Extensions of groups by braided 2-groups

EXTENSIONS OF GR OUPS BY BRAIDED 2-GR OUPS EV AN JENKINS Abstract. W e classify extensions of a group G b y a braided 2-group B as defined b y Dri nfeld, Gelaki, Ni ksh ych, and Ostrik. W e describe such extensions as homotop y classes of maps from t he classifyi ng space of G t o the classifying space of the 3-gr oup of braided B -bitorsors. The Postnik o v system of the latter space con tains a generalization of the classical Pon try agin square to the setting of lo cal coefficients, which has b een previously discussed b y Baues and more recen tly , in a setting close to ours, b y Etingof, Nikshyc h, and Ostrik. W e give an explicit co chain-lev el descri ption of this Pon tryagin square for group cohomology . 1. Introduction A classica l pro blem in homological algebr a is to classify extensions of a gr oup G by another gro up K , i.e., gro ups E equipp ed with a surjection ∂ : E ։ G and an iden tification K ∼ = ∂ − 1 ( e ). When K = A is an abelia n group, the action of an element of E o n an element of A b y conjuga tion depe nds o nly on its image in G , so extensions of G by A include the data of an action of G as automorphisms of A . If we fix such a n actio n, then extensions with that action ar e classified by H 2 ( G, A ) (see, for exa mple, [W ei95, 6.6 ]). In [DGNO10, Appendix E], a catego rification o f the notion o f a n extension o f a gro up b y an ab elian group is defined, in which the a b elia n group A is re pla ced by a br aided 2-gr oup B (see [BL04] for bas ic results a bo ut 2-gr oups, which ha ve also b een studied under the names gr-ca teg ory and ca tegorica l gr oup, a nd [JS9 3] for basic results ab out br a ided monoida l categories ). Given such an extensio n one has an underlying action of G as bra ided a uto equiv alences o f the braided 2-gro up B , and also an underlying extension of G b y the ab elian group A = π 0 ( B ) of isomorphism classes of ob jects o f B . It is natural to ask under what circumstances these tw o pieces of da ta will determine an extension of G b y B and how unique such a n extension is. In this pap er , w e show that we c an lift these data to an extension if and only if a cer ta in cohomo lo gy class in H 4 ( G, H ) v anishes, where H = π 1 ( B ) is the a b e lian group of isomorphisms of the unit ob ject in B , and the a ction of G on H is in- duced from the action of G on B . This cohomolog y class co mes from a function H 2 ( G, A ) → H 4 ( G, H ) (determined b y a fixed braided 2-gro up B and an action of G on B ) that genera lizes the classical Pont ryagin square, w hich can b e r ecov ered in our lang ua ge by taking the trivial action of G on B . If this o bstruction v a nishes, then extensions for m a torsor for H 3 ( G, H ). These res ults are para llel to the r e s ults of [ENO1 0] on the rela ted problem of classifying braided G -crossed fusion categories. In that paper, a braided G -cros sed fusion categ o ry C is view ed a s a G -indexed family of inv ertible bimo dules ov er the neutral component. Analogo usly , we will view extensions of G by a braided 2-group 1 2 EV AN JENKINS B as G -indexed families of braided bitorso rs ov er B . W e then use obstruction theory to obtain our clas s ification. Sections 2 thro ug h 4 set up the machinery of braided bitors ors. Se c tion 5 relates extensions and bitorso rs by a constructio n of Grothendieck. Section 6 con tains our classification result. Section 7 contains an explicit co chain-level description of the Pon tryagin square defined in Section 6. The a uthor would like to thank his advisor , Vladimir Drinfeld, for suggesting this topic and providing inspiration, g uidance, and careful reading o f many drafts. The a uthor w ould also like to thank Peter May , Da niel Sch¨ appi, Mike Shu lman, and Ross Street fo r helpful discussions. 