Smooth Functors vs. Differential Forms

Hamb. Beitr. Math. Nr. 297 ZMP-HH/08-04 Smo oth F unctors vs. Diﬀeren tial F orms Urs Sc hreib er an d Kon rad W aldorf Departmen t Mathematik Sc h w e rpunkt Algebra und Zahlen theorie Univ ersität Ham burg Bundesstraße 55 D–20146 Ham burg Abstract W e establish a relation b et w een smo ot h 2-functors deﬁned on the path 2-groupoid of a smo ot h manifold and diﬀeren tial forms o n this manifold. This relation ca n be understo o d as a part of a dictionary b e- t w een fundamen tal notion s from category the ory and diﬀeren tial geom - etry . W e sho w that smo oth 2-functors app ear in sever al ﬁelds, namely as connection s on (non-ab elian) gerbes , as deriv ativ es of s mo oth func- tors and as critical p oin ts in BF theory . W e demonstrate further that our dictionary provides a p ow erful to ol to discuss the transgression of geometric ob jects to lo op spaces. T able of Con ten ts In t ro duction 2 1 Review: Smoot h F unctors and 1-F orms 6 1.1 The P ath G roup oid of a Smo oth Manifold . . . . . . . . . . . 6 1.2 Diﬀeological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Equiv a lence b et w een F unctors and F orms . . . . . . . . . . . . 10 2 Smo oth 2-F unctors and Diﬀeren tial F orms 14 2.1 The P ath 2 -Group oid of a Smo oth Manifold . . . . . . . . . . 15 2.2 F rom F unctors to F o rms . . . . . . . . . . . . . . . . . . . . . 18 2.3 F rom F orms to F unctors . . . . . . . . . . . . . . . . . . . . . 30 2.4 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 Examples of Smoot h 2-F unctors 44 3.1 Connections on (non-ab elian) Gerb es . . . . . . . . . . . . . . 44 3.2 Deriv ativ es of Smo oth F unctors . . . . . . . . . . . . . . . . . 47 3.3 Classical Solutions in BF-Theory . . . . . . . . . . . . . . . . 48 4 T ransgression to Lo op Spaces 49 4.1 Generalization to Diﬀeological Spaces . . . . . . . . . . . . . . 49 4.2 Induced Structure on the Lo op Space . . . . . . . . . . . . . . 54 App endix 61 A.1 Basic 2-Category Theory . . . . . . . . . . . . . . . . . . . . . 61 A.2 Lie 2- G roups and Sm o oth Crossed Mo dules . . . . . . . . . . 65 A.3 Pro of of Lemma 2.16 . . . . . . . . . . . . . . . . . . . . . . . 69 T able of Notations 72 References 73 In tro duction The presen t article is the second of t hree articles aiming at a general and sys- tematical approac h to conne ctions on (non- a b elian) gerb es and thei r surface holonom y . In the ﬁrst article [SW09] w e hav e established an equiv alence b et w een categories of ﬁbre bundles with connection o v er a smo oth manifold X , and categories of certain functors, called transp ort functors. Let us spell out this 2 equiv a lence, r educed to trivial principal G -bundles with connection. These are just g -v alued 1-forms on X , where g is the Lie algebra of the Lie group G . Our equiv alence is then a bijection Ω 1 ( X , g ) ∼ =  Smo oth functors P 1 ( X ) → B G  b et wee n the set of g -v alued 1-forms on X and the set of smo oth f unctors b et wee n t w o group oids P 1 ( X ) and B G . One the one hand, w e ha v e the path group oid P 1 ( X ) whic h is asso ciated to the manifold X : its o b jects are the po ints of X , and the morphis ms b et w een tw o p oin ts are (thin homotopy classes of smo oth) paths b et w een these tw o p oin ts. On the other hand, w e ha v e a group oid B G whic h is asso ciated to the Lie group G : it has just one ob ject and ev ery group elemen t acts as an automorphisms of this ob ject. The notation B G is de v oted to the fact that the geometric realization of its nerv e is the classifying space B G of the group G . No w, the functors F : P 1 ( X ) / / B G we ha ve on the righ t hand side of the ab o v e bijection assign gro up elemen ts F ( γ ) to paths γ in X ; this assignmen t is smo oth in a s ense that can b e expressed in terms of smoo t h maps b etw een smo oth manifolds. F or the con v enience of the reader, w e rev iew this relation b et wee n smo oth functors and diﬀeren tial forms in Section 1. In the presen t article w e generalize the ab ov e bijection b et w een smo oth functors and 1-forms t o smo oth 2-functors and 2 - forms. The aim of this generalization is m ultiple, but for a start w e w ant to giv e a n impression ho w the generalized bijection lo oks like . The ﬁrst step is the g eneralization of the categories P 1 ( X ) and B G to appropriate 2-categories. On the one hand, we introduce the path 2-group oid P 2 ( X ) of a smo oth manifold X by adding 2-morphisms to the path group o id P 1 ( X ) . These 2-morphisms are (thin homotop y classe s of ) smo oth homotopies b et w een smo oth paths in X . On the other hand, we infer that the gr o up G whic h w as presen t b efore has to b e replaced b y a (strict) 2-group: basically , t his is a group ob ject in categories, i.e. a category with additional structure. The concept of 2- groups can b e reﬁned to Lie 2-gro ups; suc h a Lie 2-gro up G underlies the generalized relation w e are after. W e form a 2-category B G mimic king the same idea w e used for the category B G : it has just one ob ject and the Hom-category of this ob ject is the category G . Equipped with these generalized 2-categories, w e consider 2-functors F : P 2 ( X ) / / B G , and it ev en mak es p erfectly sense to qualify some as smo oth 2-functors. As b efore, the smo othness can b e expressed in terms of smo oth maps b et w een smo oth manifolds (Deﬁnition 2.5). 3 In order to explore whic h kind of diﬀeren tial fo r m corresp onds to a smoot h 2-functor F : P 2 ( X ) / / B G , we put the abstract concept of a 2-gr o up in a more familiar setting. A ccording to Bro wn and Sp encer [BS76], a 2-gro up G is equiv alen t to a crossed mo dule: a structure introduced by Whitehead [Whi46] consisting of t w o ordinary g r o ups G and H , a group homomorphism t : H / / G and a compatible action of G on H . Simi larly , a Lie 2- group correspo nds to Lie groups G and H and smo oth additional structure. W e denote the Lie algebras of the tw o Lie gro ups G and H b y g and h , resp ec- tiv ely . The ﬁrst result o f this article ( Prop osition 2.8) is that the smo oth 2-functor F deﬁnes a g - v alued 1- f orm A and an h -v alued 2-f o rm B on X that are related b y d A + [ A ∧ A ] = t ∗ ( B ) . Here, w e hav e the ordinary curv ature 2-for m of A on the left hand side, and t ∗ is the Lie algebra homomorphism induced by t . The t w o diﬀeren tial forms A and B con tain in fact all information ab out the 2- f unctor F they came from: w e describ e an explicit pro cedure ho w to in tegrate t wo forms A and B (satisfying the ab o ve condition) to a smo o th 2-functor F : P 2 ( X ) / / B G . This in tegration in v o lves iterated solutions of ordinary diﬀeren tia l eq uations. The main result of this article (Theorem 2.21) is that w e obtain a bijection  Smo oth 2-functors F : P 2 ( X ) → B G  ∼ = ( ( A, B ) ∈ Ω 1 ( X , g ) × Ω 2 ( X , h ) with d A + [ A ∧ A ] = t ∗ ( B ) ) . This is the announced generalization of the relation b et we en smo oth 1- functors and diﬀeren tial 1-forms f rom [SW09]. Besides, we also explore the geometric structure that corres p onds to morphisms (pseudonatural transfor- mations) and 2 - morphisms (mo diﬁcations) b etw een 2-functors. The deriv a- tion of all the r elations that are imposed on t his structure tak es a large part of this article, and is collected in Section 2. In Section 3 we try to con vince the reader that smo oth 2-functors are implicitly prese n t in v arious ﬁelds, and w e describe the impact of our new bijection to these ﬁelds. W e giv e three examples. The ﬁrst example are connections on (p ossibly non-ab elian) g erb es. As men tioned a t the b eginning of this in tro duction, ordinary smoo th functors corresp ond to connections on trivial principal bundles. W e claim here that smo oth 2- f unctors corresp ond in the same w ay to connections on trivial gerb es . F or ab elian gerb es, suc h connections hav e been studied b y Brylinsk i o n shea v es of group oids [Bry93], a nd b y Murra y on bundle gerb es [Mur96]. In b oth cases, our claim sho ws to b e true. Similar results for ab elian gerb es ha v e 4 b een obtained in [MP02]. Connections on a certain class o f (p ossibly) non- ab elian gerb es hav e b een in tro duced by Breen a nd Messing [BM05]. Their connection – considered on a trivial gerb e – is a pair ( A, B ) of a 1-form and a 2-form like they arise from a smo oth 2-functor in the wa y outlined ab o v e. In terestingly , the tw o forms of a Breen-Mes sing connection are not necess arily r elated to each other, in con trast to the tw o forms coming fro m a smo oth 2-functor. W e argue that this diﬀerence is related to the ques tion, whether suc h connections induce a notion of surfac e holonom y . Moreo ver, certain higher gauge theories can b e described by pairs ( A, B ) of diﬀeren tial forms with v alues in the Lie algebras belonging to the t wo Lie groups of a crossed mo dule, and ev en the relation betw een A and B w e f o und here is already presen t in this con text [GP04]. Since higher gauge theories are naturally related to connections on gerb es, a further link b et wee n smo o th 2-functors and connections on gerb es is presen t. A deep er disc ussion of connections o n non- tr ivial gerbes and their surface holonom y is the con ten t of the third part [SW A] in our serie s of articles. The second example of smoo t h 2-functors we w an t to giv e are deriv a- tiv e 2-functors. F or an y Lie group G , a smo oth functor F : P 1 ( X ) / / B G determines a smo oth 2-functor d F : P 2 ( X ) / / B E G , where E G is a Lie 2-group introduced b y Segal as a mo del for the univ ersal G -bundle E G [Seg68]. Using our dictionary b et wee n smo oth functors and diﬀeren tial fo rms for b oth F and d F we see that the functor F corresponds to a trivial principal G -bundle with connection ω ov er X , while the deriv ativ e 2- functor d F induces a 2-form B ∈ Ω 2 ( X , g ) . W e pro v e that B is the curv ature of the connec tion ω , so that the relation b et wee n F and d F implies a relation b et wee n the holonom y of ω and its curv ature. W e sh o w that this establishes a new pro of of the so-called non-ab elian Stok es’ Theorem (Theorem 3 .4). The third example whe re smo oth 2-functors a rise is a certain top ological ﬁeld theory whic h is called BF-theory due to the prese nce of t w o ﬁelds B and F . These ﬁelds are 2-f orms with v alues in the Lie algebra g of a Lie group G ; actually F = d A + [ A ∧ A ] is the curv ature of a 1-form A . W e pro v e that the critical p oints of the BF action functional are those pairs ( A, B ) satisfying the relation described ab o v e. In other w o rds, smoo th 2-functors arise as the classical solutions of the ﬁeld equations of BF-theory (Prop osition 3.7). Section 4 is dev oted to the follo wing observ at io n: ev ery elemen t in the lo op space LX of a smo oth manifold X can b e understo o d as a particular morphism in the path group oid P 1 ( X ) , and also as a particu lar 1-morphism in the path 2-group oid P 2 ( X ) . This w ay , functors on the path g roup oid, and 5 2-functors on the path 2-gro up oid are in trinsically related to structure on the lo op space of X . First w e observ e that the structure on the lo op space whic h is induced b y a smo oth functor F : P 1 ( X ) / / B G is a smo oth function LX / / G , and t hat this function is nothing bu t the holonom y of the (trivial) principal G -bundle with connection associated to F . Then w e pro v e that the structure whic h is induced b y a smo oth 2-functor F : P 2 ( X ) / / B G is a smo oth functor P 1 ( LX ) / / Λ B G , where Λ B G is a certain category constructe d from the 2-group oid B G . In order to b e able to sp eak ab out s mo oth functors on the lo op sp ace, w e w ork with the canonical diﬀeology o n LX : this is a structure whic h generalizes a smo oth manifold structure and is more suitable f or mapping spaces. W e extend the relation b etw een functors and 1-forms from [SW09] to diﬀeological spaces (Theorem 4.7), and prov e that the ab o ve smo oth functor on the path group oid of the lo op space c orresp onds to the follo wing structure: a smo o t h function LX / / G , a 1-fo rm A F ∈ Ω 1 ( LX, g ) and a 1-form ϕ F ∈ Ω 1 ( LX, h ) . Denoting b y ( A, B ) the diﬀeren tial forms that b elong to the smo oth 2- functor F w e started with, w e deriv e an explicit relation b et w een the dif- feren tial forms ( A, B ) on X and ( A F , ϕ F ) on LX (Prop osition 4.10). This relation inv olv es in tegration a long the ﬁbre, and admits an outlo ok on the question, what the tra nsgression of a non-ab elian gerb e ov er X to the lo op space LX is. Finally , w e hav e included an App endix in whic h w e review basic notio ns from 2 -category theory and imp o rt a n t deﬁnitions and examples related to Lie 2-g roups and smo oth crossed mo dules. F or the con v enience of the reader, there is also a table of notation. 1 Review: Smo oth F unctors and 1-F orms In this section we review some relev an t deﬁnitions and results from [SW09] and references therein. 1.1 The P ath Group oid of a Smo oth Manifol d In the top o lo g ical category , the basic idea o f the pat h group oid is v ery simple: for a top o logical space X , it is t he category whose o b jects are the p oin ts of X , and whose morphisms are ho moto p y classes of contin uous paths in X . F or smo oth manifolds, one considers s m o oth p aths : these are smo o t h maps γ : [0 , 1] / / X with sitting instan ts, i.e. a n um b er 0 < ǫ < 1 2 with γ ( t ) = γ (0) 6 for 0 ≤ t < ǫ and γ ( t ) = γ (1) for 1 − ǫ < t ≤ 1 . The set of smo oth paths in X is denoted by P X . The sitting instants assure that t w o paths γ : x / / y and γ ′ : y / / z can b e comp osed to a new path γ ′ ◦ γ : x / / z . How ev er, the comp osition of paths is not asso ciativ e, so that a category can only b e deﬁned using certain quotien ts of P X as its morphisms. There are essen tially three w a ys to deﬁne such quotien ts. The ﬁrst is to tak e reparameterization classes P 0 X := P X/ ∼ 0 , where γ 1 ∼ 0 γ 2 if there exists an orien ta tion-preserving diﬀeomorphism ϕ of [0 , 1] suc h tha t γ 2 = γ 1 ◦ ϕ . The second wa y is to tak e thin homotop y classes, P 1 X := P X/ ∼ 1 : Deﬁnition 1.1. Two p aths γ 1 , γ 2 : x / / y ar e c a l le d thin h omotopy e quiva - lent, denote d γ 1 ∼ 1 γ 2 , if ther e exis ts a smo oth m a p h : [0 , 1] 2 / / M such that (1) it has sitting instants: ther e exists a numb er 0 < ǫ < 1 2 with a) h ( s, t ) = x f o r 0 ≤ t < ǫ an d h ( s, t ) = y for 1 − ǫ < t ≤ 1 . b) h ( s, t ) = γ 1 ( t ) for 0 ≤ s < ǫ a nd h ( s, t ) = γ 2 ( t ) for 1 − ǫ < s ≤ 1 . (2) the diﬀer ential of h has at most r ank 1. The third w ay is to tak e homotopy classes P 2 X := P X / ∼ 2 just like in Deﬁnition 1.1 but without condition (2). Notice that there are pro jections P X / / P 0 X / / P 1 X / / P 2 X (1.1) and that the ab ov e-men tioned composition o f paths induces w ell-deﬁned com- p ositions on a ll P i X . W e denote b y id x the constan t path at a p oin t x . In P 0 X w e ha v e γ ◦ id x ∼ 0 γ ∼ 0 id y ◦ γ and ( γ 3 ◦ γ 2 ) ◦ γ 1 ∼ 0 γ 3 ◦ ( γ 2 ◦ γ 1 ) ; (1.2) these are the axioms of a category with ob jects X and morphis ms P 0 X . W e further de note b y γ − 1 : y / / x t he path γ − 1 ( t ) := γ (1 − t ) . In P 1 X w e ha v e additionally to (1.2) γ − 1 ◦ γ ∼ 1 id x for any path γ : x / / y , s o that the corresp o nding category with morphis ms P 1 X is eve n a group o id. This group oid is denoted by P 1 ( X ) and called the p ath gr o up oid of X . The group oid Π 1 ( X ) with morphisms P 2 X is w ell-kno wn as the fundamental gr oup oid of the smo oth manifold X . All these categories are compatible with smo oth maps b et w een smoo th manifolds in the sense that an y smo oth map f : X / / Y induces maps f ∗ : P i X / / P i Y , and that these maps furnish functors b et w een the resp ectiv e categories. 7 Remark 1.2. The group oids P 1 ( X ) and Π 1 ( X ) are imp ortant f or parallel transp ort in a ﬁbre bundle with connection ov er X in the sense that an y suc h bundle deﬁnes a functor tra : P 1 ( X ) / / T , where T is a category in whic h the ﬁbres of the bundle are ob jects. If the con- nection is ﬂat, this functor factors through the f undamen tal group oid Π 1 ( X ) . More on the relation b etw een functors and connections in ﬁbre bundles can b e found in Section 3.1 and in [SW09]. 1.2 Diﬀeolo gical Spaces A Lie c ate gory is a category S whose sets S 0 of ob jects and S 1 of morphisms are smo oth manifolds, whose target and source maps are surjectiv e submer- sions, whose iden tit y map is an em b edding and whose comp osition is smo oth. A functor F : S / / T b et w een Lie categories S and T is called s mo oth , if its assignmen ts F 0 : S 0 / / T 0 and F 1 : S 1 / / T 1 are smo oth maps. The path group oid P 1 ( X ) of a smo oth manifold is, how ev er, not a Lie category , since its set of morphisms P 1 X has not the structure of a smo oth manifold. In [SW09] we ha ve instead equipp ed P 1 X with a diﬀe olo gy , a structure that generalizes a smo o th manifold structure [Che77 , Sou81]. This diﬀeology on P 1 X has b een introduced in [CP94]. F or the con v enience of the reader let us recall the basic deﬁnitions (see also App endix A.2 of [SW09]). Deﬁnition 1.3. A diﬀe olo gi c al sp ac e is a set X to gether w i th a c ol le ction of plots: maps c : U / / X e ach of them de ﬁ ne d on an op en s ubset U ⊂ R k for any k ∈ N 0 , such that thr e e axioms ar e satisﬁe d: (D1) for any plot c : U / / X and any smo oth function f : V / / U also c ◦ f is a plot. (D2) every c ons tant map c : U / / X is a plot. (D3) if f : U / / X is a map deﬁ ne d o n U ⊂ R k and { U i } i ∈ I is a n op en c over of U for wh i c h al l r estrictions f | U i ar e p l o ts of X , then also f is a plot. A diﬀe olo gic a l map b etwe en diﬀe olo gic al s p ac es X and Y is a map f : X / / Y such that for every plot c : U / / X of X the map f ◦ c : U / / Y is a p lot of Y . The set of a l l diﬀ e olo gic al m a ps is denote d b y D ∞ ( X , Y ) . 8 An y smo oth manifold is a diﬀeological space, whose plots are all smo oth maps deﬁned on all op en subsets of R k , fo r a ll k . If M and N are smo oth manifolds, a map f : M / / N is diﬀeological if and only if it is smo oth. In other w ords, diﬀeological spaces and maps form a categor y D ∞ that contains the category C ∞ of smooth manifolds ( without b oundary) as a ful l sub cate- gory . Besides from smo oth manifolds, w e hav e three further examples of sets with a canonical diﬀeology: 1. If X and Y a re diﬀeological spaces, the set D ∞ ( X , Y ) of diﬀeological maps b et wee n X and Y is a diﬀeological space in the follo wing w ay: a map c : U / / D ∞ ( X , Y ) is a plot if and only if for any plot c ′ : V / / X of X the comp osite U × V c × c ′ / / D ∞ ( X , Y ) × X ev / / Y is a plot of Y . Here, ev denotes the ev aluation map ev( f , x ) := f ( x ) . 