Categorified central extensions, etale Lie 2-groups and Lies Third Theorem for locally exponential Lie algebras

Categoriﬁed cen tral extensi ons, ´ etale Lie 2-groups and Lie’s Third Theorem for lo cally exp onen ti al Lie algebras Christoph W o c k el ∗ christoph@ wockel.eu Abstract Lie’s Third Theorem, asserting that each ﬁnite-dimensional Lie algebra is the Lie algebra of a Lie group , fails in inﬁnite dimensions. The mo dern account on this ph enomenon is the integ ration p roblem for central extensions of inﬁnite-dimensional Lie algebras, which in turn is phrased in terms of an integra tion p rocedu re for Lie algebra co cycles. This paper remedies the obstructions for integrating cocycles and central extensions f rom Lie algebras to Lie groups by generalising th e integr ating ob jects. Those ob jects ob ey the maximal coherence that one can exp ect. Moreo ver, w e show that they are the universal ones for the integra tion p rob lem. The main application of this result is that a Mack ey-complete lo cally exp onential Lie algebra (e.g., a Banac h –Lie algebra) integ rates to a Lie 2-group in the sense that there is a natural Lie funct or from certain Lie 2-groups to Lie algebras, send ing the integrating Lie 2-group to an isomorphic Lie algebra. MSC: 22E65, 58H05, 55N20 Keywords: inﬁnite-d imensioal Lie 2-group ; central extension; co cycle; integratio n of co cy- cles; Lie’s Third Theorem In tro duction This pap er sets out to reso lve obstructions for in tegrating Lie alg ebras and central extensions of them. It is a celebr ated theor em that each ﬁnite-dimensional Lie alg ebra is the Lie algebra of a Lie gr oup, which is known as Lie ’s Third Theorem. It was prov en by Lie in a lo cal versions and in full strength b y ´ Elie Car tan (cf. [ Car30 ] a nd references therein). It has also been ´ Elie Cartan who ﬁrst remar ked in [ Ca r36 ] that one may also us e the fact that π 2 ( G ) v anishes for any ﬁnite-dimensional Lie g r oup 1 to prove Lie’s Third Theo rem. If G is inﬁnite-dimensio na l, then π 2 ( G ) do es not v anish a ny more, for instance for C ∞ ( S 1 , SU(2)) or P U ( H ). This was used by v an Est and Korthag en in [ EK64 ] to construct an example of a Banach–Lie algebr a which cannot be the Lie algebra of a Lie group (cf. [ DL66 ] for the cor resp onding constr uction for P U ( H )). How ever, t here is a large class of inﬁnite-dimens io nal Lie algebra s which in tegrate to a lo c al Lie group, namely lo cally e x po nential Lie algebras . In particular, all Banach–Lie algebras b elong to this class . The non-ex is tence of a (global) Lie gro up integrating a lo cally exp onential Lie algebra may thus b e rega r ded as the obstructio n a gainst the co rresp onding lo cal Lie g roup to e nla rge to a (global) Lie group. This is why a Lie alg e bra, which is the Lie algebra of a (global) Lie gr oup is o ften called enlar ge able , whilst a Lie algebra is calle d inte gr able if it is the Lie algebr a of a lo cal Lie gro up (cf. [ Nee06 ]). ∗ F ac hbereich Mathematik, Bundesstr. 55, D- 20146 Hamb urg, Germany 1 originally , Cartan’s condition was that the ﬁrst tw o Betti nu mbers v anish 1 The most sophisticated to o l for the analy sis of e nlargeability o f lo ca lly exp onential Lie al- gebras is Neeb’s machinery for in tegrating central extensio ns of inﬁnite-dimensio nal Lie gr oups, developed in [ Nee02 ]: if z → b g → g is a cent ral e x tension of Lie algebras and G is a 1-connected Lie gr oup with Lie alg ebra g , then z → b g → g integrates to a central extension of G if and only if the p erio d gro up pe r b g ( π 2 ( G )) ⊆ z is discr ete. A v ariant of this theor y (cf. [ Nee06 , Sect. VI.1]) applies in particular to a lo cally exp onential Lie a lgebra g , since z ( g ) → g → g ad is a (generalised) central extension a nd ther e alwa ys exists a 1 -connected Lie gr o up G ad with L ( G ad ) = g ad . Thus the obstruction for g to b e no n- enlargea ble is the non- discreetness of p er g ( π 2 ( G ad )) ⊆ z ( g ). If g is ﬁnite-dimensiona l, then π 2 ( G ad ) v anishes and Lie’s Thir d Theorem is immediate. F rom this po int of view the theorem seems to b e mer e ly a ho mo topy-theoretic a ccident. Enlarging lo cal groups a nd in tegrating central extensions ob ey a common pattern. The ob- struction for enlarg ing a lo ca l (Lie) gr o up to a globa l o ne is a n ass o ciativity constra int , which is coupled to top olo gical prop erties of the lo cal gr oup (cf. [ Smi50 ], [ Est6 2 a ], [ Est62b ], [ Olv96 ] a nd [ BR08 ]). In g eneral, globa l as s o ciativity c a nnot b e achiev ed a s the counterexamples, ment ioned ab ov e, show. In the in tegration pr oblem for co cyc les the obstr uction for p er b g ( π 2 ( G )) ⊆ z to b e discrete ensures that a co cycle condition holds for a cer tain univ ersal in tegrating co cycle. The ups hot of this pap er is that one may r elax global as so ciativity and co cycle conditions at the same time b y intro ducing more gener a lised but st il l c oher ent ob jects, like generalis e d g roup co cycles and 2 -groups. It is or ganised as follows. In the ﬁrst sectio n we line out an integration pro cedure for Lie algebr a co cycles to g eneralised, lo ca lly smo oth Lie gro up co cycles. This is the central idea of this pap er, a ll other results w ill build o n this. The main achiev e men t of this section is the following Theorem. If z is Mackey-c omplete and g is a Lie algebr a with simply c onne cte d Lie gr oup G , then e ach c ontinu ous Lie algebr a c o cycle ω : g × g → z i nte gr ates to a gener alise d c o cycle on G . Mor e over, the gener alise d c o cycle that we c onstruct is universal for gener alise d c o cycles inte gr ating ω . The r emaining sections des crib e interpretations of the r esults of the ﬁrst sectio n. The s econd describ es an interpretation in the languag e of lo op prolonga tions. It is dis cussed which asp ects cannot b e cov ered by lo o p prolong ations, which then leads to an interpretation in the la ng uage of 2-g roups. This is done in s ections three and four and the corr esp onding extension theor y is int ro duced in section ﬁv e. It is describ ed whic h rˆ ole ´ etalness plays in this s e tting, a nd this section even tually results in the seco nd ma in res ult of this pape r . Theorem. If g is the Lie algebr a of the simply c onn e cte d Lie gr oup G , then e ach top olo gic al ly split c entr al extension z → b g → g with Mackey-c omplete z inte gr ates t o a smo oth gener alise d c entr al extension of ´ etale Lie 2-gr oups. After having worked this out we apply the pr evious r esults to the (generalis ed) central ex- tension z ( g ) → g → g ad for g lo cally exp onential in or der to obtain our version of Lie’s Thir d Theorem in the next section: Theorem. If g is a Mackey-c omplete lo c al ly exp onent ial Lie algebr a, then ther e exists an ´ etale Lie 2-gr oup G such that L ( G ) is isomorphic to g . In the end we indicate some direc tions for further resear ch and give some deta ils on lo cally conv ex Lie gro ups in an app endix. There exist ma n y links to neighbouring topics, which we will mention throughout the tex t. Amongst those a re integrabilit y q uestions for Lie algebr oids (Remar k V.7 ), String group mo dels (Example IV.10 ), diﬀeolog ical Lie g r oups (Remar k VI I.1 ) and connections on categor iﬁed bundles and n -plectic geometry (Remar k VII.2 ). Since many of them need concepts and notation that we provide in the text we r e fr ain from s ummarising them here. 2 Ac kno wledgemen ts The author enjoy ed a scholarship in the Graduier tenkolleg 149 3 “Mathema tis che Strukturen in der moder nen Q uantenph ysik” (G¨ ottingen) and was supp orted by the SFB 676 “Particles, Strings and the Ear ly Univ erse” (Hamburg) while ca rrying o ut the work o n this pap er. He is gr ateful to Chenchang Zhu for ma ny discussions on the integration pro cedur e fo r Lie a lgebras and Lie algebroids . He also wishes to thank Ka rl-Hermann Neeb for providing references c o ncerning the classical version of Lie’s Third Theorem and p o intin g o ut Example VI.4 . Thanks go als o to Sven S. Porst for c o unt less disc ussions a nd pro of-rea ding parts of the ma nu script. Last, but no t lea st, the author is grateful to Urs Schreiber for conversations o n 2 -groups a nd related topics. The author also wan ts to thank the referee for a thorough job, helping in par ticular to improv e the presentation of the pap er. Con v en tions F or us a manifold is a Hausdorﬀ space, which is lo cally ho meomorphic to op en subsets of some lo cally conv ex space such that the co ordinate changes are diﬀeomorphisms. A Lie g roup is a group, which also is a manifold such that the g roup op era tions are smo o th (cf. Deﬁnition A.1 for details o n this). F or M , N s mo oth manifolds and f : M → N s mo oth, T f : T M → T N denotes the tangent map of f . If M , N and f are po inted, then d f : T ∗ M → T ∗ N denotes the diﬀerential in the base p oint. Mor eov er , if d f v anishes, then one may deﬁne d 2 f : T ∗ M × T ∗ M → T ∗ N in terms of lo cal co ordina tes, where the v anishing of d f implies the indep endence of the choice of a chart. F or us, a lo cally smo oth map on a p ointed ma nifold is a map which is smo o th on some op en neighbourho o d of the base- p o int (a nd not on an op en neighbourho o d of each p oint). Unless stated other wise, G shall alwa ys b e a 1- connected Lie gro up with Lie alg ebra g , which we usually identify with T e G . Moreover, z shall alwa y s denote a Mack ey-complete lo cally conv ex space (in par ticula r, in tegrals of smo o th functions from standa rd simplices to z alwa ys exist, cf. Remark A.6 ) and Z will denote the ab elian Lie g r oup z / Γ for some discrete subgro up Γ (in so me situations we will cho ose Γ explicitly , but in g e ne r al any discrete subgr oup is ﬁne). Unless sta ted otherwise, q : z → Z will denote the canonica l quotient map. If A is an a be lia n Lie group wher e G ac ts on (tr iv ially if nothing els e is s aid), then we deﬁne C n ( G, A ) := { f : G n → A : f ( g 1 , ..., g n ) = 0 if g i = e for some i and f is s mo oth on s o me ne ig hbourho o d of ( e, ..., e ) ∈ G n } , the gro up of normalise d lo cally smo oth A -v a lued n -co chains on G . Note that this implies in particular d f ( e, ..., e )( v 1 , ..., v n ) = n X i =1 d f ( e, ..., e )(0 , ..., v i , ..., 0) = 0 . On C n ( G, A ) we denote b y d gp : C n ( G, A ) → C n +1 ( G, A ) , d gp f ( g 0 , ...g n ) = g 0 .f ( g 1 , ..., g n ) − n − 1 X i =0 ( − 1) i f ( g 0 , ..., g i g i +1 , ..., g n ) − ( − 1) n f ( g 0 , ..., g n − 1 ) the o r dinary gr oup diﬀerential (we will also use this formula for d gp f in more ge ner al situa tio n, where A do es not carry a Lie gr oup structure and f do es not ob ey any smo othness co nditio n). If f ∈ C 2 ( G, A ), d gp f = 0 , and G a cts trivially , then ( a, g ) · ( b, h ) = ( a + b + f ( g , h ) , g h ) (1) deﬁnes a group structure on A × G , whic h we denote by A × f G . 3 W e deno te by ∆ ( n ) ⊆ R n the sta ndard n -simplex, whic h we v iew a s a manifo ld with co rners. F or a Hausdor ﬀ space X , C n ( X ) = h C (∆ ( n ) , X ) i Z denotes the group of s ingular n -chains in X ov er Z and ∂ : C n ( X ) → C n − 1 ( X ) the cor resp onding singular diﬀerential. Moreover, Z n ( X ) and B n ( X ) denote the co r resp onding cyc les and b oundaries and H n ( X ) the singula r homo logy of X . F or α, α ′ ∈ C (∆ ( n ) , G ), α + α ′ and − α always r efer to the additive structure in C n ( G ) whilst α · α ′ and α − 1 alwa ys refer to the (po int-wise) gr oup structures on the Lie group C (∆ ( n ) , G ). Moreov er, G ac ts by left multiplication on C n ( G ) and w e take this mo dule structure into account when using d gp for C n ( G )-v alued mappings. If C is a small categ ory , then C 0 and C 1 are the sets of o b jects and morphisms. The structure maps of C are always denoted by s, t, id and ◦ . If F : C → D is a functor , then F 0 : C 0 → D 0 and F 1 : C 1 → D 1 are the c orresp o nding maps o n the s et o f ob jects a nd morphis ms . L ikewise, if α : F ⇒ F ′ is a natural transforma tion, then we use the sa me letter to denote the corresp o nding map α : C 0 → D 1 . The set o f is o morphism cla sses o f C is denoted by π 0 ( C ) a nd π 0 ( F ) is the induced map π 0 ( C ) → π 0 ( D ). Almost a ll categ ories that we will encounter in this article will be group oids, i.e., categor ies in which ea ch mo rphism is inv ertible. I In tegrating c o cycles This section describ es the principal co ns truction of this pap er. It is an integration pro cedur e for Lie algebra co cycles and generalises the approach from [ Nee02 ]. The main achiev ement will b e to ov erc ome the obstr uction from [ Nee0 2 ] for the aforementioned integration pr o cedure by passing from group co cycles with co eﬃcients in a n ab elian Lie g roup to group co cycles with co eﬃcients in a complex of ab elian Lie gro ups. W e ﬁr st r ecall the setting and the results from [ Nee02 ]. Deﬁnition I.1. Let g b e a top ologica l Lie alg e br a and z b e a top o lo gical vector s pa ce. A Lie algebra c o cycle is a contin uous bilinear map ω : g × g → z sa tisfying ω ( x, y ) = − ω ( y , x ) and ω ([ x, y ] , z ) + ω ([ y , z ] , x ) + ω ([ z , x ] , y ) = 0 . The co cycle ω is said to be a c ob oundary if ther e exists a contin uous linear map b : g → z with ω ( x, y ) = b ([ x, y ]). The vector space o f co cy cles is denoted by Z 2 c ( g , z ) a nd the space of cob oundaries B 2 c ( g , z ) is a s ub space of Z 2 c ( g , z ). The vector space H 2 c ( g , z ) := Z 2 c ( g , z ) /B 2 c ( g , z ) is called the (second contin uous) Lie algebr a c ohomolo gy of g with co eﬃcients in z . Two co cycles ω and ω ′ are called e quivalent if [ ω ] = [ ω ′ ] in H 2 c ( g , z ). Remark I. 2. Lie algebra coho mology is a concept that unfolds its impo rtance in particular when consider ing inﬁnite-dimensional Lie algebras . F or instance, Whitehead’s lemma [ Jac62 , Thm. I I I.13] a sserts that H 2 c ( g , z ) v anishes if g and z are ﬁnite-dimensio nal and g is semi-simple. The imp or tance of H 2 c ( g , z ) comes fr om the fact that it c lassiﬁes (top o logically split) 2 c entr al extensions of top olo gical Lie alg ebras, i.e. short exact sequences z → b g q − → g for which there exists a contin uous and linear right inv erse of q [ Nee06 , Pro p. V.2.10 ]. In inﬁnite dimensions a prominent exa mple for a non- tr ivial H 2 c ( g , z ) c omes from g = C ∞ ( S 1 , k ), z = R and the Kac- Mo o dy co cycle ω h· , ·i : g × g → R , ( f , g ) 7→ Z S 1 h f ( t ) , g ′ ( t ) i dt, (2) where h· , ·i is the Killing form of the ﬁnite-dimensiona l simple Lie a lgebra k . In [ Nee02 ] it is describ ed how Lie algebra co cycles may be integrated to (lo cally smo oth) gr oup co cycles. W e shall now introduce slightly more gener a l ob jects (cf. [ Bre92 , Sect. 2]) cov er ing in particular the (loca lly smo o th) group co cycles fro m [ Nee02 ] (se e a lso [ WX91 ] or [ TW87 ] for other o ccurrences of this concept). 