2. Torsors for 2-gr oups W e can define tor sors for 2-groups m uch the sa me as w e do for groups. W e first review the notion o f modules for monoidal categ o ries. A mono idal categor y C ca n b e viewed as a one-ob ject bicategor y , whic h we will denote by C [1]. This ca n b e view ed as the “delo oping ” of C , a notio n we will revisit in Section 6. F or now, we no te only that strong monoida l functor s b etw een monoidal ca tegories corres po nd to pseudofunctors b etw een their delo o pings, a nd that C [1] op = ( C rev )[1], where C rev denotes the category C with r eversed tensor pro duct. Definition 2.1. Let C be a monoidal category . A left (resp. right) m o dule (o r mo dule category ) ov er C is a pseudofunctor X : C [1] → Cat (resp. X : C [1 ] op → Cat). By a buse of notation, we denote the image of the unique ob ject of C [1] by X . W e denote by C Mo d a nd Mo d C the 2 - categor ies of left and rig ht C -mo dules, resp ectively . W e will write a left a ction of a n element c ∈ C on an elemen t x ∈ X by c ⊲ x , and a right a ction as x ⊳ c . F or the rest of this s ubs e ction, w e will only deal with left C - mo dules. Completely analog ous definitions and pro ofs work with “left” everywhere replaced by “right.” Lemma 2.2. L et F : X → Y b e a morph ism of left C -mo dules, and supp ose further that F is an e quivalenc e of c ate gories. Then F is an e quivalenc e of C -mo dules. Pr o of. The forg etful 2-functor C Mo d → Cat is monadic, a nd hence it r eflects ad- joint equiv alences (see [CMV02]).  Recall that a 2 -group is a mono idal category in which all ob jects a nd mor phisms are in vertible. Definition 2.3. Let G be a 2-group. A left G -module X is a left G -torsor if X is no nempt y a nd the characteris tic map χ = ( a, π 2 ) : G × X → X × X is an equiv alence. Definition 2.4. The trivial l e ft G -mo dule , deno ted by G , is G equipp ed with the action of left multiplication. Definition 2.5. The ac tio n of G on X is essentially simp l y transitive if, for every x ∈ X , the map G → X , g 7→ g ⊲ x is a n equiv alence of catego ries. Theorem 2.6. L et G b e a 2-gr oup, X a left G -mo dule. The fol lowing ar e e quivalent. (1) X is a left G -torsor. EXTENSIONS OF GROUPS BY BRAIDED 2-GROUPS 3 (2) X is nonempty, and the action of G on X is essential ly simply tr ansitive. (3) X is e quivalent to the trivial left G -mo dule. Pr o of. (1) ⇒ (2): Fix x ∈ X . W e will sho w that the map F : g 7→ g ⊲ x is full, faithful, and ess ent ially surjective. If x ′ ∈ X , essential sur jectivit y o f the characteris tic map implies that ther e exists g ∈ G , x ′′ ∼ = x suc h that g ⊲ x ′′ ∼ = x ′ . It follows that g ⊲ x ∼ = g ⊲ x ′′ ∼ = x ′ . Thus, F is essentially s urjective. If g , g ′ ∈ G , φ 1 , φ 2 : g → g ′ , such that φ 1 ⊲ x = φ 2 ⊲ x : g ⊲ x → g ′ ⊲ x , then ( φ 1 ⊲ x, x ) = ( φ 2 ⊲ x, x ) : ( g ⊲ x, x ) → ( g ′ ⊲ x, x ), so faithfulness of the characteristic map implies that ( φ 1 , x ) = ( φ 2 , x ), and hence φ 1 = φ 2 . Thus, F is faithful. Finally , if g , g ′ ∈ G , and f : g ⊲ x → g ′ ⊲ x , then fullness of the characteristic map gives a map ( φ, id) : ( g , x ) → ( g ′ , x ) suc h that φ ⊲ id : g ⊲ x → g ′ ⊲ x is f . Th us, F is full. (2) ⇒ (3): The map g 7→ g ⊲ x is a map G → X of G -modules and an eq uiv alence of ca tegories, so by Lemma 2.2, it is an equiv alence of G -mo dules. (3) ⇒ (1): It suffices to show that the tr ivial left G -module is a left G -torso r. The map ( a, π 2 ) : G × G → G × G has quasi-inverse ( a ◦ ( − − 1 ) , π 2 ), so G is a le ft G -tor sor.  Corollary 2.7. Any morphism F : X → Y of left G -t orsors is an e quivalenc e. Pr o of. Fix x ∈ X . Then the following diag ram commutes up to isomorphis m. G g 7→ g ⊲x           g 7→ g ⊲F ( x )   > > > > > > > X F / / Y By Theorem 2.6, the top t wo arrows are equiv alences, so F is an equiv alence.  Prop ositi o n 2 . 8. The au t o e quivalenc e 2-gr oup of a left G -torsor is e quivalent to G . Pr o of. By Theor em 2 .6, every G -torso r is equiv alent to the trivia l left G -torsor, so it suffices to show this for the trivial left G -tor sor G . W e get a monoida l functor G → Aut G ( G ) b y sending g ∈ G to x 7→ xg − 1 . W e get a monoida l functor Aut G ( G ) → G by sending f ∈ Aut G ( G ) to f ( e ) − 1 . The comp osition G → Aut G ( G ) → G is isomorphic to the identit y; for the other direction, w e note tha t if f ( e ) − 1 ∼ = g , then f ( x ) ∼ = xx − 1 f ( x ) ∼ = xf ( e ) ∼ = xg − 1 is an isomo rphism of G -module morphisms.  3. Bitorsors for 2-gr oups W e can also de fine bitorso rs for 2- groups muc h the sa me a s we do for groups. W e b egin here with the notion of bimo dules for monoidal categor ies. Definition 3 .1. Let C and D b e monoidal categories . A ( C , D ) -bi m o dule is a ( D rev × C )- mo dule. W e think of C as ac ting on the left and D as acting on the right. W e denote b y C Mo d D the 2-ca teg ory of ( C , D )-bimo dules . 