Principal 2-bundles and their gauge 2-groups

Principal 2-bundles and their gauge 2-groups Christoph W o c k el ∗ christoph@ wockel.eu Abstract In this pap er w e introduce principal 2-bund les and sho w how they are classiﬁed by non-ab elian ˇ Cec h cohomology . Moreo ver, we sho w th at their gauge 2-groups can be describ ed by 2-group-v alued fun ctors, muc h like in classical bun dle th eory . Using this, we show that, under some mild requirements, th ese gauge 2-groups possess a n atural smooth structure. In t he last section w e p rovide some explicit examples. MSC: 55R65, 22E65, 81T13 In tro duction This pap er gives a precise description of globally deﬁned geometric o b jects, which are classiﬁed by non- abelian ˇ Cech cohomology . The g e neral philosophy is to realise these g e o metric structures as categoriﬁed principal bundles, i.e., principal bundles, wher e sets are replace d b y categories, ma ps by functors and commuting diagr ams by natural equiv alences of functors (satisfying cano nic a l coherence conditions ). The cont ro l on these catego riﬁed geo metric s tructures is the amount of natural transfor mations tha t one allows to diﬀer from the ident ity . F or instance, allowing only identit y transformations in the deﬁnition of a categor iﬁe d gro up b elow makes it accessible a s a cr ossed mo dule and thus in ter ms of or dinary group theory . The main enrichm ent t hat comes from categoriﬁca tion is the existence of “higher morphisms” b etw een morphisms, that ar e not present in the set- theoretical setup. Two prominent examples amongst are homotopies betw een contin uo us maps and bimo dule morphisms b e t ween bimo dules (as morphisms betw een rings o r C ∗ -algebra s). These highe r morphisms lea d to very rich struc- ture, b ecause it allows a more ﬂexible concept of inv ertible morphisms, namely inv er tibilit y “up to higher morphisms”. In the tw o exa mples ment ioned, these are the concepts of homotopy eq uiv alence and Mor ita equiv a lence. The main idea of this pap er is to repre s en t non-ab elian co ho mology classes as semi-strict principal 2-bundles, i.e., smo oth 2-space s with a lo cally trivial strict action of a strict Lie 2- group. This pa rticular sub class o f pr incipal 2-bundles ∗ Mathematisc hes Institut, Bunsenstrae 3-5, 37073 G¨ ottingen, German y 1 is q uite easily accessible, w hile the theory in its full gener alit y is m uch more inv o lv ed (cf. [Ba r06]). In the ﬁrst section, w e develop the concept of principa l 2-bundles fro m ﬁrst principles and show very precisely in the second section, how semi-strict pr incipal 2-bundles are classiﬁed by non- abelian ˇ Cech cohomolog y: Theorem. If G is a strict Lie 2-gr oup and M is a smo oth manifold, then semi- strict princip al G -2-bund les over M ar e classiﬁe d up t o Morita e quivalenc e by ˇ H ( M , G ) . In addition, we provide a g eometric wa y to think of the ba nd of a semi-stric t principal 2-bundle. The initial ideas exp osed in this sections ar e not new, the earliest reference we found was [Ded60]. What is in scop e is a clear and down to earth developmen t of the idea in order to make the sub ject e a sily a ccessible. The pap er also aims at op ening the sub ject to inﬁnite-dimensional Lie theory , and w e completely neglect the gauge theoretic motiv ation of the theory (cf. [BS07], [SW08b] and [SW08a] for this). The treatmen t o f symmetry g roups of principal 2-bundles in the thir d section is a ﬁrs t example. There we show how to identify the automorphism 2-g roup of a semi- s trict pr incipal 2 -bundle with 2- group v alued functors a nd use this to put smo oth structures o n these automorphism 2- groups. The ma in r esult of the third section is the following. Theorem. L et P → M b e a semi-strict princip al G -2-bund le. Assume that M is c omp act, that G is lo c al ly exp onential, and that the action of G on P is princip al. T hen C ∞ ( P , G Ad ) G is a lo c al ly exp onential strict Lie 2-gr oup with strict Lie 2-algebr a C ∞ ( P , L ( G ) ad ) G . In the classical setup of principal bundles, given a s smo oth manifolds o n which Lie groups a ct lo cally trivially , this isomorphism of gaug e transfor mations and group v alued functions was the dawning of the glo ba l formulation of gaug e theories in terms of principal bundles, which lead to its fancy developmen ts. The last section treats ex amples, in particular bundle gerb es (or gro upoid extensions, much like [LGSX09, Sect. 2], [GS0 8, Sect. 3 ], [Mo e02, Sect. 4]). This section is not exhaustive, it sho uld give an intuitiv e idea for relating bundle gerb es a nd principal 2-bundles and give some further examples. In the end, we provided a short app endix with so me bas ic co ncepts of lo cally conv ex Lie theory . In co mparison to many o ther exp ositions of this s ub ject the principal bun- dles that we consider are glo bally deﬁned ob jects, co nsidered as smo oth 2-spaces, together with a lo cally trivial smo oth action of a 2- g roup. Most o f the other approaches consider more general base spaces than we do by allowing hypercov- ers, a rbitrary s urjectiv e submersio ns or even mo re g e neral Lie gro upoids. This text only treats ˇ Cech group oids as surjective submersions. So me glo ba l a spects of constructing higher bundles can be found in [LGSX09], [GS08, Ex. 2.19], [Mo e02, Th. 3.1] and [Br e94, Sec t. 2.7], pa rt of which served a s a mo tiv ation for our construction. How ever, there is no reference known to the author, which treats principa l 2 -bundles as smo oth 2-sp ac es with a lo c al ly t rivia l action of a structure 2-g roup. The ob jects that co me closes t to wha t we have in mind a re 2 bundle gerb es, as trea ted in [Mur 96], [ACJ05], [SW08c]. Another globa l way for des cribing principal bundles is in terms of their transp ort 2 -functors (cf. [SW08b] for ordinary principal bundles and [SW08a] for principal 2-bundles ), which is clos ely r elated to our approach. Although ther e frequently ex ist more sys tema tic approaches to the things we presen t, we av oid the intro duction of more gener a l frameworks (suc h as int erna l categ ories, 2- categories, etc.). This is done to keep the text quic kly accessible. Notation: F or a small ca tegory C , w e shall write C 0 and C 1 for its sets of ob jects and morphisms. If F : C → D is a functor, then F 0 and F 1 de- notes the maps on ob jects and morphisms. If F , G : C → D are functors and α x : F ( x ) → G ( x ) is natural, then we also write α : F ⇒ G if w e wan t to emphasise thinking of α as a ma p α : C 0 → D 1 . Mo reov er, we write ∆ C (or shortly ∆ if C is understo o d) for the diagona l embedding C → C × C . Unless stated other wise, all ca tegories are a ssumed to b e small. I Principal 2 -bund les In this section we intro duce pr incipal 2- bundles. F or those r eaders who wonder what the 2 refer s to: throughout this pap er we are w or king in the 2- category of catego ries, wher e ob jects a re given b y categ o ries, mo rphisms by functor s betw een categor ies a nd 2-mor phisms by natural transfor mations betw een func- tors. It is not the case that the things we call 2-something ar e 2-catego ries by themselves (just as a set is not a categ ory but an ob ject in the categor y o f sets ). Although the rep eated term “. . . such that there exist natura l equiv a- lences. . . ” is quite annoying w e state it explicitly every time it o ccurs, b ecause it is the so urce o f the a dditio na l struc tur e in the theory (compa red to o rdinary bundle theor y), whic h deserves to b e p ointed out. O n the other hand, the o c- curring coherence conditions in terms of commutativ e diagrams can safely b e neglected at ﬁr s t reading, b ecause they will a ll b e trivially sa tisﬁed later on. def:2-grou p Deﬁnition I. 1. A we ak 2-gr oup is a monoidal categ o ry , in which each mo r- phism is inv ertible a nd each o b ject is weakly inv ertible. W e sp ell this out for conv enience. It is given by a ca teg ory G together with a multiplication functor ⊗ : G × G → G (mostly written as g · h := ⊗ ( g , h )), an o b ject 1 of G a nd natural equiv alences α g,h , k : ( g · h ) · k → g · ( h · k ) , 3 l g : 1 · g → g and r g : g · 1 → g , such that the diagr ams ( g · h ) · ( k · l ) α g,h,k · l + + X X X X X X X X X X X X X (( g · h ) · k ) · l α g · h,k ,l 3 3 f f f f f f f f f f f f f α g,h,k · i d l # # F F F F F F F F F g · ( h · ( k · l )) ( g · ( h · k )) · l α g,h · k ,l / / g · (( h · k ) · l ) id g · α ( h, k , l ) ; ; x x x x x x x x x and ( g · 1 ) · h r g · id h # # F F F F F F F F α g, 1 ,h / / g · ( 1 · h ) id g · l h { { x x x x x x x x g · h commute. Mo r eov er , we require that each morphis m is in vertible and that for each o b ject x there exis ts a n ob ject x such that x · x and x · x are isomorphic to 1 . A 2-gr oup is a weak 2-group, toge ther with a c oher ent choic e of (weak) inv er ses, given by a n additio na l functor ι : G → G and na tural equiv alence s i g : g · ι ( g ) → 1 and e g : ι ( g ) · g → 1 , s uc h that ( g · ι ( g )) · g α g,ι ( g ) ,g   i g · id g / / 1 · g l g / / g id g   g · ( ι ( g ) · g ) id g · e g / / g · 1 r g / / g commutes. Morphisms of 2-g roups ar e deﬁned to b e weakly monoidal functors of the underlying monoidal catego ry (cf. [BL0 4]). Our main reference for 2- groups is [BL04], where our 2- groups are called coherent 2- groups. As also mentioned in [BL04], this is what is also ca lled a (coherent) ca tegory with group str ucture (cf. [La p83], [Ulb81]). Example I.2. (cf. [BL04, Ex. 34 ]) Let C b e a category and A ut w ( C ) b e the category of equiv alenc e s of C . Then A ut w ( C ) is a weak 2-gro up with re spect to comp osition o f functors and na tur al transforma tions. There is also a cohere n t version o f this 2 -group, cf. [BL04, Ex . 35]. Example I.3. Let G be a g roup, A b e ab elian and f : G × G × G → A b e a group co cycle, i.e., we have f ( g h, k, l ) + f ( g , h, k l ) = f ( g , h, k ) + f ( g , hk, l ) + f ( h, k , l ) (1) eqn:groupC ocycle for g , h, k , l ∈ G . Then we deﬁne a categor y G f by setting O b( G f ) := G and Hom( g , g ′ ) =  A if g = g ′ ∅ else 4 with the comp osition coming fr om group multip licatio n in A . Then ( g a − → g ) · ( h b − → h ) := g h ab − → g h deﬁnes a multiplication functor on G f and ( g , h, k ) 7→  g · h · k f ( g,h, k ) − − − − − → g · h · k  deﬁnes a a natural equiv alence. That this natura l eq uiv alences make the dia- grams from Deﬁnition I.1 c o mm utes is equiv alent to (1). Thus, G f is a weak 2-gro up and ea ch 2-gro up is equiv alent to such a 2-g r oup (cf. [BL04, Se c t. 8.3]). Deﬁnition I. 4. A smo oth 2-sp ac e is a s mall catego ry M such that M 0 , M 1 and M 1 s × t M 1 are smo oth manifolds a nd all structure maps a r e smo oth. A smo oth functor F : M → M ′ betw een smo oth 2 - spaces is a functor such that F 0 and F 1 are smo oth maps. Likewise, a smo oth natu r al tr ansformation α : F ⇒ G b et ween smo oth functors is a natural tr ansformation which is smo oth as a map M 0 → M 1 . Even tually , a smo oth e quivalenc e be tw een smo oth 2- s paces M and M ′ is a s mo oth fu nctor F : M → M ′ such that there exis t a smooth functor G : M ′ → M and smo oth natura l eq uiv alences G ◦ F ⇒ id M and F ◦ G ⇒ id M ′ . In o ur co n text, a manifold refers to a Hausdo rﬀ spa ce, which is lo cally ho me- omorphic to op en subs e ts o f a lo cally conv ex s pa ce such tha t the c o or dinate changes are smo oth (cf. App endix A). This deﬁnition of a smo oth 2 -space is a bit more rigid than the concept used frequently in the literature for it r e quires each s pace really to b e a smo oth manifold and not just a s mooth (or diﬀeolog- ical) space (cf. [BS07], [BH08]). F or our pres en t aim, this c oncept of a smo oth 2-space suﬃces. Mo st of the smo oth 2 -spaces that app ear in this a rticle a r e in fact Lie gr o upoids, but there is no need to r estrict to L ie g roupo ids a prior i. Example I.5. The easiest example of a smo oth 2 -space is simply a smoo th manifold as space of ob jects with only identit y mor phisms and the obvious structure ma ps. Deﬁnition I. 6. A (strong) Lie 2-gr oup is a 2- g roup which is a smo oth 2-space at the same time, such that the functors and natura l equiv alences o ccurring in the deﬁnition o f a 2-g roup are s mooth. In genera l, it is quite restrictive to requir e a ll functors a nd natur a l eq uiv - alences to b e smo oth (cf. [W oc 08, Sect. 2] and [Hen0 8]). How ever, the ma jor part of this pap er deals with strict 2 -groups, where the deﬁnition is a ppropriate (cf. [W o c08, Sect. 2]). W e now consider how smo oth 2-gr oups may act on s mooth 2-spaces . def:G-2-sp ace Deﬁnition I.7. Let G b e a Lie 2 -group. Then a smo oth G -2-sp ac e is a smo oth 2-space M together with a smo oth action, i.e., a s mooth functor ρ : M × G → M 5 (mostly written as x.g := ρ ( x, g )) and smo oth natural eq uiv alences ν : ρ ◦ ( ρ × id G ) ⇒ ρ ◦ (id M ×⊗ ) a nd ξ : ρ ◦ (id M × 1 ) ⇒ id M such that the diagr ams ( x.g ) . ( h · k ) ν x,g ,h · k , , X X X X X X X X X X X X X (( x.g ) .h ) .k ν x.g ,h,k 2 2 f f f f f f f f f f f f f ν x,g ,h . id k # # G G G G G G G G G x. ( g · ( h · k )) ( x. ( g · h )) · k ν x,g · h,k / / x. (( g · h ) · k ) id x .