2 Our central extensions of Lie algebras are alwa ys assumed to b e topologically spli t, but diﬀerent authors follow diﬀerent con ven tions for this. 4 Deﬁnition I.3. Let G b e an ar bitrary Lie gr o up a nd A τ − → Z be a mor phism of ab elia n Lie groups. Then a gener alise d gr oup c o cycle on G with co eﬃcients A τ − → Z (shor tly called genera lised co cycle if the setting is understo o d) consists of tw o maps F ∈ C 2 ( G, Z ) and Θ ∈ C 3 ( G, A ) such that d gp F = τ ◦ Θ (3) d gp Θ = 0 . (4) A morphism of generalis ed c o cycles ( ϕ, ψ ) : ( F , Θ) → ( F ′ , Θ ′ ) consists o f t w o maps ϕ ∈ C 1 ( G, Z ) and ψ ∈ C 2 ( G, A ) such that F = F ′ + d gp ϕ + τ ◦ ψ and Θ = Θ ′ + d gp ψ . F urthermore, a 2-morphism γ : ( ϕ, ψ ) ⇒ ( ϕ ′ , ψ ′ ) b etw een t wo morphisms of g eneralised co cycles is given by a map γ ∈ C 1 ( G, A ) such that ψ = ψ ′ + d gp γ . If we view discrete groups as zero-dimensio na l Lie groups , then the preceding deﬁnition als o yields the c oncept of generalised co cy cles without any smo o thness as sumptions. That is why we do not explicitly distinguish b etw een co cy cles with or without smo othness conditions. In this pap er we shall mostly deal with the case that A is a discrete g roup. This implies that Θ v anishes on some identit y neighbourho o d for smo o th maps are in particular contin uous. Remark I. 4. The previous deﬁnition reduces to the cas e of lo cally s mo oth cohomolo gy H 2 ( G, Z ) (cf. [ Nee02 , Def. 4.4]) if we consider g e ne r alised co cycles mo dulo morphisms with c o eﬃcients 0 → Z . Gener a lised co cycles with (0 → Z )-coeﬃcients will sometimes be ca lled 2-c o cycles (or simply co cy c les if the dimensio n is understo o d) with c o eﬃcients (or v a lues in) Z . Similar to the case of Lie algebr a s, H 2 ( G, Z ) clas s iﬁes c entr al extensions of Lie gr oups , i.e., s ho rt exact sequences Z → b G → G p ossessing smo o th lo cal s ections 3 (see [ Nee02 , Pro p. 4.2] a nd E xample IV.2 ). Note also that a genera lised co cy cle ( F, Θ) yields a 2 - co cycle q ◦ F with v alues in Z / im( τ ) provided im( τ ) is disc r ete. In this case, we call q ◦ F the b and of ( F , Θ). If we take co eﬃcients A → 0 and c o nsider g eneralised co cy cles mo dulo morphisms, then this yields the corr esp onding higher lo cally smo o th co homology H 3 ( G, A ) (cf. [ Nee06 , Def. V.2 .5 ]). Generalised coc ycles with ( A → 0)-co eﬃcie nts will sometimes a lso b e called 3-c o cycles (or simply co cycles if the dimension is understo o d) with c o eﬃcients (or v alues in) A . W e ar e now heading for a description of the integration pro cedur e from [ Nee0 2 ]. In order to do so, we give a slightly more conceptual constr uction in the following tw o lemmata that we will use la ter on in our gener a lised construction. They de s crib e the simplicial part of the pro cedur e for enlarg ing lo ca l gro up co cyc le s to global ones (cf. [ EK64 ]). V ariant s o f this construction ar e implicitly hidden in [ Igl95 ] and [ BM93 ]. How ever, none of the ab ov e authors relates tho se co cycles to (lo cally smo oth) group cohomo logy . Recall that our assumption is that G is a 1-co nnected Lie gr oup with Lie a lgebra g . Lemma I.5. Assu me that ther e exist maps α : G → C ∞ (∆ (1) , G ) and β : G 2 → C ∞ (∆ (2) , G ) such that α e ≡ e, α g (0) = e, α g (1) = g , β g,g ( s, t ) = α g ( s + t ) and (5) ∂ β g,h = α g + g .α h − α gh . (6) Then d gp β : G 3 → C 2 ( G ) takes values in Z 2 ( G ) and we have d gp Θ β = 0 if we set Θ β := q ◦ d gp β with q : Z 2 ( G ) → H 2 ( G ) ∼ = π 2 ( G ) the c anonic al qu otient map. W e to ok β as sole subscript, indicating the dependence of Θ on α a nd β , for α is completely determined by β . 3 This is equiv alen t to demanding that Z → b G → G i s a l ocall y trivial principal bundle. Our central extensions of Lie groups are alwa ys assumed to b e locall y trivial principal bundles, but as ab ov e, diﬀerent authors follow diﬀeren t conv entions for this 5 Pro of. F rom ( 6 ) it follows directly that ∂ ( d gp β )( g , h, k ) = ∂ ( g .β h,k ) − ∂ β gh, k + ∂ β g,h k − ∂ β g,h v anishes and thus Θ β takes v a lue s in Z 2 ( G ). That Θ β ∈ C 3 ( G, π 2 ( G )) follows fro m β g,g ( s, t ) = α g ( s + t ), for then Θ β ( g , h, k ) is null-homotopic if one o f g , h or k equa ls e . Moreover, d gp Θ β = 0 follows from d 2 gp = 0. Lemma I.6. If α ′ , β ′ is another p air of maps satisfying ( 5 ) and ( 6 ) , then ther e ex ists a map γ : G → C ∞ (∆ (2) , G ) with ∂ γ g = α g + g .α e − α ′ g . Mor e over, b γ ( g , h ) = β g,h − β ′ g,h − d gp γ t akes values in Z 2 ( G ) and satisﬁ es Θ β = Θ β ′ + q ◦ d gp b γ . Pro of. Since G is simply connected, the map γ ex is ts by [ Nee02 , P rop. 5 .6]. Jus t a s ab ov e it is chec ked that b γ takes v alues in Z 2 ( G ). Mo r eov er , d 2 gp = 0 yields q ◦ d gp b γ = q ◦ d gp ( β − β ′ ) = Θ β − Θ β ′ . That Θ β is a co cycle is a ctually trivial since we wrote is as a co bo undary of the g roup cochain β . The p oint her e is that it ta kes v alues in the m uch smaller subgr oup Z 2 ( G ) and as co cycle with v alues in this gr oup it is not tr ivial. Its pro jection to π 2 ( G ) is even the other extreme, namely universal , at least for discrete ab elian gr oups (see [ PW ] and Example IV.2 ). In ge neral, the maps α and β that we a re going to choo se for our construction are pretty arbitrar y . How ever, when ﬁxing a chart a round the identit y then there e x ists a ca nonical choice for α g and β g,h if g and h are “clos e ” to the identit y: Lemma I.7. L et ϕ : U → e U ⊆ g b e a chart with ϕ ( e ) = 0 , ϕ ( U ) c onvex and e V ⊆ e U op en and c onvex such that e ∈ V := ϕ − 1 ( e V ) and V · V ⊆ U . F or g ∈ U we set ˜ g := ϕ ( g ) and s et ˜ g ∗ ˜ h = f g h for g , h ∈ V . If we deﬁne α g ( t ) = ϕ − 1 ( t · ˜ g ) , (7) β g,h ( t, s ) = ϕ − 1  t ( ˜ g ∗ s ˜ h ) + s ( ˜ g ∗ (1 − t ) ˜ h )  (8) for g , h ∈ V , then these assignment s c an b e ext en de d to mappings α and β satisfying ( 5 ) and ( 6 ) . Mor e over, if for a diﬀer ent chart ϕ ′ we set ¯ g := ϕ ′ ( g ) and γ g ( s, t ) = ϕ − 1  s (1 − t ) t + s ϕ ( ϕ ′− 1 (( t + s ) ¯ g ))) + t (1 + s ) ˜ g  (9) for g ∈ V ∩ V ′ , then this assignment c an b e extende d to a map γ : G → C ∞ ∗ (∆ (2) , G ) , satisfying ∂ γ g = α g − α ′ g . In additio n, if W ⊆ V with W · W ⊆ V , t hen Θ β | W × W × W and b γ | W × W ar e smo oth. Pro of. It is ea sily chec ked that α g and β g,h deﬁned as in ( 7 ) and ( 8 ) sa tisfy ( 5 ) and ( 6 ). Since G is co nnected, we may choose fo r each g ∈ G \ U some α g ∈ C ∞ (∆ (1) , G ) with α g (0) = e and α g (1) = g . F or g , h ∈ G with g / ∈ V o r h / ∈ V , α g + g .α h − α gh is in Z 2 ( G ), and thus there exists some β g,h ∈ C ∞ (∆ (2) , G ) with ∂ β g,h = α g + g .α h − α gh bec ause G is s imply co nnected (cf. [ Nee02 , Prop. 5.6 ]). Moreov er, we may choo se β g,g ( s, t ) = α ( s + t ). It is immediate that for γ g as deﬁned in ( 9 ) we hav e ∂ γ g = α g − α ′ g . Since G is simply connected, we may cho ose for ea ch g / ∈ V ∩ V ′ some γ g with ∂ γ g = α g − α ′ g . The rest is immediate. W e now co me to the descr iption of the int egration pro cedure from [ Nee02 ] 4 . 4 V arian ts of this pro cedure for the case of Kac-Mo ody cen tral exte nsions can b e found, for instance, in [ PS86 ], [ Mic87 ], [ Br y93 ] and [ MS03 ]. Implicitly , the co cycles that we shall construct here are already apparen t in their constructions. 6 Remark I. 8. Asso cia ted to each Lie algebra co cycle ω : g × g → z is its p erio d homomorphism per ω : π 2 ( G ) → z . This is given on (piecewise) smo oth re pr esentativ es by [ σ ] 7→ R σ ω l , where ω l is the le ft-inv ariant z -v alued 2-form o n G with ω l ( e ) = ω (cf. [ Nee02 ] or [ W oc0 9 ] for the fact tha t each homotopy class contains a smo oth repres e n tative and [ Nee0 2 ] for the fact R σ ω l is independent on the choice of a repres e n tative). W e deﬁne F ω ,β : G × G → z by F ω ,β ( g , h ) := Z β g,h ω l , (10) where β : G 2 → C ∞ ∗ (∆ (2) , G ) is the map from Lemma I.7 applied to a chart ϕ with dϕ ( e ) = id g . Since β ( g , g ) a nd β ( e, g ) are null-homotopic, it follows that F ω ,β ( g , e ) = F ω ,β ( e, g ) = 0. That ( F ω ,β , Θ β ) is a generalis ed co cycle with co eﬃcients π 2 ( G ) p er ω − − − → z follows from d gp Θ β = 0, from d gp F ( g , h, k ) = F ω ,β ( h, k ) − F ω ,β ( g h, k ) + F ω ,β ( g , hk ) − F ω ,β ( g , h ) = Z β h,k ω l − Z β gh,k ω l + Z β g,hk ω l − Z β g,h ω l = Z g.β h,k − β gh,k + β g,hk − β g,h ω l = p er ω (Θ β ( g , h, k )) (11) and fro m the fact that the maps V × V ∋ ( g , h ) 7→ β g,h and C ∞ (∆ (2) , G ) ∋ β 7→ R β ω l are smo oth. Since Neeb only considers 2-co cycles (and no gener alised co cycles), he is force d 5 to co nsider equation ( 1 1 ) mo dulo Π ω and thus obtains a 2-co cy cle f ω ,β ( g , h ) := [ F ω ,β ( g , h )] with v alues in Z ω := z / Π ω . The dr awbac k is o f course that he needs to as sume that Π ω is discrete in order to consider Z ω as a Lie gr oup (see Rema rk VII.1 for a prop osal on how to use diﬀeological Lie groups in this context). W e will now make precise in which sense a gro up co cycle may “integrate” a Lie algebr a co cycle. Recall that o ur standing assumption is that g is the Lie a lgebra o f G , that z is an arbitrar y Mackey-complete lo cally convex spa ce and that Z = z / Γ for Γ ≤ z discr ete. Lemma I.9. L et A b e discr ete and A τ − → Z b e a morphism of ab elian Lie gr ou ps. If F : G 2 → Z and Θ : G 3 → A is a gener alise d c o cycle, then dF ( e , e ) vanishes and we get a Lie algebr a c o cycle L ( F ) : g × g → z , ( x, y ) 7→ d 2 F (( x, 0) , (0 , y )) − d 2 F (( y , 0) , (0 , x )) Pro of. Let U ⊆ G be an identit y neighbour ho o d such that F | U × U and Θ | U × U × U are smo oth maps. F rom F ( e, g ) = F ( g , e ) = 0 it follo ws tha t dF ( e, e ) v anishes. Moreover, Θ | U × U × U v anishes since it is in particular contin uous and A is discr ete. Thus F ( g , h ) + F ( g h, k ) − F ( g , hk ) − F ( h, k ) = 0 for g , h, k in U . Since the computation of L ( F ) in [ Nee02 , Lem. 4.6] only depends on the v alues of F o n U × U , the same calcula tion s hows the cla im. Deﬁnition I.10. A g eneralised co cyc le ( F , Θ) a s in the pr evious lemma is s a id to inte gr ate a z -v alued Lie-alg ebra co cycle ω if L ( F ) is equiv alent to ω . Theorem I.11. The gener alise d c o cycle ( F ω ,β , Θ β ) fr om Remark I.8 inte gr ates ω . Pro of. Since F | V × V coincides with the function f : V × V → z in [ Nee02 , Lem. 6.2 ], asso ciated to the co cycle ω and the smo o th maps σ g,h : ∆ (2) → G fro m [ Nee02 , Le m. 6 .2] coincide w ith β g,h as deﬁned in Lemma I.7 , [ Nee02 , Lem. 6.2] shows L ( F )( x, y ) = d 2 F (( x, 0) , (0 , y )) − d 2 F (( y , 0) , (0 , x )) = ω ( x, y ) . 5 F rom this discussion it is clear that it is suﬃ cient to divide out Π ω := p er ω ( π 2 ( G )) in or der to ensure the cocycle ident it y . F rom Lemma I.15 is follows that this is als o necessary 7 W e now arg ue that the g eneralised co cycle ( F Ω ,β , Θ β ) do e s essentially no t depe nd o n the choices that we made. Remark I. 12. The construction in Rema rk I.8 and the preceding pr o of dep ends on the actual choice o f the map β : G × G → C ∞ (∆ (2) , G ), which in turn dep ends on the choice of a chart ϕ . How ever, fo r tw o diﬀerent choices the res ulting co cycle s Θ β and Θ β ′ are equiv alent by Lemma I.5 and Lemma I.7 . Moreover, if γ : G → C ∞ ∗ (∆ (2) , G ) is the corr esp onding map a s deﬁned in Lemma I.7 , then we obtain a mor phism ( ϕ, ψ ) : ( F ω ,β , Θ β ) → ( F ω ,β ′ , Θ β ′ ), g iven by ψ ( g , h ) = [ b γ ( g , h )] and ϕ ( g ) = Z γ g ω l . If tw o Lie algebra co cycles ω and ω ′ are equiv alent, then ω ( x, y ) = ω ′ ( x, y ) + b ([ x, y ]) for b : g → z linear a nd co nt inu ous. This lea ds to Z β g,h ( ω − ω ′ ) l = Z β g,h d ( b l ) = Z ∂ β g,h b l = Z α g b l + Z g.