4 EV AN JENKINS Definition 3.2 . L e t C , D , a nd E be 2 -groups , a nd let X and Y b e ( C , D )- and ( D , E )- bimo dules, r esp ectively . The tensor pro duct X × D Y of X and Y is defined to b e the bicolimit o f the diagram X × D × D × Y X × D × Y ⇒ X × Y , where C acts on the left o n each left fac tor, and E acts on the right on each right factor. Here, the arrows cor resp ond to the pro duct of D and the v a rious actions of D on X and Y . In other words, a ( C , E )-bimodule map X × D Y → Z is a map F : X × Y → Z equipp e d with a family of isomo rphisms φ d : F ( x ⊳ d, y ) ∼ = → F ( x, d ⊲ y ) na tur al in d and compatible with the pr o duct in D . The following imp orta nt “folklor e” r esult is en tirely for ma l, a lthough the author do es not b elieve a reference exists. Theorem 3 .3. L et C b e a monoida l c ate gory. Then C -bimo dules form a monoidal 2-c ate gory. Pr o of. W e g ive only the v aguest o utline of a pro of. W e will identify C -bimodules with endo-pseudo functors of the preshea f 2-catego ry [( C [1]) op , Cat] that pr eserve all weighted bicolimits. F or the rele v ant bicateg o rical definitions, see [Str80] and [Str87]. Given a pseudofunctor b X : [( C [1]) op , Cat] → [( C [1]) op , Cat] preserving weigh ted bicolimits, w e r e strict via the bica tegorica l Y o ne da em b edding to get a pseudo- functor X ′ : C [1] → [( C [1 ]) op , Cat], which corres po nds under adjunction to a C - bimo dule X : ( C [1]) op × C [1] → Ca t. In the other directio n, given a mo dule X : ( C [1]) op × C [1] → Cat, we cons truct a weight ed bico limit-preserving pseudo- functor b X : [( C [1]) op , Cat] → [( C [1]) op , Cat] by b X ( Y )( ∗ ) = bico lim( Y ( − ) , X ( ∗ , − )) . These tw o constr uc tio ns give qua si-inv erse equiv alences of 2-catego ries, and take the tensor pro duct of bimo dules to the comp osition of pseudofunctor s. Since pseudo- functors for m a (str ict) mono idal 2-c ategory , we may transp ort this structure to the 2-ca teg ory o f C -bimo dules to get a monoidal 2-catego r y .  Next, we pro ve a coherence result for bimo dules. Definition 3.4. Let C b e a strict monoidal catego ry . A s trict C -bimo dule is a category X equipp ed with a strict 2-functor C × C op → Aut( X ). Definition 3.5. Let C b e a strict monoidal category , and let X a nd Y b e strict C -bimodules . A strict morphism fro m X to Y is a strict 2-natural tr ansformation X ⇒ Y . Theorem 3.6 . L et C b e a m onoidal c ate gory, and let X b e a C -bimo dule. Then ther e is a strict C -bimo dule st X over st C and an e quivalenc e X → st X that is e quivariant with r esp e ct to the st rictific ation maps C → s t C . Pr o of. W e conside r the bica tegory C X defined as follows. • C X has tw o o b jects, whic h we will denote A and B . • End( A ) = E nd( B ) = C . • Hom( A , B ) = X , while Hom( B , A ) = ∅ . • Compo sition is given by the left and right actions of C o n X . EXTENSIONS OF GROUPS BY BRAIDED 2-GROUPS 5 W e now app eal to the co herence theo rem for bicatego ries (see, for ex ample, [MLP85, Section 2 ]) to get a stric t 2-c a tegory st C X with the s a me ob ject set as C X and a biequiv alence F : C X → s t C X . The bieq uiv alence F induces monoidal equiv alences End( A ) → st End( A ) = st C , End( B ) → st End( B ) = s t C , and X → st X compat- ible with the le ft and right actions.  Definition 3.7. Let G b e a (strict) 2-g roup. A G -bimodule X is a G -bitorsor if X is a tor sor separa tely with resp ect to b oth the left action and the r ight a ction. Prop ositi o n 3.8. L et X and Y b e G -bitorsors. Then X × G Y is a G -bitorsor. Pr o of. By Theorem 2.6, Y is equiv ale nt to G as a left G -torsor. It follows that as a left G -mo dule, X × G Y ∼ = X × G G ∼ = X ∼ = G , so X × G Y is a left G -torsor. An ident ical ar gument s hows that it is a righ t G - torsor .  