The inner automorphism 3-group of a strict 2-group

The inner automorphism 3-group of a strict 2-group Da vid Mic hael Rob erts and Urs Sc hreiber Octob er 25, 2018 Abstract Any group G giv es rise to a 2-group of in n er automorphisms, INN( G ). It is an old result b y Segal that the nerve of this is the u niv ersal G - bundle. W e discuss that, si milarly , for ev ery 2-group G (2) there is a 3-group INN( G (2) ) and a slightly smaller 3-group INN 0 ( G (2) ) of inner automorphisms. W e describ e these for G (2) any strict 2-group, discuss how I NN 0 ( G (2) ) can b e understoo d as arising from the mapping cone of the identit y on G (2) and show that its und erlying 2-gro up oid structure fits into a short exact sequence G (2) / / INN 0 ( G (2) ) / / B G (2) . As a co nsequence, INN 0 ( G (2) ) encodes the properties of the universal G (2) 2-bundle. Con ten ts 1 Intr o duction 2 2 n -Groups i n term s of groups 8 2.1 Conv en tions for strict 2-gr oups and cross ed mo dules . . . . . . . 8 2.2 3-Groups and 2-cros sed modules . . . . . . . . . . . . . . . . . . 11 2.3 Mapping cones of crossed modules . . . . . . . . . . . . . . . . . 12 3 Main resul ts 13 3.1 The exact sequence G (2) → INN 0 ( G (2) ) → B G (2) . . . . . . . . . 13 3.2 INN 0 ( G (2) ) from a ma pping cone . . . . . . . . . . . . . . . . . . 14 4 Inner automorphism ( n + 1) -groups 14 4.1 Exact sequences of strict 2-group oids . . . . . . . . . . . . . . . . 15 4.2 T angent 2-catego ries . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Inner automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Inner automorphism 2-groups . . . . . . . . . . . . . . . . . . . . 22 4.5 Inner automorphism 3-groups . . . . . . . . . . . . . . . . . . . . . 22 1 5 The 3-group INN 0 ( G (2) ) 26 5.1 Ob jects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2.1 Comp osition . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2.2 Pro duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 2-Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.1 Comp osition . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.2 Pro duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 Prop erties of INN 0 ( G (2) ) 34 6.1 Structure morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.1.1 Strictness as a 2 -group oid . . . . . . . . . . . . . . . . . . 34 6.2 T rivializability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.3 Univ ersality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.4 The corresp onding 2-crossed module . . . . . . . . . . . . . . . . 3 6 6.4.1 Relation to the mapping cone o f H → G . . . . . . . . . . 43 7 Universal n -bundles 44 7.1 Univ ersal 1-bundles in terms of INN ( G ) . . . . . . . . . . . . . . 44 7.2 Univ ersal 2-bundles in terms of INN 0 ( G (2) ) . . . . . . . . . . . . 46 7.3 Relation to simplicial bundles . . . . . . . . . . . . . . . . . . . . 48 7.3.1 T angent categor ie s and d ´ ecalage . . . . . . . . . . . . . . 48 7.3.2 Univ ersal simplicial bundles . . . . . . . . . . . . . . . . . 50 A C ros sed squar es 52 1 In tro duction The theor y o f gr oups and their principal fib er bundles gener alizes to that of categoric al groups and their categorical principal fiber bundles . In fact, using higher ca tegories, one has for each integer n the notion of n -gr oups and their principal n -bundles. The reader may hav e encountered principal 2-bundles mostly in the la nguage of (nona b elian) gerb es, whic h are to 2-bundles essentially like s hea ves are to ordinary bundles. The c o ncept of a 2 -bundle pro per is describ ed in [6 , 5]. These n -bundles are certainly interesting a lr eady in their own right. One crucial motiv ation for consider ing them comes fro m the study of n -dimensional quantum field theory . In this case one is int erested in n -dimensional analogs of the concept of parallel tr anspo r t in fib er bundles with connection [3 , 29, 30]. In that context a curious phenomenon o ccurs: whenev er one inv estigates n -dimensional quantum field theory governed by an n -gro up G ( n ) , it turns out [28] that the situa tion is g o v erned by an ( n + 1)-g roup a sso ciated to G ( n ) . In fact, it is appropr iate to call this ( n + 1)-group INN 0 ( G ( n ) ), b ecause, as the notation sug gests, it is r e la ted to inner automo rphisms of the or iginal n -group G ( n ) . 2 One of o ur aims here is to define w ha t inner automorphisms o f a 2-gr oup ar e and to give a concise definition a s w ell as a detailed description of the 3-g roup INN 0 ( G (2) ) for any stric t 2- group G (2) . W e then prove that INN 0 ( G (2) ) has a couple of r ather peculiar prop erties; it is c ontr actible (equiv alen t, as a 2-gro up, to the trivial 2-group), a nd fits into a s hort exact se quence G (2)   / / INN 0 ( G (2) ) / / / / B G (2) of 2-g roup o ids. T o appreciate this result, it is helpful to first consider the analogo us statement for ordinar y gro ups. The statement for ordi nary groups. F o r any o rdinary gr oup G , v ar ious constructions of in terest, like that of the universal G -bundle, are closely r e lated to a ce rtain g roup o id determined by G . There are several differen t w ays to think of this gro upoid. The simplest way to desc r ibe its structure is to say that it is the co discrete group oid over the elements o f G , na mely the gr oupoid whose ob jects are the elements of G and which has precisely one morphism from an y element to an y o ther . The relev a nce o f this gr oupoid is better under stoo d b y thinking of it as the action group oid G//G of the a ction o f G on itself by left m ultiplication. As such, we may w r ite an y of its morphisms as g h / / hg for g , h ∈ G and h g b eing the pro duct o f h and g in G . While this way of thinking ab out our gr o upoid already makes it more plau- sible that it is related to G -a ctions and hence pos sibly to G -bundles, one more prop erty remains to b e made manifest: there is a lso a monoida l structure on our group oid. F o r an y tw o morphisms, g 1 h 1 / / h 1 g 1 and g 2 h 2 / / h 2 g 2 , we can form the product mor phism g 2 g 1 h 2 Ad g 2 ( h 1 ) / / h 2 g 2 h 1 g 1 , and this a ssignmen t is functorial in b oth arguments. Moreover, to every mor- phism g h / / hg there is a morphism g − 1 Ad g − 1 ( h ) − 1 / / ( hg ) − 1 , 3 which is its inv erse with resp ect to this pr o duct op eration. This makes G// G a s trict 2-gro up [2]. A helpful wa y to make the 2- group structure o n G//G more manifest is to relate it to inner automorphisms of G . T o se e this, consider ano ther gr oupoid ca no nically asso ciated to any gro up G , namely the group oid B G = { • g / / • | g ∈ G } which has a single ob ject • , o ne mor phism for each element of G and whe r e comp osition of mor phisms is just the pro duct in the group. Automorphisms a : B G → B G (i.e. inv ertible functors) of this group oid are nothing but g roup automorphisms of G . But now there are also isomorphisms betw een tw o such morphisms a and a ′ , namely natural transformations: B G a a ′ > > B G   . This way for e very o r dinary g roup G we hav e not just its ordinary gr oup o f automorphisms, but actually a 2 -group A UT( G ) := Aut Cat ( B G ) . This is a gro upoid, whose ob jects are group automorphisms of G . The 2-gr oup structure on this gro upoid is manifest from the hor izon tal c o mpositio n of the natural transfor mations ab ov e. Hence the or dinary automo r phism gr oup of G is the group of ob jects of AU T( G ). By writing out the definition of a natural transformation, one sees that there is a mor phism b et w een tw o ob jects in AUT( G ) whenever the tw o underlying ordinary a uto morphisms o f G differ by conjugatio n with an element of G . It follows in pa rticular that the inner a utomorphisms of G corr espond to tho s e autofunctors of B G whic h ar e iso morphic to the identit y: B G Id Ad g > > B G ≃ g   . Therefore c o nsider the group oid INN( G ): its o b jects are pairs , consisting of an automor phisms to gether with a trans formation co nnecting it to the iden tit y . 