2. Subsets Y of a diﬀeological space X are diﬀeological: its plots are those plots of X whose image is con tained in Y . 3. If X is a diﬀeological space, Y is a set and p : X / / Y is a map, Y b ecomes a diﬀeological space whose plots are t hose maps c : U / / Y f o r whic h there exis ts a cov er of U b y op en sets U α and plots c α : U α / / X of X suc h that c | U α = p ◦ c α . Equipped with these examples, the sets P i X we ha v e deﬁned in Section 1.1 b ecome diﬀeological spaces in the follo wing w a y . The set P X is a sub- set of the diﬀeological space D ∞ ([0 , 1] , X ) , and hence a diﬀeological space. Then we consider one of the pro jections pr i : P X / / P i X fr o m (1.1) t o ei- ther reparameterization classes, t hin homotop y classes or homoto p y classes. A ccording to the third example a b o ve, all the sets P i X b ecome diﬀeologi- cal spaces. W e also ha v e examples o f diﬀeological maps: if f : X / / Y is a smo oth map b et wee n smo oth manifolds, the induced maps f ∗ : P i X / / P i Y are all diﬀeological. The most imp ortan t question for us will b e, whether a map P i X / / M from one of these diﬀeological spaces to a smo oth manifold M – regarded as a diﬀeological space – is diﬀeological. F rom the deﬁnitions ab o v e one can deduce the fo llo wing result, and the reader is free to tak e it either as a res ult from the bac kground of diﬀeological spaces, or as a deﬁnition. 9 Lemma 1.4 ([SW09], Prop osition A.7 i) ) . A map f : P i X / / M is di ﬀe o- lo gic al, if a n d only if for every k ∈ N 0 , every op en subse t U ⊂ R k and every map c : U / / P X such that the c omp o s i te U × [0 , 1] c × id / / P X × [0 , 1] ev / / X is smo oth, also the map U c / / P X pr i / / P i X f / / M is smo oth. No w we can study smo oth f unctors F : P 1 ( X ) / / S to a Lie category S : on ob jects F : X / / S 0 is a smo oth map and on morphisms F : P 1 X / / S 1 is a diﬀeological map. Similarly , if η : F / / F ′ is a natural transformation b et wee n tw o smo oth functors, it is called smo oth natural transformation, if its components η ( x ) ∈ S 1 furnish a sm o oth map X / / S 1 . Smo oth functors F and smo oth natural transformations η form a category F unct ∞ ( P 1 ( X ) , S ) . Remark 1.5. Concerning Deﬁnition 1.3 of a diﬀeological space, sev eral dif- feren t conv en tions for plots are common. F or example, in o rder to deal prop erly with manifolds with b oundary or corners, it is more conv enien t to consider plots being deﬁned on conv ex subse ts U ⊂ R k rather than op en ones [Bae07]. Suc h questions do not aﬀect the results of this article, since w e consider either maps deﬁned on manifolds with out b oundary or maps w hic h are constan t near the b oundary o f a manifold, for example paths with sitting instan ts. 1.3 Equiv alence b et w een F unctors and F orms In [SW09] w e ha v e established an isomorphism b et w een tw o categories, F unct ∞ ( P 1 ( X ) , B G ) ∼ = Z 1 X ( G ) ∞ . (1.3) Both categories depend on a smo oth manifold X and a Lie group G . O n the left hand side w e hav e the category of smoo th functors from the path group oid P 1 ( X ) of X to the Lie g r o up oid B G . W e recall from the in tro duction that the Lie group oid B G has one ob ject, and its set of morphisms is the Lie gro up G . The comp osition is deﬁned b y g 2 ◦ g 1 := g 2 g 1 . On the right hand side w e ha v e a category Z 1 X ( G ) ∞ deﬁned as follo ws: its ob jects are 1- forms A ∈ Ω 1 ( X , g ) 10 with v alues in the Lie algebra g of G , and a morphism g : A / / A ′ is a smo oth function g : X / / G suc h that A ′ = Ad g ( A ) − g ∗ ¯ θ , (1.4) where ¯ θ is the right in v ariant Maurer-Cartan f orm on G . The iden tit y mor- phism is the constant function g = 1 and the composition is the m ultiplication of functions, g 2 ◦ g 1 := g 2 g 1 . The equiv a lence (1.3) can be giv en explicitly in b oth directions: there ar e tw o functors F unct ∞ ( P 1 ( X ) , B G ) D ) ) Z 1 X ( G ) ∞ P i i whose de ﬁnitions w e shall revie w in the follo wing. Sev eral details and pro ofs will b e skipped and can b e fo und in [SW09]. Giv en a smo oth functor F : P 1 ( X ) / / B G , a 1-form A ∈ Ω 1 ( X , g ) is deﬁned in the follow ing three steps: 1. F or a p oin t x ∈ X and a tangen t v ector v ∈ T x X , w e c ho ose a smo oth curv e Γ : R / / X with Γ(0) = x and ˙ Γ(0) = v . Let γ R ( t 0 , t ) b e the (up to thin homotop y unique) path in R that go es from t 0 to t , regarded as a map γ R : R 2 / / P 1 R . 2. One can sho w (App endix B.4 in [SW09]) that the comp osite F Γ := F ◦ Γ ∗ ◦ γ R : R 2 / / G (1.5) is a smo oth map with F Γ ( t 0 , t 0 ) = 1 for all t 0 ∈ R . W e deﬁne A x ( v ) := − d d t     0 F Γ (0 , t ) ∈ g . (1.6) 3. One can then verify that t he v alue A x ( v ) is indep enden t o f the c hoice of Γ (Lemma B.2 in [SW09]), and that the assignmen t A : T X / / g is smo oth and linear (Lemma B.3 in [SW09]). This deﬁnes the 1-form D ( F ) := A . The comp onents of a smo oth natural transformation ρ : F / / F ′ form by deﬁnition a smo oth map D ( ρ ) := g : X / / G . Let again Γ : R / / X b e a smo oth curv e and F Γ and F ′ Γ the functions (1.5) asso ciated to the functors F and F ′ , and let g Γ := g ◦ Γ . The naturalit y of ρ implies the equation g Γ ( t ) · F Γ (0 , t ) = F ′ Γ (0 , t ) · g Γ (0) , 11 whose deriv ativ e ev aluated at t = 0 sho ws (1.4) for A and A ′ the 1-f orms deﬁned by F Γ and F ′ Γ , resp ectiv ely . Hence, D ( ρ ) is a morphism in Z 1 X ( G ) ∞ ; this deﬁnes the functor D . Con ve rsely , consider a 1-form A ∈ Ω 1 ( X , g ) . Then, a smooth functor F : P 1 ( X ) / / B G is deﬁned in the following wa y: 1. Let γ b e a path in X , whic h w e extend trivially to R by γ ( t ) := γ (0) for t < 0 a nd γ ( t ) := γ (1) for t > 1 . W e p ose the initial v alue problem ∂ ∂ t u γ ( t ) = − d r u ( t ) | 1  A γ ( t )  d γ d t  and u ( t 0 ) = 1 (1.7) for a smo oth function u : R / / G and ﬁxed t 0 ∈ R . Here d r u ( t ) is the diﬀeren tial of the m ultiplication with u ( t ) from the righ t. 2. The initial v alue problem (1.7) has a unique solution f A,γ ( t 0 , t ) , f rom whic h w e deﬁne a map F : P X / / G : γ  / / f A,γ (0 , 1) . (1.8) W e remark that this map is sometimes referred to as the „path-ordered expo nen tial“ F ( γ ) = P exp  Z γ A  . (1.9) 3. One can show that the map F is independen t of the thin homotop y class of γ (Prop osition 4.3 in [SW09]), and that it f actors through a smo oth map F : P 1 X / / G (Lemma 4.5 in [SW09]). It resp ects the comp osition o f paths so that w e hav e deﬁned a smo oth functor P ( A ) := F . F or a smo oth function g : X / / G considered as a morphism g : A / / A ′ b et wee n tw o 1-forms A, A ′ ∈ Ω 1 ( X , g ) we need to deﬁne an asso ciated smo oth natural transformation ρ = D ( g ) : F / / F ′ b et wee n the asso ciated functors. The comp onent of ρ a t x ∈ X is deﬁnes as g ( x ) . One can then show that g ( y ) · f A,γ ( t 0 , t ) · g ( x ) − 1 solv es the initial v alue problem (1.7 ) for A ′ and γ ( Lemma 4.2 in [SW09]), whic h implies the naturality of the natural transformation ρ . This deﬁnes the functor P . Theorem 1.6 (Prop osition 4.7 in [SW09]) . L et X b e a smo oth m anifold and G b e a Lie gr oup. The two functors D and P satisfy D ◦ P = id Z 1 X ( G ) ∞ and P ◦ D = id F unct ∞ ( P 1 ( X ) , B G ) , in p articular, they form an isomorphism of c ate gories . 12 W e giv e a short sk etc h of the pro of. If w e start with a 1-form A ∈ Ω 1 ( X , g ) , we shall test the 1-form D ( P ( A )) at a p oin t x ∈ X and a t a ngen t v ector v ∈ T x X . Let Γ : R / / X b e a curv e in X w ith x = Γ(0) and v = ˙ Γ(0) . If we further denote γ τ := Γ ∗ ( γ R (0 , τ ) ) ∈ P X w e hav e − D ( P ( A )) | x ( v ) ( 1.6 ) = ∂ ∂ τ     0 P ( A ) Γ (0 , τ ) ( 1.5 ) = ∂ ∂ τ     0 P ( A )( γ τ ) ( 1.8 ) = ∂ ∂ τ     0 f A,γ τ (0 , 1) Here, f A,γ τ denotes the unique solution of the initial v alue problem (1.7) for γ τ . A uniqueness argument sho ws f A,γ τ ( t 0 , t ) = f A,γ 1 ( τ t 0 , τ t ) , so that ∂ ∂ τ f A,γ τ (0 , t )     τ =0 ,t =1 = ∂ ∂ t f A,γ 1 (0 , t )     t =0 = − A p ( v ) , this yields D ( P ( A )) = A . On the other hand, if F : P 1 ( X ) / / B G is a s mo oth functor, w e test the functor P ( D ( F )) on a path γ in X . By (1 .8), P ( D ( F ))( γ ) = f D ( F ) ,γ (0 , 1) where f D ( F ) ,γ is the solution of the initial v alue problem (1.7) for the 1-form D ( F ) and the path γ . Due to the deﬁnition (1.6) of D ( F ) b y the function F γ : R 2 / / G w e ha v e ( γ ∗ D ( F )) t  ∂ ∂ t  = − ∂ ∂ τ     τ =0 F γ ( t, t + τ ) . Since F is a functor, F γ ( x, z ) = F γ ( y , z ) F γ ( x, y ) . Bo th together show that F γ also solv es the initial v alue problem, so that, b y uniqueness , f D ( F ) ,γ (0 , 1) = F γ (0 , 1) = F ( γ ) . This sho ws P ( D ( F )) = F . Remark ably , there is not m uc h structure that is preserv ed b y the functors P and D (unless the Lie group G is ab elian). F o r example, sums a nd negativ es of diﬀeren tial forms, or pro ducts and in ve rses of smo oth functors are all not preserv ed. W e only kno w the following fact: Prop osition 1.7. Th e functors P and D ar e c omp atible with pul lb acks along a smo oth map f : X / / Y b etwe en smo oth manifolds X and Y , i.e. P ( f ∗ A ) = f ∗ P ( A ) and D ( f ∗ F ) = f ∗ D ( F ) for a 1-form A ∈ Ω 1 ( Y , g ) and a smo oth functor F : P 1 ( Y ) / / B G , and similarly for morphisms. 13 Here w e ha ve used the notation f ∗ F for the functor F ◦ f ∗ , where f ∗ is the induced map on path groupoids. Prop osition 1.7 follo ws in a straigh tforw ard w ay from the naturalit y of the deﬁnitions of the functors D and P . 2 Smo oth 2-F unctors and Diﬀeren tial F orms In t his section w e generalize Theorem 1.6 – the eq uiv alence b etw een 1-forms and smo o th functors – to 2-functors. The basic 2-categorical notions suc h as 2-categor ies, 2- f unctors, pseudonatural transformations and mo diﬁcations are summarized in App endix A.1; f o r the reader familiar with these notions it is imp ortant to notice that all 2 -categories and 2-functors in this article are assumed to b e strict without further notice. The ﬁrst step in concerns the path group oid P 1 ( X ) that was presen t in Theorem 1.6: in Se ction 2.1 w e de ﬁne the path 2-gro up oid P 2 ( X ) asso ciated to a smo o t h manifold X . Instead of the Lie group G that was presen t in Theorem 1.6 w e use a (strict) Lie 2 -group G . In the same w a y that a category B G is asso ciated t o any Lie group G , a 2-category B G is a ssociated to an y Lie 2-group G , and the 2-functors w e consider are of the fo rm F : P 2 ( X ) / / B G . A conv enien t and concrete wa y to deal with Lie 2-gro ups is pro vided b y crossed mo dules [Whi46, BS76]. Their deﬁnition, their relation to Lie 2- groups, and the asso ciated 2-categories B G ar e describ ed in App endix A.2. The announced generalization of Theorem 1.6 is w ork ed out in three steps: in Section 2.2 we extract diﬀeren tial forms from 2-functors, pseudonatural transformations and mo diﬁcations. W e deriv e conditions on the extracted diﬀeren tial f o rms that lead us straigh tforw ardly to an appropriate general- ization Z 2 X ( G ) ∞ of the category Z 1 X ( G ) ∞ that was presen t in Theorem 1.6. The goal of Section 2.2 is that extracting diﬀeren tial forms furnishes 2-functor D : F unct ∞ ( P 2 ( X ) , B G ) / / Z 2 X ( G ) ∞ . In Section 2.3 w e in tro duce a 2-functor P : Z 2 X ( G ) ∞ / / F unct ∞ ( P 2 ( X ) , B G ) in the opp osite direction, that reconstructs 2-functors, pseudonatural trans- formations and mo diﬁcations from giv en diﬀeren tial forms. Finally , w e prov e in Section 2.4 t he main result of this article, namely that the 2- f unctors D and P establish an isomorphism of 2- categories. 14 2.1 The P ath 2- Group oid of a Smo oth Manifo l d As mentioned in the in tro duction, the path 2-group oid is obtained b y adding 2-morphisms to the path group oid P 1 ( X ) . These 2-morphisms are smo oth homotopies in the sens e of De ﬁnition 1.1 without the restriction (2) o n their rank, explicitly: Deﬁnition 2.1. L et γ 0 , γ 1 : x / / y b e p aths in X . A bigon Σ : γ 0 + 3 γ 1 is a smo oth map Σ : [0 , 1] 2 / / X such that ther e exists a numb er 0 < ǫ < 1 2 with a) Σ( s, t ) = x for 0 ≤ t < ǫ an d Σ( s, t ) = y fo r 1 − ǫ < t ≤ 1 . b) Σ( s, t ) = γ 0 ( t ) for 0 ≤ s < ǫ a nd Σ( s, t ) = γ 1 ( t ) for 1 − ǫ < s ≤ 1 . W e denote the set of bigons in X b y B X . Bigons can b e comp osed in t wo w ays : If Σ : γ 1 + 3 γ 2 and Σ ′ : γ 2 + 3 γ 3 are bigons, w e hav e a new bigon Σ ′ • Σ : γ 1 + 3 γ 3 deﬁned b y (Σ ′ • Σ)( s, t ) = ( Σ(2 s, t ) for 0 ≤ s < 1 2 Σ ′ (2 s − 1 , t ) for 1 2 ≤ s ≤ 1 ; (2.1) and for tw o bigons Σ 1 : γ 1 + 3 γ ′ 1 and Σ 2 : γ 2 + 3 γ ′ 2 suc h that γ 1 (1) = γ 2 (0) , w e ha v e another new bigon Σ 2 ◦ Σ 1 : γ 2 ◦ γ 1 + 3 γ ′ 2 ◦ γ ′ 1 deﬁned b y (Σ 2 ◦ Σ 1 )( s, t ) := ( Σ 1 ( s, 2 t ) for 0 ≤ t < 1 2 Σ 2 ( s, 2 t − 1) for 1 2 ≤ t ≤ 1 . (2.2) Due to the sitting instan ts, the new maps (2.1) a nd (2 .2) are aga in smo ot h and ha v e sitting instan ts. Lik e in the case of paths, there are sev eral equiv alence relations o n the set B X of bigons in X , starting with reparameterization classes, and con tin ued b y a ladder of t yp es of homotopy cl asses, graded b y an upp er b ound for the rank of the homotopies. The corresp onding quotien t spaces a re denoted b y B X / / B 0 X / / B 1 X / / B 2 X / / B 3 X . In this article w e are only inte rested in B 2 X = B X/ ∼ 2 . Deﬁnition 2.2. Two bigons Σ : γ 0 + 3 γ 1 and Σ ′ : γ ′ 0 + 3 γ ′ 1 ar e c a l le d thin homoto py e quivalent , den o te d Σ ∼ 2 Σ ′ , if ther e exists a smo oth map h : [0 , 1] 3 / / X such that (1) it has sitting instants: ther e exists a numb er 0 < ǫ < 1 2 with 15 a) h ( r, s, t ) = x for 0 ≤ t < ǫ and h ( r , s, t ) = y for 1 − ǫ < t ≤ 1 . b) h ( r, s, t ) = h ( r , 0 , t ) for 0 ≤ s < ǫ and h ( r, s, t ) = h ( r , 1 , t ) for 1 − ǫ < s ≤ 1 . c) h ( r, s, t ) = Σ( s, t ) for al l 0 ≤ r < ǫ and h ( r, s, t ) = Σ ′ ( s, t ) for al l 1 − ǫ < r ≤ 1 . (2) the diﬀer ential of h satisﬁes a) rank(d h | ( r ,s,t ) ) ≤ 2 for al l r , s, t ∈ [0 , 1] , a nd b) rank(d h | ( r ,i,t ) ) ≤ 1 for i = 0 , 1 ﬁ xe d. Condition (1) assures that thin homotop y is an equiv alence relation on B X . Condition (2b) asserts that t w o thin homotop y equiv alen t bigons Σ : γ 0 + 3 γ 1 and Σ ′ : γ ′ 0 + 3 γ ′ 1 start and end on thin homotopy equiv alen t paths γ 0 ∼ 1 γ ′ 0 and γ 1 ∼ 1 γ ′ 1 . The composition ◦ of t w o bigons deﬁned ab ov e clearly induces a w ell-deﬁned comp osition on B 2 X . F or the comp osition • this is more in volv ed: let Σ : γ 1 + 3 γ 2 and Σ ′ : γ ′ 2 + 3 γ 3 b e t w o bigons suc h that γ 2 ∼ 1 γ ′ 2 . Let h : [0 , 1] / / X b e any thin homotop y b et we en γ 2 and γ ′ 2 ; this is a particular bigon h : γ 2 + 3 γ ′ 2 . No w w e deﬁne the comp osition of the corresp o nding classes in B 2 X b y [Σ ′ ] ∼ 2 • [Σ] ∼ 2 := [Σ ′ • h • Σ] ∼ 2 . The pro of that this is independen t of the ch oice of h r equires a tec hnical computation carried out in [MP10]. Another imp ortant fact is that the t wo comp ositions ◦ and • are compatible with eac h other in the sense t hat (Σ ′ 1 • Σ ′ 2 ) ◦ (Σ 1 • Σ 2 ) ∼ 2 (Σ ′ 1 ◦ Σ 1 ) • (Σ ′ 2 ◦ Σ 2 ) (2.3) whenev er all these comp ositions are w ell-deﬁned. Deﬁnition 2.3. The p ath 2-gr oup oid P 2 ( X ) of a smo oth manifold X is the 2- c ate gory whose obje cts ar e the p oints o f X , wh ose se t of 1-morphisms is P 1 X , and w h ose set of 2-morphisms is B 2 X . Horizo ntal a n d vertic al c omp osition ar e given by ◦ and • , and the identities ar e the identity p ath id x : x / / x and the identity bigon id γ : γ + 3 γ deﬁne d by id γ ( s, t ) := γ ( t ) . The axioms of a 2-category (see Deﬁnition A.1) are satisﬁed: Axiom (C1) follow s from the second equation in (1 .2 ) , axiom (C2) follo ws from the ﬁrst equation in (1.2) and from an elemen tary construction of homotopies Σ • id γ 1 ∼ 2 Σ ∼ 2 id γ 2 • Σ . Axiom (C3) is (2.3). It is also clear the the category P 2 ( X ) is indeed a group oid. 16 F or parallel tra nspor t along surfaces, the path 2-group oid pla ys the same role the path gro up o id P 1 ( X ) plays for parallel transp ort along curv es (see Section 1.1): the corresp onding geometric ob jects a re (w eak) 2- functors tra : P 2 ( X ) / / T in to some 2-category T , as outlined in Section 6 of [SW09]. A detailed discuss ion of these 2-functors will follow in [SW A]. In exactly the same w ay as w e ha v e diﬀeological maps P i X / / M w e ha v e diﬀeological maps from all the equiv alence classes B i X o f bigons in X to smo oth manifolds M . Analogous to Lemma 1.4, w e hav e Lemma 2.4. A map f : B i X / / M i s diﬀe ol o gic al if and only if fo r every k ∈ N 0 , every op en subset U ⊂ R k and every map c : U / / B X such that the c omp osite U × [0 , 1] 2 c × id / / B X × [0 , 1] 2 ev / / X is smo oth, also the map U c / / B X pr i / / B i X f / / M is smo oth. This admits to deﬁne smo oth 2-functors deﬁned on the path 2-g roup oid of X with v alues in s mo oth 2-c ate gories S : 2 - categories for whic h ob jects S 0 , 1-morphisms S 1 and 2-morphisms S 2 are smo oth manifolds and all struc ture maps are smo oth. Deﬁnition 2.5. A 2-functor F : P 2 ( X ) / / S fr o m the p ath 2-gr oup oid of a smo oth manifo l d X to a smo oth 2-c ate gory S is c al le d smo oth, if 1. on obje cts, F : X / / S 0 is smo oth. 2. on 1-morphi sms, F : P 1 X / / S 1 is di ﬀ e olo gic al (s e e L emm a 1.4) . 3. on 2-morphi sms, F : B 2 X / / S 2 is diﬀe olo gi c al (se e L emma 2.4). F or the deﬁnitions of morphisms b et w een 2-functors, the pseudonatu- ral transformations, and morphisms b etw een those, the mo diﬁcations, we refer the reader again to App endix A.1. A pseudonatural tra nsformation ρ : F / / F ′ is called sm o oth , if its comp onents ρ ( x ) ∈ S 1 at ob jects x ∈ X furnish a smo oth map X / / S 1 , and its comp onen ts ρ ( γ ) ∈ S 2 at 1-morphisms γ ∈ P 1 X furnish a diﬀeological map P 1 X / / S 2 . Similarly , a mo diﬁcation A : ρ + 3 ρ ′ is called smo oth, if its compo nents A ( x ) ∈ S 2 from a smo oth map X / / S 2 . Summarizing, these structures form a 2-category F unct ∞ ( P 2 ( X ) , S ) . 