W e can cons truct nontrivial (strict) G -bitorsors as follows. Given Φ ∈ Aut( G ), we define a bitors or G Φ to b e G as a left G -torso r. The rig ht action is g iven by G op − 1 → G Φ → G ∼ = → Aut G ( G ), where the last ar row comes fro m Pro p o sition 2.8. Explicitly , ( g , g ′ ) ∈ G × G op acts by x 7→ g x Φ( g ′ ). Our next g o al is to show tha t every G -bitor sor is equiv alent to one of the ab ove form. T o do this, we in tro duce the adjoin t auto equiv alence of X . F or a given x ∈ X , we define a 2-gr o up Γ G , X ,x consisting of triples ( g , g ′ , φ ), where g , g ′ ∈ G a nd φ : g ⊲ x ∼ = → x ⊳ g ′ . A morphism of triples ( g 1 , g ′ 1 , φ 1 ) → ( g 2 , g ′ 2 , φ 2 ) is a pair of mor phisms g 1 → g 2 , g ′ 1 → g ′ 2 that intert wine φ 1 and φ 2 . The tensor product on Γ G , X ,x is g iven by ( g 1 , g ′ 1 , φ 1 ) ⊗ ( g 2 , g ′ 2 , φ 2 ) = ( g 1 ⊗ g 2 , g ′ 1 ⊗ g ′ 2 , φ 1 ⊲ ⊳ φ 2 ), where φ 1 ⊲ ⊳ φ 2 = ( φ 1 g ′ 2 ) ◦ ( g 1 φ 2 ). Theore m 3.6 guarantees that this pro duct is coherently asso ciative. There are tw o natur al pro jections π 1 , π 2 : Γ G , X ,x → G . Each o f these is an equiv alence, as for a ny g ∈ G , there is a unique (up to uniq ue is omorphism) g ′ ∈ G with g ⊲ x ∼ = x ⊳ g ′ , and the freedom in c ho osing this isomorphism is precisely the automorphism group of g itself. Definition 3.9. The adjoint auto equiv alence of the pair ( X , x ) is the auto e - quiv alence of G given b y Ad X ,x = π 1 ◦ π − 1 2 . Prop ositi o n 3.10. L et X b e a G -bitorsor, x ∈ X . Then X ∼ = G Ad X ,x . Pr o of. W e de fine a functor G Ad X ,x → X b y g 7→ g ⊲ x . This is a map o f left G -modules by definition. The compatibility with the rig ht action is g iven by the isomorphism ( g ⊗ Ad X ,x ( g ′ )) ⊲ x ∼ = → g ⊲ (Ad X ,x ( g ′ ) ⊲ x ) ∼ = → g ⊲ ( x ⊳ g ′ ) ∼ = → ( g ⊲ x ) ⊳ g ′ . This map of bitorso rs is necessarily an equiv alence by Corollary 2.7.  Prop ositi o n 3.11. L et Φ , Ψ ∈ Aut( G ) . Then G Φ × G G Ψ ∼ = G Ψ ◦ Φ . Pr o of. W e define a map G × G → G by ( g , g ′ ) 7→ g ⊗ Φ( g ′ ). This ma p ex tends to a G - bimo dule map G Φ × G Ψ → G Ψ ◦ Φ and descends to a G -bitorso r map G Φ × G G Ψ → G Ψ ◦ Φ . This map is an equiv alence by Corollar y 2 .7.  6 EV AN JENKINS Definition 3.12. Let X b e a ( G , H )-bimo dule. The opp osite bimo dule X − 1 is the ( H , G )-bimo dule with the sa me ob jects as X but where the action factors through the isomor phis m ( − ) − 1 : H × G rev ∼ = → G × H rev . Prop ositi o n 3.13. If Φ ∈ Aut( G ) , then ( G Φ ) − 1 ∼ = G Φ − 1 . Pr o of. The map Φ : G → G extends to a n equiv a lence G Φ − 1 → ( G Φ ) − 1 of G -bitorsor s.  Prop ositi o n 3.14. L et G b e a 2-gr oup, and let X b e a G -bitorsor. Th en X × G X − 1 ∼ = G ∼ = X − 1 × G X Pr o of. This result fo llows immediately from Corollary 3 .10, Prop osition 3.11, and Prop ositio n 3.13.  Prop ositi o n 3.15. L et G b e a 2-gr oup, and let X b e a G -bitorsor. Then Ad : X → Aut ⊗ ( G ) , x 7→ Ad X ,x is a morphism of G -bimo dules, wher e G acts on Aut ⊗ ( G ) via left and right multiplic ation by inner auto e quivalenc es. Pr o of. W e must construct na tural iso morphisms Ad X ,g⊲x ( h ) ∼ = → g − 1 Ad X ,x ( h ) g and Ad X ,x⊳g ( h ) ∼ = → Ad X ,x ( g − 1 hg ). F or the former, we construct an equiv alence Γ G , X ,g ⊲x → Γ G , X ,x by ( h, h ′ , φ ) 7→ ( g − 1 hg , h ′ , g − 1 φ ). This equiv a lence fits in to the following commutativ e diagram, which g ives the desire d isomorphism of functors. Γ G , X ,g ⊲x ∼ = { { x x x x x x x x x ∼ =   ∼ =   4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 G h 7→ ghg − 1   Γ G , X ,x ∼ = { { w w w w w w w w w ∼ = # # G G G G G G G G G G G The la tter isomo rphism is constructed simila r ly . W e construct an equiv a lence Γ G , X ,x⊳g → G G , X ,x by ( h, h ′ , φ ) 7→ ( h, g h ′ g − 1 , φg − 1 ). This equiv alence fits into the following co mm utative diagram, whic h g ives the desired isomor phism of functors. Γ G , X ,x⊳g ∼ =                   ∼ =   ∼ = # # F F F F F F F F F Γ G , X ,x ∼ = { { w w w w w w w w w ∼ = # # G G G G G G G G G G h 7→ g − 1 hg   G G  4. Bitorsors and Braided Bitorsors for Braided 2-Groups F or general g roups G , the adjoint automo rphism asso ciated with a G -bitorsor X is not unique; changing the basepo int x ∈ X will modify the automor phism by an inner automorphism. If G is ab elian, how ever, there are no nontrivial inner auto- morphisms, and the adjoint automorphism is unique. A similar phenomenon holds EXTENSIONS OF GROUPS BY BRAIDED 2-GROUPS 7 in the 2-group setting; in particular, bra ided 2 -groups are eq uipp ed with a ca noni- cal trivialization of a ny inner automorphism, and so the a djo int autoequiv alence is unique. Prop ositi o n 4 .1. L et B b e a br aide d 2-gr oup, X a B - bitorsor, and x ∈ X . Then the auto e quivalenc e Ad X ,x is indep endent of the choic e of x up to u nique isomorphism. Pr o of. The braiding on B provides a trivialization of the rig ht action of B on Aut ⊗ ( B ) g iven in Prop o sition 3.15. Thus, the imag e of x ∈ X under Ad is in- depe ndent of x up to unique isomorphism.  