α h b l − Z α gh b l by Stokes Theor em. W e thus obtain a mor phism ( ϕ, ψ ) : ( F ω ,β , Θ β ) → ( F ω ′ ,β , Θ β ) with ψ ≡ 0 and ϕ ( g ) = Z α g b l . W e conclude this s ection with showing that the co c y cle ( F ω ,β , Θ β ) we constructed here is universal for genera lis ed co cycles that integrate ω . This may b e seen a s a substitute fo r the exact sequence [ Nee02 , Thm. 7.12] 0 → Ext Lie ( G, Z ) L − → H 2 c ( g , z ) P − → Hom( π 2 ( G ) , Z ) . The ne x t lemma is the generalisatio n o f the injectivity o f L (cf. [ Nee02 , Prop. 7.4]) for not necessarily discrete subgro ups Γ ⊆ z . Lemma I.13. L et F ∈ C 2 ( G, z ) b e such that d gp F vanishes on some identity neighb ourho o d of G 3 , d gp F t akes values in some sub gr oup Γ of z and L ( F ) is trivial as a Lie algebr a c o cycle. Then ther e exists ϕ ∈ C 1 ( G, z ) such that F − d gp ϕ vanishes on some identity neighb ourho o d and takes values in Γ on G × G . Pro of. First no te that L ( F ) a ctually deﬁnes a Lie algebr a co cycle by the same a rgument as in Lemma I.9 . Since L ( F ) is trivia l, there exists a contin uous a nd linea r ma p χ : g → z such tha t L ( F ) = χ ([ · , · ]). Let U, V ⊆ G b e con tr actible iden tit y neighbourho o ds such that d gp F | U × U × U v anishes, F | U × U is smo oth and V 2 ⊆ U . Then ( z × U, z × V , µ F , (0 , e )) w ith µ F (( z , g ) , ( w, h )) := ( z + w + F ( g , h ) , g h ) is a lo cal Lie group with Lie algebra z ⊕ L ( F ) g . Since L ( F ) = χ ([ · , · ]), we hav e that z ⊕ L ( F ) g ∋ ( z , x ) 7→ z + χ ( x ) ∈ z deﬁnes a homo morphism of Lie a lgebras . This we ma y integrate to a homomor phism of lo cal Lie groups, given by ( z , g ) 7→ z + ϕ ( g ). By shrinking V if necessary we may assume that ϕ is deﬁned on V . That this map is a homomo rphism implies tha t F − d gp ϕ v anishes on V × V . Since d gp F tak es v alues in Γ, we hav e that f := q ◦ F : G × G → z / Γ is a group co cycle (where q : z → z / Γ is the cano nica l pro jection) and thus ( z / Γ) × f G is a g roup (which ac tually is top ological, but not Hausdorﬀ in gener al). Now f ϕ : V → ( z / Γ) × f G, g 7→ ( q ( ϕ ( g )) , g ) 8 satisﬁes f ϕ ( g ) · f ϕ ( h ) = f ϕ ( g · h ) wherever deﬁned. Since G is 1 -connected, f ϕ extends 6 to a unique group homomorphism. This extension is given by g 7→ ( ϕ ′ ( g ) , g ) for some function ϕ ′ : G → z / Γ. Moreov er, ϕ ′ extends q ◦ ϕ and sa tisﬁes f − d gp ϕ ′ ≡ 0. If we choo s e a lift s : z / Γ → z with q (0) 7→ 0, then g 7→ s ( ϕ ′ ( g )) for g / ∈ V extends ϕ to a ll o f G with the desired prop erties . Remark I. 14. The previous pro of easily adapts to the case where G is not simply connected. One may constr uct ϕ as in the prev ious pro of, but if G is just connected, f ϕ do es not necessar ily extend. How ever, it deter mines a homomor phism e G → ( z / Γ) × f G , whe r e e G is the 1-c onnected cov er of G . Restricting this homomor phism to π 1 ( G ) yields a homomorphism π 1 ( G ) → z / Γ. If this is trivial, f ϕ in fact e xtends to a homomorphism and the argument car ries over. The following lemma is our version of [ Nee02 , Thm. 7.9] for non-discr ete Γ. Lemma I.15. If F ∈ C 2 ( G, z ) is such that d gp F vanishes on some identity neighb ourho o d, d gp F takes values in some sub gr oup Γ of z and L ( F ) is e quivalent t o ω as a Lie algebr a c o cycle, then per ω ( π 2 ( G )) ⊆ Γ . Pro of. Since p er ω do es only dep end on the co homology class of ω (cf. [ Nee02 , Rem. 5.9 ]), we may assume that L ( F ) = ω . Set Θ := d gp F and let U, V ⊆ G b e op en and contractible ident it y neighbourho o ds such tha t F | U × U is smo oth, Θ | U × U × U v anishes and V · V ⊆ U . F or e a ch g ∈ G we deﬁne κ g ∈ Ω 1 ( g V , z ) by κ g ( w x ) = d 2 F g − 1 · x ( x − 1 .w x ) for w x ∈ T x g V , where d 2 F g ( w h ) := dF (0 g , w h ) for g , h ∈ U and w h ∈ T h U . This is smo o th for F | U × U is smo oth and a str aight for ward c o mputation shows dκ g = ω l   gV . F or g , h ∈ G with g V ∩ hV 6 = ∅ we hav e g − 1 h ∈ U . Thus d gp F ( g − 1 h, h − 1 x, x − 1 η ( t )) v anishes for η ( t ) ∈ g V ∩ h V a nd this implies ( κ g − κ h )( w x ) = d 2 F g − 1 · h ( h − 1 .w x ) . If α : [0 , 1] → g V ∩ hV is smo oth, then this in turn yields Z α κ g − κ h = Z 1 0 d 2 F g − 1 h ( h − 1 . ˙ α ( t )) dt = F ( g − 1 h, h − 1 α (1)) − F ( g − 1 h, h − 1 α (0)) = F ( h, h − 1 α (1)) − F ( g , g − 1 α (1)) + Θ( g , g − 1 h, h − 1 α (1)) − F ( h, h − 1 α (0)) + F ( g , g − 1 α (0)) − Θ( g , g − 1 h, h − 1 α (0)) . Now let [ σ ] ∈ π 2 ( G ) b e repr esented by a smoo th map σ : [0 , 1 ] 2 → G such that σ maps a neighbourho o d of ∂ [0 , 1 ] 2 to { e } . Then there exists some n ∈ N such that for i, j ∈ { 0 , ..., n − 1 } σ  i n , i + 1 n  ×  j n , j + 1 n  ⊆ g ij V for some g ij ∈ G . W e deno te by σ ij the restriction of σ to [ i n , i +1 n ] × [ j n , j +1 n ]. Then per ω ([ σ ]) = Z σ ω l = n − 1 X i,j =0 Z σ ij ω l = n − 1 X i,j =0 Z σ ij dκ g ij = n − 1 X i,j =0 Z ∂ σ ij κ g ij (12) by Stokes Theorem. W e par ametrise the intersection σ ij ∩ σ i +1 j by µ ij ( t ) := σ ( i +1 n , j + t n ) and σ ij ∩ σ ij +1 by ν ij ( t ) = σ ( i + t n , j +1 n ). In particular, w e have the identities µ i 0 (0) = µ i n − 1 (1) = e µ ij (1) = ν ij (1) µ ij (1) = ν i +1 j (0) ν 0 j (0) = ν n − 1 j (1) = e µ i j +1 (0) = ν ij (1) µ i j +1 (0) = ν i +1 j (0) . 6 The group on the right do es not need to b e top ological for this, cf. [ HM98 , Cor. A.2.26] 9 Since σ | ∂ [0 , 1] 2 v anishes, the integrals along ∂ σ ij ∩ ∂ σ in ( 12 ) v anish and we thus hav e per ω ([ σ ]) + Γ =   n − 1 X i,j =0 Z µ ij κ g ij − κ g i +1 j − n − 1 X i,j =0 Z ν ij κ g ij − κ g ij +1   + Γ = n − 1 X i,j =0 F ( g i +1 j , g − 1 i +1 j µ ij (1)) . . . . . . . . . . . . . . . . . . . . . . − F ( g ij , g − 1 ij µ ij (1)) . . . . . . . . . + n − 1 X i,j =0 − F ( g i +1 j , g − 1 i +1 j µ ij (0)) + F ( g ij , g − 1 ij µ ij (0))+ n − 1 X i,j =0 − F ( g ij +1 , g − 1 ij +1 ν ij (1)) + F ( g ij , g − 1 ij ν ij (1)) . . . . . . . . . + n − 1 X i,j =0 F ( g ij +1 , g − 1 ij +1 ν ij (0)) − F ( g ij , g − 1 ij ν ij (0)) . . . . . . . . . . . . . . . . . ! + Γ . F ro m the a bove identities it follows that the corr e sp o ndingly underlined ter ms cancel out. Thus per ω ([ σ ]) is contained in Γ. Lemma I.16. L et ( F ′ , Θ ′ ) b e a gener alise d c o cycle on G that inte gr ates ω (cf. Deﬁnit ion I.10 ). Assume that p ∈ Hom( π 2 ( G ) , A ) and ψ ∈ C 2 ( G, A ) ar e such that p ◦ Θ β = Θ ′ + d gp ψ . Then the fol lowing ar e e qu ivalent. i) τ ◦ p = q ◦ p er ω . ii) d gp ( q ◦ F ω ,β − F ′ − τ ◦ ψ ) = 0 . iii) Ther e exists ϕ ∈ C 1 ( G, Z ) such that q ◦ F ω ,β = F ′ + d gp ϕ + τ ◦ ψ . Pro of. W e ﬁst note that q : z → Z is a cov ering map and thus ther e exists a section s : Z → z such that s (0) = 0 and s is smo oth on some zero neighbo urho o d. i) ⇒ ii) : W e s e t F ♯ := s ◦ ( F ′ + τ ◦ ψ ). By Lemma I .1 3 there exists ϕ ∈ C 1 ( G, z ) such that ξ := F ω ,β − F ♯ − d gp ϕ v anishes on some identit y neig hbourho o d and takes v alues in q − 1 ( A ). T o show the asser tio n it suﬃces to show that d gp ( q ◦ ξ ) = 0 . This follows fro m τ ◦ p ◦ Θ β = q ◦ p er ω ◦ Θ β ⇒ τ ◦ (Θ ′ + d gp ψ ) = q ◦ d gp F ω ,β ⇒ d gp F ′ + τ ◦ d gp ψ = d gp ( q ◦ ξ + F ′ + τ ◦ ψ ) . ii) ⇒ iii) : Applying Lemma I.13 to s ◦ ( q ◦ F ω ,β − F ′ − τ ◦ ψ ) g ives ϕ ′ ∈ C 1 ( G, z ) such that s ◦ ( q ◦ F ω ,β − F ′ − τ ◦ ψ ) − d gp ϕ ′ has v alues in Γ. This implies q ◦ F ω ,β − F ′ − τ ◦ ψ − d gp ϕ = 0 if we set ϕ := q ◦ ϕ ′ . iii) ⇒ i) : By [ PW ], im(Θ β ) generates π 2 ( G ). Thus the claim follows from q ◦ p er ω ◦ Θ β = q ◦ d gp F ω ,β = d gp ( F ′ + τ ◦ ψ ) = τ ◦ (Θ ′ + d gp ψ ) = τ ◦ p ◦ Θ β . Prop ositi o n I.17. L et ( F ′ , Θ ′ ) b e a gener alise d c o cycle on G that int e gr ates ω (cf. Deﬁnition I.10 ). Then ther e exists a unique p : π 2 ( G ) → A such that p ◦ Θ β = Θ ′ + d gp ψ for some ψ ∈ C 2 ( G, A ) . Mor e over, τ ◦ p = q ◦ p er ω . 10 Pro of. Since Θ β is universal for discr ete g r oups, ther e e x ists a unique p Θ ′ ∈ Hom( π 2 ( G ) , A ) such that [ p Θ ′ ◦ Θ β ] = [Θ ′ ] (cf. [ PW ]), which is equiv alent to p ◦ Θ β = Θ ′ + d gp ψ fo r so me ψ ∈ C 2 ( G, A ). Set p := p Θ ′ . W e will s how that τ ◦ p = q ◦ p er ω and recall the construction of p Θ ′ for this sake. F or H an ab elian gr oup and a n arbitra ry co cycle f : G 3 → H , v anishing o n some iden tit y neighbourho o d U × U × U , w e construct an H -v alued ˇ Cech-2 co cycle as follows. T ak e V ⊆ U a symmetric op en identit y neig hbourho o d such that V 2 ⊆ U . Then the sets V g := g V for g ∈ G form an op en cov er of G and η ( f , V ) g,h ,k : V g ∩ V h ∩ V k → H , x 7→ − f ( g , g − 1 h, g − 1 k ) − f ( g − 1 h, h − 1 k , k − 1 x ) is smo oth since g − 1 h , g − 1 k , h − 1 k are ele men ts o f U if V g ∩ V h ∩ V k 6 = ∅ . Moreover, it follows from d gp f ( g , g − 1 h, h − 1 k , k − 1 x ) = 0 that η ( f , V ) g,h ,k ( x ) = − f ( g , g − 1 h, h − 1 x ) − f ( h, h − 1 k , k − 1 x ) + f ( g , g − 1 k , k − 1 x ) , showing that η ( f , V ) g,h ,k constitutes a ˇ Cech 2-co cycle on G . The clas s [ η ( f )] ∈ ˇ H 2 ( G, A ) of this co cycle only dep ends on the eq uiv alence class of f in H 3 ( G, A ). Since P G is contractible, this transgre sses to an H -v alued ˇ Cech 1-co cycle on Ω G , giving rise to a cov ering H → d Ω G → Ω G o f Ω G . Cho osing a base p oint in d Ω G turns its connected co mp o nent d Ω G 0 int o a central extension H → d Ω G 0 → Ω G of L ie groups (cf. [ HM98 , App. 2]). Thus there exists a c overing morphism P : g Ω G → d Ω G 0 , where g Ω G is the universal covering of Ω G (note tha t Ω G is connected since G is assumed to b e simply connected). The restriction o f P to the subgroup π 2 ( G ) ∼ = π 1 (Ω G ) ⊆ g Ω G then gives the homomo rphism p f . In particular, we note that p f only depends on [ η ( f )]. F ro m this it fo llows that τ ◦ p = q ◦ p er ω if and only if [ η ( τ ◦ p ◦ Θ β )] = [ η ( q ◦ p er ω ◦ Θ β )]. In order to s how the latter we assume that τ ◦ p ◦ Θ β and q ◦ p er ω ◦ Θ β are s mo oth when res tricted to U × U × U and o bs erve that [ p ◦ Θ β ] = [Θ ′ ] implies η ( τ ◦ p ◦ Θ β , V ) ∼ η ( τ ◦ Θ ′ , V ). Now τ ◦ Θ ′ = d gp F ′ implies η ( τ ◦ Θ ′ , V ) g,h ,k ( x ) = − τ (Θ ′ ( g , g − 1 h, g − 1 k ) − Θ ′ ( g − 1 h, h − 1 k , k − 1 x )) = − d gp F ′ ( g , g − 1 h, h − 1 k ) − d gp F ′ ( g − 1 h, h − 1 k , k − 1 x ) = F ′ ( g , g − 1 h ) + F ′ ( h, h − 1 k ) − F ′ ( g , g − 1 k ) −  F ′ ( g − 1 h, h − 1 x ) + F ′ ( h − 1 k , k − 1 x ) − F ′ ( g − 1 k , k − 1 x )  . This is equiv alent to the co cycle V g ∩ V h ∩ V k ∋ x 7→ −  F ′ ( g − 1 h, h − 1 x ) + F ′ ( h − 1 k , k − 1 x ) − F ′ ( g − 1 k , k − 1 x )  (13) since V g ∩ V h ∋ x 7→ F ′ ( g , g − 1 h ) ∈ Z is a 1- co chain. Similarly , p er ω ◦ Θ β = d gp F ω ,β implies that η ( q ◦ p er ω ◦ Θ β , V ) is equiv alent to the co c ycle V g ∩ V h ∩ V k ∋ x 7→ − q  F ω ,β ( g − 1 h, h − 1 x ) + F ω ,β ( h − 1 k , k − 1 x ) − F ω ,β ( g − 1 k , k − 1 x )  . (14) Since F ω ,β and F ′ bo th integrate ω , their restr ic tio ns to some op en neighbo ur ho o d (which we may still assume to b e U ) a r e equiv alent as lo cal co cycles . An y lo cal gro up co b oundary b et ween them gives rise to a cob oundary b etw een the ˇ Cech co cycles ( 13 ) and ( 14 ). Corollary I.18. The gener alise d c o cycle ( F ω ,β , Θ β ) fr om Re mark I.8 is un iversal for gener alise d c o cycles inte gr ating ω . This me ans that if ( F ′ , Θ ′ ) inte gr ates ω (cf. Deﬁnition I.10 ), then ther e exists some p ∈ Hom( π 2 ( G ) , A ) and a morphism ( ϕ, ψ ) : ( q ◦ F ω ,β , p ◦ Θ β ) → ( F ′ , Θ ′ ) of gener- alise d c o cycles. Mor e over, the existenc e of ( ϕ, ψ ) determines p uniquely. 