17 2.2 F r om F unctors to F orms As we explain in App endix A.2 that the 2- category B G asso ciated to a Lie 2-group G whic h is represen ted b y a smooth crossed mo dule ( G, H , t, α ) has one ob ject, the set of morphisms is G and the set of 2 -morphisms is the semi- direct pro duct G ⋉ H , where G acts on H via a smo oth map α : G × H / / H . The guideline how to extract diﬀeren tial forms from smooth 2-functors is the same as review ed in Section 1 .3 : we ev aluate the Lie g ro up-v alued functors on certain paths, obtain Lie group-v alued maps, and tak e their deriv ativ e. 2.2.1 Extracting F orms I: 2-F unctors Here w e start with a giv en smo oth 2-functor F : P 2 ( X ) / / B G . Clearly , F restricted to ob jects and 1-morphisms is just a smo oth 1-functor F 0 , 1 : P 1 ( X ) / / B G . By Theorem 1.6 it corresp onds to a g -v alued 1- form A on X . F rom the remaining map F 2 : B 2 X / / G ⋉ H w e now deﬁne an h -v alued 2- f orm B on X . Its deﬁnition is p oint wise: le t x ∈ X b e a p oint and v 1 , v 2 ∈ T x X b e tangen t v ectors. W e ch o ose a smo oth map Γ : R 2 / / X with x = Γ(0 ) and v 1 = d d s     s =0 Γ( s, 0) and v 2 = d d t     t =0 Γ(0 , t ) . (2.4) Note that in R 2 there is only o ne thin homotop y class of bigons b etw een eac h t w o ﬁxed paths. In particular, we ha ve a canonical f amily Σ R : R 2 / / B 2 R 2 , where Σ R ( s, t ) := (0 , 0) / /   (0 , t ) w  w w w w w w w w w w w w w w w w   ( s, 0) / / ( s, t ) . (2.5) W e use this canonical family of bigons to pro duce a map F Γ := p H ◦ F 2 ◦ Γ ∗ ◦ Σ R : R 2 / / H (2.6) where p H : G ⋉ H / / H is the pro jection to the second factor. Lemma 2.6. The map F Γ : R 2 / / H is smo oth. F urthermor e, its se c ond mixe d derivative evaluate d at 0 ∈ R 2 is a wel l-deﬁne d element in the Lie algebr a h of H , and is indep enden t of the choic e of Γ , i.e. if Γ 0 , Γ 1 : R 2 / / X ar e smo oth maps with Γ 0 (0) = Γ 1 (0) = x and b oth sa tisfying (2.4), then ∂ 2 F Γ 0 ∂ s∂ t     (0 , 0) = ∂ 2 F Γ 1 ∂ s∂ t     (0 , 0) . (2.7) 18 Pro of. The smo othness of F Γ follo ws f r o m the smo othness of the 2-functor F as explained in Section 1.2: the relev ant ev aluation map R 2 × [0 , 1] 2 Γ ∗ ◦ Σ R × id / / B X × [0 , 1] 2 ev / / X is smo oth. Since F is smo oth on 2-morphisms, F ◦ Γ ∗ ◦ Σ R is smo oth. Next w e not ice t hat F Γ (0 , t ) = F Γ ( s, 0) = 1 for all s, t ∈ R , so that the second mixed deriv ativ e natura lly t a k es v alues in h . Now w e consider the t w o families Σ k := (Γ k ) ∗ ◦ Σ R : R 2 / / B 2 X of bigons in X . W e w ork in a suitable op en neighborho o d V ⊂ R 2 of the origin, and are going to construct a homotop y H : V × [0 , 1] / / B X with H ( x, y , k ) = Σ k ( x, y ) for k = 0 , 1 . Then w e will show that H factors through the smo oth map f : V × [0 , 1] / / Z : ( x, y , α )  / / ( x, y , ( x 2 + y 2 ) α ) , where Z := { ( x, y , z ) ∈ V × [0 , 1] | 0 ≤ z ≤ x 2 + y 2 } , and another smo o t h map B : Z / / B X . Applying the chain rule to the decomposition H = B ◦ f giv es ∂ 2 F Γ k ∂ s∂ t     (0 , 0) = ∂ 2 ∂ s∂ t     (0 , 0) ( F ◦ Σ k ) = ∂ 2 ∂ s∂ t     (0 , 0 ,k ) ( F ◦ B ◦ f ) = H( F ◦ B ) | f (0 , 0 ,k ) ∂ f ∂ x     (0 , 0 ,k ) , ∂ f ∂ y     (0 , 0 ,k ) ! + D ( F ◦ B ) | f (0 , 0 ,k ) ∂ 2 f ∂ x∂ y     (0 , 0 ,k ) ! , where H( F ◦ B ) denotes the Hesse matrix of F ◦ B , considered as a symme tric, ﬁbre-wise bilinear form T Z × Z T Z / / h . By construction of the map f , the latter expression is indep enden t of k . In order to construct H and B , w e w o rk in a c hart that iden tiﬁes V with an op en neigh b orho o d of x , and f o rm the „linear in terp olation“ h : V × [0 , 1 ] × [0 , 1] 2 / / X : ( x, y , α, s, t )  / / Σ 0 ( x, y ) ( s , t ) + α · d ( x, y , s, t ) , where the „diﬀerence“ d is give n b y d : V × [0 , 1] 2 / / X : ( x, y , s, t )  / / Σ 1 ( x, y ) ( s , t ) − Σ 0 ( x, y ) ( s , t ) . Then, w e set H ( x, y , α )( s, t ) := h ( x, y , s, t, α ) . Next w e construct the map B . The coincidence of the v alues and the ﬁrst deriv ative s of the maps Γ 0 , Γ 1 19 at (0 , 0) imply via Hadamard’s lemma that there exist smo oth maps a, b, c : V × [0 , 1] 2 / / X suc h that d ( x, y , s, t ) = x 2 · a ( x, y , s, t ) + y 2 · b ( x, y , s, t ) + 2 xy · c ( x, y , s, t ) . No w w e change to p olar co o rdinates. W e denote b y U ⊂ R ≥ 0 × [0 , 2 π ) a suitable op en neigh b orho o d of (0 , 0) so that the coo rdinate transformation τ : ( r, φ )  / / ( r · cos φ, r · sin φ ) is a map τ : U / / V . Then, w e get d p ( r , φ, s, t ) := d ( τ ( r, φ ) , s, t ) = r 2 · ˜ d p ( r , φ, s, t ) with ˜ d p ( r , φ, s, t ) := cos 2 φ · a ( τ ( r, φ ) , s, t ) + sin φ cos φ · b ( τ ( r , φ ) , s, t ) + sin 2 φ · c ( τ ( r, φ ) , s, t ) deﬁning a smo oth map ˜ d p : U × [0 , 1] 2 / / X . W e lo ok at the smoo t h map b p : U × [0 , 1] × [0 , 1] 2 / / X : ( r , φ, α, s, t )  / / Σ 0 ( τ ( r, φ ) , s, t ) + α · ˜ d p ( r , φ, s, t ) . W e consider Z p := { ( r , φ, z ) ∈ U × [0 , 1] | 0 ≤ z ≤ r 2 } , and claim that the restriction of b p to Z p × [0 , 1] 2 transforms bac k into a smo oth map in c artesian co ordinates. T o see this, it suﬃces to notice that the term α · ˜ d p ( r , φ, s, t ) and all its r - deriv ativ es v anish at r = 0 , since then also α = 0 . This wa y we obtain a smo oth map b : Z × [0 , 1] 2 / / X satisfying b ( τ ( r, φ ) , α, s, t ) = b p ( r , φ, α , s, t ) for all ( r, φ ) ∈ U . Finally , w e set B ( x, y , α )( s, t ) := b ( x, y , s, t, α ) . It is straightforw ard to see that B ( x, y , α ) is a bigon, and the smo othness o f b implies the smo ot hness of the map B : Z / / B X . W e calculate for ( r, φ ) ∈ U : ( B ◦ f )( τ ( r, φ ) , α )( s, t ) = b p ( r , φ, r 2 α, s, t ) = Σ 0 ( τ ( r, φ ) , s, t ) + α · r 2 · ˜ d p ( r , φ, s, t ) = Σ 0 ( τ ( r, φ ) , s, t ) + α · d ( τ ( r, φ ) , s, t ) = h ( τ ( r, φ ) , s, t, α ) . Since the co ordinate transformation is surjectiv e, this sho ws the claimed coincidence B ◦ f = H .  With Lemma 2.6 w e hav e now extracted a w ell-deﬁned map α F : T X × X T X / / h : ( x, v 1 , v 2 )  / / − ∂ 2 F Γ ∂ s∂ t     (0 , 0) (2.8) from the 2-functor F . 20 Lemma 2.7. Th e map α F has the fol lowing pr op erties: (a) for ﬁxe d x ∈ X , it is antisymmetric and biline ar. (b) it is smo oth. Pro of. T o pro ve (a), let ¯ Γ( s, t ) := Γ( t, s ) , a nd let F Γ and F ¯ Γ the corre- sp onding smo oth maps (2.6). Due to the p erm utation, the deriv ativ es (2.7) yield the v alues fo r α F ( x, v 1 , v 2 ) a nd α F ( x, v 2 , v 1 ) , respectiv ely . Note that ¯ Γ ∗ ◦ Σ R = Γ ∗ ◦ Σ − 1 R , where Σ − 1 R is the 2-morphism in v erse to the 2- morphism (2.5) under vertical comp osition. Since the 2-functor F se nds in v erse 2- morphisms to inv erse group elemen ts, w e hav e F ¯ Γ = F − 1 Γ . Hence, by taking deriv ativ es, w e get α F ( x, v 2 , v 1 ) = − α F ( x, v 1 , v 2 ) . It remains to show that α F ( v 1 + λv ′ 1 , v 2 ) = α F ( v 1 , v 2 ) + λα F ( v ′ 1 , v 2 ) . If Γ a nd Γ ′ are smo oth functions for the tangen t v ectors ( v 1 , v 2 ) a nd ( v ′ 1 , v 2 ) , respectiv ely , w e use a c hart φ : U / / X of a neigh b ourho o d of x with φ (0) = x and construct a smo oth function ˜ Γ : ( − ǫ, ǫ ) 2 / / X b y ˜ Γ( s, t ) := φ ( φ − 1 (Γ( s, t )) + λφ − 1 Γ ′ ( s, t )) where ǫ has to ch osen small enough. It is easy to see that ˜ Γ(0 , 0) = p and that v 1 + λv ′ 1 = d d s     s =0 ˜ Γ( s, 0) and v 2 = d d t     t =0 ˜ Γ(0 , t ) . (2.9) On the other hand, ˜ Γ ∗ (Σ R ( s, t )) = φ ∗ ( φ − 1 ∗ (Γ ∗ (Σ R ( s, t ))) + λφ − 1 ∗ (Γ ′ ∗ (Σ R ( s, t )))) , where w e ha ve used that bigons in R 2 can b e added and m ultiplied with scalars. This sho ws that ∂ 2 F ˜ Γ ∂ s∂ t     (0 , 0) = ∂ 2 F Γ ∂ s∂ t     (0 , 0) + λ ∂ 2 F Γ ′ ∂ s∂ t     (0 , 0) ; together with (2.9) this pro v es that α F is bilinear. T o prov e (b), let φ : U / / X b e a c hart of X with an op en subset U ⊂ R n . The induced c hart φ T X : U × R n / / T X of the tangent bundle sends a p oin t ( u, v ) ∈ U × R n to d φ | u ( v ) ∈ T φ ( u ) X . W e show that U × R n × R n φ [2] T X / / T X × X T X α F / / h is a smo oth map. F or this purp ose, let c : U × R n × R n × R 2 / / B X b e deﬁned b y c ( x, v 1 , v 2 , σ, τ )( s , t ) := φ ( u + β ( s σ ) v 1 + β ( tτ ) v 2 ) , where β : 21 [0 , 1] / / [0 , 1] is a smooth map with β (0) = 0 and β (1 ) = 1 a nd with sitting instan ts. The map c dep ends obv iously smo o t hly on all parameters, so that f c := p 2 ◦ F ◦ c : U × R n × R n × R 2 / / H is a smo oth function. Notice that Γ u,v 1 ,v 2 ( s, t ) := c ( u, v 1 , v 2 , s, t )(1 , 1) deﬁnes a smo oth map with the prop erties Γ(0 , 0) = φ ( u ) , ∂ Γ ∂ s     (0 , 0) = d φ | u ( v 1 ) and ∂ Γ ∂ t     (0 , 0) = d φ | u ( v 2 ) . It is furthermore still related to c b y (Γ u,v 1 ,v 2 ) ∗ (Σ R ( s, t )) = c ( u , v 1 , v 2 , s, t ) . No w, ( α F ◦ φ [2] T X )( x, v 1 , v 2 ) = α F ( φ ( u ) , d φ | u ( v 1 ) , d φ | u ( v 2 )) = − ∂ 2 ∂ s∂ t     (0 , 0) ( p 2 ◦ F ◦ (Γ u,v 1 ,v 2 ) ∗ ◦ Σ R )( s, t ) = − ∂ 2 ∂ s∂ t     (0 , 0) f c ( x, v 1 , v 2 , s, t ) . The last expression is, in particular, smo oth in x , v 1 and v 2 .  All together, B x ( v 1 , v 2 ) := α F ( x, v 1 , v 2 ) deﬁnes an h -v alued 2-form B ∈ Ω 2 ( X , h ) on X , whic h is canonically asso ci- ated to the smo oth 2-functor F . Prop osition 2.8. L et F : P 2 ( X ) / / B G b e a smo oth 2-functor, and A ∈ Ω 1 ( X , g ) and B ∈ Ω 2 ( X , h ) b e the c orr esp ondin g diﬀer ential forms. Then, d A + [ A ∧ A ] = t ∗ ( B ) , (2.10) wher e t ∗ := d t | 1 : h / / g is the Lie algeb r a homo m orphism induc e d fr om the Lie gr oup homomorphism t which is p art of the cr o sse d mo dule c orr esp o n ding to the Lie 2-gr oup G . Pro of. W e consider again the bigon Σ R ( s, t ) and the asso ciated smo oth map F Γ : R 2 / / H f r o m (2.6). If we denote b y γ 1 ( s, t ) the source path and b y γ 2 ( s, t ) the target path of Σ R ( s, t ) , w e obtain further smo oth maps f i := F ◦ Γ ∗ ◦ γ i : R 2 / / G . 22 Note that eac h of these t w o paths can b e decomp osed in to horizontal and v ertical paths, γ 1 ( s, t ) = γ v 1 ( s, t ) ◦ γ h 1 ( t ) a nd γ 2 ( s, t ) = γ h 2 ( s, t ) ◦ γ v 2 ( s ) , and that this decomp osition induces accordan t decompo sitions of the f unctions f i , namely f 1 ( s, t ) = f v 1 ( s, t ) · f h 1 ( t ) and f 2 ( s, t ) = f h 2 ( s, t ) · f v 2 ( s ) . W e recall from (1.6) that the 1-form A ∈ Ω 1 ( X , g ) is related to these functions b y A x ( v 1 ) = − ∂ f v i ∂ s     (0 , 0) and A x ( v 2 ) = − ∂ f h i ∂ t     (0 , 0) for i = 1 , 2 . Now we emplo y the target-matc hing-condition (A.5) for the 2-morphism F (Σ) : f 2 = ( t ◦ F Γ ) · f 1 (2.11) as functions from R 2 to G . The second partial deriv ativ es of the func tions f i are at (0 , 0) : ∂ 2 f 1 ∂ s∂ t     (0 , 0) = ∂ 2 f v 1 ∂ s∂ t     (0 , 0) + A x ( v 1 ) A x ( v 2 ) ∂ 2 f 2 ∂ s∂ t     (0 , 0) = ∂ 2 f h 2 ∂ s∂ t     (0 , 0) + A x ( v 2 ) A x ( v 1 ) Here notice that f k (0 , t ) = 1 and f k ( s, 0) = 1 so t hat the second deriv ativ es naturally take v alues in g . Also, w e write X Y := d m | (1 , 1) ( X , Y ) ∈ g f o r X , Y ∈ g , whic h is – in a faithful matrix rep resen tation – j ust the product of matrices. The ﬁrst deriv ativ es v anish, ∂ F Γ ∂ t     (0 , 0) = ∂ F Γ ∂ s     (0 , 0) = 0 , (2.12) b ecause F is constan t on f amilies o f ide n tit y bigons. Summarizing, equation (2.11) b ecomes ∂ 2 f h 2 ∂ s∂ t     (0 , 0) + A x ( v 2 ) A x ( v 1 ) = − d t | 1 ( B x ( v 1 , v 2 )) + ∂ 2 f v 1 ∂ s∂ t     (0 , 0) + A x ( v 1 ) A x ( v 2 ) , this implies the claimed equalit y .  2.2.2 Extracting F orms I I: P seudonatural T ransformations No w w e discuss a smo oth pseudonatural transformation ρ : F / / F ′ 23 b et wee n tw o smo oth 2-functors F, F ′ : P 2 ( X ) / / B G . The comp onen ts of ρ are a s mo oth map g : X / / G and a diﬀeological map ρ 1 : P 1 X / / G ⋉ H . Notice that ρ 1 is not functorial, i.e. ρ 1 ( γ 2 ◦ γ 1 ) is in general not the pro duct of ρ 1 ( γ 2 ) and ρ 1 ( γ 1 ) . In order to remedy this problem, w e construct another map ˜ ρ : P 1 X / / G ⋉ H f r o m ρ that will b e functorial. W e denote the pro jection of ρ 1 to H b y ρ H := p H ◦ ρ 1 : P 1 X / / H . Then w e deﬁne ˜ ρ ( γ ) := ( F ′ ( γ ) , ρ H ( γ ) − 1 ) . (2.13) Lemma 2.9. ˜ ρ deﬁne s a sm o oth functor ˜ ρ : P 1 ( X ) / / B ( G ⋉ H ) . Pro of. F or our conv en tion concerning the semi-direct pro duct, we refer the reader the equ ation (A.3) in Appendix A.2. With this con ve n tion, axiom (T1) of the pseudonatural transformation ρ infers f o r t wo composable paths γ 1 and γ 2 that α ( F ′ ( γ 2 ) , ρ H ( γ 1 )) ρ H ( γ 2 ) = ρ H ( γ 2 ◦ γ 1 ) . (2.14) Then, the pro duct of ˜ ρ ( γ 2 ) with ˜ ρ ( γ 1 ) in the semi-direct pro duct G ⋉ H is ( F ′ ( γ 2 ) , ρ H ( γ 2 ) − 1 ) · ( F ′ ( γ 1 ) , ρ H ( γ 1 ) − 1 ) (A.3) = ( F ′ ( γ 2 ) F ′ ( γ 1 ) , ρ H ( γ 2 ) − 1 α ( F ′ ( γ 2 ) , ρ H ( γ 1 ) − 1 )) (2.14) = ( F ′ ( γ 2 ◦ γ 1 ) , ρ H ( γ 2 ◦ γ 1 ) − 1 ) , and th us equal to ˜ ρ ( γ 2 ◦ γ 1 ) . S ince F ′ (id x ) = 1 , equation (2.14) also sho ws that ˜ ρ (id x ) = (1 , 1) . Th us, ˜ ρ is a functor. Its smo othness is clear from the deﬁnition.  By Theorem 1.6, the smoo t h functor ˜ ρ corresp onds to a 1- form with v alues in g ⋉ h , whic h in turn giv es b y pro jection in to the t w o summands an h -v alued 1-form ϕ ∈ Ω 1 ( X , h ) and a g -v alued 1-form. The ladder iden tiﬁes (due to the deﬁnition of ˜ ρ ) with the 1-f o rm A ′ that corresp onds to the f unctor F ′ . Summarizing, the smo oth pseudonatural transformation ρ deﬁnes a smo oth function g : X / / G and a 1-form ϕ ∈ Ω 1 ( X , h ) . Prop osition 2.10. L et F , F ′ : P 2 ( X ) / / B G b e smo oth 2-functors w ith as- so ciate d 1-forms A, A ′ ∈ Ω 1 ( X , g ) and 2-f o rms B , B ′ ∈ Ω 2 ( X , h ) r esp e ctively. The smo oth function g : X / / G an d the 1-form ϕ ∈ Ω 1 ( X , h ) ex tr acte d fr om a smo oth pseudonatur al tr ansformation ρ : F / / F ′ satisfy the r el a tions A ′ + t ∗ ( ϕ ) = Ad g ( A ) − g ∗ ¯ θ (2.1 5 ) B ′ + α ∗ ( A ′ ∧ ϕ ) + d ϕ + [ ϕ ∧ ϕ ] = ( α g ) ∗ ( B ) . (2.16) 24 In (2.15), ¯ θ is the right i n variant Maur er-Cartan form on G . In (2.16), A ′ ∧ ϕ is a 2-fo rm w ith va lues i n h ⊕ g , which is sent by the line ar map α ∗ to a 2-form with values in h . Pro of. Lik e in the pro of of Prop osition 2.8 w e emplo y the target-matc hing condition (A.5) for the comp onen t ∗ F ( γ ) / / g ( x )   ∗ ρ ( γ )         {          g ( y )   ∗ F ′ ( γ ) / / ∗ of the pseudonatural transformation ρ at 1-morphism γ : x / / y in P 2 ( X ) . F or this purpo se w e c ho ose a smo oth curv e Γ : R / / X through a p oin t x := Γ(0) and consider the asso ciated tangent vec tor v ∈ T x X . With the standard path γ R ( t ) in the real line from 0 to t we form from the 2-functors the smo oth maps f := F ◦ Γ ∗ ◦ γ R : R / / G and f ′ := F ′ ◦ Γ ∗ ◦ γ R : R / / G and from the pseudonatural transformation the smo oth maps ˜ g := ρ ◦ Γ : R / / G and h := ρ H ◦ Γ ∗ ◦ γ R : R / / H . (2.17) The condition w e w ant to emplo y then b ecomes f ′ ( t ) · ˜ g (0) = t ( h ( t )) · ˜ g ( t ) · f ( t ) . (2.18) If w e tak e the deﬁnition of the function g : X / / G and the 1-forms A , A ′ and ϕ in to accoun t, namely g ( x ) = ˜ g (0) and A x ( v ) = − ∂ f ∂ t     t =0 , A ′ x ( v ) = − ∂ f ′ ∂ t     t =0 and ϕ x ( v ) = − ∂ ∂ t     t =0 h − 1 , the deriv ativ e of this equation at zero yields − A ′ x ( v ) · g ( x ) = d t | 1 ( ϕ x ( v )) · g ( x ) + d g x ( v ) − g ( x ) · A x ( v ) , this implies equation (2.18) that we had to sho w. Here, the sym b ol · stands for the deriv ativ es of left or righ t m ultiplication. T o pro v e the second equation w e use axiom (T2) o f the pseudonatu- ral transformation ρ , namely the compatibilit y with 2- morphisms. F o r a 25 2-morphism Σ in P 2 ( X ) , that w e tak e of the form x 1 γ h 1 / / γ v 2   y 1 Σ } } } } } } } } } } z  } } } } } } } } γ v 1   x 2 γ h 2 / / y 2 this axiom requires F ( x 1 ) F ( γ h 1 ) / / ρ ( x 1 ) } } { { { { { { F ( y 1 ) ρ ( γ h 1 ) i i i i i i i i i i i i i i i i p x i i i i i i i i i i i i i i F ( γ v 1 )   ρ ( y 1 ) | | } } | | F ′ ( x 1 ) F ′ ( γ h 1 ) / / F ′ ( γ v 2 )   F ′ ( y 1 ) F ′ (Σ) { { { { { { { { { { { { { { { { y  { { { { { { { { { { { { { { F ′ ( γ v 1 )   F ( y 2 ) ρ ( γ v 1 ) B B B B ] e B B B B ρ ( y 2 ) } } | | | | | | F ′ ( x 2 ) F ′ ( γ h 2 ) / / F ′ ( y 2 ) = F ( x 1 ) F ( γ v 2 )   F ( γ h 1 ) / / ρ ( x 1 ) } } { { { { { { F ( y 1 ) F (Σ) | | | | | | | | | | | | | | | | y  | | | | | | | | | | | | | | F ( γ v 1 )   F ′ ( x 1 ) F ′ ( γ v 2 )   F ( x 2 ) ρ ( γ v 2 ) C C C C ] e C C C C ρ ( x 2 ) { { } } { { F ( γ h 2 ) / / F ( y 2 ) ρ ( γ h 2 ) i i i i i i i i i i i i i i i i p x i i i i i i i i i i i i i i ρ ( y 2 ) } } | | | | | | F ′ ( x 2 ) F ′ ( γ h 2 ) / / F ′ ( y 2 ) With a c hoice of a smo oth map Γ : R 2 / / X w e can pullbac k these diagrams to R 2 and us e the standard bigon Σ R ( s, t ) . W e us e the smo oth functions F Γ , f 1 and f 2 deﬁned b y the 2-functor F as describ ed in t he pro o f of Propo sition 2.8, and the analogous functions F ′ Γ , f ′ 1 and f ′ 2 for the 2-functor F ′ . F rom the pseud onatural transformation ρ we further obtain a function ˜ g := ρ ◦ Γ : R 2 / / X a nd functions h h i := ρ H ◦ Γ ∗ ◦ γ h i and h v i := ρ H ◦ Γ ∗ ◦ γ v i . Now w e ha v e f (0 , 0) f h 1 ( t ) / / ˜ g (0 , 0) } } { { { { { { { f (0 , t ) h h 1 ( t ) h h h h h h h h h h h h h h h h p x h h h h h h h h h h h h h h f v 1 ( s,t )   ˜ g (0 ,t ) | | ~ ~ | | f ′ (0 , 0) f ′ h 1 ( t ) / / f ′ v 2 ( t )   f ′ (0 , t ) F ′ Γ ( s,t ) { { { { { { { { { { { { { { { { y  { { { { { { { { { { { { { { f ′ v 1 ( s,t )   f ( s, t ) h v 1 ( s,t ) B B B B ] e B B B B ˜ g ( s,t ) ~ ~ | | | | | | f ′ ( s, 0) f ′ h 2 ( s,t ) / / f ′ ( s, t ) = f (0 , 0) f v 2 ( s )   f h 1 ( t ) / / g (0 , 0) } } { { { { { { { f (0 , t ) F Γ ( s,t ) | | | | | | | | | | | | | | | | y  | | | | | | | | | | | | | | f v 1 ( s,t )   f ′ (0 , 0) f ′ v 2 ( s )   f ( s, 0) h v 2 ( s ) C C C C ] e C C C C ˜ g ( s, 0) { { } } { { f h 2 ( s,t ) / / f ( s, t ) h h 2 ( s,t ) i i i i i i i i i i i i i i p x i i i i i i i i i i i i ˜ g ( s,t ) ~ ~ | | | | | | f ′ ( s, 0) f ′ h 2 ( s,t ) / / f ′ ( s, t ) Using the rules (A.6) and (A.7) fo r v ertical a nd horizonal compo sition in B G , 26 the ab o v e diagram b oils dow n to the equation F ′ Γ ( s, t ) · α ( f ′ v 1 ( s, t ) , h h 1 ( t )) · h v 1 ( s, t ) = α ( f ′ h 2 ( s, t ) , h v 2 ( s )) · h h 2 ( s, t ) · α ( ˜ g ( s, t ) , F Γ ( s, t )) . W e no w tak e the second mixed deriv ativ e and ev aluate at (0 , 0) . F or the ev aluation w e use the prop erties of the 2-functor F that imply – on the lev el of 2- morphisms – f (0 , 0) = 1 and – on the lev el of 1-morphisms – f h 1 (0) = f v 1 (0 , 0) = f v 2 (0) = f h 2 (0 , 0) = 1 . The same rules hold for F ′ . Similarly , the prop erties of the functor ˜ ρ giv e additionally h h 1 (0) = h v 1 (0 , 0) = 1 and h v 2 (0) = h h 2 (0 , 0) = 1 . T o compute the deriv ativ e o f the terms that con tain α , it is con v enien t to use the rule d α | ( g, h ) ( X , Y ) = d α h | g ( X ) + d α g | h ( Y ) (2.19) where α g : H / / H a nd α h : G / / H a re o bta ined from α by ﬁxing o ne of the tw o parameters, and the diﬀeren tia ls on the righ t hand side are tak en only with resp ect to the remaining parameter. Finally , w e use (2.12). The result of the computation is (in notation in tro duced in the pro of of Prop osition 2.8) ∂ 2 F ′ Γ ∂ s∂ t + d α | (1 , 1)  ∂ f ′ v 1 ∂ s , ∂ h h 1 ∂ t  + ∂ h h 1 ∂ t · ∂ h v 1 ∂ s + ∂ 2 h v 1 ∂ s∂ t = d α | (1 , 1)  ∂ f ′ h 2 ∂ t , ∂ h v 2 ∂ s  + ∂ h v 2 ∂ s · ∂ h h 2 ∂ t + ∂ 2 h h 2 ∂ s∂ t + d α ˜ g (0 , 0) | 1  ∂ 2 F Γ ∂ s∂ t  Express ed by diﬀerential forms, this giv es − B ′ x ( v 1 , v 2 ) − α ∗ ( A ′ x ( v 1 ) , ϕ x ( v 2 )) + ϕ x ( v 2 ) ϕ x ( v 1 ) + ∂ 2 h v 1 ∂ s∂ t = − α ∗ ( A ′ x ( v 2 ) , ϕ x ( v 1 )) + ϕ x ( v 1 ) ϕ x ( v 2 ) + ∂ 2 h h 2 ∂ s∂ t − ( α g ) ∗ ( B ) , whic h yields the second equalit y .  