4 A morphism from ( g, Ad g ) to ( gh , Ad gh ) is a co mmuting triangle Ad g h   Id B G g 0 0 hg . . Ad hg . This is again ex actly the g roupo id G//G which we are discuss ing INN( G ) = G//G . In this formulation the natural no tion of comp osition of group automo rphisms nicely explains the monoidal structure on G//G . Notice tha t INN( G ) remembers the cent er of the group. W e will discuss that it sits inside an exa ct seq uence 1 → Z ( G ) → INN( G ) → AUT( G ) → OUT( G ) → 1 of 2-groups, and tha t this is what genera lizes to higher n . If we think of the g roup G just as a dis c rete catego ry , whose o b jects are the elements of G and which ha s only identit y mor phisms, then there is an obvious monomorphic functor G → INN( G ) . Moreov er, there is an obvious epimorphic functor INN( G ) → B G from our group oid to the group G , but no w with the latter rega r ded as a catego ry with a single o b ject. This simply forg ets the so urce and tar g et lab els and r ecalls only the group ele ment whic h is acting. These tw o functors are such that the image of the fo r mer is precise ly the collection of morphisms which get sent to the iden tit y morphism by the latter. Therefore w e sa y that we ha v e a short exact sequence G → INN( G ) → B G (1) of group oids. Notice that G and INN( G ) are gro upoids which are also 2-gr oups (the first one, b eing an ordinar y group, is a dege nerate case o f a 2-gr oup), and that the morphism G → INN( G ) is also a mor phism of 2-gr oups. But B G is in g eneral just a g roupo id without monoidal structure – it has the structur e of a 2- g roup if and only if G is abelian. Even though all this is rather e le mentary , the exa ct sequence (1) is impo r- tant. W e ca n a pply the functor | · | to our s equence, which takes the nerve o f 5 a category a nd then forms the geometric realization. Note that when K is a 2- group, | K | is a top ological group. Under this functor, (1) becomes the univ ersal G -bundle G → E G → B G, even when G is a top ological or Lie group. The fact that B G ≃ | B G | is the very definition of the classifying space B G o f a gro up G in [31 ]. That E G ≃ | INN( G ) | is co n tractible follows from the existence of a n equiv alence of group oids INN( G ) ∼ → ∗ . Finally , the inclusion G → INN( G ) together with the monoidal structure on INN( G ) gives the free G -action of G on E G whose q uo - tien t is exa ctly B G . The obser v ation that | INN( G ) | is a mo del for E G is or ig - inally due to Seg al [31], who proved the remaining no n trivial statement: that | INN( G ) | → B G is lo cally trivial when G is a well-pointed gr oup. 1 Our first main r esult is the higher categ orical analog ue of (1 ), obtained by starting with a strict 2 -group G (2) in place of the ordinary gr oup G . Here w e do not co nsider geometr ic realiza tio ns of our ca teg ories and 2- categorie s (for more on that see the clo s ely related article [4] as well as [27]) but instead fo cus on the existence of thes e sequences of 1- and 2-g roupo ids. W e comment o n further asp ects of the topic of universal n -bundles in § 7. Mor e details will be given in [2 6]. The form ulation in terms of cross ed mo dules. F or many purp oses, like doing explicit computations a nd for apply ing the r ic h too lbox of simplicia l meth- o ds, it is p ossible (and useful!) to express n -groups in ter ms of n -term co mplexes of ordina ry gro ups with extra s tructure on them. F o r instanc e strict 2 -groups are w ell kno wn to be equiv alen t to crossed mo dules of tw o o rdinary groups: one describ es the group of ob jects, the other the group of mor phisms of the 2-gro up. This pattern co n tin ues, but ther e is a bifurcation o f c onstructions, a ll of which are (homotopy) equiv alen t. Sufficien tly strict 3-g r oups – “Gray gro ups ” – are des cribed by 2-cro s sed mo dules, which inv olv e three ordinary groups form- ing a normal complex, and also b y cross e d sq ua res, which lo ok lik e cross e d mo d- ules of c rossed mo dules of gr oups. W e will pr imarily use the for mer, and only men tion c rossed squares when we ca nnot av oid it. 2 The way w e use these tw o mo dels can b e illustrated in o ne lower ca teg orical dimension by comparing the map G id / / G to the cr osse d mo dule G id / / G Ad / / Aut( G ) 1 That i s, the inclusion of the ident it y elemen t is a closed cofibration. 2 As one go es to higher categorical dimensions (whic h we do not do here), there are multiple directions i n which to extend the relev an t diagrams, so there are a n umber of different mo dels for n -groups. There is a sort of nonab elian Dold-Kan theorem, due to Carrasco and Cegarra [11], which can be used to ch aracterise n -groups by n -term complexes of (poss i bly nonabelian) groups with the structure they call a hypercrossed complex. 6 using that map. The cross ed mo dule ca n b e thought o f as the mapping cone (=homotopy quotient) of the identit y map. The translation be tw een n -g roups and their corres ponding n - term complexes of o rdinary groups sheds light on b oth o f these p oints of vie w. The analog ue of our statement ab out the 3- group INN 0 ( G (2) ) is our second ma in result: the complex of g roups des cribing INN 0 ( G (2) ) is the mapping cone of the iden tit y on the complex of g roups describing G (2) itself. This fac t was anticipated from considerations in the theor y of Lie n - a lgebras [28], where the Lie ( n +1)-algebr a c orresp onding to a Lie ( n +1)-gro up I NN 0 ( G ( n ) ) has proven to be crucial for under standing connections with v alues in L ie n - algebras . Ther e o ne finds that inner der iv ation Lie ( n + 1)-algebra s gov ern Lie ( n + 1 )-algebras of Chern-Simons t ype. The fact that INN 0 ( G (2) ) arises from a mapping cone of the ide ntit y is crucial in this context. The plan of this article. The main c o n ten t of this w ork is as follo ws – first a concise and natural definition of inner automorphisms of 2-gr oups, relating them to the full automor phism ( n + 1 )-group and to the categorical c en ter. Then w e apply this definition to the cas e that G (2) is a strict 2-g roup a nd work o ut in full detail what INN 0 ( G (2) ) lo oks like, i.e. how the v ar io us compo sition oper ations work, thus extracting its desc r iption in terms of co mplexes of ordinary groups. W e state and prov e the main prop erties of INN 0 ( G (2) ). The plan of our discussion is as follows. • In pa rt 2 we recall the relation b et w een 2- and 3-groups and crossed mod- ules of ordinar y groups. This serves to set up our conven tion fo r the precise c hoice of iden tification of 2-gr o up morphisms with ordinary g roup elements. • In par t 3 we s tate our tw o main res ults. • In par t 4 we define inner automorphism n -groups and pro ve some impor- tant gener al prop erties of them. • In par t 5 we apply our definition of inner automorphisms to an arbitrary strict 2-gro up G (2) , to form the 3-group INN 0 ( G (2) ). W e then w ork out in detail the description of INN 0 ( G (2) ) in terms of ordinary groups, spelling out the nature of the v arious compo sition a nd pro duct op erations. • In pa rt 6 we state and prov e the main prop erties of INN 0 ( G (2) ), including our t w o main r esults. • In part 7 we clo se by indicating in more deta il how inner automor phism ( n + 1)-g roups play the ro le of universal n -bundles. W e also rela te our construction here to analogo us simplicial constructions. W e are gra teful to J im Stasheff for helpful discussions and fo r emphasizing the imp o rtance of the ma pping cone construc tio n in the present context. W e 7 profited fro m gene r al discus sion with Danny Stevenson and thank him for his help o n the references to Sega l’s work. W e also thank Christoph Sc hw e ig ert and Zoran ˇ Skoda for helpful comments on the manuscript, Tim Porter for r e minding us of Norr ie’s w ork. W e a re grateful to T o dd T rimble for discussio n ab out the simplicial asp ects of our constr uction a nd its relation to d ´ ecalage, a nd to an anonymous refere e for making the remark whic h appears in 7.3.2. DMR is suppor ted by an Australian P ostgraduate Award. 