Integrating morphisms of Lie 2-algebras

INTEGRA TING MORPHISMS OF LIE 2 -ALGEBRAS BEHRANG NOOHI Abstract. W e sho w ho w to in tegrate a weak morphism of Lie algebra crossed- modules to a weak morphism of Lie 2-groups. T o do so we dev elop a theory of butterflies for 2-term L ∞ -algebras. In particular, we obtain a new description of the bicategory of 2-term L ∞ -algebras. W e use butterflies to give a functorial construction of conn ected co v ers of Lie 2- groups. W e also discuss the notion of homotop y fib er of a morphism of 2-term L ∞ -algebras. 1. Introduction In this pap er, we tac kle t w o main problems in the Lie theory of 2-groups: 1) int egrating weak morphisms of Lie 2-algebr as to weak morphisms of Lie 2-gro ups; 2) functor ial construction of connected cov ers of Lie 2-g roups. As w e will see, the latter pla ys a n important role in the solution of the former. Let us explain (1) a nd (2) in detail and outline o ur solution to them. Pr o blem (1). A w eak morphism f : H → G of Lie 2-groups gives r ise to a w eak morphism of Lie 2-a lgebras Lie f : Lie H → Lie G . (If we reg ard Lie H and Lie G as 2-term L ∞ -algebra s, Lie f is then a morphism of 2-term L ∞ -algebra s in the sense of Definition 2.5.) Pro blem (1) can b e sta ted as fo llows: given a mo rphism F : Lie H → Lie G of Lie 2-algebras can we integrate it t o a weak morphism Int F : H → G o f Lie 2-gro ups? W e a nswer this question affir matively by the follo wing theor em (see Theo rem 9.2 for a more precise statement). Theorem 1.1. L et G and H b e (st rict) Lie 2-gr oups. Su pp ose that H is 2-c onne cte d (Definition 7.2). Then, to give a we ak m orphism f : H → G is e quivalent to giving a morphism of Lie 2-alg ebr as Lie f : Lie H → Lie G . The same thing is true for 2-morphisms. This theor em is the 2-g roup version of the well-known fact in Lie theor y that a Lie homomorphism f : H → G is uniquely given b y its effect on Lie algebras Lie f : Lie H → Lie G , whenever H is 1-connected. It implies the following (see Corollar y 9.3). Theorem 1.2 . The bifunctor Lie : LieXM → LieAlgXM has a left adjoint Int : LieAlgXM → LieXM . Here, LieXM is the bicategor y of Lie crossed- mo dules a nd weak morphisms, and LieAlgXM is the bicategor y of Lie algebra cros s ed-mo dules and weak morphisms. 1 1 Note that LieXM is natu rally biequiv alent to the 2-categ ory of stri ct Li e 2-groups and w eak morphisms, and this is in turn biequiv al en t to the 2-category of strict Lie group stac ks. The bicat- egory LieAlgXM is naturally biequiv alent to the full sub 2-category of the 2-category 2T ermL ∞ of 2-term L ∞ -algebras consisting of str i ct 2-term L ∞ -algebras, and this is in turn biequiv alent to the 2-catego ry of 2-term dglas. 1 2 BEHRANG NOOHI The bifunctor In t takes a Lie cro ssed-mo dule to the unique 2-connected (strict) Lie 2-g roup that integrates it. When r e s tricted to the full sub ca tegory Lie ⊂ LieAlgXM of Lie a lgebras, it coincides with the standard integration functor which sends a Lie algebra V to the simply- connected Lie group Int V with Lie algebr a V . The problem of int egrating L ∞ -algebra s ha s been studied in [Ge] and [He], where they sho w how to integrate a n L ∞ -algebra (to a simplicial manifold). T he fo cus of these tw o pap ers, howev er , is different from ours in that we b eg in we fixe d Lie 2 - groups H and G and s tudy the pr oblem of integrating a morphism of Lie 2-alg ebras Lie H → Lie G . The pr oblem o f integrating morphisms of Lie 2-algebras in the case where the source is a Lie algebra h as b een studied in [ZhZa] using the clas s ical approach via paths and so lving PDEs. Our a pproach circum ven ts the necess ity to use pa ths and solving PDEs and is mor e fo r mal, making it co mpletely functorial (hence applicable in other circumsta nces) and explicit. Pr oblem (2). F or a Lie group G , its 0-th and 1-st connected cov er s G h 0 i and G h 1 i , which ar e again Lie groups, play an important role in Lie theory . W e observe that for (strict) Lie 2-groups one needs to go o ne step further, i.e., one needs to consider the 2-nd connected cov er as w ell. W e prov e the following theorem. Theorem 1.3. F or n = 0 , 1 , 2 , t her e ar e bifunct ors ( − ) h n i : LieXM → LieXM sending a Lie cr osse d-mo dules G to its n -t h c onne cte d c over. These bifunctors c ome with natur al t r ansformations q n : ( − ) h n i ⇒ id s uch that, for every G , q n : G h n i → G induc es isomorphisms on π i for i ≥ n + 1 . F u r t hermor e, ( − ) h n i is ri ght adjoint to the inclusion of the ful l sub bic ate gory of n -c onne cte d Lie cr osse d-mo dules in LieXM . The ab ov e theor em is essentially the co nten t of Sections 7 – 8. W e will b e esp e- cially interested in the 2-connected cov er G h 2 i b ecause, as sugges ted b y Theorem 1.2, it seems to b e the correct r eplacement for the universal cover of a Lie group in the Lie theory of 2-gr oups. Metho d. T o solv e (1) and (2) we employ the machinery of butt erflies , which we belie ve is of indep endent interest. Roughly sp ea king, a butterfly (Definition 3.1) betw een 2- term L ∞ -algebra s is a Lie algebr a theo retic version of a Mor ita mor - phism. W e use butterflies to give a new descr iption of the 2 -categor y 2T ermL ∞ of 2-term L ∞ -algebra s in tr o duced in [BaCr]. The adv antage of using butter flie s is t wofold. On the o ne ha nd, butterflies do aw ay with cum b er some co c ycle form ulas and are muc h easier to manipulate. On the other hand, given the dia grammatic nature of butterflies, they a r e b etter adapted to geometric s ituations; this is what allows us to prov e Theorem 1.2. Butterflies for 2- ter m L ∞ -algebra s parallel the corre sp onding theory fo r Lie 2- groups developed in [No 3] (see § 9.6 therein) and [AlNo1]. In fact, taking Lie alg ebras conv er ts a butterfly in Lie gr oups to a butterfly in Lie algebr as ( § 9). This allows us to study weak morphisms of Lie 2-groups using butterflies b etw e e n 2- term L ∞ - algebras , thereby reducing the problem to o ne ab out ex tensions of Lie alg ebras. With Theorem 1.2 at ha nd, we ex p ec t that this provide a co nv enient framework for studying weak morphisms of Lie 2-gro ups. Or ganization of t he p ap er. Sections 2 – 5 are devoted to setting up the machinery of butterflies a nd constructing the bicatego ry 2T ermL ♭ ∞ of 2-ter m L ∞ -algebra s and butterflies. W e show that 2T ermL ♭ ∞ is biequiv alent to the Baez-Cra ns 2 - category 2T ermL ∞ of 2-term L ∞ -algebra s. In § 4 we discuss the homotopy fi b er INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 3 of a morphism of 2-term L ∞ -algebra s. The homoto py fib er is the Lie alge br a counterpart of what we called the homotopy fib er of a weak morphism of Lie 2- groups in [No3], § 9.4. The homo topy fib er of f mea sures the deviatio n of f fro m being an equiv alenc e a nd it sits in a natur a l exact triangle which gives r ise to a 7- term long exact sequence. The homotopy fib e r comes with a rich structure consisting o f v arious br ack e ts and J a cobiator s (see § 4.1). W e are no t aware whether this structure has b een previo us ly studied. It is presumably some kind of a Lie algebra v ersion of what is ca lled a ‘crossed-mo dule in gr o up oids’ in [BrGi]. In Sec tion 6 we review Lie 2-gro ups and weak morphisms (butterflies) o f Lie 2-gro ups. Sections 7 – 8 are devoted to the s olution of Problem (2). F or a Lie crossed- mo dules G w e define its n -th connected covers G h n i , for n ≤ 2, and show that they are functorial and have the expected a djunction prop er ty . In Section 9 we solve P roblem (1) by pr oving Theorems 1.2 and 1.1. The proo fs rely on the solution o f Pr o blem (2) given in Sections 7 – 8 and the theory of butterflies developed in Sections 2 – 5. Ac knowledgemen t. I thank Gusta vo Granja, Christoph W o ck el and Chenchang Zhu for helpful conv er sations o n the sub ject of this paper . I also thank F ernando Muro, Jim Sta sheff, Tim Porter, Dmitry Royten b e r g and Mar c o Zambon for making useful comments on an earlier version o f the pap er. Contents 1. Int ro duction 1 2. 