2.2.3 Extracting F orms I I I: Mo diﬁcations Let us no w consider a smo oth mo diﬁcation A : ρ 1 + 3 ρ 2 b et wee n smo o t h pseudonatural transformations ρ 1 , ρ 2 : F / / F ′ b et wee n tw o smo oth 2-functors F , F ′ : P 2 ( X ) / / B G . Its comp onen ts furnish a smo ot h map X / / G ⋉ H . W e denote its pro jection on the second factor b y a : X / / H . 27 Prop osition 2.11. L et F , F ′ : P 2 ( X ) / / B G b e smo oth 2-functors w ith as- so ciate d 1-forms A, A ′ ∈ Ω 1 ( X , g ) , let ρ 1 , ρ 2 : F / / F ′ b e smo oth pseudonat- ur al tr ansformations with asso ciate d smo oth functions g 1 , g 2 : X / / G and 1-forms ϕ 1 , ϕ 2 ∈ Ω 1 ( X , h ) . T h en, the smo o th map a : X / / H asso cia te d to a smo oth mo diﬁc ation A : ρ 1 + 3 ρ 2 satisﬁes g 2 = ( t ◦ a ) · g 1 and ϕ 2 + ( r − 1 a ◦ α a ) ∗ ( A ′ ) = Ad a ( ϕ 1 ) − a ∗ ¯ θ , (2.20 ) wher e r a ( x ) : H / / H is the multiplic ation with a ( x ) fr om the right. Pro of. In the same w ay as b efore w e c ho ose a smo oth map Γ : R / / X with Γ(0) =: x and ˙ Γ(0) =: v ∈ T x X and consider the smo oth functions f Γ , f ′ Γ : R / / G fro m (2.6), the smo oth functions ˜ g 1 , ˜ g 2 : R / / G and h 1 , h 2 : R / / H from (2.17), and de ﬁne an additional smo oth function a Γ := a ◦ Γ : R / / H with a Γ (0) = a ( x ) . The target-matching condition (A.5) f or the 2-morphism f Γ (0) ˜ g 1 (0) # # ˜ g 2 (0) ; ; a Γ (0)   f ′ Γ (0) in B G ob viously giv es us the ﬁrst equation. The axiom for the mo diﬁcation A implies α ( f ′ γ ( t ) , a Γ (0)) · h 1 ( t ) = h 2 ( t ) · a Γ ( t ) . The ﬁrst deriv ativ e ev aluated at 0 giv es ( α a Γ (0) ) ∗  ∂ f ′ γ ∂ t     0  h 1 (0)+ α ( f ′ γ (0) , a Γ (0)) · ∂ h 1 ∂ t     0 = ∂ h 2 ∂ t     0 · a Γ (0)+ h 2 (0) · ∂ a Γ ∂ t     0 With f ′ γ (0) = h 1 (0) = h 2 (0) = 1 this yields ( α a ( x ) ) ∗ ( − A ′ ) + a ( x ) · ϕ 1 | x ( v ) = ϕ 2 | x ( v ) · a ( x ) + d a | x ( v ) whic h is the second equation w e had to prov e.  2.2.4 Summary of Section 2.2 In order to obtain a prec ise relation betw een smo oth 2- functors and diﬀeren- tial fo r ms, we deﬁne a 2-category whic h is adapted to the relations w e ha v e found in Prop ositions 2.8, 2.10 and 2.11. 28 Deﬁnition 2.12. L et G b e a Lie 2-gr o up, ( G, H , t, α ) the c orr esp onding smo oth cr osse d mo dule, and X a smo oth manifold. W e deﬁne the fol low - ing 2-c ate gory Z 2 X ( G ) ∞ : 1. A n obje ct is a p ai r ( A, B ) of a 1-form A ∈ Ω 1 ( X , g ) and a 2-fo rm B ∈ Ω 2 ( X , h ) which satisfy the r elation (2.10): d A + [ A ∧ A ] = t ∗ ( B ) . 2. A 1-morphism ( g , ϕ ) : ( A, B ) / / ( A ′ , B ′ ) is a smo oth map g : X / / G and a 1-form ϕ ∈ Ω 1 ( X , h ) that satisfy the r elations (2.15) and (2.16): A ′ + t ∗ ( ϕ ) = Ad g ( A ) − g ∗ ¯ θ B ′ + α ∗ ( A ′ ∧ ϕ ) + d ϕ + [ ϕ ∧ ϕ ] = ( α g ) ∗ ( B ) . The c omp osition of 1-morphisms ( A, B ) ( g 1 ,ϕ 1 ) / / ( A ′ , B ′ ) g 2 ,ϕ 2 / / ( A ′′ , B ′′ ) is given by the map g 2 g 1 : X / / G and the 1-form ( α g 2 ) ∗ ( ϕ 1 ) + ϕ 2 , wher e α g : H / / H is the a ction of G on H with ﬁxe d g . The identity 1-morphism is given by g = 1 an d ϕ = 0 . 3. A 2-m orphism a : ( g 1 , ϕ 1 ) + 3 ( g 2 , ϕ 2 ) is a smo oth map a : X / / H that satisﬁes (2.20): g 2 = ( t ◦ a ) · g 1 and ϕ 2 + ( r − 1 a ◦ α a ) ∗ ( A ′ ) = Ad a ( ϕ 1 ) − a ∗ ¯ θ . The vertic al c omp osi tion ( g , ϕ ) a 1 + 3 ( g ′ , ϕ ′ ) a 2 + 3 ( g ′′ , ϕ ′′ ) is given by a 2 a 1 . The ho riz ontal c om p osition is ( A, B ) ( g 1 ,ϕ 1 )   ( g ′ 1 ,ϕ ′ 1 ) A A a 1   ( A ′ , B ′ ) ( g 2 ,ϕ 2 )   ( g ′ 2 ,ϕ ′ 2 ) @ @ a 2   ( A ′′ , B ′′ ) = ( A, B ) ( g 2 g 1 , ( α g 2 ) ∗ ( ϕ 1 )+ ϕ 2 ) $ $ ( g ′ 2 g ′ 1 , ( α g ′ 2 ) ∗ ( ϕ ′ 1 )+ ϕ ′ 2 ) : : a 2 α ( g 2 ,a 1 )   ( A ′′ , B ′′ ) , and the identity 2-morphism is given by a = 1 . 29 It is straigh tforw ard to che c k that this deﬁnition give s indeed a 2-category . In the Sections 2.2.1 , 2.2.2 a nd 2.2.3 ab ov e w e ha v e collected the structure of a 2-functor D : F unct ∞ ( P 2 ( X ) , B G ) / / Z 2 X ( G ) ∞ . Let us c hec k that the axioms o f a 2-functor are satisﬁed. Horizon ta l and v ertical comp ositions of 2-morphisms are resp ected b ecause these are just smo oth maps a : X / / H whic h b ecome m ultiplied in exactly the same w ay in b oth 2-categories. It remains to c hec k the compatibilit y with the comp osition of 1-morphisms, i.e. we ha ve t o sho w that D ( ρ 2 ◦ ρ 1 ) = D ( ρ 2 ) ◦ D ( ρ 1 ) for smo o th pseudonatural transformations ρ 1 : F / / F ′ and ρ 2 : F ′ / / F ′′ . Let ( g i , ϕ i ) := D ( ρ i ) for i = 1 , 2 . Ac cording to the deﬁnition (A.1) of the comp osition of pseudonatural transformations, the comp onen t of ρ 2 ◦ ρ 1 at an ob ject x ∈ X is g 2 ( x ) g 1 ( x ) ∈ G , and its compo nen t at a 1 - morphism γ : x / / y is ρ 2 ( γ ) · α ( g 2 ( y ) , ρ 1 ( γ )) ∈ H . If w e consider the smooth functions ˜ g 1 , ˜ g 2 : R / / G and h 1 , h 2 : R / / H asso ciated to ρ 1 and ρ 2 lik e in (2.17), the 1-f orm associated t o ρ 2 ◦ ρ 1 is, at x := Γ(0) and v := ˙ Γ(0) and using (2.19), − d d t     0 α ( ˜ g 2 ( t ) , h 1 ( t ) − 1 ) h 2 ( t ) − 1 = − d α ˜ g 2 (0) | h 1 (0))  ∂ h − 1 1 ∂ t     0  h 2 (0) − 1 − α ( ˜ g 2 (0) , h 1 (0) − 1 ) ∂ h − 1 2 ∂ t     0 = ( α g 2 ( x ) ) ∗ ( ϕ 1 | x ( v )) + ϕ 2 | x ( v ) , this is exactly the rule for horizon tal comp osition of 1-mor phisms in Z 2 X ( G ) ∞ . 2.3 F r om F orms to F unctors In this section w e in tro duce a 2-functor P : Z 2 X ( G ) ∞ / / F unct ∞ ( P 2 ( X ) , B G ) that go es in the direction opposite to the 2-functor D deﬁned in Section 2.2. The principle here is to p ose initial v alue problems go v erned b y diﬀerential forms. Their unique solutions deﬁne smo oth 2- f unctors, smo oth pseudonat- ural transformations and smo oth mo diﬁcations. 30 2.3.1 Reconstruction I : 2-F unctors Here we consider a give n 1-form A ∈ Ω 1 ( X , g ) and a giv en 2- form B ∈ Ω 2 ( X , h ) that satisfy the condition from Prop osition 2.8, d A + [ A ∧ A ] = t ∗ ( B ) . (2 .2 1 ) By Theorem 1.6 the 1-form A deﬁ nes a smo oth functor F A : P 1 ( X ) / / B G . Our aim is no w to deﬁne a map k A,B : B 2 X / / H suc h that F A and k A,B together de ﬁne a smo oth 2-functor F : P 2 ( X ) / / B G , w hic h is dedicated to b e the image of the pair ( A, B ) under the 2- functor P w e wan t to deﬁne. F or our con v ention concerning the semi -direct pro duct, we refer the reader again to equation (A.3) in App endix A.2. In order to ﬁnd the correct deﬁnition of k A,B w e lo ok at the target- matc hing condition F A ( γ 1 ) = t ( k A,B (Σ)) · F A ( γ 0 ) (2.22) that has to b e satisﬁed for an y bigon Σ : γ 0 + 3 γ 1 . F or tec hnical reasons w e consider Σ : [0 , 1] 2 / / X to b e extended trivially o v er all of R 2 , i.e. Σ( s, t ) =          γ 0 (0) = γ 1 (0) for t < 0 γ 0 (1) = γ 1 (1) for t > 1 γ 0 ( t ) for s < 0 and 0 ≤ t ≤ 1 γ 1 ( t ) for s > 1 and 0 ≤ t ≤ 1 . Let τ s 0 ( s, t ) b e the closed path in R 2 that runs coun ter-clo ck wise around the rectangle spanned b y ( s 0 , 0) and ( s 0 + s, t ) , and let the smo oth function u A,s 0 : R 2 / / G b e deﬁned b y u A,s 0 ( s, t ) := F A (Σ ∗ ( τ s 0 ( s, t ))) . F or this function, w e recall Lemma 2.13 (Lemma B.1 in [SW09]) . (a) u A, 0 (1 , 1) = F A ( γ − 1 0 ◦ γ 1 ) (b) u A,s 0 ( s, 1) = u A,s 0 ( s ′ , 1) · u A,s 0 + s ′ ( s − s ′ , 1) (c) ∂ ∂ s ∂ ∂ t u A,s 0     (0 ,t ) = − Ad − 1 F A ( γ s 0 ,t ) (Σ ∗ K ) ( s 0 ,t )  ∂ ∂ s , ∂ ∂ t  with γ s,t the p a th deﬁne d by γ s,t ( τ ) := Σ( s, τ t ) and K the curvatur e 2-form K := d A + [ A ∧ A ] ∈ Ω 2 ( X , g ) . 31 The function u A,s 0 is interes ting for us because b y (a) u A, 0 (1 , 1) coincides up to conjugation with the image of the group elemen t k A,B (Σ) ∈ H we w an t to determine under the homomorphism t . The mu ltiplicativ e prop erty (b) sho ws that the smo oth function f : R / / G deﬁned b y f ( σ ) := u A, 0 ( σ , 1) solv es the initial v alue problem ∂ ∂ σ f ( σ ) = d l f ( σ ) | 1  ∂ ∂ s     0 u A,σ ( s, 1)  and f (0) = 1 . (2.23) This initial v alue problem is go v erned b y the 1-f orm ∂ ∂ s     0 u A,σ ( s, 1) = Z 1 0 d t ( ∂ ∂ s ∂ ∂ t u A,σ     (0 ,t ) ) (c) = − Z 1 0 d t Ad − 1 F A ( γ σ,t )  (Σ ∗ K ) ( σ ,t )  ∂ ∂ s , ∂ ∂ t  . (2.24) Here, Ad − 1 F A ( γ − , − ) ◦ Σ ∗ K is a g -v alued 2-form on [0 , 1] 2 , and we ha v e just p erformed a ﬁbre in tegra tion o v er the second factor [0 , 1] . The result is a g -v alued 1-form on [0 , 1] . This f o rm actually lies in the image of t ∗ , ( t ◦ ( α F A ( γ − , − ) − 1 ) ∗ (Σ ∗ B ) ( 2.21 ) = Ad − 1 F A ( γ − , − ) (Σ ∗ K ) . (2.25) W e are thu s forced to consider the 1-f o rm A Σ := − Z [0 , 1] ( α F A ( γ − , − ) − 1 ) ∗ (Σ ∗ B ) ∈ Ω 1 ([0 , 1] , h ) . (2.26) Due to the sitting instants o f Σ , w e can equiv alen tly sp eak of a 1- form on R whic h v anishes outside of [0 , 1] . Now w e use again Theorem 1.6 and obtain a smooth functor F A Σ : P 1 R / / H . Since P 1 R can b e identiﬁe d with R × R (compare L emma 4.1 in [SW09]) this is just a smo oth function f Σ : R 2 / / H . The purp ose of these deﬁnitions is, that b y ( 2 .24) and (2.25) the smo oth function f : R / / G : σ  / / t ( f Σ (0 , σ )) − 1 solv es the initial v alue problem (2.23). Th us, b y uniqueness t ( f Σ (0 , σ )) − 1 = u A, 0 ( σ , 1) . If w e now deﬁne k A,B : B X / / H : Σ  / / α ( F A ( γ 0 ) , f Σ (0 , 1) − 1 ) (2.27) for γ 0 the source path of the bigon Σ w e ha v e ac hiev ed t ( k A,B (Σ)) = F A ( γ 0 ) · t ( f Σ (0 , 1)) − 1 · F A ( γ 0 ) − 1 (a) = F A ( γ 1 ) · F A ( γ 0 ) − 1 ; (2 .2 8 ) this is the required target-matc hing condition (2.22). Another indication that the map k A,B is w e hav e fo und is t he correct one is the follo wing 32 Prop osition 2.14. The map k A,B : B X / / H is diﬀe olo gi c al. F or any smo oth map Γ : R 2 / / X w i th x := Γ(0) , v 1 := ∂ Γ ∂ s and v 2 := ∂ Γ ∂ t , we have − ∂ 2 ∂ s∂ t     (0 , 0) k A,B (Γ ∗ Σ R ( s, t )) = B x ( v 1 , v 2 ) . Pro of. Assume that c : U / / B X is a map from an op en sub set U ⊂ R n to B X suc h that the ev aluation c ( u )( s, t ) ∈ X is smooth on all of U × [0 , 1] 2 . Hence, the diﬀeren tial form A c ( u ) from (2.26) dep ends smo othly on u ∈ U , and so do es the solution f c ( u ) : R 2 / / H of the diﬀeren tial equation go v erned b y A c ( u ) . This implies that k A,B ◦ pr ◦ c : U / / H is smo oth, so that k A,B is diﬀeological b y Lemma 1.4. No w w e consider U = R 2 and c := Γ ∗ ◦ Σ R the standard bigon (2.5), so that k A,B ◦ c : R 2 / / H is a smo oth map. In order to compute the deriv ativ e ∂ ∂ s     0 k A,B ( c ( s, t )) = ∂ ∂ s     0 α ( F A (Γ ∗ γ 0 ( s, t )) , f c ( s,t ) (0 , 1) − 1 ) w e observ e that A c ( s,t ) | σ = σ A c (1 ,t ) | σs . F o r the solutions of the corresp ond- ing diﬀeren tial equations we obtain by a uniquene ss argumen t f c ( s,t ) (0 , σ ) = f c (1 ,t ) (0 , sσ ) . W e compute ∂ ∂ s     0 f c ( s,t ) (0 , 1) − 1 = − ∂ ∂ s     0 f c (1 ,t ) (0 , s ) = A c (1 ,t ) | 0  ∂ ∂ s  = − Z 1 0 d τ  α F A (Γ ∗ γ 0 ,tτ )  ∗ ( c (1 , t ) ∗ B ) (0 ,τ )  ∂ ∂ s , ∂ ∂ τ  = − Z t 0 d τ ′  α F A (Γ ∗ γ 0 ,τ ′ )  ∗ (Γ ∗ B ) (0 ,τ ′ )  ∂ ∂ s , ∂ ∂ τ ′  (2.29) In t he last ste p w e ha ve p erformed an in tegral transformation and use d that c (1 , 1) = Γ . Finally ∂ 2 ∂ s∂ t     0 k A,B ( c ( s, t )) = ∂ ∂ t     0 ( α F A (Γ ∗ γ 0 (0 ,t )) ) ∗  ∂ ∂ s     0 f c ( s,t ) (0 , 1) − 1  = ∂ 2 ∂ s∂ t     0 f c ( s,t ) (0 , 1) − 1 ( 2.29 ) = − (Γ ∗ B ) | 0  ∂ ∂ s , ∂ ∂ t  = − B x ( v 1 , v 2 ) . 33 In the ﬁrst line w e hav e used (2.19) and that f c (0 ,t ) (0 , 1) = 1 ∈ H , so that the diﬀeren tial of α 1 : G / / H is the zero map.  The next thing w e would lik e to kno w ab out the map k A,B is its com- patibilit y with the horizon tal and v ertical comp osition of bigons in X . Con- cerning the ve rtical comp osition, this will b e straigh tf o rw ard, but for the horizon ta l comp osition w e hav e to in tro duce ﬁrstly an auxiliary horizon ta l comp osition and to c hec k the compatibilit y of k A,B with this one. T o deﬁne this auxiliary horizontal comp osition, we consider t w o bigons Σ 1 : γ 1 + 3 γ ′ 1 and Σ 2 : γ 2 + 3 γ ′ 2 , with γ 1 , γ ′ 1 : x / / y and γ 2 , γ ′ 2 : y / / z . The result will b e a bigon Σ 2 ∗ Σ 1 : γ 2 ◦ γ 1 + 3 γ ′ 2 ◦ γ ′ 1 . W e deﬁne a map p : [0 , 1] 2 / / [0 , 1] 2 b y p ( s, t ) :=          (0 , t ) for 0 ≤ t < 1 2 and 0 ≤ s < 1 2 (2 s, t ) for 1 2 ≤ t ≤ 1 and 0 ≤ s < 1 2 (2 s − 1 , t ) for 0 ≤ t < 1 2 and 1 2 ≤ s ≤ 1 (1 , t ) for 1 2 ≤ t ≤ 1 and 1 2 ≤ s ≤ 1 see Figure 1. This map p is not smo oth, but its composition with Σ 2 ◦ Σ 1 is, p :  / / Figure 1: A useful deformatio n of the unit square. due to the sitting instan ts of the bigons Σ 1 and Σ 2 . W e deﬁne Σ 2 ∗ Σ 1 := (Σ 2 ◦ Σ 1 ) ◦ p to b e this smo oth map, this deﬁnes the auxiliary horizontal comp osition of Σ 1 and Σ 2 . Lemma 2.15. The map k A,B : B X / / H r esp e cts the vertic al c omp osition of bigons in the sense that k A,B (id γ ) = 1 and k A,B (Σ 2 • Σ 1 ) = k A,B (Σ 2 ) · k A,B (Σ 1 ) 34 for any p ath γ ∈ P X and any two vertic al l y c omp osable bigon s Σ 1 and Σ 2 . It r esp e cts the auxiliary horizontal c omp osition ∗ in the sense that k A,B (Σ 2 ∗ Σ 1 ) = k A,B (Σ 2 ) · α ( F A ( γ 2 ) , k A,B (Σ 1 )) for any two horizontal ly c omp osable bigons Σ 1 : γ 1 + 3 γ ′ 1 and Σ 2 : γ 2 + 3 γ ′ 2 . Pro of. Concerning the ve rtical compo sition, the iden tit y bigon id γ : γ + 3 γ has the 1-form A id γ = 0 , so that f Σ (0 , σ ) is constan t. Hence, k A,B (id γ ) = 1 . Now let Σ 1 : γ 0 + 3 γ 1 and Σ 2 : γ 1 + 3 γ 2 b e tw o bigons. F or the 1-fo rm (2.26) asso ciated to the bigon Σ 2 • Σ 1 w e ﬁnd 1 2 A Σ 2 • Σ 1 | σ = ( A Σ 1 | 2 σ for 0 ≤ σ ≤ 1 2 A Σ 2 | 2 σ − 1 for 1 2 ≤ σ ≤ 1 and accordingly f Σ 2 • Σ 1 (0 , 1) = f Σ 2 • Σ 1  1 2 , 1  · f Σ 2 • Σ 1  0 , 1 2  = f Σ 2 (0 , 1) · f Σ 1 (0 , 1) . (2.30) A short calculation then sho ws that k A,B (Σ 2 • Σ 1 ) ( 2.27 ) = α ( F A ( γ 0 ) , f Σ 2 • Σ 1 (0 , 1) − 1 ) ( 2.30 ) = α ( F A ( γ 0 ) , f Σ 1 (0 , 1) − 1 · f Σ 2 (0 , 1) − 1 ) = α ( F A ( γ 0 ) · t ( f Σ 1 (0 , 1)) − 1 , f Σ 2 (0 , 1) − 1 ) · α ( F A ( γ 0 ) , f Σ 1 (0 , 1) − 1 ) (a) = α ( F A ( γ 1 ) , f Σ 2 (0 , 1) − 1 ) · α ( F A ( γ 0 ) , f Σ 1 (0 , 1) − 1 ) (2.27) = k A,B (Σ 2 ) · k A,B (Σ 1 ) . In the step in the middle w e hav e used the axioms of a crossed mo dule, namely that α ( g , h 1 h 2 ) = α ( g , α ( t ( h 1 ) , h 2 ) · h 1 ) = α ( g · t ( h 1 ) , h 2 ) · α ( g , h 1 ) for all g ∈ G and h 1 , h 2 ∈ H . Concerning the auxiliary horizon tal comp o sition, w e obtain for the 1- form (2.26) asso ciated to the bigon Σ 2 ∗ Σ 1 1 2 A Σ 2 ∗ Σ 1 | σ = ( Ad − 1 F A ( γ 1 ) ( A Σ 2 | 2 σ ) for 0 ≤ σ ≤ 1 2 A Σ 1 | 2 σ − 1 for 1 2 ≤ σ ≤ 1 35 and accordingly f Σ 2 ∗ Σ 1 (0 , 1) = f Σ 2 ∗ Σ 1  1 2 , 1  · f Σ 2 ∗ Σ 1  0 , 1 2  = f Σ 1 (0 , 1) · α ( F A ( γ 1 ) − 1 , f Σ 2 (0 , 1)) . (2.31) Then w e obtain k A,B (Σ 2 ∗ Σ 1 ) ( 2.27 ) = α ( F A ( γ 2 ◦ γ 1 ) , f Σ 2 ∗ Σ 1 (0 , 1) − 1 ) ( 2.31 ) = α ( F A ( γ 2 ◦ γ 1 ) , α ( F A ( γ 1 ) − 1 , f Σ 2 (0 , 1)) − 1 · f Σ 1 (0 , 1) − 1 ) ( 2.27 ) = k A,B (Σ 2 ) · α ( F A ( γ 2 ) , k A,B (Σ 1 )) this yields the required iden tit y .  Before w e come to the original horizon tal comp osition of bigons it is con ven ien t to sho w ﬁrst the follo wing Lemma 2.16. F or thin homotopy e quivalent bigons Σ ∼ 2 Σ ′ we have k A,B (Σ) = k A,B (Σ ′ ) . W e hav e mo ve d the pro of of this lemma to Appendix A.3. Then it follows that k A,B factors through B 2 X , B X pr 2 / / B 2 X / / H Since pr 2 is surjectiv e, t he map B 2 X / / H is uniquely determined, and b y Prop osition 2.14, it is diﬀeological. W e denote this unique diﬀeological map also b y k A,B : B 2 X / / H . Prop osition 2.17. The assignme nt F : x γ   γ ′ B B Σ   y  / / ∗ F A ( γ )   F A ( γ ′ ) C C k A,B (Σ)   ∗ (2.32) deﬁnes a smo oth 2-functor F : P 2 ( X ) / / B G . Pro of. Since F A is a smoo th functor, w e ha v e nothing to sho w for 1- morphisms. On 2-morphism s, the assignmen t k A,B is smo oth b y Prop osition 36 2.14. By Lemma 2.15 it further respects the v ertical comp osition. Concerning the horizon ta l comp osition, notice that h : [0 , 1] × [0 , 1] 2 / / X : ( r , s, t )  / / (Σ 2 ◦ Σ 2 )( r p + (1 − r )id [0 , 1] 2 )( s, t ) , deﬁnes a homotop y b et w een Σ 2 ∗ Σ 1 and Σ 2 ◦ Σ 1 , and since its ra nk is limited b y dimensional reasons to 2, this homoto p y is thin. Then, b y Lemmata 2.15 and 2.16 w e ha v e k A,B (Σ 2 ◦ Σ 1 ) = k A,B (Σ 2 ∗ Σ 1 ) = k A,B (Σ 2 ) · α ( F A ( γ 2 ) , k A,B (Σ 1 )) . (2.33) Th us, the 2-functor F resp ects the horizontal composition.  