α ( g , h, k ) ; ; w w w w w w w w w and ( x. 1 ) .g ξ x . id g # # G G G G G G G G ν x, 1 ,g / / x. ( 1 · g ) id x .l g { { w w w w w w w w x.g commute. A morphism b et ween G - 2-spaces is a smo oth functor F : M → M ′ , and a smo oth na tural equiv a lence σ F : F ◦ ρ ⇒ ρ ′ ◦ ( F × id G ) such that F ( x ) . ( g · h ) σ F x,g · h + + X X X X X X X X X X X X X ( F ( x ) .g ) .h ν ′ x,g ,h 3 3 f f f f f f f f f f f f σ F x,g . id h " " E E E E E E E E E F ( x. ( g · h )) F ( x.g ) .h σ F x.g ,h / / F (( x.g ) .h ) F ( ν x,g ,h ) ; ; w w w w w w w w w commutes. A 2-morphism be t ween t wo morphisms F , F ′ : M → M ′ of G -2- spaces is a smo oth natura l transformatio n τ : F ⇒ F ′ , s uc h that the diagra m F ( x ) .g σ F x,g   τ x . id g / / F ′ ( x ) .g σ F ′ x,g   F ( x.g ) τ x.g / / F ′ ( x.g ) commutes. An e quivalenc e of s mooth G -2-spaces is a mo r phism F : M → M ′ such that there exists a morphism F ′ : M ′ → M and 2-mor phisms F ◦ F ′ ⇒ id M ′ and F ′ ◦ F ⇒ id M . In this ca se, F ′ is called a we ak inverse of F . Since we shall only consider smo oth actions of Lie 2-gro ups on smo oth 2- spaces we suppres s this adjective in the sequel. W e are now ready to deﬁne principal bundles, who se base is a 2-spac e with only ident ity morphisms. def:princi pal-2-bundle Deﬁnition I.8. Let G b e a Lie 2 -group and M be a smo oth manifold (viewed as a smo oth 2 -space with only ide ntit y mo rphisms). A princip al G -2-bu nd le ov er M is a lo cally trivia l G -2- space ov er M . More pre c isely , it is a smo oth G -2- space P , together with a smo oth functor π : P → M , s uc h that there exist a n ope n cov er ( U i ) i ∈ I of M and equiv a lences Φ i : P | U i → U i × G o f G -2-spaces (where 6 G acts on U i × G by right m ultiplication on the second factor). Moreov er, we require π | U i = pr 1 ◦ Φ i and π ◦ Φ i = pr 1 on the nose for a weak inverse Φ i of Φ i . V ar ious dia grams, emerging fro m the natural equiv alences , a re r equired to commute to ensur e coherence (cf. [Bar 06 , Sect. 2.5]). A morphism o f principal G -2-bundles ov er M is a morphism Φ : P → P ′ of G -2-spaces satisfying π ′ ◦ Φ = π , and a 2- morphism b et ween tw o morphisms of principal G -2-bundles is a 2-mo rphism o f the underlying morphisms of strict G -2 -spaces. As ab ov e, v a rious dia grams are required to commute (cf. [Bar06, Sect. 2 .5]). W e suppr ess an explicit statement of the coherence conditions for br e vit y . W e do not need them in the sequel, as we shall res trict to ca ses where mo s t natural equiv alences a re required to b e the identit y tra nsformation. Note tha t in the previous sense, a princ ipa l G -2-bundle is “lo cally trivia l”, i.e., each P | U i is equiv alent to U i × G . In particula r, each principal G -2-bundle is a Lie gro upoid. Lemma I.9. Princip al G -2-bund les over M , to gether with their morphisms and smo oth natur al e quivalenc es b etwe en morphisms form a 2-c ate gory 2- B un ( M , G ) . Pro of. It is ea sily chec ked, that 2- B un ( M , G ) actually is a sub-2-categ ory of the 2 -category of sma ll categorie s , functors and natura l tr ansformations. I I Classiﬁcation of principal 2-bu ndles b y ˇ Cec h cohomology So far, w e hav e clariﬁed the categ oriﬁcation pro cedure for pr incipal bundles. W e now stick to more sp eciﬁc examples of these bundles, which are class iﬁed by non-ab elian ˇ Cech co ho mology . The idea is to strictify everything that co ncerns the ac tio n in case of a strict structure gr oup. Strictiﬁca tion means for us to require natur a l transforma tions to b e the identit y . Deﬁnition I I.1. A st rict 2-gr ou p is a 2-g r oup G , where all natural equiv alences betw een functors o ccurr ing in the deﬁnition of a 2- group are the identit y . A morphism of str ict 2 -groups is a weak mo noidal functor F : G → G ′ with F ( g · h ) = F ( G ) · F ( h ). W e promised to keep the reader aw ay from 2-catego ries. How ever, man y formulae and calculations b ecome intuitiv ely understa ndable in a gra phical rep- resentation, which we shortly outline in the following rema rk. The r e a der who wan ts to neglect this repr esen tation may do so, we sha ll provide at ea c h stag e explicit formulae. rem:2-grou pAs2-category Remark I I.2. F or a diagr ammatic interpretation of v arious formulae and a r- guments, it is convenien t to view a strict 2 -group G no t only as a categor y , but also a s a 2 -category with one o b ject. The re a der unfamiliar with 2-categ ories 7 may under stand this as the as socia tion of an a rrow • g / / • betw een one ﬁxe d ob ject • , which we assign to each ob ject g o f G , and the asso ciation of a 2- arrow • g % % g ′ 9 9 • h   betw een the ar rows g a nd g ′ , which we as s ign to each mor phism h : g → g ′ in G . Then comp osition in G is depicted by the vertical comp osition • g   g ′ / / g ′′ _ _ • h   h ′   = • g   g ′′ _ _ • h ′ ◦ h   of 2-arr o ws. These diagr ams sho uld cause no confusion with the kind of dia - grams from the previous section, wher e ob jects were r epresented by p oin ts and morphisms were represented by arrows. The latter kind of diagra ms will not o ccur any mo r e in the s e q uel. The multiplication functor on ob jects is then depicted by the hor izon tal concatenation • g / / • g / / • = • g · g / / • of arrows. On morphisms, multiplication is depicted by the horizontal conca te- nation • g % % g ′ 9 9 • g % % g ′ 9 9 • h   h   = • g · g ( ( g ′ · g ′ 6 6 • h · h   of 2-a rrows. W e sha ll o nly deal with str ict 2-g roups in the following text, and there are many diﬀerent wa ys to describ e them. In [BL0 4, p. 3] one ﬁnds the following list (which is explained in detail in [Por08 ]). A stric t 2-gr oup is list:diffe rentDescriptio nOf2-groups • a strict monoidal catego ry in which all ob jects and mor phisms are inv ert- ible, • a strict 2- category with one ob ject in which all 1-morphisms a nd 2- morphisms a re inv ertible, • a gr oup ob ject in ca tegories (also ca lled a c ate goric al gr oup ), 8 • a ca teg ory ob ject in gr oups (also ca lled internal c ate gory in g roups), • a cr o ssed mo dule. F or the prese nt text, the in terpr e tation of a strict 2-gr o up as a cr ossed mo d- ule (and vice versa) will b e of central interest, so we r ecall this co ncept and relate it to 2-gr oups. Deﬁnition I I.3. A cr osse d mo dule consists of tw o g roups G, H , an actio n α : G → Aut( H ) and a homomor phism β : H → G satisfying β ( α ( g ) .h ) = g · β ( h ) · g − 1 (2) eqn:crosse dModule_equiva ri a n c e α ( β ( h )) .h ′ = h · h ′ · h − 1 . (3) eqn:crosse dModule_peiffe r rem:2-grou psFromCrossedM odules Remark I I.4. F r om a cr ossed mo dule, one can build a 2 -group a s follows, cf. [Por08 ], [Ba r06, P r op. 16], [FB02] and [Mac98, Sect. XI I.8]. The s et of o b jects is G a nd the set of mo r phisms is H ⋊ G . E ac h element ( h, g ) ∈ H ⋊ G deﬁnes a mor phism from g to β ( h ) g . This is graphica lly r e presen ted by a 2-ar row • g & & β ( h ) · g 8 8 • . h   Comp osition in the categor y is given by ( h ′ , β ( h ) · g ) ◦ ( h, g ) := ( h ′ · h, g ). This is depicted by deﬁning • g β ( h ) · g / / β ( h ′ ) · β ( h ) · g > > • h   h ′   := • g β ( h ′ · h ) · g > > • . h ′ · h   (4) eqn:2-grou psFromCrossedM od u l e s 1 Consequently , ( e, g ) deﬁnes the identit y of g . One easily checks that the spa ce of co mposa ble pairs of mor phisms is H ⋊ ( H ⋊ G ), where H ⋊ G acts on H by ( h, g ) .h ′ = α ( β ( h ) · g ) .h ′ and on this space, co mposition is given by the homomorphism ( h ′ , ( h, g )) 7→ ( h ′ · h, g ). S imilar ly o ne shows that the s pace of comp osable tr iples o f morphis ms is H ⋊ ( H ⋊ ( H ⋊ G )) and the asso ciativity in H yields the a ssoc ia tivit y of comp osition. The m ultiplication functor is determined by the group mult iplication in H ⋊ G . O n ob jects, it is given by multiplication in G and is thus depicted by • g / / • g / / • := • g · g / / • . On mor phisms, it is g iv en by multiplication in H ⋊ G and thus depicted by • g % % β ( h ) · g 9 9 • g % % β ( h ) · g 9 9 • h   h   := • g · g ( ( β ( h ) · g · β ( h ) · g 6 6 • h · ( g . h )   9 (note that the target of ( h · ( g .h ) , g · g ) is in fac t β ( h ) · g · β ( h ) · g by (2)). Likewise, the inv ersion functor is determined b y in version in H ⋊ G . All this together deﬁnes a strict 2-g roup. The r ev erse cons truction is also s traightf orward. F or a stric t 2-gr oup one chec k s that ob jects and morphisms a re g roups themselves and that all structure maps are gr oup homomor phisms. Then one sets H to be the kernel of the source map and G to b e the spa ce of o b jects. Then G a cts on H by g .h = id g · h · id g − 1 and β is given by the restr ic tion of the targ et map to H . Historically , cr ossed mo dules arose ﬁrst in the work o f Whitehead o n ho- motopy 2 -t yp es [Whi46 ], and the equiv alence of cr ossed mo dules and 2 -groups was established by B rown and Sp encer in [BS76] and by Lo day in [Lo d82]. A detailed exp osition of the eq uiv alence of the 2-catego r ies of strict 2-gr oups a nd of cro ssed mo dules is given in [Por08]. rem:whiske ring Remark I I.5 . T o put more in volv ed dia grams a s the ones o ccurring in the previous re ma rk in to formulae, one has to repla c e • g / / • g   β ( h ) · g ? ? • h   by • g · g   g · β ( h ) · g ? ? • g .h   (5) eqn:whiske ringFromTheLef t on a • with at le ast one incoming tw o outgoing 1- arrows and on a • with at least o ne outgoing and tw o incoming 1-a rrows o ne has to replace • g   β ( h ) · g ? ? • g / / • h   by • g · g   β ( h ) · g · g _ _ • . h   (6) eqn:whiske ringFromTheRig ht This pro cedure is called whiskering and cor resp o nds exactly to m ultiplication of ( h, g ) ∈ H ⋊ G with ( e, ¯ g ) from the left (in (5) ) and fro m the r igh t (in (6)). By p erforming these substitutions one ends up in a diagram which depicts comp ositions o f 2 - arrows, whic h can b e p e r formed as in (4). That v ario us wa ys of doing these s ubstitutions yield the same re sult is enco ded in the a xioms of a crossed mo dule. W e shall s e e how this works in practise in Deﬁnition I I.12. Deﬁnition I I.6. A smo oth cr osse d mo dule is a crossed mo dule ( G, H , α, β ) suc h that G, H ar e Lie groups, β is smo oth a nd the ma p G × H → H , ( g , h ) 7→ α ( g ) .h is smo oth. Remark I I.7. Given a smo oth cr o ssed mo dule, the pull-back of compo sable pairs of morphis m is H ⋊ ( H ⋊ G ) and the space of c o mposa ble triples of mor- phisms is H ⋊ ( H ⋊ ( H ⋊ G )). F ro m the description in Remar k I I.4 it follows that a ll structure maps are s mo oth and th us the corresp onding 2 -group actually is a Lie 2-gr oup. 10 