2 n -Groups in terms of groups Sufficien tly strict n - g roups a r e equiv alent to certain structures – cro ssed modules and generalizatio ns theoreof – inv olving just collec tions o f ordinary groups with certain structure on them. 2.1 Con v entions for strict 2-groups and crossed mo dules An ordinary gr oup G may b e reg arded a s a one ob ject ca tegory . If we regard G as such a ca teg ory , we write B G in order to emphasize that we are thinking of the monoidal 0-category G a s a one ob ject 1-catego r y . This w ay we obtain a notion of n -groups fro m any notion of n - c ategories: an n -group G ( n ) is a monoidal ( n − 1 )-category such that when rega rded as a one- ob ject ( n )- category B G ( n ) it beco mes a one-o b ject n - group oid. An n -gr o upoid is an n -category all who se k -mor phisms are equiv alences, for a ll 1 ≤ k ≤ n . Here we sha ll b e co ncerned with strict 2-gr oups and with 3-groups which are Gray-categor ie s. A strict 2- group G (2) is one such that B G (2) is a strict one-ob ject 2-gr oupoid. A Gray-group oid is a 3-g roupo id which is strict ex c ept for the exc hange la w o f 2-mor phisms. The s ta ndard r eference for 2-g r oups is [2]. A discussio n of Gr ay-gr oupoids useful for our con text is in [22]. W e co me to Gray-groups in 2.2. It is well known that s trict 2-g r oups are equiv alen t to cro ssed mo dules of ordinary groups . This w as first es ta blished in [1 0]. The rela tio n to categ ory ob jects in groups was also discussed in [23]. The notion of a crossed module is originally due to [32]. Definition 1. A crossed m o dule o f groups is a diagr am H t / / G α / / Aut( H ) in Gr p such that H t   ? ? ? ? ? ? ? ? Ad / / Aut( H ) G α ; ; w w w w w w w w w 8 and G × H Id × t / / α   G × G Ad   H t / / G . Definition 2. A strict 2 -group G (2) is any of the fol lowing e quivalent entities • a gr oup obje ct in Ca t • a c ate gory obje ct in Grp • a strict 2-gr oup oid with a single obje ct A detailed discussion can be found in [2]. One identifies • G is the gro up of ob jects of G (2) . • H is the gro up of morphism of G (2) starting at the iden tit y ob ject. • t : H → G is the target homomorphism so that h : Id → t ( h ) for all h ∈ H . • α : G → Aut( H ) is conjugation with iden tit y morphisms: Ad Id g ( Id h / / t ( h ) ) = Id α ( h )( h ) / / t ( α ( g )( h )) for all g ∈ G , h ∈ H . W e often abbrev iate g h := α ( g )( h ) . Beyond that there are 2 × 2 choices to be ma de when identifying a strict 2-gro up G (2) with a crossed module o f groups . The first choice to be made is in which order to multiply elemen ts in G . F or • g 1 / / • and • g 2 / / • t wo morphisms in B G (2) , w e can either set • g 1 / / • g 2 / / • := • g 1 g 2 / / • (F) or • g 1 / / • g 2 / / • := • g 2 g 1 / / • (B) . The other choice to b e made is how to describ e arbitrar y morphisms by an element in the s emidirect pro duct gro up G ⋉ H : every mor phism of G (2) may 9 be written as the pro duct of one starting at the identit y ob ject with a n iden tit y morphism on some o b ject. The choice of ordering here yields either • g   A A • h   := • Id   A A • g / / • h   (R) or • g   A A • h   := • g / / • Id   A A • h   (L) Here we choos e the conv en tion LB . This implies • g   g ′ = t ( h ) g A A • h   for all g ∈ G , h ∈ H , a s well as the following tw o equations for hor izon tal and vertical comp osition in B G (2) , ex pr essed in terms of op erations in the cro s sed mo dule • g 1   A A • g 2   A A • h 1   h 2   = • g 2 g 1   A A • h 2 g 2 h 1   and • g 1   g 2 / / g 3 E E • h 1   h 2   = • g 1   A A • h 2 h 1   . 10 2.2 3-Groups and 2-crossed mo dules As we are considering s tr ict mo dels in this pa per, we will as s ume that all 3 - groups are as str ict as possible. This mea ns they will b e one- o b ject Gray- categorie s, o r Gr ay-monoid s [15]. A Gr ay-monoid is a (strict) 2-categor y M such that the pro duct 2 -functor M ⊗ M → M uses the Gray tensor pro duct [17, 16], not the us ual Ca rtesian pr oduct o f 2- categorie s. Thus non-identit y co herence mor phisms only app ear when we use the mono idal structure on M . So from now on a “3-g r oup” G (3) will mean a 3-gro up suc h tha t regarded as a one-ob ject 3 -group oid B G (3) it is a one- ob ject Gray-group oid. Just as a 2-g roup gives r ise to a c r ossed mo dule, a 3-gr oup gives rise to a 2-cros sed module. Roug hly , this is a complex o f groups L → M → N , and a function M × M → L (2) such that L → M is a cro ssed mo dule, and (2) measures the failur e of M → N to be a cr ossed mo dule. An example is when L = 1, and then w e ha v e a crossed mo dule. The relation b et w een 3-g roups and 2- crossed mo dules was describ ed in [22]. The precise definition of a 2 - crossed mo dule is as follows, see also [14]. Definition 3. A 2-crossed mo dule is a normal c omplex of length 2 L ∂ 2 / / M ∂ 1 / / N of N -gr oups ( N acting on itself by c onjuga tion) and an N -e quivaria nt funct ion {· , ·} : M × M → L , c al le d a Peiffer lifting, satisfying these c onditions: 1. ∂ 2 { m, m ′ } = ( mm ′ m − 1 )( ∂ 1 m m ′ ) − 1 , 2. { ∂ 2 l , ∂ 2 l ′ } = [ l , l ′ ] := l l ′ l − 1 l ′− 1 3. (a) { m, m ′ m ′′ } = { m, m ′ } mm ′ m − 1 { m, m ′′ } (b) { mm ′ , m ′′ } = { m, m ′ m ′′ m ′− 1 } ∂ 1 m { m ′ , m ′′ } , 4. { m, ∂ 2 l } = ( m l )( ∂ 1 m l ) − 1 5. n { m, m ′ } = { n m, n m ′ } , 11 wher e l , l ′ ∈ L, m, m ′ , m ′′ ∈ M and n ∈ N . Here m l denotes the action M × L → L ( m, l ) 7→ m l := l { ∂ 2 l − 1 , m } . (3) A normal complex is one in which im ∂ is normal in k er ∂ fo r all differentials. It follows from these conditions that ∂ 2 : L → M is a crossed mo dule with the action (3). T o get from a 3-g roup G (3) to a 2 -crossed mo dule [22], we emulate the construction of a c rossed mo dule from a 2-gr oup: o ne identitfies • N is the group of ob jects of G (3) . • M is the group of 1-morphisms of G (3) starting at the iden tit y ob ject. • L is the g roup of 2-morphisms star ting at the identit y 1-arr o w of the ident it y ob ject • ∂ 1 : M → N is the target homomo rphism such that m : Id → ∂ 1 ( m ) for all m ∈ M . • ∂ 2 : L → M is the target homomorphism such that l : Id Id → ∂ 2 ( l ) for all l ∈ L . • The v arious a ctions arise by w his k ering, a nalogously to the case o f a 2 - group. W e w ill no t go into the pro of that this g iv es rise to a 2-cr o ssed module for all 3-gro ups, but only in the case we are consider ing. One reas o n to consider 2-cros sed mo dules is that the homotopy gr o ups o f G (3) can b e calculated as the homology of the s equence under lying the 2 -crossed module. 2.3 Mapping cones of crossed mo dules Another notion r elated to 3-gr oups [1] is cr ossed mo dules in ternal to cr ossed mo dules (more technically known as cro ssed squares, [18],[23]). More genera lly , consider a ma p φ o f crossed mo dules: Definition 4 (nonab elian mapping cone [23]) . F or H 2 φ H / / t 2   H 1 t 1   G 2 φ G / / G 1 12 a 2-term c omple x of cr osse d mo dules ( t i : H i → G i ), we say its mapping c one is the c omplex of gr oups H 2 ∂ 2 / / G 2 ⋉ H 1 ∂ 1 / / G 1 , (4) wher e ∂ 1 : ( g 2 , h 1 ) 7→ t 1 ( h 1 ) φ G ( g 2 ) and ∂ 2 : h 2 7→ ( t 2 ( h 2 ) , φ H ( h 2 ) − 1 ) . Here G 2 acts on H 1 by wa y o f the morphism φ G : G 2 → G 1 . When no structure is impos ed on φ , (4 ) is merely a complex. How ev er, if φ is a cro ssed sq ua re, the mapping cone is a 2 -crossed mo dule (or iginally shown in [1 3], but see [14]). W e will not need to define crosse d squar e s here but just note they come e q uipped with a map h : G 2 × H 1 → H 2 satisfying conditions simila r to the Peiffer lifting. The deta ils can b e found in [14], which we r ecall in our a ppendix. The only cross ed s quare we will see in this pap e r is the identit y map on a crossed mo dule H id / / t   H t   G id / / G with the structure map h : G × H → H (5) ( g , h ) 7→ h g h − 1 . (6) This co ncept of “crossed mo dules of cros s ed mo dules” is explored in Norrie’s thesis [25 ] on ‘actors ’ of cross ed mo dules, with a fo c us o n ca tegorifying gro up theory , rather than geometry . Automorphis ms of crosse d mo dules of g r oups and group oids ha v e b een discussed in [9] and [8]. 3 Main results 3.1 The exact sequence G (2) → INN 0 ( G (2) ) → B G (2) W e describ e the 3-group INN 0 ( G (2) ) for G (2) any strict 2 -group, and s how that it plays the role of the univ ersal principal G (2) -bundle in that 13 • INN 0 ( G (2) ) is equiv alent to the trivial 3-gr oup (hence “contractible”). • INN 0 ( G (2) ) fits in to the sho rt ex act se quence G (2)   / / INN 0 ( G (2) ) / / / / B G (2) of strict 2-group oids. 3.2 INN 0 ( G (2) ) from a mapping cone W e show that the 3-gr oup INN 0 ( G (2) ) comes from a 2-crossed mo dule H / / G ⋉ H / / G which is the mapping cone of H Id / / t   H t   G Id / / G , the iden tit y ma p of the cro ssed mo dule ( t : H → G ) which determines G (2) . Notice that this har monizes with the a nalogous r esult for Lie 2 - algebras discussed in [28]. 