2-term L ∞ -algebra s 4 3. Butterflies b etw een 2 - term L ∞ -algebra s 6 4. Homotopy fib er of a morphism of 2-term L ∞ -algebra s 7 4.1. Structure of the homotopy fiber 8 5. The bicategory of Lie 2-a lgebras and butterflies 9 5.1. Compo sition of a butterfly with a strict morphism 10 6. W eak morphisms o f Lie 2-groups and butterflies 10 6.1. A note on terminolog y 11 6.2. Quick r eview of Lie butterflies 11 7. Connected cov er s of a Lie 2-gro up 12 7.1. Definition of the connected co vers 13 7.2. Uniform definition of the n -connected cov er s 14 8. F unctorial pro p erties of connected covers 15 8.1. Construction of the n - th connected cov er functor 15 8.2. Effect on the comp osition of butterflies 17 8.3. Adjunction prop erty of connected cov ers 21 9. The bifunctor from Lie cr ossed-mo dules to 2-term L ∞ -algebra s 22 10. Appendix: functorial n -connected covers for n ≥ 3 25 References 26 4 BEHRANG NOOHI 2. 2 -term L ∞ -algebras In this section we review some ba sic facts ab out 2-term L ∞ -algebra s. W e fo llow the nota tio ns o f [Ba Cr] (also see [Ro ]). All mo dules are over a fixed base co mmu- tative unital ring K . Definition 2.1. A 2 -term L ∞ -algebra V consists of a linea r map ∂ : V 1 → V 0 of mo dules together with the following data: • three bilinear maps [ · , · ] : V i × V j → V i + j , i + j = 0 , 1 ; • an a nt isymmetric trilinea r map (the Jac obiator ) h· , · , ·i : V 0 × V 0 × V 0 → V 1 . These maps satisfy the following axioms for all w , x, y , z ∈ V 0 and h, k ∈ V 1 : • [ x, y ] = − [ y , x ]; • [ x, h ] = − [ h, x ]; • ∂ ([ x, h ]) = [ x, ∂ h ]; • [ ∂ h, k ] = [ h, ∂ k ]; • ∂ h x, y , z i = [ x, [ y , z ]] + [ y , [ z , x ]] + [ z , [ x, y ]]; • h x, y , ∂ h i = [ x, [ y , h ]] + [ y , [ h, x ]] + [ h, [ x, y ]]; • [ h x, y , z i , w ] − [ h w, x, y i , z ] + [ h z , w , x i , y ] − [ h y , z , w i , x ] = h [ x, y ] , z , w i + h [ z , w ] , x, y i + h [ x , z ] , w, y i + h [ w, y ] , x, z i + h [ x, w ] , y , z i + h [ y , z ] , x, w i . W e sometimes use the notation [ V 1 → V 0 ] for a 2-ter m L ∞ -algebra . Definition 2.2. The equality [ ∂ h, k ] = [ h, ∂ k ] allows us to define a brack et on V 1 by se tting [ h, k ] := [ ∂ h, k ] = [ h, ∂ k ]. Lemma 2.3 . F or the br acket define d in Definition 2.2, the failur e of the Jac obi identity is me asur e d by the e quality h ∂ h, ∂ k , ∂ h i = [ h, [ k , l ]] + [ k , [ l, h ]] + [ l , [ h, k ]] . Pr o of. Eas y .  A cross ed-mo dule in Lie algebra s is the same thing as a s trict 2-ter m L ∞ -algebra , i.e., one for whic h the Jaco biator h· , · , ·i is iden tically zero. More precisely , given a 2-term L ∞ -algebra V with zero Jaco biator we obtain, b y Lemma 2.3, a Lie a lgebra structure on V 1 , wher e the bracket is as in Definition 2.2. This makes ∂ a Lie alg e bra homomorphism. The actio n of V 0 on V 1 is the g iven brack et [ · , · ] : V 0 × V 1 → V 1 . Also, observe that a strict 2-term L ∞ -algebra is the same thing as a 2-term dgla. Definition 2.4. Let V = [ ∂ : V 1 → V 0 ] b e a 2-ter m L ∞ -algebra . W e define H 1 ( V ) := ker ∂ , H 0 ( V ) := coker ∂ . Note that H 0 ( V ) and H 1 ( V ) b oth inher it natura l Lie algebr a structures, the latter being ne c e ssarily ab elian. F ur thermore, H 1 ( V ) is naturally an H 0 ( V )-mo dule. Definition 2. 5. A morphism f : W → V of 2-term L ∞ -algebra s consists of the following data : • linear maps f i : W i → V i , i = 0 , 1, comm uting with the differ ent ials; • an antisymmetric bilinear map ε : W 0 × W 0 → V 1 . These maps satisfy the following axioms: • for every x, y ∈ W 0 , [ f 0 ( x ) , f 0 ( y )] − f 0 [ x, y ] = ∂ ε ( x, y ); • for every x ∈ W 0 and h ∈ W 1 , [ f 0 ( x ) , f 1 ( k )] − f 1 [ x, k ] = ε ( x, ∂ k ); INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 5 • for every x, y , z ∈ W 0 , h f 0 ( x ) , f 0 ( y ) , f 0 ( z ) i − f 1 ( h x, y , z i ) = ε ( x, [ y , z ]) + ε ( y , [ z , x ]) + ε ( z , [ x, y ]) + [ f 0 ( x ) , ε ( y , z ) ] + [ f 0 ( y ) , ε ( z , x ) ] + [ f 0 ( z ) , ε ( x , y )] . A morphism f : W → V of 2-term L ∞ -algebra s induce s a Lie algebra homomor- phism H 0 ( f ) : H 0 ( W ) → H 0 ( V ) and an H 0 ( f )-equiv aria nt morphism of Lie algebra mo dules H 1 ( f ) : H 1 ( W ) → H 1 ( V ). Definition 2 . 6. A morphism f : W → V o f 2 -term L ∞ -algebra s is called a n equiv- alence (or a quasi-isom orphism ) if H 0 ( f ) and H 1 ( f ) are iso mo rphisms. Definition 2.7. A morphism of 2-term L ∞ -algebra is s trict if ε is identically zero. In the case wher e V and W are crossed-mo dules in Lie a lgebras , this means that f is a (strict) morphism of cr ossed-mo dules . Definition 2.8. If f = ( f 0 , f 1 , ε ) : W → V and g = ( g 0 , g 1 , δ ) : V → U are morphisms o f 2-ter m L ∞ -algebra s, the comp osition g f is defined to b e the triple ( g 0 f 0 , g 1 f 1 , γ ), wher e γ ( x, y ) := g 1 ε ( x, y ) + δ ( f 0 ( x ) , f 0 ( y )) , x, y ∈ W 0 . Finally , we recall the definition of a trans fo rmation b etw een morphisms of 2- term L ∞ -algebra s. Up to a minor difference in sign con ven tions, it is the same as [Ro], Definition 2.20 . It is als o e q uiv alent to Definition 3 7 in the archiv e version [arXiv:math/03 0726 3 v5] of [Ba C r]. Definition 2. 9. Given mor phis ms f , g : W → V of 2-term L ∞ -algebra s, a trans- formation (or an L ∞ -homotopy ) from g to f is a linear map θ : W 0 → V 1 such that • for every x ∈ W 0 , f 0 ( x ) − g 0 ( x ) = ∂ θ ( x ); • for every h ∈ W 1 , f 1 ( h ) − g 1 ( h ) = θ ( ∂ h ); • for every x, y ∈ W 0 , [ θ ( x ) , θ ( y )] − θ ( [ x , y ]) = ε f ( x, y ) − ε g ( x, y ) + [ g 0 ( y ) , θ ( x )] + [ θ ( y ) , g 0 ( x )] . R emark 2 .10 . It may perhaps lo ok mor e na tur al to consider such a θ as a transfor- mation from f to g and not fro m g to f . (This is how it is in [Ro], Definition 2.20.) W e, how ever, choose the bac kward conv ention to b e co mpatible with corresp onding notion of transformatio n for butterflies (Definition 3.3). It is easy to see that if f and g are r e lated b y a transfor mation, then H i ( f ) = H i ( g ), i = 0 , 1. Definition 2.11. If θ is a tra nsformation from f to g a nd σ a transfor mation from g to h , their comp osition is the transfor ma tion from f to h given by the linea r map θ + σ . The following definition is the one in [BaCr], P rop osition 4.3.8. Definition 2 . 12. W e define 2T ermL ∞ to be the 2-ca tegory in which the ob jects are 2-term L ∞ -algebra s, the mor phisms are as in Definition 2 .5 and the 2-morphisms are as in Definition 2.9. 6 BEHRANG NOOHI 3. Butterflies between 2 -term L ∞ -algebras In this s e ction we intro duce the notion of a butterfly b etw een 2 -term L ∞ -algebra s and show that butterflies enco de mor phisms of 2-term L ∞ -algebra s (Pro p ositions 3.4, 3.5). A butterfly sho uld b e r egraded as an analo gue of a Morita morphism. Definition 3.1. Let V and W b e 2-term L ∞ -algebra s. A butterfly B : W → V is a commutativ e diagr am W 1 κ " " E E E   V 1 ι } } { { {   E σ | | y y y ρ ! ! C C C W 0 V 0 of mo dules in which E is endow e d with an a nt isymmetric brac ket [ · , · ] : E × E → E satisfying the following axioms: • bo th diagona l sequences are complexes and the NE-SW sequence 0 → V 1 ι − → E σ − → W 0 → 0 is shor t exa ct; • for every a, b ∈ E , ρ [ a, b ] = [ ρ ( a ) , ρ ( b )] and σ [ a , b ] = [ σ ( a ) , σ ( b )]; • for every a ∈ E , h ∈ V 1 , l ∈ W 1 , [ a, ι ( h )] = ι [ ρ ( a ) , h ] and [ a, κ ( l )] = κ [ σ ( a ) , l ]; • for every a, b, c ∈ E , ι h ρ ( a ) , ρ ( b ) , ρ ( c ) i + κ h σ ( a ) , σ ( b ) , σ ( c ) i = [ a, [ b, c ]] + [ b, [ c, a ]] + [ c, [ a, b ]] . In the case where V and W are crosse d-mo dules in Lie alg ebras (i.e., when the Jacobiato rs are iden tically zero ), the brack et on E makes it in to a Lie algebra and all the maps in the butterfly diagram beco me Lie algebr a homomo r phisms. R emark 3.2 . The map κ + ι : W 1 ⊕ V 1 → E ha s a natur