle morphism ϕ , we have to ﬁnd a morphism in H om ( G 1 , G 2 ) F P such that the t wo mo r phisms are iden tiﬁed under the equiv a lence relation on bundle gerb e 2-morphisms. W e deno te the bundles ov er Y 1 × M Y 2 corres p o nding to A and A ′ by Q and Q ′ . The reﬁnement maps a re denoted as b efore by r = ( s 1 , s 2 ) : W / / Y 1 × M Y 2 and r ′ = ( t 1 , t 2 ) : W / / Y 1 × M Y 2 . Then we obta in an isomorphism r ∗ Q / / r ∗ Q ′ ﬁbrewise o ver a p o int w ∈ W by Q | s 1 ( w ) ,s 2 ( w ) d − 1 ⊗ id / / P ∨ 2 | s 2 ( w ) ,t 2 ( w ) ⊗ P 2 | s 2 ( w ) ,t 2 ( w ) ⊗ Q | s 1 ( w ) ,s 2 ( w ) id ⊗ ϕ   P ∨ 2 | s 2 ( w ) ,t 2 ( w ) ⊗ Q ′ | t 1 ( w ) ,t 2 ( w ) ⊗ P 1 | s 1 ( w ) ,t 1 ( w ) id ⊗ β ′− 1   P ∨ 2 | s 2 ( w ) ,t 2 ( w ) ⊗ P 2 | s 2 ( w ) ,t 2 ( w ) ⊗ Q ′ | s 1 ( w ) ,s 2 ( w ) d ⊗ id / / Q ′ | s 1 ( w ) ,s 2 ( w ) (5.2.1) where d : P ∨ 2 | s 2 ( w ) ,t 2 ( w ) ⊗ P 2 | s 2 ( w ) ,t 2 ( w ) / / I 1 is the death map. O ne can use the compatibilit y condition for ϕ to show that this morphism descends to a morphism ψ : Q / / Q ′ which is a morphism in H om ( G 1 , G 2 ) F P . The tw o morphisms ( W, ψ ) a nd ( Y 1 × M Y 2 , ϕ ) are identiﬁed if their pullbacks to W × ( Y 1 × M Y 2 × M Y 1 × M Y 2 ) ( Y 1 × M Y 2 ) = { w ∈ W | r ( w ) = r ′ ( w ) } =: W 0 are equa l. On the one side, the map W 0 / / W is the inclusion and the ma p W 0 / / Y 1 × M Y 2 is equal to r . The pullback of ψ along r is by construction the map r ∗ Q / / r ∗ Q ′ from (5.2.1). On the other side, bundles x ∗ W P 1 and y ∗ W P 2 ov er W 0 hav e canonica l trivializa tio ns (Lemma 5.2 .6 (i)) under which ϕ b ecomes also equal to the morphism (5.2.1).  5.3 Classiﬁcation b y ˇ Cec h Cohomology In this section w e prov e that V ersions I ( ˇ Cech Γ-1- co cycles) and II I (Γ-bundle ge r b es) are equiv alent. F or this pur p o se, we extract a ˇ Cech co cycle from a Γ-bundle ge r b e G ov er M , a nd prov e that this pro cedure deﬁnes a bijection on the level of equiv alence cla sses (Theorem 5.3.2). First w e hav e to ensure the exis tence of appro priate op en cov ers. Lemma 5.3. 1. F or every Γ -bun d le gerb e G = ( Y , P, µ ) over M ther e exists an op en c over U = { U i } i ∈ I of M with se ctions σ i : U i / / Y , such that the princip al Γ -bun d les ( σ i × σ j ) ∗ P over U i ∩ U j ar e trivial izable. Pro of. One can choose an op en co ver such that the 2-fold in tersectio ns U i ∩ U j are con tractible. Since ev ery Lie 2- g roup is a crossed mo dule G/ /H (Remark 2.4.3), and G/ /H - bundles are ordinar y H -bundles (Lemma 2.2.9), these admit sections o ver co nt ra ctible smo oth ma nifolds. B ut a s ection is eno ugh to trivia lize the or iginal Γ-bundle (Lemma 2.2.6).  – 31 – Let G b e a Γ-bundle ger be ov er M , a nd let U = { U i } i ∈ I be an op en cov er with the prop erties of Lemma 5 .3.1. W e denote by M U the disjoin t union o f all the open sets U i , and by σ : M U / / Y the union of the sections σ i . Then, σ is a r eﬁnement of π : Y / / M , and we hav e a Γ-bundle gerb e G U ,σ that is is omorphic to G (Pro po sition 5.2.4). The principa l Γ-bundle P ij of G U ,σ ov er the comp onent U i ∩ U j is b y assumption trivializable. Thu s there exists a trivializa tio n t ij : P ij / / I f ij for s mo oth functions f ij : U i ∩ U j / / Γ 0 . W e deﬁne an isomo rphism µ ij k betw een trivial bundles such that the diagram P j k ⊗ P ij µ / / t jk ⊗ t ij   P ik t ik   I f jk ⊗ I f ij µ ijk / / I f ik is co mmutative. Now w e are in the situation of Lemma 5 .2.3, whic h implies tha t the Γ-bundle gerb e G U ,σ,t := ( M U , I f ij , µ ij k ) is s till isomo rphic to G . Combining Lemma 2 .2.8 with E xample 2.4.7 (a), we see that the is omorphisms µ ij k corres p o nd to s mo oth maps g ij k : U i ∩ U j ∩ U k / / Γ 1 such that s ( g ij k ) = f j k · f ij and t ( g ij k ) = f ik . The asso ciativity condition for µ ij k implies moreov er that g αγ δ ◦ ( g αβ γ · id f γ δ ) = g αβ δ ◦ (id f αβ · g β γ δ ). Hence, the co llection { f ij , g ij k } is a Γ-1-co cycle on M with resp ect to the op en cov er U . Theorem 5.3.2. L et M b e a smo oth manif old and let Γ b e a Lie 2-gr oup. The ab ove c onstru ction deﬁnes a bij e ction  Isomorphism classes of Γ -bund le gerb es ov er M  ∼ = ˇ H 1 ( M , Γ) . Pro of. W e claim that Γ-bundle gerb es ( M U , I f ij , µ ij k ) and ( M V , I h ij , ν ij k ) ar e isomorphic if and only if the co r resp onding Γ-1-co cycles are equiv alent. This prov es at the same time that the choices of op en cov ers and sections we hav e ma de during the co nstruction do not ma tter , that the resulting map is w ell-deﬁned on isomorphism classes, and that this map is injectiv e. Surjectivity follows by assig ning to a Γ-1-co cycle ( f ij , g ij k ) with re sp ect to so me co ver U the Γ - bundle gerb e ( M U , I f ij , µ ij k ) with µ ij k determined by Lemma 2.2.8. It remains to prov e that c la im. W e as sume A = ( Z, Q, α ) is a 1-is o morphism betw een the Γ- bundle gerb es ( M U , I f ij , µ ij k ) and ( M V , I h ij , ν ij k ). Similarly to Lemma 5.3.1 one can show tha t there exists a cov er W of M by o p en sets W i that reﬁnes both U and V , and that allows smo oth sections ω i : W i / / Z for which the Γ-bundle ω ∗ i Q is trivializa ble. In the terminolog y of the a b ove construction, choos ing a trivializa tion t : ω ∗ Q / / I h i with smo oth maps h i : W i / / Γ 0 ov er M W conv erts the isomor phism α into smo o th functions s ij : W i ∩ W j / / Γ 1 satisfying s ( s ij ) = g ′ ij · h i – 32 – and t ( s ij ) = h j · g ij . The compatibility diagram (5.1 .1) implies the remaining condition that ma kes ( h i , s ij ) an equiv alence betw een the Γ-2-co cycles ( f ij , g ij k ) and ( f ′ ij , g ′ ij k ).  6 V ersion IV: Principal 2-Bu n dles The basic idea of a smoo th 2-bundle is that it gives for ev ery p oint x in the base manifold M a Lie gro up o id P x v ar y ing s mo othly with x . Numero us diﬀeren t versions have a pp eared so far in the literature, e.g [Bar04, BS07, W o c11, SP11]. The main ob jectiv e of our version of principal 2-bundles is to make the deﬁnition of the ob jects (i.e. the 2-bundles) as simple as p ossible , while keeping their iso morphism cla sses in bijection with non-ab elian cohomology . Th us, our principal 2-bundles will b e deﬁned using strict actions of Lie 2-groups on Lie gr oup oids, and not using anafunctors. The necessary “weakness” will b e pushed in to the deﬁnition o f 1- mo rphisms. 6.1 Deﬁnition of Principal 2-Bundles As an importa nt prerequisite for principal 2-bundles w e hav e to discuss actio ns o f Lie 2 -groups on Lie group oids, a nd equiv aria nt anafunctor s. Deﬁnition 6. 1.1. L et P b e a Lie gr oup oid, and let Γ b e a Lie 2-gr oup. A smo oth right action of Γ on P is a smo oth functor R : P × Γ / / P su ch that R ( p, 1) = p and R ( ρ, id 1 ) = ρ for al l p ∈ P 0 and ρ ∈ P 1 , and su ch that the diagr am P × Γ × Γ id × m / / R × i d   P × Γ R   P × Γ m / / P of smo oth fun ctors is c ommutative (strictly, on t he nose). F or example, every Lie 2-gr oup acts on itself via multiplication. Note that due to strict com- m utativity , one has R ( R ( p, g ) , g − 1 ) = p a nd R ( R ( ρ, γ ) , i ( γ )) = ρ for a ll g ∈ Γ 0 , p ∈ P 0 , γ ∈ Γ 1 and ρ ∈ P 1 . Remark 6.1.2. This deﬁnition could b e weak ened in t wo steps. First, one could allow a natural transformatio n in the ab ove diag ram instea d of commut ativity . Seco ndly , one could allow R to b e an a na functor instead o f an or dinary functor. It turns out tha t for o ur purp oses the ab ov e deﬁnition is suﬃcient. – 33 – Deﬁnition 6.1. 3. L et X and Y b e Lie gr oup oids with smo oth actions ( R 1 , ρ 1 ) , ( R 2 , ρ 2 ) of a Lie 2-gr oup Γ . An e quivariant structur e on an anafunctor F : X / / Y is a tr ansformation X × Γ F × id   R 1 / / X λ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ x  ③ ③ ③ ③ ③ ③ ③ ③ F   Y × Γ R 2 / / Y satisfying the fo l lowing c ondition: X × Γ × Γ id × m / / R 1 × id ❑ ❑ ❑ % % ❑ ❑ F × id × id   X × Γ R 1   ❃ ❃ ❃ ❃ ❃ X × Γ λ × id s s s s s s u } s s s s R 1 / / F × id   X λ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ y  ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ F   Y × Γ × Γ R 2 × id % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ Y × Γ R 2 / / Y = X × Γ × Γ id × m / / F × id × id   X × Γ F × i d   R 1   ❄ ❄ ❄ ❄ ❄ X λ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ {  ⑧ ⑧ ⑧ ⑧ F   Y × Γ × Γ R 2 × id % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ id × m / / Y × Γ R 2 ❃ ❃   ❃ ❃ Y × Γ R 2 / / Y An anafunct or to gether with a Γ -e quivariant struct ur e is c al le d Γ -e quivariant anafunctor . In Appendix A w e tra nslate this a bstract (but eviden tly co rrect) deﬁnition of equiv a riance into more concrete ter ms involving a Γ 1 -action on the total space o f the ana functor. Deﬁnition 6.1.4. If ( F , λ ) : X / / Y and ( G, γ ) : X / / Y ar e Γ -e quivariant anafunctors, a tr ansformation η : F + 3 G is c al le d Γ -e quivariant, if t he fol lowing e quality of tra nsformation holds: X × Γ G × id   F × i d   η × id k s R 1 / / X λ s s s s s s s s s s s s s s s s u } s s s s s s s s F   Y × Γ R 2 / / Y = X × Γ G × id   R 1 / / X γ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ v ~ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ G   F _ _ η k s Y × Γ R 2 / / Y It follo ws from abstra ct nons ense in the bicategory of Lie group oids, anafunctors and transfor- mations that we hav e another bicateg ory with • o b jects: Lie gr o up oids with smo o th rig ht Γ-a ctions. • 1 -morphisms: Γ-equiv aria nt anafunctors. • 2 -morphisms: Γ-equiv aria nt transforma tions. – 34 – W e need three further notions for the deﬁnition o f a principal 2-bundle. Let M b e a s mo oth manifold, and let P be a Lie group oid. W e s ay that a smo o th functor π : P / / M dis is a su rje ctive submersion functor , if π : P 0 / / M is a sur jective submersion. Let π : P / / M dis be a sur jective submersion functor , a nd let Q b e a Lie group oid with so me smo oth functor χ : Q / / M dis . Then, the ﬁbre pro duct P × M Q is deﬁned to b e the full sub catego ry of P × Q over the submanifold P 0 × M Q 0 ⊂ P 0 × Q 0 . Deﬁnition 6.1.5. L et M b e a s m o oth manifo ld and let Γ b e a Lie 2-gr oup. (a) A princip al Γ -2-bund le over M is a Lie gr ou p oid P , a s u rje ctive submersion functor π : P / / M dis , and a smo oth right action R of Γ on P that pr eserves π , such that the smo oth functor τ := (pr 1 , R ) : P × Γ / / P × M P is a we ak e quivalenc e. (b) A 1-morphism b etwe en princip al Γ -2-bund les is a Γ -e quivariant anafunctor F : P 1 / / P 2 that r esp e cts the surje ctive submersion functors to M . (c) A 2-morphism b etwe en 1-morphisms is a Γ -e quivariant tr ansformation b etwe en these. Remark 6.1.6. (a) The condition in (a) that the action R preser ves the surjective submersion functor π means that the diagra m o f functors P × Γ R / / pr 1   P π   P π / / M dis is co mm utative. (b) The co ndition in (b) that the anafunctor F r esp ects the surjective s ubmer sion functors means in the ﬁrst place that ther e exis ts a tra nsformation P 1 F / / π 1   ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ P 2 q y ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ π 2   ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ M dis . – 35 – How ever, since the target of the anafunctors π 1 and π 2 ◦ F is the disc r ete group o id M dis , the equiv alenc e of Example 2 .3.4 applies, and implies that if such a tra nsformation exists, it is unique. Indeed, it is easy to see tha t an anafunctor F : P / / Q with anchors α l : F / / P 0 and α r : F / / Q 0 resp ects smo o th functors π : P / / M dis and χ : Q / / M dis if and o nly if π ◦ α l = χ ◦ α r . Example 6.1.7. The trivial Γ - 2-bund le ov er M is deﬁned by P := M dis × Γ , π := pr 1 , R := id M × m . Here, the smo oth functor τ even has a smo oth inv erse functor . In the following w e denote the trivial Γ-2-bundle by I . Remark 6.1.8. The principal Γ-2-bundles of Deﬁnition 6.1.5 are v ery similar to those o f Bar tels [Bar04] and W o ck el [W o c1 1], in the sense that their ﬁbres are g roup oids with a Γ-action. They o nly diﬀer in the s trictness assumptions for the action, and in the formulation of principality . Opp os ed to that, the princip al 2-gr oup bund les intro duced in [GS] are quite diﬀerent: their ﬁbres are Lie 2-gro upo ids equipp e d with a cer ta in Lie 2-gro up o id morphism to B Γ. 6.2 Prop erties of Pr incipal 2-Bundles Principal Γ - 2-bundles over M form a bicategor y denoted 2- B u n Γ ( M ). There is an evident pullback 2-functor f ∗ : 2- B un Γ ( N ) / / 2- B un Γ ( M ) asso ciated to smoo th maps f : M / / N , and these mak e 2- B un Γ a pre-2 -stack over smo oth manifolds. W e deduce the following imp or tant tw o theorems ab o ut this pr e-2-stack. The ﬁrst asserts that it actually is a 2 - stack: Theorem 6.2.1. Princip al Γ -2-bund les form a 2-stack 2 - B un Γ over smo oth manifolds. Pro of. This follows from Theorem 5.1.5 (Γ-bundle ger b es for m a 2 -stack) and Theorem 7 .1 (the equiv alenc e G r b Γ ∼ = 2- B un Γ ) w e prov e in Section 7.  Remark 6. 2.2. Similar to Remark 5.1 .6, we obtain automatica lly bica tegories 2- B un Γ ( X ) of prin- cipal Γ-2- bundles ov er Lie gro up o ids X , including bicategor ies o f e quivariant princip al Γ -2-bund les . The seco nd concerns a homomo r phism Λ : Γ / / Ω of Lie 2- groups, which induces the e x tension Λ : G r b Γ / / G r b Ω betw een 2 -stacks of bundle ger b es (Prop ositio n 5.2.1). Co mbined with the equiv alenc e G r b Γ ∼ = 2- B un Γ of Theorem 7 .1, it deﬁnes a 1 -morphism Λ : 2 - B un Γ / / 2- B un Ω – 36 – betw een 2-s ta cks of principal 2-bundles. Now we get as a direct consequence of Theo rem 5.2 .2: Theorem 6.2.3. If Λ : Γ / / Ω is a we ak e quivalenc e b etwe en Lie 2-gr oups, then t he 1-morphism Λ : 2 - B un Γ / / 2 - B un Ω is an e quivalenc e of 2-stacks. A third consequence of the equiv alence of T he o rem 7.1 in combination with Lemma 5.2.7 is Corollary 6.2.4. Every 1-morphism F : P 1 / / P 2 b etwe en princip al Γ -2-bu n d les over M is invertible. The following discussio n centers aro und lo c al triviali zability that is implicitly contained in Def- inition 6.1.5. A principal Γ-2-bundle that is isomorphic to the trivial Γ-2-bundle I introduced in Example 6.1.7 is called trivializable . A se ction of a principa l Γ-2 -bundle P ov er M is an a nafunctor S : M dis / / P such that π ◦ S = id M dis (recall that an anafunctor π ◦ S : M / / M is the same as a s mo oth ma p). One can show that every p oint x ∈ M has an op en neig hborho o d U together with a s e ction s : U dis / / P | U . Such sections can even be c hosen to b e smo oth functors, r ather than anafunctors, na mely simply a s ordinar y sections of the surjectiv e submersion π : ( P | U ) 0 / / U dis . Lemma 6.2. 5. A princip al Γ -2-bund le over M is t r ivializable if and only if it has a smo oth se ct ion. Pro of. The trivial Γ-2 -bundle I has the s e c tion S ( m ) := ( m, 1), whe r e 1 denotes the unit of Γ 0 . If P is trivializa ble, and F : I / / P is an iso morphism, then, F ◦ S is a section of P . Conversely , suppo se P has a section S : M dis / / P . Then, we g e t the anafunctor I = M dis × Γ S × id / / P × Γ R / / P . (6.2.1) It has an evident Γ-equiv a riant s tr ucture a nd resp ects the pro jections to M . According to Coro llary 6.2.4, this is suﬃcient to hav e a 1-isomo rphism.  Corollary 6.2.6. Every princip al Γ -2-bund le is lo c al ly trivializable , i.e. every p oint x ∈ M has an op en neigh b orho o d U and a 1-morphism T : I / / P | U . Remark 6.2. 7. In W o ck el’s version [W o c11] of principal 2-bundles, lo cal trivializa tions are required to be smo oth functors and to b e inv ertible a s smo oth functor s, rather tha n allo wing a nafunctors. This version turns out to b e to o restrictive in the sense that the resulting bicategory receives no 2-functor from the bicatego ry G r b Γ ( M ) of Γ-bundle gerb es that would establis h an equiv ale nce. It is also p o s sible to r eformulate our deﬁnition of principal 2- bundles in terms of lo cal tr ivializa- tions. This reformulation gives us criteria which might b e ea sier to c heck than the actual deﬁnition, similar to the case of ordinary principal bundles. Prop ositi on 6.2.8. L et P b e a Lie gr oup oid, π : P / / M dis b e a surje ctive submersion functor, and R b e a s mo oth right action of Γ on P that pr eserves π . Supp ose every p oint x ∈ M has – 37 – an op en neighb orho o d U to gether with a Γ -e quivariant anafunctor T : I / / P | U that r esp e cts the pr oje ctions. Then, π : P / / M dis is a princip al Γ -2-bund le over M . Pro of. W e only hav e to prove that the functor τ is a weak equiv ale nc e , and we use Theorem 2.3 .1 3. Since a ll mor phisms of P ha ve source a nd ta r get in the same ﬁbre o f π : P 0 / / M dis , we may chec k the t wo conditions of Theorem 2.3.1 3 lo cally , i.e. for P | U i where U i is a n ope n cov er of M . Using lo cal trivializatio ns T i : I / / P | U i , the smo oth functor τ translates int o the smo oth functor (id , pr 1 , m ) : M dis × Γ × Γ / / ( M dis × Γ) × M ( M dis × Γ). This functor is a n iso mo rphism of Lie group oids, and henc e essentially surjective and fully faithful.  7 Equiv alence b et w een Bun d le Gerb es and 2-B u ndles In this s e ction we show that V e rsions II I and IV of s mo oth Γ-gerb es are eq uiv alent in the stronge s t po ssible sense: Theorem 7. 1. F or M a smo oth manifold and Γ a Lie 2-gr oup, ther e is an e quivalenc e of bic ate gories G r b Γ ( M ) ∼ = 2 - B un Γ ( M ) b etwe en the bic ate gories of Γ -bund le gerb es and pri ncip al Γ -2-bund les over M . This e quivalenc e is natur al in M , i.e. it is an e quivalenc e b etwe en pr e-2-stacks. Since the deﬁnitions of the bicategor ies G r b Γ ( M ) and 2- B un Γ ( M ), and the ab ov e equiv alence are all natural in M , we obtain automatically an induced equiv alence for the induced bicategories ov er Lie group oids (see Rema rks 5.1.6 and 6.2.2). Corollary 7.2. F or X a Lie gr oup oid and Γ a Lie 2-gr oup, ther e is an e quivalenc e G r b Γ ( X ) ∼ = 2 - B un Γ ( X ) . The following outlines the pro o f of Theore m 7.1. In Section 7 .1 w e construct explicitly a 2- functor E M : 2- B u n Γ ( M ) / / G r b Γ ( M ). Then we use a general criterio n assuring that E M is an equiv alence of bicategories. This criterio n is s tated in Lemma B.1: it r equires (A) that E M is fully faithful on Hom-catego ries, a nd (B) to choose certain preima ges o f ob jects and 1-mo rphisms under E M . Under these circumstances, Lemma B.1 constructs an inv ers e 2- functor R M together with the r equired pseudonatural transformatio ns assuring that E M and R M form an equiv alence of bicatego ries. Condition (A) is pr ov ed as Lemma 7.1.7 in Section 7.1. The choices (B) ar e co nstructed in Section 7.2. – 38 – In order to prov e that the 2-functor s E M extend to the claimed equiv alence b etw een pr e -2- stacks, we use another cr iterion stated in Lemma B.3. The o nly additional a ssumption of Lemma B.3 is that the given 2-functors E M form a 1-morphism o f pre-2-stacks; this is proved in Prop os ition 7.1.8. Then, the inv erse 2-functor s R M obtained b efore automatically form an inverse 1-mor phism betw een pre-2 -stacks. 