4 Inner automorphism ( n + 1) -groups An automo r phism of an n -gr oup G ( n ) is simply an a utomorphism of the n - category B G ( n ) . W e wan t to say that suc h an automor phism q is inn er if it is equiv ale nt to the iden tit y automorphism B G ( n ) Id # # q ; ; B G ( n ) . ∼   Notice that w e rea lly do mean automorphisms her e, and not auto- equiv alences: we require an autmorphism to be an endo- n -functor with a strict inverse. Automorphisms (of cr ossed mo dules) connected to the identit y a ppear in definition 2.3 of [8 ] under the name “free deriv a tions”. Since the naturality diagram for the transformatio n connecting an a utomorphism to the identit y implies that this automo r phism ar is es from conjugations, we think of them as 14 inner automorphisms here and re s erve the term “ (inner) der iv ations” for the image of these automorphisms as o ne pass e s from Lie n -groups to Lie n -algebr as [28]. A us eful wa y to think o f the n -group oid of inner automor phisms is in terms of what we call “tange n t categor ies”, a slig h t v ariation of the c o ncept o f comma categorie s. T ang en t categories in general happen to live in in teresting exact sequences. In order to be a ble to talk about these, w e fir st quickly se t up a our definitions for exact sequences of str ict 2-group oids. Remem ber that we work entirely within the Gray-categor y whose ob jects are strict 2- g roup o ids, whose mo rphisms are stric t 2- functors, whose 2-morphisms are pseudonatural transformations and whose 3-mor phis ms are mo difications of these. 4.1 Exact sequences of strict 2 -group oids Inner automorphism n -gr oups turn o ut to live in interesting exact se quenc es of ( n + 1 ) -gr oups . Therefor e we wan t to talk a bout genera lizations of exac t sequences of gr oups to the world of n -group oids. Since for o ur purp oses her e only strict 2 -group oids matter , we shall be conten t with just using a de finitio n applicable to that case. Definition 5 (exact s equence of strict 2-g roupo ids) . A c olle ction of c omp osable morphisms C 0 f 1 / / C 1 f 2 / / · · · f n / / C n of strict 2-gr oup oids C i is c al le d an exact se quenc e if, as or dinary maps b etwe en sp ac es of 2-morphisms, • f 1 is inje ctive • f n is su r je ctive • the image of f i is the pr eimag e under f i +1 of the c ol le ction of al l identity 2-morphisms on identity 1-morphisms in Mor 2 ( C i +1 ) , for al l 1 ≤ i < n . In order to mak e this harmonize with our distinction b et w een n -groups G ( n ) and the corresp onding 1 -ob ject n -group oids B G ( n ) we add to that Definition 6 (exact sequences o f stric t 2- groups) . A c ol le ction of c omp osable morphisms G 0 f 1 / / G 1 f 2 / / · · · f n / / G n of strict 2-gr oups is c al le d an ex act se quenc e if the c orr esp onding chain B G 0 B f 1 / / B G 1 B f 2 / / · · · B f n / / B G n is an exact se qu enc e of strict 2-gr oup oids. 15 Remark. Ordinary exa ct sequences of groups thus precise ly co rresp ond to exact sequences of strict 2-groups a ll whos e morphisms are identities. 4.2 T angen t 2 -categories W e present a simple but useful wa y describ e 2- categories of mor phisms with coinciding so ur ce. W e find it helpful to refer to this construction as tangent c ate gories for r easons to b ecome clea r. It is not hard to see that this is the globular analog of the simplicial construction k no wn as d´ ec alage , as will be discussed more in 7.3. In the context of higher categor ies it has b een considered (ov er single ob jects) in section 3 .2 o f [22]. Definition 7. Denote by pt := {•} the st rict 2-c ate gory with a single obje ct and no n ontrivial morphisms and by I := { • ∼ / / ◦ } , the st rict 2-c ate gory c onsisting of t wo obje cts c onne cte d by a 1-isomorphism. Of course I is equiv a len t to pt – but not isomorphic. W e fix one injection i : pt   / / I i : • 7→ • once and for a ll. It is useful to think o f mo rphisms f : I → C from I to some co domain C as lab eled by the cor respo nding image of the ordi- nary point pt  _   f / / C =   I f / / C . Definition 8 (tang en t 2-bundle) . Given any strict 2 -c ate gory C , we define its tangent 2-bun d le T C ⊂ Hom 2Cat ( I , C ) 16 to b e that sub 2 -c ate gory o f morph isms fr om I into C which c ol lapses to a 0-c ate gory when pul le d b ack along the fixe d inclusion i : pt   / / I : the mor- phisms h in T C ar e al l those for which pt  _   I f   f ′ @ @ C h   = pt  _   I f / / C . The tangent 2-bund le is a disjoint u nion T C = M x ∈ Ob j( C ) T x C of tangent 2 -c ate gories at e ach obje ct x of C . In this way it is a 2 -bu nd le p : T C / / Ob j( C ) over the sp ac e of obje ct s of C . As befits a tangent bundle, the tangent 2-bundle has a canonical section e Id : O b j( C ) → T C which sends every ob ject of C to the Ident it y morphism on it. Remark. The g roupo id I plays the role of the interv al in top ology a nd under- lies the homotopy theor y for g roupo ids, as desc ribed in [2 1]. The terminology “tangent category ” finds further justification when the discussion her e is do ne for smo oth n - groups which are then sent to the co r resp onding Lie n -a lgebras: indeed, as indicated in figure 3 of [28], one finds a close re la tion b etw een maps from the interv al I , inner automor phisms, the notion of univ ersal n -bundles and tangency relations. Example (slice categories). F o r C any 1 - group oid, i.e . a strict 2-gr oupo id with only iden tit y 2-morphisms, its tangen t 1 - category is the comma categor y T C = ((Ob j( C ) ֒ → C ) ↓ Id C ) . This is the disjoin t union of all co-over categories on all o b jects of C T C = M a ∈ Ob j( C ) ( a ↓ C ) 17 Ob jects of T C are mor phis ms f : a → b in C , and mor phis ms f h / / f ′ in T C ar e commuting triangles b h   a f / / f ′ / / b ′ in C . Example (strict tangen t 2-group oids). The example which we ar e mainly int erested in is that where C is a strict 2-gro upoid. F or a any ob ject in C , an ob ject o f T a C is a morphism a q / / b . A 1-morphism in T a C is a filled triangle b f   a q / / q ′ / / b ′ F   in C . Finally , a 2-mor phism in T a C lo oks lik e b f ′   f   a q / / q ′ / / b ′ F   F ′   L + 3 . The comp osition o f these 2-morphisms is the obvious one. W e g iv e a detailed description for the case the C = B G (2) in 5. Prop osition 1. F or any strict 2 - c ate gory C , its tangent 2 - bund le T C fits into an ex act se quenc e Mor( C )   / / T C / / / / C of strict 2-c ate gories. 18 Here Mor( C ) := Disc (Mor( C )) is the 1-ca teg ory o f mor phisms of C , r egarded as a strict 2 -category with only identit y 2-morphisms. Pro of. The strict inclus ion 2-functor o n the le ft is  g h / / g ′  7→ b id   id   a g / / g ′ / / b h   h   id + 3 for g , g ′ : a → b a n y tw o parallel morphisms in C and h any 2-morphism betw een them. The strict surjection 2-functor on the right is b k   f   a q / / / / b ′ F   K   L + 3 7→ b f   k @ @ b ′ L   . The imag e of the injection is precisely the preimage under the surjection of the ident it y 2-morphism on the identit y 1 -morphisms . This means the sequence is exact.  4.3 Inner automorphisms Often, for G a n y gro up, inner and outer automorphisms ar e regar ded as sitting in a short exact s equence Inn( G ) / / Aut( G ) / / Out( G ) of ordinary groups. But we will find shor tly that w e ought to b e regarding the conjugation automorphisms b y tw o group e le men ts which differ b y an elemen t in the c en ter of the group as differ ent inner automorphis ms. So adopting this point of view for o rdinary gro ups , o ne gets instead the exact sequence Z( G ) / / Inn ′ ( G ) / / Aut( G ) / / Out( G ) . Of course this means setting Inn ′ ( G ) ≃ G , which seems to ma k e this step rather ill motiv ated. B ut it turns out that this dege ne r acy of concepts is a coin- cidence o f low dimensions a nd will be lifted a s we pass to inner automorphisms of higher groups. First recall the standard definitions of center and a utomorphism of 2- group oids: 19 Definition 9. Given any strict 2 - gr oup oi d C , • the automorphism 3 -gr oup A UT( C ) := Aut 2Cat ( C ) is the 2 -gr oup oid of isomorp hisms on C : obje cts ar e the strict and strictly invertible 2-fun ctors C f ≃ / / C , morphisms ar e pseudonatur al tr ansformations C f   f ′ B B C h   b etwe en t hese and 2-morphisms ar e mo dific ations h   h ′ = H T B B j v  ρ       f ′ y y f % % C C b etwe en those – t he pr o duct on the 3-gr oup c omes fr om t he c omp osition of autofunctors; • the c enter of C Z ( C ) := B AUT(Id C ) is the (sus p ende d) automorphism 2-gr oup of the identity on C , i.e. t he ful l sub gr oup oid of AU T( C ) on the single obje ct Id C . Example. The automor phism 2- g roup of any ordinary group G (regarded as a 2-group Disc( G ) w ith only ide ntit y morphisms) A UT( G ) := AU T( B G ) is that coming from the cro ssed mo dule G Ad / / Aut( G ) Id / / Aut( G ) . The cen ter Z ( G ) := Z ( B G ) of an y or dina ry group is indeed the ordinary c en ter of the group, reg arded as a 1-ob ject ca tegory . 