11 I I L o op prolongations In this section we pr ovide the minimal a lgebraic structure that the g eneralised co cycles from the previous sectio n yield. Ho wever, it will turn out that these a lgebraic structures do not mix very well with the underlying smo o th str uc tur es, so we will go to s lig htly a dv anced algebraic structures in the next sectio n to treat smo othness issues appr opriately . Deﬁnition I I.1. A lo op is a set X together with a ma p µ : X × X → X , ( x, y ) 7→ µ ( x, y ) =: x · y and a distinguishe d element e ∈ X such that x · e = e · x = x for all x ∈ X and such that the maps λ x , ρ x : X → X , λ x ( y ) = x · y a nd ρ x ( y ) = y · x a re bijective. A morphism b et ween tw o lo ops X and Y is a map ϕ : X → Y satisfying ϕ ( x · y ) = ϕ ( x ) · ϕ ( y ). Remark I I.2. (cf. [ EM47 ]) Let X b e a lo op. The n in genera l we do no t hav e that ( x · y ) · z equals x · ( y · z ), but since ρ ( x · y ) · z is bijectiv e, there exists a unique element A ( x, y , z ) such that x · ( y · z ) = A ( x, y , z ) · (( x · y ) · z ) . W e call the map A : X × X × X → X the asso ciator of X a nd sometimes refer to A ( x, y , z ) as the asso ciato r of ( x, y , z ). W e hav e to add the following data and a ssumptions in o rder to come fr om gener al lo ops to group c ohomolog y . Suppos e that we hav e a homomorphism ϕ : X → H for H an arbitrar y group. Then the kernel of ϕ is a no rmal sublo op 7 of X and since H is a gro up, all asso ciato rs are contained in ker( ϕ ). If we choose a sub gr oup A o f ker( ϕ ) and assume that ϕ is surjective , (15) A ( k , x, y ) = A ( x, k , y ) = e for all x, y ∈ X and k ∈ ker( ϕ ) , (16) k · A ( x, y , z ) = A ( x, y , z ) · k for all x, y , z ∈ X and k ∈ k er( ϕ ) , (17) x · a = a · x for all x ∈ X a nd a ∈ A, (18) then we c all ( A, ϕ : X → H ) a (gener al) lo op pr olongation of H by A (cf. [ EM47 , Sect. 4]). In this ca se, one can actually show that ker( ϕ ) is a sub gr oup o f X [ E M47 , (4.6 )] and tha t A a nd each asso cia tor is contained in Z (k er( ϕ )). It has be en shown in [ EM47 ] that if we assume, in addition, that a ll a sso ciator s ar e contained in A and tha t A ( x, y , k ) = e for each k ∈ K , then A factors through a map fro m H × H × H → A which is in fact a co cycle. W e will from now on assume that this is the case (and drop the adjective “ general” to indicate this in the notation). Note that the ass ignment o f a 3-co cy c les to a lo op pr olongatio n gives ris e to a gr oup is o- morphism betw een (equiv alence classes o f ) lo op pr olongatio ns and H 3 ( G, A ) (with resp ect to a suitably deﬁned pro duct, see [ EM47 ]). Lemma I I.3. L et ( F, Θ ) b e a gener alise d c o cycle on the discr ete gr oup H with c o eﬃcients A ⊆ Z , also discr ete. Then µ (( z , g ) , ( w, h )) = ( z + w + F ( g , h ) , g h ) endows Z × H with the structur e of a lo op, which we denote by Z × F H . Mor e over, if q : Z → Z/ A is the c anonic al quotient homomorphism, then ϕ : Z × F H → ( Z/ A ) × ( q ◦ F ) H , ( z , h ) 7→ ( q ( z ) , h ) deﬁnes a lo op pr olongation of ( Z/ A ) × ( q ◦ F ) H by A . The gr oup 3-c o cycle asso ciate d to t his lo op pr olongation then e quals Θ . Pro of. That Z × F H deﬁnes a lo op is dir ectly c heck ed from the deﬁnition, as well a s conditio n ( 15 ), ( 17 ) and ( 18 ). F r om ( 1 ) is follows immediately that ϕ deﬁnes a homomor phism. Since (( z , g )( w, h ))( v , k ) = ( z + w + v + F ( g , h ) + F ( g h , k ) , g hk ) = (Θ( g , h, k ) , e )  ( z + w + v + F ( h, k ) + F ( g , hk ) , g hk )  = (Θ( g , h, k ) , e )  ( z , g )(( w, h )( v , k ))  7 One should not get confused by the diﬀeren t notions of normal sublo ops, for instance in [ NS02 ] [ Pﬂ90 ], [ Bae45 ] or the one used in [ EM47 ]. One easily chec ks that they are all equiv alent, the probably easiest one is present ed, for instance, in [ KK04 ]. In particular, the usual ke rnel-epimorphism corresp ondence goes through, see [ Bae45 ] or [ NS02 , p. 14] and r eferences given there. 12 follows from d gp F = Θ, we have A (( z , g ) , ( w , h ) , ( v , k )) = (Θ( g , h , k ) , e ). Thus ( 16 ) follows from the normalisa tion conditions of ( F , Θ). F rom the ab ove eq uation it is also clear that Θ is the group 3-co cy cles asso cia ted to this lo o p pr o longation. The preceding lemma shows in par ticular that the integrating co c ycle ( F β ,ω , Θ β ) from Re- mark I.8 g ives r is e to a lo op prolonga tion and one might b e tempted to incor po rate smo othness assumptions into the game. But we s hall see now that in gener al one do es no t hav e a s mo oth structure o n z × F ω,β G which ha s z × V (the subset o n which µ alrea dy is smo oth) as an op en subset. Example I I.4. Let G = C ∞ ( S 1 , SU 2 ), g = C ∞ ( S 1 , su 2 ) a nd ω h· , ·i be the Ka c -Mo o dy co cycle from ( 2 ). If we normalis e h· , ·i such that the left-inv ariant extension of h [ · , · ] , ·i is a generator of H 3 dR (SU 2 , Z ), then it follows from the calculation in the pr o of of [ MN03 , Thm. I I I.9] that per ω h· , ·i = Z . Thus f := q ◦ F ω h· , ·i ,β is a 2-co cycle (cf. Remar k I.8 ). W e th us obtain a cen tral extension of L ie groups U (1) → b G → G with b G := U (1) × f G . F rom [ Nee02 , Pro p. 5.1 1 ] it follows that the connecting homomorphism π 2 ( G ) → π 1 ( U (1)) in the long exa ct homo to py sequence of this ﬁbration is (up to the choice of a sig n) the identit y on Z . This implies that π 2 ( b G ) = 0. Now consider the lo op R × F G with F := F ω h· , ·i ,β from the previous lemma. If there existed a top olo gy o n R × G having R × V as a n op en subs e t (for V as in Re ma rk I.8 ) and turning µ int o a g lobally smo oth ma p, then the e x act sequence Z → R × F G → b G , induced by ϕ a s in the previous lemma, would deﬁne a lo cally trivial principa l bundle ov er ˆ G . In fact, the maps g · V ∋ x 7→ (0 , g ) · (0 , g − 1 · x ) ∈ R × G would provide smo oth lo cal sections (it is her e that we use the glo ba l smo o thness of µ ). Since this bundle is in fact a cov er ing a nd b G is simply connected, this cov ering would b e trivial and b G would b e diﬀeomorphic to Z × G . But π 2 ( G ) = Z , a co ntradiction to the existence of a globa lly smo oth extension of the lo cally smo oth multiplication. Remark I I.5. It is a lwa ys the c a se that a gr oup which r estricts to a lo c al (analytic) Lie group po ssesses a compatible (global) s mo oth (analy tic) str ucture, at least if the g roup is genera ted by the lo cal group (see Theorem IV.1 fo r the precise statement). It is a well-kno wn fact that an analogous statement do e s not hold for lo ops. In fact, ea ch ﬁnite-dimensional almost smo o th lo op restricts to a s mo oth lo cal lo op on some neighbour ho o d of e . If there ex isted a compatible smo oth structur e on this lo op, then the identit y would yield a diﬀeomo rphism b et ween this s mo oth structure and the almo s t smo oth str uc tur e. This is due to the fact tha t a mor phism b etw een almost smo oth lo ops is smo oth if and only if it is so on an op en neighbourho o d of e . So ea ch almost smo o th lo op which is not a smo oth lo o p yie lds a counterexample to the g eneralisatio n o f the introductor y statement of this rema rk to lo ops. See [ NS02 , Sect. 1 .3 ] fo r details. In fact, the situation is e ven worse, there ex is t one-dimensional analytic lo ops which may not b e ex tended to a global analytic lo op [ HS90 , Lines b efore Rem. IX.6.8]. As the previo us discussion shows, we are forced to co nsider as integrating ob jects of ω lo op prolonga tions, which are compatible with the s mo oth structure o nly in an iden tit y neighbour - ho o d: Deﬁnition I I.6. A loo p prolo ngation ( A, ϕ : L → H ) is ca lled lo c al ly smo oth if L is endow ed with the structure of a smo oth lo cal lo o p. W ith this we mean that there exists a subset W containing the iden tit y , whic h is endow ed with some manifold structure suc h that L | W := ( µ − 1 ( W ) ∩ ( W × W ) , W , µ, 0) is a lo cal lo o p and all s tr ucture maps are smo oth (cf. [ NS02 , Sect. 1.1]). In particular, the ma nifold structure is part of the data. If, in a dditio n, the as s o cia- tor o f ( A, ϕ : L → H ) v a nishes o n so me identit y neighbourho o d (in W × W × W ), then we call the lo op prolonga tion lo c al ly asso ciative . 13 Of cours e, lo cal smo othness and lo cal asso ciativity make sense for lo o ps in genera l, but we shall use this concepts only for lo ops that ar e parts of a lo o p pr o longation. Lemma I I.7. If ( A, ϕ : L → H ) is a lo c al ly sm o oth lo op pr olongation and L is gener ate d by some op en V ⊆ W with V ∩ A = { e } , then L/ A c arries a Lie gr oup structure such that the quotient map q : L → L/ A r estricts to a lo c al diﬀe omorphism on some op en neighb ourho o d of e . Pro of. Since V ∩ A = { e } , q is injectiv e on V and we use it to endow q ( V ) ⊆ L/ A with a manifold structure . Clear ly , the gro up multiplication and inv ersion are lo cally smo o th on A/L with resp ect to this smo o th structur e. Since V generates L we hav e that q ( V ) generates L/ A and the asser tion follows from Theore m IV.1 . Remark I I.8. If ( A, ϕ : L → H ) is a lo cally smo oth and lo ca lly a s so ciative lo o p prolonga tion, then L | V is a lo cal Lie group for s o me op en identit y neig hbourho o d V ⊆ W . This then gives rise to a Lie algebra L ( L ), which is indep endent of the choice of V . Prop ositi o n I I.9. If g is a Mackey-c omplete lo c al ly exp onential Lie algebr a, then ther e exists a lo c al ly smo oth and lo c al ly asso ciative lo op pr olongation such that the asso ciate d Lie algebr a is isomorphi c t o g . Since the up coming se c tions are indep endent of this result, there is no harm in p ostp oning the pro of unt il the end of Section VI . I I I 2-groups W e now intro duce 2-gr oups in o der to ov e r come the discrepa nc y b etw een the globally deﬁned algebraic structure and the lo cally given smo o thness in the next section (cf. Remark IV.9 ). Deﬁnition I I I.1. A (unital) 2-gr oup is a small gro upo id G , tog e ther with a mult iplic ation func- tor ⊗ : G × G → G , an inversion functor : G → G and an ob ject 1 (frequently identiﬁed with its ident it y morphism id 1 ), together with natural isomo r phisms α g,h ,k : ( g ⊗ h ) ⊗ k → g ⊗ ( h ⊗ k ) for g , h, k ob jects of G , called asso ciators . W e require 1 = 1 , g ⊗ 1 = g = 1 ⊗ g a nd g ⊗ g = 1 = g ⊗ g on ob jects and morphisms and we require that α g,h ,k ⊗ l ◦ α g ⊗ h, k,l = (id g ⊗ α h,k,l ) ◦ α g,h ⊗ k,l ◦ ( α g,h ,k ⊗ id l ) for all g , h, k , l . The la st req uir ement is frequently r eferred to as p entagon identity . Moreover, we require that α g,h ,k is an identit y if one of g , h or k is 1 and that α g, g,g = id g and α g ,g ,g = id g . A morphism of (unital) 2-g roups is a functor F : G → G ′ , to gether with natural iso morphisms β g,h : F ( g ) ⊗ ′ F ( h ) → F ( g ⊗ h ), s atisfying F ( 1 ) = 1 ′ , F ( g ) = F ( g ), β g, 1 = id F ( g ) = β 1 ,g and β g, g = 1 ′ = β g ,g for all ob jects g of G . Here, we re quire F ( α ( g , h, k )) ◦ β g ⊗ h, k ◦ ( β g,h ⊗ ′ id k ) = β g,h ⊗ k ◦ (id F ( g ) ⊗ ′ β h,k ) ◦ α ′ ( F ( g ) , F ( h ) , F ( k )) for all g , h, k . Finally , a 2-morphism b etw e e n morphisms F and F ′ consists of natura l is o mor- phisms γ g : F ( g ) → F ′ ( g ) such that γ g ⊗ h ◦ β g,h = β ′ g,h ◦ ( γ g ⊗ ′ γ h ) . The resulting 2-ca tegory is denoted by 2-Grp . 14 W e to o k the clumsy notation for the inv er s ion functor to distinguish it explicitly fr o m the functor G → G op that maps each mo rphism to its inverse morphism. Non-unital 2- groups in volv e additional natural isomorphis ms, replacing the identities g ⊗ 1 = g = 1 ⊗ g and g ⊗ g = 1 = g ⊗ g , which are themselves req uired to ob ey certain coherence conditions. How ever, in o ur constructions these iso morphisms shall alwa ys b e identit ies which is why w e excluded them from our deﬁnition. This is also why we drop the adjective “unital” in the sequel. Remark I I I.2. 2 -gro ups a re sp ecial kinds o f monoidal categorie s (cf. [ BL04 ]), just as gr oups are sp ecial kinds of monoids. How ever, obs e r ve that a group is a monoid with a sp ecial pr op erty (existence o f inv er ses), while a 2 -group is a mo no idal catego ry with an additional structu r e (an inv ersio n functor). Our 2-gr oups are also examples of coher ent 2-groups, as considere d in [ BL04 ]. Example I I I.3. A particular imp ortant cla ss of examples form the so calle d strict 2-gr oups . They can b e characterised to be 2- groups, for which all natura l is omorphisms in Deﬁnition II I.1 are the identities. Besides this description, strict 2- groups can b e descr ibed by c r ossed mo dules as follows (in fact, the 2-ca teg ory of s trict 2-g roups is 2-equiv ale nt to the 2-categ ory of cr ossed mo dules, cf. [ Por08 ], [ FB0 2 ] or [ Lo d82 ]). A cr osse d mo dule is a morphism of groups τ : H → G (for G now an arbitrary group), together with an automo rphic actio n of G o n H such that τ ( g .h ) = g · τ ( h ) · g − 1 (19) τ ( h ) .h ′ = h · h ′ · h − 1 . (20) Note that these tw o equations force ker( τ ) to b e central in H a nd im( τ ) to be norma l in G . F r om this one can build up a 2- g roup G as follows. The o b jects are G , the mo r phisms ar e H ⋊ G and the structure maps are given by s ( h, g ) = g , t ( h, g ) = τ ( h ) · g , id g = ( e, g ), ( h ′ , τ ( h ) · g ) ◦ ( h, g ) = ( h ′ h, g ). The identit y ob ject is e and the m ultiplication and in version functor ar e given by m ultiplication and inv ers ion on the g r oups G and H ⋊ G . Then ( 19 ) and ( 20 ) ensure that this deﬁnes a functor with the desired prop erties if we set the as so ciator from Deﬁnition I I I.1 to be the ident ity . Example I I I.4. W e o bta in a slightly weak e r version of the previo us example if we a r e given instead o f a crosse d mo dule a lo o p pro longation ( A, ϕ : L → G ). F rom this we cons tr uct an (in general non-s trict) 2 -group as follows. The set o f ob jects is L , the set of mor phis ms is A × L and the structur e maps are given b y s ( a, x ) = x , t ( a, x ) = a · x and ( a, b · y ) ◦ ( b, y ) = ( a + b, y ). Then we s e t x ⊗ y = x · y and ( a, x ) ⊗ ( b, y ) = ( a · b, x · y ), which clea rly deﬁnes a functor. Since the lo op L may fail to b e asso ciative, this only deﬁnes a 2-gr oup if we introduce the as s o ciators α x,y ,z = ( A ( x, y , z ) , ( x · y ) · z ). One readily chec ks that this deﬁnes a 2 -group with the aid of the axioms of a lo op prolo ng ation from Remark II.2 . W e ﬁnish this section with a few co nstructions that will b e impo rtant later on. Remark I I I.5. E ach 2-gr oup comes along with a couple of natura l gro ups asso ciated to it. • The set of isomorphism class es π 0 ( G ) of G . Since ⊗ is a functor , it induces a ma p π 0 ( G ) × π 0 ( G ) → π 0 ( G ). This clearly deﬁnes a gr oup multiplication for isomo rphic ob jects in G bec ome equal in π 0 ( G ). • The set G 1 of mo rphisms in the full s ub ca tegory of G , generated b y 1 . On G 1 , we deﬁne a map G 1 × G 1 → G 1 , ( g , h ) 7→ g ⊗ h. If we as s ume that α g,h ,k is an identit y if one of g , h o r k is isomo r phic to 1 , 8 then this deﬁnes an asso cia tive m ultiplication on G 1 . Since f ∈ G ⇔ f ∈ G 1 , this is in fact a group. 