al 2-term L ∞ -algebra structure. Let us denote this 2 -term L ∞ -algebra b y E . The tw o pr o jections E → W and E → V are strict morphisms of 2- term L ∞ -algebra s and the former is a quasi- isomorphism. Thus, we can think o f the butterfly B as a zig-zag of stric t morphisms from W to V . Definition 3.3. Given tw o butterflies B , B ′ : W → V , a morphism of butterflies from B to B ′ is a linea r ma p E → E ′ commuting with the bra ck ets a nd a ll four structure maps of the butterfly . (Note tha t such a map E → E ′ is necessa rily an isomorphism.) A butterfly B : W → V induces a Lie algebra ho momorphism H 0 ( B ) : H 0 ( W ) → H 0 ( V ) and an H 0 ( B )-eq uiv ariant mo rphism H 1 ( B ) : H 1 ( W ) → H 1 ( V ). If B and B ′ are related by a morphism, then H i ( B ) = H i ( B ′ ), i = 0 , 1. Let f : W → V b e a morphism of 2-term L ∞ -algebra s as in Definition 2.5. Define a brack et on V 1 ⊕ W 0 by the rule [( k , x ) , ( l , y )] :=  [ k , l ] + [ f 0 ( x ) , l ] + [ k , f 0 ( y )] + ε ( x, y ) , [ x, y ]  . Define the following four maps: INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 7 • κ : W 1 → V 1 ⊕ W 0 , κ ( l ) = ( − f 1 ( l ) , ∂ l ), • ι : V 1 → V 1 ⊕ W 0 , ι ( k ) = ( k , 0), • σ : V 1 ⊕ W 0 → W 0 , σ ( k , x ) = x , • ρ : V 1 ⊕ W 0 → V 0 , ρ ( k , x ) = ∂ k + f 0 ( x ). Prop ositi on 3.4. With the br acket on V 1 ⊕ W 0 and the maps κ , ι , ρ a nd σ define d as ab ove, t he diagr am W 1 κ & & L L L L   V 1 ι y y r r r r   V 1 ⊕ W 0 σ x x r r r r ρ % % L L L L W 0 V 0 is a but t erfly (Definition 3.1). Conversely, given a but terfly as in Definition 3.1 and a line ar se ction s : W 0 → E to σ , we obtain a morphism of 2 -term L ∞ -algebr as by setting f 0 := ρs, f 1 := s∂ − κ, ε := [ s ( · ) , s ( · )] − s [ · , · ] . (In t he definition of t he last two maps we ar e using the exactness of the NE-SW se quenc e.) F u rthermor e, these t wo c onstructions ar e inverse to e ach other. Prop ositi on 3.5. Via the c onst ruction intr o duc e d in Pr op osition 3.4, t ra nsfor- mations b etwe en morphisms of 2 -t erm L ∞ -algebr as (Definition 2.9) c orr esp ond t o morphisms of butt erflies (Defin ition 3.3). In other wor ds, we have an e quivalenc e of gr oup oids b etwe en the gr oup oid of m orphisms of 2 -t erm L ∞ -algebr as fr om W to V and the gr oup oid of butterflies fr om W to V . Example 3.6 . Let V a nd W b e Lie algebr as. D efine D er ( V ) to b e the cross ed- mo dule in Lie a lgebras ∂ : V → Der( V ), where ∂ sends v ∈ V to the deriv ation [ v , · ]. Then, the equiv alence cla sses of 2-term L ∞ -algebra morphisms W → D er ( V ) are in bijection with is omorphism cla s ses of extensio ns of W b y V . H ere, W is regar ded as the 2-term L ∞ -algebra [0 → W ]. 4. Homotopy fiber of a morphism of 2 -term L ∞ -algebras W e in tr o duce the homotopy fib er (or “shifted mapping c o ne”) of a butterfly (and also of a morphismof 2-ter m L ∞ -algebra s). The homolo gies of the homo topy fiber sit in a 7-term long exa ct sequence. W e see in § 4.1 that the homotopy fib er has a rich structure consisting of v arious brack ets . Definition 4.1. Let B : W → V , W 1 κ " " E E E   V 1 ι } } { { {   E σ | | y y y ρ ! ! C C C W 0 V 0 be a butterfly . W e define the homo top y fib e r hfib( B ) of B to b e the NW-SE sequence W 1 κ − → E ρ − → V 0 . W e will think of W 1 , E a nd V 0 as sitting in deg rees 1,0 and − 1. 8 BEHRANG NOOHI The homotop y fib er measures the deviation of B from being an e quiv alence (see Remark 4.5 b elow). Prop ositi on 4.2. Mor e pr e cisely, we have a long exact se quenc e 0 / / H 1 (hfib( B )) / / H 1 ( W ) H 1 ( B ) / / H 1 ( V ) / / H 0 (hfib( B )) 54 23 76 01 / / H 0 ( W ) H 0 ( B ) / / H 0 ( V ) / / H − 1 (hfib( B )) / / 0 . Pr o of. Exer cise.  Except for H − 1 (hfib( B )), all the ter ms in the ab ove sequence are Lie algebr as and all the maps ar e Lie algebra homomor phis ms; see § 4.1 below. Corollary 4 .3. A butterfly B is an e quivalenc e (i.e., induc es isomorph isms on H 0 and H 1 ) if and only if its NW-SE se quenc e is short exact. In this c ase, t he inverse of B is obtaine d by flippi ng it along the vertic al axis. Definition 4.1 leads to the following definition. Definition 4. 4. F o r a morphism f = ( f 0 , f 1 , ε ) : W → V o f 2-term L ∞ -algebra s, we define its homotop y fiber hfib( f ) to b e the sequence W 1 ( − f 1 ,∂ ) − → V 1 ⊕ W 0 ∂ + f 0 − → V 0 . R emark 4.5 . If w e forget all the brack ets and index the terms of hfib( f ) b y 2,1 ,0, we see that hfib( f ) coincides with the co ne o f f in the derived ca tegory of chain complexes. 4.1. Structure of the homotopy fib er. The hfib( B ) co mes with some additiona l structure which w e discus s b elow. First, le t us rename the ho motopy fib er in the following way C 1 ∂ − → C 0 ∂ − → C − 1 . W e hav e the following da ta: • antisymmetric bilinear brack ets [ · , · ] i : C i × C i → C i , i = 1 , 0 , − 1; • antisymmetric bilinear brackets [ · , · ] 01 : C 0 × C 1 → C 1 , [ · , · ] 10 : C 1 × C 0 → C 1 ; • antisymmetric trilinear Jaco bia tors h· , · , ·i i : C i × C i × C i → C i +1 , i = − 1 , 0. W e denote [ · , · ] − 1 by [ · , · ]. The following axioms are satisfied: • [ · , · ] 01 = − [ · , · ] 10 ; • for every a ∈ C 0 and h ∈ C 1 , ∂ ([ a, h ] 01 ) = [ a, ∂ h ] 0 ; • for every h, k ∈ C 1 , [ h, k ] 1 = [ ∂ h, k ] 01 = [ h, ∂ k ] 10 ; • for every a, b ∈ C 0 , ∂ [ a, b ] 0 = [ ∂ a, ∂ b ]; • for every a, b, c ∈ C 0 , h ∂ a, ∂ b, ∂ c i − 1 + ∂ ( h a , b, c i 0 ) = [ a, [ b, c ] 0 ] 0 + [ b, [ c, a ] 0 ] 0 + [ c, [ a, b ] 0 ] 0 . • for every a, b ∈ C 0 and h ∈ C 1 , h a, b, ∂ h i 0 = [ a, [ b, h ] 01 ] 01 + [ b, [ h, a ] 10 ] 01 + [ h, [ a, b ] 0 ] 10 . • for every a, b, c, d ∈ C 0 , [ h a, b, c i 0 , d ] 10 − [ h d, a, b i 0 , c ] 10 + [ h c, d, a i 0 , b ] 10 − [ h b, c, d i 0 , a ] 10 = h [ a, b ] 0 , c, d i 0 + h [ c, d ] 0 , a, b i 0 + h [ a, c ] 0 , d, b i 0 + h [ d, b ] 0 , a, c i 0 + h [ a, d ] 0 , b, c i 0 + h [ b, c ] 0 , a, d i 0 . INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 9 There are natural c ha in maps V → hfib( B )[ − 1] and hfib( B ) → W which resp ect all the brackets on the nose. In fact, hfib( B ) → W → V is an ex act tr iangle in the derived ca tegory of chain complexes (note the reverse shift due to homological indexing). 5. The bica tegor y of Lie 2-algebras and butterflies Given butterflies W 1 κ " " E E E   V 1 ι } } { { {   E σ | | y y y ρ ! ! C C C W 0 V 0 V 1 κ ′ ! ! C C C   U 1 ι ′ } } { { {   F σ ′ } } { { { ρ ′ ! ! C C C V 0 U 0 we define their compo s ition to b e the butterfly W 1 ( κ, 0) $ $ I I I I I   U 1 (0 ,ι ′ ) z z u u u u u   E V 1 ⊕ V 0 F σ ◦ pr z z u u u u u ρ ′ ◦ pr $ $ I I I I I W 0 U 0 Here E V 1 ⊕ V 0 F is, by definition, the fibe r pro duct of E and F over V 0 mo dulo the diagonal image of V 1 via ( ι, κ ′ ). The brack et on it is defined co mpo nent-wise. Prop ositi on 5. 1 . W ith butterflies as morphisms, m orphisms of bu tterflies as 2 - morphisms, and c omp osition define d as ab ove, 2 -term L ∞ -algebr as form a bic ate- gory 2T ermL ♭ ∞ . F or a 2-term L ∞ -algebra V , the identit y butterfly from V to itself is defined to be V 1 κ % % K K K K   V 1 ι y y s s s s   V 1 ⊕ V 0 σ y y s s s s ρ % % K K K K V 0 V 0 Here, the brack et on V 1 ⊕ V 0 is defined by [( k , x ) , ( l , y )] :=  [ k , l ] + [ x, l ] + [ k , y ] , [ x, y ]  . The four structure maps o f the butterfly are: • κ : V 1 → V 1 ⊕ V 0 , κ ( l ) = ( − l, ∂ l ) • ι : V 1 → V 1 ⊕ V 0 , ι ( k ) = ( k , 0) • σ : V 1 ⊕ V 0 → V 0 , σ ( k , x ) = x • ρ : V 1 ⊕ V 0 → V 0 , ρ ( k , x ) = ∂ k + x Prop ositi on 5.2. The c onstruction of Pr op osition 3.4 induc es a bie quivalenc e 2T ermL ∞ ∼ = 2T ermL ♭ ∞ . 10 BEHRANG NOOHI Pr o of. Straig ht forward verification.  