20 Example. The automo rphism 3-g roup o f a strict 2-group (conceived in terms of crossed modules and 2-cro ssed mo dules) is discussed in theo rem 4 .3 o f [8]. T o these tw o standar d definitio ns , we add the following one, which is sup- po sed to b e the pro per 2-categ orical generalizatio n of the concept of inner au- tomorphisms. Definition 10 (inner a utomorphisms) . Given any strict 2 -gr ou p oid C , the t an- gent 2 -gro up oid INN( C ) := T Id C (Aut 2Cat ( C )) is c al le d the 2 -gr oup oid of inner automorphisms of C , and as su ch thought of as b eing e quipp e d with the monoidal structu r e inherite d fr om End( C ) . If the tra ns formation sta rting at the iden tit y is denoted q , it makes go o d sense to call the inner automo rphism being the targ et o f that tra nsformation Ad q : C Id   Ad q A A C q ∼   . A bigon of this for m is an ob ject in INN( C ). The pro duct of tw o such ob jects is the horizontal co mposition of these big o ns in 2Cat. W e shall sp ell this out in great detail for the case C = B G (2) in 5. Prop osition 2. F or C any strict 2-c ate gory, we have c anonic al morphisms Z( C )   / / INN( C ) / / A UT( C ) of strict 2-c ate gories whose c omp osition sends everything to the identity 2- morphism on the identity 1-morphism on the identity automorphism of C . Mor e over, t his sits inside the exact se quenc e fr om pr op osition 1 as Z ( C ) / /  _   INN( C ) / /  _   A UT( C )  _   Mor( C ) / / T C / / C , wher e C := Aut 2Cat ( C ) . Pro of. Recall that a mor phis m in Z ( C ) is a tr ansformation of the fo r m C Id   Ad q =Id A A C ∼ q   . 21 This gives the obvious inclusion Z ( G ) ֒ → INN ( G ). The morphism INN( G ) → A UT( G ) maps C Id   Ad q A A C q   7→ C Ad q / / C .  Remark. One would now w ant to de fine and construct the cokernel O UT( C ) of the morphism INN ( C ) → AUT( C ) and then sa y that Z ( C ) / / INN( C ) / / A UT( C ) / / OUT( C ) (7) is an exact sequence of 3-g roups. W e shall not consider OUT( C ) here for proper 3-gro ups. Restricted to just 2-g roups, howev er, one obtains the situatio n de- scrib ed in the next sectio n. 4.4 Inner automorphism 2 -groups Even though inner automo rphism 2-g roups of or dina ry (1-)g roups ar e just a sp ecial cas e of the inner automorphism 3-g roups of str ic t 2-groups to b e dis- cussed in the following, it may b e helpful to spell out this simpler case in detail, in order to se e ho w it connec ts with familiar examples of crossed modules. F or G any (or dinary) group, the sequence 7 is the ex a ct sequence of 2-groups 1 / / Z ( G ) / / INN( G ) / / A UT( G ) / / OUT( G ) / / 1 (1 → 1) / / (1 → Z ( G )) / / / / ( G Id → G ) / / ( G Ad → Aut( G )) / / (1 → Out( G )) / / (1 → 1) corres p onding to the exact seq uence of cros s ed mo dules giv en in the sec o nd line. Notice that there exists a lso the crossed mo dule (Inn( G ) → Aut( G )), whic h how ev er does not app ear in the ab o ve seq uenc e . In particular, this cro ssed mo dule is not the one corresp onding to our 2 -group INN( G ), which we had describ ed in deta il in the introduction. 4.5 Inner automorphism 3 -groups. Now we a pply the g eneral concept of inner automo rphisms to 2 -groups. The following definition just establishes the appropriate shorthand notation. 22 Definition 11. F or G (2) a strict 2 - gr oup, we write INN( G (2) ) := INN( B G (2) ) for its 3 -gr oup of inner au t omorph isms. In general this notation co uld be ambiguous, since one migh t want to consider the inner auto morphisms o f just the 1-gro upoid underlying G (2) . How ev er, in the present context this will never o ccur and using the ab o ve definition makes a couple of e x pressions more manifestly app ear as gener alizations o f familia r ones. Example. F o r G an ordinary group, regarded as a discrete 2- group, o ne finds that INN( G ) := T Id B G ( n Cat) ≃ T • ( B G ) is the co discrete gr oupoid over the elements of G . Its nature as a gro upoid is manifest from its realization as INN( G ) = T • ( B G ) . But it is also a (strict) 2-g roup. The monoidal structure is that coming fro m its r ealization as INN( G ) := T Id B G ( n Cat) . The cross ed mo dule corr e sponding to this strict 2-group is G Id / / G Ad / / Aut( G ) . The main p oin t o f interest for us is the g eneralization o f this fact to str ict 2-gro ups. One issue that one needs to b e a w are of then is that the ab ov e isomorphism T Id B G ( n Cat) ≃ T • ( B G ) b e comes a mere inclusion. Prop osition 3. F or G (2) any strict 2-gr oup, we have an inclusion T • B G (2) ⊂ T Id B G (2) (Aut 2Cat ( B G (2) )) of strict 2-gr oup oids. This r e alizes T • B G (2) as a sub 2-gr oup oid of T Id B G (2) (Aut 2Cat ( B G (2) )) . Pro of. The inclusion is es s en tially fixed by its action on ob jects: we define that an ob ject in T • B G (2) , which is a morphism • q / / • in B G , is s en t to the conjuga tion auto morphism Ad q : B G (2) → B G (2) • g   g ′ A A • h   7→ • q − 1 / / • g   g ′ A A • q / / • h   23 The transformation B G (2) Id " " Ad q < < B G (2) q ∼   . connecting this to the identit y is giv en by the component map ( • g / / • ) 7→ • q   g / / • q   Id u } s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s • q − 1 / / • g / / • q / / • . In g e neral one c ould consider transfo r mations whose comp onen t maps inv olve here a non-identit y 2-mo r phism. The inclusion w e ar e describing pic ks out excactly those transfor mations whose comp onent map only inv olv es identit y 2-morphisms. The crucial p oin t to realize no w is the for m of the comp onen t maps of mor - phisms Ad q Ad F   Id B G (2) q 0 0 q ′ . . Ad q ′ F   in T Id B G (2) (Aut 2Cat ( B G (2) )). The corresp onding compo nen t map equation is • g / / q   q ′   • q   • Ad q g / / f   • f   • Ad q ′ g / / • Id |                              Ad F ( g )               {              F k s = • g / / q ′   • q ′   q   @ @ @ @ @ @ @ • f          • Ad q ′ g / / • Id }                                F k s . 24 Solving this for Ad F shows that this is given by co njugation Ad F : ( • g / / • ) 7→ • q − 1   f   • f   • g / / • q / / q ′ / / • q ′− 1 E E •   F   with a morphism in T • ( B G (2) ). And e a c h such mor phism in T • ( B G (2) ) yields a morphism in T Id B G (2) (Aut 2Cat ( B G (2) )) this w a y . Finally , 2-morphisms in T Id B G (2) (Aut 2Cat ( B G (2) )) betw een these morphisms Ad F   Ad K = H T B B j v  L       Ad q ′ y y Ad q % % B G (2) B G (2) come from component ma ps • 7→ • f   k A A • L   ∈ Mor 2 ( B G (2) ) . A sufficient co ndition for these comp onen t maps to solve the re q uired condition for modifica tions o f pseudona tural tr ansformations is that they make • k   f   • q / / / / • F   K   L + 3 2-commute. But this defines a 2-morphism in T • B G (2) . And each such 2- morphism in T • ( B G (2) ) yields a 2-mo rphism in T Id B G (2) (Aut 2Cat ( C )) this wa y .  The crucial po in t is that by the embedding T • B G (2) ⊂ T Id B G (2) (Aut 2Cat ( B G (2) )) 25 the former 2 -category inherits the monoidal structure of the la tter a nd hence bec omes a 3- group in its own rig ht. This 3 -group is the ob ject of in terest here. 5 The 3-group INN 0 ( G (2) ) Definition 12 (INN 0 ( G (2) )) . F or G (2) any st rict 2-gr oup, the 3-gr oup INN 0 ( G (2) ) is, as a 2-gr oup oid, given by INN 0 ( G (2) ) := T • B G (2) and e quipp e d with the monoidal structur e inherite d fr om the emb e dding of pr op o- sition 3. W e now des cribe INN 0 ( G (2) ) for G (2) coming from the crossed mo dule H t / / G α / / Aut( H ) in mor e deta il, in pa rticular sp elling o ut the mono idal structure . W e extra ct the oper ations in the cro ssed mo dule corresp onding to the v arious compos itions in INN 0 ( G (2) ) and then finally iden tify the 2- crossed mo dule enco ded b y this. 5.1 Ob jects The ob jects of INN 0 ( G (2) ) are exa ctly the ob jects of G (2) , hence the elements of G : Ob j(INN( G (2) )) = G . The pro duct o f tw o ob jects in INN( G (2) ) is just the product in G . 