F or a strict Lie 2-g roup to deﬁne a s mo oth cro s sed mo dule it is neces sary that the k ernel of the sourc e map is a Lie subgroup (cf. Example I I.4). In ﬁnite dimensions this is always true but an inﬁnite-dimensio nal Lie group may po ssess closed subgro ups that are no Lie gro ups (cf. [Bou89, Ex. I II.8.2 ]). So in the diﬀer e n tiable se tup we ta k e s mooth cross ed mo dules a s the concept fr om the list o n page 8 r epresenting all its equiv alent descriptions. Deﬁnition I I.8. If G is a strict Lie 2 - group, then a G -2-space is called strict if all natural equiv alences b etw een functor s in the deﬁnition of a G -2 -space are identities. Likewise, a morphism betw een strict G -2- spaces is a morphism F : M → M ′ of the under lying G -2-spaces with σ F the identit y transfo rmation. A 2-morphism b etw een tw o morphisms F , F ′ : M → M ′ of strict G -2-spaces is 2-mor phis m τ : F ⇒ F ′ , satisfying τ ( x.g ) = τ ( x ) . id g . An e quivalenc e o f strict G -2-spaces is a morphism F : M → M ′ such that there exist a morphism F ′ : M ′ → M and 2-morphis ms (all of strict G -2-spaces) F ◦ F ′ ⇒ id M ′ and F ′ ◦ F ⇒ id M . In this ca se, F ′ is called a we ak inverse of F . The crucial p oin t for the following text shall b e tha t we r estrict to strict G -2 -spaces, i.e., we do not allow non-identical natural e q uiv alences to o ccur in any a xiom tha t co ncerns the action functor ρ : M × G → M . Deﬁnition I I.9. If G is a str ict Lie 2- group, then w e call a pr incipal G -2-bundle P semi-strict if G acts strictly o n P a nd the lo cal tr ivialisations may b e c hosen to be eq uiv alences of strict G -2-spaces. A morphism b et ween semi-s trict principa l G -2 -bundles P and P ′ ov er M (or a semi-st rict bun d le morphism for short) is a mo rphism Φ : P → P ′ of strict G -2 -spaces s atisfying π ′ ◦ Φ = π . Lik ewise, a 2-morphis m b et ween tw o mor phisms of semi-str ict pr incipal G -2-bundles is a 2-morphism b etw ee n the underly ing morphisms of strict G -2-spaces. Note tha t our co nc e pt of semi-strictness diﬀers from the o ne used in [Bar06], which is a normalisation req uiremen t o n the c o cycles classifying principal 2- bundles. A str ict principal G -2-bundle would r equire that all natur al equiv a- lences o ccurring in its deﬁnition can b e chosen to be the iden tity . How ever, this deﬁnition is slightly to o rig id for a treatment of no n-abelia n co homology . On the other hand, many genera lis ations are po ssible by increasing ly admitting v ario us additio nal natural equiv alenc e s to b e non-triv ial. Lemma I I.10. Semi-strict princip al G -2-bund les over M , to gether with their morphisms and 2-morphisms form a 2-c ate gory ss - 2- B un ( M , G ) . It shall turn out that semi- strict principal 2- bundles are the immediate gener- alisations of principal bundles, as long as one is interested in g e ometric str uctures deﬁned ov er ordinar y (smo oth) spaces, class iﬁed in ter ms o f ˇ Cech cohomolog y . Moreov er, semi-stric t principal 2-bundles are also linked to gerb es a s follows. rem:from-p rincipal-2-bun dles-to-gerbes Remark I I.11. In general, a gerb e o ver X is a lo cally trans itiv e a nd lo c a lly non-empty stack in gr oupo ids (cf. [Moe0 2]). This mea ns that if U 7→ F ( U ) is the underlying ﬁb ered category of the stack, then one assumes X = S { U | F ( U ) 6 = ∅} 11 and that for ob jects a, b of F ( U ), each x ∈ U has an op en neighbour ho o d V in U with at least o ne arrow a | V → b | V . Now if P is a principal G -2 -bundle with structur e group coming from the smo oth cr ossed mo dule H → Aut ( H ) (and a ssume that Aut( H ) is a Lie group, for instance if π 0 ( H ) is ﬁnitely gener ated, cf. [Bou89]), then this deﬁnes a ger be by U 7→ { sec tio ns of P | U } for U an op en subset of M (cf. [Mo e02, Thm. 3.1]). F ro m now o n w e shall assume that G is a smo oth Lie 2- g roup a rising from a smo oth cros sed mo dule ( α, β , H , G ). W e sp end the res t of this s ection on the classiﬁcation pr oblem for s emi-strict principal G - 2-bundles ov er M . It shall turn o ut that principal G -2-bundles ov e r M a re classiﬁed (in an appropria te se nse) by the non-ab elian co homology ˇ H ( M , G ). There are many treatments of no n-abelia n cohomology in the liter ature, i.e., [B a r06], [BM05], [Bre94], [Gir71], [Ded60], and our deﬁnition is essentially the same (with the usual minor conv en tional diﬀerences). Note that we did not put a degree (or dimension) to ˇ H ( M , G ), for the kind of 2-gr oup that one ta k es for G determines its deg ree in ordinar y ˇ Cech-cohomology . F or ins ta nce, ˇ H ( M , G ) = ˇ H 2 ( M , A ) if G is asso ciated to the cro s sed mo dule A → {∗ } (for A ab elian) a nd ˇ H ( M , G ) = ˇ H 1 ( M , G ) if G is as s ocia ted to the cross ed mo dule {∗} → G (for G a rbitrary). def:non-ab elianCechCohom ology Deﬁnition I I.12. An element in ˇ H ( M , G ) is r epresented b y an op en cov er ( U i ) i ∈ I of M , together with a co llection of smo oth maps g ij : U ij → G and h ij k : U ij k → H (where multiple low er indices r efer to multiple in tersec tions). These maps are r equired to sa tis fy p oin t-wise β ( h ij k ) · g ij · g j k = g ik on U ij k , i.e., • g jk   @ @ @ @ @ @ @ h ijk   • g ik / / g ij ? ?        • (7) eqn:cocyc1 and h ikl · h ij k = h ij l · ( g ij .h j kl ) o n U ij k l , i.e., • g jk / / • g kl   • g ij O O g ik      ? ?      g il / / • h ijk  # ? ? ? ? ? ? ? ? h ikl                 = • g jk / / g jl ? ? ? ? ?   ? ? ? ? ? • g kl   • g ij O O g il / / • h jk l {          h ijl   5 5 5 5 5 5 5 5 5 5 5 5 5 5 (8) eqn:cocyc2 (note that the o ccurrence of g ij in the formula is caused by whis k ering, cf. Remark I I.5). W e furthermor e r e quire g ii = e G and h ij j = h j ji = e H po in t- wise. W e call such a collection ( U i , g ij , h ij k ) a (non-ab elian, nor malised, G - v alued) c o cycle o n M . Two co cycles ( U i , g ij , h ij k ) and ( U ′ i ′ , g ′ i ′ j ′ , h ′ i ′ j ′ k ′ ) are called c ohomolo gous (or e quivalent ) if there exis t a common r eﬁnemen t ( V i ) i ∈ J (i.e., V i ⊆ U ¯ i and V i ⊆ U ′ ¯ i ′ 12 for each i ∈ J and some ¯ i ∈ I and ¯ i ′ ∈ I ′ ) and a collection of smo oth ma ps γ i : V i → G and η ij : V ij → H , satisfying p oint-wise γ i · g ′ ¯ i ′ ¯ j ′ = β ( η ij ) · g ¯ i ¯ j · γ j on V ij , i.e., • g ¯ i ¯ j / / γ i   • γ j   • g ′ ¯ i ′ ¯ j ′ / / • η ij {                (9) eqn:coboun d1 and η ik · h ¯ i ¯ j ¯ k = γ i .h ′ ¯ i ′ ¯ j ′ ¯ k ′ · η ij · g ¯ i ¯ j .η j k on V ij k , i.e., • g ¯ j ¯ k   @ @ @ @ @ @ @ • γ i   g ¯ i ¯ k / / g ¯ i ¯ j ? ? ~ ~ ~ ~ ~ ~ ~ • γ k   • g ′ ¯ i ′ ¯ k ′ / / • h ¯ i ¯ j ¯ k   η ik y  { { { { { { { { { { { { { { { { { { { { { { { { { { = • g ¯ j ¯ k B B B B B B B B γ j   • γ i   g ¯ i ¯ j ~ ~ ~ ~ ? ? ~ ~ ~ ~ • γ k   • g ′ ¯ j ′ ¯ k ′ B B B B • g ′ ¯ i ′ ¯ j ′ ~ ~ ? ? ~ ~ g ′ ¯ i ′ ¯ k ′ / / • . η ij                                 η jk z  | | | | | | | | | | | | | | h ′ ¯ i ′ ¯ j ′ ¯ k ′   (10) eqn:coboun d2 W e furthermor e require η ii = e H po in t-wise. Sometimes, ( V i , γ i , η ij ) is a lso called a c ob oundary b et ween ( U i , g ij , h ij k ) a nd ( U ′ i ′ , g ′ i ′ j ′ , h ′ i ′ j ′ k ′ ). It is ea sily chec ked b y taking reﬁnements a nd p oin t-wise pro ducts in G that this deﬁnes in fact an equiv alence rela tion and we denote by ˇ H ( M , G ) the res ulting s et of equiv alence classe s of co cycles. Note that our normalisation conditions g ii = e G , h ij j = h iij = e H and η ii = e H do not imply g ij = g − 1 j i , h ij k = h − 1 j ik and η ij = h − 1 j i as one might exp ect. Note also that one obtains the non-a belian cohomolog y as used in the texts, mentioned above, when one takes for a (connected) Lie gr o up G the crossed mo dule G → Aut( G ), induced by the conjugation ho mo morphism. Remark I I.13. The previous deﬁnition is not ar bitrary but is the natura l gen- eralisatio n o f the following idea. If G is a Lie group, then an ordinary G -v alued co cycle on M is given by an op e n cov er ( U i ) i ∈ I and smoo th g ij : U ij → G satisfying g ij g j k = g j k and g ii = e G po in t-wise. B ut this is the same as a smo oth functor from the ˇ Cech gro upoid of ( U i ) i ∈ I to the smoo th o ne-ob ject category B G . Likewise, cob oundarie s betw een co cycles a re given by natural transformatio ns b et ween the corre s ponding functors on reﬁned cov ers. If G is a Lie 2-gro up, then we can view G as a 2-ca teg ory B G with only one ob ject (cf. Remark I I.2). Mo reov er the ˇ Cech group oid can also b e viewed as a 2- category with only identit y 2-mor phis ms. Then the co cyles a rise as pseudo (or weak) 2-functor s fr o m this 2-categ ory to B G and cob oundaries as pseudonatural transfor mations b etw een them on reﬁned covers (cf. [Bor 9 4 , Sect. 7.5] o r [SW08 a, App. A] for the terminolog y). 13 The po in t of view from the previous remark also shows that co cycles actually hav e mo re categor ical struc tur e, i.e., there is a 2-categ ory of co cycles, co ming from the 2-categ ory str ucture of pseudofunctor s, pseudona tural transformations and mo diﬁcations from ˇ Cech g roupo ids to G . Ho wev er, this lies b ey ond the scop e o f the present a rticle and we s hall not ela bora te on this additiona l structure. W e shall now show how to obtain a co cycle from a pr incipal 2- bundle. rem:fromBu ndlesToCocycle s Remark I I.14 . The ar g umen t explained b elow reo ccurs freq ue ntly in the fol- lowing construction. F or U ⊆ M , ea c h morphism of s trict G -2-spaces (or stric tly equiv ar ian t functor) Ψ : U × G → U × G , which is the identit y in the ﬁrst com- po nen t, is given b y ( x, g ) 7→ ( x, g ( x ) − 1 · g ) on o b jects for a map g : U → G and ( x, ( h, g )) 7→ ( x, ( g ( x ) − 1 .h, g ( x ) − 1 · g )) on morphisms (it is determined b y its v alues on the sub category ( U × { e G } , U × ( e H , e G )) and the ar tiﬁcial inv er- sions is taken in order to match our other conv ent ions, in pa rticular (11)). If Ψ is smo oth, then g is smo oth and vice v ers a . F or tw o diﬀeren t such Ψ 1 , Ψ 2 , a 2 - morphism Ψ 1 ⇒ Ψ 2 betw een morphisms of strict G -2-spaces is given by ( x, g ) 7→ ( x, ( g 1 ( x ) − 1 .h ( x ) − 1 , g 1 ( x ) − 1 · g )) for a unique map h : U → H satisfy- ing g 2 = ( β ◦ h ) · g 1 . (11) eqn:fromBund lesToCocycles_ 1 Moreov er, each map h : U → H , satisfying (11), deﬁnes a na tural equiv alence Ψ 1 ⇒ Ψ 2 which is smo oth if and only if h is smo oth. F or a principal G -2-bundle P we now co nstruct a co cycle z ( P ) as follows. W e choo se an op en cov er ( U i ) i ∈ I and loc al trivia lisations Φ i : P | U i → U i × G , as well as weak inv erses Φ i : U i × G → P | U i and 2-mor phisms τ i : Φ i ◦ Φ i ⇒ id U i ×G and τ i : Φ i ◦ Φ i ⇒ id P | U i . F or ea c h pair i, j ∈ I , we consider the comp osition Φ j ◦ Φ i : U ij × G → U ij × G , of lo cal trivialisa tions. This is of the ab ov e fo r m and thus determined by a smo oth map g ij : U ij → G , i.e., P | U ij Φ j $ $ J J J J J J J J J U ij × G Φ i : : v v v v v v v v v g ij / / U ij × G , % % % % or, equiv alently , Φ j ( Φ i ( x, e )) = ( x, g − 1 ij ( x )). Substituting Φ i by ( g − 1 ii ◦ π ) · Φ i , we may assume that Φ i ◦ Φ i = id U i ×G on the nose and thus g ii = e G . F or each i 1 , . . . , i n ∈ I with n ≥ 2 we deﬁne the functor Ψ i 1 ...i n := Φ i n ◦ Φ i n − 1 ◦ Φ i n − 1 ◦ · · · ◦ Φ i 2 ◦ Φ i 2 ◦ Φ i 1 . On ob jects, Ψ ij k is then given by ( x, g ) 7→ ( x, (( g ij ( x ) · g j k ( x )) − 1 · g )). With τ j : Φ j ◦ Φ j ⇒ id U j ×G , co mposition of natural transfor mations yields a smo oth equiv alence Ψ ij k ⇒ Ψ ik given by a smo oth map h ij k : U ij k → H satis fying 14 ( β ◦ h ij k ) · g ij · g j k = g ik by (11), i.e., U ij k × G g jk & & L L L L L L L L L L U ij k × G g ik / / g ij 8 8 r r r r r r r r r r U ij k × G h ijk   := U ij k × G g jk   4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Φ j   P | U ijk Φ k M M M & & M M M M Φ j F F U ij k × G Φ i r r