By Lemma 4.3, a butterfly B : W → V is inv er tible (in the bic a tegorica l s e nse) if and only if its NW-SE s equence is also short exact. In this case, the in verse of B is obtained by flipping B along the vertical axis. 5.1. Co mp osi tion o f a butterfly with a strict morphism. Compo sition of butterflies takes a s impler from when one of the butterflies comes from a strict morphism. When the first morphisms is strict, say W 1 f 1 / /   V 1   W 0 f 0 / / V 0 then the comp osition is W 1 $ $ I I I   U 1 z z u u u   f ∗ 0 ( F ) f ∗ 1 ( σ ′ ) z z u u u $ $ I I I W 0 U 0 Here, f ∗ 0 ( F ) s tands for the pullback of the extension F along f 0 : W 0 → V 0 . Mo re precisely , f ∗ 0 ( F ) = W 0 ⊕ V 0 F is the fib er pr o duct. When the second morphis ms is strict, say V 1 g 1 / /   U 1   V 0 g 0 / / U 0 then the comp osition is W 1 % % K K K   U 1 g 1 , ∗ ( ι ) z z t t t t   g 1 , ∗ ( E ) y y t t t $ $ J J J J W 0 U 0 Here, g 1 , ∗ ( E ) stands for the push forward of the extension E along g 1 : V 1 → U 1 . More precisely , g 1 , ∗ ( E ) = E ⊕ V 1 U 1 is the pushout. 6. Weak morphisms of Lie 2-groups and butterflies There a re at least three equiv alent wa ys to define weak morphisms of Lie 2- groups. One w ay is to lo calize the 2-c ategory o f Lie 2-g roups and strict mo r phisms with resp e ct to equiv alences, and define w eak mor phisms to b e morphis ms in this lo calized categ ory (by definition, an equiv alence b etw een Lie 2 - groups is a morphism which induces isomorphisms on π 0 and π 1 ). INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 11 The sec ond definition is that a weak morphism of L ie 2-gr oups is a weak mor- phism (i.e., a monoida l functor) betw een the asso ciated Lie group stacks. The third definition, which is shown in [AlNo1] to b e equiv a lent to the sta ck definition, makes use of butterflies and is the sub ject of this section. It is the butterfly definition that proves to b e mos t suita ble for the study o f co nnected cov er s o f Lie 2-gr oups and also for proving our integration result (Theor em 1.2). 6.1. A note on te rm inology. Lie 2-gr oup co uld mean differ ent things to differe nt peo ple, so so me clar ification in terminology is in or der befo r e we move o n. One definition is that a Lie 2-gr oup is a crosse d- mo dule G := [ ∂ : G 1 → G 0 ] in the category of Lie groups. Although most kno wn examples of Lie 2 - groups are of this form, this is not the mos t general definition, as it is too strict . Arguably , the cor rect definition is that a Lie 2-gr oup is a differentiable group stacks, that is, a (weak) group o b ject G in the ca tegory o f differentiable sta cks. Every Lie crossed-mo dule [ G 1 → G 0 ] gives rise to a group stack [ G 0 /G 1 ], but no t every differentiable gro up stack is of this form. A differen tiable gro up stack G comes from a Lie cr ossed-mo dule if and only if it admits an a tlas ϕ : G 0 → G s uch that G 0 is a Lie gro up and ϕ is a differentiable (w eak ) homomorphism. Another de finitio n of a Lie 2-group (which is presumably equiv alen t to the stac k definition) is discussed in the App endix o f [He]. This definition is motiv a ted by the fact that a Lie 2 -gro up gives rise to a simplicial ma nifold and, conv ersely , a simplicial manifold with ce r tain fibra ncy pro pe r ties and so me conditions on its homotopy g roups should come from a Lie 2 -gro up. In this pap er w e only deal with str ic t Lie 2-groups, i.e., Lie group stacks coming from Lie crossed-mo dules. Pr esumably our theory can b e extended to a rbitrary Lie 2-gro ups. Throughout the text, all Lie groups are assumed to be finite dimensional unless otherwise stated. 6.2. Q u i c k revie w of Lie butterflies. F or mo re details o n butterflies see [No3], esp ecially § 9.6, § 10 .1, and [AlNo1]. In wha t follows, by a homo morphism of Lie groups we mea n a differentiable homomorphism. R emark 6.1 . In [No3] a nd [AlNo1 ] w e use the right-action conv ention for crossed- mo dules, while in these notes, in or der to be compatible with the existing literatur e on L ∞ -algebra s, we hav e used the left-action conv e ntion for Lie algebr a crossed- mo dules. Ther efore, for the sa ke of co nsistency , we will adopt the left-action con- ven tion for Lie cro s sed-mo dules as well. Let G and H b e a Lie cro ssed-mo dules (i.e., a cr ossed-mo dules in the categ ory of Lie groups ). A butterfly B : H → G is a commutativ e diagram H 1 κ " " D D D   G 1 ι } } z z z   E σ | | z z z ρ ! ! D D D H 0 G 0 in which b oth diago nal sequences a re c o mplexes of Lie groups, a nd the NE - SW sequence is shor t exact. W e also require that for every x ∈ E , α ∈ G 2 and β ∈ H 2 12 BEHRANG NOOHI the following e qualities hold: ι ( ρ ( x ) · α ) = xι ( α ) x − 1 , κ ( σ ( x ) · β ) = xκ ( β ) x − 1 . A butterfly betw een Lie cro ssed-mo dules can be regarded as a Morita morphism which resp ects the gro up structure s. A morphism B → B ′ of butterflies is, by definition, a homo morphism E → E ′ of Lie gr oups whic h commutes with all four structure maps of the butterflies. Note tha t such a morphism is necessar ily an isomorphism. R emark 6 .2 . F or the re ader interested in the topo logical version of the story , we remark that in the definition of a top o logical butterfly o ne needs to a ssume tha t the map σ : E → H 0 , viewed as a contin uous map of top ologica l spaces, admits lo cal sections. This is automa tic in the Lie case b ecause σ is a submersion. Thu s, with butterflies a s morphis ms, Lie cro ssed-mo dules form a bicategor y in which every 2-mor phism is a n isomorphism. W e denote this bicatego ry by LieXM . The following theor e m justifies why butterflies provide the right notion o f mor- phism. Theorem 6.3 ([AlNo1]) . The 2-c ate gory of (strict) Lie 2-gr oups and we ak mor- phisms is bie quivalent to the bic ate gory LieXM of Lie cr osse d-mo dules and butter- flies. W e recall ([No3], § 1 0.1) how comp osition o f tw o butterflies C : K → H and B : H → G is defined. Let F and E b e the Lie groups app ear ing in the c ent er of these butterflies, resp ectively . Then, the comp ositio n B ◦ C is the butterfly K 1 $ $ I I I   G 1 z z u u u   F H 1 × H 0 E z z u u u $ $ I I I K 0 G 0 where F H 1 × H 0 E is the fib er pro duct F × H 0 E mo dulo the diag onal image of H 1 . In the case where one o f the butterflies is strict, the comp ositio n takes a simpler form similar to the discussio n of § 5.1. See ([No3], § 10.2) for more details. 7. Connected covers of a Lie 2-group In this section we construct n -th connec ted cov ers G h n i of a Lie crossed-mo dule G = [ G 1 → G 0 ] for n = 0 , 1 , 2. In § 8 we prove tha t these a re functorial with resp ect to butterflies. Hence, in particular, they a re inv ariant under equiv a lence of Lie cro ssed-mo dules (Co rollar y 8.7). All Lie g roups are assumed to b e finite dimensional unless otherwise sta ted. Cave at on n otation. In this section, by the i -th homoto py g roup π n G of a top o- logical cr o ssed-mo dule G = [ ∂ : G 1 → G 0 ] we mean the i -th homotopy of the simplicial s pace asso ciated to it (equiv alently , the i -th homotopy o f the quotient stack G 0 /G 1 ]). When i = 0 , 1 , this should not b e confused with the usag e of π 0 and π 1 for coker ∂ and k er ∂ ; the t wo notations agree only when G 0 and G 1 are discr ete groups. INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 13 Recall that a map f : X → Y of to po logical spa c es is n -connected if π i f : π i X → π i Y is an iso morphism of i ≤ n and a surjection for i = n + 1 . Prop ositi on 7.1. L et G = [ G 1 → G 0 ] b e a top olo gic al cr osse d-mo dule and n ≥ 0 an inte ger. The fol lowing ar e e quivalent: i) The map ∂ is ( n − 1 ) -c onne cte d; ii) The quotient stack G := [ G 0 /G 1 ] is n - c onne cte d (in t he sense of [No1] , § 17); iii) The classifying sp ac e of G is n -c onne cte d. (We ar e viewing G as a stack and ignoring its gr oup stru ctur e.) Pr o of. The equiv alenc e of (i) a nd (ii) follows fro m the ho mo topy fib er sequence applied to the fibration of stac k s G 1 → G 0 → [ G 0 /G 1 ]. T he equiv alence of (ii) and (iii) follows from [No2], Theorem 10.5.  