8 This should foll ow from coherence , but we were not able to ﬁnd a r eference for it. How ever, all 2-groups that we encount er in this article ob ey this condition 15 • The sourc e and targ et ﬁbr es s − 1 ( 1 ) and t − 1 ( 1 ) are a subgroup of G 1 . • The endomor phis ms π 1 ( G ) := End( 1 ) = s − 1 ( 1 ) ∩ t − 1 ( 1 ) of 1 for m a subgroup of G 1 . IV Lie 2-groups In this section we shall elab or ate on our concept of smo othness o n 2- groups. Note the s ubtlety that we ca ll our o b jects of main interest Lie 2-gro ups, so we emphasise their Lie-theor etic int er- pretation. Other authors ca lls their corresp onding ob jects smo oth 2-gr oups to put an emphasis on their prop erties as gener alisations o f smo o th manifolds. Since o ur 2 -groups are internal to sets (for they are assumed to b e small ca tegories), it seems to b e natura l to work internal to ma nifolds (i.e., require sets to b e manifolds and ma ps to b e smo oth), but this turns out to b e to o r estrictive. The p ersp ective to Lie groups that we will follow for our no tio n of Lie 2-g roups is that a Lie g roup is a gr oup with a lo cally s mo oth gro up m ultiplication. W e make this precise in the following theor em. No te that Lie 2 -groups ma ke sense for smo oth spaces in a mor e gener al setting than just lo cally conv ex manifolds, but to stay clear and brief we will stick to the manifold ca s e. Theorem IV.1. L et G b e a gr ou p, U ⊆ G b e a subset c ontaining e and let U b e endowe d with a manifold st ructur e. Mor e over, assume t hat t her e exists V ⊆ U op en such that i) e ∈ V , V = V − 1 and V · V ⊆ U , ii) the maps V × V ∋ ( g , h ) 7→ g h ∈ U and V ∋ g 7→ g − 1 ∈ V ar e smo oth, iii) V gener ates G (as a monoid or, e quivalently, as a gr oup). Then ther e exists a unique Lie gr oup st ructur e on G such that the inclusion U ֒ → G is a diﬀe o- morphism on some op en identity neighb ourho o d. In p articular, V ֒ → G is a diﬀe omorphism onto its op en image and any other choic e of V satisfying t he ab ove c onditions gives the same s mo oth structur e on G . Pro of. The pr o of is standard, see for instance [ Bou98 , Pro p. I I I.1 .9 .18]. How ever, we shall rep eat the essential parts to illustrate the general idea. Let W ⊆ V be op en such that e ∈ W , W · W ⊆ V and W = W − 1 . Then we transp or t the smo oth structur e from W to g W b y left tra nslation λ g : W → g W (i.e. w e deﬁne λ g to b e a diﬀeomorphism). This is well-deﬁned since fo r g W ∩ hW 6 = ∅ we hav e h − 1 g ∈ V so that the co ordinate change λ − 1 g ( g W ∩ hW ) ∋ x 7→ λ h − 1 g ( x ) ∈ λ − 1 h ( g W ∩ hW ) is smo oth by ii) . In particular, V ⊆ G is o p en a nd V ֒ → G is a diﬀeomorphism onto its image. T o verify that the group m ultiplication is smo o th, we ﬁr st o bserve that fo r ea ch h ∈ G there exists W h ⊆ V ope n with e ∈ W h such that h − 1 W h h ⊆ V and x 7→ h − 1 xh is smo oth. In fact, the set o f a ll h ∈ G s uch that W h exists forms a sub monoid co nt aining V , which equals G by iii) . Thus, g W h × hW ∋ ( x, y ) 7→ xy = λ gh  ( h − 1 · λ − 1 g ( x ) · h ) · λ − 1 h ( y )  ∈ g hV is smo oth. A similar argument shows that inv ersion is a lso smo oth. If G is endowed with a Lie group str ucture s uch that U ֒ → G restr icts to a diﬀeomorphism on V ′ , then the r estriction of id G is smo oth on V ′ ∩ V . Since a homomorphism b etw een Lie groups is smo oth if a nd only if it so on an identit y neighbourho o d, this shows that id G is in fac t a diﬀeomorphism. This applies in particular to a p ossibly diﬀerent choice of V . 16 Note that the pr e v ious theorem tells us that the gr o up structure determines the global top ol- ogy , as so on a s the lo cal top o logy is ﬁxed. It a lso says that a Lie group may equally well be deﬁned as a gro up G , together with the smo oth structure on U such that V ⊆ U op en with the corres p o nding prop erties exist. This is a very familiar pattern in Lie theo ry a nd we shall take this p ersp ective when deﬁning Lie 2-g r oups b elow. W e hav e alrea dy seen that a s ta tement co r - resp onding to the pr eceding theo rem is not v alid for lo ops a nd we shall see b elow that non-strict 2-gro ups yet hav e a diﬀerent b ehaviour. The following example illustrates an imp ortant a pplication of the preceding theorem. Example IV.2 . Let G b e an arbitrar y co nnected Lie g r oup and let f : G × G → Z be a 2- co cycle, which is smo o th on some iden tit y neig hbourho o d U × U . This gives a group Z × f G as in ( 1 ). T a k ing V ⊆ U op en with e ∈ V , V − 1 = V and V · V ⊆ U shows that this mult iplication is smo oth when restricted to ( V × G ) 2 (a similar argument w o rks for the inv ersion, since ( a, g ) − 1 = ( − a − f ( g , g − 1 ) , g − 1 )). Thus Theo r em IV.1 yields a Lie g r oup str uctur e on Z × f G . This turns Z → Z × f G → G into a central extension o f Lie g roups, p o s sessing U ∋ x 7→ (0 , x ) ∈ Z × f G as smo oth lo ca l section . This construction applies in par ticular to the 2 -co cycle q ◦ F ω h· , ·i ,β from Example II.4 . Another imp ortant applica tio n of this construc tio n is the following. Let P G b e the space o f contin uous p ointed paths in G and G → P G , g 7→ α g be a sec tion of the map that ev aluates in 1 (the compact-op en topo logy deﬁnes in fact a Lie gro up to p o logy on P G with Lie a lg ebra P g ). Moreov er, assume that α is smo oth o n so me identit y neighbourho o d. Then Θ α ( g , h ) = [ α g + g .α h − α gh ] ∈ H 1 ( G ) ∼ = π 1 ( G ) is a g r oup co cycle for the universal cov er ing gro up e G ∼ = P G/ (Ω G ) 0 . In fact, reconstr uc ting co cycles from the central extensio n π 1 ( G ) → e G → G as in [ Nee02 , Pro p. 4.2 ] shows that Θ α is equiv alent to each of those co c ycles. W e now turn to the developmen t of o ur notion of Lie 2-group. At ﬁrst, we ne e d the concept of a smo oth 2-space. Deﬁnition IV.3. A smo oth 2-sp ac e is a (p oss ibly inﬁnite-dimensiona l) L ie group o id. This means that it is a gro upo id C such that C 0 and C 1 are endow ed with smo oth manifold struc- tures, source and targ e t maps ar e smo oth surjective s ubmer sions 9 and the other structure maps are smo oth. A smo oth funct or b etw een smo oth 2-space s is a functor whose res p ective maps on o b jects and morphisms ar e smo oth. A smo oth n atur al tr ansformation b etw e en smo oth functors is a natural transformatio n such tha t the corr esp onding map from o b jects to morphisms is smo oth. The resulting 2 -categor y is denoted by 2-M an . Two smo oth 2-spaces C a nd D ar e c alled isomorphic if there exist smo oth functors F : C → D and G : D → C such that F ◦ G = id D and G ◦ F = id C (on the nose). The corr e c t notio n of equiv a lence of smo oth 2 -spaces is Morita e quivalenc e , a more inv o lved notion than the naive one. W e shall not need this notion in this a rticle. The previo us deﬁnition takes Lie gr oup oids a s int ernal categor ies in the ca tegory of lo cally conv ex manifolds. F or mor e general purp oses as we are aiming for here this deﬁnition is insuﬃcient. The c a tegory of lo c ally conv ex ma nifo lds has bad categ orical pro p erties: it lacks pull-ba cks, quotients and in ternal ho ms, also when restricting to ﬁnite-dimensio nal ones. This can b e remedied by intro ducing smo oth 2-spaces as categor ies internal to smo o th s pa ces (als o called Chen- or diﬀeolo gical spaces), for which we refer to [ BH08 ]. The fo llowing Pro po sition is the equiv alent statement to the pr evious theo rem for st rict 2 - groups. In order to state it we ﬁst have to introduce the following no ta tion. 9 Surjectiv e submersion i n the s tr ong sense that it is a pro jection i n lo cal coor dinates. This ensures in particular that the space of comp osable morphisms C 1 × C 0 C 1 is a smo oth manifold (cf. [ NSW11 , App. A]). 17 Remark IV. 4 . Let C b e a small monoidal catego ry (e.g., a 2 -group) and V ⊆ C b e a sub categ ory . Then the monoidal sub c ate gory gener ate d by V is the smallest monoidal subca tegory containing V . Since intersections o f monoidal sub categor ies a re in turn monoidal sub categ ories, there is a unique smallest monoidal sub catego ry , which we denote by hV i . Prop ositi o n IV.5. L et G b e a strict 2-gr oup, U b e a ful l su b c ate gory c ontaining 1 and let U b e endowe d with the structu r e of a smo oth 2-sp ac e. Mor e over, assum e t hat ther e exist s a ful l sub c ate gory V ⊆ U su ch that V 0 is op en in U 0 , i) 1 ∈ V , V = V and V ⊗ V ⊆ U , ii) the functors ⊗| V ×V : V × V → U and | V : V → V ar e smo oth, iii) hV i = G . Then ther e ex ists on G the structur e of a smo oth 2-sp ac e such that and ⊗ ar e smo oth functors and the inclusion U ֒ → G r estricts to an isomorphism on some ful l s u b c ate gory V ′ with V ′ 0 ⊆ U op en. Mor e over, the smo oth structur e on G is u n ique with r esp e ct t o t hese pr op ert ies. The following pro of r elies heavily on the fact that strict 2- groups are a ctually ca tegory ob jects int ernal to the categ ory of gr oups, i.e., spaces of ob jects, morphisms and c o mp o sable mo rphisms are groups and all structure maps are gr oup homomo rphisms (cf. [ Por08 ], [ BL04 ], [ FB02 ]). Pro of. It is clear from the assumptions that U 0 ⊆ G 0 is a subset containing the identit y which is endow ed with a ma nifold structure and V 0 ⊆ U 0 is an op en subs e t satisfying the assumptions from Theorem IV.1 . This yields a smo oth structur e on G 0 . On morphisms, we hav e the smo oth manifold U 1 = s − 1 ( U 0 ) ∩ t − 1 ( U 0 ) containing id 1 and the op en subset V 1 = s − 1 ( V 0 ) ∩ t − 1 ( V 0 ). Now V 1 generates G 1 by a ssumption, so Theor em IV.1 als o yields a smo o th structure on G 1 . Moreov er, s a nd t are submersio ns since they ar e so on U 1 and the smo othness of and ⊗ is part of the co nclusion of Theor em IV.1 . The uniqueness asser tion also follows immedia tely from the one in Theorem IV.1 . It might loo k quite pro mising to exp ect a similar constructio n of globally smo oth 2-gr oup structures from lo ca lly ones also in the case of non-strict 2 -gro ups , but this exp ectation is to o optimistic. In fact, the following lemmata show that the top olog y of G 1 splits as a pro duct in this case into the part that comes from the identities a nd the arrow part. Lemma IV.6. L et G b e a 2-gr oup which is also a smo oth 2-sp ac e su ch t hat the functors , ⊗ and the asso ciator ar e smo oth. Then s − 1 ( 1 ) is a submanifold of G 1 and in p articular a Lie gr oup. Mor e over, s − 1 ( 1 ) × G 1 → G 1 , ( a, f ) 7→ a ⊗ f deﬁnes a smo oth action. Mor e over, this action is fr e e, G 1 /s − 1 ( 1 ) ∼ = G 0 as m anifolds and G 1 is a trivial smo oth princip al s − 1 ( 1 ) -bund le. Pro of. Since inv er se images of po ints under submers io ns a re submanifolds, s − 1 ( 1 ) is a Lie gro up. That the action is free follows from a ⊗ f = b ⊗ f ⇒ a ⊗ ( f ⊗ f − 1 ) | {z } = 1 = b ⊗ ( f ⊗ f − 1 ) | {z } = 1 ⇒ a = b. The source map G 1 → G 0 is s − 1 ( 1 )-inv a r iant and thus induces a smo o th map G 1 /s − 1 ( 1 ) → G 0 . The identit y map G 0 → G 1 provides a smo oth globa l section, proving the claim. Lemma IV.7. L et G b e a 2-gr oup which is also a smo oth 2-sp ac e su ch t hat the functors , ⊗ and t he asso ciator ar e smo oth. If s − 1 ( 1 ) is discr ete in the induc e d top olo gy then t he arr ow p art ( G 0 ) 3 ∋ ( g , h, k ) 7→ α ( g , h , k ) ⊗ id ( g ⊗ h ) ⊗ k ∈ s − 1 ( 1 ) ⊆ G 1 of α is lo c al ly c onst ant. 18 Pro of. This is due to the fact that smo oth maps b etw een lo ca lly co n vex manifolds are in par- ticular contin uous. The impo rtance of the previo us lemma is that we a re fo rced to work with 2-gro ups with s − 1 ( 1 ) discrete if we wan t a reasona ble interpretation of a Lie 2-gro up integrating an or dinary Lie algebra (cf. Sectio n V ). Thus it illustrates the limitation on building 2 -groups with to o many smo othness co nditio ns . How e ver, lo ca lly smo othness of the gro up mult iplication is esse ntial for passing from Lie 2 -gro ups to Lie 2-algebr as. In view o f Theo rem IV.1 and Prop osition IV.5 , the following deﬁnition seems to b e natural. Deﬁnition IV.8. A Lie 2-gr oup is a tuple ( G , U ) suc h that G is a 2-g roup a nd U is a full sub c ategory containing 1 , which is endow e d with the s tructure of a smo oth 2-space . Mor e over, there has to exist a full sub categor y V ⊆ U with V 0 ⊆ U 0 op en such that i) 1 ∈ V , V = V and V ⊗ V ⊆ U , ii) the functor s ⊗| V ×V : V × V → U and | V : V → V a re smo o th, iii) hV i = G . When working with Lie 2-g roups we will sometimes not mention U e x plicitly if it is understo o d. A morphism of Lie 2- groups is a morphis m of the underlying 2- groups such that the constituting functors and natura l transfo rmations restrict (and co-re strict) to s mo oth functors and natural transformatio ns o n some of the sub categorie s V from a b ove. Likewise, 2-morphisms betw e e n morphisms of Lie 2-gr oups are deﬁned. F or the case of a non- strict 2-g r oup our notio n of a Lie 2 -gro up do es not ﬁt with the notion used in [ BL04 , Def. 2 7], where the functors and natura l tra ns formation deﬁning the 2-g roup structure are required to b e globally smo oth. F or the reas ons explained ab ov e we ﬁnd our no tio n more natura l in the non-strict ca se (see also [ B L 0 4 , Thm. 59 ]). The obser v ation that the co nc e pt int ro duced in [ BL04 ] is sometimes inadequate has a ls o b een made b y Henr iques in [ Hen08 , Sect. 9] (the latter notion of smo oth 2-gro ups has also b een used