7.1 F r om Principal 2-Bundles to Bundle Gerb es In this se c tion we deﬁne the 2-functor E M : 2- B u n Γ ( M ) / / G r b Γ ( M ). Deﬁnition of E M on ob je cts Let P be a principal Γ-2-bundle ov er M , with pro jection π : P / / M a nd r ig ht action R of Γ o n P . The ﬁrst ingredient of the Γ- bundle gerbe E M ( P ) is the surjective submer sion π : P 0 / / M . The second ing redient is a principal Γ- bundle P over P [2] 0 . W e put P := P 1 × Γ 0 . Bundle pro jection, a nchor and Γ-action a re given, r esp ectively , by χ ( ρ, g ) := ( t ( ρ ) , R ( s ( ρ ) , g − 1 )) , α ( ρ, g ) := g and ( ρ, g ) ◦ γ := ( R ( ρ, id g − 1 · γ ) , s ( γ )). (7.1.1) These deﬁnitio ns a r e mo tiv ated b y Remark 7.1.2 b elow. Lemma 7.1.1. This deﬁnes a princip al Γ -bund le over P [2] 0 . Pro of. Firs t we chec k that χ : P / / P [2] 0 is a sur jective submer sion. Since the functor τ = (id , R ) is a weak equiv alence, w e know fro m Theor em 2.3.13 that f : ( P 0 × Γ 0 ) τ × t × t P [2] 1 / / P [2] 0 : ( p, g , ρ 1 , ρ 2 ) ✤ / / ( s ( ρ 1 ) , s ( ρ 2 )) is a sur jective submer sion. Now consider the smoo th surjectiv e map g : ( P 0 × Γ 0 ) τ × t × t P [2] 1 / / P 1 × Γ 0 : ( p, g , ρ 1 , ρ 2 ) ✤ / / ( ρ − 1 1 ◦ R ( ρ 2 , id g − 1 ) , g − 1 ). W e hav e χ ◦ g = f ; thus, χ is a s ur jective submersion. Next w e c heck that w e hav e deﬁned an action. Supp o se ( ρ, g ) ∈ P and γ ∈ Γ 1 such that α ( ρ, g ) = g = t ( γ ). Then, α (( ρ, g ) ◦ γ ) = s ( γ ). Moreov er, supp os e γ 1 , γ 2 ∈ Γ 1 with t ( γ 1 ) = g and t ( γ 2 ) = s ( γ 1 ). Then, (( ρ, g ) ◦ γ 1 ) ◦ γ 2 = ( R ( ρ, id g − 1 · γ 1 ) , s ( γ 1 )) ◦ γ 2 = ( R ( ρ, id g − 1 · γ 1 · id s ( γ 1 ) − 1 · γ 2 ) , s ( γ 2 )) = ( ρ, g ) ◦ ( γ 1 ◦ γ 2 ), – 39 – where w e ha ve used that γ 1 ◦ γ 2 = γ 1 · id s ( γ 1 ) − 1 · γ 2 in an y 2-group. It remains to chec k that the smo oth ma p ˜ τ : P α × t Γ 1 / / P χ × χ P : (( ρ, g ) , γ ) ✤ / / (( ρ, g ) , ( ρ, g ) ◦ γ ) is a diﬀeomor phism. F o r this purp o s e, we cons ider the diag r am P [2] 1 s × t   ( P 0 × Γ 0 ) × ( P 0 × Γ 0 ) τ × τ / / P [2] 0 × P [2] 0 (7.1.2) and claim that (a) N 1 := P α × t Γ 1 is a pullback o f (7.1.2), (b) N 2 := P χ × χ P is a pullbac k of (7.1.2), and (c) the unique ma p N 1 / / N 2 is ˜ τ . Thus, ˜ τ is a diﬀeo morphism. In order to pr ov e c la im (a) we use again that the functor τ = (id , R ) is a w eak equiv alence, so that by Theore m 2.3.13 the triple ( P 1 × Γ 1 , τ , s × t ) is a pullback of (7.1.2). W e consider the smo oth map ξ : N 1 / / P 1 × Γ 1 : (( ρ, g ) , γ ) ✤ / / ( R ( ρ, id g − 1 ) , γ ) which is a diﬀeo mo rphism b ecaus e ( ρ, γ ) ✤ / / (( R ( ρ, id t ( γ ) ) , t ( γ )) , γ ) is a s mo oth map whic h is inv erse to ξ . Thus, putting f 1 := τ ◦ ξ and g 1 := ( s × t ) ◦ ξ w e see that ( N 1 , f 1 , g 1 ) is a pullback o f (7.1.2). In order to prov e claim (b), we put f 2 (( ρ 1 , g 1 ) , ( ρ 2 , g 2 )) := ( R ( ρ 1 , id g − 1 1 ) , ρ 2 ) g 2 (( ρ 1 , g 1 ) , ( ρ 2 , g 2 )) := ( R ( s ( ρ ) , g − 1 1 ) , g 2 , R ( t ( ρ 1 ) , g − 1 1 ) , g 1 ), and it is straightforw ard to chec k that the cone ( N 2 , f 2 , g 2 ) mak es (7.1 .2) commutativ e. The triple ( N 2 , f 2 , g 2 ) is also universal: in order to see this suppose N ′ is any smo o th manifold with smooth maps f ′ : N ′ / / P [2] 1 and g ′ : N ′ / / ( P 0 × Γ 0 ) × ( P 0 × Γ 0 ) so that (7 .1.2) is comm utative. F o r n ∈ N ′ , we write f ′ ( n ) = ( ρ 1 , ρ 2 ) and g ′ ( n ) = ( p 1 , g 1 , p 2 , g 2 ). Then, σ ( n ) := (( R ( ρ 1 , id g − 1 2 ) , g 2 ) , ( ρ 2 , g 1 )) deﬁnes a smo oth ma p σ : N ′ / / P χ × χ P . One chec ks that f 2 ◦ σ = f ′ and g 2 ◦ σ = g ′ , and that σ is the only smo oth map satisfying these equations. This prov es that ( N 2 , f 2 , g 2 ) is a pullback. W e are left with claim (c). Here one only ha s to chec k that τ : N 1 / / N 2 satisﬁes f 2 = f 1 ◦ τ and g 2 = g 1 ◦ τ .  Remark 7.1. 2. The smo oth functor τ = (id , R ) : P × Γ / / P × M P is a weak equiv alence, and so has a canonical in verse ana functor τ − 1 (Remark 2.3.14). The anafunctor P [2] 0 ι / / P × M P c / / P × M P τ − 1 / / P × Γ pr 2 / / Γ, where c is the functor that switches the factors, corr esp onds to a principal Γ-bundle ov er P [2] 0 that is ca nonically isomorphic to the bundle P deﬁned ab ov e. – 40 – It rema ins to provide the bundle gerb e pr o duct µ : π ∗ 23 P ⊗ π ∗ 12 P / / π ∗ 13 P , which we deﬁne b y the for mula µ (( ρ 23 , g 23 ) , ( ρ 12 , g 12 )) := ( ρ 12 ◦ R ( ρ 23 , id g 12 ) , g 23 g 12 ). (7.1.3) Lemma 7.1.3. F ormula (7.1.3) deﬁnes an asso ciative isomorph ism µ : π ∗ 23 P ⊗ π ∗ 12 P / / π ∗ 13 P of princip al Γ -bund les over P [3] 0 . Pro of. First of all, we reca ll from E xample 2.4.7 (b ) that a n element in the tensor pro duct π ∗ 23 P ⊗ π ∗ 12 P is repr esented b y a triple ( p 23 , p 12 , γ ) where p 23 , p 12 ∈ P with π 1 ( χ ( p 23 )) = π 2 ( χ ( p 12 )), and α ( p 23 ) · α ( p 12 ) = t ( γ ). In (7.1.3) we refer to triples where γ = id g 23 g 12 , and this deﬁnition extends to triples with genera l γ ∈ Γ 1 by employing the equiv alence relation ( p 1 , p 2 , γ ) ∼ ( p 1 ◦ ( γ · id α ( p 2 ) − 1 ) , p 2 , id s ( γ ) ). (7.1.4) The complete formula for µ is then µ (( ρ 23 , g 23 ) , ( ρ 12 , g 12 ) , γ ) = ( ρ 12 ◦ R ( ρ 23 , id g − 1 23 · γ ) , s ( γ )) . (7.1.5) Next w e c heck that (7.1.5) is well-deﬁned under the equiv ale nce r e lation (7.1.4): µ ((( ρ 23 , g 23 ) , ( ρ 12 , g 12 ) , γ )) = ( ρ 12 ◦ R ( ρ 23 , id g − 1 23 · γ ) , s ( γ )) = ( ρ 12 ◦ R ( ρ 23 ◦ R (id R ( s ( ρ 23 ) ,g − 1 23 ) , γ · id g − 1 12 ) , id g 12 ) , s ( γ )) = µ (( ρ 23 ◦ R (id R ( s ( ρ 23 ) ,g − 1 23 ) , γ · id g − 1 12 ) , s ( γ ) g − 1 12 ) , ( ρ 12 , g 12 ) , id s ( γ ) )) = µ ((( ρ 23 , g 23 ) ◦ ( γ · id g − 1 12 ) , ( ρ 12 , g 12 ) , id s ( γ ) )). Now w e hav e shown that µ is a w ell-deﬁned map fro m π ∗ 23 P ⊗ π ∗ 12 P to π ∗ 13 P , a nd it remains to pr ov e that it is a bundle mor phism. Checking that it preserves ﬁbres and anc hor s is straightforw ar d. It remains to chec k that (7.1.5) pres erves the Γ-action. W e calculate µ ((( ρ 23 , g 23 ) , ( ρ 12 , g 12 ) , γ ) ◦ ˜ γ ) = µ (( ρ 23 , g 23 ) , ( ρ 12 , g 12 ) , γ ◦ ˜ γ ) = ( ρ 23 ◦ R ( ρ 12 , id g 12 · i ( γ ◦ ˜ γ )) , s ( ˜ γ )) = ( ρ 23 ◦ R ( R ( ρ 12 , id g 12 ) , i ( γ ) ◦ i ( ˜ γ )) , s ( ˜ γ )) = ( ρ 23 ◦ R ( R ( ρ 12 , id g 12 ) , i ( γ )) ◦ R (id R ( s ( ρ 12 ) ,g ) , i ( ˜ γ )) , s ( ˜ γ )) = ( ρ 23 ◦ R ( ρ 12 , id g 12 · i ( γ )) ◦ R (id R ( s ( ρ 12 ) ,g ) , i ( ˜ γ )) , s ( ˜ γ )) = ( ρ 23 ◦ R ( ρ 12 , id g 12 · i ( γ )) , s ( γ )) ◦ ˜ γ = µ (( ρ 23 , g 23 ) , ( ρ 12 , g 12 ) , γ ) ◦ ˜ γ . Summarizing, µ is a mor phism of Γ- bundles over P [3] 0 . The asso ciativity of µ follows directly from the deﬁnitions.  – 41 – Deﬁnition of E M on 1-morphisms W e deﬁne a 1 -morphism E M ( F ) : E M ( P ) / / E M ( P ′ ) b etw ee n Γ-bundle gerb es from a 1 -morphism F : P / / P ′ betw een principal Γ-2 -bundles. The re ﬁnement of the surjective submersions π : P / / M a nd π ′ : P ′ / / M is the ﬁbre pro duct Z := P 0 × M P ′ 0 . Its principal Γ-bundle has the total s pace Q := F × Γ 0 , and its pro jection, anch or and Γ-a ction a re given, resp ectively , b y χ ( f , g ) := ( α l ( f ) , R ( α r ( f ) , g − 1 )) , α ( f , g ) := g and ( f , g ) ◦ γ := ( ρ ( f , id g − 1 · γ ) , s ( γ )), (7.1.6) where ρ : F × Γ 1 / / F denotes the Γ 1 -action on F that c omes from the given Γ- equiv ar iant structure on F (see Appendix A). Lemma 7.1.4. This deﬁnes a princip al Γ -bund le Q over Z . Pro of. W e show ﬁrst the the pro jection χ : Q / / Z is a surjective s ubmersion. Since the functor τ ′ : P ′ × Γ / / P × M P is a weak eq uiv alence, we hav e by Theor em 2.3 .13 a pullback X / / ξ   ( P ′ 0 × Γ 0 ) R × t ( P ′ 1 × M P ′ 1 ) s ◦ pr 2   F π ′ ◦ α l ( f ) × π ′ P ′ 0 / / P ′ 0 × M P ′ 0 along the bottom map ( f , p ′ ) ✤ / / ( α r ( f ) , p ′ ), whic h is w ell-deﬁned b eca us e the anafunctor F pre- serves the pro jections to M (see Rema r k 6.1.6 (b)). In par ticular, the map ξ is a surjective sub- mersion. It is easy to se e that the smoo th map k : X / / F × Γ 0 : (( f , p ′ ) , ( p ′ 0 , g , ρ, ˜ ρ )) ✤ / / ( f ◦ ρ − 1 ◦ R ( ˜ ρ, id g − 1 ) , g − 1 ) is sur jective. Now we cons ide r the commutativ e diagr a m X ξ   k / / F × Γ 0 χ   F π ′ ◦ α l ( f ) × π ′ P 0 α l × id / / P 0 × M P ′ 0 . The sur jectivity of k and the fact that ξ and α l × id ar e surjective submersions shows that χ is o ne, to o. – 42 – Next, one chec ks (as in the pro o f of Le mma 7 .1 .1) that the Γ- action o n Q deﬁned ab ov e is well-deﬁned a nd preserves the pro jection. Then it remains to chec k that the smo oth map ξ : Q α × t Γ 1 / / Q × P 0 × M P ′ 0 Q : ( f , g , γ ) ✤ / / ( f , g , ρ ( f , id g − 1 · γ ) , s ( γ )) is a diﬀeomorphism. An in verse map is given a s follows. Giv en an e le ment ( f 1 , g 1 , f 2 , g 2 ) on the right hand s ide, we have α l ( f 1 ) = α l ( f 2 ), so that ther e ex is ts a unique element ρ ′ ∈ P ′ 1 such that f 1 ◦ ρ ′ = f 2 . One calculates that ( ρ ′ , g 2 ) and (id α r ( f 1 ) , g 1 ) are elements o f the principal Γ- bundle P ′ × Γ 0 ov er P ′ [2] 0 of Lemma 7.1.1. Thus, there e xists a unique element γ ∈ Γ 1 such that ( ρ ′ , g 2 ) = (id α r ( f 1 ) , g 1 ) ◦ γ . Clearly , t ( γ ) = g 1 and s ( γ ) = g 2 , and we hav e ρ ′ = R (id α r ( f 1 ) , id g − 1 1 · γ ). W e deﬁne ξ − 1 ( f 1 , g 1 , f 2 , g 2 ) := ( f 1 , g 1 , γ ). The calculation that ξ − 1 is an inv ers e for ξ uses prop erty (ii) o f Deﬁnition A.1 for the action ρ , and is left to the reader.  