Definition 7.2. W e sa y that a Lie crossed-mo dule G is n -connected if it satisfies the equiv alent conditions of Prop ositio n 7.1. It follo ws from Prop ositio n 7.1 tha t the no tio n of n -connected is in v ar iant under equiv alence of Lie cro ssed-mo dules. R emark 7.3 . A L ie crossed-mo dule G is 2-co nnected if and o nly if π i ∂ : π i G 1 → π i G 0 is an isomor phism for i = 0 , 1. This is b ecause π 2 of every (finite dimensional) Lie group v anishes. 7.1. Defini tion of the connected cov ers. In this subsection w e define the n -th connected cov e r of a Lie cr o ssed-mo dule for n ≤ 2. In the next section we pr ov e that these definitions are functorial with res pe c t to butterflies. In par ticular, it follows that they are inv ar iant under equiv alence of L ie cr ossed-mo dules. The discussio n of this and the ne x t section is v alid for top olog ical cr o ssed-mo dules (and also for infinite-dimensional Lie cr o ssed-mo dules) as well. The 0 - th connected co ver of G . It is easy to see that a Lie 2-g r oup G is connected if and only if it ha s a presentation by a Lie cr ossed-mo dule [ G 1 → G 0 ] with G 0 connected. (Pro of: choose an a tla s ϕ : G 0 → G such tha t ϕ is a homomorphism and G 0 is co nnected, and set G 1 := { 1 G } × G ,ϕ G 0 .) F or a g iven Lie c r ossed-mo dule G = [ G 1 → G 0 ] its 0-th connec ted cover is defined to be G h 0 i := [ ∂ − 1 ( G o 0 ) → G o 0 ] , where G o stands for the connected comp onent of the iden tity . The crosse d-mo dule G h 0 i s hould be thought of as the connected c o mp o nent of the identit y of G . There is an o bvious str ic t morphism q 0 : G h 0 i → G whic h induces isomor phis ms o n π i for i ≥ 1 (see Prop ositio n 7.5). The 1 -st connected co ver o f G . A Lie 2-group G is 1-connected if and only if it has a presentation by a Lie crossed- mo dule [ G 1 → G 0 ] with G 0 1-connected and G 1 connected. (Pro o f: choo se an atlas ϕ : G 0 → G such that ϕ is a homomorphis m and G 0 is 1 -connected, and set G 1 := { 1 G } × G ,ϕ G 0 .) F or a given Lie crossed- mo dule G = [ G 1 → G 0 ] its 1-s t connected cov er is defined to b e G h 1 i := [ L o → f G o 0 ] , where L := G 1 × G 0 f G o 0 and ˜ stands for universal co v er. Ther e is an o bvious strict morphism q 1 : G h 1 i → G whic h factors through q 0 and induces iso morphisms on π i for i ≥ 2 (see Prop os ition 7.5). 14 BEHRANG NOOHI The 2 -nd connected co ver of G . A Lie 2-gro up G is 2-connected if and only if it has a presentation b y a Lie cro ssed-mo dule [ G 1 → G 0 ] with G 0 and G 1 bo th 1- connected. (Pro o f: choo se an atlas ϕ : G 0 → G such that ϕ is a homomorphis m and G 0 is 1 -connected, and set G 1 := { 1 G } × G ,ϕ G 0 .) F or a given Lie crossed- mo dule G = [ G 1 → G 0 ] its 2-nd co nnec ted cover is defined to be G h 2 i := [ f L o → f G o 0 ] , where L is as in th e previous part. There is an obvious str ict morphism q 2 : G h 2 i → G w hich facto rs thro ug h q 1 and induces isomor phisms on π i for i ≥ 3 (see Pr op o- sition 7.5). R emark 7 .4 . Note that a 1-connected Lie group is automatica lly 2-connected. The same is not true for Lie 2 -gro ups. 7.2. Unifo rm defini tion of the n -connected co vers. In or der to b e av oid rep- etition in the co ns tructions and argumen ts given in the next sectio n, w e phra se the definition of G h n i in a unifor m manner for n = 0 , 1 , 2, and s ing le out the main prop erties of the connected covers q n : G h n i → G whic h will be needed in the next section. 2 Our discussion will be v a lid for topolog ical crossed-mo dules (and also for infinite-dimensional Lie cr o ssed-mo dules) as well. First off, we need functorial n -connected covers q n : G h n i → G for n = 0 , 1 , 2 . 3 W e set G h− 1 i = G . W e take G h 0 i := G o and G h 1 i = G h 2 i = f G o , where G o means connected comp onent of the identit y . (In the ca se where G is a top olog ical group, or an infinite dimensional L ie group, one has to make a different choice for G h 2 i ; see Remark 7.6.) F or a cro ssed-mo dule G = [ G 1 → G 0 ] we define G h n i to be G h n i := [ ∂ : L h n − 1 i → G 0 h n i ] , where L := G 1 × G 0 ,q n G 0 h n i , and ∂ = pr 2 ◦ q n − 1 . T he action of G 0 h n i on L h n − 1 i is defined as follows. There is a n action of G 0 h n i on L defined c o mp o nent wise (on the first comp onent it is obtained, via q n , from the action of G 0 on G 1 and on the second comp onent it is given by right co njugation). By functoriality of the n - th connected co ver constr uction (applied to L ), this action lifts to L h n − 1 i . F o r G h n i to be a crossed-mo dule, we use the following property: ( ⋆ 0) F or every x ∈ G h n − 1 i , the a ction of q n − 1 ( x ) ∈ G on G h n − 1 i obtained (b y functoriality) from the co njugation action o f q n − 1 ( x ) on G is equal to conjugation by x . There is a strict mor phism of cross e d-mo dules q n : G h n i → G defined by L h n − 1 i pr 1 ◦ q n − 1 / / ∂   G 1 ∂   G 0 h n i q n / / G 0 2 Apart from i mproving the clarit y of pro ofs in th e next section, there is ano ther purpose for singling out prop erties of con nected co vers in the form of ax ioms ⋆ : i n con texts other than Lie crossed-mo dul es, it may b e possible to arrange for the axioms ⋆ for, say , other v alues of n , or by using differen t construct ions for G h n i . In s uc h cases, our pro ofs apply verbat im. 3 This, in fact, can b e arranged for any n i n the category of topological groups. INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 15 W e will a lso need the following prop erty: ( ⋆ 1) The map q n − 1 : G h n − 1 i → G admits lo cal sections near ev er y p o int in its image (hence is a fibr ation with op en-clo sed image). Prop ositi on 7.5. F or i ≤ n , we have π i ( G h n i ) = { 0 } . F or i ≥ n + 1 t he morp hism q n : G h n i → G induc es isomorph isms π i ( q n ) : π i ( G h n i ) → π i ( G ) . Pr o of. Consider the commutative diag ram L h n − 1 i ∂ / / pr 1 ◦ q n − 1   G 0 h n i / / q n   G h n i q n   G 1 ∂ / / G 0 / / G Both rows are fibr a tions of crossed- mo dules (so , induce fibra tions on the classifying spaces). The first claim follows by applying the fib er homotopy exact s e quence to the first ro w. F o r the second claim use the fact that L → G 1 is a fibr ation (b ecause of ⋆ 1) with the same fiber as q n : G 0 h n i → G 0 , and apply the fib er ho motopy e xact sequence to the tw o rows of the ab ov e diagram (together with Five Lemma).  R emark 7.6 . In the definition of G h n i = [ L h n − 1 i → G 0 h n i ], the fact that G 0 h n i is an n -connec ted cov e r of G 0 is no t rea lly needed. All w e need is to ha ve a functoria l replacement q : G ′ → G such that q is a fibration and π i G ′ is trivial for i ≤ n . F o r instnace, w e could tak e G ′ to be the gro up P a th 1 ( G ) of paths originating at 1. (In the finite dimensional Lie con text, how ever, this would not b e a suitable choice as Path 1 ( G ) is infinite dimensional. Tha t is why w e chose G h 2 i := f G o instead.) 