in [ SP 11 ]). Howev er, the pr evious deﬁnition cov er s str ict L ie 2-g roups by Pr op osition IV.5 (cf. [ BC04 ], [ BL04 ], [ W oc1 1 ]). Mo reov er, it leads to a lo c ally smo oth g roup structur e o n the gro up π 0 ( G ) (in the appropria te catego ry where π 0 ( G ) is a smo oth space ) and th us a globa lly smo oth group structure thereon b y Theo rem IV.1 . W e thus may interpret our Lie 2 -groups as a ca tegoriﬁed version of a Lie gr oup, muc h like Lie group oids are categor iﬁed manifolds. Remark IV. 9 . It is the fact that 2-gr oups form a 2- c a tegory which makes them mor e interesting then lo op pro longations. Thu s 2-gr oups a llow for the notion of equiv a lence, which is w eaker than isomorphism. Mor eov er , the category of smo o th 2-spa ces also a llows for a w e aker notion of morphisms, the so-called Hilsum-Sk andalis morphisms (also called spans or weak morphisms or bibundles) b etw een Lie gro upo ids. Making use of this conce pt, one can show that certa in Lie 2-gro ups in the ab ove sense ar e in fact eq uiv a lent to weak gro up o b jects in the weak 2 -catego r y of Lie gr oup oids with morphisms the a fo rementioned Hilsum-Sk anda lis mo rphisms (the la tter are called st acky Lie gr oups in [ Blo08 ]). This applies in particular to the 2- groups that we will construct in Theor e m V.16 (cf. Remark VI I.2 a nd [ ZW ]) a nd to the o ne from the following example (cf. [ SP11 ]). Although it do es not play a r ˆ ole in the main theme o f the pape r , we pr esent the following example for it illustrates the use and simplicity o f o ur co ncept of Lie 2-gr o ups. Example IV.1 0 . Let G be compact, simple and simply connected. Then h [ · , · ] , ·i is a Lie- algebra 3 -co cycle o n g , where h· , ·i denotes the Killing form of g . Under this assumptions the left- inv ar iant extension h [ · , · ] , ·i l is a g enerator of H 3 dR ( G, Z ) ∼ = Z . Consequently , the corres po nding per io d homomor phism per h [ · , · ] , ·i : π 3 ( G ) → R , [ σ ] 7→ Z σ h [ · , · ] , ·i l 19 (cf. [ Nee06 , Def. V.2.12]) has image Z . Now the maps α : G → C ∞ (∆ (1) , G ) and β : G 2 → C ∞ (∆ (2) , G ) from Lemma I.7 are accompanied b y an additional map γ : G 3 → C ∞ (∆ (3) , G ) satisfying ∂ γ g,h ,k = g .β h,k − β gh, k + β g,h k − β g,h (21) for the ab ove ass umptions on G imply that it is 2 -connected. Mor eov er , one can choose γ g,h ,k to depe nd smo othly on ( g , h, k ) in some neighbourho o d of ( e, e, e ), simila r to α and β in equations ( 7 ) and ( 8 ). T he n w e s et ϕ γ : G 3 → U (1) = R / Z , ( g , h, k ) 7→ exp Z γ g,h,k h [ · , · ] , ·i l ! where exp : R → R / Z is the ca no nical quo tien t map. This deﬁnes a 3-co cycle since d gp ( ϕ γ )( g , h, k , l ) = Z ( d gp γ )( g ,h, k,l ) h [ · , · ] , ·i l ∈ Z , which in turn follows from ( d gp γ )( g , h, k , l ) ∈ Z 3 ( G ), similar to Remark I.8 . This is in fact a lo cally smo oth 3 - co cycle and by [ Nee06 , Thm. V.2.6] we may diﬀerentiate this co cycle to g et back the Lie algebr a 3-co cycle h [ · , · ] , ·i . Similar to the argument from Remark I.12 we see that the cohomolog y class o f ϕ γ do es not dep end on the choice of γ , as long as ( 21 ) is fulﬁlled (that is w hy we drop the subscript from now on). F rom this co cycle we get a 2- group G G by setting ( G G ) 0 to b e G and ( G G ) 1 to b e U (1) × G with source a nd target map equal to the pr o jection to G and comp ositio n of mor phism induced by the group str ucture on U (1). The monoidal structure is given by the group m ultiplication in G (o n ob jects) and in U (1) × G (on morphisms) and the asso ciator is given by α g,h ,k = ( ϕ ( g , h, k ) , g hk ). Now ϕ is smo oth on U × U × U for U ⊆ G some op en identit y neighbourho o d. Cho osing some V ⊆ U o p en with e ∈ V , V = V − 1 and V 2 ⊆ U one directly chec ks tha t a ll requir ement s from Deﬁnition IV.8 a re satisﬁed. This turns G G int o a Lie 2-gro up. The natura l gener a lisation of the diﬀer e n tiation pro cess describ ed in the nex t se c tio n enables one to diﬀerentiate G G to a Lie 2-a lgebra. Since the diﬀerentiation of ϕ is the 3 -co cycle h [ · , · ] , ·i , this Lie 2-alge br a is the non-stric t Lie 2 -algebr a determined by the 3 -co cycle h [ · , · ] , ·i (cf. [ BC04 ]). This is (one mo del for) the string Lie 2-a lgebra, and the Lie 2 -gro up G G would thus b e another mo del for the string 2- group (cf. [ Sto96 ], [ ST04 ], [ BCSS07 ], [ Hen08 ], [ SP11 ] or [ NSW11 ]). There is certainly muc h more to say ab out this Lie 2-gr oup (cf. Remar k VII.2 ), but this lies b eyond the scop e of the present pap er. V Categoriﬁed cen tral extensions and ´ eta le Lie 2-groups In this section w e deﬁne central extensio ns of (Lie) 2- groups and show how they arise from generalise d co c y cles (cf. Remark V.3 ), for the more gener al setting see [ Bre9 2 ]. In particular , we seek for an interpretation o f the integrating co cy cle fr om Theorem I.11 in ter ms o f central extensions. In the ﬁrs t part of the section we shall describ e the route from gener alised co cycles to g e ner- alised cen tral extensio ns. The second part ela bo rates on the ba sic no tions o f Lie theory for Lie 2-gro ups and central extensions . O ur p ersp ective will b e that central extensions are describ ed by g roup cohomolog y , see [ Mac63 , Sect. IV.3] for ordina ry gr oups, [ AM08 ] for genera lis ations and [ Nee02 ] for the sp e cialisation to top ologica l and Lie groups . The approach to Schreier-like inv ar iants for extensions of gr o up o ids in [ BBF05 ] does not ﬁt into our situation, for our sequences of group oids shall not b e bijectively on o b jects. 20 Remark V. 1. In o rder to match the following deﬁnition with the situation of extensio ns of groups r ecall that a short e xact sequence A i − → B j − → C is a sequence o f order t wo such that the diagram A i / /   B j   ∗ / / C (22) is at the same time a pullback ( i injective and im( i ) ⊆ ker( j )) and a pushout ( j surjective and ker( j ) ⊆ im( i )). In the case that we are working with a strict 2-ca tegory we have to replace the (or dinary) pullback by a 2-pullba ck and likewise replace a pushout by a 2 -pushout. If X f − → Z and Y g − → Z are morphisms in a 2- categor y , then a 2-pul lb ack consis ts of an o b ject, de no ted X × Z Y , 1- morphisms X × Z Y p − → X a nd X × Z Y q − → Y and a 2-isomor phis m ϕ : f ◦ p ⇒ g ◦ q such tha t the diagram X × Z Y p / / q   X f   Y g / / Z ϕ s { p p p p p p p p p p p p p p has the following universal pr op erty: F or a ny ob ject W that co mes equipp ed with morphis ms W m − → X and W n − → Y and a 2- isomorphism ψ : f ◦ m ⇒ g ◦ n there exists a morphism s : W → X × Z Y and 2 -isomorphis ms ξ : m ⇒ p ◦ s and ζ : q ◦ s ⇒ n such that W s $ $ m # # n $ $ X × Z Y p / / q   X f   Y g / / Z ϕ s { p p p p p p p p p p p p p p ζ w  w w w w ξ       = W m   n ! ! X f   Y g / / Z ψ y  | | | | | | | | | | . Moreov er, given another morphism W s ′ − → X × Z Y and 2-is omorphisms α : p ◦ s ⇒ p ◦ s ′ and β : q ◦ s ⇒ q ◦ s ′ such that W s / / s ′   X × Z Y p / / q   X f   X × Z Y q / / Y g / / Z ϕ t | q q q q q q q q q q q q q q β s { p p p p p p p p p p p p p p p p p p = W s ′   s / / X × Z Y p   X × Z Y p / / q   X f   Y g / / Z ϕ q y k k k k k k k k k k k k k k α q y k k k k k k k k k k k k k k , there has to b e a unique 2-iso mo rphism χ : s ⇒ s ′ such that W s ) ) s ′ 5 5 X × Z Y p / / X χ   = W p ◦ s & & p ◦ s ′ 8 8 X α   and W s ) ) s ′ 5 5 X × Z Y q / / X χ   = W q ◦ s & & q ◦ s ′ 8 8 X β   . Along the same lines, one deﬁnes 2-pushouts . Deﬁnition V.2. (cf. [ SP11 , Def. 6 6 ]) If τ : A → Z is a morphism o f ab elian gro ups (viewed as a cr ossed mo dule for the trivial actio n of Z on A ), then we denote the asso cia ted 2 -group from Example I I I.3 by Z τ , which we also call a strict ab elian 2-gr oup . F or an arbitra ry gro up G denote by G the 2-gr oup with ob jects g ∈ G , only identit y mo r- phisms a nd the 2- group structure induced by multiplication in G . Then we deﬁne a n ab elian 21 extension of G by Z τ to b e a s equence of 2-gro ups Z τ i − → b G q − → G such tha t q ◦ i is the co nstant functor 1 and that the diagra m Z τ i / /   b G q   ∗ / / G id 1 {            (23) is a 2-pullback and a 2 -pushout in the 2-ca tegory 2-Grp . Such an extensio n is ca lled c ent r al if the t wo functors Z τ × b G → b G , ( z , g ) 7→ g ⊗ ( i ( z ) ⊗ g ) and Z τ × b G → b G , ( z , g ) 7→ i ( z ) (24) are naturally isomorphic. Note that the fact that G has only iden tit y morphisms e nfo r ces us to put id 1 int o the 2 -cell of the ab ov e diagr a m. Remark V. 3. Let G b e a discrete gro up and A, Z be discre te a b elian g r oups. F or ( F, Θ ) a generalise d gr o up co cycle with co eﬃcients in τ : A → Z , giv e n by Θ ∈ C 3 ( G, A ) and F ∈ C 2 ( G, A ) satisfying ( 3 ) and ( 4 ), the following assig nmen t deﬁnes a 2-gr oup b G ( F, Θ) . The catego ry b G ( F, Θ) is given by Ob j( b G ( F, Θ) ) = Z × G, s ( a, x, g ) = ( x, g ) , t ( a, x, g ) = ( τ ( a ) + x, g ) Mor( b G ( F, Θ) ) = A × Z × G, id ( x,g ) = (0 , x, g ) , ( a, x, g ) ◦ ( b , y , g ) = ( a + b, y , g ) and the multiplication functor by ( a, x, g ) ⊗ ( b, y , h ) = ( a + b, x + y + F ( g , h ) , g h ) . Since F satisﬁes the co cycle identit y only up to cor rection by Θ, this assignmen t deﬁnes a monoidal categor y if we deﬁne 1 = (0 , e ) and α ( x,g ) , ( y ,h ) , ( z ,k ) = (Θ( g , h, k ) , x + y + z + F ( g , h ) + F ( g h , k ) , g hk ) . W e clearly have 1 ⊗ g = g = g ⊗ 1 , the source- target matching co ndition of α is equiv a lent to d gp F = τ ◦ Θ and the p entagon iden tit y is equiv alent to d gp Θ = 0. Moreover, ( a, x, g ) = ( − a, − x − F ( g , g − 1 ) , g − 1 ) deﬁnes an inversion functor on b G ( F, Θ) , tur ning it into a 2-g r oup. In addition to the 2-gr oup structure on b G ( F, Θ) , we hav e canonica l functors Z τ i − → b G ( F, Θ) and b G ( F, Θ) q − → G . Lemma V.4. In the sett ing of the pr evious r emark, Z τ i − → b G ( F, Θ) q − → G is a c entr al extension of G by Z τ . Pro of. W e ﬁrst notice that the functors fr om ( 24 ) are actually eq ual in this case, s o the ex tension will b e central. W e abbre viate Z := Z τ and G = b G ( F, Θ) . Ass ume that m : W → G is given such that q ◦ m = 1 . Then on ob jects we hav e that m 0 ( w ) ∈ q − 1 0 ( 1 ) = Z × e G ∼ = Z 0 and on morphisms we hav e m 1 ( v ) ∈ q − 1 1 ( 1 ) = A × Z × { e G } ∼ = Z 0 . So m factors (on the nose) thro ugh a morphism s : W → Z , i.e., we may cho ose ξ : i ◦ s ⇒ m (and of course also ζ : ∗ ◦ s ⇒ ∗ ) to b e the iden tit y natural transfo r mations. Moreov e r, if we have s ′ : W → Z and 2-iso mo rphisms α : i ◦ s ⇒ i ◦ s ′ such that q 1 ( α ( w )) = id 1 , then α ( w ) ∈ q − 1 1 ( 1 ) = A × Z × { e G } ∼ = Z 1 so tha t α factors through a 2-isomor phism χ : s ⇒ s ′ which obviously satisﬁes the r equirements. This shows that Z τ i − → b G ( F, Θ) q − → G is a 2 -pull back. Along similar lines one shows that it als o is a 2-pusho ut. 22 W e will now consider extensio ns of Lie 2-g r oups. Note that in the case o f Lie g roups (in [ Nee02 ]) or in the setting of smo oth 2 -groups (in [ SP 11 ]) there is an additional requirement on a sequence A i − → B q − → C besides that the diag ram fro m ( 22 ), resp ectively ( 23 ), is a (2- )pullback and a (2-)pusho ut. F o r Lie group extensions one r equires the existence of a smo oth lo cal section (this then implies that B → C is a lo cally triv ial principa l A -bundle) a nd in [ SP11 ] it is r equired that A i − → B q − → C is an A -ger be over C . In our treatment we res trict from now on to ´ etale Lie 2-groups, a concept that we a re heading for now. T his concept will b e tailored to ﬁt our Lie theoretic needs. Remark V. 5. Similarly to the concept of a smo oth 2-space (and smo o th functors a nd natural transformatio ns, cf. Deﬁnition IV.3 ), one deﬁnes (top olog ical) ve ct or 2-sp ac es to be internal categorie s in lo cally conv ex vector spaces, i.e., small ca tegories such tha t a ll sets o ccur ring in the deﬁnition o f a small categor y are loc ally co nvex spa ces, all structure maps are contin uous linear maps a nd source and ta rget ar e pro jections. Likewise, linear functors and na tural transformatio ns are deﬁned internally , deﬁning the 2-catego ry 2-V ect . There is a natur al functor T from the categor y Man pt (of p ointed manifolds with smo oth base-p oint preser ving maps) to the catego ry V ect (of top olog ical vector spaces with contin uous linear maps), sending manifolds to the tange n t spa ces at the bas e-p oint and s mo oth maps to their diﬀerentials at the bas e-p oint. Since this functor preserves pull-backs, it maps categor ies, functors and natural transforma tio n in Man pt to ones in V ect a nd thus deﬁnes a 2 - functor T : 2-Man pt → 2-V ect . If we wan t to enfor ce T to take v alues in V ect instead o f 2 -V ect , then we need a canonica l ident iﬁcation of T ( M ) 0 and T ( M ) 1 . This is the case if M is ´ etale , as deﬁned b elow. Deﬁnition V.6. A s mo oth 2 -space is called ´ etale if all structure maps are lo cal diﬀeomorphisms. A Lie 2 -group ( G , U ) is ca lled ´ etale if U is an ´ eta le 2-space. Morphis ms and 2-mor phis ms for ´ etale Lie 2-gr oups ar e deﬁned to be morphisms and 2-mor phis ms o f Lie 2-groups . The cor resp onding 2-catego ry is denoted by Lie2-Grp ´ et . Most o f the Lie 2-g roups that we sha ll e nc o unter in this a r ticle ar e ´ etale. Note that the diﬀerentials of lo cal diﬀeomorphisms give canonical