W e will abuse notatio n a nd denote by Ad X the a uto equiv alence Ad X ,x . W e denote by B T B the 3-g roup of B -bitors ors, and by Aut ⊗ ( B ) the 2-gr oup of (not necessarily braided) auto eq uiv alences of B . Corollary 4.2. The assignment X 7→ Ad X defines a monoidal pseudofunctor Ad : B T B → Aut ⊗ ( B ) . Let A = π 0 ( B ). Then ther e is a monoidal pseudofunctor π 0 : B T B → A T A . Theorem 4.3. The maps Ad : B T B → Aut ⊗ ( B ) and π 0 : B T B → A T A identify Aut ⊗ ( B ) × Aut A A T A with the monoida l 1-trun c ation B T ≤ 1 B of B T B . Pr o of. It s uffices to no te that specifying an auto equiv alence of B Φ is determined, up to isomorphism, by a choice of x ∈ B Φ (the imag e of the unit o b ject e ), or equiv alently , a n automor phism of the underlying A -bitorso r, and an automorphism of Φ.  Definition 4.4 . A B -bitorso r X is braided if the auto eq uiv alence Ad X is braided. W e denote by B e T B the full sub-3-g r oup of the 3-g roup B T B consisting of br aided bitorsors , a nd by Aut e ⊗ ( B ) the 2- g roup of braided auto equiv a lences of B . Corollary 4.5. The maps Ad : B e T B → Aut e ⊗ ( B ) and π 0 : B e T B → A T A identify Aut e ⊗ ( B ) × Aut A A T A with the monoida l 1-trun c ation B e T ≤ 1 B of B e T B . Prop ositi o n 4.6. L et B b e a br aide d 2-gr oup with π 1 ( B ) = H . Then π 2 ( B e T B ) ∼ = H , and t he action of π 0 ( B e T B ) on π 2 factors thr ough the natur al map π 0 ( B e T B ) → Aut( H ) . Pr o of. The first part fo llows fro m Pro po sition 2.8 and the e x istence of the bra iding. The second part follows from Prop os ition 3.11.  5. Extensions and bitorsor s Having develop ed the theory of braided bitorsor s, we now turn to the problem of lifting group extensio ns to 2-group extensions . W e sta r t with a review of gro up extensions. In [SGA72, Exp os´ e VI I], Gro thendiec k describ es extensions of g roups in ter ms of G -bitorsor s . W e summarize his r esults as follows. Theorem 5.1 (Grothendieck) . L et G and K b e gr oups. Isomorphism classes of extensions of G by K ar e in bije ction with isomorphism classes of monoidal functors fr om the discr ete monoidal c ate gory G to the 2-gr oup K T K of K -bitorsors. If K = A is ab elian, the action of G on A in such an extension is give n by the c omp osite map G → A T A Ad → Aut( A ) . 8 EV AN JENKINS Grothendieck’s cor resp ondence is g iven by ass igning to an extension 1 → K → E ∂ → G → 1 the functor g 7→ E g = ∂ − 1 ( g ). Definition 5.2. Let G be a gro up, and let K be a 2-g roup. An extension of G by K is a 2 -group E equipp ed with a surjective monoida l functor ∂ : E → G a nd an ident ification of K with ∂ − 1 ( e ) as 2-gr oups. W e have the following stra ightforw ard analo gue of Gr o thendieck’s theorem in this setting. Theorem 5.3. L et G b e a gr oup, and let K b e a 2-gr oup. Equivalenc e classes of extensions of G by K ar e in bije ction with e quivalenc e classes of monoida l pseudo- functors fr om the discr ete monoidal c ate gory G to the 3-gr oup K T K . The a nalogue of an a belia n gro up in o ur setting is a braided 2-gr oup. The no tion of an extensio n of a group by a braided 2- group was defined in [DGNO10] as follows. Definition 5.4. Let G be a gro up, and le t B b e a braided 2-gro up. Supp ose we are given an action a : G → Aut e ⊗ ( B ) of G as braided auto equiv alences o f B . An extension of G by B with action a is an underlying 2-group extension E equipped with an isomorphism b etw ee n a ◦ ∂ : E → Aut ⊗ ( B ) and the a djoint action of E on B ∼ = ∂ − 1 ( e ) which r e stricts to the trivialization of the adjoin t action of B on itself given by the braiding . W e re c a ll the following equiv a lent c haracter iz ation of suc h extensions. Prop ositi o n 5.5 ([DGNO10, Pr op osition E.10]) . L et 1 → B → E → G → 1 b e an or dinary extension of 2-gr oups with B br aide d. This ext ension c an b e given the structur