The next step in the deﬁnitio n of the 1 -morphism E ( F ) is to deﬁne the bundle morphism β : P ′ ⊗ ζ ∗ 1 Q / / ζ ∗ 2 Q ⊗ P ov er Z × M Z . W e use the notation of Ex ample 2.4.7 (b) for elements of tensor pro ducts o f principal Γ-bundles; in this nota tion, the mo rphism β in the ﬁbre ov er a p oint (( p 1 , p ′ 1 ) , ( p 2 , p ′ 2 )) ∈ Z × M Z is g iven by β : (( ρ ′ , g ′ ) , ( f , g ) , γ ) ✤ / / (( ˜ f , g ′ g h ) , ( ˜ ρ, h − 1 ) , γ ), where h ∈ Γ 0 and ˜ ρ ∈ P ′ 1 are c hosen suc h that s ( ˜ ρ ) = R ( p 2 , h − 1 ) and t ( ˜ ρ ) = p 1 , and ˜ f := ρ ( ˜ ρ − 1 ◦ f ◦ R ( ρ ′ , id g ) , id h ). (7.1.7) Lemma 7.1.5. This deﬁnes an isomorphism b etwe en princi p al Γ -bund les. Pro of. The existence of c hoices of ˜ ρ, h follows becaus e the functor τ ′ : P ′ × Γ / / P ′ × M P ′ is smo othly ess entially s urjective (Theorem 2.3.13); in particular, o ne can c ho ose them lo cally in a smo oth way . W e claim that the equiv alence relation on ζ ∗ 2 Q ⊗ P iden tiﬁes diﬀerent choices; th us, we hav e a well-deﬁned smo o th map. In order to pr ov e this claim, we a ssume o ther c hoices ˜ ρ ′ , h ′ . The pairs ( ˜ ρ, h − 1 ) and ( ˜ ρ ′ , h ′− 1 ) are elemen ts in the principal Γ-bundle P ′ ov er P ′ 0 × M P ′ 0 and sit o ver the same ﬁbre; thus, there exists a unique ˜ γ ∈ Γ 1 such that ( ˜ ρ, h − 1 ) ◦ ˜ γ = ( ˜ ρ ′ , h ′− 1 ), in par ticular, R ( ˜ ρ, id h · ˜ γ ) = ˜ ρ ′ . Now we hav e (( ˜ f , g ′ g h ) , ( ˜ ρ, h − 1 ) , γ ) = (( ˜ f , g ′ g h ) , ( ˜ ρ, h − 1 ) , (id t ( γ ) · i ( ˜ γ ) · ˜ γ ) ◦ γ ) ∼ (( ˜ f , g ′ g h ) ◦ (id t ( γ ) · i ( ˜ γ )) , ( ˜ ρ, h − 1 ) ◦ ˜ γ , γ ) so that it suﬃces to calculate ( ˜ f , g ′ g h ) ◦ (id t ( γ ) · i ( ˜ γ )) = ( ρ ( ˜ f , id h − 1 · i ( ˜ γ )) , g ′ g h ′ ) = ( ρ ( ˜ ρ − 1 ◦ f ◦ R ( ρ ′ , id g ) , i ( ˜ γ )) , g ′ g h ′ ) = ( ρ ( R ( ˜ ρ − 1 , i ( ˜ γ ) · id h ′− 1 ) ◦ f ◦ R ( ρ ′ , id g ) , id h ′ ) , g ′ g h ′ ), – 43 – where the las t step use s the compatibility c o ndition for ρ from Deﬁnition A.1 (ii). I n any 2-gr oup, we hav e i ( ˜ γ ) · id s ( ˜ γ ) = (id t ( ˜ γ ) − 1 · ˜ γ ) − 1 , in which case the last line is exa ctly the formula (7.1 .7) for the pair ( ˜ ρ ′ , h ′ ). Next we check that β is w ell-deﬁned under the eq uiv alence relation on the tenso r pro duct P ′ ⊗ ζ ∗ 1 Q . W e hav e x := (( ρ ′ , g ′ ) , ( f , g ) , ( γ 1 · γ 2 ) ◦ γ ) ∼ (( ρ ′ , g ′ ) ◦ γ 1 , ( f , g ) ◦ γ 2 , γ ) =: x ′ for γ 1 , γ 2 ∈ Γ 1 such that t ( γ 1 ) = g ′ , t ( γ 2 ) = g and s ( γ 1 ) s ( γ 2 ) = t ( γ ). T aking adv antage of the fact that we can make the same choice o f ( ˜ ρ, h ) for b oth repre s entativ es x a nd x ′ , it is straightforward to show that β ( x ) = β ( x ′ ). Finally , it is obvious from the deﬁnition o f β that it is a nchor-preserving and Γ- equiv ar iant.  In order to show that the triple ( Z, Q , β ) de ﬁnes a 1-mo rphism betw een bundle gerb es , it remains to v erify that the bundle isomorphism β is compa tible the with the bundle ger be pro ducts µ 1 and µ 2 in the sens e of diagram (5.1.1). This is s traightforward to do and left for the reader . Deﬁnition of E M on 2-morphisms, comp o sitors and unitors Let F 1 , F 2 : P / / P ′ be 1 -morphisms b etw een principal Γ-bundles ov er M , and let η : F + 3 G be a 2- morphism. Betw een the Γ-bundles Q 1 and Q 2 , which live ov er the same common r eﬁnement Z = P 0 × M P ′ 0 , we ﬁnd immediately the smoo th map η : Q 1 / / Q 2 : ( f 1 , g ) ✤ / / ( η ( f 1 ) , g ) which is easily veriﬁed to b e a bundle mo r phism. Its compa tibility with the bundle morphisms β 1 and β 2 in the sense of the simpliﬁed dia gram (5.1.4) is a lso ea sy to check. Thus, we hav e deﬁned a 2-morphism E M ( η ) : E M ( F 1 ) + 3 E M ( F 2 ). The comp ositor for 1-morphisms F 1 : P / / P ′ and F 2 : P ′ / / P ′′ is a bundle gerbe 2- morphism c F 1 ,F 2 : E M ( F 2 ◦ F 1 ) / / E M ( F 2 ) ◦ E M ( F 1 ). Employing the above constr uctio ns, the 1 -morphism E M ( F 2 ◦ F 1 ) is deﬁned on the common reﬁne- men t Z 12 := P 0 × M P ′′ 0 and has the Γ-bundle Q 12 = ( F 1 × P ′ 0 F 2 ) / P ′ 1 × Γ 0 , wher e as the 1 - morphism E M ( F 2 ) ◦ E M ( F 1 ) is deﬁned on the common reﬁnement Z := P 0 × M P ′ 0 × M P ′′ 0 and has the Γ -bundle Q 2 ⊗ Q 1 with Q k = F k × Γ 0 . The comp os ito r c F 1 ,F 2 is deﬁned ov er the reﬁnemen t Z with the obvious reﬁnement maps pr 13 : Z / / Z 12 and id : Z / / Z making diag ram (5.1 .3) c o mmut ative. It is thus a bundle mor phism c F 1 ,F 2 : pr ∗ 13 Q 12 / / Q 2 ⊗ Q 1 . F or elements in a tensor product of Γ-bundles we use the notation of Example 2 .4.7 (b). Then, w e deﬁne c F 1 ,F 2 by (( p, p ′ , p ′′ ) , ( f 1 , f 2 , g )) ✤ / / (( ρ 2 ( ˜ ρ − 1 ◦ f 2 , id h ) , g h ) , ( f 1 ◦ ˜ ρ, h − 1 ) , id g ), (7.1.8) – 44 – where h ∈ Γ 0 and ˜ ρ : R ( p ′ , h − 1 ) / / α r ( f 1 ) = α l ( f 2 ) are chosen in the sa me way as in the pro o f of Lemma 7.1.5. The assignment (7.1 .8) do es not dep end on the choices of h and ˜ ρ , nor on the choice of the r epresentativ e ( f 1 , f 2 ) in ( F 1 × P ′ 0 F 2 ) / P ′ 1 . It is obvious that (7.1 .8) is anchor-preser ving, a nd its Γ-equiv ar iance can be seen by c ho osing ( ˜ ρ, h ) in order to co mpute c F 1 ,F 2 (( p, p ′ , p ′′ ) , ( f 1 , f 2 , g )) and ( ˜ ρ ′ , h ) with ˜ ρ ′ := R ( ˜ ρ, id g − 1 · γ − 1 ) in order to compute c F 1 ,F 2 ((( p, p ′ , p ′′ ) , ( f 1 , f 2 , g )) ◦ γ ). In order to complete the construction of the bundle ger be 2 -morphism c F 1 ,F 2 we hav e to prov e that the bundle mo r phism c F 1 ,F 2 is co mpatible with the isomor phisms β 12 of E M ( F 2 ◦ F 1 ) and (id ⊗ β 1 ) ◦ ( β 2 ⊗ id) of E M ( F 2 ) ◦ E M ( F 1 ) in the s ense of diagr am (5.1.4). W e star t with a n element (( ρ ′′ , g ′′ ) , ( f 12 , g )) ∈ E M ( P ′′ ) ⊗ ζ ∗ 1 Q 12 , where f 12 = ( f 1 , f 2 ). W e ha ve β 12 (( ρ ′′ , g ′′ ) , ( f 12 , g )) = ( f f 12 , g ′′ g h, ˜ ρ , h − 1 ) upo n choo sing ( ˜ ρ, h ) as required in the deﬁnition of E M ( F 2 ◦ F 1 ). W riting f f 12 = ( ˜ f 1 , ˜ f 2 ) further w e hav e ( ζ ∗ 2 c F 1 ,F 2 ⊗ id )( f f 12 , g ′′ g h, ˜ ρ , h − 1 ) = ( ρ 2 ( ˜ ρ − 1 2 ◦ ˜ f 2 , id h 2 ) , g ′′ g hh 2 , ˜ f 1 ◦ ˜ ρ 2 , h − 1 2 , ˜ ρ, h − 1 ) (7.1.9) upo n choosing appr opriate ( ˜ ρ 2 , h 2 ) as req uired in the deﬁnition o f c F 1 ,F 2 . This is the result of the clo ckwise comp osition of dia gram (5 .1 .4). Counter-clo ckwise, we ﬁr st g e t (id ⊗ ζ ∗ 1 c F 1 ,F 2 )(( ρ ′′ , g ′′ ) , ( f 12 , g )) = ( ρ ′′ , g ′′ , f ′′ , g h 1 , f ′ , h − 1 1 ) for choices ( ˜ ρ 1 , h 1 ), where f ′′ := ρ 2 ( ˜ ρ − 1 1 ◦ f 2 , id h 1 ) and f ′ := f 1 ◦ ˜ ρ 1 . Next we apply the isomorphism β 2 of E M ( F 2 ) and get ( β 2 ⊗ id)( ρ ′′ , g ′′ , f ′′ , g h 1 , f ′ 1 , h − 1 1 ) = ( f f ′′ , g ′′ g hh 2 , ˆ ρ, ˆ h − 1 , f ′ 1 , h − 1 1 ) where we hav e used the c hoices ( ˆ ρ , ˆ h ) deﬁned b y ˆ ρ := R ( ˜ ρ − 1 1 , h 1 ) ◦ R ( ˜ ρ 2 , h − 1 h 1 ) and ˆ h := h − 1 1 hh 2 . The last step is to apply the isomorphis m β 1 of E M ( F 2 ) whic h gives (id ⊗ β 1 )( f f ′′ , g ′′ g hh 2 , ˆ ρ, ˆ h − 1 , f ′ 1 , h − 1 1 ) = ( f f ′′ , g ′′ g hh 2 , e f ′ , h − 1 2 , ˜ ρ, h − 1 ), (7.1.10) where we hav e used the choices ( ˜ ρ, h ) from ab ov e. Comparing (7.1.9) a nd (7.1 .1 0), we have obvious coincidence in a ll but the ﬁrs t and the third co mpo nents. F or these r emaining factors, coincidence follows from the deﬁnitions of the v ario us v a riables. Finally , we have to c onstruct unitors. The unito r for a principal Γ- 2 -bundle P ov er M is a bundle gerb e 2 - morphism u P : E M (id P ) + 3 id E M ( P ) . Abstractly , one can ass o ciate to id E M ( P ) the 1-mo rphism id F P E M ( P ) constructed in the pro of of Lemma 5.2.8, and then notice that id F P E M ( P ) and E M (id P ) are canonically 2-isomor phic. In more concrete terms, the unitor u P has the reﬁnement W := P [3] 0 with the surjective submer sions r := pr 12 – 45 – and r ′ := pr 3 to the reﬁnements Z = P [2] 0 and Z ′ = P 0 of the 1 -morphisms E M (id P ) and id E M ( P ) , resp ectively . The rele v ant maps x W and y W are pr 13 and pr 23 , resp ectively . The pr incipal Γ- bundle of the 1-morphism id E M ( P ) is the tr iv ial bundle Q ′ = I 1 . W e c la im that the principal Γ- bundle Q of E M (id P ) is the bundle P o f the bundle gerb e E M ( P ). Indeed, the fo r mulae (7.1.6) re duce for the ident ity a nafunctor id P to those o f (7.1.1). Now, the bundle iso morphism of the unitor u P is y ∗ W P ⊗ r ∗ Q = pr ∗ 23 P ⊗ pr ∗ 12 P µ / / pr ∗ 13 P ∼ = r ′ ∗ Q ′ ⊗ x ∗ W P , where µ is the bundle gerb e pro duct of E M ( P ). The co mm utativity of dia gram (5 .1.2) follows from the asso ciativity o f µ . Prop ositi on 7.1. 