identiﬁcations of the ta ngent spaces at the base-p oints. Thus T ( M ) is in fact a vector space for ´ etale M . W e shall make this precise for Lie 2-gro ups be low. Note also tha t s − 1 ( 1 ) is discre te in an ´ etale Lie 2 -group. In particula r, Lemma IV.7 applies to ´ e ta le L ie 2- groups with globally smo oth group op erations . Remark V. 7. The pass age from genera l Lie 2- groups to ´ etale o nes will have an eﬀect that ha s also b een used in [ TZ06 ] for the solution of the int egration pr oblem of (ﬁnite-dimensional) Lie algebroids (cf. [ CF03 ]). It is a lo ng -standing observ ation that Lie algebro ids integrate to lo cal Lie group [ Pra6 8 ], but in genera l the integrating Lie gr o up o id may not b e e nla rged to a g lobal Lie gr oup oid. The r easons for this failure is esse n tially the non- dis creteness of the image of a p erio d map (cf. [ CF03 , Sect. 3.2 a nd Th. 4.1 ]) for w hich [ CF03 , E x. 3.7] and [ TZ06 , Ex. 1] give examples, very close to the int egration problem that this article deals with. In [ TZ06 ] the int egrating ob jects ar e Weinstein gr oup oids , which a re a lso ´ eta le and categ o riﬁed replacements of Lie g r oup oids. The result fr om [ TZ06 ] can also b e seen a s integrating Lie algebroids to lo cally deﬁned Lie g roup oids and then so lve the asso cia tivit y-constra int by passing to W einstein group oids. In the same spirit, ´ etale Lie 2 -gro ups will b e the in tegrating ob jects for integration lo cally exp onential Lie alg ebras. Deﬁnition V.8. Let G b e an arbitrar y Lie gro up a nd τ : A → Z be a mor phism of ab elia n Lie groups with discr ete A . Then a smo oth gener alise d c entr al extension (s.g.c.e.) is a sequence Z τ i − → b G q − → G of ´ etale Lie 2-g roups such that p ◦ i = 1 and that the diag ram ( 23 ) is a 2- pullba ck and a 2-pushout in Lie2-Grp ´ et . Moreov er, we dema nd that there exists a smo oth functor 23 q : U → b G sa tisfying q ◦ s = id U , where U ⊆ G is some op en identit y neighbour ho o d and that the functors Z τ × b G → b G , ( z , g ) 7→ g ⊗ ( i ( z ) ⊗ g ) and Z τ × b G → b G , ( z , g ) 7→ i ( z ) are smo othly isomor phic when restricted to so me neig hbourho o d of (0 , e G ) ∈ ( Z τ × G ) 0 . The r equirement o n a s.g .c.e. to b e a sequence in ´ etale Lie 2-gr oup will e na ble us to take an easy way to central extens io ns of Lie algebr as (cf. Pro po sition V.14 ). The ´ etalness is not crucial for the deﬁnition to make sens e , the s ame deﬁnition o f course also works in Lie2-Grp . The following lemma is immediate fro m the deﬁnitions. Lemma V.9. If ( F, Θ) is a gener alise d c o cycle on G with c o eﬃcients τ : A → Z and A is discr ete, then t he 2-gr oup b G ( F, Θ) fr om R emark V.3 is c anonic al ly an ´ et ale Lie 2-gr oup and Z τ i − → b G ( F, Θ) q − → G is a s.g.c.e. The following prop osition describ es the wa y back from genera lised ce n tral extensions to or- dinary ones. It is the ca tegorical version o f the discreteness condition for per ω ( π 2 ( G )) from [ Nee02 ]. Prop ositi o n V.10. L et Z τ i − → b G q − → G b e a s.g.c.e. such that τ ( A ) ⊆ Z is discr ete. Then π 0 ( Z τ ) and π 0 ( b G ) c arry Lie gr oup structur es with mo del ling sp ac e z and z × g (r esp e ctively), turning π 0 ( Z τ ) π 0 ( i ) − − − → π 0 ( b G ) π 0 ( q ) − − − → G (25) into a c en tr al extension of Lie gr oups. Pro of. First we no te that π 0 ( Z τ ) ∼ = Z/τ ( A ) has a natural L ie gr oup structur e with Lie a lgebra z . Let s : U → b G b e a smo oth sec tion of q . Then ( π 0 ( q )) − 1 ( U ) ∼ = π 0 ( Z τ ) × U a s a set a nd we endow ( π 0 ( q )) − 1 ( U ) with the smo oth str uc tur e making this identiﬁcation a diﬀeomor phism. Since the gro up multip lication on b G is smo o th on an op en subc a tegory containing 1 , the g roup m ultiplication in π 0 ( b G ) is lo cally smo oth. Since U generates G , π 0 ( q ) − 1 ( U ) generates π 0 ( b G ) a nd the assertio n follows from Theore m IV.1 . Deﬁnition V.11. The induced cen tral ex tens ion ( 25 ) is called the b and of Z τ i − → b G q − → G . Corollary V.12. If ω : g × g → z is a Lie algebr a c o cycle and ( F , Θ) is a gener alise d c o cycle which inte gra tes ω (cf. Deﬁnition V.15 ) and if τ ( A ) ⊆ Z is discr ete, then the b and of b G ( F, Θ) is a c entr al extension Z/τ ( A ) → b G → G inte gr ating z → z ⊕ ω g → g . Pro of. T o see that the band of b G ( F, Θ) int egrates z → z ⊕ ω g → g we ﬁrst observe that for q : Z → Z/τ ( A ) the canonica l quotient map T q ( e ) : z = T e Z → T e ( Z/τ ( A )) is a n is omorphism for τ ( A ) is discr ete. Using this to identify z with T e ( Z/τ ( A )) the c laim follows from L ( q ◦ F ) = T q ( e ) ◦ L ( F ). Remark V. 13. W e now derive a Lie algebr a canonically a sso ciated to each ´ etale Lie group G . W e ﬁrst show that the as s o ciator α is trivia l on so me neighbour ho o d of 1 . Since G is ´ etale, the identit y map G 0 → G 1 is a lo cal inv ers e aro und 1 for b o th, s and t . Since α 1 , 1 , 1 = 1 , w e th us have id ◦ t ◦ α = id ◦ s ◦ α , which implies s ◦ α = t ◦ α on so me neighbourho o d of 1 . No w m ultiplying α ( g , h, k ) with id ( g ⊗ h ) ⊗ k deﬁnes a map with v alues in s − 1 ( 1 ), which is contin uous on some iden tit y neighbour ho o d and thus constantly 1 . Since α x, x,x is an identit y for each x and α is natural, all of this implies id ( g ⊗ h ) ⊗ k = ( α g,h ,k ⊗ id ( g ⊗ h ) ⊗ k ) ⊗ id ( g ⊗ h ) ⊗ k = α g,h ,k 24 on some iden tit y neighbourho o d, which yields ( g ⊗ h ) ⊗ k = g ⊗ ( h ⊗ k ) for g , h, k fro m some neighbour ho o d of 1 . Thus the multiplication functor deﬁnes on G 0 the structure o f a lo cal Lie gro up a nd induces on T 1 G 0 a Lie brack et. W e denote this Lie algebra by L ( G ). A simila r argument as ab ove s hows that for a mor phism F : G → G ′ of Lie 2-g roups, where G and G ′ are ´ eta le, we hav e F ( g ) ⊗ ′ F ( h ) = F ( g ⊗ h ) for g , h from some iden tit y neighbourho o d. Thu s F 0 induces a morphism of lo cal Lie groups and thus of Lie algebr as L ( F ) : L ( G ) → L ( G ′ ). Likewise, a 2 - morphisms θ betw een t w o such morphisms has to b e the identit y on some identit y neighbourho o d. Summarising, L : Lie2-Grp ´ et → LieAlg deﬁnes a 2-functor to the category of Lie alg ebras, considered as a 2 -categor y with only identit y 2-morphisms. Prop ositi o n V.14. L et Z τ i − → b G q − → G b e a s.g.c.e. Then L ( Z τ ) L ( i ) − − → L ( b G ) L ( q ) − − − → L ( G ) (26) is a c ent r al extension of Lie algebr as. Pro of. That ( 26 ) is sho rt ex a ct follows the fact that the 2-functor L pr eserves 2-limits, which turn in to ordinar y limits in LieAlg . The diﬀerential o f a section (o n ob jects) of Z τ i − → b G q − → G provides a linear and contin uo us section of ( 26 ). Deﬁnition V.15. F or a s.g.c.e. Z τ i − → b G q − → G its derive d cen tral extension is the cen tral extension ( 26 ). If z → b g → g is a to po logically split central e x tension, then it is said to inte gr ate to a smo o th genera lised central extension if there ex ists a s.g .c.e. suc h that its derived cent ral extension is equiv alent to z → b g → g . Theorem V.16. If g is the Lie algebr a of the simply c onne cte d Lie gr oup G , then e ach top olo g- ic al ly split c entr al extension z → b g → g inte gr ates to a smo oth gener alise d c entr al ex tension of ´ et ale Lie 2-gr oups. Pro of. W e may a ssume that b g is equiv a le ntly given by a Lie alg ebra co cycle ω : g × g → z , which we integrate to a Z p er ω -v alued c o cycle ( F ω ,β , Θ β ) by Theo r em I.11 for so me appr opriate choice of β . T he n Lemma V.9 yields a s.g.c.e. Z p er ω → b G ( F ω,β , Θ β ) → G . Let U, V ⊆ G b e op en identit y neighbour ho o ds such that F | U × U and Θ | U × U × U are smo oth and V · V ⊆ U . T o calcula te the derived central extension w e consider the restric tio n o f the m ultiplication functor m to the full sub categ ory with ob jects in z × U , where it is given by m 0 (( z , g ) , ( w, h )) = ( z + w + F ω ,β ( g , h ) , g h ) on ob jects. By the deﬁnition of the Lie bracket of a loca l Lie group, the Lie bra cket on L ( b G ( F ω,β , Θ β ) ) is given by (( z , x ) , ( w , y )) 7→ ( L ( F ω ,β )( x, y ) , [ x, y ]) and L ( F ω ,β ) = ω shows the claim. W e thus recover the classica l c a se of central ex tensions by passing from a genera lis ed central extension to its band in the case that p er ω ( π 2 ( G )) ⊆ z is discrete. Moreover, we c a n interpret the pro of o f the previous theorem as ﬁrst passing to a 2-co nnected cov er of G a nd then solve a trivial integration problem in the following s ense. 25 Remark V. 17. Let β : G 2 → C ∞ ∗ (∆ (2) , G ) b e the map from Lemma I.7 , a pplied to a chart ϕ with dϕ ( e ) = id g and Θ β : G 3 → π 2 ( G ) be the corr e sp onding gro up 3-co cyc le from Lemma I.5 . Then Θ β determines a n (in gener a l non-strict) Lie 2-g roup G := e G (0 , Θ) (cf. Le mma V.9 ), which we interpret as a n appropriate version o f a 2 -connected cov er of G (cf. [ PW ]). In par ticular, we hav e a smo oth generalis ed central ex tens ion B π 2 ( G ) → G → G , where B π 2 ( G ) is the strict Lie 2-gr o up, asso c iated to the cro s sed mo dule π 2 ( G ) → { ∗} . Now b G := b G ( F ω,β , Θ β ) can b e see n a s a central extension z → b G → G (when generalising central extensions of Lie 2-g r oups to non-´ etale ones in the obvious way). Summar ising, we hav e the commutativ e diagra m z / /   z / /   {∗}   Z p er ω / /   b G / /   G   B π 2 ( G ) / / G / / G with ex a ct rows and co lumns. Since G is a n ´ etale Lie 2-gr oup with Lie algebr a g , one may int erpret z → b G → G also as a central extension of ´ etale 2-gro ups int egrating z → b g → g . VI Lie’s Third Theorem W e conclude this pap er with the following gene r alisation of Lie’s Third Theore m. W e brieﬂy recall deﬁnitions and some basic facts. Deﬁnition VI.1. A lo cally c o nv ex Lie algebr a g is said to be lo cally exp onential if there exists a circular conv ex op en ze ro neighbourho o d U ⊆ g and an o p en subs e t D ⊆ U × U on which there exists a smo oth map m U : D → U, ( x, y ) 7→ x ∗ y such that ( D , U, m U , 0) is a lo cal Lie group and such that the following ho lds. i) F or x ∈ U and | t | , | s | , | t + s | ≤ 1 , we have ( tx, sx ) ∈ D with tx ∗ sx = ( t + s ) x . ii) The seco nd o rder term in the T aylor e xpansion o f m U in 0 is b ( x, y ) = 1 2 [ x, y ]. Remark VI. 2 . (cf. [ Nee06 , E x. IV.2.4]) All Ba nach-Lie alg e bras are lo cally exp onential, as well as all Lie algebra s of lo cally exp onential Lie gr o ups. Theorem VI.3. ([ Ne e06 , Thm. IV.3.8]) L et g b e a lo c al ly exp onential Lie algebr a. Th en the adjoint gr oup G ad ≤ Aut( g ) c arries the stru ctur e of a lo c al ly exp onential Lie gr oup whose Lie algebr a is g ad := g / z ( g ) . The route to Lie’s Third Theo r em seems to b e clea r, simply integrate z ( g ) incl − − → g q − → g ad . But the latter need not b e top olog ically s plit, as the following example shows. Example VI.4 . Let F ≤ E := ℓ p ( N ) for some 1 < p < 2 be a non-complemented, in par ticular inﬁnite-dimensional subspace, i.e., there exists no contin uous pro jectio n E → F . W e choo se a linearly indep endent sequence ( e n ) n ∈ N in F . Moreov er , we choo se a linearly indep endent 26 sequence ( a n ) n ∈ N in F ⊥ such that span { a n } is dense in F ⊥ and for each a n another linea rly independent b n ∈ F ⊥ . Having ﬁxed this we set [ x, y ] := ∞ X n =1 1 2 n ( a n ( x ) b n ( y ) − a n ( y ) b n ( x )) e n . Since a n ( e m ) = b n ( e m ) = 0 we have that [[ x, y ] , z ] + [[ y , z ] , x ] + [[ z , x ] , y ] = 0 and thus [ · , · ] deﬁnes a Lie brack et on E . An element x ∈ E is in the ce ntre precisely if the ma p [ x, · ] is trivial. This is the ca se if x ∈ F . O n the other ha nd, if x / ∈ F , then a n ( x ) 6 = 0 for at least one n ∈ N . F o r each 0 6 = y ∈ ker( a n ) we have y / ∈ k er( b n ) and th us [ x, y ] 6 = 0. This shows x / ∈ F ⇒ [ x, · ] 6 = 0 and th us F is the centre of E . A pro cedure similar to consider ing gener alised central extensions as in [ Nee06 , Sect. VI.1 ] now remedies this failure. Theorem VI.5. If g is a Mackey-c omplete lo c al ly exp onential Lie algebr a, t hen ther e exists an ´ et ale Lie 2-gr oup G such that L ( G ) is isomorphic to g . Pro of. W e consider g ad := g / z ( g ) a nd the ma p g × g → g , ( x, y ) 7→ [ x, y ]. This map v anishes if x ∈ z ( g ) or y ∈ z ( g ) a nd thus induces a contin uous co cycle ω g : g ad × g ad → | g | , where | g | deno tes the Ma ckey-complete lo cally co nvex space underlying g . This integrates by Theorem I.11 to a generalise d co cycle ( F ω g , Θ), which in turn g ives rise to an ´ etale Lie 2-g roup b G ( F ω g , Θ) with se t of ob jects | g | × G ad . Moreov er, we may assume that F ω g is smo oth on V × V and V = V − 1 . The exp onential function exp g ad : g ad → G ad restricts to a diﬀeomorphis m on some op en zer o neighbourho o d U ⊆ g ad and we may assume that exp( U ) ⊆ V . W e now want to construct a lo cal exp onential function for | g | ⊕ ω g g ad and for this ﬁrst deﬁne γ x ( t ) := exp g ad ( tx ) for x ∈ g ad and for x ∈ U and t ∈ [0 , 1] we set z x ( t ) := − Z t 0 T F ω g  0 γ x ( s ) − 1 , d du     u = s γ x ( u )  ds, where 0 γ x ( s ) − 1 denotes the zero element in T γ x ( s ) − 1 G ad . Note that the integral exists s ince | g | is Mack ey- complete. With this we set η ( z ,x ) : [0 , 1] → | g | × G ad , η ( z ,x ) ( t ) := ( tz + z x ( t ) , γ x ( t )) and observe that ˙ z x ( t ) = − T F ω g (0 γ x ( t ) − 1 , ˙ γ x ( t )) implies d dt     t =0 η ( z ,x ) ( t 0 ) − 1 η ( z ,x ) ( t 0 + t ) = ( z , x ) for t 0 ∈ (0 , 1). Thus exp : | g | × U → | g | × exp g ad ( U ) ⊆ | g | × G ad , ( z , x ) 7→ ( z + z x (1) , exp g ad ( x )) . deﬁnes a lo cal exp o nential function, which is a diﬀeomo rphism since ( z , x ) 7→ ( z , exp g ad ( x )) is one. Now g is isomorphic to the clo sed ideal { ( x, q ( x )) : x ∈ g } of | g | ⊕ ω g g ad , and th us ex p restr icts to a diﬀeomor phism of ( | g | × U ) ∩ g o nto W := ( | g | × exp g ad ( U )) ∩ exp( g ). Note that we have in pa r ticular z ( g ) × exp g ad ( U ) ⊆ W . W e deﬁne G to b e the monoidal sub categor y of b G ( F ω g , Θ) , generated by the full sub catego ry W deter mined by W . It r e mains to check that G deﬁned this way ac tua lly is a n ´ etale Lie 2-gro up with Lie alg ebra g . W e hav e W 0 = W (by deﬁnition) and W 1 = π 2 ( G ad ) × W , s inc e p er ω g : π 2 ( G ad ) → | g | takes v alues in z ( g ) ⊆ g by [ Nee06 , Th. VI.1.6.] a nd z ( g ) × exp g ad ( U ) ⊆ W . With the r estricted structure ma ps this clear ly is an ´ etale 2- space and thus ( G , W ) is an ´ etale Lie 2 -group. Since exp is a lo cal exp onential function it is a lso clear that the Lie algebr a , ass o ciated to the lo cal gro up ( µ − 1 ( W ) , W , µ, (0 , e )) (with µ (( z , x ) , ( w, y )) = ( z + w + F ω g ( x, y ) , xy )) is isomo rphic to g . Thus L ( G ) ∼ = g . 