e of a br aide d extension if and only if e ach Ad x : B → B is br aide d for x ∈ E , and such a structur e is unique up to unique isomorphi sm. This prop ositio n implies that the fib ers of a bra ided extension are precisely braided B - bitorsor s, so we can c hara cterize braided extensions as follo ws. Corollary 5.6. Equivalenc e cla sses of br aide d extensions of G by B ar e in bije ct ion with e quivalenc e classes of monoida l pseudofunctors G → B e T B . This problem of classifying braided extensions can be na turally broken up into smaller pro blems. First, we can limit ours elves to cla ssifying ex tensions with a fixed a ction a : G → Aut e ⊗ ( B ). Secondly , since every extension o f 2- groups has an under lying group extension, we can study those braided extensions living ab ov e a fixed group extensio n. Of course, these t wo res tr ictions must b e compatible: bo th actions o f G o n B and extensions of G by A = π 0 ( B ) have an under lying automorphism of A , and these must agree. Thus, sp ecifying these tw o restrictions is the same as sp ecifying a monoidal functor from G to Aut e ⊗ ( B ) × Aut A A T A . By Corollar y 4 .5, this is the same as g iving a monoidal functor fro m G to B e T ≤ 1 B . Lifting the data of a gr oup e xtension and an action of G o n B to an extension of G by B is thus equiv alent to lifting a monoidal functor G → B e T ≤ 1 B to a monoidal pseudofunctor G → B e T B . W e will descr ib e a cohomo logical obstruction to such a lifting in the nex t section. EXTENSIONS OF GROUPS BY BRAIDED 2-GROUPS 9 6. Obstr uctions to lifting extensions Grothendieck’s homo topy hypothesis, first formulated in the manuscript [Gr o83], argues that n -gr oup oids should be “the same as” homotopy n -t yp es. One wa y to make this sta tement a bit mor e precise is to say that, for any r easona ble notion of n - group oid, there sho uld be a simplicial nerve functor that tak es n -g roup oids to Kan complexes whose ho mo topy groups v anish a bove lev el n (which we will henceforth refer to simply as “ n -t yp es”), and this nerve functor should b e quasi-inv erse to the fundamen tal n -gr oup oid functor. In o ur current setting, we c an make this precise as follows. W e may “delo op” all of our gro ups , 2-gro ups, a nd 3 -groups and functors b etw e en them to obtain one- ob ject gro up o ids, 2-gro upo ids, and 3-g roup oids. W e will aga in deno te b y C [1] the delo oping of a gr o up (or 2-gro up or 3-g r oup) C . It is shown in [Ber 99] that there is a simplicia l nerve functor for 3 -group oids . W e will henceforth abuse notation and cons ider the deloopings C [1] themselves a s n -g roup oids and Ka n complexes int erchangeably . W e note that π i ( C [1]) = π i − 1 ( C ). The nerve of a braided 2-gr oup B is a 2- fold loop space, so we ca n talk not only ab out the delo oping B [1] but also the double delo oping B [2]. Ab elian gro ups, being infinite lo o p spaces, ca n b e delo o pe d arbitra rily many times; we will deno te b y A [ n ] the n th deloo ping of an abe lian group A (i.e., the Eilenber g-Mac Lane space K ( A, n ).) As a warm-up, we w ill g ive a cohomolo gical int erpre tation of Grothendieck’s The- orem 5.1. Cla ssifying extensions of G by A with sp ecified action G [1] → Aut( A )[1] is equiv alent, by Grothendieck’s result, to c lassifying lifts as in the follo wing diagram. A T A [1]   G [1] : : t t t t t / / Aut( A )[1] W e note that A T A [1] is a very specia l space: it is a 2-type with π 1 = Aut A and π 2 = A , and the action of Aut A on A is the usual o ne . Thus, it is the space b K ( A, 2) that classifies arbitrary (i.e., not ne c essarily simple) fibrations with fib er K ( A, 1). In particular, there is a univ ersa l fibration K ( A, 1) → P → b K ( A, 2) (whic h was co nstructed in [Rob72] and g e neralizes the path-space fibra tion over K ( A, 2)) from whic h every fibration with fiber K ( A, 1) is a pullback. Similarly , the spaces b K ( A, n ) admit universal fibrations with fib er K ( A, n − 1). In analogy with our notation of A [ n ] fo r K ( A, n ), we will write b A [ n ] for b K ( A, n ). So roughly s p ea king, b A [2] is the classifying space for A -bitor s ors, and we ca n view an ex tens io n of G b y A as a family of A -bito r sors on G [1], which comes via pullback fro m b A [2]. Given an action G [1] → Aut( A )[1], homotopy cla sses of lifts to G [1] → b A [2] are pr ecisely e le ments o f H 2 ( G, A ) with the given action, so we r e c ov e r the cohomolo gical classification of group extensions . W e no w re tur n to the setting of braided 2 - groups. W e again let B be a braided 2-gro up with π 0 ( B ) = A and π 1 ( B ) = H . In analogy with the case o f groups, w e denote b y b B [2] the (delo oping of ) the 3-group of br aide d B -bitor sors. By Corollary 4.5, the following is a homotopy pullbac k square. 