5.2 Morphisms The morphisms g → h in INN( G (2) ) are Mor(INN 0 ( G (2) )) =                    • f   • g / / h / / • F                 f , g , h ∈ G, F ∈ H h = t ( F ) f g                    = { ( f , F ; g ) | f , g ∈ G, F ∈ H } . 26 5.2.1 Comp osition The compos itio n of tw o suc h mor phisms • f 1   • q 1 / / q ′′ / / q ′ / / • f 2   • F 1         F 2         . is in terms o f g roup lab els given by • f 1   • q / / / / / / • f 2   • F 1         F 2         = • f 2 f 1   • q / / / / • F 2 f 2 F 1   . 5.2.2 Pro duct Horizontal compos itio n of a utomorphisms B G (2) → B G (2) gives the pr o duct in the 3-group INN( G (2) ) Left whiskering of pseudonatura l transfor ma tions B G (2) Ad g / / B G (2) Ad q " " Ad q ′ < < B G (2) Ad F   amounts to the op eration • f   • q / / / / • F   7→ • f   • g / / • q / / / / • F   = • f   • qg / / / / • F   on the corresp onding tr ia ngles. 27 Right whiskering of pseudona tural transformations B G (2) Ad q " " Ad q ′ < < B G (2) Ad q / / B G (2) Ad F   amounts to the op eration • f   • q / / / / • F   7→ • f   g / / = • gf g − 1         • q / / / / • g / / • F   = • gf g − 1   • gq / / / / • g F   on the corresp onding tr ia ngles. Since 2Cat is a Gray-category , the horizo n tal co mposition of pseudonatura l transformatio ns B G (2) Ad q 1 " " Ad q ′ 1 < < B G (2) Ad q 2 " " Ad q ′ 2 < < B G (2) Ad F 1   Ad F 2   is am biguous. W e shall agree to read this a s B G (2) Ad q 1 " " Ad q ′ 1 < < B G (2) Ad q 2 " " B G (2) Ad F 1   B G (2) Ad q ′ 1 < < B G (2) Ad q 2 " " Ad q ′ 2 < < B G (2) Ad F 2   . 28 The corresp onding o pera tio n on triangles lab elled in the crossed mo dule is • f 1   • q 1 / / / / • f 2   • q 2 / / / / F 1   • F 2   = • q 2 f 1 q − 1 2   • f 1   q 2 / / = • q 1 / / / / • f 2   • q 2 / / / / F 1   • F 2   = • q 2 f 1 q − 1 2   • q 2 q 1 / / / / / / • f 2   • q 2 F 1         F 2         = • f 2 q 2 f 1 q − 1 2   • q 1 / / / / • F 2 f 2 q 2 F 1   The non- ide ntitcal isomor phis m which relates this to the other p ossible way to ev aluate the hor izon tal comp osition of pseudo natural transfo rmations gives rise to the Peiffer lifting of the cor r espo nding 2- c rossed mo dule. This is disc us sed in 6.4. 29 5.3 2-Morphisms The 2-morphisms in INN 0 ( G (2) ) are giv en by diagrams • k   f   • q / / / / • F   K   L + 3 . In terms of the group lab els this means that L ∈ H sa tisfies L = K − 1 F . (8) 5.3.1 Comp osition The horizontal comp osition of 2-morphisms in INN 0 ( G (2) ) is given by • k 1   f 1   • q 1 / / q 2 / / • F 1   K 1   L 1 + 3 k 2   f 2   • / / / / • F 2   K 2   L 2 + 3 = • k 2 k 1   f 2 f 1   • q 1 / / / / • G   J   L 2 f 2 L 1 + 3 G = F 2 f 2 F 1 , J = K 2 k 2 K 1 and v ertical compo sition by •     f   • q / / / / • F     L 1 + 3 L 2 + 3 = •   f   • q / / / / • F     L 2 L 1 + 3 (Notice that these co mpositions do go horizo n tally and vertically , resp ec- tively , once we rotate suc h that the bigons hav e the standar d or ien tation.) 30 Notice that whiskering along 1-morphisms f   f ′ = H T B B j v  L       Ad q ′ y y Ad q % % B G (2) B G (2) Ad q 2 t t Ad q 1 * * g / / g ′ / / acts on the co mponent maps a s • g   • k   f   • q / / / / p 6 6 • F   K   L + 3 G         = • kg   f g   • p / / / / • F ′   K ′   L + 3 F ′ = F f G, K ′ = K k G and • k   f   • q / / / / ) ) • g   F   K   L + 3 G           = • kg   f g   • p / / / / • F ′   K ′   g L + 3 F ′ = G g F , K ′ = G k K . 31 There is one more t ype of whiskering p ossible with 2-morphisms, f   k = H T B B j v  L       Ad q ′ y y Ad q % % B G (2) B G (2) B G (2) B G (2) % % y y g + 3 # # { { g ′ + 3 , which acts in the following wa y on the comp onen ts: • g   • q / / / / • k   f   • p / / / / G   • F   K   L + 3 = • k Ad p g   f Ad p g   • pq / / / / • F ′   K ′   L ′ + 3 , where F ′ = F f p G, K ′ = K kp G, L ′ = kp G − 1 L f p G. and • k   f   • p / / / / • g   • q / / / / G   • F   K   L + 3 = • g Ad q k   g Ad q f   • qp / / / / • F ′   K ′   gq L + 3 , 32 where F ′ = G gq F, K ′ = G gq K. An impor tan t case of this is: • k   f   • id / / / / • F   K   L + 3 = • f   • id / / t ( F ) f / / • kf − 1   id   • id / / id / / F   • id   L − 1   L + 3 . 5.3.2 Pro duct The whisk ering along ob jects f   k = H T B B j v  L       Ad q ′ y y Ad q % % B G (2) B G (2) B G (2) B G (2)     gives the pro duct of ob jects with 2-mor phis ms in the 3- group INN( G (2) ). Its action on 2-morphis ms , which we hav e alrea dy disucssed, extends in a simple wa y to 3-morphis ms: left whiskering alo ng an ob ject acts as • k   f   • q / / / / • F   K   L + 3 7→ • k   f   • g / / • q / / / / • F   K   L + 3 , 33 while right whiskering a long an ob ject acts as • k   f   • q / / / / • F   K   L + 3 7→ • k   f   g / / • gk g − 1   0 *    gf g − 1      * 0 • q / / / / • g / / • F   K   L + 3 . T o calculate the pro duct of a pair of 2-mor phisms, we use the fact that a 2-morphism is uniquely determined b y its so urce and targ et. • k 1   f 1   • q 1 / / / / • k 2   f 2   • q 2 / / / / F 1   K 1   L 1 + 3 • F 2   K 2   L 2 + 3 = • k 2 Ad q 2 k 1   f 2 Ad q 2 f 1   • q 2 q 1 / / / / • F ′   K ′   L ′ + 3 , with F ′ = F 2 f 2 q 2 F 1 K ′ = K 2 k 2 q 2 K 1 L ′ = L 2 f 2 q 2 L 1 6 Prop erties of INN 0 ( G (2) ) 6.1 Structure morphisms W e have defined B (INN 0 ( G (2) )) essentially as a sub 3 - category of 2 Cat. The latter is a Gray-category , in that it is a 3 -category which is strict except for the exchange law for c o mpositio n of 2-morphisms. Accor dingly , a lso B (INN 0 ( G (2) )) is strict except for the exchange la w for 2-mo rphisms. This means that as a mere 2-g roupo id (forgetting the mo no idal structure) INN 0 ( G (2) ) is strict. 6.1.1 Strictness as a 2-group oid Prop osition 4. The underlying 2-gr oup oid of INN 0 ( G (2) ) is strict. 34 Pro of. This follows from the r ules for horizontal and vertical comp osition of 2 -morphisms in INN 0 ( G (2) ) – displayed in 5.3.1 – and the fact that G (2) itself is a s trict 2-group, b y a ssumption.  But the pro duct 2-functor on INN 0 ( G (2) ) res p ects horizontal comp osition in INN 0 ( G (2) ) o nly weakly . In the lang uage of 2 -groups, this corre sponds to a failure of the Peiffer iden tit y 6.2 T rivializabilit y Prop osition 5. The 2-gr oup oid INN 0 ( G (2) ) is c onne cte d, π 0 (INN 0 ( G (2) )) = 1 . Pro of. F or any tw o ob jects q and q ′ there is the morphism • q ′ q − 1   • q / / q ′ / / • id    Prop osition 6. The H om-gr oup oi ds of the 2-c ate gory INN 0 ( G (2) ) ar e c o di s- cr ete, me aning t hat they have pr e cisely one morphism for every or der e d p ai r of obje cts. Pro of. By equation (8) ther e is at most one 2 -morphism betw een a n y par- allel pa ir of morphisms in INN 0 ( G (2) ). F or there to be any suc h 2-mo r phism at all, the tw o gro up elements f and k in the diagr am ab ov e (8) have to satisfy k f − 1 ∈ im( t ). But by using the source-targe t matc hing condition for F and K one readily sees that this is always the case.  Theorem 1 . The 3-gr oup INN 0 ( G (2) ) is e qu ivale nt to the trivial 3-gr oup. If G (2) is a Lie 2-gr oup, then INN 0 ( G (2) ) is e quiva lent t o the trivial Lie 3-gr oup even as a Lie 3-gr oup. Pro of. Equiv alence o f 3-gr oups G (3) , G ′ (3) is, by definition, that o f the cor- resp onding 1-o b ject 3-gro upoids B G (3) , B G ′ (3) . F or showing equiv a lence with the trivial 3 -group, it suffices to exhibit a pseudonatur a l transforma tio n of 3- functors id B (INN 0 ( G (2) )) → I B (INN 0 ( G (2) )) , where I B (INN 0 ( G (2) )) sends e verything to the ident it y on the single ob ject of B INN 0 ( G (2) ). Such a tra nsformation is obtained by sending the single ob ject 35 to the identit y 1-morphism on tha t ob ject and s ending a n y 1 -morphism q to the 2 - morphism q → id fro m prop 5. By pro p 6 this implies the existence o f a unique assignment of a 3-morphis m to any 2- mo rphism such that w e do indeed obtain the comp onent map of a pseudonatural tr a nsformation of 3-functor s. By construction, this is clearly s mooth when G (2) is Lie.  6.3 Univ ersalit y Theorem 2. We have a short exact se quenc e of strict 2-gr oup oids G (2)   / / INN 0 ( G (2) ) / / / / B G (2) . Pro of. This is just pro position 1, after noticing that Mor( B G (2) ) = G (2) .  So the strict inclusion 2-functor on the left is  g h / / g ′  7→ • id   id   • g / / g ′ / / • h   h   id + 3 , while the strict surjection 2-functor on the rig h t is • k   f   • q / / / / • F   K   L + 3 7→ • f   k A A • L   . 