r r 8 8 r r r g ij E E                     g ik / / U ij k × G . τ j   R R R R l l l l $ $ $ $ Due to (3), we thus have Φ k ( τ j (Φ i ( x, e ))) = ( g ik ( x ) − 1 .h ij k ( x ) − 1 , ( g ij ( x ) · g j k ( x )) − 1 ) . (12) eqn:fromBund lesToCocycles_ 2 F ro m Φ i ◦ Φ i = id U i ×G it follows that the a bov e natural transfor mation is the ident ity if tw o neig h b o uring indices agr ee and thus that h iij = h ij j = e H . Moreov er, comp osition of na tural tra nsformations yields smo oth equiv a lences Ψ ij k l ⇒ Ψ ij l and Ψ ij k l ⇒ Ψ ikl , a nd the comp ositions Ψ ij k l ⇒ Ψ ij l ⇒ Ψ il and Ψ ij k l ⇒ Ψ ikl ⇒ Ψ il coincide (cf. [Bor 94, Prop. 1.3.5 ]). Sp elling this out leads to h ikl · h ij k = h ij l · ( g ij .h j kl ). The corre s ponding diagram can be o btained from plugging the diagr am, deﬁning h ij k , into (8). The co cycle z ( P ) dep ends on the choices of ( U i ) i ∈ I , Φ i , Φ i and τ i . How ever, a sp ecial cas e of the following ar gumen t shows tha t tw o diﬀerent such choices lead to cohomolo gous co cycles. Assume for the moment that we hav e ﬁxed s ome c hoice o f the previo us data for each P . Given a morphism Φ : P → P ′ of s emi-strict pr incipal G -2-bundles ov er M , we shall show that the co cycles z ( P ) a nd z ( P ′ ), constructed from that da ta, are coho mologous. Fir s t, we ch o ose a co mmon reﬁnement ( V i ) i ∈ J of ( U i ) i ∈ I and ( U ′ i ′ ) i ′ ∈ I ′ , i.e., V i ⊆ U ¯ i and V i ⊆ U ′ ¯ i ′ for each i ∈ J and some ¯ i ∈ I and ¯ i ′ ∈ I ′ . Clear ly , all the choices o f Φ ¯ i ’s and τ ¯ i ’s restrict to V i . F or each i ∈ J , we c onsider the mo rphism Φ ′ ¯ i ′ ◦ Φ ◦ Φ ¯ i : V i × G → V i × G . Since π ′ ◦ Φ = π , this is of the form descr ibed initially and th us determined by a smo oth ma p γ i : V i → G , i.e., P | V i Φ / / P ′ | V i Φ ′ ¯ i ′ $ $ I I I I I I I I I V i × G Φ ¯ i ; ; w w w w w w w w w γ i / / V i × G . " " " " " " As ab ov e, τ j and τ ′ ¯ i − 1 give rise to a smo oth natural equiv a lence Φ ′ ¯ j ′ ◦ Φ ◦ Φ ¯ j | {z } = γ j ◦ Φ ¯ j ◦ Φ ¯ i | {z } = g ¯ i ¯ j ⇒ Φ ′ ¯ j ′ ◦ Φ ′ ¯ i ′ | {z } = g ′ ¯ i ′ ¯ j ′ ◦ Φ ′ ¯ i ′ ◦ Φ ◦ Φ ¯ i | {z } = γ i 15 given by a smoo th map η ij : V ij → H satisfying β ( η ij ) · g ¯ i ¯ j · γ j = γ i · g ′ ¯ i ′ ¯ j ′ by (11), i.e., V ij × G γ j / / V ij × G V ij × G g ¯ i ¯ j O O γ i / / V ij × G g ′ ¯ i ′ ¯ j ′ O O η ij ! ) J J J J J J J J J J J J J J J J J J := V ij × G γ j / / Φ ¯ j   V ij × G P | V ij Φ / / Φ ¯ j W W P ′ | V ij Φ ′ ¯ j ′ t t t : : t t t t Φ ′ ¯ i ′ * * V ij × G γ i / / g ¯ i ¯ j O O Φ ¯ i v v v : : v v v V ij × G . Φ ′ ¯ i ′ l l g ′ ¯ i ′ ¯ j ′ O O       τ ¯ j H H H H H H ( H H H H τ ′ ¯ i ′ − 1 J J J J ( J J J J Now the comp ositions Φ ′ ¯ k ′ ◦ Φ ◦ Φ ¯ k ◦ Φ ¯ k ◦ Φ ¯ j ◦ Φ ¯ j ◦ Φ ¯ i h ¯ i ¯ j ¯ k + 3 Φ ′ ¯ k ′ ◦ Φ ◦ Φ ¯ k ◦ Φ ¯ k ◦ Φ ¯ i η ik + 3 Φ ′ ¯ k ′ ◦ Φ ′ ¯ i ′ ◦ Φ ′ ¯ i ′ ◦ Φ ◦ Φ ¯ i and Φ ′ ¯ k ′ ◦ Φ ◦ Φ ¯ k ◦ Φ ¯ k ◦ Φ ¯ j ◦ Φ ¯ j ◦ Φ ¯ i η jk + 3 Φ ′ ¯ k ′ ◦ Φ ′ ¯ j ′ ◦ Φ ′ ¯ j ′ ◦ Φ ◦ Φ ¯ j ◦ Φ ¯ j ◦ Φ ¯ i η ij + 3 Φ ′ ¯ k ′ ◦ Φ ′ ¯ j ′ ◦ Φ ′ ¯ j ′ ◦ Φ ′ ¯ i ′ ◦ Φ ′ ¯ i ′ ◦ Φ ◦ Φ ¯ i h ′ ¯ i ′ ¯ j ′ ¯ k ′ + 3 Φ ′ ¯ k ′ ◦ Φ ′ ¯ i ′ ◦ Φ ′ ¯ i ′ ◦ Φ ◦ Φ ¯ i coincide (cf. [Bor 94, Prop. 1.3 .5]). Sp elling this out leads to η ik · h ¯ i ¯ j ¯ k = γ i .h ′ ¯ i ′ ¯ j ′ ¯ k ′ · η ij · g ¯ i ¯ j .η j k The cor resp o nding diagra m ca n b e o btained from plugging the diag rams, deﬁn- ing h ij k and η ij int o (10). F ro m the prev ious a rgument it a lso fo llows that diﬀerent choices of ( U i ) i ∈ I , Φ i , Φ i , τ i and τ i in the co ns truction of z ( P ) lead to coho mologous co cycles. In fact, if w e apply the constr uction to the identit y of P , then the resulting γ i : U i → G and η ij : U ij → H yield the desired cob oundary . prop:fromB undlesToCocycl es Prop osition I I.15. The c onst ruction fr om t he pr evious r emark assigns t o e ach semi-strict princip al G -2-bund le P an element [ z ( P )] ∈ ˇ H ( M , G ) . Mor e over, if for P and P ′ ther e exists a semi-strict bund le morphism Φ : P → P ′ , then [ z ( P )] = [ z ( P ′ )] . What is s pecial ab out the cons truction in Remark I I.14 is tha t we c an con- struct a pr incipal 2-bundle P z out of a given 2-co cycle z = ( U i , g ij , h ij k ) with [ z ] = [ z ( P )]. T o illus trate the co nstruction we ﬁrst reca ll that for a g iv en g roup v alued co cycle g ij : U ij → G a corr esponding principal bundle is given by P = [ i ∈ I { i } × U i × G/ ∼ , where ( i, x, g ij ( x ) · g ) ∼ ( j, x, g ). The main idea is to mo dify this co nstruction by int ro ducing a re ﬁned identiﬁcation. Thinking catego rically , this means that we 16 do not only remember tha t ob jects a re isomor phic (equiv a len t), but a lso tr ac k consistently the diﬀerent isomo rphisms that may exist. Identif ying iso mo rphic ob jects in P z then leads to the co nstruction b elow of the under lying principal G/β ( H )-bundle (cf. Coro llary I I.25). Mor eo ver, the following construction shall be tailored to ge neralise the fact that for an ordinary principal bundle, the ﬁbres of the pro jection map are equiv alent to the structure g r oup G a s rig h t G -spaces . rem:fromCo cyclesToBundle s Remark I I.16 . Given a 2-co cycle z = ( U i , g ij , h ij k ), we deﬁne the categor y P z by Ob( P c ) = [ i ∈ I { i } × U i × G Mor( P c ) = [ i,j ∈ I { ( i, j ) } × U ij × H × G with the obvious smo oth structure. It shall b e c le ar in the sequel that all maps are smo oth with res pect to this structure, so we shall not comment o n this any more. W e want the structure maps to make the identiﬁcation of ( i, x, g ) ( i,j,x,h,g ) − − − − − − → ( j, x, g ′ ) with • g / / g ij   @ @ @ @ @ @ @ • • g ′ ? ?        h   (the latter dia gram is in G ) an equiv alence of c a tegories. W e thus req uire g ′ = g − 1 ij · β ( h ) · g . Conseq uen tly , the space of comp osable morphisms is ` U ij k × H × G , which we also endow with the obvious smo oth s tr ucture. Then comp osition in P z is deﬁned by setting the comp osition ( i, x, g ) ( i,j,x,h,g ) − − − − − − → ( j, x, g − 1 ij · β ( h ) · g | {z } =: g ′ ) ( j,k,x,h ′ ,g ′ ) − − − − − − − → ( k , x, g − 1 j k · β ( h ′ ) · g ′ ) to b e ( i, x, g ) ( i,k,x,h ijk · ( g ij .h ′ ) · h,g ) − − − − − − − − − − − − − − − → ( k , x, g − 1 ik · β ( h ij k · g ij .h ′ · h ) · g ) , i.e., it is induced by the co mposition • g ij ? ? ? ? ?   ? ? ? ? ? g / / g ik   / / / / / / / / / / / / / / / / / / / / / • • g ′     ? ?     g jk   • g ′′ G G                      h   h ′                   h ijk i q \ \ \ \ \ \ (13) eqn:fromCo cyclesToBundle s1 17 in G . That this deﬁnition sa tisﬁes source -target match ing follows from β ( h ij k ) · g ij · g j k = g ik . Moreov er, ( i , i , x, e, g ) deﬁnes the identit y of ( i, x, g ) and ( i, j, x, h, g ) − 1 = ( j, i , x, g − 1 ij . (( h · h ij i ) − 1 ) , g − 1 ij · β ( h ) · g ). That comp osition is asso ciative follows fro m h ikl · h ij k = h ij l · g ij .h j kl (a corresp onding equality of diagrams may b e o btained from sticking together (13) and (8)). Thus we obtain a smo oth 2 -space P z with a n o b vious morphism P z → M . The right actio n of G on P c is g iv en b y ( i, x, g ) . g := ( i, x, g · g ) on ob jects and by ( i, j, x, h, g ) . ( h, g ) := ( i, j, x, h · ( g .h ) , g · g ) on mor phisms, i.e., it is induced by the horizontal comp osition • g % % g ij · g ′ 9 9 • g % % g ′ 9 9 • h   h   = • g · g ) ) g ij · g ′ · g ′ 5 5 • . h · ( g . h )   W e w ant the lo cal triv ia lisations Φ i to b e given by the canonical inclusion U i × G → P z and Φ i to be tailore d such that the natura l equiv alences τ i : Φ i ◦ Φ i ⇒ id P z | U i are g iv en by ( j, x, g ) 7→  ( i, x, g ij · g ) ( i,j,x,e,g ij · g ) / / ( j, x, g )  . W e thus s et Φ i : P z | U i → U i × G to b e induced by the assig nmen t ( i, x, g ij g ) ( i,j,x,e,g ij g )   ( i,i,x,h ∗ ,g ij g ) / / ( i, x, g ik g − 1 j k β ( h ) g ) ( j, x, g ) ( j,k,x,h, g ) / / ( k , x, g − 1 j k β ( h ) g ) ( i,k,x,e,g ik g − 1 jk β ( h ) g ) − 1 O O 7→ • ( x,g − 1 ji g ) ( ( ( x, ( g jk g ki ) − 1 g ) 6 6 • ( x,h ∗ ,g − 1 ji g )   with h ∗ = h ij k · g ij .h , i.e., Φ i is deﬁned by ( j, x, g ) 7→ ( x, g ij · g ) ( j, k , x, h, g ) 7→ ( x, h ij k · g ij .h, g ij · g ) . In fact, Φ i ◦ Φ i = id U i ×G on the nose a nd τ i : id P c | U i ⇒ Φ i ◦ Φ i is then given by ( j, x, g ) 7→ ( i, j, x, e, g ij g ). prop:fromC ocyclesToBundl es Prop osition I I.17. F or e ach G -value d c o cycle z on M , the princip al G -2- bund le P z over M , c onstructe d in the pr evious r emark, has [ z ( P z )] = [ z ] . Mor e- over, if z is e quivalent t o z ′ , then ther e exists a semi-strict princip al G -2-bu n d le P over M , and two semi-strict bund le morphisms Φ : P → P z and Φ ′ : P → P z ′ . Pro of. Applying the co nstruction of z ( P z ) fro m Remark I I.14 to the bundle P z , constructed in Remark I I.16 (and cho osing in this cons tr uction ( U i ) i ∈ I , Φ i , 18 Φ i , τ i and τ i as deﬁned in Re ma rk I I.1 6) shows that with these choices we have in fac t z ( P z ) = z . In order to verify the second claim w e ﬁrst consider the case of tw o coho- mologous co cycles z = ( U i , g ij , h ij k ) and z ′ = ( U ′ i ′ , g ′ i ′ j ′ , h i ′ j ′ k ′ ), where ( U i ) i ∈ I is a reﬁnement of ( U ′ i ′ ) i ′ ∈ I ′ . I.e., we hav e U i ⊆ U ′ ¯ i ′ for some ¯ i ∈ I ′ , and the cob oundary is given by smo oth maps γ i : U i → G a nd η ij : U ij → H . In this case we set P to b e P z , Φ to b e id P z , and it remains to construct Φ ′ : P z → P z ′ as follows. First, (9) induces • g ij   @ @ @ @ @ @ • γ − 1 i ? ? ~ ~ ~ ~ ~ ~ g ′ ¯ i ¯ j   @ @ @ @ @ @ • • γ − 1 j ? ? ~ ~ ~ ~ ~ ~ γ − 1 i .η ij   := • γ j   @ @ @ @ @ @ • γ − 1 i / / • g ij ? ? ~ ~ ~ ~ ~ ~ γ i   @ @ @ @ @ @ • γ − 1 j / / • • g ′ ¯ i ¯ j ? ? ~ ~ ~ ~ ~ ~ η ij   and this induces Φ ′ by the ass ig nmen t • g / / g ij $ $ I I I I I I I I I I • • g ′ : : u u u u u u u u u u h   7→ • γ − 1 i / / g ′ ¯ i ¯ j $ $ I I I I I I I I I I • g / / g ij I I I $ $ I I I • • γ − 1 j / / • g ′ : : u u u u u u u u u u h   γ − 1 i .η ij   (14) eqn:fromCocy clesToBundles2 that is Φ ′ ( i, x, g ) = ( ¯ i, x, γ − 1 i · g ) Φ ′ ( i, j, x, h, g ) = ( ¯ i, ¯ j , x, η ij · ( γ − 1 i .h ) , γ − 1 i · g ) . That Φ ′ satisﬁes sour ce-target matching follows from γ i · g ′ ¯ i ′ ¯ j ′ = β ( η ij ) · g ij · γ j and that Φ ′ preserves identities follows from η ii = e . Tha t Φ ′ commutes with comp osition follows from η ik · h ij k = γ i .h ′ ¯ i ′ ¯ j ′ ¯ k ′ · η ij · g ij .