2.3.2 Reconstruction I I: Pseudonatural T ransformations Here w e consider a 1-morphism ( g , ϕ ) : ( A, B ) / / ( A ′ , B ′ ) in the 2-category Z 2 X ( G ) ∞ , i.e. a smoo th map g : X / / G and a 1-fo rm ϕ ∈ Ω 1 ( X , h ) that satisfy the relations from Prop osition 2.10, A ′ + t ∗ ( ϕ ) = Ad g ( A ) − g ∗ ¯ θ (2.3 4 ) B ′ + α ∗ ( A ′ ∧ ϕ ) + d ϕ + [ ϕ ∧ ϕ ] = ( α g ) ∗ ( B ) . (2.35) The 1- forms A ′ and ϕ deﬁne a 1-fo rm ( A ′ , ϕ ) ∈ Ω 1 ( X , g ⋉ h ) , and th us b y Theorem 1.6 a smo oth functor ˜ ρ : P 1 ( X ) / / B ( G ⋉ H ) . W e denote its pro jection to H b y h : P 1 X / / H . W e w an t to deﬁne a smo oth pseudonat- ural transformation ρ : F / / F ′ b et wee n the 2-functors F := P ( A, B ) and F ′ := P ( A ′ , B ′ ) b y ρ : x γ / / y 7− → ∗ F ( γ ) / / g ( x )   ∗ g ( y )   h ( γ ) − 1         {          ∗ F ′ ( γ ) / / ∗ . (2.36) W e ha v e to sho w Lemma 2.18. Th e tar get-matchin g c ond ition F ′ ( γ ) · g ( x ) = t ( h ( γ ) − 1 ) · g ( y ) · F ( γ ) (2.37) for the 2-morphis m h ( γ ) − 1 is satisﬁe d. 37 Pro of. W e recall that F ( γ ) , F ′ ( γ ) and h ( γ ) are v a lues of solutions f γ , f ′ γ : R / / G and h γ : R / / H of initial v alue problems . W e sho w that f ′ γ (0 , t ) = t ( h γ ( t ) − 1 ) · g ( γ ( t )) · f γ (0 , t ) · g ( γ (0)) − 1 =: β ( t ) whic h giv es for t = 1 equation (2.34). F or this purp ose, w e sho w that β ( t ) satisﬁes t he initial v alue problem fo r f ′ γ . The initial condition β (0) = 1 is satisﬁed. Notice that with p := γ ( t ) and v := ˙ γ ( t ) ∂ h γ ( t ) ∂ t = − d r h γ ( t ) | 1 ( ϕ p ( v )) − ( α h γ ( t ) ) ∗ ( A ′ p ( v )) (2.38) so that – using Axiom 2a) of the crossed mo dule – ∂ ∂ t t ( h γ ( t ) − 1 ) = d t | h γ ( t ) − 1  ∂ h γ ( t ) − 1 ∂ t  =  Ad − 1 t ( h γ ( t ))  t ∗ ( ϕ p ( v )) + A ′ p ( v )  − A ′ p ( v )  · t ( h γ ( t ) − 1 ) . Then w e compute ∂ β ∂ t =  Ad − 1 t ( h γ ( t ))  t ∗ ( ϕ p ( v )) + A ′ p ( v ) + g ∗ ¯ θ | p ( v ) − Ad g ( A p ( v ))  − A ′ p ( v )  · β ( t ) . Using equation (2.34), the right hand side b ecomes − A ′ p ( v ) β ( t ) . Hence, β ( t ) solve s the same initial v alue problem as f ′ γ (0 , t ) . By uniqueness, b oth functions coincide.  It remains to ch ec k t ha t the axioms of a pseudonatural t ra nsformation are satisﬁed. Axiom (T1) follow s from the fact that ˜ ρ is a functor by the same arg umen ts as giv en in the pro of of Lemma 2.9. F or axiom (T2) w e ha v e to prov e Lemma 2.19. Th e 2-mo rphism h ( γ ) s atisﬁes F ′ (Σ) · h − 1 ( γ 0 ) = h − 1 ( γ 1 ) · α ( g ( y ) , F (Σ)) for any bigon Σ : γ 0 + 3 γ 1 . Pro of. W e recall that F (Σ) = k A,B (Σ) = α ( F ( γ ) , f Σ (0 , 1) − 1 ) , where f Σ (0 , s ) is the solution of a initial v alue problem go v erned by a 1- form A Σ . F or F ( γ ′ ) the same is true with primed quan tities. W e deﬁne the notion γ s ( t ) := Σ( s, t ) consisten t with γ 0 and γ 1 . Then, the equation f ′ Σ (0 , s ) = α ( F ′ ( γ 0 ) − 1 , h ( γ 0 ) − 1 · α ( g ( y ) · F ( γ 0 ) , f Σ (0 , s )) · h ( γ s )) := κ ( s ) , 38 ev aluated for s = 1 , is the equation w e ha ve to prov e. Lik e in the pro o f of Lemma 2.18 w e show that κ ( s ) solv es the initial v alue problem for f ′ Σ (0 , s ) . In a ﬁrst step, the deriv ativ e ∂ κ/∂ s can b e written as d r κ ( s ) | 1 X ( s ) where X ( s ) ∈ h is X ( s ) = ( α F ′ ( γ 0 ) − 1 ) ∗  − Ad − 1 h ( γ 0 ) ( α g ( y ) F ( γ 0 ) ) ∗ A Σ | s  ∂ ∂ s  + Ad h ( γ 0 ) − 1 α ( g ( y ) F ( γ 0 ) ,f Σ (0 ,s ))  ∂ h ( γ s ) ∂ s h ( γ s ) − 1  = − ( α g ( x ) ) ∗  A Σ | s  ∂ ∂ s  + ( α F ′ ( γ − 1 s ) ) ∗ h ( γ s ) − 1 ∂ h ( γ s ) ∂ s . In the second line w e ha v e used the target matc hing conditions (2.22) and (2.37). With the deﬁnition (2.26) and aga in (2.37), the ﬁrst summand b e- comes − ( α g ( x ) ) ∗ A Σ | s  ∂ ∂ s  = Z 1 0 d t ( α F ′ ( γ s,t ) − 1 ) ∗ Ad − 1 h ( γ s,t ) W ( s, t ) . where w e ha v e written W ( s, t ) := Σ ∗ (( α g ) ∗ ( B )) ( s,t )  ∂ ∂ s , ∂ ∂ t  ∈ h . T o compute the second summand, we recall from Section 2.3.1 the deﬁnition of t he path τ s 0 ( s, t ) that runs coun ter-clo ckw ise around the rectangle spanned b y ( s 0 , 0) and ( s 0 + s, t ) . W e consider the smo oth f unction u s 0 : R 2 / / G ⋉ H b e deﬁned b y u s 0 ( s, t ) := ˜ ρ (Σ ∗ ( τ s 0 ( s, t ))) , where ˜ ρ is the smo oth functor correspo nding to the 1-form ( A ′ , ϕ ) ∈ Ω 1 ( X , g ⋉ h ) w e started with. F or this smo oth function, we recall Lemma 2.13 (a), here h ( γ − 1 0 ◦ γ s ) = p H ( u 0 ( s, 1)) . F urthermore, w e hav e ∂ ∂ s u 0 ( s, 1) (b) = u 0 ( s, 1) · ∂ ∂ σ     0 u s ( σ , 1) = u 0 ( s, 1) · Z 1 0 d t ∂ ∂ σ ∂ ∂ t     (0 ,t ) u s ( σ , t ) (c) = − u 0 ( s, 1) · Z 1 0 d t Ad − 1 ˜ ρ ( γ s,t ) (Σ ∗ K ) ( s,t )  ∂ ∂ s , ∂ ∂ t  . (2.39) In t he last line, K = ( K A ′ , K ϕ ) is the curv ature 2 -form of the 1-form ( A ′ , ϕ ) , consisting of K A ′ = d A ′ + [ A ′ ∧ A ′ ] ( 2.21 ) = t ∗ ◦ B ′ and K ϕ = α ∗ ( A ′ ∧ ϕ ) + d ϕ + [ ϕ ∧ ϕ ] . 39 If w e write Y ( s, t ) for Σ ∗ K ϕ ev aluated at ( s, t ) , and similarly Z ( s, t ) for Σ ∗ B ′ , the adjoin t action in (2.39) on the semidirect pro duct g ⋉ h is Ad − 1 ( g, h ) ( t ∗ ( Z ) , Y ) =  Ad − 1 g ( t ∗ ( Z )) , ( α g − 1 ) ∗  Ad − 1 h ( Y + Z ) − Z  . With Y + Z = W from (2.35), the pro jection of (2.39 ) to h b ecomes ∂ h ( γ − 1 0 ◦ γ s ) ∂ s = − h ( γ − 1 0 ◦ γ s ) · ( α F ′ ( γ − 1 0 ◦ γ s ) ) ∗  Z 1 0 d t ( α F ′ ( γ s,t ) − 1 ) ∗  Ad − 1 h ( γ s,t ) ( W ( s, t )) − Z ( s, t )   (2.40) Then, with h ( γ − 1 0 ◦ γ s ) = h ( γ − 1 0 ) α ( F ′ ( γ 0 ) − 1 , h ( γ s )) , w e ha v e summarizing X ( s ) = Z 1 0 d t ( α F ′ ( γ s,t ) − 1 ) ∗ Z ( s, t ) ( 2.26 ) = −A ′ Σ | s  ∂ ∂ s  . This sho ws κ ( s ) = f ′ Σ (0 , s ) .  2.3.3 Reconstruction I I I: Mo diﬁcations W e consider a 2-morphism a : ( g , ϕ ) + 3 ( g ′ , ϕ ′ ) in the 2-category Z 2 X ( G ) ∞ , b et w een t w o 1-morphisms ( g , ϕ ) and ( g ′ , ϕ ′ ) from ( A, B ) to ( A ′ , B ′ ) . This is a smo oth map a : X / / H that satisﬁes (2.20): g 2 = ( t ◦ a ) · g 1 and ϕ 2 + ( r − 1 a ◦ α a ) ∗ ( A ′ ) = Ad a ( ϕ 1 ) − a ∗ ¯ θ . (2 .41) W e w an t to deﬁne a smo oth mo diﬁcation A : ρ + 3 ρ ′ b et wee n the pseudo- natural transformations ρ := P ( g , ϕ ) and ρ ′ := P ( g ′ , ϕ ′ ) . W e deﬁne A : x  / / ∗ g ( x )   g ′ ( x ) C C a ( x )   ∗ . The targ et-matc hing condition for the 2-morphism f ( x ) is ob viously satisﬁed due to the ﬁrst equation in (2.41). The axiom f or the mo diﬁcation A is 40 Lemma 2.20. Th e 2-mo rphism a ( x ) sa tisﬁes α ( F ′ ( γ ) , a ( x )) · h ( γ ) − 1 = h ′ ( γ ) − 1 · a ( y ) for al l p aths γ ∈ P X . Pro of. W e rewrite the equation as h ′ γ ( t ) = a ( γ ( t )) · h γ ( t ) · α ( f ′ γ (0 , t ) , a ( x ) − 1 ) := λ ( t ) whic h we will prov e by sho wing that λ ( t ) satisﬁes the same initial v alue problem as h ′ γ ( t ) , namely (2.38): ∂ h ′ γ ( t ) ∂ t = − d r h ′ γ ( t ) | 1 ( ϕ ′ p ( v )) − ( α h ′ γ ( t ) ) ∗ ( A ′ p ( v )) (2.42) for p := γ ( t ) and v := ˙ γ ( t ) . A straigh tforw ard calculation sho ws that ∂ λ ∂ t = − d r λ ( t ) | 1  − ( a ∗ ¯ θ ) p ( v ) + Ad a ( p ) ( ϕ 1 | p ( v )) − ( r − 1 a ( p ) ◦ α a ( p ) ) ∗ ( A ′ p ( v ))  − ( α λ ( t ) ) ∗ ( A ′ p ( v )) . F or this calculation, one tw ice has to use the identit y ( α h 1 h 2 ) ∗ ( X ) = d r h 2 | h 1 ( α h 1 ) ∗ ( X ) + d l h 1 | h 2 ( α h 2 ) ∗ ( X ) . (2.43) Using then the second equation o f (2.41), we ha v e sho wn that λ ( t ) satisﬁes the diﬀeren tial equation (2.42). Thu s, λ ( t ) = h ′ γ ( t ) .  2.3.4 Summary of Section 2.2 Ab o v e w e hav e collected the structure of a 2-functor P : Z 2 X ( G ) ∞ / / F unct ∞ ( P 2 ( X ) , B G ) . Let us c heck that the axioms of a 2-functor are satisﬁed. Lik e in Section 2.2.4, horizon ta l and vertic al composition o f 2- morphisms is resp ected b ecause they are deﬁned on b oth sides in the same w a y for the same H - v alued functions. It remains to c hec k the compatibilit y with the comp osition of 1- morphisms, P (( g 2 , ϕ 2 ) ◦ ( g 1 , ϕ 1 )) := P ( g 2 , g 1 , ( α g 2 ) ∗ ( ϕ 1 ) + ϕ 2 ) = P ( g 2 , ϕ 2 ) ◦ P ( g 1 , ϕ 1 ) for 1-morphisms ( g i , ϕ i ) : ( A i , B i ) / / ( A i +1 , B i +1 ) . F or the comp onen ts at ob jects x ∈ X , this equality is clear. W e recall the comp onen t of P ( g i , ϕ i ) 41 at a path γ ∈ P X is a 2-mor phism in B G giv en according to (2.3 6) b y a group elemen t h ( γ ) − 1 , where h ( γ ) = h i (1) f or h i ( t ) the solution of the initial v alue problem (2.38). Similar, the comp onent o f P ( g 2 , g 1 , ˜ ϕ ) at γ with ˜ ϕ := ( α g 2 ) ∗ ◦ ϕ 1 + ϕ 2 is ˜ h ( γ ) − 1 , where ˜ h ( γ ) = ˜ h (1) for ˜ h ( t ) the solution of the initial v alue problem ∂ ˜ h ( t ) ∂ t = − d r ˜ h ( t ) | 1 ( ˜ ϕ 2 | γ ( t ) ( v t )) − ( α ˜ h ( t ) ) ∗ ( A 3 | γ ( t ) ( v t )) (2.44) with v t := ˙ γ ( t ) . A ccording to the deﬁnition of t he comp osition o f pseudonat- ural transformations, the equation w e ha v e to prov e no w follow s from ˜ h ( t ) = α ( g 2 ( γ ( t )) , h 1 ( t )) · h 2 ( t ) =: ζ ( t ) (2.45) ev aluated at t = 1 , and w e pro ve (2.45) b y showin g that ζ ( t ) solv es (2.44). A straigh tforw ard calculation similar to the one p erformed in the pro of o f Lemma 2.20, using (2.43) and (2.3 4) for ( g 2 , ϕ 2 ) , sho ws that this is indeed the case. 2.4 Main Theorem W e hav e so far deﬁned t w o 2- functors D and P whic h g o from smo oth 2- functors to diﬀeren tial forms, a nd fro m diﬀeren tia l forms bac k to smooth 2-functors. Here we prov e the main theorem of this article: Theorem 2.21. T he 2-func tors D : F unct ∞ ( P 2 ( X ) , B G ) / / Z 2 X ( G ) ∞ fr om Se ction 2.2 and P : Z 2 X ( G ) ∞ / / F unct ∞ ( P 2 ( X ) , B G ) fr om Se ction 2.3 satisfy D ◦ P = id Z 2 X ( G ) ∞ and P ◦ D = id F unct ∞ ( P 2 ( X ) , B G ) (2.46) and form henc e an isomorphism of 2-c ate go rie s. Pro of. W e start with an ob ject ( A, B ) in Z 2 X ( G ) ∞ , i.e. a 1-form A ∈ Ω 1 ( X , g ) and a 2-form B ∈ Ω 2 ( X , h ) suc h that d A + [ A ∧ A ] = t ∗ ◦ B . W e let ( A ′ , B ′ ) := D ( P ( A, B ) ) b e the diﬀeren tial forms extracted from the reconstructed 2-functor F := P ( A, B ) . By Theorem 1.6 w e ha v e A ′ = A . No w w e test the 2-form B ′ at a p o int x ∈ X and at tangent v ectors v 1 , v 2 ∈ 42 T x X . Let Γ : R 2 / / X b e a smo oth map with x = Γ(0) , v 1 = ∂ Γ ∂ s   0 and v 2 = ∂ Γ ∂ t   0 . W e only hav e to summarize B ′ x ( v 1 , v 2 ) ( 2.8 ) = − ∂ 2 ∂ s∂ t     0 P ( A, B )(Γ ∗ Σ R ( s, t )) ( 2.32 ) = − ∂ 2 ∂ s∂ t     0 k A,B (Γ ∗ Σ R ( s, t )) = B x ( v 1 , v 2 ) , where the last equalit y has b een sho wn in Prop osition 2.1 4. Con ve rsely , let F : P 2 ( X ) / / B G b e a smo oth 2-functor, and let F ′ := P ( D ( F )) . By Theorem 1.6 it is clear that F ′ ( x ) = F ( x ) and F ′ ( γ ) = F ( γ ) for ev ery p oint x ∈ X and ev ery path γ ∈ P X . F or a bigon Σ ∈ B 2 X w e recall that F ′ (Σ) = k D ( F ) (Σ) = α ( F ( γ 0 ) , f ′ Σ (0 , 1) − 1 ) , (2.47) where f ′ Σ is the solution of the initial v alue problem ∂ f ′ Σ (0 , s ) ∂ s = − d r f ′ Σ (0 ,s ) ( X ( s )) and f ′ Σ (0 , 0) = 1 . (2.48) This initial v alue problem is go v erned b y X ( s ) ∈ h , whic h is giv en by the 1-form A Σ from (2.26), namely X ( s ) := A Σ | s  ∂ ∂ s  := − Z 1 0 d t ( α F ( γ s,t ) − 1 ) ∗ (Σ ∗ B ) ( s,t )  ∂ ∂ s , ∂ ∂ t  , and B is the 2-f orm in ( A, B ) = D ( F ) . W e deﬁne a bigon Σ s,t ( σ , τ ) b y Σ s,t ( σ , τ )( s ′ , t ′ ) := Σ( s + β ( σ s ′ ) , t + β ( τ t ′ )) , where β is some ﬁxed smo oth map β : [0 , 1] / / [0 , 1] with β ( 0) = 0 and β (1) = 1 and with sitting instan ts. W e notice from ( 2 .47) and (2.48 ) that F ′ (Σ 0 , 0 ( s, 1)) is the unique solution of the initial v alue problem ∂ ∂ s F ′ (Σ 0 , 0 ( s, 1)) = F ′ (Σ 0 , 0 ( s, 1)) · d α F ( γ 0 ) ( X ( s )) and F ′ (Σ 0 , 0 (0 , 1)) = 1 . In the follo wing w e pro v e that F (Σ 0 , 0 ( s, 1)) also solv es this initial v alue prob- lem, so that in particular F ′ (Σ) = F ′ (Σ 0 , 0 (1 , 1)) = F (Σ 0 , 0 (1 , 1)) = F (Σ) follo ws, and w e hav e P ( D ( F )) = F . T o sho w that F (Σ 0 , 0 ( s, 1)) is a solution w e compute ∂ ∂ s F (Σ 0 , 0 ( s, 1)) = F (Σ 0 , 0 ( s, 1))d α F ( γ 0 ) α F ( γ s ) − 1  ∂ ∂ σ     0 F (Σ s, 0 ( σ , 1))  43 and then α F ( γ s ) − 1  ∂ ∂ σ     0 F (Σ s, 0 ( σ , 1))  = Z 1 0 d t ( α F ( γ s,t ) − 1 ) ∗ ∂ 2 ∂ σ ∂ τ     0 α ( F ( γ s,t + τ ) − 1 , F (Σ s, 0 ( σ , t + τ ))) . T o compute the deriv ativ e w e decompose Σ s, 0 ( σ , t + τ ) in tw o bigons Σ s, 0 ( σ , t ) and Σ s,t ( σ , τ ) and obtain ∂ 2 ∂ σ ∂ τ     0 α ( F ( γ s,t + τ ) − 1 , F (Σ s, 0 ( σ , t + τ ))) = − (Σ ∗ B ) s,t  ∂ ∂ s , ∂ ∂ t  . No w, the three last equations sh o w that F (Σ 0 , 0 ( s, 1)) solv es the ab o v e initial v alue problem. So fa r w e ha v e pro v ed equations (2.46) o n the lev el of ob j ects. On the lev el of 1-morphisms, it is a consequenc e o f Theorem 1 .6: for a pseudo- natural tr a nsformation ρ : F / / F ′ with comp onen ts g : X / / G and ρ H : P 1 X / / H w e ha v e P ( D ( ρ )) ( 2.13 ) = P ( g , D ( F ′ , ρ − 1 H )) ( 2.36 ) = ( g , P ( D ( F ′ , ρ − 1 H )) − 1 ) Th . 1.6 = ( g , ρ H ) = ρ , and con v ersely , for a 1- mor phism ( g , ϕ ) : ( A, B ) / / ( A ′ , B ′ ) in Z 2 X ( G ) ∞ , D ( P ( g , ϕ )) ( 2.36 ) = D ( g , P ( A ′ , ϕ ) − 1 ) ( 2.13 ) =  g , D   P ( A ′ , ϕ ) − 1  − 1  Th . 1.6 = ( g , ϕ ) . Finally , on the lev el o f 2-morphisms, whic h are on b oth sides just the same H -v alued functions on X , there is nothing to sho w.  3 Examples of S mo oth 2-F unctors W e giv e three examples of situations where smo oth 2-f unctors are prese n t. 3.1 Connections on (non-ab el ian) Gerb e s Let us ﬁrst recall from [SW09] what connections on ordinary principal bun- dles hav e to do with ordinary functors. F or G a Lie gro up, we denote b y G - T or the category whose ob jects ar e smo oth manifolds with transitiv e, free and smoo th G -action from the righ t, and whose morphisms are G -equiv arian t smo oth maps. The functor whic h regards G itself as a G -space is denoted 44 b y i G : B G / / G - T or . If γ : x / / y is a pa t h in X , any principal G -bundle P pro vides us with ob jects P x and P y of G - T o r , namely its ﬁbres ov er the endp o in ts of γ . F urthermore, a connection ∇ on P deﬁnes a morphism τ γ : P x / / P y in G - T or , namely the parallel transp ort along γ . W ell-know n prop erties of parallel transp ort assure that the assignmen ts x  / / P x and γ  / / τ γ deﬁne a functor tra P , ∇ : P 1 ( X ) / / G - T or . The main result of [SW09] is the c haracterization o f f unctors obtained lik e this among all functors F : P 1 ( X ) / / G - T or . They are c hara cterized b y the follo wing deﬁning prop ert y of a tr ansp ort functor : there exists a surjectiv e submersion π : Y / / M and a smo oth functor triv : P 1 ( Y ) / / B G suc h that the functors π ∗ F and i G ◦ triv are (with additional conditions w e skip here) naturally equiv alen t. In other w ords, transp ort functors a re lo c a l ly smo ot h functors. These transp ort functors form a category T rans 1 B G ( X , G - T or) , and w e ha v e Theorem 3.1 ([SW09], Theorem 5.4) . The assignment of a functor t ra P to a princip al G -b und l e P with c onne ction over X d e ﬁnes a surje ctive e quivalenc e of c ate gories Bun ∇ G ( X ) ∼ = T rans 1 B G ( X , G - T or) . Under this equiv alence, triv ial principal G -bundles with connection corre- sp ond to glo b al ly smo oth functors, i.e. functors tra : P 1 ( X ) / / G - T or with tra = i G ◦ triv for a smo oth functor triv : P 1 ( X ) / / B G . T rivial izable prin- cipal G -bundles with connection corresp ond to functors whic h a re naturally equiv a len t to globa lly smo oth functors (again with additional assumptions on the natural equiv alence ). W e think that the concept of transp ort functors is a dequate to b e cate- goriﬁed and to capture all a sp ects of connections on 2-bundles, in particular gerb es. W e anticipate the follo wing results of [SW A]: 1. Gerb es with connection o v er X hav e structure 2-g roups G . 2. A trivial G -gerb e with connection ov er X is a smo oth 2-functor F : P 2 ( X ) / / B G . Let us test these assertions in tw o examples. 45 Example 3.2. W e consider the Lie 2-group G = B U ( 1 ) from Example A.8. The corresp onding B U (1) -gerb es are also kno wn as ab elian gerb es, or U (1 ) - gerb es. No w, a trivial B U (1) - gerb e with connection ov er X is b y the ab o v e assertion and Theorem 2.21 nothing but a 2-fo rm B ∈ Ω 2 ( X ) . Ab elian gerb es with connection can b e realized con v enien tly b y bund le gerb es [Mur96]. In this con text it is w ell-know n that a connection o n a trivial bundle gerb e is indeed just a 2-f o rm, see, e.g., [W al07]. Example 3.3. Let H b e a connected Lie gro up. W e denote b y aut ( H ) the Lie algebra o f the Lie gr o up Aut( H ) of Lie group automorphisms of H . W e consider the Lie 2-group G = A UT( H ) from Example A.10. By the ab o v e assertion and Theorem 2.21 , a trivial A UT ( G ) -gerb e with connection ov er X is a pair ( A, B ) of a 1-fo rm A ∈ Ω 1 ( X , aut ( H )) and a 2-form B ∈ Ω 2 ( X , h ) suc h that d A + [ A ∧ A ] = ad( B ) , (3.1) where ad : h / / aut ( H ) : X  / / ad X . A UT ( H ) -gerb es are also kno wn a s H -gerb es 1 in the sens e o f Breen and Messing [BM05]. There, a connection on a trivial H -g erb e is a pair ( A, B ) just as in Example 3 .3 but without the condition (3.1 ) . This diﬀerence lies at the heart of a question N. Hitc hin p osed at the VBAC-mee ting in Bad Honnef in June 200 7 a f ter a talk by L. Breen, namely if it is p ossible to deﬁne a surface holonom y fro m a connection on an H -gerb e. Let us presume that “ a surface holonomy ” is at least a 2- functorial assignmen t, i.e. is describ ed b y a 2- functor on the path 2-group oid. This assump tion is supp orted by the approa ch via “ 2-holonomies” [MP10], a s w ell as b y tr ans p ort 2-func tors [SW A ]. Similar considerations hav e also b een made for ordinary holonom y [CP94, SW09]. Then, the following three statemen ts on a Breen-Mess ing connection ( A, B ) on a trivial H -gerb e o ver X are equiv alen t: (a) it deﬁnes a surface holonomy . (b) it satisﬁes condition (3.1), d A + [ A ∧ A ] = ad( B ) . (c) there exists a smo oth 2- functor F : P 2 ( X ) / / B A UT ( H ) suc h that ( A, B ) = D ( F ) . 1 W e have to re ma rk that a U (1) -g e rbe in the sense of Example 3.2 is not the s ame as an H -ge r be for H = U (1) in the sense of Br een and Messing. The diﬀerence b ecomes clear if one uses the classiﬁcation of ger bes by L ie 2 - g roups w e hav e prop osed here: we hav e B U (1) -gerb es on one s ide but AUT( U (1)) -ger bes on the o ther. Indeed, B U (1) is o nly a sub-2-gr o up of AUT( U (1)) . 46 A detailed discussion of surface holonomies tha t also co ve rs non-trivial H - gerb es, is p ostp oned to [SW A]. 3.2 Deriv ati v es of Smo oth F unctors In App endix A.2 we ha v e review ed the Lie 2-group E G asso ciated to an y Lie group G . F o r an y func tor F : P 1 ( X ) / / B G there is an asso ciated 2-functor d F : P 2 ( X ) / / B E G that we call the derivative 2-functor of F . It sends a 1-morphism γ ∈ P 1 X to the imag e F ( γ ) ∈ G of γ under the functor F . This determines d F com- pletely , since the Lie 2-group o id B E G has only one ob ject, and precis ely only one 2-mor phism betw een an y t w o ﬁxed 1- morphisms. It will b e in teresting to determine the unique 2- morphism d F (Σ) asso ciated to a bigon Σ : γ 1 + 3 γ 2 explicitly . F or this purp ose, w e denote by ∂ Σ the 1-morphism γ − 1 1 ◦ γ 2 . Then, w e obtain directly from the deﬁnitions: Theorem 3.4 (The non-ab elian Stok es’ Theorem for functors) . L et G b e a Lie gr oup and let F : P 1 ( X ) / / B G b e a functor. Then, d F (Σ) = F ( ∂ Σ) for any bigon Σ ∈ B 2 X . In o rder to understand wh y w e call this iden tit y Stok es’ Theorem, notice that if the functor F is smo oth, also its deriv ativ e 2- functor d F is smo o t h. Then, w e ha v e asso ciated diﬀeren tial forms: Lemma 3.5. L et A ∈ Ω 1 ( X , g ) b e the 1-form as s o ciate d to the sm o oth functor F , and let B ∈ Ω 2 ( X , g ) b e the 2-form asso ciate d to its derivative 2- functor d F by The or em 2.21. Then, B = [ A ∧ A ] + d A . Pro of. W e recall that there is also the 1-for m A ′ ∈ Ω 1 ( X , g ) asso ciated to the 2-functor d F , and that by Prop osition 2 .8 B = [ A ′ ∧ A ′ ] + d A ′ , since t is the iden tity in the crossed mo dule that deﬁnes E G . F urthermore, since d F ( γ ) = F ( γ ) , w e ha v e A = A ′ .  