6. The assignments E M for obje cts, 1-morphisms and 2-morphisms, to gether with the c omp ositors and unitors deﬁne d ab ove, deﬁne a 2-functor E M : 2 - B u n Γ ( M ) / / G r b Γ ( M ) . Pro of. A list of a xioms for a 2-functor with the same conven tio ns a s w e use here ca n be found in [SW, Appendix A]. The ﬁr st ax iom r equires that the 2 -functor E M resp ects the vertical comp os ition of 2-morphisms – this follows immediately from the deﬁnition. The second axio m r equires tha t the co mp o sitors resp ect the horizon tal comp ositio n o f 2- morphisms. T o see this, let F 1 , F ′ 1 : P / / P ′ and F 2 , F ′ 2 : P ′ / / P ′′ be 1-mor phisms be tw een principal Γ-2-bundles, and let η 1 : F 1 + 3 F ′ 1 and η 2 : F 2 + 3 F ′ 2 be 2-morphisms. Then, the diagram E M ( F 2 ◦ F 1 ) c F 1 ,F 2   E M ( η 1 ◦ η 2 ) + 3 E M ( F ′ 2 ◦ F ′ 1 ) c F ′ 1 ,F ′ 2   E M ( F 2 ) ◦ E M ( F 1 ) E M ( η 1 ) ◦ E M ( η 2 ) + 3 E M ( F ′ 2 ) ◦ E M ( F ′ 1 ) has to commute. Indeed, in order to compute c F 1 ,F 2 and c F ′ 1 ,F ′ 2 one ca n mak e the s ame choice of ( ˜ ρ, h ), be c ause the transfor mations η and η 2 preserve the a nchors. Then, commutativit y follows from the fact that η 1 and η 2 commute with the gr oup oid actions and the Γ 1 -action accor ding to Deﬁnition A.1. The third a xiom describ es the c ompatibility o f the comp ositor s with the comp osition of 1 - morphisms in the sense that the dia g ram E M ( F 3 ◦ F 2 ◦ F 1 ) c F 2 ◦ F 1 ,F 3 + 3 c F 3 ◦ F 2 ,F 1   E M ( F 3 ) ◦ E M ( F 2 ◦ F 1 ) id ◦ c F 2 ,F 1   E M ( F 3 ◦ F 2 ) ◦ E M ( F 1 ) c F 3 ,F 2 ◦ id + 3 E M ( F 3 ) ◦ E M ( F 2 ) ◦ E M ( F 1 ). – 46 – is commut ative. In order to verify this, one star ts with an element ( f 1 , f 2 , f 3 , g ) in E M ( F 3 ◦ F 2 ◦ F 1 ). In order to go clo ckwise, one c ho os es pairs ( ˜ ρ 12 , 3 , h 12 , 3 ) a nd ( ˜ ρ 1 , 2 , h 1 , 2 ) a nd gets fro m the deﬁnitions CW = (( ρ 3 ( ˜ ρ − 1 12 , 3 ◦ f 3 , id h 12 , 3 ) , g h 12 , 3 ) , ( ρ 2 ( ˜ ρ − 1 1 , 2 ◦ f 2 ◦ ˜ ρ 12 , 3 , id h 1 , 2 ) , h − 1 12 , 3 h 1 , 2 ) , ( f 1 ◦ ˜ ρ 1 , 2 , h − 1 1 , 2 )). Counter-clockwise, one can c ho os e ﬁrstly again the pair ( ˜ ρ 1 , 2 , h 1 , 2 ) and then the pair ( ˜ ρ 2 , 3 , h 2 , 3 ) with ˜ ρ 2 , 3 = R ( ˜ ρ 12 , 3 , id h 1 , 2 ) and h 2 , 3 = h − 1 1 , 2 h 12 , 3 . Then, one gets CCW = (( ρ 3 ( ˜ ρ − 1 2 , 3 ◦ ρ 3 ( f 3 , id h 1 , 2 ) , id h 2 , 3 ) , g h 1 , 2 h 2 , 3 ) , ( ρ 2 ( ˜ ρ − 1 1 , 2 ◦ f 2 , id h 1 , 2 ) ◦ ˜ ρ 2 , 3 , h − 1 2 , 3 ) , ( f 1 ◦ ˜ ρ 1 , 2 , h − 1 1 , 2 )), where o ne has to use for mula (A.2) for the Γ 1 -action on the co mpo sition o f equiv aria nt anafunctors. Using the deﬁnitions of h 2 , 3 and ˜ ρ 2 , 3 as w ell as the axiom of Deﬁnition A.1 (ii) one c an s how that CW = CCW . The fourth and last axio m requir es that comp ositor s a nd unitors are compatible with each other in the sens e that for e a ch 1 -morphism F : P / / P ′ the 2-morphisms E M ( F ) ∼ = E M ( F ◦ id P ) c id P ,F + 3 E M ( F ) ◦ E M (id P ) id ◦ u P + 3 E M ( F ) ◦ id E M ( P ) ∼ = E M ( F ) and E M ( F ) ∼ = E M (id P ′ ◦ F ) c F, id P ′ + 3 E M (id P ′ ) ◦ E M ( F ) u P ′ ◦ id + 3 id E M ( P ′ ) ◦ E M ( F ) ∼ = E M ( F ) are the iden tity 2-morphisms. W e pr ove this for the ﬁrst one and leav e the second as a n exercise. Using the de ﬁnitio ns, we see that the 2-morphism has the reﬁnemen t W := P 0 × M P 0 × M P ′ 0 with r = pr 13 and r ′ = pr 23 . The maps x W : W / / P 0 × M P 0 and y W : W / / P ′ 0 × M P ′ 0 are pr 12 and ∆ ◦ pr 3 , resp ectively , where ∆ is the diagona l map. Its bundle mo rphism is a morphism ϕ : pr ∗ 13 Q / / pr ∗ 23 Q ⊗ pr ∗ 12 P , where Q = F × Γ 0 is the principal Γ-bundle o f E M ( F ), and P = P 1 × Γ 0 is the principal Γ-bundle of E M ( P ). Over a p o int ( p 1 , p 2 , p ′ ) and ( f , g ) ∈ pr ∗ 13 Q , i.e. α l ( f ) = p 1 and R ( α r ( f ) , g − 1 ) = p ′ , the bundle morphism ϕ is given by ( f , g ) ✤ / / ( ρ ( ˜ ρ − 1 ◦ f , id h ) , g h, ˜ ρ, h − 1 ), where h ∈ Γ 0 , and ˜ ρ ∈ P 1 with s ( ˜ ρ ) = R ( p 2 , h − 1 ) and t ( ˜ ρ ) = α l ( f ). W e have to compare ( W, ϕ ) with the identit y 2 -morphism of E M ( F ), which has the reﬁnement Z with r = r ′ = id and the ident ity bundle morphism. Accor ding to the equiv alence r elation o n bundle g e r b e 2-mor phisms we hav e to ev aluate ϕ over a p oint w ∈ W with r ( w ) = r ′ ( w ), i.e. w is of the for m w = ( p, p, p ′ ). Here we ca n choose h = 1 and ˜ ρ = id p , in which case we hav e ϕ ( f , g ) = (( f , g ) , (id p , 1)). This is indeed the iden tity on Q .  – 47 – Prop erties of the 2-functor E M F or the pro of o f Theorem 7.1 we pr ovide the following tw o statemen ts. Lemma 7.1.7. The 2-fu n ctor E M is ful ly faithful on H om-c ate gories. Pro of. Let P , P ′ be principa l Γ-2 -bundles over M , a nd let F 1 , F 2 : P / / P ′ be 1- morphisms. By Lemma 5.2.8 ev ery 2-morphis m η : E M ( F 1 ) + 3 E M ( F 2 ) can be repr e sented b y one whose reﬁnement is P 0 × M P ′ 0 , so that its bundle isomor phism is η : Q 1 / / Q 2 , where Q k := F k × Γ for k = 1 , 2 . W e can read oﬀ a map η : F 1 / / F 2 , and it is e a sy to see that this is a 2-morphism η : F 1 + 3 F 2 . This pro cedure is clear ly inv erse to the 2-functor E M on 2-morphisms.  Prop ositi on 7. 1.8. The 2-funct ors E M form a 1-morph ism b etwe en pr e-2-stacks. Pro of. F or a smo oth map f : M / / N , we hav e to lo ok at the dia gram 2- B un Γ ( N ) E N   f ∗ / / 2- B un Γ ( M ) E M   G r b Γ ( N ) f ∗ / / G r b Γ ( M ) of 2-functors . F or P a principal Γ-2-bundle ov er N , the Γ-bundle ger b e E M ( f ∗ P ) has the s ur jective submersion pr 1 : Y := M × N P 0 / / M , the principa l Γ-bundle P := M × N P 1 × Γ 0 ov er Y [2] , and a bundle gerb e pro duct µ deﬁned as in (7.1.3) that ignor es the M -factor. On the other hand, the Γ- bundle gerb e f ∗ E N ( P ) has the same surjective s ubmersion, and – up to canonical ident iﬁcatio ns b etw een ﬁbre pro ducts – the same Γ- bundle and the s ame bundle gerb e pr o duct. These cano nica l identiﬁcations make up a pse udonatural trans fo rmation that renders the ab ov e diagram comm utative.  7.2 F r om Bundle Ger b es to Principal 2-Bundles W e now provide the data w e will feed int o Lemma B.1 in order to pro duce a 2 -functor R M : G r b Γ ( M ) / / 2- B un Γ ( M ) that is inv erse to the 2-functor E M constructed in the previous section. These data are: 1. A principa l Γ-2 -bundle R G for eac h Γ-bundle gerb e G over M . 2. A 1- is omorphism A G : G / / E M ( R G ) for each Γ-bundle ger b e G ov er M . 3. A 1-isomorphism R A : P / / P ′ and a 2-isomorphism η A : A + 3 E M ( R A ) for all principal Γ-2-bundles P , P ′ ov er M and all bundle ger b e 1-morphis ms A : E M ( P ) / / E M ( P ′ ). – 48 – Construction of the principal Γ -2-bundle R G W e assume that G consists of a surjective submersio n π : Y / / M , a pr incipal Γ- bundle P o ver Y [2] and a bundle gerb e pro duct µ . Let α : P / / Γ 0 be the anchor of P , and let χ : P / / Y [2] be the bundle pro jection. The Lie gr o up oid P of the principa l 2 - bundle R G is deﬁned by P 0 := Y × Γ 0 and P 1 := P × Γ 0 ; source ma p, target ma ps, a nd comp o s ition a re giv en b y , r esp ectively , s ( p, g ) := ( π 2 ( χ ( p )) , g ) , t ( p , g ) := ( π 1 ( χ ( p )) , α ( p ) − 1 · g ) and ( p 2 , g 2 ) ◦ ( p 1 , g 1 ) := ( µ ( p 1 , p 2 ) , g 1 ). (7.2.1) The identit y morphism of a n ob ject ( y , g ) ∈ P 0 is ( t y , g ) ∈ P 1 , wher e t y denotes the unit element in P ov er the point ( y , y ), see Lemma 5.2 .5. The inv ers e o f a morphism ( p, g ) ∈ P 1 is ( i ( p ) , α ( p ) − 1 g ), where i : P / / P is the map from Lemma 5.2.5. The bundle pro jection is π ( y , g ) := π ( y ). The action is g iven on ob jects and morphisms b y R 0 (( y , g ) , g ′ ) := ( y , g g ′ ) and R 1 (( p, g ) , γ ) :=  p ◦  id g · γ · id t ( γ ) − 1 g − 1 α ( p )  , g · s ( γ )  . (7.2.2 ) Lemma 7.2.1. This deﬁnes a functor R : P × Γ / / P , and R is an action of Γ on P . Pro of. W e assume that t : H / / G is a smooth crossed module, and that Γ is the Lie 2-group asso ciated to it, see E xample 2.4.2 a nd Remar k 2.4.3. Then we use the co rresp ondence b etw een principal Γ-bundles and principal H -bundles with H -anti-equiv ariant maps to G of Lemma 2.2.9. W riting γ = ( h, g ′ ), w e ha ve R 1 (( p, g ) , γ ) = ( p ⋆ g h, g g ′ ). With this simple formula a t hand it is straightforward to s how that R resp ects so urce and tar- get maps and satis ﬁe s the axiom of an a ction. F o r the comp osition, w e assume co mpo sable ( p 2 , g 2 ) , ( p 1 , g 1 ) ∈ P 1 , i.e. g 2 = α ( p 1 ) − 1 g 1 , and comp osable ( h 2 , g ′ 2 ) , ( h 1 , g ′ 1 ) ∈ Γ 1 , i.e. g ′ 2 = t ( h 1 ) g ′ 1 . Then w e ha ve R (( p 2 , g 2 ) ◦ ( p 1 , g 1 ) , ( h 2 , g ′ 2 ) ◦ ( h 1 , g ′ 1 )) = R (( µ ( p 1 , p 2 ) , g 1 ) , ( h 2 h 1 , g ′ 1 )) = ( µ ( p 1 , p 2 ) ⋆ g 1 ( h 2 h 1 ) , g 1 g ′ 1 ) = ( µ ( p 1 ⋆ g 1 h 2 , p 2 ) ⋆ g 1 h 1 , g 1 g ′ 1 ) = ( µ ( p 1 , p 2 ⋆ g 2 h 2 ) ⋆ g 1 h 1 , g 1 g ′ 1 ) = ( µ ( p 1 ⋆ g 1 h 1 , p 2 ⋆ g 2 h 2 ) , g 1 g ′ 1 ) = ( p 2 ⋆ g 2 h 2 , g 2 g ′ 2 ) ◦ ( p 1 ⋆ g 1 h 1 , g 1 g ′ 1 ) = R (( p 2 , g 2 ) , ( h 2 , g ′ 2 )) ◦ R (( p 1 , g 1 ) , ( h 1 , g ′ 1 )), – 49 – ﬁnishing the pr o of.  