6.4 The corresp onding 2-crossed mo dule W e now extract the structure of a 2-cr ossed mo dule from INN 0 ( G (2) ). First, let Mor I 1 = Mor 1 (INN 0 ( G (2) )) | s − 1 (Id) and Mor I 2 = Mor 2 (INN 0 ( G (2) )) | s − 1 (id Id ) be s ubg roups o f the 1- and 2-morphisms of INN 0 ( G (2) ) resp ectiv ely . 36 Prop osition 7. The gr oup of 1-morphi sms in INN 0 ( G (2) ) starting at the iden- tity obje ct form t he semidir e ct pr o duct gr oup Mor I 1 = G ⋉ H under the identitfic atio n • f   • Id / / / / • F   7→ ( f , F ) in t hat • f 1   • Id / / / / • f 2   • Id / / / / F 1   • F 2   = • f 2 f 1   • Id / / / / • F 2 f 2 F 1   . Pro of. Use co mposition in B G (2) .  W e have the obvious g roup homomor phism which is just the restriction of the target map ∂ 1 : Mor I 1 → O b j := Ob j(INN 0 ( G (2) )) given by ∂ 1 : • f   • Id / / / / • F   7→ t ( F ) f . This and the following constructions ar e to be compared with definition 4. There is an ob vious action o n Mor I 1 : 37 • f   • id / / / / • F   7→ • f   g / / = • gf g − 1         • g − 1 / / • id / / / / • g / / • F   = • gf g − 1   • id / / / / • g F   (9) This action a lmost gives us a cr o ssed mo dule Mor I 1 → O b j. But not quite, since the Peiffer ident it y holds only up to 3 - is omorphism. T o see this, let g = ∂ 1          • h   • id / / g / / • H            = t ( H ) h. F or the Peiffer identit y to hold w e nee d the a c tion (9) to b e equal to the adjoin t action of the 2-cell ( h, H ; id ). T o see that this fails, firs t notice that the inv erse of the approriate 2-cell considered as an element in the gro up Mor I 1 is            • h   • id / / t ( H ) h / / • H              − 1 = • h − 1   • id / / h − 1 t ( H ) − 1 / / • h − 1 H − 1   . Therefore the conjugation is 38 • h − 1   • id / / g − 1 / / • f   • id / / t ( F ) f / / • h   h − 1 H − 1   • id / / g / / • F   H   = • h − 1   • f   • id / / g − 1 / / • f   id / / = • id / / t ( F ) f / / • h   h − 1 H − 1   • id / / g / / • F   H   = • h − 1   • h − 1   id / / = • id / / g − 1 / / • hf   • id / / gt ( F ) f / / h − 1 H − 1   • H h F   = • hf h − 1   • id / / gt ( F ) f g − 1 / / • H − 1 h F hf h − 1 H   (10) 6 = • gf g − 1   • id / / gt ( F ) f g − 1 / / • g F   (11) 39 Though the Peiffer identit y do es not hold, b oth a c tions give rise to 2-cells with the sa me source a nd target, a nd hence define a 3 -cell P . Denote the 2-cell (10) by ( c, C ; id ) and the 2-cell (11) by ( a, A ; id ) (for c onjugation a nd a ctio n resp ectively). • a   c   • id / / / / • C   A   P + 3 Then P = A − 1 C = g F − 1  H h F hf h − 1 H − 1  = g F − 1  t ( H ) h F H hf h − 1 H − 1  = g F − 1  g F H hf h − 1 H − 1  = H hf h − 1 H − 1 How ev er, what we really wan t is the Peiffer lifting, which will be a 3-cell with source the iden tit y 2-cell. Hence, Prop osition 8. The gr oup of 2-morphi sms in INN 0 ( G (2) ) starting at the iden- tity arr ow on the identity obje ct form the gr oup Mor I 2 = H under the identitfic atio n • t ( L )   id   • id / / / / • id     L + 3 7→ L in t hat 40 • t ( L 1 )   id   • id / / / / • t ( L 2 )   id   • id / / / / • id     L 1 + 3 id     L 2 + 3 = • t ( L 2 L 1 )   id   • id / / / / • id     L 2 L 1 + 3 Pro of. Use the multiplication of 2 -morphisms.  So, we whisker the 3-cell ( P ; a, A ; id) ab o ve with the in verse of ( a, A ; id): • a − 1   • id / / / / • c   a   • id / / / / a − 1 A − 1   • A   C   P + 3 = • t ( P )= ca − 1   id   • id / / / / • id     P + 3 , and the bac k fa ce is neces sarily P − 1 . Definition 13 (Peiffer lifting) . Define the map {· , ·} : Mor I 1 × Mor I 1 → Mor I 2 by                  • h   • id / / / / • H   , • f   • id / / / / • F                    = • t ( P )   id   • id / / / / • id     P + 3 , P = H hf h − 1 H − 1 Now define the homomorphism ∂ 2 : Mor I 2 → Mor I 1 41 by ∂ 2 : • t ( L )   id   • id / / / / • id     L + 3 7→ • t ( L )   • Id / / Id / / • L − 1   , which is a gain the restriction of the target map. Note there is an action of Ob j on Mor I 2 : • t ( L )   id   • id / / / / • id     L + 3 7→ • t ( L )   id   g / / • gt ( L ) g − 1   0 *    id      * 0 • g / / • id / / / / • g / / • id     L + 3 = • t ( g L )   id   • id / / / / • id     g L + 3 (12) Clearly ∂ 2 ◦ ∂ 1 is the constant ma p at the identit y , a nd im ∂ 2 is a no rmal subgroup of k er ∂ 1 , so Mor I 2 ∂ 2 / / Mor I 1 ∂ 1 / / Ob j (13) is a seq ue nce . W e let the action of Ob j on the other two g roups b e as described ab o ve in (9) and (12), a nd the maps ∂ 2 and ∂ 1 are clear ly equiv ar ian t for this action. Prop osition 9. The map { · , ·} do es inde e d satisfy the pr op ert ies of a Peiffer lifting, and (13) is a 2- cr osse d mo dule. Pro of. The fir st condition holds by definition, the seco nd and the last one are easy to check. The o thers are tedious. It is easy , using the cros sed module prop erties of H → G , to calculate that the actions of Mor I 1 on Mor I 2 as defined from INN 0 ( G (2) ) and as defined via {· , ·} are the same.  Since im ∂ 2 = ker ∂ 1 , ∂ 2 is injective and ∂ 1 is o n to, this shows that (13) has trivial homolog y and provides us with another pro of that INN 0 ( G (2) ) is contractible. 42 6.4.1 Relation to the mapping cone of H → G Given a crossed square L f / / u   M v   N g / / P with structure map h : N × M → L , Conduch ´ e [1 4]giv es the Peiffer lifting o f the mapping cone L / / N ⋉ M / / P as { ( g , h ) , ( k , l ) } = h ( g k g − 1 , h ) . Recall from 2.3 that the identit y map o n t : H → G is a cross ed square with h ( g , h ) = h g h − 1 , so the mapping cone is a 2-cr ossed mo dule H ∂ 2 / / G ⋉ H ∂ 1 / / G, where d 2 ( h ) = ( t ( h ) , h − 1 ) , d 1 ( g , h ) = t ( h ) g , and with P eiffer lifting { ( g 1 , h 1 ) , ( g 2 , h 2 ) } = h 1 g 1 g 2 g − 1 1 h − 1 1 . which is what w e found fo r INN 0 ( G (2) ). More precisely , Definition 14. A morphism ψ of 2-cr osse d m o dules is a map of the underlying c omplexes L 1 ∂ 2 / / ψ L   M 1 ∂ 1 / / ψ M   N 1 ψ N   L 2 ∂ 2 / / M 2 ∂ 1 / / N 2 such t hat ψ L , ψ M and ψ N ar e e quivaria nt for the N - and M -actions, and { ψ M ( · ) , ψ M ( · ) } 2 = ψ L ( {· , ·} 1 ) . 43 Using prop ositions 7 a nd 8, we have a map Mor I 2 ∂ 2 / / ≃   Mor I 1 ∂ 1 / / ≃   Ob j ≃   H d 2 / / G ⋉ H d 1 / / G and the actions and P eiffer lifting a gree, s o Prop osition 10 . The 2-cr osse d mo dule asso ciate d to INN 0 ( G (2) ) is isomo rphic to the mapping c one of the identity map on the cr osse d mo dule asso ciate d to G (2) . 7 Univ ersal n -bund les In order to put the relev ance of the 3-group INN 0 ( G (2) ) in p ersp ective, we further illuminate o ur statemen t, 3.1, that INN 0 ( G (2) ) plays the r ole of the universal G (2) -bund le . An exhaustive discussion will b e giv en elsewhere. 7.1 Univ ersal 1-bundles in terms of INN( G ) Let π : Y → X b e a g oo d cov er of a spac e X a nd write Y [2] := Y × X Y fo r the corres p onding g roupo id. Definition 15 ( G -co cycles) . A G -(1-)c o cycle on X is a functor g : Y [2] → B G . This functor can be unders too d as arising fr om a choice π ∗ P t ∼ / / Y × G of trivializ ation o f a principal r igh t G -bundle P → X (which is essentially just a map to G ) as g := π ∗ 2 t ◦ π ∗ 1 t − 1 , by noticing that G -equiv ar ian t is omorphisms G → G are in bijection with elements of G g ( x, y ) : h 7→ g ( x, y ) h acting from the left. 44 Observ ation 1 ( G -bundles as mo rphisms of sequences of g roupo ids) . Given a G -c o cycle on X as ab ove, its pul lb ack along the exact se quenc e G / / INN( G ) / / B G , which we write as Y × G / /   Y [2] × g INN( G ) / /   Y [2] / / g   X   G / / INN( G ) / / B G / / {•} , pr o duc es the bund le of gr oup oids Y [2] × g INN( G ) / / Y [2] which plays the r ole of the total sp ac e of the G -bund le classifie d by g . This should b e compar ed with the simplicial construc tio ns desc r ibed, for instance, in [20]. Remark. Using the fact that INN( G ) is a 2- group, a nd using the injection G → INN( G ) w e na turally obtain the G -a c tion on Y [2] × g INN( G ). Remark. Notice that this is closely related to the in tegrated At iyah sequenc e Ad P / / P × G P / / X × X of group oids ov er X × X coming from the G -principal bundle P → X : Ad P / / P × G P / / X × X Y × G / /   Y [2] × g INN( G ) / /   Y [2] / / g   X   G / / INN( G ) / / B G / / {•} . W e now mak e precise in which sense, in turn, Y [2] × g INN( G ) pla ys the ro le of the total space of the G -bundle characterized b y the co cycle g . 