W e ha v e review ed in Section 3.1 that a smo oth functor F : P 1 ( X ) / / B G correspo nds t o trivial principal G - bundle P with connection ω in the sense that tra P ,ω = i G ◦ F . By Lemma 3.5, the 2- form B determined by the 2- functor d F is the curv ature of this connection ω . Moreov er, the holonomy 47 of ω around an y closed path γ (iden tiﬁed with a group elemen t) is giv en b y F ( γ ) . If γ is of the form γ = ∂ Σ for an y bigon Σ , Theorem 3.4 implies Hol ∇ ( ∂ Σ) = F ( ∂ Σ) = d F (Σ) ; this is a relation b et w een the holonomy and the curvatur e of a connection on a (trivial) principal G -bundle. F urther restricted to the case that the bigo n Σ is of the form Σ : id x + 3 γ for a closed path γ : x / / x , w e hav e Corollary 3.6. L et ω b e a c onne c tion on a trivial princip al G -b und le of curvatur e K , and let γ b e a c ontr actible lo op at x ∈ X . Then, Hol ω ( γ ) = P exp Z 1 0 A Σ = P exp Z 1 0 d s  Z 1 0 d t Ad − 1 τ ( γ s,t ) K | Σ( s,t )  ∂ Σ ∂ s , ∂ Σ ∂ t  wher e Σ : id x + 3 γ is any choic e of a smo oth c ontr action of γ to its b ase p oint, the gr oup element τ ( γ s,t ) ∈ G is the p ar al lel tr ansp ort of the c onne ction ω alo n g the p a th γ s,t , and the p ath-or der e d ex p onential P exp indic ates the unique solution of the r esp e ctive initial value pr oblem , like in (1.9). Exactly the same for mula can b een found in [AF G 99], deriv ed by com- pletely diﬀeren t metho ds. In the ab elian case o f G = U (1) Corollary 3.6 b oils do wn to the w ell-kno wn iden tity Hol ω ( γ ) = exp  i Z Σ K  for a surface Σ with b oundary γ = ∂ Σ . 3.3 Classical Soluti ons i n BF-Theory F our-dimensional BF theory is a top ological ﬁeld theory on a four- dimensional, compact, oriente d smo oth manifold X , see e.g. [Bae96]. Usu- ally , it in v olves a symmetric, non-degenerate inv aria nt bilinear form h− , −i on the Lie algebra g of a Lie gr o up G , and the ﬁelds are pairs ( A, B ) of a 1-form A ∈ Ω 1 ( X , g ) and a 2-form B ∈ Ω 2 ( X , g ) . W e infer that a naturally generalized setup in whic h BF theory should b e considered, is a Lie 2-group G , i.e. a smoot h crossed mo dule ( G, H , t, α ) , together with the inv ariant form h− , −i on the L ie algebra of G . Other generalizations hav e b een prop osed in [GPP08]. The ﬁelds are no w pairs ( A, B ) of a 1-form A ∈ Ω 1 ( X , g ) and a 2-form B ∈ Ω 2 ( X , h ) , and the action is, with F A := d A + [ A ∧ A ] and β A,B := F A − t ∗ B , S ( A, B ) := 1 2 Z X h β A,B ∧ β A,B i . (3.2) 48 Express ed in terms of A and B , this is S ( A, B ) = 1 2 Z X h F A ∧ F A i − Z X h t ∗ B ∧ F A i + 1 2 Z X h t ∗ B ∧ t ∗ B i ; (3.3) these terms can b e iden tiﬁed a s: a top ological Y ang-Mills term, the “ original” BF-term and a so-called cosmological term. The v ariation of this a ction giv es δ S δ A = 0 ⇔ t ∗ d B + A ∧ t ∗ B = 0 δ S δ B = 0 ⇔ β A,B = 0 . W e notice that the second equation implies the ﬁrst, so that the critical p oin t of S ( A, B ) are exactly those with β A,B = 0 . It follo ws further that the top ological Y ang-Mills term, whic h is usually not presen t in BF theory , has no inﬂuence on the critical p oin ts. Since pairs ( A, B ) with β A,B = 0 correspo nd b y Theorem 2.21 to smo oth 2-functors F : P 2 ( X ) / / B G , w e ha v e Prop osition 3.7. The critic al p oin ts o f the BF action (3.2) ar e exa c tly the smo oth 2-functors F : P 2 ( X ) / / B G . 4 T ransgression to Lo op Spaces In this section w e use the observ ation that 2-functors deﬁned on the path 2- group oid P 2 ( X ) of a sm o oth manifold X induce struc ture on the loop space LX := C ∞ ( S 1 , X ) , since lo ops are particular 1-morphisms in P 2 ( X ) . In order to understand this structure prop erly , we equip LX with the canonical diﬀeology of LX = D ∞ ( S 1 , X ) , see Section 1.2. In Section 4.1 w e generalize the relation b et wee n smo oth functors and diﬀeren tial forms (Theorem 1.6) from smo oth manifolds X to arbitrary diﬀeological spaces, in particular to LX . In Section 4.2 w e com bine this generalized statemen t on LX with Theorem 2.21 on X . 4.1 Generalization to Diﬀeologi cal Spaces In order to describe a generalization of The orem 1.6 from smo oth manifolds to diﬀeological spaces, w e ﬁrst ha v e to deﬁne the path group oid P 1 ( X ) of a diﬀeological space X . W e will see that almost all deﬁnitions w e ga ve for X a smo oth manifold pass through; only the notion of thin homotop y has to b e adapted. 49 So, a p ath in X is a diﬀeological map γ : [0 , 1] / / X with sitting instan ts. As describ ed in Section 1.2 the set P X of paths can b e considered as a subse t of the diﬀeological space D ∞ ((0 , 1) , X ) , and is hence itself a diﬀeological space. By axiom (D2) fo r diﬀeological spaces, the constan t path id x at a p oin t x ∈ X is diﬀeological. Let us further sho w exemplarily Lemma 4.1. The c omp o sition γ 2 ◦ γ 1 of two p aths γ 1 : x / / y and γ 2 : y / / z is a gain a p ath. Pro of. Notice that if γ : [0 , 1] / / X is a path a nd U ⊂ [0 , 1] is op en, then γ | U : U / / X is a plot of X . T o see that the composition γ 2 ◦ γ 1 (whic h is deﬁned in the same w a y a s for smo o th manifolds) is diﬀeological, let U ⊂ [0 , 1] b e op en, let ǫ i b e a sitting instan t of γ i , and let U 1 := U ∩ (0 , 1 2 ) , U 2 := U ∩ ( 1 2 − ǫ 1 , 1 2 + ǫ 2 ) and U 3 := U ∩ ( 1 2 , 1) . These are op en sets that co v er U , furtherm ore, ( γ 2 ◦ γ 1 ) | U i = γ i | U i for i = 1 , 3 are plots of X and ( γ 2 ◦ γ 1 ) | U 2 is constan t a nd hence also a plot of X by axiom (D2). Hence, ( γ 2 ◦ γ 1 ) | U is a plot of X b y axiom (D3).  W e leav e it to the reader to prov e that the inv erse γ − 1 of a path γ is aga in a path. Next w e ha v e to deﬁne thin homotop y for paths in a diﬀeological space. F or this purp ose, w e ﬁrst giv e a reform ulation of a thin homotopy o n smo oth manifolds, whic h generalizes b etter to diﬀeological spaces. Lemma 4.2. L et X and Y b e sm o oth man ifolds and f : X / / Y b e a s mo oth map. The r ank of the diﬀer ential of f is b ounde d a b ove by a numb er k ∈ N if and only if the pul lb ack of every ( k + 1) -form ω ∈ Ω k +1 ( Y ) along f vanishe s. Pro of. Assu me that the rank of the diﬀeren tial of f is at most k ev erywhere. Then, f ∗ ω = 0 f or all ω ∈ Ω k +1 ( Y ) . Con vers ely , assume that f ∗ ω = 0 for all ω ∈ Ω k +1 ( Y ) . Assume further that there exists a p o int p ∈ X suc h that d f | p has rank k ′ > k . Then, there exist v ectors v 1 , ..., v k ′ ∈ T p X suc h that their images w i := d f | p ( v i ) are linear indep enden t. Using a chart of a neigh b ourho o d of f ( p ) one can construct a k ′ -form ω ∈ Ω k ′ ( Y ) such that ω f ( p ) ( w 1 , ..., w k ′ ) is non- zero. Since this is equal to ( f ∗ ω ) p ( v 1 , ..., v k ′ ) , w e ha v e a con tradiction to the assumption that f ∗ ω = 0 .  W e th us ha v e reform ulated restrictions on the rank of the diﬀeren tial o f a smo oth function in terms of pullbac ks of diﬀeren tial forms. Now w e generalize to diﬀeological spaces. 50 Deﬁnition 4.3. L et X b e a di ﬀ e olo gic al sp a c e. A diﬀer en tial k -form o n X is a family of k -forms ω c ∈ Ω k ( U ) for every plot c : U / / X , such that ω c 1 = f ∗ ω c 2 for every smo oth map f : U 1 / / U 2 with c 2 ◦ f = c 1 . Notice that the k - forms on a diﬀeological space X form a v ector space Ω k ( X ) , and that the w edge pro duct a nd the exterior deriv ativ e generalize naturally to diﬀeren tial forms o n diﬀeological spaces . F urthermore, it is clear that a diﬀeren tial form ω on a smooth manifold X induces a diﬀeren tial form on X regarded as a diﬀeological space: for a c hart φ : U / / X of X one tak es ω φ := φ ∗ ω . W e ha v e also a v ery simple deﬁnition of pullback s of diﬀeren t ia l forms on diﬀeological spaces along diﬀeological maps f : X / / Y b et we en diﬀeological spaces X and Y : the pullbac k f ∗ ω o f a k -form ω = { ω c } on Y is the k -form on X deﬁned by ( f ∗ ω ) c := ω f ◦ c for eve ry plot c of X . Here it is imp ortant that f ◦ c , since f w a s supp osed to b e diﬀeological, is a plot of Y . In particular, if Y is a smo ot h manifold, f ◦ c : U / / Y is a smo oth map and ( f ∗ ω ) c = ( f ◦ c ) ∗ ω . Deﬁnition 4.4. Two p aths γ 0 : x / / y and γ 1 : x / / y in a diﬀ e olo gic al sp ac e X ar e c al le d thin h o motopy e quivalent, if ther e exists a diﬀe olo gic al map h : [0 , 1] 2 / / X with s i tting instants as describ e d in (1) of Deﬁnition 1.1, such that the pul lb ack h ∗ ω of every 2-form ω ∈ Ω 2 ( X ) vani shes. By Lemma 4.2 it is clear that for X a smo ot h manifold Deﬁnition 4.4 is equiv alen t to Deﬁnition 1.1. By argumen ts similar to those giv en in the pro of of Lemma 4.1 one can sho w that Deﬁnition 4.4 deﬁnes an equiv alence relation ∼ 1 on the diﬀeological space P X of paths in X , so that the set of equiv a lence classes P 1 X := P X/ ∼ 1 is again a diﬀeological space. This will b e the set of morphisms of the path group o id P 1 ( X ) w e are going to deﬁne. In t he follo wing lem ma w e pro v e that the axioms of a groupo id are satisﬁed. Lemma 4.5. L et X b e a diﬀ e olo gic al s p ac e. F or a p ath γ : x / / y we h a ve γ − 1 ◦ γ ∼ 1 id x and id y ◦ γ ∼ 1 γ ∼ 1 γ ◦ id x . F o r thr e e p aths γ 1 : x / / y , γ 2 : y / / z and γ 3 : z / / w we have γ 1 ◦ ( γ 2 ◦ γ 3 ) ∼ 1 ( γ 1 ◦ γ 2 ) ◦ γ 3 . 51 Pro of. W e prov e γ − 1 ◦ γ ∼ 1 id x ; the remaining equiv alenc es can b e sho wn analogously . W e c ho ose the standard homotop y: this is, for some smo oth map β : [0 , 1] / / [0 , 1] with β (0) = 0 and β (1) = 1 and with sitting instan ts, the map h : [0 , 1] 2 / / X : ( s, t )  / / ( γ (2 β ( s ) t ) 0 ≤ t ≤ 1 2 γ (2 β ( s )( 1 − t )) 1 2 < t ≤ 1 . This map has sitting ins tan ts. T o se e that it is diﬀeological, w e use the same tric k as in the pro of of Lemma 4.1, i.e. we co v er (0 , 1) 2 with V γ := (0 , 1) × (0 , 1 2 ) , V γ − 1 := (0 , 1) × ( 1 2 , 1) and V ǫ := (0 , 1) × ( 1 2 − ǫ, 1 2 + ǫ ) for ǫ a sitting instan t γ , and accordingly an y op en subset U ⊂ [0 , 1] 2 b y V γ ∩ U , V γ − 1 ∩ U and V ǫ ∩ U . No w, h | V γ ∩ U = ( γ ◦ m β ) | V γ ∩ U (4.1) with m β ( s, t ) := 2 β ( s ) t ; it is th us the comp osition of a plot with a smo oth map and hence by axiom (D1) a plot. Similarly h | V γ − 1 ∩ U and h V ǫ ∩ U are plots. This sho ws that h | U is co v ered b y plots a nd th us itself a plot. This implies that h is diﬀeological. It remains to show that the pullback h ∗ ω of ev ery 2 -form ω ∈ Ω 2 ( X ) v anishes. This follo ws fro m the fact that h restricted to eac h of the subse ts V γ , V γ − 1 and V ǫ is either constan t or factors as in (4.1) through the one-dimensional manifold [0 , 1] via γ or γ − 1 , resp ectiv ely .  This ﬁnishes the deﬁnitions of the path group oid P 1 ( X ) of a diﬀeological space X . It is clear that one no w can consider smo oth functors F : P 1 ( X ) / / S in to an y Lie category S lik e b efore: the maps F 0 : X / / S 0 on ob jects a nd F 1 : P 1 X / / S 1 on morphisms ha v e to b e diﬀeological maps. F urther tow ards a generalization of Theorem 1.6 w e hav e to generalize the category Z 1 X ( G ) ∞ in tro duced in Deﬁ nition 2.12 from a smo o th manifold X to a diﬀeological space. Notice that Deﬁnition 4 .3 extends naturally to g -v alued diﬀeren tial for ms on diﬀeological spaces. No w, for a diﬀeological space X an ob ject in Z 1 X ( G ) ∞ is a g -v alued 1-form A = { A c } on X . A morphism g : A / / A ′ is a diﬀeological map g : X / / G suc h that for any plot c : U / / X and the asso ciated smo oth map g c := g ◦ c : U / / X A ′ c = Ad g c ( A c ) − g ∗ c ¯ θ . (4.2) 52 The functor D from Section 1.3 generalizes straightforw ardly to a functor D : F unct ∞ ( P 1 ( X ) , B G ) / / Z 1 X ( G ) ∞ for an y diﬀeological space X : • Let F : P 1 ( X ) / / B G b e a smo oth functor. F or an y plot c : U / / X of X (whic h is its elf a diﬀeological map), the pullbac k c ∗ F is a smo oth functor c ∗ F : P 1 ( U ) / / B G deﬁned on the path group oid of the smo ot h manifold U . Hence A c := D ( c ∗ F ) ∈ Ω 1 ( U, g ) is a 1-for m. If c ′ : U ′ / / X is another plot a nd f : U / / U ′ is a smo oth map with c = c ′ ◦ f , w e hav e b y Prop osition 1.7 A c = D ( c ∗ F ) = f ∗ D ( c ′∗ F ) = f ∗ A c ′ . • Let ρ : F / / F ′ b e a smo oth na t ura l transformation. Its comp o nents furnish a diﬀeological map g : X / / G . F or any plot c : U / / X , w e ha v e g c := ρ ◦ c = D ( c ∗ ρ ) : U / / X , hence, since D is a functor, (4.2) is satisﬁed. The extension of the in vers e functor P to diﬀeological spaces is sligh tly more in volv ed. Let A = { A c } ∈ Ω 1 ( X , g ) b e a 1- f orm on the diﬀeological space X . F or eve ry plot c : U / / X w e obta in a smo o t h functor F c := P ( A c ) . In particular, since eve ry path γ : [0 , 1] / / X deﬁnes a plot γ | (0 , 1) , w e ha v e functors F γ deﬁned on the path g roup oid of the op en interv al (0 , 1) . Let ǫ s,t ∈ P 1 ((0 , 1)) b e the path in (0 , 1) that go es from s + ǫ to t − ǫ , where ǫ is a sitting instan t of γ . Then, we deﬁne a map F : P 1 X / / G : γ  / / F γ ( ǫ 0 , 1 ) . Lemma 4.6. Th is deﬁn es a s m o oth functor F : P 1 ( X ) / / B G . Pro of. T o see that F : P 1 X / / G is diﬀeological, w e hav e to sho w tha t for ev ery plot c : U / / P 1 X the comp osite F ◦ c : U / / G is a smo oth map. Since w e can c hec k smo othness lo cally , w e may assume that c = pr ◦ c ′ for a plot c ′ : U / / P X and the pro jection pr : P X / / P 1 X . The relev an t ev aluation map ˜ c : U × (0 , 1) / / X giv en b y U × (0 , 1) c ′ × id / / P X × (0 , 1) ev / / X is a plot of X . Hence, w e ha ve a smo oth functor F ˜ c : P 1 ( U × (0 , 1)) / / B G . With the map i u : (0 , 1) / / U × (0 , 1) : t  / / ( u, t ) w e ha v e a plot ˜ c ◦ i u and accordingly A c ′ ( u ) = i ∗ u A ˜ c for all u ∈ U . Then, b y Prop osition 1.7, F ( c ( u )) = i ∗ u F ˜ c ( ǫ 0 , 1 ) = F ˜ c (( i u ) ∗ ǫ 0 , 1 ) . 53 Since U / / P 1 ( U × (0 , 1 )) : u  / / ( i u ) ∗ ǫ 1 , 2 is a diﬀeological map, and F ˜ c is diﬀeological, w e ha v e sho wn that F ◦ c is smo oth. The compatibilit y of F with the comp osition of paths follows from F γ ′ ◦ γ ( ǫ 0 , 1 ) = F γ ′ ◦ γ ( ǫ 1 / 2 , 1 ) · F γ ′ ◦ γ ( ǫ 0 , 1 / 2 ) = F γ ′ ( ǫ 0 , 1 ) · F γ ( ǫ 0 , 1 ) . (4.3) F or the last step of (4 .3 ), w e show the equalit y F γ ( ǫ 0 , 1 ) = F γ ′ ◦ γ ( ǫ 0 , 1 / 2 ) ; the one for γ ′ go es analog o usly . Indeed, consi der the inclusion ι 1 : [0 , 1] / / [0 , 1] deﬁned b y ι 1 ( t ) := 1 2 t . Pulling bac k A γ ′ ◦ γ along ι 1 and using Prop osition 1.7 w e get F γ = ι ∗ 1 F γ ′ ◦ γ . Ev aluating this on t he path ǫ 0 , 1 , and using that ( ι 1 ) ∗ ( ǫ 0 , 1 ) = ǫ 0 , 1 / 2 in P 1 ((0 , 1)) sho ws the claim.  No w the followin g theorem f ollo ws from Theorem 1.6 applied to func tors and forms on the co domain U of each plot c : U / / X of X . Theorem 4.7. L et X b e a diﬀe olo gi c al sp ac e and G a Lie gr oup. The func- tors D : F unct ∞ ( P 1 ( X ) , B G ) / / Z 1 X ( G ) ∞ and P : Z 1 X ( G ) ∞ / / F unct ∞ ( P 1 ( X ) , B G ) satisfy D ◦ P = id Z 1 X ( G ) ∞ and P ◦ D = id F unct( P 1 ( X ) , B G ) , and ar e henc e isomorphis ms of c ate gories. 4.2 Induced Structure on the Lo op Spa ce In this section w e discuss the diﬀeological space LX = D ∞ ( S 1 , X ) , where X is a smo oth manifold. In order to formalize the relation b et we en functors deﬁned on the path group oid P 1 ( X ) a nd structure on LX w e are going to explore w e in tro duce t w o constructions. Firstly , w e denote for any category T b y Λ T := T 1 set of morphisms in T . A ccordingly , for a functor F : S / / T , we call its induced map on morphisms Λ F : Λ S / / Λ T . Clearly , if F was a diﬀeological functor, Λ F is a diﬀeological map. Secondly , we in tro duce a diﬀeological map ℓ : LX / / Λ P 1 ( X ) . (4.4) Its deﬁnition is not completely ob vious since lo ops ha v e no sitting instants. W e ﬁx some smooth map β : [0 , 1] / / [0 , 1] with β (0) = 0 and β (1) = 1 a nd 54 with sitting instan ts. W e ha v e a smo oth map e β : [0 , 1] / / S 1 deﬁned b y e β ( t ) := e 2 π i β ( t ) and accordingly a diﬀeological map ℓ β : LX / / P X : τ  / / τ ◦ e β . W e deﬁne ℓ := pr ◦ ℓ β , where pr : P X / / P 1 X is the pro jection to thin homotop y classes. This map ℓ is diﬀeological and indeed independen t of the c hoice of β : for another ch oice β ′ and some τ ∈ LX w e ﬁnd a thin homoto p y ℓ β ( τ ) ∼ 1 ℓ β ′ ( τ ) for example by h : [0 , 1] 2 / / X : ( s, t )  / / τ  e 2 π i( β ( s ) β ′ ( t )+(1 − β ( s )) β ( t ))  ; (4.5) this map is diﬀeological, has sitting instan ts and is eviden tly thin, since it factors through S 1 . No w, hav ing the t w o deﬁnitions Λ and ℓ at hand, fo r F : P 1 ( X ) / / T a smo oth functor, Λ F ◦ ℓ : LX / / Λ T (4.6) is a diﬀeological map on the lo op space. A particular situation arises if the category T = B G f o r a Lie group G . In this case Λ B G = G . W e hav e no w obtained a map H 1 :  Smo oth functors F : P 1 ( X ) → B G  / / D ∞ ( LX, G ) . ( 4 .7) This map is o f course w ell-know n: as mentione d in Section 3.1, a smo oth functor F : P 1 ( X ) / / B G corresp onds to a (trivial) principal G - bundle P with connection ω ov er X , in suc h a w ay t hat the parallel transp ort along a path γ in X is g iven b y m ultiplication with F ( γ ) . F or a lo op τ ∈ LX , understoo d as a path ℓ ( τ ) , this means H 1 ( F )( τ ) = F ( ℓ ( τ )) = Hol ω ( τ ) , so that H 1 ( F ) is nothing but the holonom y o f the connection ω around γ . In t he follo wing w e explore whic h structure on the lo op space LX is induced fr o m a smo oth 2- f unctor F : P 2 ( X ) / / B G . T o start with, w e generalize the t w o constructions Λ and ℓ w e ha ve describ ed b efore, to 2- categories. Deﬁnition 4.8. L et T b e a 2 -c ate gory. W e deﬁne a c ate gory Λ T as fol lows: the obje cts ar e the 1-morphis m s T 1 of T , and the morphisms b etwe en two 55 obje cts f : X f / / Y f and g : X g / / Y g ar e triples ( x, y , ϕ ) ∈ T 1 × T 1 × T 2 of 1-morphisms x : X f / / X g and y : Y f / / Y g and of a 2-morphis m X f f   x / / X g g   ϕ } } } } } } } } } } z  } } } } } } } } Y f y / / Y g . The c om p osition in Λ T is putting these squar es next to e ac h o ther, and the identity of an obje ct f : X / / Y is the triple (id X , id Y , id f ) . Clearly , if the se ts T 1 and T 2 of the 2-category T are diﬀeological spaces, the o b jects and morphisms of Λ T fo r m also diﬀeological spaces. F o r F : S / / T a 2-functor, w e ha v e an asso ciated functor Λ F : Λ S / / Λ T , whic h just acts as F o n 1-morphisms and 2- morphisms of S . If the 2-functor F is diﬀeological, the f unctor Λ F is also diﬀeological. Next w e gene ralize the diﬀeological map ℓ in tro duced ab ov e to a diﬀeological functor ℓ : P 1 ( LX ) / / Λ P 2 ( X ) . On ob jects, it is just the map ℓ from (4.4), regarding a lo o p τ ∈ LX as a particular path in X , i.e. as an ob ject in Λ P 2 ( X ) . T o deﬁne ℓ on morphisms, let γ b e a path in LX , i.e. a diﬀeological map γ : [0 , 1] / / D ∞ ( S 1 , X ) with sitting instan ts. W e ha v e an asso ciated smo oth map m γ : R 2 / / X deﬁned b y m γ ( s, t ) := γ ( t )(e 2 π i s ) , where w e assume γ to b e trivially extended to R in the usual w ay . Using the standard bigo n Σ R ( s, t ) ∈ B 2 R 2 from (2.5), w e ha v e a bigon m ( γ ) := m γ ∗ (Σ R (1 , 1)) ∈ B 2 X (4.8) asso ciated to the path γ , and thus a w ell-deﬁned map m : P LX / / B 2 X . Lemma 4.9. Th e map m : P L X / / B 2 X is diﬀe olo gic al . Pro of. W e ha ve to sho w that for a ny plot c : U / / P LX the comp osite m ◦ c is a plot of B 2 X . This means that, for a ﬁxed represen tative Σ ∈ B X of Σ R (1 , 1) , and an y op en subset W ⊂ [0 , 1] 2 , the asso ciated map U × W c × id / / P LX × W m ∗ Σ × id / / B X × W ev / / X (4.9) has to b e smo o th. Let us deﬁne the op en in terv als V := p 2 (Σ( W )) and V ′ := p 1 (Σ( W )) for p i : [0 , 1] 2 / / [0 , 1] the canonical pro j ections, and consider the 56 c hart ϕ : V ′ / / S 1 : s  / / e 2 π i s of S 1 . Going through all inv olv ed deﬁnitions sho ws that the map (4.9) coincides with the comp osite U × W id × Σ / / U × V × V ′ c ′ / / X (4.10) where c ′ is giv en b y U × V × V ′ c × id × id / / D ∞ ([0 , 1] , X ) × V × V ′ ev × ϕ / / LX × S 1 ev / / X No w, (4.10) is the comp osition of tw o smo oth maps, where c ′ is smo oth b ecause c w as supp osed to b e a plot of P LX ⊂ D ∞ ([0 , 1] , X ) .  