It is obvious that the action R preserves the pro jection π . Th us, in or der to complete the construction o f the principal 2-bundle R G it remains to show that the functor τ = (pr 1 , R ) is a weak equiv alence. This is the conten t of the following tw o lemmata in connection with Theorem 2.3.13. Lemma 7.2.2. τ is smo othly essential ly surje ctive. Pro of. The condition we hav e to chec k is whether or not the map ( Y × Γ 0 × Γ 0 ) τ × t (( P × Γ 0 ) × M ( P × Γ 0 )) ( s × s ) ◦ pr 2 / / ( Y × Γ 0 ) × M ( Y × Γ 0 ) is a surjective submersion. The left hand s ide is diﬀeomorphic to ( P × Γ 0 ) π 1 × π 1 ( P × Γ 0 ) via pr 2 , so that this is equiv alent to chec king that s × s : ( P × Γ 0 ) π 1 ◦ χ × π 1 ◦ χ ( P × Γ 0 ) / / ( Y × Γ 0 ) × M ( Y × Γ 0 ) is a surjective submers ion. Since the Γ 0 -factors are just sp ectators, this is in turn equiv alent to chec king that ( π 2 × π 2 ) ◦ ( χ × χ ) : P π 1 ◦ χ × π 1 ◦ χ P / / Y [2] is a sur jective submer sion. It ﬁts into the pullback dia gram P π 1 ◦ χ × π 1 ◦ χ P   / / χ × χ   P × P χ × χ   Y [2] π 1 × π 1 Y [2] π 2 × π 2     / / Y [2] × Y [2] π 2 × π 2   Y [2]   / / Y × Y which has a surjective submersion o n the r ight hand side; he nce , als o the map on the left hand s ide m ust b e a s ur jective s ubmersion.  Lemma 7.2.3. τ is smo othly ful ly fa ithful. Pro of. W e assume a smo oth manifold N with tw o s mo oth ma ps f : N / / ( P 0 × Γ 0 ) × ( P 0 × Γ 0 ) and g : N / / P 1 × M P 1 – 50 – such that the diagra m N f   g / / P 1 × M P 1 s × t   ( P 0 × Γ 0 ) × ( P 0 × Γ 0 ) τ × τ / / ( P 0 × M P 0 ) × ( P 0 × M P 0 ) is co mm utative. F or a ﬁxed po int n ∈ N we put (( p 1 , g 1 ) , ( p 2 , g 2 )) := g ( n ) ∈ ( P × Γ 0 ) × M ( P × Γ 0 ) and (( y , g , ˜ g ) , ( y ′ , g ′ , ˜ g ′ )) := f ( n ) ∈ ( Y × Γ 0 × Γ 0 ) × ( Y × Γ 0 × Γ 0 ). The commutativit y of the diagram implies χ ( p 1 ) = χ ( p 2 ) = ( y ′ , y ), so that there exists γ ′ ∈ Γ 1 with p 2 = p 1 ◦ γ ′ . W e deﬁne γ := id g − 1 1 · γ ′ · id α ( p 2 ) − 1 g 2 , whic h yields a morphism γ ∈ Γ 1 satisfying τ ( p 1 , g 1 , γ ) = ( p 1 , g 1 , p 2 , g 2 ) = g ( n ). On the other hand, we chec k that ( s ( p 1 , g 1 , γ ) , t ( p 1 , g 1 , γ )) = ( π 2 ( p 1 ) , g 1 , s ( γ ) , π 1 ( p 1 ) , α ( p 1 ) − 1 g 1 , t ( γ )) = f ( n ), using tha t s ( γ ) = g − 1 1 g 2 and t ( γ ) = g − 1 1 α ( p 1 ) α ( p 2 ) − 1 g 2 . Summar izing, we have deﬁned a smo oth map σ : N / / P 1 × Γ 1 : n ✤ / / ( p 1 , g 1 , γ ) such that τ ◦ σ = g and ( s × t ) ◦ σ = f . Now let σ ′ : N / / P 1 × Γ 1 be another such map, and let σ ′ ( n ) =: ( p ′ 1 , g ′ 1 , γ ′ ). The condition that τ ( σ ( n )) = g ( n ) = τ ( σ ′ ( n )) shows immediately that p 1 = p ′ 1 and g 1 = g ′ 1 , and then that p 1 ◦ γ = p 1 ◦ γ ′ . But s ince the Γ-action on P is principal, w e hav e γ = γ ′ . This shows σ = σ ′ . Summariz ing, P 1 × Γ 1 is a pullback.  Example 7.2 . 4. Supp ose Γ = B U(1) (see E xample 2.1 .1 (b)) and suppose G is a Γ-bundle gerb e ov er M , also known as a U(1)-bundle ge r b e, see E xample 5.1.7. Then, the asso ciated principal B U(1)- 2-bundle R G has the group oid P with P 0 = Y and P 1 = P , sourc e and target maps s = π 2 ◦ χ a nd t = π 1 ◦ χ , and compo sition p 2 ◦ p 1 = µ ( p 1 , p 2 ). The a ction o f B U(1) on P is trivial on the level of ob jects and the giv en U(1)-a ction on P o n the le vel of morphisms. The sa me applies for g eneral ab elian Lie g roups A instead of U(1) . Construction of the 1-isomorphism A G : G / / E M ( R G ) The Γ-bundle gerb e E M ( R G ) has the surjective submersio n ˜ Y := Y × Γ 0 with ˜ π ( y , g ) := π ( y ). The total space o f its Γ-bundle ˜ P is ˜ P := P × Γ 0 × Γ 0 ; it has the anchor α ( p, g , h ) = h , the bundle pro jection ˜ χ : ˜ P / / ˜ Y [2] : ( p, g , h ) ✤ / / (( π 1 ( χ ( p )) , α ( p ) − 1 g ) , ( π 2 ( χ ( p )) , g h − 1 )), – 51 – the Γ-action is ( p, g , h ) ◦ γ (7.1.1) = (( p, g ) ◦ R (( t π 2 ( χ ( p )) , g h − 1 ) , γ ) , s ( γ )) (7.2.2) = (( p, g ) ◦ ( t π 2 ( χ ( p )) ◦ (id gh − 1 · γ · id g − 1 ) , g h − 1 s ( γ )) , s ( γ )) (7.2.1) = ( µ ( t π 2 ( χ ( p )) ◦  id gh − 1 · γ · id g − 1  , p ) , g h − 1 s ( γ ) , s ( γ )) (2.4.3) = ( p ◦  id gh − 1 · γ · id g − 1 α ( p )  , g h − 1 s ( γ ) , s ( γ )), and its bundle gerb e pro duct ˜ µ is given by ˜ µ (( p 23 , g 23 , h 23 ) , ( p 12 , g 12 , h 12 )) (7.1.3) = (( p 12 , g 12 ) ◦ R (( p 23 , g 23 ) , id h 12 ) , h 23 h 12 ) (7.2.2) = (( p 12 , g 12 ) ◦ ( p 23 , g 23 h 12 ) , h 23 h 12 ) (7.2.1) = ( µ ( p 23 , p 12 ) , g 23 h 12 , h 23 h 12 ). In o r der to co mpare the bundle gerb es G a nd E M ( R G ) we consider the smo oth ma ps σ : Y / / Y × Γ 0 and ˜ σ : P / / ˜ P that are deﬁned b y σ ( y ) := ( y , 1) and ˜ σ ( p ) := ( p, α ( p ) , α ( p )). Lemma 7.2.5. ˜ σ deﬁnes an isomorphism ˜ σ : P / / ( σ × σ ) ∗ ˜ P of Γ -bu nd les over Y [2] . Mor e over, the diagr am π ∗ 23 P ⊗ π ∗ 12 P µ   ˜ σ ⊗ ˜ σ / / ˜ π ∗ 23 ˜ P ⊗ ˜ π ∗ 12 ˜ P ˜ µ   π ∗ 13 P ˜ σ / / ˜ π ∗ 13 ˜ P is c ommutative. Pro of. F or the ﬁrs t part it suﬃces to pr ov e that ˜ σ is Γ-equiv ariant, pre s erves the anchors, and that the diagram P χ   ˜ σ / / ˜ P ˜ χ   Y [2] σ × σ / / ˜ Y [2] is commutativ e. Indeed, the commut ativity of the diagram is o bvious, and also that the anchors are preserved. F or the Γ-equiv ar iance, we hav e ˜ σ ( p ◦ γ ) = ( p ◦ γ , s ( γ ) , s ( γ )) = ( p, α ( p ) , α ( p )) ◦ γ = ˜ σ ( p ) ◦ γ . Finally , we calculate ˜ µ (( p 23 , α ( p 23 ) , α ( p 23 )) , ( p 12 , α ( p 12 ) , α ( p 12 ))) = ( µ ( p 23 , p 12 ) , α ( p 23 ) α ( p 12 ) , α ( p 23 ) α ( p 12 )) = ( µ ( p 23 , p 12 ) , α ( µ ( p 23 , p 12 )) , α ( µ ( p 23 , p 12 ))) – 52 – which shows the commutativit y of the diag ram.  Via Lemma 5 .2.7 the bundle mo rphism ˜ σ de ﬁnes the required 1 -morphism A G , and L e mma 5.2.3 guarantees that A G is a 1- iso morphism. Construction of the 1-morphism R A : P / / P ′ Let A : E M ( P ) / / E M ( P ′ ) b e a 1-mor phism betw een Γ- bundle gerb es obta ine d fro m principal Γ-2-bundles P a nd P ′ ov er M . By Lemma 5.2.8 we can assume that A consists of a principal Γ-bundle χ : Q / / Z with Z = P 0 × M P ′ 0 , and some isomor phism β ov er Z [2] . F or prepar ation, w e consider the ﬁbr e pr o ducts Z r := P 0 × M P ′ [2] 0 and Z l := P [2] 0 × M P ′ 0 with the obvious embeddings ι l : Z l / / Z and ι r : Z r / / Z obtained by doubling elements. T ogether with the trivialization of Corollar y 5.2.6, the pullbac ks o f β along ι l and ι r yield bundle mo rphisms β l := ι ∗ l β : pr ∗ 13 Q / / pr ∗ 23 Q ⊗ pr ∗ 12 P and β r := ι ∗ r β : pr ∗ 23 P ′ ⊗ pr ∗ 12 Q / / pr ∗ 13 Q , where P := P 1 × Γ 0 and P ′ := P ′ × Γ 0 are the principal Γ - bundles of the Γ-bundle ger be s E M ( P ) and E M ( P ′ ), respe c tively . Lemma 7.2.6. The bu nd le morphi sms β l and β r have the fo l lowi ng pr op erties: (i) Th ey c ommute wi th e ach other in these sense that the diagr am P ′ p ′ 1 ,p ′ 2 ⊗ Q p 1 ,p ′ 1 β ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ) ) ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ β r / / id ⊗ β l   Q p 1 ,p ′ 2 β l   P ′ p ′ 1 ,p ′ 2 ⊗ Q p 2 ,p ′ 1 ⊗ P p 1 ,p 2 β r ⊗ id / / Q p 2 ,p ′ 2 ⊗ P p 1 ,p 2 is c ommutative for all (( p 1 , p ′ 1 ) , ( p 2 , p ′ 2 )) ∈ Z [2] . (ii) β l is c omp atible with the bund le gerb e pr o duct µ in the sense that β l | p 1 ,p 3 ,p ′ = (id ⊗ µ p 1 ,p 2 ,p 3 ) ◦ ( β l | p 2 ,p 3 ,p ′ ⊗ id) ◦ β l | p 1 ,p 2 ,p ′ for al l ( p 1 , p 2 , p 3 , p ′ ) ∈ P [3] 0 × M P ′ 0 . (iii) β r is c omp atible with the bund le gerb e pr o duct µ ′ in the sense that β r | p,p ′ 1 ,p ′ 3 ◦ ( µ ′ p ′ 1 ,p ′ 2 ,p ′ 3 ⊗ id) = β r | p,p ′ 2 ,p ′ 3 ◦ (id ⊗ β r | p,p ′ 1 ,p ′ 2 ) for al l ( p, p ′ 1 , p ′ 2 , p ′ 3 ) ∈ P 0 × M P ′ [3] 0 . – 53 – Pro of. The iden tities (ii) and (iii) follow b y restr icting the co mm utative diagra m (5.1.1) to the submanifolds P [3] 0 × M P ′ 0 and P 0 × M P ′ [3] 0 of Z [3] , resp ectively . Similarly , the comm utativity of the t wo triangula r s ubdia grams in (i) follows b y restricting (5 .1.1) along appropriate embeddings Z [2] / / Z [3] .  