8. Functorial proper ties of co nnected covers F or n ≤ 2 w e prov e that our definition of the n -th connected co ver G h n i of a Lie crossed- mo dules is functor ial in Lie butterflies and sa tisfies the exp ected adjunction prop erty (P rop osition 8.8). W e will ne e d the following pro p e rty of the connected cov er s: ( ⋆ 2) F or any homomor phism f : H → G , suc h that π i f : π i H → π i G is an isomorphism for 0 ≤ i ≤ n − 1, the dia gram H h n − 1 i f h n − 1 i / / q n − 1   G h n − 1 i q n − 1   H f / / G is cartes ian. 8.1. Co nstruction of the n -th connected co v er functor. Consider the Lie butterfly B : H → G , H 1 κ " " D D D   G 1 ι } } z z z   E σ | | z z z ρ ! ! D D D H 0 G 0 16 BEHRANG NOOHI The butterfly B h n i : H h n i → G h n i is defined to b e the dia gram L H h n − 1 i κ n ( ( Q Q Q   L G h n − 1 i ι n v v n n n   F h n − 1 i σ n v v m m m m m ρ n ( ( P P P P P H 0 h n i G 0 h n i Let us expla in what the terms a ppe a ring in this diagram ar e. The groups L G and L H are what we called L in the definition of the n -connected cover (see § 7.2). F o r example, L H = H 1 × H 0 H 0 h n i . The Lie gr oup F app earing in the center of the butterfly is defined to b e F := H 0 h n i × H 0 E × G 0 G 0 h n i . The maps ρ n and σ n are obtained by comp o s ing q n − 1 : F h n − 1 i → F with the corres p o nding pro jections. The map κ n is obtained by a pplying the functoriality of ( − ) h n − 1 i to (pr 2 , κ ◦ pr 1 , 1) : L H → F . Definition of ι n is less trivia l and is given in the next paragraphs. W e need to show that th e k er nel o f σ n : F h n − 1 i → H 0 h n i is naturally isomor phic to L G h n − 1 i . There is an equiv alent w ay o f defining F which is somewhat mo re illuminating. Set K := H 0 h n i × H 0 E . Let σ ′ : K → H 0 h n i be the fir st pr o jection map and ρ ′ : K → G 0 the second pro jection map comp ose d with ρ . Then, F = K × ρ ′ ,G 0 G 0 h n i . Now, o bserve that w e ha ve a shor t exact sequence 1 → G 1 α − → K σ ′ − → H 0 h n i → 1 . Therefore, we have a cartesian diagram L G   β / / pr 1   F pr 1   G 1   α / / K and the sequence 1 → L G β − → F σ ′ ◦ pr 1 − → H 0 h n i → 1 is sho rt exact. (Exactness at the right end follows fro m ( ⋆ 1) and the fact that H 0 h n i is connected.) A homoto py fib er seque nc e ar gument applied to this short exact sequenc e sho ws that α induces isomorphis ms π i G 1 → π i K , for 0 ≤ i < n . By ( ⋆ 2) we have a cartesian diagram L G h n − 1 i   β h n − 1 i / / q n − 1   F h n − 1 i q n − 1   L G   β / / F INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 17 Therefore, 1 → L G h n − 1 i β h n − 1 i − → F h n − 1 i σ n − → H 0 h n i → 1 is s ho rt e xact, where σ n := σ ′ ◦ pr 1 ◦ q n − 1 . (Exa c tness a t the right end follows from ( ⋆ 1) and the fact that H 0 h n i is connected.) Setting ι n := β h n − 1 i completes the construction of o ur butterfly diagra m. The equiv a riance axio ms for this butterfly follow from the functoriality of the ( n − 1)-th connected co ver. R emark 8 .1 . In the cas e wher e we hav e a str ic t morphism f : H → G , we can define a natural strict morphism f h n i : H h n i → G h n i compo ne nt wise. It is natural to ask whether this mor phism coincides with the one we constructed ab ov e using butter- flies. The answer is yes. The pr o of uses the the following prop er t y o f connected cov er s: ( ⋆ 3) If G is an n -connected gro up acting on H , then (id , q n − 1 ) : G ⋉ H h n − 1 i → G ⋉ H is th e ( n − 1)-co nnec ted cov e r of G ⋉ H . That is, the map (id , q n − 1 ) is isomor phic to the q n − 1 map of G ⋉ H . 8.2. E ff ect on the comp osi tion of butterflies . The pro of that the construction of the previous subse ction r esp ects comp osition of butterflies is somewhat in tr icate. W e will only consider Lie butterflies and assume that 0 ≤ n ≤ 2, but the exact sa me pro ofs apply v erbatim to to p o logical butterflies (and als o to infinite dimensional Lie butterflies). W e b egin with a few lemmas. Lemma 8.2. L et m ≥ 0 b e an inte ger. Consider a homotopy c artesian diagr am of top olo gic al sp ac es X h / /   Y f   Z g / / W Supp ose that W is ( m + 1) -c onne cte d and Z is m -c onne cte d. Then, h induc es isomorphi sms π i h : π i X → π i Y for i ≤ m . Pr o of. The connectivity a ssumptions on Z and T imply that the homotopy fiber of g is m -connected. Since the diagram is ho motopy cartesia n, the same is true for the homotopy fiber of h . A homo topy fib er ex act sequence implies the cla im.  Corollary 8.3. L et f : Y → W and g : Z → W b e homomorphisms of Lie gr oups and supp ose that W is ( m + 1) - c onne cte d. Supp ose that either f or g is a fibr ation (e.g., surje ctive). Then, we have n atur al isomorph isms Z h m i × W Y h m i ∼ =  Z h m i × W Y ) h m i ∼ =  Z × W Y h m i ) h m i . In p articular, al l thr e e gr oups ar e m - c onne cte d. Pr o of. W e prov e the first equality . Apply Lemma 8.2 to the diagra m X h / /   Y f   Z h m i g ◦ q m / / W 18 BEHRANG NOOHI where X := Z h m i × W Y . The dia gram is homotopy cartes ian b ecause either f o r g ◦ q m is a fibration. Now apply ( ⋆ 2) to h = pr 2 : Z h m i × W Y → Y .  The next lemma is the tec hnical core of this subsection. Lemma 8.4. Consider the c ommutative diagr am X h / / k   Y f   Z g / / W of Lie gr oups. Supp ose t hat W acts on X so that [ f ◦ h : X → W ] is a Lie cr osse d- mo dule. Also, su pp ose tha t the induc e d action of Y on X via f makes t he map h Y - e quivariant (the action of Y on itself b eing the right c onjugation). Assume the same thing fo r the induc e d action of Z on X via g . Supp ose that W is ( m + 1) -c onne cte d, f is surje ctive, and that k is close d inje ctive normal with ( m + 1) -c onne cte d c okernel. Then, the se quenc e 1 / / X h m i ( k h m i ,h h m i ) / / Z h m i × W Y h m i u / /  Z X × W Y  h m i / / 1 is short exact. Here , u is the c omp osition ( q m , id) h m i ◦ φ , wher e φ : Z h m i × W Y h m i ∼ − → ( Z h m i × W Y ) h m i is the isomorphism of Cor ol lary 8.3. (F or the definition of Z X × W Y se e the end of § 6.). In other wor ds, we have a natur al isomorph ism Z h m i X h m i × W Y h m i ∼ =  Z X × W Y  h m i . Pr o of. W e start with the short exact seque nc e 1 → X → Z × W Y → Z X × W Y → 1 , which is essen tially the definition of Z X × W Y . F rom it we c o nstruct the exact seque nce 1 → X h m i α − → Z h m i × W Y ( q m , id) − → Z X × W Y with α : X h m i → Z h m i × W Y is ( k h m i , h ◦ q m ). T o see why this sequence is exa ct, we ca lculate the kernel of the homomorphism ( q m , id) : Z h m i × W Y → Z × W Y : X × Z × W Y  Z h m i × W Y  ∼ = X × Z × W Y  ( Z × W Y ) × Z Z h m i  ∼ = X × Z Z h m i ∼ = X h m i . F or the fir s t equality w e have used Z h m i × W Y ∼ = ( Z × W Y ) × Z Z h m i . F or the la s t equa lit y w e ha ve used ( ⋆ 2) for k : X → Z . (Note that, since coker k is ( m + 1)-connec ted, k : X → Z induces is omorphisms on π i for all i ≤ m .) Observe tha t the last map in the ab ov e sequence is a fibration with (o pe n- closed) image I ⊆ Z X × W Y . This fibra tion has an m -connected kernel X h m i , so, using the INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 19 homotopy fib er exa ct sequence, we s e e that it induces isomorphisms o n π i for i ≤ m . By ( ⋆ 2) we g et the car tesian square ( Z h m i × W Y ) h m i q m   ( q m , id) h m i / /  Z X × W Y  h m i q m   Z h m i × W Y ( q m , id) / / I (Note that I h m i =  Z X × W Y  h m i b ecause the m -th co nnected cov er only dep ends on the connected comp onent of the identit y , which is co ntained in I .) Pre c omp osing the top row with the isomo rphism φ : Z h m i × W Y h m i ∼ − → ( Z h m i × W Y ) h m i of Corollar y 8.3, and calling the compo sition u as in the sta tement of the lemma, we find the following co mm utative diagr a m in which the s q uare on the right is ca rtesian 1 / / X h m i ( k h m i ,h h m i ) / / id   Z h m i × W Y h m i (id ,q m )   u / /  Z X × W Y  h m i q m   / / 1 1 / / X h m i ( k h m i ,h ◦ q m ) / / Z h m i × W Y ( q m , id) / / I / / 1 Since the b o ttom row is short ex act, so is the top r ow. Pro o f of the lemma is complete.  W e need o ne more technical lemma. Lemma 8.5. Consider a c ommutative diag r am X h / / k   Y f   Z g / / W of top olo gic al gr oups. Supp ose that W acts on X so that [ f ◦ h : X → W ] is a top olo gic al cr osse d-mo dule. Also, supp ose that the induc e d action of Y on X via f makes the map h Y -e quivariant (the action of Y on itself b eing the right c onjuga- tion). Assu me the same thing for t he induc e d action of Z on X via g . Supp ose that f is surje ct ive. L et α : W ′ → W b e a homomorphism with normal image, and denote its c okernel by W 0 . Denote the pul lb ack of the ab ove diagr am along α by adding prime sup erscripts. Denote the images of X and Z in W 0 by X 0 and Z 0 , r esp e ctively. (Note that X 0 is normal in W 0 .) Then, the se quenc e 1 → Z ′ X ′ × W ′ Y ′ → Z X × W Y → Z 0 /X 0 → 1 is exact. In p articular, if the image of W ′ is op en in W , then Z ′ X ′ × W ′ Y ′ is a union of c onne cte d c omp onents of Z X × W Y . 20 BEHRANG NOOHI Pr o of. The pro of is elemen ta ry group theory .  