10 EV AN JENKINS b B [2] ≤ 2 / /   b A [2]   Aut e ⊗ ( B )[1] / / Aut( A )[1] The data of a n extension of G by A a nd a braided action of G on B with compatible underlying actions o f G on A is th us given by a homotop y c la ss of maps G [1] → b B [2] ≤ 2 . W e w is h to know when this lifts to a map G [1] → b B [2]. Since b B [2] is a 3-type, the truncation map b B [2] → b B [2] ≤ 2 is a fibration with fiber π 3 ( b B [2])[3] = H [3] by Pro po sition 4.6. This fibra tion is a pullback from the universal fibration ov er b H [4]. Thus, the obs truction to lifting an extension lies in H 4 ( G, H ), where the action of G o n H is the specified one b y Pr op osition 4.6. W e will call the map H 2 ( G, A ) → H 4 ( G, H ) that a ssigns to a class [ ω ] ∈ H 2 ( G, A ) the corresp onding o bstruction in H 4 ( G, H ) the P on try agi n square , and we will denote it by Pon tr B ,a ([ ω ]), where a : G → Aut e ⊗ ( B ) is the action of G on B . W e hav e thus prov ed the following classificatio n result. Theorem 6.1. L et G b e a gr oup, and let B b e a br aide d 2-gr oup. Supp ose we ar e given an action a : G → Aut e ⊗ ( B ) and an extension E of G by π 0 ( B ) = A such that the underlying actions of G on A c oming fr om a and E agr e e. This data lifts to an extension of G by B if and only if Pontr B ,a ([ ω ]) = 0 , and if this obstruction vanishes, such ex t ensions form a torsor for H 3 ( G, H ) . When the a ction of G on B is trivial (corresp onding to the case of c entr al exten- sions ), w e hav e the following dia g ram, wher e a ll squares ar e (homotopy) pullbacks. H [3]   H [3]   B [2]   / / b B [2]   A [2]   / / b B [2] ≤ 2   / / b A [2]   ∗ / / Aut e ⊗ ( B )[1] / / Aut( A )[1] F rom this diagra m it follows that the fibr ation of H [3] ov er A [2] defining B [2] pulls back from H [4] via the map we have defined. Thus, our definition o f Pontry agin square reduces to the clas s ical Pon tryagin squar e A [2] → H [4] cor resp onding to the quadratic map A → H coming fro m the braided 2-gr oup B (i.e., the Whitehead half-square map π 2 ( B [2]) → π 3 ( B [2])). 7. Cochain-level description of the P ontr y agin square In the pre vious se c tion, we defined the Pon tryagin squar e as a “parametrized cohomolog y oper ation” Aut e ⊗ ( B )[1] × Aut( A )[1] b A [2] → b H [4]. In order to give an EXTENSIONS OF GROUPS BY BRAIDED 2-GROUPS 11 explicit description of this op er a tion in ter ms o f gr oup co c ha ins, we need first to understand the structur e of Aut e ⊗ ( B ). W e recall (see [JS86] or [JS9 3, Section 3] for definitions and details) that the data of a braided 2 -group with π 0 = A and π 1 = H may be pr esented skeletally as an ab elian 3-co cycle ( A, H, h, c ); t wo such ab elia n 3 - co cycles descr ibe the same braided 2-gr oup if and only if their difference is an ab elian 3-co bo undary . Prop ositi o n 7.1 . L et B b e the br aide d 2-gr oup c orr esp onding to the ab elian 3- c o cycle ( A, H, h, c ) . Then the 2-gr oup Aut e ⊗ ( B ) c an b e describ e d as fol lows. • An obje ct c onsists of a triple ( φ, ψ , k ) , wher e φ ∈ Aut( A ) , ψ ∈ Aut( H ) , and k : A × A → H is a normalize d 2-c o chain such that dk = ψ ◦ h − h ◦ φ 3 , and (7.1.1) ψ ◦ c + k = k ◦ τ + c ◦ φ 2 , (7.1.2) wher e τ : A × A → A × A switches the two factors. • A morphism ( φ, ψ , k ) → ( φ ′ , ψ ′ , k ′ ) c onsists of a normalize d 1-c o chain η : A → H su ch that dη = k − k ′ . Comp osition of morphisms is given by addition of 1-c o chains. • The tensor pr o duct on obj e cts is given by ( φ, ψ , k ) ◦ ( φ ′ , ψ ′ , k ′ ) = ( φ ◦ φ ′ , ψ ◦ ψ ′ , k ◦ ( φ ′ ) 2 + ψ ◦ k ′ ) . This pr o duct is strictly asso ciative. Pr o of. See [JS86, Pro p o sition 14].  