Now we ar e in po sition to deﬁne the anafunctor R A . Firs t, we consider the left a ction β 0 : Γ 0 × Q / / Q : ( g , q ) ✤ / / β r ((id , g ) , q ) that satisﬁes α ( β 0 ( g , q )) = g α ( q ). The action β 0 is pr o p erly disco nt inuous and free b eca use β r is a bundle isomorphism. The quotient F := Q/ Γ 0 is the to ta l spa ce o f the ana functor R A we want to construct. Left and right anchors of an element q ∈ F with χ ( q ) = ( p, p ′ ) are given by α l ( q ) := p and α r ( q ) := R ( p ′ , α ( q )). The actions are deﬁned b y ρ l ( ρ, q ) := β − 1 l ( q , ( ρ, 1)) a nd ρ r ( q , ρ ′ ) := β r (( R ( ρ ′ , id α ( q ) − 1 ) , 1) , q ). The left action is inv a riant under the action β 0 bec ause of Lemma 7 .2.6 (i). F or the r ight action, inv ar iance follows fro m Lemma 7.2.6 (ii) a nd the identit y µ ′ (( R ( ρ ′ , id α ( q ) − 1 g − 1 ) , 1) , (id , g )) (7.1.3) = µ ′ ((id , g ) , ( R ( ρ ′ , id α ( q ) − 1 ) , 1)). Lemma 7.2.7. The ab ove formulas deﬁne an anafunct or F : P / / P ′ . Pro of. The compatibility b etw ee n anc hor s and actions is ea sy to chec k. The axio m for the a ctions ρ l and ρ r follows from Lemma 7.2.6 (ii) and (iii). Le mma 7.2.6 (i) shows that the actions commute. It remains to prov e tha t α l : F / / P 0 is a principal P ′ -bundle. Since α l is a co mpo sition of surjective submersions, we o nly have to show that the map τ : F α r × t P ′ / / F α l × α l F : ( q , ρ ′ ) ✤ / / ( q , ρ r ( q , ρ ′ )) is a diﬀeomorphism. W e construct an in verse ma p τ − 1 as follows. F or ( q 1 , q 2 ) with χ ( q 1 ) = ( p, p ′ ) and χ ( q 2 ) = ( p, ˜ p ′ ), c ho ose a representativ e (( ˜ ρ ′ , g ′ ) , ˜ q ) := β r | − 1 p,p ′ , ˜ p ′ ( q 2 ). Such choices can be ma de lo cally in a smo oth wa y , and the result will not dep end o n them. W e hav e χ ( ˜ q ) = ( p, p ′ ) that that there exists a unique γ ∈ Γ 1 such that q 1 = ˜ q ◦ γ . Now we put τ − 1 ( q 1 , q 2 ) := ( q 1 , R ( ˜ ρ ′ , γ − 1 )). – 54 – The ca lculation of τ − 1 ◦ τ is s traightforw ar d. F or the calculation of ( τ ◦ τ − 1 )( q 1 , q 2 ) we hav e to compute in the seco nd comp onent β r (( R ( ˜ ρ ′ , γ − 1 · id α ( q 1 ) − 1 ) , 1) , q 1 ) = β r (( R ( ˜ ρ ′ , γ − 1 · id α ( q 1 ) − 1 ) , 1) ◦ ( γ · id α ( ˜ q ) − 1 ) , ˜ q ) = β r (( ˜ ρ ′ , α ( q 1 ) α ( ˜ q ) − 1 ) , ˜ q ) = β 0 ( α ( q 1 ) α ( ˜ q ) − 1 g ′− 1 , β r (( ˜ ρ ′ , g ′ ) , ˜ q )) = β 0 ( α ( q 1 ) α ( ˜ q ) − 1 g ′− 1 , q 2 ), and this is equiv alent to q 2 .  In order to promote the anafunctor F to a 1-morphism betw een principal 2- bundles, we have to do t wo things: we ha ve to check that F co mm utes with the pr o jections of the bundle P 1 and P 2 , and w e hav e to co ns truct a Γ- equiv ar iant structure on F . F or the ﬁrst point we use Rema rk 6.1.6 (b), whose criterion π ◦ α l = π ◦ α r is cle arly satisﬁed. F or the seco nd p oint we pro vide a smo oth action ρ : F × Γ 1 / / F in the sense of Deﬁnition A.1 and use Le mma A.2, which provides a co nstruction of a Γ-equiv ariant structure. The action is deﬁned by ρ ( q , γ ) := β − 1 l ( q ◦ (id α ( q ) · γ · id t ( γ ) − 1 ) , (id R ( α l ( q ) ,t ( γ )) , t ( γ ))). (7.2.3) Lemma 7.2.8. This deﬁnes a smo oth action of Γ 1 on F in the sense of Deﬁnition A.1. Pro of. Smo o thness is clea r fr om the deﬁnition. The ident ity ρ ( ρ ( q , γ 1 ) , γ 2 ) = β − 1 l ( q ◦ (id α ( q ) · γ 1 · γ 2 · id t ( γ 2 ) − 1 t ( γ 1 ) − 1 ) , (id , t ( γ 1 · γ 2 ))) = ρ ( q , γ 1 · γ 2 ) follows from the deﬁnition and the tw o ident ities α ( ρ ( q , γ )) = α ( q ) s ( γ ) and ( γ 1 · id t ( γ 1 ) − 1 ) · (id s ( γ 1 ) · γ 2 · id t ( γ 2 ) − 1 t ( γ 1 ) − 1 ) = γ 1 · γ 2 · id t ( γ 2 ) − 1 t ( γ 1 ) − 1 . (7.2.4) The latter can easily b e veriﬁed up on s ubstituting a cro ssed module for Γ. Checking condition (i) of Deﬁnition A.1 just uses the deﬁnitions. W e c heck condition (ii) in tw o steps. First w e prove the ident ity ρ ( ρ l ( ρ, q ) , γ l ◦ γ ) = ρ l ( R ( ρ, γ l ) , ρ ( q , γ )). The main ingre dient is the decomp osition id α ( q ) · ( γ l ◦ γ ) · id t ( γ l ) − 1 = (id α ( q ) · γ · id t ( γ ) − 1 ) ◦ (id α ( q ) s ( γ ) t ( γ ) − 1 · γ l · id t ( γ l ) − 1 ) (7.2.5) that can e.g. b e veriﬁed in the cross ed mo dule language. Now w e compute ρ ( ρ l ( ρ, q ) , γ l ◦ γ ) = β − 1 l ( q ◦ (id α ( q ) · ( γ l ◦ γ ) · id t ( γ l ) − 1 ) , ( R ( ρ, t ( γ l )) , t ( γ l ))) (7.2.5) = β − 1 l ( q ◦ (id α ( q ) · γ · id t ( γ ) − 1 ) , ( R ( ρ, γ l ) , t ( γ l ))) = ρ l ( R ( ρ, γ l ) , ρ ( q , γ )). – 55 – The second step is to show the identit y ρ ( ρ r ( q , ρ ′ ) , γ ◦ γ r ) = ρ r ( ρ ( q , γ ) , R ( ρ ′ , γ r )). Here we use the decomp osition id α ( q ) · ( γ ◦ γ r ) · id t ( γ ) − 1 = (id α ( q ) · γ · id t ( γ ) − 1 ) ◦ (id α ( q ) · γ r · id t ( γ ) − 1 ). (7.2.6) Then w e compute ρ ( ρ r ( q , ρ ′ ) , γ ◦ γ r ) = β − 1 l ( β r (( R ( ρ ′ , id α ( q ) − 1 ) , 1) , q ◦ (id α ( q ) · ( γ ◦ γ r ) · id t ( γ ) − 1 )) , (id , t ( γ ))) (7.2.6) = β − 1 l ( β r (( R ( ρ ′ , γ r · id s ( γ ) − 1 α ( q ) − 1 ) , 1) , β 0 ( α ( q ) s ( γ r ) s ( γ ) − 1 α ( q ) − 1 , q ◦ (id α ( q ) · γ · id t ( γ ) − 1 ))) , (id , t ( γ ))) (7.2.4) = β − 1 l ( β r (( R ( ρ ′ , γ r · id α ( ρ ( q,γ )) − 1 )) , q ◦ (id α ( q ) · γ · id t ( γ ) − 1 )) , (id , t ( γ ))) = ρ r ( ρ ( q , γ ) , R ( ρ ′ , γ r )), where w e ha ve employ ed the equiv alence r elation on F that was g enerated b y the action of β 0 .  Construction of a 2-isomorphis m η A : A + 3 E M ( R A ) W e ma y aga in a s sume that the common re ﬁnement of A is the ﬁbre pr o duct P 0 × M P ′ 0 ; other wise, the pro of o f Lemma 5.2 .8 provides a 2-isomo rphism be t ween A and one of these. Now, A and E M ( R A ) ha ve the same common reﬁnemen t, and η A is given by the map η : Q / / F × Γ 0 : q ✤ / / ( q , α ( q )). This is obviously smo o th and resp ects the pro jections to the ba se: if χ ( q ) = ( p, p ′ ), then χ ( q , α ( q )) (7.1.6) = ( α l ( q ) , R ( α r ( q ) , α ( q ) − 1 )) = ( p, p ′ ). F urther, it resp ects the Γ - actions: η ( q ◦ γ ) = ( q ◦ γ , s ( γ )) = β − 1 l ( q ◦ γ , (id , 1)) (7.2.3) = ( ρ ( q , id α ( q ) − 1 · γ ) , s ( γ )) (7.1.6) = η ( q ) ◦ γ , so that η is a bundle mor phism. It remains to verify the comm utativity of the compatibility dia gram (5.1.4). Let (( ρ ′ , g ′ ) , q ′ ) ∈ P ′ ⊗ ζ ∗ 1 Q , a nd let ( q , ( ρ, g )) ∈ ζ ∗ 2 Q ⊗ P be a repre s entativ e for β (( ρ ′ , g ′ ) , q ′ ). In particular, we hav e α ( q ) g = g ′ α ( q ′ ), since β r is a nchor-preserving. Then, we get clo ckwise ( η ⊗ id)( β (( ρ ′ , g ′ ) , q ′ )) = (( q , α ( q )) , ( ρ, g )). (7.2.7) – 56 – Counter-clockwise, we hav e to use the is omorphism o f Lemma 7.1.5 that w e call ˜ β here. Then, ˜ β ((id ⊗ η )(( ρ ′ , g ′ ) , q ′ )) = ˜ β (( ρ ′ , g ′ ) , ( q ′ , α ( q ′ ))) = (( ˜ q , g ′ α ( q ′ ) g − 1 ) , ( ρ, g )) (7.2.8) where the choices ( ˜ ρ, h ) we hav e to mak e for the deﬁnition of ˜ β are here ( ρ, g − 1 ), and ˜ q is deﬁned in (7.1.7), which gives her e ˜ q = β − 1 l ( β r (( ρ ′ , 1) , q ′ ) , ( R ( ρ − 1 , id g − 1 ) , g − 1 )). Comparing (7.2.7) and (7.2.8) it r emains to prove q = ˜ q in F . As F w as the quotient o f Q by the action β 0 , it suﬃces to hav e β 0 ( g ′ , ˜ q ) (i) = β − 1 l ( β r ((id , g ′ ) , β r (( ρ ′ , 1) , q ′ )) , ( R ( ρ − 1 , id g − 1 ) , g − 1 )) (iii) = β − 1 l ( β r (( ρ ′ , g ′ ) , q ′ ) , ( R ( ρ − 1 , id g − 1 ) , g − 1 )) = β − 1 l ( β − 1 l ( q , ( ρ, g )) , ( R ( ρ − 1 , id g − 1 ) , g − 1 )) (ii) = β − 1 l ( q , (id , 1)) = q . This ﬁnishes the constructio n of the 2-isomorphism η A . A Equiv arian t Anafunctors and Gr ou p Actions In this section we are c o ncerned with a Lie 2-gro up Γ and L ie group oids X and Y with actions R 1 : X × Γ / / X and R 2 : Y × Γ / / Y . Deﬁnition A.1. An action of the 2-gr oup Γ on an anafunctor F : X / / Y is an or dinary smo oth action ρ : F × Γ 1 / / F of the gr oup Γ 1 on the total sp ac e F that (i) pr eserves the anchors in the sense t hat the di agr ams F × Γ 1 α l × t   ρ / / F α l   X 0 × Γ 0 R 1 / / X 0 and F × Γ 1 ρ / / α r × s   F α r   Y 0 × Γ 0 R 2 / / Y 0 ar e c ommutative. (ii) is c omp atible wi th the Γ -actions in the sense t hat t he identity ρ ( χ ◦ f ◦ η, γ l ◦ γ ◦ γ r ) = R 1 ( χ, γ l ) ◦ ρ ( f , γ ) ◦ R 2 ( η , γ r ) holds for al l appr opriately c omp osable χ ∈ X 1 , η ∈ Y 1 , f ∈ F , and γ l , γ , γ r ∈ Γ 1 . – 57 – If F 1 , F 2 : X / / Y ar e anafunct ors with Γ -action, a tr ansformation η : F 1 + 3 F 2 is c al le d Γ - e quivariant if the map η : F 1 / / F 2 b etwe en total sp ac es is Γ 1 -e qu ivariant in t he or dinary sense. Anafunctors X / / Y with Γ-a ctions together with Γ-equiv ar iant trans fo rmations form a group oid A na ∞ Γ ( X , Y ). O n the other hand, there is ano ther g roup oid Γ- A na ∞ ( X , Y ) co nsisting of Γ-equiv a riant ana functors (Deﬁnition 6.1 .3) a nd Γ-eq uiv ar ia nt transfor mations (Deﬁnition 6.1.4). Lemma A.2. The c ate gories A na ∞ Γ ( X , Y ) and Γ - A na ∞ ( X , Y ) ar e c anonic al ly isomorphic. Pro of. W e construct a functor E : A na ∞ Γ ( X , Y ) / / Γ- A na ∞ ( X , Y ). (A.1) Let F : X / / Y be an anafunctor with Γ-ac tion ρ . W e sha ll deﬁne a transformation λ ρ : F ◦ R 1 + 3 R 2 ◦ ( F × id). First o f all, the comp osite X × Γ R 1 / / X F / / Y is g iven by the total space ( X 0 × Γ 0 ) R 1 × α l F , left a nd r ig ht anchors send an element ( x, g , f ) to ( x, g ) and α r ( f ), res pe c tively , and the a ctions ar e ( χ, γ ) ◦ ( x, g , f ) = ( t ( χ ) , t ( γ ) , R 1 ( χ, γ ) ◦ f ) and ( x, g , f ) ◦ η = ( x, g , f ◦ η ). On the other hand, the c omp osite X × Γ F × id / / Y × Γ R 2 / / Y is g iven by the tota l space (( F × Γ 1 ) R 2 ◦ ( α r × s ) × t Y 1 ) / ∼ with the equiv alence r e la tion ( f ◦ η ′ , γ ◦ γ ′ , η ) ∼ ( f , γ , R 2 ( η ′ , γ ′ ) ◦ η ). The left and rig ht anchors send an element ( f , γ , η ) to ( α l ( f ) , t ( γ )) and s ( η ), resp ectively , and the actions are ( χ, γ ′ ) ◦ ( f , γ , η ) = ( χ ◦ f , γ ′ ◦ γ , η ) a nd ( f , γ , η ) ◦ η ′ = ( f , γ , η ◦ η ′ ). The in verse of the following ma p will deﬁne the transformation λ : ( F × Γ 1 ) R 2 ◦ ( α r × s ) × t Y 1 / / ( X 0 × Γ 0 ) R 1 × α l F : ( f , γ , η ) ✤ / / ( α l ( f ) , t ( γ ) , ρ ( f , γ ) ◦ η ). Condition (i) ens ures that this map ends in the co rrect ﬁbre pr o duct, and co ndition (ii) a s sures that it is well-deﬁned under the equiv alence relation ∼ . The left anchors are automatically r e s p e cted, and the rig ht anc hors require co nditio n (i). Similar ly , the left action is resp ected automatically , – 58 – and the rig ht actions due to co ndition (ii). The axiom for a transformation is satisﬁed because ρ is a group action. This deﬁnes the functor E on ob jects. On morphisms, it is straigh tforward to chec k that the conditions on b oth hand sides coincide; in particular , E is full a nd faithful. In order to prov e that the functor E is an isomorphism, we start with a given Γ-equiv ar iant structure λ on the anafunctor F . T he n, an action ρ : F × Γ 1 / / F is deﬁned b y ( f , γ ) ✤ / / pr 3 ( λ − 1 ( f , γ , id R 2 ( α r ( f ) ,s ( γ )) )) with pr 3 : ( X 0 × Γ 0 ) R 1 × α l F / / F the pro jection. The axiom for a n a ction is satisﬁed due to the ident ity λ ob eys. It is str a ightforw ard to v erify conditions (i) and (ii) of Deﬁnition A.1. T o clos e the pro o f it suﬃces to notice that the tw o pro ce dures we hav e deﬁned a re (strictly) in verse to eac h other.  W e are also concerned with the compo sition of anafunctors with Γ-actio n. Supp ose that Z is a third Lie groupo id with a Γ-action R 3 , and F : X / / Y a nd G : Y / / Z are anafunctors with Γ-actions ρ : F × Γ 1 / / F and τ : G × Γ 1 / / G . Then, the comp osition G ◦ F is equipped with the Γ-action deﬁned by ( F × Y 0 G ) × Γ 1 / / ( F × Y 0 G ) : (( f , g ) , γ ) ✤ / / ( ρ ( f , γ ) , τ ( g , id s ( γ ) )). (A.2) W e leav e it to the reader to chec k Lemma A.3. L et X , Y and Z b e Lie gr oup oids with Γ -actions R 1 , R 2 and R 3 . (a) L et F : X / / Y and G : Y / / Z b e Γ -e quivariant anafunctors. If Γ -e quivariant structure s on F and G c orr esp ond to Γ 1 -actions under the isomorph ism of L emm a A .2, then the Γ -e quivariant structur e on the c omp osite F ◦ G c orr esp onds to the Γ 1 -action deﬁne d ab ove. (b) The isomorph ism of L emma A.2 identiﬁes the t rivial Γ -e quivariant struct u r e on the identity anafunctor id : X / / X with the Γ 1 -action R 1 : X 1 × Γ 1 / / X 1 on its t otal sp ac e X . B Constructing Equiv alences b e t w een 2-Stac k s Let C b e a bicatego ry (w e a ssume that a sso ciator s and uniﬁers are invertible 2-mor phisms). W e ﬁx the following terminolo g y: a 1- iso morphism f : X 1 / / X 2 in C always includes the data o f an inv erse 1-mor phism ¯ f : X 2 / / X 1 and of 2-is omorphisms i : ¯ f ◦ f + 3 id and j : id + 3 f ◦ ¯ f satisfying the zigzag iden tities. Let D be another bicatego ry . A 2- functor F : C / / D is ass umed to have invertible comp ositor s and unitors. The following lemma is certainly “well-known”, although we have not b een able to ﬁnd a r efer- ence for ex actly this statement. – 59 – Lemma B.1. L et F : C / / D b e a 2-functor that is ful ly faithful on Hom-c ate gories. Supp ose one has chosen: 1. for every obje ct Y ∈ D an obje ct G Y ∈ C and a 1-isomorp hism ξ Y : Y / / F ( G Y ) . 2. for al l obje ct s X 1 , X 2 ∈ C and al l 1-morphisms g : F ( X 1 ) / / F ( X 2 ) , a 1-morph ism G g : X 1 / / X 2 in C to gether with a 2-isomorphism η g : g + 3 F ( G g ) . 1 Then, ther e is a 2-functor G : D / / C and pseudonatur al e quivalenc es a : id D + 3 F ◦ G and b : G ◦ F + 3 id C . In p articular, F is an e qu ivalenc e of bic ate gories. Pro of. W e recall our conv ention co nc e r ning 1-isomo rphisms: the 1-isomo rphisms ξ Y include choices of inv erse 1-morphisms ¯ ξ Y together with 2-iso morphisms i Y : ¯ ξ Y ◦ ξ Y + 3 id and j Y : id + 3 ξ Y ◦ ¯ ξ Y satisfying the zig zag identities. First we explicitly construct the 2-functor G . O n o b jects, we put G ( Y ) := G Y . W e use the notation ˜ g := ( ξ Y 2 ◦ g ) ◦ ¯ ξ Y 1 for all 1- morphisms g : Y 1 / / Y 2 in D , and deﬁne G ( g ) = G ˜ g . If g , g ′ : Y 1 / / Y 2 are 1-mor phisms, and ψ : g + 3 g ′ is a 2-morphism, we consider the 2-morphism ˜ ψ deﬁned by F ( G ˜ g ) η − 1 ˜ g + 3 ( ξ Y 2 ◦ g ) ◦ ¯ ξ Y 1 (id ◦ ψ ) ◦ i d + 3 ( ξ Y 2 ◦ g ′ ) ◦ ¯ ξ Y 1 η ˜ g ′ + 3 F ( G ˜ g ′ ). Since F is fully faithful on 2-morphisms , we may choose the unique 2-mor phis m G ( ψ ) : G ( g ) + 3 G ( g ′ ) such that F ( G ( ψ )) = ˜ ψ . In order to deﬁne the co mp o sitor of G we lo ok at 1- morphisms g 12 : Y 1 / / Y 2 and g 23 : Y 2 / / Y 3 . W e consider the 2 - morphism F ( G ( g 23 ) ◦ G ( g 12 )) c − 1 G ( g 12 ) ,G ( g 23 ) + 3 F ( G ˜ g 23 ) ◦ F ( G ˜ g 12 ) η − 1 ˜ g 23 ◦ η − 1 ˜ g 12   (( ξ Y 3 ◦ g 23 ) ◦ ¯ ξ Y 2 ) ◦ (( ξ Y 2 ◦ g 12 ) ◦ ¯ ξ Y 1 ) a,i Y 2   ( ξ Y 3 ◦ ( g 23 ◦ g 12 )) ◦ ¯ ξ Y 1 η ^ g 23 ◦ g 12 + 3 F ( G ( g 23 ◦ g 12 )); its unique pr eimage under the 2-functor F is the compo sitor c g 12 ,g 23 : G ( g 23 ) ◦ G ( g 12 ) + 3 G ( g 23 ◦ g 12 ). In order to deﬁne the unitor of G w e consider an ob ject Y ∈ D and lo ok at the 2-morphism F ( G (id Y )) η − 1 g id Y + 3 ( ξ Y ◦ id Y ) ◦ ¯ ξ Y l ξ Y ,j − 1 Y + 3 id F ( G ( Y )) u − 1 G ( Y ) + 3 F (id G ( Y ) ). 1 More accurately we should write G X 1 ,X 2 ,g and η X 1 ,X 2 ,g , but we will suppress X 1 and X 2 in the notation. – 60 – Its unique preimag e under the 2 -functor F is the unitor u Y : G (id Y ) + 3 id G ( Y ) . The s e c ond step is to verify the axioms of a 2-functor. This is s imple but extr e mely tedious and can only b e left as an ex ercise. The third step is to construct the pseudonatural transformation a : id D + 3 F ◦ G . Its compo nent at a n ob ject Y in D is the 1-mo r phism a ( Y ) := ξ Y : Y / / F ( G ( Y )). Its comp onent at a 1- morphism g : Y 1 / / Y 2 is the 2 -morphism a ( g ) deﬁned by a ( Y 2 ) ◦ g ξ Y 2 ◦ g id ◦ l − 1 ξ Y 2 ◦ g   ( ξ Y 2 ◦ g ) ◦ id a,i − 1 Y 2   (( ξ Y 2 ◦ g ) ◦ ¯ ξ Y 1 ) ◦ ξ Y 1 η ˜ g ◦ id   F ( G ˜ g ) ◦ ξ Y 1 F ( G ( g ) ) ◦ a ( Y 1 ). There are tw o axio ms a pseudonatura l transfor mation has to satis fy , and their pro ofs ar e aga in left as an exe r cise. It is easy to see that a is a pseudonatur al e quivalenc e , with an inv erse transforma tio n given by ¯ a ( Y ) := ¯ ξ Y . The fourth and las t step is to c o nstruct the pseudona tural transformation b : G ◦ F + 3 id C . Its comp onent at an ob ject X is b ( X ) := G ¯ ξ F ( X ) : G ( F ( X )) / / X . Its comp onent at a 1-mor phism f : X 2 / / X 2 is the 2-mor phism b ( f ) : b ( X 2 ) ◦ G ( F ( f )) + 3 f ◦ b ( X 1 ) given as the unique preimag e under F o f the 2-morphis m F ( b ( X 2 ) ◦ G ( F ( f ))) c − 1 + 3 F ( b ( X 2 )) ◦ F ( G ( F ( f ))) η − 1 ¯ ξ F ( X 2 ) ◦ η − 1 F ( f )   ¯ ξ F ( X 2 ) ◦ (( ξ F ( X 2 ) ◦ F ( f )) ◦ ξ F ( X 1 ) ) a,i F ( X 2 ) ,r   F ( f ) ◦ ¯ ξ F ( X 1 ) id F ( f ) ◦ η ¯ ξ F ( X 1 )   F ( f ) ◦ F ( b ( X 1 )) c + 3 F ( f ◦ b ( X 1 )). The pro ofs of the axio ms are again le ft for the rea der, and aga in it is easy to see that b is a pseudonatural e quivalenc e with an in verse trans formation g iven by ¯ b ( X ) := G ξ F ( X ) .  As a consequence of Lemma B.1 we obtain the certa inly well-known result: – 61 – Corollary B. 2 . L et F : C / / D b e essent ial ly surje ctive, and an e quivalenc e on al l Hom- c ate gories. Then, F is an e quivalenc e of bic ate gories. Since w e work with 2 - stacks ov er ma nifo lds , we nee d the following “sta cky” extension of Lemma B.1. F or a pre-2- stack C , w e denote by C M the 2-ca tegory it ass o ciates to a smo oth ma nifold M , a nd by ψ ∗ : C N / / C M the 2- functor it asso ciates to a smo oth map ψ : M / / N . The pseudonatural equiv alenc e s ψ ∗ ◦ ϕ ∗ ∼ = ( ϕ ◦ ψ ) ∗ will b e suppr essed from the nota tion in the follo wing. If C and D are pre-2- stacks, a 1-morphism F : C / / D asso c iates 2-functors F M : C M / / D M to a smo oth manifold M , pseudonatura l equiv a lences F ψ : ψ ∗ ◦ F N / / F M ◦ ψ ∗ to smo oth maps ψ : M / / N , and certain modiﬁca tions F ψ ,ϕ that control the r elation betw een F ψ and F ϕ for compos able maps ψ and ϕ . Lemma B. 3. Su pp ose C and D ar e pr e-2-stacks over smo oth manifolds, and F : C / / D is a 1-morphism. Supp ose that for every smo oth manif old M 1. the assu mptions of L emma B.1 for the 2-functor F M ar e satisﬁe d and 2. the data ( G Y , ξ Y ) and ( G g , η g ) is ch osen for al l obje cts Y a nd 1-morph isms g in D M . Then, ther e is a 1-morphism G : D / / C of pr e-2-stacks to gether with 2-iso morphisms a : F ◦ G + 3 id D and b : G ◦ F + 3 id C such that for every smo oth manifold M t he 2-functor G M and the pseudonatur al tr ansformations a M and b M ar e the ones of L emma B. 1. In p articular, F is an e quivalenc e of pr e- 2-s t acks. F or the pr o of one constructs the requir ed pse udo natural eq uiv alences G ψ and the mo diﬁcations G ψ ,ϕ from the given ones , F ψ and F ψ ,ϕ , resp ectively , in a similar wa y as explained in the pro of of Lemma B.1. 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Four Equivalent Versions of Non-Abelian Gerbes

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