η j k (a co rresp onding equal- it y of diag rams may b e obtained fro m plugging together (1 4) and (10)). It is obvious from the deﬁnition that Φ c omm utes strictly with the previously deﬁned action of G . Summarising , Φ deﬁnes a strictly eq uiv ariant functor P z → P z ′ and, mor e o ver, a morphism of pr incipal 2-bundles. In the case that the cob oundary is given by a common reﬁnemen t ( V i ) i ∈ I , which is pr o perly ﬁner than ( U i ) i ∈ I and ( U ′ i ′ ) i ′ ∈ I ′ , w e pr o ceed as follows. As- sume that V i ⊆ U ¯ i and tha t V i ⊆ U ′ ¯ i ′ . Then we restrict g ¯ i ¯ j and g ′ ¯ i ′ ¯ j ′ to V ij , as well as h ¯ i ¯ j ¯ k and h ¯ i ′ ¯ j ′ ¯ k ′ to V ¯ i ′ ¯ j ′ ¯ k ′ . This yields reﬁned co cycles z and z ′ with the corr esponding bundles P z , P z ′ and the cano nical inclusion Φ : P z → P z . Since ( V i ) i ∈ I is a reﬁnement of itself, the pr evious construction yields mor- phisms P z → P z ′ and we set Φ ′ to b e the compo sition of this morphism with the inclusio n P z ′ → P z ′ . rem:morphi smsAndOpenCove rs Remark I I.18. As the previo us pro position sugges ts, ˇ H ( M , G ) do es no t clas- sify bundles in the class ical sense, for bundle mor phis ms b et ween diﬀerent bun- dles may not b e in vertible. F or instance, ea ch cov er U = ( U i ) i ∈ I of M gives 19 rise to a co cycle with v alues in the trivial 2-gr oup. F or tw o diﬀerent cov ers , all these co cycles a re coho mo logous, but one gets a morphism P U → P U ′ (where P U , as cons tructed in Rema rk I I.16, is simply the ˇ Cech-groupo id asso ciated to U ) if and only if U is a reﬁnement of U ′ . How ever, if we s tart with a bundle P , extract a cla s sifying co cycle z ( P ) as in Prop osition II.1 5 (given by the choice of a cov er ( U i ) i ∈ I , trivia lisations Φ i , Φ i and natur a l tr ansformations τ i ) and reconstruct a bundle P z ( P ) as in Prop osition I I.17, then w e always hav e a mo rphism ` Φ i : P z ( P ) → P . On ob jects, this morphism is given by ( i, x, g ) 7→ Φ i ( x, g ) a nd on mor phisms b y the equiv ar ian t extension of  ( i, x, g ) ( i,j,x,e,e ) − − − − − − → ( j, x, g − 1 ij )  7→  Φ i ( x, e ) τ j (Φ i ( x,e )) − 1 − − − − − − − − − → Φ j ( x, g − 1 ij )  (15) eqn:morphism sAndopenCovers 1 (with the nota tio n from Remark II.1 4). That (15) sa tis ﬁe s source - target match- ing follows fro m the deﬁnition of g ij (in the pro of o f Prop osition II.1 5 ), implying Φ j ◦ Φ j ◦ Φ i ( x, e ) = Φ j ( x, g − 1 ij ) on o b jects. Tha t (15) is also co mpatible with comp osition follows from the fact that ` Φ i is equiv ar ian t and from Φ k  τ j  Φ i ( x, e )  − 1 ◦ τ k  Φ j ( x, g − 1 ij )  − 1  = Φ k  τ k  Φ i ( x, e )  − 1 · ( h ij k , e )  , which can b e veriﬁed dir ectly with the aid of (1 2). Since (15) is obviously smo oth and equiv ariant by its deﬁnition, it deﬁnes a morphism P z ( P ) → P . Moreov er, it follows from the fact that G 1 acts free ly on P 1 that this functor is faithful. Prop osition I I.19. The smo oth functor ` Φ i : P z → P , deﬁne d by (15) is a we ak e quivalenc e (or str ong Morita e quivalenc e) of the underlying Lie gr oup oids. Pro of. W e hav e to show that i) the map { (( i, x, g ) , f ) ∈ ( P z ) 0 × P 1 : Φ i ( x, g ) = s ( f ) } | {z } := Q ev − → P 0 , (( i, x, g ) , f ) 7→ t ( f ) admits lo cal inv erses a nd tha t ii) the diagram S { ( i, j ) } × U ij × H × G   / / P 1   ( S { i } × U i × G ) × ( S { j } × U j × G ) / / P 0 × P 0 is a pull-ba ck. 20 T o show i), we choos e a lo cal inv erse of the ta rget ma p in a lo cal trivia lisation and trans form it to P . In fact, it is eas ily chec ked that for (( i, x, g ) , f ) ∈ Q the map π − 1 ( U i ) 0 ∋ y 7→  ( i, x ( y ) , β ( h − 1 f ) g ( y )) | {z } ∈ ( P z ) 0 , τ i ( y ) ◦ Φ i ( x ( y ) , ( h f , β ( h − 1 f ) g ( y ))) ◦ τ i (Φ i ( x ( y ) , β ( h − 1 f ) g ( y ))) − 1 | {z } ∈P 1  , where x ( y ) and g ( y ) denote the co mponents of Φ i ( y ) and h f is deﬁned by Φ i ( f ) = ( x, h f , h − 1 f g ( t ( f ))), deﬁnes a lo cal (left) inv erse for ev , mapping t ( f ) to (( i, x, g ) , f ). In order to check ii), we verify the universal pro perty directly . If ( i, x, g ), ( j, y , k ) and f ar e given, such that f is a morphism from Φ i ( x, g ) to Φ j ( y , k ), then f is a mor phism in π − 1 ( U i ) and in π − 1 ( U j ), for bo th sub categories ar e full. Thus Φ i ( f ) is a mo r phism from (Φ i ◦ Φ i )( x, g ) = ( x, g ) to (Φ i ◦ Φ j )( y , k ) = ( y , g j i ( y ) − 1 k ). F rom this it follows that x = y and that Φ i ( f ) = ( x, ( h f , g )) with β ( h f ) g = g − 1 j i ( x ) k . Th us ( i, j, x, h ij i ( x ) − 1 h f , g ) is a mo rphism in P z , which maps to ( i, x, g ), ( j, x, k ) a nd f under the co rresp onding maps. This morphism is unique, b ecause ` Φ i is faithful. Clearly , if ( i, x, g ), ( j, x, k ) and f dep end contin uo usly o n some parameter , then this morphism do es also. Deﬁnition I I.20 . Two semi- strict pr incipal G - 2-bundles P and P ′ ov er M a re said to b e Morita e quivalent if there exists a third such bundle P and a diagram P Φ      Φ ′ @ @ @ @ P P ′ for semi- strict bundle morphisms Φ and Φ ′ . Lemma I I.21. Morita e quivalenc e of bun d les is in fact an e quivalenc e r elation. Pro of. Supp ose tha t we are g iv en a diagr am P Φ       Φ ′   ? ? ? ? P Φ       ¯ Φ ′ @ @ @ @ P P ′ P ′′ implemen ting Morita equiv ale nce s b etw een P and P ′ , and b et ween P ′ and P ′′ . Then Prop osition I I.15 implies that [ z ( P )] = [ z ( P )] a nd th us there ex- ists by Prop osition II.17 a bundle Q , together with morphisms Q → P z ( P ) and Q → P z ( P ) . With the construction from Remark I I.18 we can ﬁll in the 21 morphisms in the diagra m P z ( P )   Q o o ! ! C C C C C } } { { { { { / / P z ( P )   P Φ | | x x x x x P ¯ Φ ′ # # H H H H H P P ′′ showing the claim. With this sa id, Prop osition I I.15 and Pr opo sition II.1 7 now imply the fol- lowing clas s iﬁcation theorem. Theorem I I.22. If G is a strict Lie 2-gr oup and M is a smo oth manifold, t hen semi-strict princip al G -2-bund les over M ar e classiﬁe d up to Morita e quivalenc e by ˇ H ( M , G ) . cor:Morita EquivalentLieG roupoids Corollary I I.23. If P and P ′ ar e Morita e quivalent as princip al 2-bun d les, then the underlying Lie gr oup oids ar e also Morita e quivalent. W e conclude this section with a couple of remarks on the classiﬁcation result. Remark I I.24. Cor ollary I I.23 shows in par ticular that the Mo rita equiv a lence class [ P ] of a principal 2-bundle gives rise to a Mor ita equiv alence c la ss o f the underlying Lie group oid a nd th us deter mines a smo oth stack. Moreover, the Lie 2-gro up G determines a gr oup sta c k [ G ]. In fact, the Lie gr oupoid underlying G can b e given the structur e of a stacky Lie gr oup [Blo 08] b y turning the structure morphisms into bibundles as in [Blo08, Sect. 4.6], and this stacky Lie g roup gives a gr oup sta c k, cf. [Blo0 8]. T og ether with the mo r phism [ π ] : [ P ] → [ M ], the right [ G ]-action on [ P ] and the existence of lo cal trivialisations give ris e to something like a principal bundle in the 2-ca tegory of smo oth stacks. One could hav e started our inv estiga tion with a rigo urous deﬁnition of this conc e pt a nd then purs uing a cla ssiﬁcation of those principal bundles in ter ms of non-ab elian cohomolog y . This would also hav e lead to a classiﬁca tion in terms of no n- abelia n cohomo logy by very similar arguments. F rom this p oin t of v iew it seems natural that no n-abelia n cohomol- ogy can classify principal 2- bundles o nly up to Morita equiv alence. Howev e r , our a pproach is mo re direct a nd in mo r e down-to-earth terms. rem:BandOf AGerbe Remark I I.25. Let ( G, H , α, β ) b e a smoo th cross ed mo dule such that β ( H ) is a no rmal split Lie subgro up of G a nd let G b e the asso ciated Lie 2-gro up. Then G/ β ( H ) carr ies a natur a l Lie group s tructure (cf. [Nee07, Def. 2.1 ]) a nd the pro jection map G → G/β ( H ) is s mo oth. If P is a semi-strict principal G -2- bundle, then we o bta in fro m this a principa l G/β ( H )-bundle P by identifying isomorphic ob jects in P , i.e., we deﬁne P to b e Ob( P ) / ∼ , wher e p ∼ p ′ if there exists a morphism betw een p and p ′ . Then P inher its naturally a G/β ( H )- action, given by [ p ] . [ g ] := [ p.g ] (where the dot betw een p and g refers to the G -2 -space structure on P ) and we endow P with the quotient top ology fro m 22 Ob( P ). Since P → M (where M is viewed as a catego ry with o nly identit y morphisms) maps iso morphic ob jects to the s ame e le men t in M , this functor induces a map P → M (wher e M is viewed as a spa c e). If ( U i ) i ∈ I is an op en cov er such that there exis t trivia lisations Φ i : P | U i → U i × G , then Φ i induces an G/β ( H )-equiv ariant bijective ma p P | U i → U i × G/β ( H ) and we use this map to endow P w ith a smo o th str ucture. That this is in fact well-deﬁned follows from the fact that the co ordinate changes are then induced by the smo oth maps U ij → G/β ( H ), x 7→ [ g ij ( x )], where g ij : U ij → G is deduce d from Φ i and Φ i as in Rema r k I I.14. This turns P into a principa l G/β ( H )-bundle, whic h we call the b and of P . rem:groupo idExtensions Remark II. 26. Another approach to assig n diﬀerential geometric da ta to non- ab elian ˇ Cech c ohomology is to realise classes in ˇ H ( M , G ) by Mo r ita equiv alence classes of Lie gro upoid extensions, as o utlined in [LGSX09]. In par ticula r, we recov er [LGSX09, 3.14] from the a bov e classiﬁcatio n by considering the c r ossed mo dule H → G := Aut( H ) (for a ﬁnite-dimensional H with π 0 ( H ) ﬁnite, say). F or a non-a belian ˇ Cech co cycle z , P rop osition I I.17 yields a 2-bundle P z . Now G acts on the manifolds of o b jects and morphism o f P z and s inc e this actio n is ob viously principal and all the str ucture maps of P z are compatible with the G -action, we hav e an induced Lie gro upoid P z /G , with ob jects ` U i and morphisms ` U ij × H . Mo reov er, the description of the comp osition in P z shows that P z /G is exa c tly the extens io n of gr oupoids fr o m [LGSX09, Prop. 3 .14]. How ever, P z /G is n ot Mor ita equiv a len t to P z . This ca n b e seen for M = {∗} , where P z is the a ction gr o upoid o f H , acting via β on G and P z is the group oid with one ob ject and automorphism group H . Clear ly , P z /G is transitive while P z is not. On the other hand, there is a n extens io n o f Lie g r oupo ids , cano nically as soci- ated to each principal 2-bundle, for an ar bitr ary ﬁnite-dimensio nal crossed mo d- ule from now on. F o r this we no te that the s trong e q uiv alence π − 1 ( U i ) ∼ = U i × G yields a weak equiv a lence [MM03, Prop. 5.1 1] and we thus hav e Mor( π − 1 ( U i )) ∼ = { ( p, p ′ , ( x, ( h, g ))) ∈ P 0 × P 0 × ( U i × H ⋊ G ) : Φ i ( p ) = ( x, g ) , Φ i ( p ′ ) = ( x, β ( h ) · g ) } from the pull-back condition in the deﬁnition of weak equiv a lences. Mor eo ver, the ab ov e diﬀeomorphism is in fact ( H ⋊ G )-equiv a riant a nd we thus see that the action of ker ( β ) on Mor( π − 1 ( U i )) is principa l. W e thus have an asso ciated extension (iden tities in P z ) · ker( β )   / / P 1 / /     P 1 / ker ( β )     P 0 P 0 P 0 , (16) eqn:LieGro upoidExtension Fr o m C o c y c l e of Lie gr oupoids. Note also , that the co nstruction of the band of a Lie gro upoid extension from [LGSX09] diﬀer s from the co nstruction in Remark I I.25, fo r the band there is a 23 principal bundle ov er the spac e of ob jects of the consider ed Lie group oid, while the band that we co nstruct is a principal bundle over the quotient P 0 / P 1 , if it exists a s a manifold. It would be interesting to under stand the exact co rresp ondence b etw een our approach and [LGSX09] in more detail. Remark I I.27. If β ( H ) is a split Lie s ubgroup, so that K := G/β ( H ) is aga in a smo oth Lie group, then each non-ab elian co cycle determines a smo oth K - v alued 1-co cycle k ij : U ij → K and b ecause cohomolog ous co cycles are mapp ed to coho mologous 1-co cycles, we th us get a ma p Q : ˇ H ( M , G ) → ˇ H 1 ( M , K ) (realised on bundles by the preceding constructio n). This map is surjective, fo r each U ij is contractible, a nd thus each map U ij → K = G/β ( H ) has a lift to G . The ﬁbr es of this map then clas sify semi-strict principal G -2-bundles with a ﬁxed underlying band. In par ticular, if H is ab elian, the ﬁbre of the trivial band (i.e., all g ij take v alues in β ( H )) is iso morphic to H 2 ( M , H ). Remark I I.28. One can also deﬁne a top ological version ˇ H top ( M , G ) of ˇ H ( M , G ), where a ll o ccurring maps g ij , h ij k , γ i and η ij are r equired to b e contin uo us r ather than smo oth. The same classiﬁc a tion go es throug h along the same lines for top ological G -2 -bundles (for G a top ological 2 - group) ov er an ar - bitrary top ological s pa ce M . If M is para compact, then ˇ H top ( M , G ) stands in bijection with the set of ho mo top y classes [ M , B | N G | ] and co nsequen tly with ˇ H 1 top ( M , | N G | ), where | N G | is a top ologic al group, a ssoc iated to G (the g e omet- ric realisa tion of the nerve o f the ca tegory G ). This has been shown in [BS08], cf. also [J ur05] (note that G is alwa ys well-pointed in our case, for we are only deal- ing with Lie gro ups). In particula r, ˇ H top ( M , G ) is tr ivial if M is paracompa ct and co n tractible. This shows that for paraco mpact ﬁnite-dimensiona l M , one can always assume that bundles a re tr ivialised ov er a ﬁxed go o d cov er and o ne do es not run into the pro ble ms describ ed in Remar k II.1 8. A similar a pproach as in [MW09] should yield the same for ˇ H ( M , G ). rem:classi ficationForCen tralExtensions Remark I I.29. Assume that β is s urjectiv e (i.e., assume that β is a central extension) and set ˇ H := ( {∗} , ker( β )). Then H ( M , G ) is is o morphic (as a s et) to ˇ H ( M , H ) for pa racompact a nd ﬁnite-dimensio na l M . In fact, if ( g ij , h ij k ) is a non-ab elian co cycle, then we deﬁne a co homologous co cycle a s follows. First, we assume w.l.o .g. that ea c h U ij is contractible, s o that g ij lifts to η ij : U ij → H (assuming η ii ≡ e H ), a nd we set γ i to b e constantly e G . Then (9) and (10) deﬁne a cohomo logous co cycle ( g ′ ij , h ′ ij k ) and fro m (7) it follows that g ′ ij is also constantly e G and thus h ′ ij k takes v alues in ker( β ). Th us the cano nical map ˇ H ( M , H ) → H ( M , G ) is sur jectiv e and the injectivit y follows similar ly . Consequently , principal G 2-bundles are classiﬁed (up to Morita equiv alenc e ) by H 2 ( M , ker( β )) if β is sur jective. 