W e are now r eady to prove that o ur construction of n - connected covers is func- torial, that is, it r esp ects comp osition of butterflies. Prop ositi on 8.6. L et C : K → H and B : H → G b e Lie but terflies, and let B ◦ C : K → G b e their c omp osition. Then, ther e is a n at u r al isomorphism of butterflies B h n i ◦ C h n i ⇒ ( B ◦ C ) h n i which makes the assignment G 7→ G h n i a bif unctor fr om the bic ate gory LieXM of Lie cr osse d-mo dules and butt erflies to itself. Pr o of. Let C a nd B b e given by K 1 κ " " F F F   H 1 ι | | x x x   E C σ | | x x x ρ " " F F F K 0 H 0 H 1 κ ′ " " F F F   G 1 ι ′ | | x x x   E B σ ′ | | x x x ρ ′ " " F F F H 0 G 0 resp ectively . Then, the comp os ition B ◦ C is the butterfly K 1 % % L L L   G 1 y y r r r   E C H 1 × H 0 E B y y r r r % % L L L K 0 G 0 Recall the notation of § 8.1: F C = K 0 h n i × K 0 E C × H 0 H 0 h n i and F B = H 0 h n i × H 0 E B × G 0 G 0 h n i . The group app earing in the ce nter of the butterfly B h n i ◦ C h n i is F C h n − 1 i L H h n − 1 i × H 0 h n i F B h n − 1 i . The group app earing in the ce nter of the butterfly ( B ◦ C ) h n i is F B ◦ C h n − 1 i =  K 0 h n i × K 0 ( E C H 1 × H 0 E B ) × G 0 G 0 h n i  h n − 1 i . W e show that there is a natural isomo r phism fr o m the for mer to the la tter. F o r this, first we apply Lemma 8.4 with m = n − 1 and X = L H , Z = F C , Y = F B , and W = H 0 h n i to get F C h n − 1 i L H h n − 1 i × H 0 h n i F B h n − 1 i ∼ = ( F C L H × H 0 h n i F B ) h n − 1 i . It is now enough to construct a natural isomorphism F C L H × H 0 h n i F B → F B ◦ C , that is, ( K 0 h n i × K 0 E C × H 0 H 0 h n i ) L H × H 0 h n i ( H 0 h n i × H 0 E B × G 0 G 0 h n i ) INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 21 − →  K 0 h n i × K 0 ( E C H 1 × H 0 E B ) × G 0 G 0 h n i  . This, how ever, ma y not be the case. More precise ly , there is suc h a natural homo- morphism, but it is not necessar ily a n isomorphism. It is, howev er, an isomorphis m betw een the connected comp one nts of the identit y elements (and that is enough fo r our purpo ses). T o see this, use Lemma 8.5 with X = H 1 , Z = K 0 h n i × K 0 E C , Y = E B × G 0 G 0 h n i , W = H 0 , W ′ = H 0 h n i , and α = q n . (Recall that L H = H 1 × H 0 H 0 h n i .) Her e we are using the fact that α = q n : H 0 h n i → H 0 surjects onto the connected compone nt of the identit y element in H 0 . W e omit the verification that the isomorphism B h n i ◦ C h n i ⇒ ( B ◦ C ) h n i resp ects isomorphisms of butterflies and that it co mm utes with the as s o ciator isomorphisms in LieXM .  The prop osition is v a lid in the top olo gical setting a s w e ll, and the pro of is iden- tical. Corollary 8.7. L et f : H → G b e an e quivalenc e of Lie cr osse d-m o dules. Then the induc e d morphism f h n i : H h n i → G h n i , n = 0 , 1 , 2 , is also an e quivalenc e of Lie cr osse d-mo dules. 8.3. Adjunction prop erty of conne cted co v ers . W e show that n -connected cov er s of Lie cross ed-mo dules satisfy the exp ected adjunction pro p erty , namely , that a weak mor phism f : H → G fro m an n -co nnected Lie cr o ssed-mo dule H uniquely factors throug h q n : G h n i → G (Prop osition 8.8). As in the previous sectio n, we will ass ume that n ≤ 2 . What we say re mains v alid fo r topolo gical cr ossed-mo dules (and also for infinite dimensional Lie crossed- mo dules). W e will us e the adjunction prop er ty for gro ups: ( ⋆ 4) F or any homomor phism f : H → G with H ( n − 1)- connected, f factors uniquely throug h q n − 1 : H h n − 1 i → H . Prop ositi on 8.8. L et G and H b e Lie cr osse d-mo dules, and su pp ose that H is n -c onne ct e d (D efinition 7.2 ). Then, the morphism q = q n : G h n i → G induc es an e quivalenc e of hom-gr oup oids q ∗ : LieXM ( H , G h n i ) ∼ − → LieXM ( H , G ) . Pr o of. W e cons truct an inv ers e functor (quasi- inv erse, to b e prec ise) to q ∗ . The con- struction is very s imilar to the constructio n of the n -connected cov er of a butterfly given in the previous subsection. Since H = [ H 1 → H 0 ] is n -co nnec ted, we may assume that H 0 is n - connected and H 1 is ( n − 1 )- connected (this was discussed in § 7.1). Consider a butterfly B H 1 κ " " D D D   G 1 ι } } z z z   E σ | | z z z ρ ! ! D D D H 0 G 0 in LieXM ( H , G ). Define F := E × G 0 G 0 h n i . Let τ : F → H 0 be σ ◦ pr 1 . Since τ is a (lo cally trivia l) fibration and H 0 is connected, τ is surjective. O n the other ha nd, 22 BEHRANG NOOHI ker τ is the inv er se imag e of ι ( G 1 ) under the pr o jection pr 1 : F → E ; this is ex actly G 1 × G 0 G 0 h n i = L G . T ha t is, we ha ve a short exact sequence 1 → L G β − → F τ − → H 0 → 1 . It follows fr o m ( ⋆ 2) applied to β that the sequence 1 → L G h n − 1 i β h n − 1 i − → F h n − 1 i τ ◦ q n − 1 − → H 0 → 1 is also short exact. Define the butterfly B ′ to b e H 1 κ ′ % % K K K K K   L G h n − 1 i β h n − 1 i w w n n n n   F h n − 1 i τ ◦ q n − 1 y y s s s s s  ' ' P P P P P H 0 G 0 h n i where  = pr 2 ◦ q n − 1 and κ ′ is o btained by the a djunction pro p e rty ( ⋆ 4) applied to q n − 1 : F h n − 1 i → F . It is easy to verify that B 7→ B ′ is a n inv erse to q ∗ . (F or this, use the fact that E is the pushout o f F h n − 1 i a long pr 1 ◦ q n − 1 : L G h n − 1 i → G 1 and apply [No3], § 10.2.)  Corollary 8.9. F or n = 0 , 1 , 2 , the inclusion of the ful l sub bic ate gory of LieXM c onsisting of n -c onne ct e d Lie cr osse d-mo dules is left adjoint to the n -c onne cte d c over bifunctor ( − ) h n i : Lie XM → LieXM . 9. The bifunctor fr om Lie crossed-modules to 2 -term L ∞ -algebras In this section we pr ov e our main integration results for weak mo rphisms of 2- term L ∞ -algebra s (Theor em 9 .2 and Corollary 9.3). Throughout the section, we fix the base ring to b e R or C . All Lie groups are finite dimensional (real or complex, resp ectively). W e b egin with a simple lemma. Lemma 9 .1. L et H , K and K ′ b e c onne cte d Lie gr oups. Supp ose t hat H acts on K and K ′ by au t omorphisms and let f : K → K ′ b e a Lie homomorphi sm. If the induc e d map Lie f : Lie K → Lie K ′ is H -e quivariant t hen so is f itself. Pr o of. This follows from the fact that if tw o g r oup ho mo morphisms induce the same map on Lie algebr as then they are equal.  Let LieAlgXM b e the full sub bicateg ory of 2T ermL ♭ ∞ consisting of strict 2-term L ∞ -algebra s (i.e., Lie algebra cro s sed-mo dules). Theorem 9.2 . T aking Lie algebr as induc es a bifunctor Lie : LieXM → 2T ermL ♭ ∞ . The bifunctor Lie factors thr ough and essent ial ly surje cts onto LieAl gXM . F ur- thermor e, for H , G ∈ LieXM , the induc e d functor Lie : LieXM ( H , G ) → 2T ermL ♭ ∞ (Lie H , Lie G ) on hom-gr oup oids is INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 23 i ) faithful, if H is c onne cte d; ii ) ful ly faithful, if H is 1-c onne cte d; iii ) an e qu ivalenc e, if H is 2-c onne ct e d. Pr o of. That Lie : LieXM → 2 T ermL ♭ ∞ is a bifunctor follows fro m the fact that taking Lie algebra s is exact and commutes with fiber pr o ducts of Lie groups. Pr o of of ( i ). Let G = [ G 1 → G 0 ] and H = [ H 1 → H 0 ]. Let B , B ′ : H → G be tw o butterflies. Since H is connected, we may assume that H 0 is co nnected (see § 7.1). Denote the NE-SW shor t exact sequences for B and B ′ by 0 → G 1 → E → H 0 → 0 , 0 → G 1 → E ′ → H 0 → 0 . Consider t wo is omorphisms B ⇒ B ′ , given by Φ , Ψ : E → E ′ , such that Lie Φ = Lie Ψ : Lie E → Lie E ′ . Then, Φ and Ψ a re equal on the c onnected comp onent E o and also on G 1 . Since H 0 is connected, E o and G 1 generate E , so Φ a nd Ψ are equal on the whole E . Pr o of of ( i i ). Notation b e ing as in the pre v ious part, we may a ssume that H 0 is connected and simply-c o nnected and H 1 is connected (se e § 7.1). Consider an isomorphism Lie B ⇒ Lie B ′ given by f : Lie E → L ie E ′ . W e show that f integrates to Φ : E → E ′ . Let ˜ E → E be the universal cover of E . Integrate f to a homo morphism ˜ Φ : ˜ E → E ′ . C o