27 Pro of. (of Prop osition II. 9 ) T he s et of ob jects o f the ´ etale Lie 2- group G constructed in the previo us theorem give rise to a lo o p, which r estricts to a lo cally smo oth lo op on so me op en neighbourho o d of 1 . Since the Lie 2- group is ´ etale, this lo ca lly smo oth loo p is also lo ca lly asso ciative. Moreover, the Lie algebra ass o ciated to this lo cally smo oth a nd lo cally asso ciative lo op coincides with L ( G ) and thus is is omorphic to g . VI I Prosp ects W e tried to develop a completed a ccount on the integration of inﬁnite-dimensional Lie algebr a s to Lie 2-gr o ups. In o rder to do so we dropp ed s ome topics that may be at hand which w e shortly line out in this section. Most of them deser ve to b e worked out ser iously . Remark VI I.1 (Diﬀ e ological Lie Groups). The pro blem that one enco unters when trying to integrate central extensions of inﬁnite-dimensio nal Lie algebra s to Lie groups is that o ne has to factor out subgroups from lo ca lly conv ex spaces that may b e not disc r ete. This has to b e done to ensure that the co cycle condition for a certa in universal co c y cle holds. How ever, one may resolve this problem by e nla rging the categ ory of smo o th manifolds to a category in which this quotient exists. F o r instance , the ca teg ory of diﬀeologica l spaces (o r mor e general smo oth s paces, cf. [ BH08 ]) has this prop erty . F r om our co cycle ( F ω ,β , Θ β ), integrating a given Lie algebra co cycle ω , one obtains an ordinar y gr oup co cycle q ◦ F ω ,β , which is in gener al (lo cally) smo oth as a map betw een diﬀeolo g ical spaces, beca use the quotient map q : z → z / Π ω is s mo oth, no matter whether Π ω := per ω ( π 2 ( G )) is discre te or not. With the corr e sp onding version of Theo rem IV.1 for diﬀeologica l spaces one thus co nstructs a diﬀeologic a l group b G ω and z / Π ω → b G ω → G is a candidate for a central extension of diﬀeolo g ical g roups, integrating z → z ⊕ ω g → g . The c rucial p oint here would b e to s e t up the notion of a Lie functor fro m diﬀeolo gical spaces to vector spaces such that it takes z / Π ω to z , e ven if Π ω is not discrete (such a thing should exist according to [ Sch08 ]) . In [ Igl95 ], a similar construction has b een done in o rder to obtain a preq uantisation for an arbitrar y symplectic manifold ( M , ω ), with not necessa rily in tegral [ ω ] ∈ H 2 dR ( M ). In particular , prequantisation ca n be per formed by dir ectly passing to the dual of the Lie a lgebra, witho ut constructing a Lie algebr a at ﬁrst 10 [ Igl08 ]. Remark VI I.2 (Diﬀ e ren ti al Geo metry of Generalised E xtens ions). O ne p ersp ective to the in tegration pr o cedure for central extensio ns of Lie alg ebras is to ﬁnd a Lie gr oup extension as a principal bundle with a pres crib ed curv ature. It sho uld b e p os sible to develop such a p oint of view also for smo oth gener alised central extensions , a s imilar p ersp ective ha s b een taken, for instance, b y Schommer-Pries [ SP11 ]. On the level of co cy cles, the passage is quite clear . F or a co cy cle f : G × G → Z , smo o th on U × U , the central extension Z → Z × f G → G is a principal bundle, describ ed by the transgre ssed ˇ Cech co cycle γ g,h : g V ∩ hV → Z, x 7→ f ( g , g − 1 x ) − f ( h, h − 1 x ) where V ⊆ U is an op en identit y neig h b o urho o d with V · V ⊆ U . Tha t γ g,h is smo oth follows from f ( g , g − 1 x ) − f ( h, h − 1 x ) = f ( g − 1 h, h − 1 x ) − f ( g , g − 1 h ) 10 Ev en if there is a Lie algebra around, there is a prior i no canonical dual space, asso ciated to it, for the usual topologies on dual s paces ar e not go o d enough (cf. [ Nee06 ]). So it is more natural to pass directly to the dual. 28 and from g − 1 h ∈ U if g V ∩ hV 6 = ∅ . F or a generalis ed co cycle ( F , Θ) the transgresse d non-ab elian ˇ Cech co cycle is acco r dingly g iven by γ g,h : g V ∩ hV → Z, x 7→ F ( g , g − 1 x ) − F ( h, h − 1 x ) − τ (Θ( g , g − 1 h, h − 1 x )) and η g,h ,k : g V ∩ hV ∩ k V → Z , x 7→ − Θ( g , g − 1 h, h − 1 x ) − Θ ( h, h − 1 k , k − 1 x ) + Θ ( g , g − 1 k , k − 1 x ) . This yields a principal Z -2-bundle P (ov er G ) [ W oc11 ], whic h is as a gro upo id (without any additional structure) equiv alent to G F, Θ . Applied to the string co cycle ϕ from Example IV.10 this 2-bundle is the prequantisation for the 2- plectic manifold ( G, h [ · , · ] , ·i ) [ Bry 93 ] and [ BHR10 ]. In general, the interpretation of P a s a bundle with connectio n is b e a bit more tricky sinc e the principal bundle π 0 ( P ) sho uld admit curv ature (in fancy terms, w e want the fake cur v ature not to v a nish). The theory of higher bundles with connectio n is b e ing developed at the moment (cf. [ SW08 ], [ NW11 ], [ Sch11 ], refer ences there in and [ W al10 ] for the case of group extensions). Remark VI I.3 (Non- Lo cally Exp onential Li e Algebras). One may w onder whether a sim- ilar theorem a s our version for Lie’s Third Theor e m is also in reach for non- lo cally exp onential Lie alge br as. T o our b est knowledge it would b e unlikely to exp e ct a similar res ult in this direc- tion, for the algebra ic pr op erties of non-lo ca lly e xp o nential Lie algebra s couple very har dly to their lo ca l Lie gr o ups (if they exist at all). F or instance, Lemp ert pr ov ed that V ( M ) C is even not inte gr able for any compact ma nifo ld M (cf. [ Lem97 ]), which relies on more in volv ed arguments as the counterexample of v an Est and Ko r thagen in [ E K64 ]. Remark VI I.4 (Hi g her Lie Algebras and Lie Alg ebroids). In a sense, we performed a similar integration pro ce dur e as Henriques in [ Hen08 ]. It th us s e e ms to b e promising to ca rry this analogy further to in tegrate even inﬁnite-dimensional Lie 2-a lg ebras o r to enla rge Henriques’ pro cedure b eyond the Ba nach-case. Since the obstructions for integrating lo cally exp onential Lie algebras and ﬁnite-dimensio na l Lie alg ebroids [ CF03 ] s e em to b e the sa me, a n integration pro ce- dure for spe cial cla sses of inﬁnite-dimensional Lie algebroids (e.g. Banach-Lie algebroids ) as in [ TZ06 ] is quite likely . Remark VI I.5 (Stac ky Lie groups). Our deﬁnition of a Lie 2 - group is somewhat weaker than one w ould exp ect at ﬁrst. Howev er, if o ne leaves the world of manifolds and co ns iders Lie group oids as presentations of diﬀerentiable stacks, then we ex pe c t tha t L ie 2 -groups as de ﬁned ab ov e lead to stacky Lie gro ups in the sense of [ Blo08 ]. Problem: If G is a Lie 2-g roup (in the sense of Deﬁnition IV.8 ), do es there exis t a stacky Lie group (in the sense of [ Blo08 ]) o r alternatively a smo oth gr o up stack H such that the underly- ing 2-gro ups are e quiv alent a nd the smo oth stacks ar e equiv alent in a “neighbourho o d” of the ident it y? If this is the case, can this co rresp ondence b e pr omoted to an equiv alence of the cor- resp onding 2-ca teg ories? One p oss ible way to obtain this would be to follow the usage of the ass o ciativity of the group m ultiplication through T he o rem IV.1 . The co ordina tes of the Lie gr oup structur e on G would yield a L ie group oid, the multiplication on G a Hilsum-Sk andalis mor phism descr ibing the multiplication morphism b e t ween the stacky Lie gr o ups and the usage o f the asso cia tivity , ﬁnally , the asso ciator 2-morphism. The ab ov e problem seems to b e s olv able since in the cases know to the author an ad-ho c construction yields stacky Lie groups from Lie 2-gro ups. F or the String 2-gro up from E xample IV.10 this is the construction of Schommer-Pries in [ SP11 ] and for the 2 - groups b G ( F ω,β , Θ β ) and e G (0 , Θ β ) from Theor em V.16 and Remark V.17 this is carr ied out in [ Z W ]. Mo reov e r, it would b e desirable to work out a L ie theo ry o f sta cky Lie g roups directly in the cor r ect categ orical se tup. 29 A App endix: Diﬀeren tial calculus on lo cally c onv ex spaces W e pr ovide some ba ckground material on lo cally conv ex Lie groups and their Lie algebra s in this app endix. Deﬁnition A.1. Let X and Y b e a lo cally conv ex spac es and U ⊆ X b e o p e n. Then f : U → Y is diﬀer entiable or C 1 if it is c ontin uous, for each v ∈ X the diﬀerential quotient d f ( x ) .v := lim h → 0 f ( x + hv ) − f ( x ) h exists and if the map d f : U × X → Y is co nt inuous. If n > 1 we inductively deﬁne f to b e C n if it is C 1 and d f is C n − 1 and to be C ∞ or smo oth if it is C n . W e say that f is C ∞ or smo oth if f is C n for a ll n ∈ N 0 . W e denote the c o rresp onding spaces of maps by C n ( U, Y ) and C ∞ ( U, Y ). A (lo cally conv ex ) Lie gr oup is a gr oup which is a smo oth manifold mo delle d on a lo cally conv ex space such that the gr oup o pe r ations ar e smo oth. A lo cally conv ex Lie a lg ebra is a L ie algebra, whose underlying vector space is lo c a lly convex and whose Lie brack et is contin uous. Remark A. 2. W e hav e the chain rule d ( g ◦ f )( x ) .v = dg ( f ( x )) . ( d f ( x ) .v ) and the identities d 2 f ( x )( v , w ) = pr 2 ( d ( T f )( x, v ) . ( w , 0)) (more precisely d ( T f )( x, v )( w , 0) = ( d f ( x ) .w, d 2 f ( x )( v , w )) ) and d ( T f )( x, v )( w , w ′ ) = d ( T f )(( w, 0) + (0 , w ′ )) = ( d f ( x ) .w, d 2 f ( x )( v , w )) + (0 , d f ( x ) .w ′ ) . This implies the “chain rule” for d 2 f : d 2 ( g ◦ f )( x ) . ( v, w ) = d 2 g ( f ( x ))( d f ( x ) .v , d f ( x ) .w ) + dg ( f ( x )) .d 2 f ( x )( v , w ) . (27) If M is a ma nifo ld and we ta ke the deﬁnition of the tangent bundle T M := [ i ∈ I { i } × ϕ i ( U i ) × X ! / ∼ with ( i, ϕ i ( x ) , v ) ∼ ( i ′ , ϕ i ′ ( x ) , d ( ϕ i ′ ◦ ϕ − 1 i )( ϕ i ( x )) .v ) if x ∈ U i ∩ U i ′ , then the map d 2 f : ( T x M ) 2 → T f ( x ) N , [ i, ϕ i ( x ) , v ] , [ i, ϕ i ( x ) , w ] 7→ [ j, ψ j ( f ( x )) , d 2 ( ψ j ◦ f ◦ ϕ − 1 i )( ϕ i ( x ))( v, w )] is well-deﬁned acco r ing to ( 27 ). Deﬁnition A.3. Let G b e a lo cally conv ex Lie group. The group G is sa id to have an exp o- nential function if for ea ch x ∈ g the initial v a lue pr oblem γ (0) = e, γ ′ ( t ) = T λ γ ( t ) ( e ) .x has a solution γ x ∈ C ∞ ( R , G ) and the function exp G : g → G, x 7→ γ x (1) is smo o th. F urthermore, if there exists a zero neig h b o urho o d W ⊆ g suc h that exp G | W is a diﬀeomorphism onto some op en identit y neig h b o urho o d o f G , then G is said to b e lo c al ly exp onential . 30 Lemma A.4. If G and G ′ ar e lo c al ly c onvex Lie gr oups with exp onential function, then for e ach morphism α : G → G ′ of Lie gr oups and the induc e d morphism dα ( e ) : g → g ′ of Lie algebr as, the diagr am G α − − − − → G ′ x   exp G x   exp G ′ g dα ( e ) − − − − → g ′ c ommutes. Remark A. 5. The F undamental Theorem of Calculus fo r lo cally co nv ex spaces (cf. [ Gl¨ o02 a , Thm. 1.5]) yields that a lo cally convex Lie g roup G can hav e a t mo s t one exp onential function (cf. [ Nee06 , Lem. I I.3.5]). Typical exa mples o f lo cally exp onential Lie groups a re Bana ch-Lie gr oups (by the existence of solutions of diﬀer e n tial equations and the in verse mapping theor e m, cf. [ Lan99 ]) and gr oups of smo oth a nd contin uous mappings from c ompact manifolds in to lo cally exp onential groups ([ Gl¨ o02 b , Sect. 3 .2], [ W oc0 6 ]). How ever, diﬀeomorphism gro ups of compact manifolds are never lo cally exp onential (cf. [ Nee06 , Ex. I I.5.1 3]) a nd direct limit Lie gr oups not alwa ys (cf. [ Gl¨ o 05 , Rem. 4 .7]). F or a detailed tr eatment of lo cally exp onential Lie gro ups and their s tr ucture theory we refer to [ Nee06 , Sect. IV]. Remark A. 6. Let X be a lo cally co nv ex space. Then X is sa id to b e Mack ey- complete if each Mack ey-Cauch y sequence co n verges in X (cf. [ KM97 , Sect. I.2]). In particular, sequentially complete spaces are Ma ck ey-co mplete. The main re ason fo r working with this weaker co ncept of c o mpleteness is that it ensures the e xistence of (weak) int egrals o f smo o th curves (cf. [ KM97 , Thm. I.2.14 ]), even for no n- complete spaces. 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