24 Remark I I.30. There is also a wa y of unders ta nding the construction in Prop osition I I.17, given by a c o nstruction o f 2- bundles as quotients of equiv a- lence 2- r elations as in [Ba r06, P rop. 22]. Let U = ( U i ) i ∈ I be an op en cover o f M . W e set Y := ` U i and π 1 : Y → M , ( i, x ) 7→ x . Then Y [ n ] := Y × M · · · × M Y (n-fold ﬁbre pro duct) is the disjoint union of n -fold intersections of the U i and we deno te by U [ n ] the corr esponding category with o nly iden tity mo r phisms. Moreover, we have canonical pro jections π n 1 ...n k : Y [ n ] → Y [ n − k ] for k < n , which we iden tify with the cor respo nding functors π n 1 ...n k : U [ n ] → U [ n − k ] . A non- abelia n co cycle c = ( g ij , h ij k ), with underlying op en cov er U , deﬁnes what is called a 2-tr a nsition in [Bar 06, Sect. 2 .5.1]. The functor g : U [2] → G (called 2-map in [Bar06]) is g iven by the smo oth map g : Y [2] → G , (( i, j ) , x ) 7→ g ij ( x ), a nd the na tural isomor phism γ : µ ◦ ( g × g ) ◦ ( π 01 × π 12 ) ⇒ g ◦ π 02 is then g iv en by Y [3] ∋ (( i, j, k ) , x ) 7→ ( h ij k ( x ) , g ij ( x ) · g j k ( x )) ∈ H ⋊ G . This 2-transitio n is semi-str ic t in the sense of [Bar0 6 ] (i.e., the natura l µ ◦ g ◦ ι ⇒ 1 for ι : U ֒ → U [2], ( i, x ) 7→ (( i, i ) , x ) is the identit y), for g ii ≡ e G in our s etting. Note that γ is a na tural iso morphism b ecause of co ndition (7) a nd the co herence, required in [Bar 06, Sect. 2 .5.1] is condition (8)). In [Bar0 6 , Prop. 2 2 ], the bundle is cons tr ucted fro m the 2-tr ansition ( g , γ ) by taking the quotient of the categor y U [2] × G by an equiv a lence 2-r elation, determined by ( g , γ ). This equiv alence 2-relation is a categor iﬁed version o f an equiv a lence relation, expressed pur e ly in arr o w-theor etical ter ms (cf. [Bar 06, 1.1.4 and 2.1.4]). This equiv alence 2-r elation is deter mined by tw o functors U [2] × G → U × G , one given by π 1 × id G and the o ther one by (id U × ρ ) ◦ (id U × g × id G ) ◦ ι × id G . One readily chec ks that these t wo functors are what is called jointly 2-monic in [Ba r06], for na tural equiv a lences ar e basic ally given by H -v alued mappings, allowing lifts of natural equiv alences to b e constr ucted directly . The 2 -reﬂexivity map is given by ι × id G (and identities a s natural isomorphisms, for our 2 - transition is semi-s tr ict). The 2-kernel pair of π 2 × id G , π 1 × id G is simply U [3] × G π 23 × id G − − − − − → U [2] × G π 12 × id G   y   y π 1 × id G U [2] × G π 2 × id G − − − − − → U × G . The E uclideanness functor is given b y π 13 × id G and the ﬁrst e q uiv alences in the Euclideanness condition is trivia l and the second o ne is given by γ (we choos e the 2-kernel pair to b e deﬁned by π 1 and π 0 so tha t it ﬁts with the us ual notion of an equiv alence r elation). 25 It can b e chec ked that the catego r y P c is a quotient of this equiv alence relation by the inclusion U × G ֒ → P c (cf. [Ba r06, Sect. 2.1.4]) and thus realis es the bundle co nstructed in [Bar0 6 , Pro p. 22]. W e leav e the details as an exercise. F ro m this construction one sees immediately that the quotient exists in the category of smo oth manifo lds . I I I Gauge 2-groups In the cla s sical setup, a gauge transfor mation o f a principal bundle is a bundle self-equiv a le nce a nd a ll gauge transfo rmations fo rm a group under comp osition. Likewise, in the catego riﬁed cas e the vertical self-e q uiv alences form a category (as functors and natura l transformatio ns) which is in fact a weak 2- g roup with resp ect to the natural comp ositions. W e will show that this weak 2-gro up is in fact equiv alent to a naturally given s trict 2-gro up. Moreover, we show that under so me mild co nditions, this strict 2-gr oup carries natura lly the structure of a strict Lie 2- g roup. As in the previous sectio n, the fact that we o nly consider str ic t actions shall b e the crucial po in t to make the ideas work. Unless stated otherwise, we assume throug hout this section that G is a s trict Lie 2-g roup arising fro m the smo oth cro s sed mo dule ( α, β , G, H ) and that P is a semi-strict principal 2 -bundle ov er the smo oth manifold M . W e will identify M with the s mooth 2-space it deter mines by adding o nly identit y morphis ms . Remark I I I.1. W e conside r the ca tegory Aut( P ) G , whose ob jects a re mo r - phisms F : P → P o f principal G -2-bundles and whose mor phisms are 2 - morphisms α : F ⇒ G (cf. Deﬁnition I.8 ). This is a weak 2-gr oup with resp ect to comp osition of functors a nd natural equiv alenc e s (cf. [BL04, E x. 34]). W e call this weak 2-gro up the gauge 2-gr oup o f P . W e shall ma k e this w eak 2 -group more accessible by showing that it is equiv- alent to a strict 2-gr oup. Initially , we star t with the mo st simple case. Prop osition I I I.2. The c ate gory Aut( G ) G of strictly e quivariant endofunctors of G is e quivalent to G . Pro of. E ac h F ∈ Aut( G ) G is given b y a functor F : G → G satisfying F ( g · g ′ ) = F ( g ) · g ′ on ob jects and F (( h, g ) · ( h ′ , g ′ )) = F (( h, g )) · ( h ′ , g ′ ) on morphisms. F ro m this it follows tha t F ( h, g ) = ( k 1 · k 2 .h, k 2 · g ), where F (( e, e )) = ( k 1 , k 2 ), and the compatibility with the structure ma ps yie lds k 1 = e . Likewise, a natural equiv alence b et ween such functors is uniq ue ly given by its v alue at e G , which is an element of H . Recall that for a category C , we denote b y ∆ : C → C × C the diagonal embedding and ∆ 0 denotes its map on ob jects. lem:mappin gGroupIsA2-Gro up Lemma I I I.3. L et M b e an arbitr ary smo oth 2-sp ac e. Then t he c ate gory C ∞ ( M , G ) of smo oth functors fr om M to G is a 2-gr oup with r esp e ct to the 26 monoidal functor µ ∗ , given on obje cts by µ ∗ ( F, G ) := µ ◦ ( F × G ) ◦ ∆ and on morphisms by µ ∗ ( α, β ) := µ 1 ◦ ( α × β ) ◦ ∆ 0 . Mo r e over, if G is strict, then C ∞ ( M , G ) is so. Pro of. T o check that µ ∗ ( α, β ) is a natural transformatio n from µ ∗ ( F, F ′ ) to µ ∗ ( G, G ′ ), o ne computes that it coincides with the horizo n tal comp osition of the na tural transfor mations (id : µ ⇒ µ ) ◦ (( α, β ) : ( F, F ′ ) ⇒ ( G, G ′ )) ◦ (id : ∆ ⇒ ∆) . The rest is obvious. rem:push-f orwardCrossedM odule Remark I I I.4. O ne can easily rea d o ﬀ from Le mma I I I.3 the cr ossed mo dule ( α ∗ , β ∗ , G ∗ , H ∗ ), that yields C ∞ ( M , G ) as s tr ict 2- g roup (cf. Rema rk I I.4). The ob jects of C ∞ ( M , G ) form a s e t M or ( M , G ) (as a subse t of C ∞ ( M 0 , G 0 ) × C ∞ ( M 1 , G 1 )) and µ ∗ deﬁnes a gr o up multiplication on this set. Th us we set G ∗ = M or ( M , G ). Moreov er, it is easily chec ked that M or ( M , G ) is a subg roup of C ∞ ( M 0 , G 0 ) × C ∞ ( M 1 , G 1 ). Likewise, the mor phis ms in C ∞ ( M , G ) for m a set 2- M or ( M , G ) a nd µ ∗ deﬁnes a gro up m ultiplication on this se t. Ag a in, one can interpret 2- M or ( M , G ) a s a subgro up 2- M or ( M , G ) ≤ M or ( M , G ) × C ∞ ( M 0 , G 1 ) × M or ( M , G ) , with (( F, α, G ) ∈ 2- M or ( M , G )) : ⇔ ( α : F ⇒ G ). W e set H ∗ to be the kernel of the source map as a s ubgroup of C ∞ ( M 0 , G 1 ) × M or ( M , G ). Then the homomorphism β ∗ : H ∗ → G ∗ is the pro jectio n to the second comp onent and the a ction α ∗ of G ∗ on H ∗ is the conjugatio n action on the second comp onen t. If M = ( M , M ) has only identit y morphisms, then M or ( M , G ) ∼ = C ∞ ( M , G 0 ) and 2- M or ( M , G ) ∼ = C ∞ ( M , G 1 ). F ro m this it fol- lows that C ∞ ( M , G ) is asso ciated to the push-for w ard cr ossed mo dule ( α ∗ , β ∗ , C ∞ ( M , G ) , C ∞ ( M , H )), where α ∗ and β ∗ are the point-wise applica- tions of α and β . The following prop osition can b e unders too d as an ins tance of the fac t that in the classical case, a bundle endomor phism (cov ering the identit y on the ba se) of a principa l bundle is a utomatically inv e r tible, and thus bundle endomor phis ms form a gr oup. This ca n b est b e veriﬁed by viewing bundle maps as smo oth group-v alued maps on the total space. How e ver, note that morphisms b et ween distinct pr incipal bundles need not b e invertible (cf. Remark I I.18). prop:gauge 2-GroupIsMappi ng2-Group Prop o sition I I I.5. The we ak 2-gr oup Aut( P ) G of self-e quivalenc es of P is e quivalent, as a we ak 2-gr oup, to C ∞ ( P , G Ad ) G , the strict 2-gr oup of morphisms of G -2-sp ac es, wher e G Ad denotes G with the c onjugation action fr om the right. Pro of. The existence of s trictly e q uiv ariant lo cal tr ivialisations imply tha t P x := π − 1 ( x ) is eq uiv alent to G . Then the usual reaso ning gives Aut( P x , P x ) G ∼ = F un ( P x , G Ad ) G . Since self-equiv a lences preser v e the s ubcatego ries P x , each ob- ject in Aut ( P ) G is thus given by a str ic tly eq uiv ariant functor γ F : P → G Ad . 27 That this functor is in fact s mooth can b e s een in lo cal co ordinates. Likewise, each smo o th 2-mor phism α : F ⇒ G b etw een morphis ms F a nd G of G -2 -spaces is given by a smo oth equiv a riant map η α : P 0 → H ⋊ G . It is readily chec ked that F 7→ γ F and α 7→ η α deﬁnes a mo noidal functor from Aut( P ) G to C ∞ ( P , G ) G . The inv erse functor fro m C ∞ ( P , G ) G to Aut( P ) G is obviously g iven by γ 7→ F γ on o b jects and η 7→ α η on morphisms, where F γ = ρ ◦ (id P × γ ) ◦ ∆ P and α η ( p 0 ) = id p 0 · η ( p 0 ). Remark I I I.6. F or a semi-s trict principal 2-bundle P c , given by a non-ab elian co cycle c = ( h ij k , g ij ), we c a n als o interpret the equiv a lence Aut( P c ) G ∼ = C ∞ ( P c , G ) G as follows. As we hav e s een in the pro of o f P ropo sition I I.15, e ac h self-equiv a le nce o f P c gives r ise to a n equiv a lence of c , given by smo oth maps γ i : U i → G a nd η ij : U ij → H , ob eying (9)-(10) and normalisation. This deﬁnes a smo oth functor P c → G , given o n o b jects by ( i, x, g ) 7→ g · γ i ( x ) a nd on mor phisms by (( i, j ) , x, ( h, g )) 7→ ( γ j ( x ) − 1 . ( h · η ij ( x )) , g · γ j ( x )). W e now turn to the smo othness conditions on Aut( P ) G . W e w ill endow all spaces of contin uo us maps with the s mo oth C ∞ top ology , i.e., if M and N are smo oth manifolds, then we endow C ∞ ( M , N ) with the initial top ology with resp ect to C ∞ ( M , N ) → ∞ Y k =0 C ( T k M , T k N ) , f 7→ ( T k f ) k ∈ N 0 (where C ( T k M , T k N ) is equipp ed with the co mpact-ope n topo lo gy). This is the top ology on spaces of smo oth functions used in [W o c07] and [NW09 ], whose results we shall cite in the sequel in or der to establish Lie 2-g roup str uctures on gauge 2- groups. Prop osition I I I.7. If M is c omp act, then C ∞ ( M , G ) is a Lie 2-gr ou p, which is asso ciate d to the smo oth cr osse d mo dule ( α ∗ , β ∗ , C ∞ ( M , G ) , C ∞ ( M , H )) . Pro of. W e hav e already se