nsider the diagr am ˜ E β   ˜ Φ   γ   9 9 9 9 9 9 9 9 9 9 9 9 9 9 G α   δ 9 9 s s s s s s s s s s E Φ   σ % % L L L L L L L L L 0 / / G 1 r r r r r r r r r 9 9 % % K K K K K K K K K H 0 / / 0 E ′ 9 9 r r r r r r r r r Here G is the kernel o f γ := σβ : ˜ E → H 0 . Note that G ∼ = G 1 × E ˜ E . That is, G is the pullback of ˜ E along the ma p G 1 → E . Since π i H 0 = 0 for i = 1 , 2 , a fib er homotopy exact s e quence arg ument shows that π 1 G 1 → π 1 E is an iso morphism. Hence, G is the universal cov er of G 1 and, in particular , is connected. If we apply Lie to the a b ov e diagra m, we obtain a co mmu tative diagra m of Lie algebr as. Ther efore, sinc e all the groups in volv ed ar e co nnected, the origina l diagram of Lie groups is also co mm utative. Since the to p left square is cartesian, δ induces an isomorphism δ : ker α → ker β . Commutativit y of the diag r am implies then that ˜ Φ v anishes on ker β . Ther efore, ˜ Φ induces a homomorphism Φ : E → E ′ which makes the diagra m comm ute. By lo oking at the cor resp onding Lie a lgebra maps, w e s e e that if f commutes with the other tw o maps o f the butterflies, then so does Φ. Tha t is, Φ is indeed a morphism of butterflies from B to B ′ . 24 BEHRANG NOOHI Pr o of of ( iii ). W e ma y assume that H 0 and H 1 are co nnected and simply-connected (see § 7.1). In view of the previous part, we hav e to s how that every butterfly B : Lie H → L ie G , Lie H 1 κ $ $ I I I I   Lie G 1 ι z z u u u u   E σ z z u u u u ρ $ $ I I I I Lie H 0 Lie G 0 int egrates to a butterfly Int B : H → G . Let Int E b e the simply-connected Lie group whose Lie alg ebra is E . Let G b e the kernel of Int σ : Int E → H 0 . Since π i H 0 = 0, i = 0 , 1 , 2, a n easy homotop y fib er exact sequence argumen t implies that G is connected and simply-co nnected. W e ident ify the Lie alg ebras of G and G 1 via ι : Lie G 1 → E and r egard them as equal. Since G is simply-connected and G 1 is connected, we have a natural isomorphism ¯ ι : G 1 → G/ N for some discr e te central subgroup N ⊆ G . W e claim that N is a no r mal subgr oup of In t E . T o prove this, we compare the conjugation action of In t E on G with the actio n of Int E on G 1 obtained via In t ρ : In t E → G 0 . (The latter is the in tegration of the Lie algebra homo morphism ρ : E → L ie G 0 .) The equiv ar iance axiom of the butterfly for the ma p ρ , plus the fact that ¯ ι − 1 ◦ pr : G → G 1 induces the identit y map o n the Lie alg ebras, implies (Lemma 9.1) that ¯ ι − 1 ◦ pr : G → G 1 is Int E -equiv ar iant. Ther efore, its kernel N is inv ar iant under the conjugation actio n of In t E . That is, N ⊆ Int E is normal. An ar gument simila r to the one used in the previous part shows that the map Int ρ : Int E → G 0 v anishes o n N . Mo r e precis ely , r ep eat the same a r gument with the diagra m Int E   Int ρ   Int σ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ G α   δ 7 7 n n n n n n n n n n n (In t E ) / N ¯ ρ   ¯ σ ' ' O O O O O O O O O O 0 / / G 1 ¯ ι o o o o o o o o o o 7 7 ∂ ' ' P P P P P P P P P P P P H 0 / / 0 G 0 Thu s, we obtain a n induced homomor phism ¯ ρ : (In t E ) / N → G 0 . Denote the map (In t E ) / N → H 0 induced from In t σ by ¯ σ . Collecting what w e ha ve s o far, w e obtain a partial butterfly dia gram G 1 ¯ ι x x q q q q   (In t E ) / N ¯ σ x x q q q q ¯ ρ & & M M M M H 0 G 0 (Observe that a pplying Lie to this partial butterfly g ives us back the corr esp onding po rtion of the original butterfly B .) Finally , using the fact that H 1 is connected INTEGRA TING MORPHIS MS OF LIE 2-ALGEBRAS 25 and simply-connected, we can complete the butter fly by in tegr a ting κ to ¯ κ : H 1 → (In t E ) / N . It is ea sily verified that the resulting diagra m s atisfies the butterfly axioms – this is the sought after butterfly Int B : H → G . The pro o f is co mplete.  Let V 7→ Int V denote the functor which ass o ciates to a L ie a lgebra V the corres p o nding co nnected simply-co nnec ted Lie group. F or a cro s sed-mo dule in Lie algebras V = [ V 1 → V 0 ] w e denote the corres p o nding cr ossed-mo dule in Lie gro ups by Int V := [Int V 1 → In t V 0 ]. Corollary 9.3 . The bifunctor Int : LieAlgXM → LieXM is left adjoint to the bifunctor Lie : LieXM → LieAlg XM . Pr o of. This follows immediately from Theorem 9.2.  R emark 9 .4 . P resumably , the adjunction of Corollar y 9 .3 can b e extended to Int : LieAl gXM  _   ⇌ LieXM : Lie  _   Int : 2T ermL ∞ ⇌ Lie2Gp : Lie Here, by Lie2 Gp we mean the 2-ca tegory of L ie group stacks. The inclus io n on the right is given by the fully faithful bifunctor LieXM → Lie2Gp , [ G 1 → G 0 ] 7→ [ G 0 /G 1 ] . 10. Appendix: functorial n -connected covers f or n ≥ 3 Axioms ( ⋆ 1-4) discussed in § 7 - 8 hav e a c ertain iter ative prop erty which we would like to p oint out in this app endix. T o simplify the no tation, we will replace n − 1 by m . W e saw in § 7- 8 that, for m ≤ 1 , the sta nda rd choices for the m -connec ted cov er functors ( − ) h m i on the category of topolo gical groups automatically satisfy ( ⋆ 1-4). Using this we constructed our m -connec ted cov er bifu nctor ( − ) h m i on the bicategory of top ologic a l (or Lie) cr ossed-mo dules for m ≤ 2. It can b e sho wn that these bifunctors ag a in satisfy (a categorifie d version) of ( ⋆ 1-4). A magic se ems to have o ccurred here: we mana ged to ra ise m from 1 to 2! This may sound contradictory , as we do not to exp ect to ha ve a functorial 2-connected cov er functor ( − ) h 2 i on the ca tegory of top ologica l groups which sa tisfies either the pullback property ( ⋆ 2) or the a djunction prop erty ( ⋆ 4). This a pparent contradiction is expla ined by noticing that our definition of ( − ) h 2 i indeed y ields a cr ossed-mo dule, even if the input is a top olo gical group. Mo re precisely , for a top ologica l gro up G , we get G h 2 i = [ f L o → G ′ ] , where q : G ′ → G is a choice of a 2-connected replacement for G and L = k er q . (F or example, take G ′ = Path 1 ( G ), the s pace of pa ths starting at 1 ; se e Remar k 7.6.) It is also interesting to note that, for differe nt choices of 2-co nnec ted re pla cement 26 BEHRANG NOOHI q : G ′ → G , the resulting crosse d-mo dules G h 2 i a re ca nonically (up to a unique isomorphism of butterflies) eq uiv a lent. The upshot of this discussion is that, 2-connected covers of top olog ical gr oups seem to more na turally exist a s top olo gical crossed- mo dules. Another implication is that w e can no w itera te the pro ce ss. F or ex ample, w e get a functorial constr uction of a 3- connected cover G h 3 i o f a top olog ical gr oup G as a 2-cro s sed-mo dule, and this (essentially unique) construction enjoys a c ategorified version of ( ⋆ 1-4). This seems to hint at the following general philo sophy: for a ny m ≤ k + 1, there should b e a (essentially unqiue) definition of m -connected covers G h m i for top ologica l k -cr ossed-mo dules G whic h enjoys a categorified v ersion of ( ⋆ 1-4). W e p oint out that the notion o f k -cros sed-mo dule exists for k ≤ 3 (se e [Con] and [ArKuUs]). The butterfly construction of the tricateg ory o f 2-crossed-mo dules (and week morphisms) is b eing developed in [AlNo2]. F or higher v alues of k the simplicial appro ach is p erhaps a b etter alternative, as k - c r ossed-mo dules tend to bec ome immensly complicated as k inc r eases. R emark 10.1 . The ab ov e discussion a pplies to the case where we replace top olog ical groups with infinite dimensional Lie groups. References [AlNo1] E. Aldrov andi, B. Noohi, Butterflies I: morphisms of 2-gr oup stacks , Adv. Math. 221 (2009), no. 3, 687–773. [AlNo2] E. Aldrov andi, B. No ohi, Butterflies III: 2-butterflies and 2-gro up stacks , in preparation. [ArKuUs] Z. Arv as, T. S. K uzpinari , E. ¨ O. Usl u, Thr e e cr osse d mo dules , arX iv:0812.4685v1 [math.CT]. [BaCr] J. Baez, A. Crans, Higher-dimensional algebr a. VI. Lie 2-algebra s , Theory Appl. Categ. 12 (2004), 492–538. [BrGi] R. 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