W e sho w next that the bigon m ( γ ) ∈ B 2 X do es not depend on the thin homotop y class of the pa th γ . F or this purp ose, let h : [0 , 1] / / LX b e a thin homotopy b et w een t w o paths γ , γ ′ ∈ P LX , and let Σ ∈ B X b e a represen tativ e for the bigon Σ R (1 , 1) . W e hav e an asso ciated map m h : [0 , 1] 3 / / X deﬁned b y m h ( r , s, t ) := h ( r, t )(e 2 π i s ) . Then, the map H : [0 , 1] 3 / / X : ( r, s, t )  / / m h ( r , Σ( s, t )) is a thin homotop y b etw een m γ ∗ Σ and m γ ′ ∗ (Σ) . Hence, w e ha v e o bta ined a diﬀeological map m : P 1 LX / / B 2 X . Going through the deﬁnitions, o ne ﬁnds that this bigon m ( γ ) has ( up to thin homoto p y) the follo wing target and source paths: γ (0)(1) ℓ ( γ (0))   pr( b ◦ γ ) / / γ (1)(1) ℓ ( γ (1))   m ( γ ) q q q q q q q q q q q q t | q q q q q q q q q q γ (0)(1) pr( b ◦ γ ) / / γ (1)(1) , (4.11) where b : LX / / X is the base p oint ev aluation. Hence, the triple ℓ ( γ ) := (pr( b ◦ γ ) , pr( b ◦ γ ) , m ( γ )) is a morphism in Λ P 2 ( X ) . The comp osition of paths in LX is r esp ected in the sense that ℓ ( γ 2 ◦ γ 1 ) = ℓ ( γ 2 ) ◦ ℓ ( γ 1 ) where the latter is the comp osition in Λ P 2 ( X ) . Thu s, we ha v e completely deﬁned the diﬀeolog ical functor ℓ : P 1 ( LX ) / / Λ P 2 ( X ) . If no w F : P 2 ( X ) / / T is a smo ot h 2-functor, w e obtain an asso ciated smo oth functor Λ F ◦ ℓ : P 1 ( LX ) / / Λ T , 57 generalizing the map (4 .6 ) . In the imp orta n t case that T = B G for G a Lie 2-group, the group oid Λ B G is – follow ing a notion of [M ac87] – trivializable: a group oid is called trivializable , if it is equiv alen t to a group oid of the form Gr S,N = S × B N for a set S regarded as a category with only iden tit y morphisms, and a group N ; these group oids are called trivial . Explicitly , the ob jects of Gr S,N are the elemen ts of S , the Hom-set b et w een tw o ob jects s 1 and s 2 is N if s 1 = s 2 and empt y else, and the comp osition is m ultiplication in N . In our case w e ﬁnd suc h an equiv alence tr : Λ B G / / Gr G,G ⋉ H as follo ws: on ob jects, tr is just the iden tity on G . A morphism ∗ g 1   x / / ∗ g 2   h           {          ∗ y / / ∗ . in Λ B G is sen t to the morphism ( y , h − 1 ) in Gr G,G ⋉ H , the in v erse b eing nece s- sary in order to respect the comp osition. The functor tr : Λ B G / / Gr G,G ⋉ H is an equiv alence of categories: the inv erse functor sends a morphism ( y , h ) : g 1 / / g 2 in Gr G,G ⋉ H to the triple ( g − 1 2 t ( h ) y g 1 , y , h − 1 ) . No w, w e ha v e constructed a smo oth functor tr ◦ Λ F ◦ ℓ : P 1 ( LX ) / / Gr G,G ⋉ H . A ccording to the direct pro duct structure of the target Lie group o id, this functor splits in 1. a diﬀeological function h F : LX / / G and 2. a smo oth functor P 1 ( LX ) / / B ( G ⋉ H ) , wh ic h in turn corresp onds b y Theorem 4.7 and pro jection to the factors to (a) a 1-fo rm A F ∈ Ω 1 ( LX, g ) and (b) a 1-form ϕ F ∈ Ω 1 ( LX, h ) . Summarizing, w e hav e, f o r an y smo oth manifold X and an y Lie 2 -group G , a map H 2 :  Smo oth 2-functors F : P 2 ( X ) → B G  / / D ∞ ( LX, G ) × Ω 1 ( LX, g ) × Ω 1 ( LX, h ) . This map generalizes the map H 1 from (4.7) from smo oth functors to smo oth 2-functors. Let us describe the image H 2 ( F ) = ( h F , A F , ϕ F ) . The diﬀeolog- ical function h F : LX / / G is clearly h F = H 1 ( F 0 , 1 ) for the restriction F 0 , 1 of F to ob jects and 1-morphisms. The diﬀeren tial forms A F and ϕ F can b e c haracterized as described in the follow ing prop osition, also see Figure 2. 58 Prop osition 4.10. L et F : P 2 ( X ) / / B G b e a smo oth 2-functor, let A ∈ Ω 1 ( X , g ) and B ∈ Ω 2 ( X , h ) the c o rr esp onding d iﬀer ential forms on X , and let A F and B F the diﬀer ential forms on the lo op sp ac e determine d by H 2 ( F ) . Then A F = b ∗ A an d B F = Z S 1 ( α F ◦ γ ) ∗ (ev ∗ B ) , wher e b : LX / / X is the pr oje ction to the b ase p oint, ev : LX × S 1 / / X is the evaluation m a p, γ : LX × S 1 / / P 1 X ass i g ns to a lo op τ ∈ LX an d z ∈ S 1 the p ath obtaine d by p arsing the lo op τ fr o m z to 1 c ounter clo ckwise, and α F ◦ γ : H / / H i s the action o f the cr o s s e d mo dule G alo ng the m ap F ◦ γ : LX × S 1 / / G . Pro of. Let c : U / / LX b e a plot of LX and let Γ : R / / U b e a sm o oth curv e. Using all in v olved deﬁnitions w e obtain ( A F ) c | Γ(0)  ∂ Γ ∂ t     0  = − d d t     0 ( p G ◦ c ∗ (tr ◦ Λ F ◦ ℓ ) ◦ Γ ∗ ◦ γ R )(0 , t ) ( b ∗ A ) c | Γ(0)  ∂ Γ ∂ t     0  = − d d t     0 (( b ◦ c ) ∗ F ◦ Γ ∗ ◦ γ R )(0 , t ) . Then one observ es that F ◦ b ∗ = p G ◦ tr ◦ Λ F ◦ ℓ as maps from P 1 LX to G ; this sho ws the ﬁrst equalit y . In o rder to pro of the second equalit y we still use the plot c and the smo oth curv e Γ , and consider the path γ t := c ∗ (Γ ∗ ( γ R (0 , t ))) ∈ P 1 LX , where γ R ( s, t ) ∈ P 1 R is the standard path from s to t . W e ha v e ( ϕ F ) c | Γ(0)  ∂ Γ ∂ t     0  = − d d t     0 ( p H ◦ c ∗ (tr ◦ Λ F ◦ ℓ ) ◦ Γ ∗ ◦ γ R )(0 , t ) = − d d t     0 p H ( F ( m ( γ t ))) − 1 = d d t     0 p H ( F ( m γ t ∗ Σ R (1 , 1))) , where w e ha ve used the deﬁnitions of the functor ℓ and the map m f r o m (4.8). Let us remark that for the bigo n Σ Γ ( s, t ) := m γ 1 ∗ Σ R ( s, t ) w e hav e Σ Γ (1 , t ) = m γ t ∗ Σ R (1 , 1) , so that w e ma y write d d t     0 p H ( F ( m γ t ∗ Σ R (1 , 1))) = Z 1 0 d θ ∂ 2 ∂ ρ∂ t     0 p H ( F (Σ Γ 1 − θ − ρ ( θ + ρ, t ))) . (4.12) A b y no w standard calculation sho ws that p H ( F (Σ Γ 1 − θ − ρ ( θ + ρ, t ))) = p H ( F (Σ Γ 1 − θ ( θ , t ))) · α ( F ( γ ( c (Γ( t )) , e − 2 π i θ )) , F (Σ Γ 1 − θ − ρ ( ρ, t ))) 59 where we ha v e used the map γ : LX × S 1 / / P 1 X . No w the deriv ativ e in (4.12) b ecomes ∂ 2 ∂ ρ∂ t     0 p H ( F (Σ Γ 1 − θ − ρ ( θ + ρ, t ))) = − ( α F ( γ ( c (Γ( t )) , e − 2 π i θ ) ) ∗  ∂ 2 ∂ ρ∂ t     0 F (Σ Γ 1 − θ ( ρ, t ))  . Let us no w induce from the plot c : U / / LX of LX a plot of LX × S 1 , namely the map ˜ c : U × (0 , 1) / / LX × S 1 : ( u, θ )  / / ( c ( u ) , e 2 π i θ ) . W e ha v e m γ 1 (1 − θ, 0) = (ev ◦ ˜ c )(Γ(0) , − θ ) ∈ X and the tangen t ve ctors d d t     0 m γ 1 (1 − θ, t ) = d(ev ◦ ˜ c ) | (Γ(0) , − θ )  ∂ Γ ∂ t     0  d d ρ     1 − θ m γ 1 ( ρ, 0) = d(ev ◦ ˜ c ) | (Γ(0) , − θ )  ∂ ∂ θ  . By Prop osition 2.14 w e hence hav e ∂ 2 ∂ ρ∂ t     0 F (Σ Γ 1 − θ ( ρ, t )) = − (ev ∗ B ) ˜ c | (Γ(0) , − θ )  ∂ ∂ θ , ∂ Γ ∂ t     0  . Putting all pieces together and transforming θ  / / − θ , w e ha v e sho wn ( ϕ F ) c | Γ(0)  ∂ Γ ∂ t     0  = Z 0 − 1 d θ ( α ( F ◦ γ )( ˜ c (Γ(0) ,θ )) ) ∗ (ev ∗ B ) ˜ c | (Γ(0) ,θ )  ∂ ∂ θ , ∂ Γ ∂ t     0  this is the announced ﬁbre in tegral written in the plot ˜ c of LX × S 1 .  T o conclude, let us discuss the case G = B U (1) . A smooth 2-functor F : P 2 ( X ) / / B B U (1) induces a smo o th functor tr ◦ Λ F ◦ ℓ : P 1 ( LX ) / / B U (1) , (4.13) since Λ B B U (1 ) = B U (1) ; the functor tr : B U (1) / / B U (1) just in v erts group elemen ts. The image of F under H 2 is hence j ust a 1-f orm ϕ F ∈ Ω 1 ( X ) , and this 1-for m is b y Prop osition 4.10 just the ordinary ﬁbre in tegral ϕ F = Z S 1 ev ∗ B . (4.14) Let us now in terpret the 2-f unctor F as a trivial ab elian gerb e G with con- nection ov er X (see Example 3.2 in Section 3.1), and the asso ciated functor 60 F _ Theorem 2.21    / / tr ◦ Λ F ◦ ℓ _ Theorem 4.7   ( A, B )  b ∗ × R S 1 ( α F ◦ γ ) ∗ ◦ ev ∗ / / ( A F , ϕ F ) Figure 2: A diagram for manipulations on a smo oth 2-functor F : P 2 ( X ) → B G , whose comm utativit y is Prop osition 4.10. The ﬁrst ro w consists of functors, and the second row of diﬀeren tial for ms. The ﬁrst column con tains structure on X , and the second one structure on LX . (4.13) as a trivial principal U (1) -bundle L with connection ϕ F o v er the lo op space LX , see Theorem 3.1 . Equation (4.14) sho ws that the line bundle L is the line bundle ov er the lo op space obtained b y tr ansgr ess ion from the gerb e G . T ransgression of ab elian g erb es as so far b een realized in ma ny w ays , for example in [Gaw 88, Bry93, GT01, SWB], and w e ha v e seen here that F  / / tr ◦ Λ F ◦ ℓ is just another w ay to realize transgressi on. It has one impo rtan t adv antage compared to all the ab o ve previous realizations: it w o r ks also for non-ab elian gerb es. A further discussion is p o stpo ned to the up coming article [SW A ]. App endix A.1 Basic 2-Category Theory In this art icle w e o nly consider strict 2-categories, 2-f unctors, in v erse 1- morphisms etc., in con t r a st to the general case. W e only use the qualiﬁer “ strict” in t his section and omit it elsew here. A general reference is [P ow 90]. Deﬁnition A.1. A (smal l) 2 - c ate gory c onsists of a set of obje cts, f o r e ach p air ( X , Y ) of obje c ts a set of 1 - m orphisms den o te d f : X / / Y and for e ach p air ( f , g ) o f 1-morphisms f , g : X / / Y a set of 2-morphism s denote d ϕ : f + 3 g , to gether with the fol lowi n g structur e: 1. F or e very p air ( f , g ) of 1-morphisms f : X / / Y and g : Y / / Z , a 1-morphism g ◦ f : X / / Y , c a l l e d the c omp osition o f f and g . 61 2. F or eve ry o b je ct X , a 1-morphism id X : X / / X , c al le d the identity 1-morphism of X . 3. F or every p ai r ( ϕ, ψ ) of 2-morphisms ϕ : f + 3 g an d ψ : g + 3 h , a 2-morphism ψ • ϕ : f + 3 h , c al le d the vertic al c om p osition of ϕ and ψ . 4. F or every 1-morphism f , a 2-morphism id f : f + 3 f , c al le d the iden- tity 2-morphism of f . 5. F or every triple ( X , Y , Z ) of obje cts, 1-morphisms f , f ′ : X / / Y and g , g ′ : Y / / Z , and every p ai r ( ϕ , ψ ) of 2-morphi s ms ϕ : f + 3 f ′ and ψ : g + 3 g ′ , a 2-mo rp hism ψ ◦ ϕ : g ◦ f + 3 g ′ ◦ f ′ , c al le d the horizon tal c omp osition of ϕ and ψ . This structur e has to satisfy the fol lowing list of axioms : (C1) The c omp osition of 1 - m orphisms and vertic al a n d horizontal c om p osi- tion of 2-morphisms ar e asso ciative. (C2) The i d entity 1-morphis m s ar e units with r esp e ct to the c omp osition of 1- morphisms, and iden tity 2-m orphisms ar e units with r esp e ct to ve rtic al c omp osition, i.e. ϕ • id f = id g • ϕ for every 2-morphism ϕ : f + 3 g . Horizontal c omp o sition pr eserves the identity 2-morphisms in the sense that id g ◦ id f = id g ◦ f . (C3) Horizontal and vertic al c omp o s i tions ar e c omp a tible in the sense that ( ψ 1 • ψ 2 ) ◦ ( ϕ 1 • ϕ 2 ) = ( ψ 1 ◦ ϕ 1 ) • ( ψ 2 ◦ ϕ 2 ) whenever these c omp ositions ar e wel l-deﬁn e d. The axioms of a strict 2-category allo w to use pasting diagrams for 2- morphisms: ev ery pasting diagram correspo nds to a unique 2- morphism. In a 2-category , a 2-morphism Σ : γ 1 + 3 γ 2 is called invertible o r 2-isomorphism if there exists another 2-morphism Σ − 1 : γ 2 + 3 γ 1 suc h that Σ − 1 • Σ = id γ 1 and Σ • Σ − 1 = id γ 2 . In this case, Σ − 1 is uniquely determined a nd called the inverse of Σ . A 1- morphism γ : x / / y is called strictly invertible or strict 1-isomorphism , if there exists another 1-morphism ¯ γ : y / / x suc h that id x = ¯ γ ◦ γ and γ ◦ ¯ γ = id y . A 2-category in whic h ev ery 1-morphism is strictly in v ertible is called a strict 2-gr oup oid . T o relate tw o 2-categories, w e use the following deﬁnition of a 2-functor, whic h is analogous to a functor b et w een categories. 62 Deﬁnition A.2. L et S and T b e two strict 2-c ate gories. A strict 2-functor F : S / / T is an assignm ent F : X f   g A A ϕ   Y 7− → F ( X ) F ( f ) # # F ( g ) ; ; F ( ϕ )   F ( Y ) such that (F1) The vertic al c om p osition is r esp e cte d in the sen se that F ( ψ • ϕ ) = F ( ψ ) • F ( ϕ ) and F (id f ) = id F ( f ) for al l c omp osable 2-morphisms ϕ and ψ , and any 1- m orphism f . (F2) The c omp os i tion of 1-morph i s ms is r esp e c te d in the sense that F ( g ) ◦ F ( f ) = F ( g ◦ h ) for al l c omp os able 1- m orphisms f and g , an d the horizontal c omp osi tion of 2-morphism s is r esp e cte d in the sense that F ( ψ ) ◦ F ( ϕ ) = F ( ψ ◦ ϕ ) for al l horizontal ly c omp osable 2-morphisms ϕ and ψ . T o compare 2-functors, w e use the notion of a pseudonatural transforma- tion, whic h generalizes a natural transformation b et w een functors. Deﬁnition A.3. L et F 1 and F 2 b e two strict 2-functors b oth fr om S to T . A pseudonatur al tr ansformation ρ : F 1 / / F 2 is an assignment ρ : X f / / Y 7− → F 1 ( X ) F 1 ( f ) / / ρ ( X )   F 1 ( Y ) ρ ( Y )   ρ ( f ) v v v v v v v v v v v ~ v v v v v v v v v v F 2 ( X ) F 2 ( f ) / / F 2 ( Y ) , of a 2-isomorp hism ρ ( f ) in T to e ach 1- m orphism f : X / / Y in S such that two axioms ar e satisﬁe d: 63 (T1) The c omp osi tion of 1-morph i s m s in S is r esp e cte d: F 1 ( X ) F 1 ( f ) / / ρ ( X )   F 1 ( Y ) F 1 ( g ) / / ρ ( Y )   ρ ( f ) v v v v v v v v v v w  v v v v v v v v F 1 ( Z ) ρ ( g ) w w w w w w w w w w w  w w w w w w w w ρ ( Z )   F 2 ( X ) F 2 ( f ) / / F 2 ( Y ) F 2 ( g ) / / F 2 ( Z ) = F 1 ( X ) F 1 ( g ◦ f ) / / ρ ( X )   F 1 ( Z ) ρ ( Z )   ρ ( g ◦ f ) v v v v v v v v v v v ~ v v v v v v v v v v F 2 ( X ) F 2 ( g ◦ f ) / / F 2 ( Z ) . (T2) It is c om p atible with 2-mo rphisms: F 1 ( X ) F 1 ( f ) / / ρ ( X )   F 1 ( Y ) ρ ( Y )   ρ ( f ) v v v v v v v v v v w  v v v v v v v v F 2 ( X ) F 2 ( g ) F F F 2 ( f ) / / F 2 ( Y ) F 2 ( ϕ )   = F 1 ( x ) F 1 ( f )   F 1 ( g ) / / F 1 ( ϕ )   ρ ( X )   F 1 ( Y ) ρ ( Y )   ρ ( g ) v v v v v v v v v v v ~ v v v v v v v v v v F 2 ( X ) F 2 ( g ) / / F 2 ( Y ) . It follow s that ρ (id X ) = id ρ ( X ) for ev ery ob ject X in S . Pseudonatural transformations ρ 1 : F 1 / / F 2 and ρ 2 : F 2 / / F 3 can naturally b e comp osed to a pseudonatural transformation ρ 2 ◦ ρ 1 : F 1 / / F 3 : ρ 2 ◦ ρ 1 : X f / / Y 7− → F 1 ( X ) F 1 ( f ) / / ρ 1 ( X )   F 1 ( Y ) ρ 1 ( Y )   ρ 1 ( f ) v v v v v v v v v v v ~ v v v v v v v v v v F 2 ( X ) ρ 2 ( X )   F 2 ( f ) / / F 2 ( Y ) ρ 2 ( Y )   ρ 2 ( f ) v v v v v v v v v v v ~ v v v v v v v v v v F 3 ( X ) F 3 ( f ) / / F 3 ( Y ) . (A.1) W e need one more deﬁnition for situations where w e hav e tw o pseudonatural transformations. Deﬁnition A .4. L et F 1 , F 2 : S / / T b e two strict 2-functors and let ρ 1 , ρ 2 : F 1 / / F 2 b e pseudonatur al tr ansformations. A mo diﬁc ation A : ρ 1 + 3 ρ 2 is an assignme n t A : X 7− → F 1 ( X ) ρ 1 ( X ) $ $ ρ 2 ( X ) : : A ( X )   F 2 ( Y ) 64 of a 2-morph i s m A ( X ) in T to an y o b je ct X in S which satisﬁes F 1 ( X ) ρ 2 ( X ) ( ( F 1 ( f ) / / ρ 1 ( X )   k s A ( X ) F 1 ( Y ) ρ 1 ( y )   ρ 1 ( f ) v v v v v v v v v v w  v v v v v v v v F 2 ( X ) F 2 ( f ) / / F 2 ( Y ) = F 1 ( X ) F 1 ( f ) / / ρ 2 ( X )   F 1 ( Y ) ρ 1 ( X ) v v ρ 2 ( y )   A ( Y ) k s ρ 2 ( f ) v v v v v v v v v v w  v v v v v v v v F 2 ( X ) F 2 ( f ) / / F 2 ( Y ) Horizon ta l and v ertical comp ositions of 2-morphisms in T induce accor- dan t comp ositions on mo diﬁcations. F or t w o ﬁxed strict 2- categories S and T , w e recognize the follo wing structures: 1. F or t w o strict 2-functors F 1 , F 2 : S / / T , the pseudonatural transfor- mations ρ : F 1 / / F 2 together with mo diﬁcations a nd their v ertical comp osition, form a category Hom( F 1 , F 2 ) . 