e n in Rema rk I I I.4 that C ∞ ( M , G ) is asso ciated to the push-forward cr ossed mo dule ( α ∗ , β ∗ , C ∞ ( M , G ) , C ∞ ( M , H )). This is actually a smo oth cro ssed mo dule, the o nly non-trivial thing to chec k is the smo othness of the a c tion of C ∞ ( M , G ) on C ∞ ( M , H ). But this follows from the smo othness of parameter- dependent push-forward maps (cf. [Gl¨ o0 2b, P rop. 3.1 0] and [W o c06, Prop. 28]) and the fact that automorphic ac tio ns need only b e smo oth on unit neighbourho o ds in o r der to b e globally smo oth. Before coming to the main result o f this section, we hav e to provide some Lie theory for strict Lie 2-gro ups by hand. Remark I I I.8. W e br ie ﬂy r ecall str ict Lie 2-algebr as [BC04]. The deﬁnition is analogo us to that of a strict Lie 2-gr oup as a categ ory in Lie gro ups. First, a 2- vector space is a categ ory , in which all spaces are vector spaces a nd all structure maps are linear. A strict Lie 2 - algebra is then a 2-vector space G , together with 28 a functor [ · , · ] : G × G → G , which is required to b e linea r and skew symmetric on ob jects and morphisms a nd which satisﬁes the Ja cobi identit y [ x, [ y , z ]] = [[ x, y ] , z ] + [ y , [ x, z ]] on ob jects and morphisms. Coming from strict Lie 2 -groups, there is a natura l wa y to asso ciate a strict Lie 2 -algebra to a strict Lie 2-g r oup by applying the Lie functor G 7→ T e ( G ), f 7→ T f ( e ). This works, b ecause this functor preser v es pull- ba c ks and th us a ll categoric al s tructures (cf. [BC0 4, Prop. 5.6 ]). If G is a Lie 2-gr oup, then we denote by L ( G ) the strict Lie 2-alge bra one obtains in this way . W e hav e the same in terplay b et ween cros s ed mo dules of Lie alg ebras and strict Lie 2- algebras as in the case of strict Lie 2-gro ups. A cr osse d mo dule (of Lie a lgebras) consis ts o f tw o Lie alg ebras g , h , and action ˙ α : g → der ( h ) and a homomorphism ˙ β : h → g satisfying ˙ β ( ˙ α ( x ) .y ) = [ x, ˙ β ( y )] and ˙ α ( ˙ β ( x )) .y = [ x, y ]. T o such a cros sed mo dule o ne can asso ciate the Lie 2-a lgebra ( g , h ⋊ g ) with s ( x, y ) = y , t ( x, y ) = ˙ β ( x ) + y , ( z , ˙ β ( x ) + y ) ◦ ( x, y ) = ( z + x, y ) a nd [ · , · ] given by the Lie-bra c ket on g a nd h ⋊ g . Moreover, one checks readily that if G is a ssoc ia ted to ( α, β , G, H ), then L ( G ) is asso ciated to the derived crossed mo dule ( ˙ α, ˙ β , h , g ). thm:LiesTh irdTheorem Theorem I I I.9. If G is a strict Lie 2-algebr a with ﬁnite-dimensional obje ct- and morphism sp ac e, then ther e exists a st rict Lie 2-gr ou p G su ch t hat L ( G ) is isomorphi c to G . Pro of. Ther e is a functor from Lie alg ebras to s imply co nnec ted Lie groups, which is adjoint to the Lie functor. This functor a lso preser ves pull-backs a nd applied to the spaces of ob jects and morphisms a nd to the str ucture maps o f a strict Lie 2- algebra pro duces a strict Lie 2 -group. Remark I I I.10. If G is a strict Lie 2-gr o up with strict Lie 2- algebra G , then we a lso have a strict 2-action A d : G × G → G of G on G . This is g iv en on ob jects and morphisms by ( x, g ) 7→ Ad( g − 1 ) .x , where Ad is the o rdinary adjoint ac tio n. That this deﬁnes a functor follows fr om Ad( ϕ ( g )) . ˙ ϕ ( x ) = ˙ ϕ (Ad( g ) .x ) for e ac h homomorphism ϕ of Lie g roups. W e denote the co rresp onding G -2- spaces by G Ad and G ad . The Lie 2- algebra which is of particular int eres t in this s e ction is the follow- ing. Prop osition I I I.11. If M is a s trict G -2-sp ac e, then C ∞ ( M , L ( G ) ad ) G , the c ate gory of morphisms of G -2-sp ac es fr om M t o L ( G ) ad is a strict Lie 2-algebr a. The functor [ · , · ] is given by the p oint- wise applic ation of the functor in L ( G ) as in L emma III.3. Pro of. W e set G := L ( G ), K := C ∞ ( M , G ) G and identify K 0 with a subset of the lo cally c o n vex Lie algebr a l := C ∞ ( M 0 , G 0 ) × C ∞ ( M 1 , G 1 ). The requirement 29 on ( ξ , ν ) to deﬁne a functor may be expres sed in terms of p oint ev aluations and linear ma ps, for insta nce, the co mpatibilit y with the so urce map is s G ( ν ( x )) = ξ ( s M ( x )) for a ll x ∈ M 1 . The sa me a rgumen t applies for the r equirement on a functor to b e G -equiv ar ian t, and thus K 0 is a clo s ed subalgebra in l . In the sa me wa y , w e may view K 1 as a closed subalg ebra of K 0 × C ∞ ( M 0 , G 1 ) × K 0 , with ( F , α, G ) ∈ Mor( C ∞ ( M , G )) if and only if α : F ⇒ G is a smo oth na tural equiv a le nce. The str ucture maps are given by pro jections, embeddings and push-for w ards by contin uous linear ma ppings and thus all c o n- tin uous algebra morphisms. The crucial to ol in the descriptio n of the Lie gro up structure on C ∞ ( P , G Ad ) G shall b e the exp onential functions on G and H ⋊ G in the cas e that it provides charts for the Lie group structures (i.e., if G a nd H ⋊ G ar e lo c al ly ex p onential , cf. App endix A). Lemma I I I.12. If G is a strict Lie 2-gr oup, such that its gr oup of obje cts and morphisms p ossess an exp onential function, then these functions deﬁne a smo oth functor E xp : L ( G ) → G . Pro of. F or each homomorphis m ϕ : G 1 → G 1 betw een Lie g roups with exp o- nent ial function, the dia gram G 1 ϕ − − − − → G 2 exp 1 x   exp 2 x   g 1 L ( ϕ ) − − − − → g 2 (17) eqn:expone ntialMapCommut es W i t h M o r p h i s m s commutes. Since all requirements on E xp to deﬁne a functor can b e phras ed in such diag rams, the a ssertions follows. rem:equiva rianceOf2-Morp hisms Remark I I I.13. L e t M b e a s trict G -2-space. If the 2-mor phism α : F ⇒ G betw een the morphisms F , G of G -2-spaces is viewed as a map α : M 0 → H ⋊ G , then α = ( α, F 0 ) for some α : M 0 → H and G sa tisﬁes satis ﬁes G 0 = ( β ◦ α ) · F 0 and G 1 = (( α ◦ t M ) · F 1 · ( α ◦ s M ) − 1 , G 0 ◦ s M ) . Thu s G is uniquely determined by F and α . If F and α are is strictly equiv aria n t, then so is G s ince α ∈ C ∞ ( M 0 , H ) G by deﬁnition. F or classica l principal bundles, the compactnes s of the base ma nifold and the lo cal exp onen tiality of the str ucture group ensure the exis tence of Lie group structures on ga uge trans fo rmation gr oups (cf. [W o c07]). W e shall follow s imila r ideas here a nd call a strict Lie 2- group lo c al ly exp onential if its Lie groups of ob jects a nd mo rphisms are so. 30 thm:gauge2 -GroupIsLie2-G roup Theorem I I I.14. As s u me that M is c omp act, that G is lo c al ly exp onential, and that t he actions of G 1 on P 1 and of G 0 on P 0 ar e princip al. Then C ∞ ( P , G Ad ) G is a lo c al ly exp onential strict Lie 2-gr oup with strict Lie 2-algebr a C ∞ ( P , L ( G ) ad ) G . Pro of. The pr oof works simila r as in the case for classical principal bundles in [W o c07]. W e set K := C ∞ ( P , G ) G , K := C ∞ ( P , L ( G )) G and deno te by K 0 , K 0 and K 1 , K 1 the co rresp onding space s of ob jects a nd morphisms. Then we hav e K 0 = { ( γ , η ) ∈ C ∞ ( P 0 , G ) G × C ∞ ( P 1 , H ⋊ G ) H ⋊ G : η ◦ i P = i G ◦ γ γ ◦ s P = s G ◦ η , γ ◦ t P = t G ◦ η , η ◦ c P = c G ◦ ( η s × t η ) } . Since the conditions on ( γ , η ) to b e in K 0 can all be phrased in terms o f ev al- uation maps on P 0 and P 1 , it follows that K 0 is a closed subgro up o f L := C ∞ ( P 0 , G ) G × C ∞ ( P 1 , H ⋊ G ) H ⋊ G , endow ed with the C ∞ -top ology . Similarly , we obta in K 0 as a closed subalgebr a o f l := C ∞ ( P 0 , g ) G × C ∞ ( P 1 , h ⋊ g ) H ⋊ G . Now L is a lo cally co n vex, lo cally e x ponential Lie group, mo delled on l (cf. [W o c07, Thm. 1.11]), beca use the actions o f G o n P 0 and of H ⋊ G on P 1 are free a nd lo cally trivial (cf. Remark I I.26). The exp onen tial function for this Lie group is then given by l ∋ ( ξ , ν ) 7→ (exp G ◦ ξ , exp H ⋊ G ◦ ν ) ∈ L, which restric ts to a diﬀeomorphism on so me zer o neighbo urhoo d in l . Since this is the same a s the comp osition o f the exp onen tial functor E xp with ( ξ , ν ), this ex ponential function res tricts to a ma p from K 0 to K 0 . It follows from the construction o f the Lie group structure on L that this map restr icts to a bijective map of an o p en zer o neighbourho o d in K 0 to an op en unit neighbourho o d in K 0 . The compa tibilit y with the s tructure maps of P , L ( G ) and G ca n b e chec ked by rep eated use of (17) in lo cal co ordinates, for all structure maps of L ( G ) and G are (by the co nstruction of L ( G )) given by mo rphisms of Lie algebra s and Lie groups, co mm uting with the resp ectively ex p onential functions. F or instance exp G ◦ ξ ◦ s P = s G ◦ exp H ⋊ G ◦ ν ⇔ ξ ◦ s P = s L ( G ) ◦ ν. if ξ and ν hav e re presen tatives in loca l co ordinates, which take v a lues in op en zero neighbourho o ds of g a nd h ⋊ g , on whic h the exp onen tial functions restricts resp ectively to a diﬀeo morphism. Thus K 0 is a clo s ed Lie subgr oup of L . In the same manner, one constructs K 1 as a clo sed s ubgroup of K 0 × C ∞ ( P 0 , H ⋊ G ) G × K 0 with ( F, α, G ) ∈ K 1 if and only if α : F ⇒ G is a smo oth natural eq uiv alence (cf. Remark II I.13 and [NW09, Th. A.1], [W o c07, Thm. 1.11 ] for the L ie g roup s tructure o n C ∞ ( P 0 , H ⋊ G ) G ). The exp onential functions on K 0 and C ∞ ( P 0 , H ⋊ G ) H ⋊ G induce an exp onential function o n K 1 and a s b efore, K 1 is a clo sed Lie subgr o up. The structure ma ps a re given by pro jections, em b eddings and push-forwards by Lie g roup mor phisms and thus they a ll are mor phis ms of lo cally c o n vex Lie groups . Corollary I I I.15. If we endow the 2-gr oup C ∞ ( P , G Ad ) G with the Lie 2-gr oup structur e fr om the pr evious t he or em, then the natur al action turns P into a smo oth C ∞ ( P , G Ad ) G -2-sp ac e. 31 Pro of. This follows fro m the fact that ev alua tion ma ps are s mo oth in the C ∞ - top ology . rem:crosse dModuleFromGau ge2-Group Remark I I I.16. T aking Remark II I.13 in to account, one obtains that C ∞ ( P , G Ad ) G is asso ciated to the smo oth crossed mo dule ( α ∗ , β ∗ , C ∞ ( P 0 , H ) G , M or ( P , G ) G ) with β ∗ ( α ) = ( β ◦ α , ( α ◦ t ) · ( α ◦ s ) − 1 ) ( α ∗ ( γ 0 , γ 1 ) . α ) ( p 0 ) = γ 0 ( p 0 ) .α ( p 0 ) . Remark I I I.17. In [Gom0 6], ther e ar e constructed c e n tral ex tensions of ga uge groups of (higher) ab elian gerb es by the use of the cup-pr oduct in smo oth Deligne (hyper-)co homology H n +2 ( M , Z ∞ D ( n + 2)). There the ter m gauge trans - formation is used for H n +1 ( M , Z ∞ D ( n + 1)). It would be very int eres ting to explore the connection to our appro ac h in or de r to g et more gener al central ex- tensions, for C ∞ ( P , G ) G in the non-ab elian case (cf. Example IV.3 and [NW09]). IV Examples In this section, we provide some classe s of ex a mples of principal 2- bundles. This ﬁrst ex ample is a n analogous constructio n of the simply connected cover of a connec ted manifold M as a π 1 ( M )-principal bundle. It constructs for a simply connected ma nifold N a principal B π 2 ( N )-2-bundle, where B π 2 ( N ) is the (discrete) 2-g roup asso ciated to the cro s sed mo dule π 2 ( N ) → {∗} . F or brevity , w e sha ll r e s trict to the case where the manifold N actually is a Lie group G (not necessa rily ﬁnite-dimensiona l, so π 2 ( G ) 6 = 0 in gener al). It a lready app eared implicitly a t many places in the literatur e (e.g . in [BM9 4] and [Ig l95]) but, as far as the author knows, it has not b een w or ked out in terms of principal 2-bundles. It has the correct universal pr oper t y for calling it the 2-connected cov er of G (cf. [PW09]). Example IV.1. Le t G b e a 1-connected Lie gr oup. F or each g ∈ G we choose a contin uo us path from γ g from e to g , such that γ e ≡ e and γ g depe nds con- tin uously on g on so me unit neighbo urhoo d. Mor e o ver, since G is 1 -connected, we ﬁnd for each pair ( g , h ) ∈ G 2 a contin uous map η g,h : ∆ 2 → G with ∂ η g,h = γ g + g .