W e ca n now describ e actions o f G as braided a uto equiv alences of B that lift g iven actions of G as a utomorphisms of A a nd H . Corollary 7.2. Fix actions of G on A and H . An action of G as br aide d auto e- quivalenc es of B that induc es these two actio ns is given by the fol low ing data. • F or e ach g ∈ G , we have a n ormalize d 2-c o chain k g : A × A → H satisfying (7.1.1) and (7.1.2) . • F or e ach p air g 1 , g 2 ∈ G , we have a normalize d 1-c o chain θ g 1 ,g 2 : A → H such that dθ g 1 ,g 2 = k g 1 g 2 − k g 1 ◦ ( g 2 ⊲ − ) 2 − g 1 ⊲ k g 2 , and θ g 1 g 2 ,g 3 + θ g 1 ,g 2 ◦ ( g 3 ⊲ − ) = θ g 1 ,g 2 g 3 + g 1 ⊲ θ g 2 ,g 3 . An isomorphism ( k , θ ) ∼ = ⇒ ( k ′ , θ ′ ) of such a ctions is given by the d ata of, for e ach g ∈ G , a normalize d 1-c o chain η g : A → H su ch t hat dη g = k ′ g − k g , and such t hat for e ach p air g 1 , g 2 ∈ G , θ ′ g 1 ,g 2 − θ g 1 ,g 2 = η g 1 g 2 − η g 1 ◦ ( g 2 ⊲ − ) − g 1 ⊲ η g 2 . Since Aut e ⊗ ( B )[1] × Aut( A )[1] b A [2] is the 2 - truncation of b B [2], maps from G [1] to this 2-truncation c an be presented as quintu ples ( φ, ψ , k , θ, ω ), where φ and ψ are the actions of G on A and H , k a nd θ ar e the data described in Coro llary 7 .2, and ω : G × G → A is a 2- co cycle for G with co e fficie nts in A . W e wish to wr ite down an o bs truction co cyc le in Z 4 ( G, H ) representing the Pon tryagin squa re in ter ms o f the data ( A, H , h, c ) of the braided 2-gr oup B and ( φ, ψ , k , θ , ω ) of a map G [1] → b B [2] ≤ 2 . 12 EV AN JENKINS Prop ositi o n 7.3 . L et B b e a br aide d 2-gr oup with data ( A, H, h, c ) , a : G → Aut e ⊗ ( B ) an action with data ( φ, ψ, k, θ ) , and [ ω ] ∈ H 2 φ ( G, A ) . The Pontryagin squar e Pon tr B ,a ([ ω ]) is r epr esente d by the 4-c o cycle π ( g 1 , g 2 , g 3 , g 4 ) = c ( ω ( g 1 , g 2 ) , g 1 g 2 ⊲ ω ( g 3 , g 4 )) + h ( g 1 g 2 ⊲ ω ( g 3 , g 4 ) , ω ( g 1 , g 2 ) , ω ( g 1 g 2 , g 3 g 4 )) − h ( g 1 g 2 ⊲ ω ( g 3 , g 4 ) , g 1 ⊲ ω ( g 2 , g 3 g 4 ) , ω ( g 1 , g 2 g 3 g 4 )) + h ( g 1 ⊲ ω ( g 2 , g 3 ) , g 1 ⊲ ω ( g 2 g 3 , g 4 ) , ω ( g 1 , g 2 g 3 g 4 )) − h ( g 1 ⊲ ω ( g 2 , g 3 ) , ω ( g 1 , g 2 g 3 ) , ω ( g 1 g 2 g 3 , g 4 )) + h ( ω ( g 1 , g 2 ) , ω ( g 1 g 2 , g 3 ) , ω ( g 1 g 2 g 3 , g 4 )) − h ( ω ( g 1 , g 2 ) , g 1 g 2 ⊲ ω ( g 3 , g 4 ) , ω ( g 1 g 2 , g 3 g 4 )) + θ g 1 ,g 2 ( ω ( g 3 , g 4 )) − k g 1 ( g 2 ⊲ ω ( g 3 , g 4 ) , ω ( g 2 , g 3 g 4 )) + k g 1 ( ω ( g 2 , g 3 ) , ω ( g 2 g 3 , g 4 )) . Pr o of. The co cycle ω : G × G → A describ es an “ extension with section” : an extension 0 → A → E ∂ → G → 1 equipp ed with a section s : G → E . The m ultiplication on E is descr ibe d by ω : we hav e ( a 1 + s ( g 1 )) · ( a 2 + s ( g 2 )) = a 1 + g 1 ⊲ a 2 + ω ( g 1 , g 2 ) + s ( g 1 g 2 ) . The co cycle condition ensures that this multip lication is asso ciative. The co cy cle ω lifts uniquely to Ω : G × G → B . W e would like to under stand when this co cycle describ es an “extension with sec tion” of G by B , but this requires additional data: a co llection of iso morphisms Υ g 1 ,g 2 ,g 3 : Ω( g 1 , g 2 ) ⊗ Ω( g 1 g 2 , g 3 ) ⇒ g 1 ⊲ Ω( g 2 , g 3 )) ⊗ Ω( g 1 , g 2 g 3 ) , which corresp onds to the asso ciator . Such isomorphisms always exist b e c ause ω is a co cycle. In order to b e a true extension, this a sso ciator must satisfy the p entagon identit y , which tra nslates to the co mm utativity of the following diagram. Ω( g 1 , g 2 ) ⊗ ( g 1 g 2 ⊲ Ω( g 3 , g 4 )) ⊗ Ω( g 1 g 2 , g 3 g 4 ) c ⊗ id / / ( g 1 g 2 ⊲ Ω( g 3 , g 4 )) ⊗ Ω( g 1 , g 2 ) ⊗ Ω( g 1 g 2 , g 3 g 4 ) id ⊗ Υ   Ω( g 1 , g 2 ) ⊗ Ω( g 1 g 2 , g 3 ) ⊗ Ω( g 1 g 2 g 3 , g 4 ) id ⊗ Υ O O Υ ⊗ id   g 1 ⊲ ( g 2 ⊲ Ω( g 3 , g 4 ) ⊗ Ω( g 2 , g 3 g 4 )) ⊗ Ω( g 1 , g 2 g 3 g 4 ) ( g 1 ⊲ Ω( g 2 , g 3 )) ⊗ Ω( g 1 , g 2 g 3 ) ⊗ Ω( g 1 g 2 g 3 , g 4 ) id ⊗ Υ / / g 1 ⊲ (Ω( g 2 , g 3 ) ⊗ Ω( g 2 g 3 , g 4 )) ⊗ Ω( g 1 , g 2 g 3 g 4 ) g 1 ⊲ Υ ⊗ id O O T ransla ting this diagr a m into the skeletal s etting y ields the desired for mula for π (plus a cob ounda r y corresp onding to Υ).  Remark 7.4. It is not a priori clear, without our earlier homotopy theor etic discussion, that the co chain π is even a co cycle, m uch less that is w ell-defined (up to changing it by a cob ounda r y) indep endently of the choice of rigidifying skeletal da ta. Indeed, direct computatio ns of these facts at the level of co chains are extremely lengthy . 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