2. Ev en more, stric t 2-functors from S to T , together with pseudonatural transformations and their mo diﬁcations, and the assignmen ts ◦ and • as deﬁned ab ov e, form a strict 2-category F unct( S, T ) . Deﬁnition A.5. L et S and T b e strict 2- c ate gories. Strict 2-functors F : S / / T and G : T / / S ar e c al le d isomorph isms of 2-c ate gories, if G ◦ F = id S and F ◦ G = id T . A.2 Li e 2-Groups and Smo oth Crossed Mo dul es An y strict monoidal category ( G , ⊠ , 1 ) deﬁnes a 2-category B G : it has a single ob ject, the 1-morphisms are the ob jects of G and the 2-morphisms are the morphisms of G . The horizon tal comp osition is giv en by the t ensor functor ⊠ , and t he v ertical comp osition is the comp osition in G . The iden tit y 1-morphism of the single ob ject is the tensor unit 1 , and the iden tity 2- morphism of a 1-morphism X is just the identit y morphism id X of t he ob ject X in G . The axioms for the 2-category B G follo w from the prop erties of the tensor functor ⊠ . In the follo wing, w e enhance this construction b y t w o features. First, w e assume that G is a group oid and that we ha v e an additional functor i : G / / G whic h is an in v erse to the tensor functor ⊠ in the sense that X ⊠ i ( X ) = 1 = i ( X ) ⊠ X and f ⊠ i ( f ) = id 1 = i ( f ) ⊠ f (A.2) 65 for all ob jects X and a ll morphisms f in G . In this case the 2- category B G is ev en a 2- g roup oid. Secondly , we assume that G is a Lie category , and that the functors ⊠ and i are smo oth. Then, B G is a Lie 2-group oid. Deﬁnition A.6. A Lie 2 -gr oup is a strict monoi dal Lie c ate gory ( G , ⊠ , 1 ) to gether with a smo oth functor i : G / / G such that (A.2) i s satisﬁe d. W e denote the Lie 2-group oid asso ciated t o a Lie 2-group G b y B G . An imp ortan t source of Lie 2-groups are smo oth crossed mo dules. Deﬁnition A.7. A smo oth cr osse d mo dule is a c ol le ction ( G, H , t, α ) of Lie gr oups G and H , and of a Lie gr oup homomorphi sm t : H / / G and a smo oth map α : G × H / / H , such that 1. α is a left a c tion of G on H b y Lie g r oup homomorphis ms, i. e . the smo oth map α g : H / / H d e ﬁne d b y α g ( h ) := α ( g , h ) a) is a Lie gr oup homom orphism for al l g ∈ G . b) satisﬁes α 1 = id H and α g g ′ = α g ◦ α g ′ for al l g , g ′ ∈ G . 2. α and t ar e c omp atible in the fol lowing two ways: a) t ( α ( g , h )) = g t ( h ) g − 1 for al l g ∈ G and h ∈ H . b) α ( t ( h ) , x ) = hxh − 1 for al l h, x ∈ H . An y smo oth crossed mo dule ( G, H, t, α ) deﬁne s a Lie 2-group ( G , ⊠ , 1 , i ) in the follo wing w a y . The category G : W e put Ob j( G ) := G and Mor( G ) := G ⋉ H , t he semi- direct pro duct of G and H deﬁned b y α , explicitly ( g 2 , h 2 ) · ( g 1 , h 1 ) := ( g 2 g 1 , h 2 α ( g 2 , h 1 )) . (A.3) An elemen t ( g , h ) ∈ Mor( G ) is considered as a morphism from g to t ( h ) g . The comp o sition is giv en by ( g ′ , h ′ ) ◦ ( g , h ) := ( g , h ′ h ) , (A.4) where g ′ = t ( h ) g . It is obv iously asso ciativ e, a nd the identit y mor- phisms are id g = ( g , 1 ) . All these deﬁnitions are smo oth, so that G is a Lie category . 66 The monoidal structure ( ⊠ , 1 ) : The functor ⊠ : G × G / / G is deﬁned on ob j ects b y g 2 ⊠ g 1 := g 2 g 1 and on morphism s by the pro duct (A.3). By axiom 2.a), the morphisms hav e the correct target. It resp ects iden tity morphisms, ( g 2 , 1) ⊠ ( g 1 , 1) = ( g 2 g 1 , 1) and b y axiom 2.b) the comp osition (( g ′ 2 , h ′ 2 ) ⊠ ( g ′ 1 , h ′ 1 )) ◦ (( g 2 , h 2 ) ⊠ ( g 1 , h 1 )) = (( g ′ 2 , h ′ 2 ) ◦ ( g 2 , h 2 )) ⊠ (( g ′ 1 , h ′ 1 ) ◦ ( g 1 , h 1 )) . It is also strictly asso ciativ e and the ob ject 1 := 1 ∈ G is a left and righ t unit. The functor i : The functor i : G / / G is deﬁned on o b jects by i ( g ) := g − 1 and on morphisms b y i ( g , h ) := ( g − 1 , α ( g − 1 , h − 1 )) . It resp ects sources and targets b y axiom 2.a), the iden tities and by a xioms 1.a) and 2.b) the comp osition. It is also smo oth and satisﬁes the condition (A.2). No w w e hav e completely deﬁned the Lie 2-gro up associated to a smo oth crossed mo dule. Indeed, it is a we ll-kno wn fact [BS76], also see [BL04] for a review, that ev ery Lie 2- group arises – up to a certain notion of equiv a lence – from a smo oth crossed mo dule in this w ay . Let us also write down the Lie 2 - group oid B G asso ciated the the Lie 2- group G coming from a smo oth crossed mo dule ( G, H, t, α ) . A 2-morphism is a morphism ( g , h ) ∈ Mor( G ) , denoted as ∗ g   g ′ C C h   ∗ with g ′ = t ( h ) g . (A.5) The ladder equation is also called the tar get-matching - c ondition for the 2- morphism ( g , h ) . The v ertical comp osition is according to (A.4) ∗ g ′ / / g   g ′′ E E h   h ′   ∗ = ∗ g   g ′′ C C h ′ h   ∗ (A.6) 67 with g ′ = t ( h ) g a nd g ′′ = t ( h ′ ) g ′ = t ( h ′ h ) g , and the horizontal comp osition is according to (A.3) ∗ g 1   g ′ 1 C C h 1   ∗ g 2   g ′ 2 C C h 2   ∗ = ∗ g 2 g 1   g ′ 2 g ′ 1 A A h 2 α ( g 2 ,h 1 )   ∗ (A.7) The construction of Lie 2-groups from smo oth crossed mo dules is conv e- nien t to discuss basic examples. Example A.8. Let A b e an ab elian Lie group. W e deﬁne a smooth crossed mo dule by taking G = { 1 } the trivial group and H := A . This ﬁxes the maps to t ( a ) := 1 and α (1 , a ) := a . All axioms a r e satisﬁed in a trivial manner exc ept axiom 2.b), whic h is satisﬁ ed only b ecause A is ab elian. The asso ciated Lie 2-group G is denoted by B A , and the asso ciated Lie 2 - group oid b y B B A . Example A.9. Let G b e an y Lie group. W e obtain a smoo t h crossed mo dule b y taking H := G , t = id a nd α ( g , h ) := g hg − 1 . The asso ciated Lie 2-gro up, whic h also underlies the construction of a geometric realization of E G [Seg68 ] is here denoted b y E G . It can b e inte rpreted as the in ner automorphism 2- gr oup o f G [RS08], and its Lie algebra pla ys a n imp ortan t role in [SSS09]. Let us brieﬂy exhibit the details of the asso ciated Lie 2-group oid E B G . It has one ob jects, and the set of 1-morphisms is G with the usual comp o sition g 2 ◦ g 1 := g 2 g 1 . Betw een ev ery pair ( g , g ′ ) of 1-morphisms there is a unique 2-morphism ∗ g   g ′ C C h   ∗ determined b y h := g ′ g − 1 . Example A.10. Let H b e a connected Lie g roup. The group of Lie group automorphisms of H is again a Lie group G := Aut( H ) [O V91 ]. T ogether with t ( h )( x ) := hxh − 1 and α ( ϕ, h ) := ϕ ( h ) , we ha v e deﬁned a smo oth crossed mo dule whose asso ciated Lie 2-group G is denoted by A UT( H ) . 68 A.3 Pro of of Le mma 2.1 6 In this section w e sho w that the map k A,B : B X / / H deﬁned for the construction of a smoo th 2-functor from t w o diﬀeren tial forms A ∈ Ω 1 ( x, g ) and B ∈ Ω 2 ( X , h ) , only depends on the thin homotop y class of a bigon Σ ∈ B X . W e ﬁrst start with a general homotopy h : [0 , 1] × [0 , 1 ] 2 / / X b et w een t w o bigons Σ 1 and Σ 2 , i.e h has the prop erties from Deﬁnition 2.2 except condition 2a) which constrains the rank of its diﬀeren tial. W e shall represen t the surface of the unit cub e [0 , 1] 3 on whic h h is deﬁned as a bigon in R 3 . F or this purp ose, w e deﬁne t w o paths µ ( r, s, t ) and ν ( r , s, t ) in R 3 going from 0 ∈ R 3 to ( r , s, t ) . With the notation introduced in F igure 3 these paths are (0 , 0 , 0) γ hl γ lo               (0 , s, 0) Σ l ! ) J J J J J J γ ov / / γ lv        (0 , s, t ) Σ v t | γ r v   ( r , 0 , 0) γ ul         _ _ _ γ hu / / ( r , 0 , t ) γ r u              Σ u 7 ? ( r , s, 0) h h h h h h h h h h h h h h h h h h γ uv / / ( r , s, t ) (0 , 0 , 0)     γ ho / / γ lo ~ ~ } } } } } } } } } (0 , 0 , t ) Σ o i i i i i i i i i i i i i i i i i i i i p x i i i i i i i i i i i i i i i i i i i i γ or            γ hr   (0 , s, 0) γ ov / / γ hl   5 = r r r r r r r r r r r r r r (0 , s, t ) γ r v   ( r , 0 , 0) Σ h γ hu / / _ _ _ ( r , 0 , t ) Σ r @ @ @ @ @ @ [ c @ @ @ @ @ @ @ @ γ r u            ( r , s, t ) Figure 3: The unit cub e view ed as t w o bigons: Λ 1 : µ = > ν on the right hand side , and Λ 2 : ν → µ on the left hand side. µ ( r , s, t ) := γ r u ◦ γ hu ◦ γ hl and ν ( r, s, t ) := γ vr ◦ γ ov ◦ γ lo . Bet wee n these paths w e ha ve t w o bigons Λ 1 ( r , s, t ) : µ ( r, s, t ) + 3 ν ( r, s, t ) and Λ 2 ( r , s, t ) : ν ( r , s, t ) + 3 µ ( r , s, t ) deﬁned b y Λ 1 := (id γ r v ∗ Σ o ) • (Σ r ∗ id γ ho ) • (id γ r u ∗ Σ h ) (A.8) 69 and Λ 2 := (Σ u ∗ id γ hl ) • (id γ uv ∗ Σ l ) • (Σ v ∗ id γ lo ) (A.9) The v ertical comp osition Λ ( r , s, t ) := Λ 2 ( r , s, t ) • Λ 1 ( r , s, t ) is then a bigon whose image is the surface of the cub e. Notice that the tw o bigons Σ 1 and Σ 2 w e started with can b e found on the top and o n the b ottom of the unit cub e, i.e. Σ 1 = h ∗ Σ o and Σ 2 = h ∗ (Σ u ) − 1 . W e ev aluate the map k A,B on the bigon h ∗ (Λ( r , s, t )) , deﬁning a smo oth function u : [0 , 1] 3 / / H . Since k A,B is b y Lemma 2.15 compatible with t he v ertical comp osition and the auxiliary horizon ta l comp osition ∗ w e get from (A.8) and (A.9) u ( r , s, t ) = k A,B ( h ∗ Σ u ) · α ( F A ( h ∗ γ uv ) , k A,B ( h ∗ Σ l )) · k A,B ( h ∗ Σ v ) · α ( F A ( h ∗ γ r v ) , k A,B ( h ∗ Σ o )) · k A,B ( h ∗ Σ r ) · α ( F A ( h ∗ γ r u ) , k A,B ( h ∗ Σ h )) . (A.10 ) On the right hand side w e ha v e omitted the argumen ts ( r , s, t ) f o r simplicit y . In the follo wing we use a bigon Λ r 0 ( r , s, t ) that is shifted b y r 0 along the r -axis with resp ect to the bigon Λ( r , s, t ) from Figure 3, to whic h it reduces fo r r 0 = 0 . A ccordingly , w e ha ve a smo oth function u r 0 : [0 , 1] 3 / / H additionally dep ending on the shift r 0 . In the same w a y , w e hav e a smo oth function u r 0 ,s 0 ,t 0 : [0 , 1] 3 / / H asso ciated to a bigon Λ r 0 ,s 0 ,t 0 that is additionally shifted b y s 0 and t 0 along the s -axis and the t -axis, resp ectiv ely . Lemma A.11. The smo oth function u r 0 : [0 , 1] 3 / / H has the fol lo wing pr op erties: (a) u 0 (1 , 1 , 1) = k A,B (Σ 2 ) − 1 · k A,B (Σ 1 ) . (b) u r 0 ( r , 1 , 1) = u r 0 + r ′ ( r − r ′ , 1 , 1) · u r 0 ( r ′ , 1 , 1) . (c) u r 0 ( r , s + σ , 1) = H 1 ( r 0 , r , s ) · u r 0 ,s, 0 ( r , σ, 1) · H 2 ( r 0 , r , s ) . (d) 1 3 ∂ 2 ∂ r ∂ σ     0 ∂ ∂ t u r 0 ,s, 0 ( r , σ, t ) = ( h ∗ K ) ( r 0 ,s,t )  ∂ ∂ r , ∂ ∂ s , ∂ ∂ t  . In (c), H 1 and H 2 ar e c ertain H -value d smo oth functions that do not dep end on σ . In (d), the 3-form K ∈ Ω 3 ( X , h ) is given by K := d B + α ∗ ( A ∧ B ) . Pro of. Condition 1 of Deﬁnition 2.2 for the homotop y h implie s the v an- ishing of the group elemen ts k A,B ( h ∗ Σ r ) , k A,B ( h ∗ Σ l ) and F A ( h ∗ γ r v ) , F A ( h ∗ γ r u ) in the pro duct (A.10), so that k A,B ( h ∗ Λ(1 , 1 , 1 ) ) = k A,B (Σ 2 ) − 1 · k A,B ( h ∗ Σ v ) · k A,B (Σ 1 ) · k A,B ( h ∗ Σ h ) . ( A.11 ) 70 By condition 2 b), the bigons h ∗ Σ v : γ ′ 1 + 3 γ ′ 2 and h ∗ Σ h : γ 1 + 3 γ 2 are thin homotopies b et w een paths, i.e. the rank of their diﬀeren tials is less or equal to 1. Accordingly , A h ∗ Σ v = A h ∗ Σ h = 0 and hence k A,B ( h ∗ Σ v ) = k A,B ( h ∗ Σ h ) = 1 . No w, assertion (a) follo ws from (A.11). The same v anishing argumen ts sho w that u x ( y , 1 , 1) = k A,B ( h ∗ Σ u x + y (1 , 1)) · k A,B ( h ∗ Σ o x (1 , 1)) . (A.12) Using formula (A.12) b y setting ( x, y ) to ( r 0 , r ) , ( r 0 + r ′ , r − r ′ ) and ( r 0 , r ′ ) , respectiv ely , show s assertion (b). Still the same argumen ts show that u r 0 , 0 , 0 ( r , s + σ , 1) = k A,B h ∗ Σ u r 0 + r, 0 , 0 ( s, 1) · u r 0 ,s, 0 ( r , σ, 1) · k A,B h ∗ Σ v r 0 ,s, 0 ( r , 1) · k A,B h ∗ Σ o r 0 , 0 , 0 ( s, 1) , whic h is assertion (c) iden tifying H 1 ( r 0 , r , s ) = k A,B h ∗ Σ u r 0 + r, 0 , 0 ( s, 1) H 2 ( r 0 , r , s ) = k A,B h ∗ Σ v r 0 ,s, 0 ( r , 1) · k A,B h ∗ Σ o r 0 , 0 , 0 ( s, 1) . F or (d), w e write down f orm ula (A.10) for u r 0 ,s, 0 ( r , σ, t + τ ) and decomp ose bigons of length t + τ into tw o bigo ns of length t a nd τ , resp ectiv ely . This giv es u r 0 ,s, 0 ( r , σ, t + τ ) = k A,B h ∗ Σ u r 0 + r,s,t ( σ , τ ) · α ( k A,B h ∗ γ uv r 0 + r,s 0 + σ ,t ( τ ) , k A,B h ∗ Σ u r 0 + r,s, 0 ( σ , t )) · α ( k A,B h ∗ γ uv r 0 + r,s + σ , t ( τ ) , k A,B h ∗ Σ v r 0 ,s + σ, 0 ( r , t )) · k A,B h ∗ Σ v r 0 ,s + σ,t ( r , τ ) · α  k A,B h ∗ γ r v r 0 ,s + σ,t + τ ( r ) , α ( k A,B h ∗ γ or r 0 ,s + σ,t ( τ ) , k A,B h ∗ Σ o r 0 ,s, 0 ( σ , t )) · k A,B h ∗ Σ o r 0 ,s,t ( σ , τ )  · k A,B h ∗ Σ l r 0 ,s,t + τ ( r , σ ) · α ( k A,B h ∗ γ r u r 0 + r,s,t + τ ( σ ) , k A,B h ∗ Σ h r 0 ,s,t ( r , τ ) · α ( k A,B h ∗ γ hu r 0 + r,s,t ( τ ) , k A,B h ∗ Σ h r 0 ,s, 0 ( r , t ))) . No w we tak e the deriv ativ e of this expression b y the three v ariables r , σ and τ , ev aluate at zero and use Prop osition 2.14 in order to iden tify second deriv ativ es of k A,B with the 2-form B . This give s ∂ 3 ∂ r ∂ σ ∂ τ     0 u r 0 ,s, 0 ( r , σ, t + τ ) = 3d B h ( r 0 ,s,t ) ( v r , v s , v t ) + α ∗ ( A r 0 ,s,t ( v r ) , B r 0 ,s,t ( v s , v t )) + α ∗ ( A r 0 ,s,t ( v s ) , B r 0 ,s,t ( v t , v r )) + α ∗ ( A r 0 ,s,t ( v t ) , B r 0 ,s,t ( v r , v s )) . Using the an tisymmetry of B , this sho ws (d).  71 Remark A.12. The 3-form K = d B + α ∗ ( B ∧ A ) that drops out in (d) has to b e in terpreted as the curv ature of the connection ( A, B ) o n a trivial, (non-ab elian) gerb e, see Section 3.1. No w, if the homotopy h is thin, i.e. satisﬁes condition 2a) of Deﬁnition 2.2, w e ha v e b y (d) ∂ 2 ∂ r ∂ σ     0 u r 0 ,s, 0 ( r , σ, 1) = Z 1 0 d t ∂ 2 ∂ r ∂ σ     0 ∂ ∂ t u r 0 ,s, 0 ( r , σ, t ) = 0 . P erforming this tric k once more, w e obtain ∂ ∂ r     0 u r 0 ( r , 1 , 1) = Z 1 0 d s ∂ ∂ r     0 ∂ ∂ s u r 0 ( r , s, 1) (c) = Z 1 0 d sH 1 ( r 0 , r , s ) ·  ∂ 2 ∂ r ∂ σ     0 u r 0 ,s, 0 ( r , σ, 1)  · H 2 ( r 0 , r , s ) = 0 The m ultiplicativ e prop erty (b) transfers this result to all v alues of r , ∂ ∂ r     r 0 u 0 ( r , 1 , 1) = ∂ ∂ r     0 u r 0 ( r , 1 , 1) · u 0 ( r 0 , 1 , 1) = 0 . This means that the function u 0 ( r , 1 , 1) is constan t, and th us determined b y its v alue at r = 0 , 1 = u 0 (0 , 1 , 1) = u 0 (1 , 1 , 1) (a) = k A,B (Σ 2 ) − 1 · k A,B (Σ 1 ) . This sho ws that k A,B tak es the same v alues on thin homotopic bigons Σ 1 and Σ 2 . T able of Notations A UT ( H ) the automorphism 2-group of a Lie group H . P ag e 68 B G the Lie group oid with one o b ject associated to a Lie group G . P ag e 10 B G the Lie 2- g roup oid with o ne ob ject asso ciated to a Lie 2-group G . P ag e 18 B X the diﬀeological space of bigons in X . P ag e 15 72 B 2 X the diﬀeological space o f thin homotopy classes of bigons in X . P ag e 15 D the 2 -functor that extracts diﬀeren tial forms from smo oth 2-functors. P ag e 30 D ∞ the category of diﬀeological spaces. P ag e 9 E G the inner automorphism 2-group asso ciated to a Lie group G . P ag e 68 F unct ∞ the category of smo oth functors b et w een Lie cate- gories. P ag e 10 LX the lo op space D ∞ ( S 1 , X ) of a diﬀeological space X . P age 54 Λ the functor that mak es a category out of a 2- category . P ag e 54 ℓ t he functor that regards a path in the lo op space of X as a bigon in X . P ag e 56 P X the diﬀeolog ical space of smo o th paths (with sitting instan ts) in X . P ag e 7 P 1 X the diﬀeological space o f thin homotopy classes of paths in X . P ag e 7 P 1 ( X ) the path group oid of X . P ag e 7 P 2 ( X ) the path 2-group oid of X . P ag e 16 P the 2-functor that in tegrat es diﬀeren tial forms to smo oth 2-functors. P ag e 30 Z 1 X ( G ) ∞ the category of G -connections on X . P ag e 10 Z 2 X ( G ) ∞ the 2-category of G -connections on X . P ag e 29 References [AF G99] O. Alv arez, L. F erreira and J. S. Guillén, In tegrable Theories in an y Dimension: a P erspectiv e, in T rends in Theoretical Ph ysics I I , v olume 484 of AIP Conference Pro ceedings , pages 81 – 97, 1999. [Bae96] J. C. Baez, 4-Dimensional BF Theory as a T op ological Quantum Field Theory , Lett. Math. Ph ys. 38 , 129–1 43 (1996). [Bae07] J. C. Baez, Quantiz ation a nd Cohomology , (2007), Lecture Notes, UC Riv erside. 73 [BL04] J. C. Baez and A. D. Lauda, Highe r-dimensional Algebra V: 2- Groups , Theory Appl. Categ. 12 , 423 – 491 (2004). [BM05] L. Breen and W. Messing, Diﬀeren t ia l Geometry of Gerb es , Adv. Math. 198 (2), 732–84 6 (2005). [Bry93] J.-L. Brylinski, Lo op spaces, Characteristic Classes and G eometric Quan t ization , v olume 107 of Progr. Math. , Birkhäuser, 199 3. [BS76] R. Brown and C. B. Sp encer, G -Group oids, crossed Mo dules, and the classify ing Space of a top ological G roup , Pro c. K on. Ak ad. v. W et. 79 , 296–3 02 (1976). [Che77] K.-T. Chen, Iterated P ath Inte grals , Bull. Amer. Math. So c. 83 , 831–879 (1977). [CP94] A. Caetano and R. F. Pic k en, An axiomatic Deﬁnition of Holonom y , In t. J. Math. 5 (6), 835–84 8 (1994). [Ga w88] K. Ga w¸ edzki, T op olog ical Ac tions in t w o -dimensional Quan tum Field Theories, in Non-p erturbative Quan tum Field Theory , edited b y G. Hoo ft, A. Jaﬀe, G . Mac k, K. Mitter and R. Stora, pages 101– 142, Plen um Press, 1988. [GP04] F. Girelli and H. Pfeiﬀer, Higher Gauge Theory - diﬀeren tial v ersus in tegral F ormul ation , J. Math. Phys . 45 , 3949–3971 (2004). [GPP08] F . Girelli, H. Pfeiﬀer and E. M. Popescu, T op ological Higher Gauge Theory - f r o m BF to BF CG Theory , J. Math. Ph ys. 49 (3), 2503 (2008). [GT01] K. Go mi and Y. T erash ima, Higher-dimensional parallel T ransp orts , Math. Res. Lett. 8 , 25–33 (2001). [Mac87] K. C. H. Mac k enzie, Lie Gro up o ids and Lie Algebroids in Diﬀeren tial Geometry , v olume 124 of London Math. So c. Lecture Note Ser. , Cam bridge Univ. Press, 1987. [MP02] M. Mack aay a nd R. Pic ken, Holonom y and para llel T ransp ort fo r ab elian Gerb es , Adv . Math. 170 (2 ), 287 –339 (2002). [MP10] J. F. Martins and R. F . Pic k en, On tw o-Dimensional Holonomy , T rans. Amer. Math. So c. 362 (11), 565 7–5695 (2010). [Mur96] M. K. Murray , Bundle Gerb es , J. Lond. Math. So c. 54 , 403–416 (1996). 74 [O V91] A. L. Onishc hik and E. B. Vinberg, I. F oundations of Lie Theory , in Lie Groups and Lie Algebras I , edited b y A. L. O nishc hik, v olume 20 of Encyclopaedia of Mathematical Sciences , Springer, 1991. [P ow90] J. P o w er, A 2-categorical pasting theorem , J. Algebra 129 (2), 439– 445 (1990). [RS08] D. Rob erts and U. Sc hreib er, The inner Automorphism 3-Gro up of a strict 2-Gro up , J. Homotopy Relat. Struct. 3 (1), 193–244 (2008 ). [Seg68] G. Segal, Classifying Spaces and Sp ectral Sequences , Publ. Math. Inst. Hautes Études Sci. 34 , 105–11 2 (1968). [Sou81] J.-M. Souriau, Group es diﬀéren tiels, in Lecture Notes in Math. , v olume 836, pages 91–128, Springer, 1981 . [SSS09] H. Sati, U. Sc hreib er and J. D. Stasheﬀ, L- inﬁnit y Algebra Connec- tions a nd Applications t o String- and Chern-Simons n-T ransport, in Quan tum Field Theory , edited b y B. F auser, J. T olksdorf and E. Zeidler, pages 303–4 24, Birkh äuser, 200 9 . [SW A] U. Sc hreib er and K . W aldorf, Connections on non-ab elian Gerb es and their Holonom y , preprin t. [SWB] C. Sc h w eigert and K. W aldorf, Gerb es and Lie G roups, in D ev elop- men ts and T rends in Inﬁnite-Dimensional Lie The ory , edite d b y K.- H. Neeb and A. Pianzola, v o lume 600 of Progr. Math. , Birkhäuser, to app ear. [SW09] U. Sch reib er and K. W aldorf, P arallel T ransp ort and F unctors , J. Homotop y Relat. Struct. 4 , 187–2 44 (2009). [W al07] K. W aldorf, More Morphisms b et w een Bundle Gerb es , Theory Appl. Categ. 18 (9), 240–27 3 (2007). [Whi46] J. H. C. Whitehead, Note on a previous P ap er en titled „On a dding Relations to Homotop y Groups“ , Ann. of Math. 47 , 806–810 (1946 ). 75

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