γ g − 1 h − γ h , where the sum o n the r igh t is taken in the group of singular 1-chains of G . Again, we a ssume σ e,e ≡ e and that σ g,h depe nds c o n tinuously on g , h o n some unit neig hbo urhoo d U of G . With these choices we now set η g,h , k : g V ∩ hV ∩ k V → π 2 ( G ) , x 7→ [ σ e,g,h + σ e,h,k − σ e,g,k + g .σ e,g − 1 h,g − 1 k | {z } tetrahedr on with vertices e,g, h ,k ]+ [ g . ( σ e,g − 1 h,g − 1 k + σ e,g − 1 k,k − 1 x − σ e,g − 1 h,g − 1 x + g − 1 h.σ e,h − 1 k,h − 1 x ) | {z } tetrahedr on with vertices g, h,k,x ] , 32 where V ⊆ U is an op en symmetric unit neighbo ur hoo d with V 2 ⊆ U . If g V ∩ hV ∩ k V 6 = ∅ , then g − 1 h , g − 1 k and h − 1 k are e lemen ts of V 2 ⊆ U . Since σ is contin uous on U the v alue o f η ghk do es not dep end on x , and η ghk is constant and in pa rticular smo oth. It is easily veriﬁed from the ab o ve pr esen tation that η ghk + η gk l = η ghl + η hkl (note that this follows from the ab ov e formula only bec ause we added the second tetrahedr on, whic h does not con tribute to the class in π 2 ( G )). Since ( V g ) g ∈ G with V g := g V covers G , ( V g , ∗ , η g,h , k ) deﬁnes a ˇ Cech co cycle with v a lues in π 2 ( G ) and thus a principal B π 2 ( G )-2-bundle, wher e B π 2 ( G ) deno tes the 2-group asso ciated to the crossed mo dule π 2 ( G ) → {∗ } . It is fairly easy to chec k that diﬀerent choices of the ab ove data lead to c o homologous co cycles. The r emainder of this section dea ls with (lifting) bundle ge r bes. ex:lifting Gerbes Example IV.2. W e brieﬂy recall (ab elian) bundle g erbes a s intro duced in [Mur96] (cf. [A CJ05], [SW08c]). The class of gerbes that connect mo st nat- urally with o ur principa l 2-bundles are lifting gerb es, so we will r estrict to this class (the general case can easily b e adapted). Let ( α, β , H , G ) b e a smo oth crossed mo dule a nd π : P → M b e a principal G -bundle. Mor eo ver, we assume that A := ker ( β ) ֒ → H ։ G is a central extension (i.e., w e assume β to b e surjective, cf. [Nee07, Sect. 3]). There is a ca nonical map f : P × M P → G , de ter mined by p = p ′ · f ( p, p ′ ) and we consider the pull-back principal A -bundle Q := f ∗ ( H ) o ver P × M P . The question that one is in terested in is whether the A -action on Q extends to an H -action, turning Q in to a principa l H -bundle ov er M (cf. [LGW08], [Nee0 6a]). With G = ( G, H ⋊ G ), o ne c a n co ok up a princ ipa l G -2-bundle P as follows. W e deﬁne ob jects and mor phisms by Ob( P ) = P Mor( P ) = Q, where we identify Q with { (( p, p ′ ) , h ) ∈ P × P × G : p = p ′ · β ( h ) } . Sour c e and the tar get map a re given by s (( p, p ′ ) , h ) = p ′ , t (( p, p ′ ) , h ) = p , a nd comp osition of morphisms is then deﬁned by (( p ′′ , p ) , h ′ ) ◦ (( p, p ′ ) , h ) = (( p ′′ , p ′ ) , h · h ′ ) (the order o f h and h ′ is impo rtan t for (( p ′′ , p ′ ) , h · h ′ ) to be in Q aga in). In or der to match this with the bundle g erbes deﬁned in [Mur96], no te that this may also be v ie w ed as a A -bundle mo rphism π ∗ 13 ( Q ) → π ∗ 12 ( Q ) × π ∗ 23 ( Q ) / A, where A acts on the rig h t hand side via the embedding A ֒ → A × A , a 7→ ( a, a − 1 ) and π ij : P × M P × M P → P × M P are the v a rious p ossible pro jections. Fixing lo cal trivialisatio ns π × g i : P | U i → U i × G for an o pen cov er ( U i ) i ∈ I deﬁnes the functors Φ i : P | U i → U i × G on ob jects and on morphisms we set Φ i (( p, p ′ ) , h ) = ( π ( p ) , ( g i ( p ′ ) .h, g i ( p ′ ))). The a ction o f G on P is given b y the given G - a ction on o b jects a nd by ((( p, p ′ ) , h ′ ) , ( h, g )) 7→ (( p · β ( h ) · g , p ′ · g ) , g − 1 . ( h ′ · h )) 33 on mor phisms. Note that the band of this bundle is the trivial bundle over M , b ecause β is sur jectiv e. In particular, it ha s nothing to do with the appa ren t bundle P , which serves o nly a s a mea ningless intermediate space. The outer action G/β ( H ) → Out( H ) is trivial and lifting gerb es a re classiﬁed by H 2 ( M , A ) (cf. Remark I I.29). The fo llo wing example illustr ates the close int erplay b etw een g roups of sec- tions in L ie gro up bundles (cf. [NW09 ]) a nd gauge g roups o f pr incipal bundles (cf. [W o c07]). ex:lieGrou pBundle Example IV. 3. Cons ide r the cross ed mo dule A ut ( H ), given by the conjuga- tion morphism H → Aut( H ) and the na tural action of Aut( H ) on H . Assuming that H is ﬁnite-dimensional and π 0 ( H ) is ﬁnitely gener ated, Aut( H ) b ecomes a Lie gr o up, mo delled o n der( h ) (cf. [Bou89]). Moreover, we ass ume that we are given a smo oth action λ : G → Aut( H ) for Lie group G , such that the action lifts to a homomo rphism ϕ : G → H ♭ := H / Z ( H ) (this is the case , e.g., if G is connected and h is semi-simple, for then the derived a ction lifts b y [Hel78, Prop. I I.6.4]). A Lie gr o up bundle now arises from a principa l G -bundle P → M by taking the as s ocia ted bundle P × G H . The group of s ections of this bundle is isomo rphic to the equiv ar ian t mapping g roup C ∞ ( P, H ) G . Considering the asso ciated prin- cipal H ♭ -bundle P ♭ = P × ϕ H ♭ , o ne c a n ask whether this principal H ♭ -bundle lifts to a principal H -bundle P ♯ . In this case, one ha s C ∞ ( P ♯ , H ) H ∼ = C ∞ ( P ♭ , H ) H ♭ ∼ = C ∞ ( P, H ) G (as one can see in lo cal co ordinates), and the gr oup of sections is actually the gauge gr oup o f P ♯ . In general, we a ssoc iate to λ the pull-ba c k central ex tension ϕ ∗ ( H ) → G and thus obtain a strict 2-g roup G . Then we ass ociate to P ♭ the pr incipal G -2- bundle P o f the lifting gerb e asso ciated to the central extension Z ( H ) ֒ → H ։ H ♭ and o btain C ∞ ( P , G ) G as its ga ug e 2-gro up. F r om Rema rk I I I.16 we see that C ∞ ( P , G ) G is asso ciated to a crosse d mo dule ( α ∗ , β ∗ , H ∗ , G ∗ ) with H ∗ = C ∞ ( P ♭ , H ) H ♭ ∼ = C ∞ ( P, H ) G and G ∗ ≤ C ∞ ( P ♭ , H ♭ ) H ♭ × C ∞ ( Q, H ⋊ H ♭ ) H ⋊ H ♭ . F ro m the compatibility with the structur e maps of P is fo llo ws tha t ( γ 0 , ( γ 1 , γ 0 ◦ s P )) ∈ G ∗ ⇔ γ 1 ∈ C ∞ ( Q, Z ( H )) H ♭ (note that γ 0 ◦ s P = γ 0 ◦ t P , b ecause s P ( p ) = t P ( p ) · h with h ∈ Z ( H )) and thus G ∗ ∼ = C ∞ ( P ♭ , H ♭ ) H ♭ × C ∞ ( M , Z ( H )). Thus C ∞ ( P , G ) G is in general as s ocia ted to the cros sed mo dule C ∞ ( P ♭ , H ) H ♭ → C ∞ ( P ♭ , H ♭ ) H ♭ × C ∞ ( M , Z ( H )) , γ 7→ ( q ◦ γ , e H ) . with the obvious p oin t-wise action. 34 Example IV.4. An instance of the prev io us example is given by consider- ing the e x tension Gau( P ) ֒ → Aut( P ) ։ Diﬀ ( M ) P (cf. [W o c07]) for a ﬁnite- dimensional principal K -bundle P → M , deﬁning a cross ed module by the conjugation action of Aut( P ). If P = M × K is tr iv ial, then H := Gau( P ) ∼ = C ∞ ( M , K ) and Aut ( P ) ∼ = H ⋊ Diﬀ ( M ) and if K is compac t and simple, then Aut( P ) is an op en subg roup o f Aut( H ) (cf. [G ¨ un09]). Then a given actio n G → Aut( H ) lifts to H ♭ , for instanc e , if G is connected and the induced actio n g → V ( M ) on the base spa ce is trivial. A App endix: Diﬀeren tial calculus on lo cally con ve x spaces app:differ entialCalculus OnSpacesOfMappings W e pr o vide some background material on spaces of mappings a nd their Lie group structur e in this a pp endix. def:diffca lcOnLocallyCon vexSpaces Deﬁnition A.1. Let X and Y b e a lo cally co n vex vector space s and U ⊆ X be op e n. The n f : U → Y is diﬀer en t iable or C 1 if it is cont inuous, for each v ∈ X the diﬀerential q uotien t d f ( x ) .v := lim h → 0 f ( x + hv ) − f ( x ) h exists and if the ma p d f : U × X → Y is contin uous. If n > 1 we inductively deﬁne f to b e C n if it is C 1 and d f is C n − 1 and to be C ∞ or smo oth if it is C n . W e s a y that f is C ∞ or smo oth if f is C n for all n ∈ N 0 . W e denote the co rresp onding space s of maps by C n ( U, Y ) and C ∞ ( U, Y ). A (lo cally conv ex) Lie gr oup is a group which is a smo oth manifold mo delled on a lo cally co n vex space such that the g r oup op erations ar e smo oth. prop:local DescriptionsOf LieGroups Prop o sition A.2. Le t G b e a gr oup with a lo c al ly c onvex manifold structur e on some subset U ⊆ G with e ∈ U . F urt hermor e, assum e that ther e ex ists V ⊆ U op en such that e ∈ V , V V ⊆ U , V = V − 1 and i) V × V → U , ( g , h ) 7→ g h is smo oth, ii) V → V , g 7→ g − 1 is smo oth, iii) for al l g ∈ G , ther e exists an op en un it n eigh b ourho o d W ⊆ U such that g − 1 W g ⊆ U and t he map W → U , h 7→ g − 1 hg is smo oth. Then ther e exists a un ique lo c al ly c onvex manifold stru ctur e on G which turns G into a Lie gr oup, such t hat V is an op en su bmanifo ld of G . def:expone ntialfunction Deﬁnition A.3. Let G b e a lo cally conv ex Lie gr oup. The group G is sa id to hav e an exp onential fu nction if for each x ∈ g the initial v a lue problem γ (0) = e, γ ′ ( t ) = T λ γ ( t ) ( e ) .x 35 has a solution γ x ∈ C ∞ ( R , G ) and the function exp G : g → G, x 7→ γ x (1) is smo oth. F urthermo r e, if there exists a zero neighbour hoo d W ⊆ g such that exp G | W is a diﬀeomo rphism onto so me o pen unit neighbourho o d of G , then G is sa id to b e lo c al ly exp onential . lem:interc hangeOfActions OnGroupAndAlgebra Lemma A.4. If G and G ′ ar e lo c al ly c onvex Lie gr oups with exp onential func- tion, t hen for e ach morphism α : G → G ′ of Lie gr oups and the induc e d mor- phism dα ( e ) : g → g ′ of Lie algebr as, the diagr am G α − − − − → G ′ x   exp G x   exp G ′ g dα ( e ) − − − − → g ′ c ommut es. rem:banach LieGroupsAreLo callyExponential Remark A.5. The F undamen tal Theorem of Ca lculus for lo cally co n vex spa ces (cf. [Gl¨ o0 2a, Th. 1.5]) yields that a lo cally conv ex Lie group G can hav e at most one exp onential function (cf. [Nee06 b, Lem. I I.3.5]). Typical examples of lo cally exp onen tial Lie g r oups are Banach-Lie gro ups (b y the existence of solutions o f diﬀerential e q uations and the in verse mapping theo- rem, cf. [La n99]) a nd groups of s mooth a nd co n tinuous mappings fro m co mpact manifolds into lo cally exp onen tial g roups ([Gl¨ o02b, Sect. 3.2], [W o c06]). How- ever, diﬀeomo rphism gr oups of c o mpact manifolds are never lo cally e xponential (cf. [Nee06b, Ex. I I.5.13]) and direct limit Lie groups no t alwa ys (cf. [Gl¨ o 05, Rem. 4.7 ]). F o r a detailed treatment of lo cally e x ponential Lie gr oups a nd their structure theo ry we refer to [Nee06b, Sect. IV]. Remark A.6 . The most interesting examples of inﬁnite-dimensional Lie groups for this ar ticle shall be groups of smo oth mappings C ∞ ( M , G ) from a compa ct manifo ld M (p ossibly with b oundary) to an arbitrar y Lie gr oup G . These groups po ssess na tur al Lie g roup structur e s if one endows them with the initial top ology with resp ect to the embedding C ∞ ( M , G ) ֒ → Y k ∈ N 0 C ( T k M , T k G ) c , γ 7→ ( T k γ ) k ∈ N 0 . The Lie algebra is C ∞ ( M , g ) (with the above top o logy), whe r e g is the Lie algebra of G (a ll spa ces are endow ed with p oint-wise op erations). Details can be found in [Gl¨ o02b, Sect. 3.2 ] and [W o c06, Sect. 4]. Ac kno wledgemen ts The a uthor would lik e to thank Thomas Sc hick, Peter Ar ndt, Ulric h Pennig, Sven Porst, and Ales s andro F ermi for v ario us us e ful dis cussions and Alessan- dro F ermi for pr oof-r eading the manuscript. Moreover, he would like to thank 36 Ieke Mo erdijk for wr iting the excellen t in tro duction [Mo e02], which clariﬁed the sub ject essentially and suppo rted many idea s of this pap er. Urs Schreib e r gav e a couple of very useful comments on v ario us formation stages, providing the background a nd global picture and pro of-read the pap er thoro ughly . Even tually , Chris Schommer-Pries p ointed o ut Remark I I.18 and he r eb y help ed correcting an err or in a prev ious version of the pap er. The work on this article w as ﬁnancially suppo rted by the Graduier tenk olleg “Grupp en und Geometrie”. References [A CJ05 ] Paolo Asch ieri, Luigi Cant ini, and Branislav Jurˇ co . Nona belian bun- dle g erbes, their diﬀerential geometry a nd gauge theory . Comm. 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