45 T o reobtain the G -bundle P → X from the group oid Y [2] × g INN( G ) we form the pushout of Y [2] × g INN( G ) target / / source   Y × G Y × G . (14) Prop osition 11. If g is the c o cycle classifying a G -bund le P on X , then the pushout of 14 is (up to isomorphism) that very G -bu nd le P . Pro of. Consider the square Y [2] × g INN( G ) target / / source   Y × G t − 1   π ∗ P   Y × G t − 1 / / π ∗ P / / P , where t : π ∗ P ∼ / / Y × G is the lo cal trivia lization of P whic h gives rise to the transition function g . Then the diagram c o mm utes by the v ery definition of g . Since t is an isomo rphism and since π ∗ P → P is lo cally an iso morphism, it follows that this is the univ ersal pushout.  7.2 Univ ersal 2-bundles in terms of INN 0 ( G (2) ) Now let G (2) be an y strict 2-group. Le t Y [3] be the 2-gro up oid whose 2- morphisms are triples of lifts to Y of p oin ts in X . A principal G (2) -2-bundle [6, 5] has lo cal trivializatio ns characterized b y 2- functors g : Y [3] → B G (2) . Definition 16 ( G (2) -co cycles) . A G (2) -(2-)c o cycle on X is a 2-functor g : Y [3] → B G (2) . (Instead of 2-functors on Y [3] one could use pseudo functors on Y [2] .) As b efore, we ca n pull these back along o ur exact se q uence of 2-g roupo ids 3.1 G (2) / / INN( G (2) ) / / B G (2) 46 to obtain Y × G (2) / /   Y [3] × g INN( G (2) ) / /   Y [3] / / g   X   G (2) / / INN( G (2) ) / / B G (2) / / {•} . W e rec o nstruct the total 2-space of the 2-bundle by forming the weak pushout of Y [3] × g INN( G (2) ) target / / source   Y × G (2) Y × G (2) . (15) Here “source” and “target” are defined rela tiv e to the inclusion Y × G (2) ֒ → Y [2] × g INN( G (2) ) . This means that for a g iv en 1-mor phism • f := g ( x,y )   x   • q 1 / / q 2 / / • y F               in Y [3] × g INN( G (2) ) (for any x, y ∈ Y with π ( x ) = π ( y ) and for g ( x, y ) the corres p onding comp onen t o f the given 2-co cycle) which we may equiv alently rewrite as • f   x   • q 1 / / f − 1 q 2 E E • y f − 1 F   + + + + 47 the source in this s ense is • x • q 1 / / f − 1 q 2 E E f − 1 F   + + + + , regar ded as a morphism in Y × G (2) , while the target is • f q 1   q 2 / / • y F       regar ded as a morphism in Y × G (2) . This w ay the transition function g ( x, y ) a cts on the copies of G (2) which app ear as the trivialize d fib ers of the G (2) -bundle. Bartels [6][pro of of prop. 22 ] gives a reconstr uc tio n of total spac e of principal 2-bundle from their 2-co cycles which is c losely re la ted to Y [3] × g INN( G (2) ). As an ano n ymous referee po in ted out, our cons tr uction in the ca s e o f 1- groups is related to the universal co ver. Since B G is a (mo del of a) connec ted 1-type, and INN 0 ( G ) is co n tractible, it can b e co nsidered as (a mo del of ) the universal cov er. Indeed, o ne of the motiv ations for the fir st author was to understand 2-connected ‘universal co vers’ for 2-types. The connections with such a notion will be treated in [26], as w ell as g eneralisations. 7.3 Relation to simplicial bundles Our considera tions can b e translated, along the nerve or double nerve func- tor, to the w orld of simplicia l sets. Under this translation one finds that the tangent categ ory co nstruction co rresp onds to the simplicial o pera tion k no wn as d´ eca lage, and INN 0 ( G ) cor respo nds to the simplicial se t denoted W G . It do es not a ppear to b e w ell known that a group structure can be put o n W G , and one wa y of seeing this for simplicial gro ups whic h ar e nerv es of 2-gro ups is via our construction. There is a g eneral descr iptio n o f this group s tructure bypassing INN 0 ( G ) altogether [27]. 7.3.1 T angen t categories and d´ ecalage F or C any categor y , notice that a seq uence of k comp osable morphisms a / / b / / c / / d x _ _ ? ? ? ? ? ? ? ? O O ? ?         7 7 o o o o o o o o o o o o o o 48 in T C is, since all triangles co mm ute, the sa me a s a sequence a / / b / / c / / d x _ _ ? ? ? ? ? ? ? ? of k + 1 comp osable morphisms in C . T o formalize this obse r v ation, let ∆ denote, as usual the simplicial ca tegory whose ob jects are the catgories [ n ] = { 0 → 1 → · · · → n } for all n ∈ N and whose mo rphisms ar e the functors b e t w een these. W r ite [∆ op , Set] for the category of simplicia l sets. Denote b y [( − ) + 1] : ∆ op → ∆ op the obvious functor whic h acts on o b jects as [ n ] 7→ [ n + 1 ] (shifts everything up b y one). The induced map on simplicial sets, [( − ) + 1] ∗ : [∆ op , Set] → [∆ op , Set] is ca lled d´ ec al age [1 9] and is denoted Dec 1 . F rom a mor e p edestrian v iewpoint, Dec 1 strips off the first face and firs t degeneracy map 3 from each le v el of a simplicial ob ject X , r e-indexes the res t and mov es the sets of s implices down one lev el: (Dec 1 X ) n = X n +1 . Prop osition 12. T he tangent c ate gory c onst ruction fr om definition 8 is taken by the nerve functor N : Cat → [∆ op , Set] t o the d´ ec alage c onstruction in that we have a we akly c ommuting squar e Cat T   N / / [∆ op , Set] [( − )+1] ∗   Cat N / / [∆ op , Set] ≃ x  y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y . 3 It is a matte r of con v en tion that the first face and degeneracy maps are remov ed. The d ´ ecalage is sometimes defined by remo ving the last face and degeneracy maps, but our tangen t category construction is related to the former con v en tion. 49 Pro of. This is not hard to see by chasing ex plic it elements through this diagram. A little more a bstractly , we see as follows that the assignment of ( n + 1 )- simplices in C to n -simplices in T C is functor ial. Notice that n -simplices in T C a re commuting s quares [ n ] / /   I × [ n ]   [0] / / C but the pushout of this co - cone is [ n + 1]: [ n ] / /   I × [ n ] f   [0] / / [ n + 1 ] , where f : ( ◦ , 0) / / ( ◦ , 1) / / ( ◦ , 2) / / ( ◦ , n ) ( • , 0) / / O O ( • , 1) / / O O ( • , 2) / / O O ( • , n ) O O 7→ ( ◦ , 0) / / ( ◦ , 1) / / ( ◦ , 2) / / ( ◦ , k ) ( • , 0) ] ] < < < < < < < O O A A        8 8 q q q q q q q q q q q 7→ ( ◦ , 0) / / ( ◦ , 1) / / ( ◦ , 2) / / ( ◦ , n ) ( • , 0) ] ] < < < < < < < . Hence w e functoria lly assig n ( n + 1)-simplices in C to n -simplices in T C b y using the univ ersality of the pushout: [ n ] / /   I × [ n ]   f z z u u u u u u u u u [ n + 1 ] ! $ $ I I I I I I I I I I [0] / / < < y y y y y y y y y C .  7.3.2 Univ ersal si mplicial bundl es If G is a simplicial group in sets, there is a notion of principa l bundle int ernal to s Set [24], a nd for s uc h bundles there is a cla ssifying simplicial s e t W G completely 50 analogo us to the case of top ological bundles. As such, there is a contractible simplicial set W G which is the total space of a simplicial bundle W G   W G and in fact W G ≃ Dec 1 W G . It is a short calculation to show that W G = N B G a nd W G = N INN( G ) when G is a constant simplicial gro up. T o recover a similar result for strict 2-gro ups (in Set), w e recall that fo r 2-categor ies there is a functor N called the double nerve , which for ms a bisimplicia l s et whose ge o metric r ealization is calle d the classifying space of the 2 -category . Definition 1 7. R e c alling that st rict 2-gr oups ar e t he same as c ate gories internal to gr oups, the double nerve N G (2) of a strict 2-gr oup G (2) is define d to b e its image u nder N : C a t(Gp) N / / [∆ op , Gp] / / [∆ op , [∆ op , Set]] , with gr oups c onside r e d as one-obje ct gr oup oids. Explicitly , let G (2) be the strict 2 - group c o ming from the crosse d mo dule t : H → G . Recalling that then Ob j( G (2) ) = G and Mor( G (2) ) = H ⋊ G we find that N G (2) is the bisimplicial set given by ( N G (2) ) 0 n = {·} ( N G (2) ) 1 n = ( N G (2) ) n =    G, n = 0; H ⋊ G n = 1; ( H ⋊ G ) × G n · · · × G ( H ⋊ G ) , n > 1 ( N G (2) ) kn = ( N G (2) ) 1 n × k · · · × ( N G (2) ) 1 n k > 1 . So fo r eac h n , ( N G (2) ) • n is the nerv e of the group of sequences of n comp osable morphisms in G (2) . Applying Dec 1 to ea c h of these, i.e. forming the bisimplicial set Dec 1 N G (2) in the image of Dec 1 N : C a t( Gp) N / / [∆ op , Gp] / / [∆ op , [∆ op , Set]] [∆ op , Dec 1 ] / / [∆ op , [∆ op , Set]] we o btain a surjection Dec 1 N G (2) → N G (2) whose kernel is the bisimplicial set which ha s just the fiber, namely the group of sequences of n -mor phisms, in each row: ( N ′ G (2) ) kn := ( N G (2) ) n ∀ k ∈ N . Hence w e ha v e an exact sequence N ′ G (2) → Dec 1 N G (2) → N G (2) . 51 The realiza tion | · | of a bisimplicial set is the or dinary rea lization o f the dia gonal simplicial space n 7→ ( N G (2) ) nn . It is hence clea r that | N ′ G (2) | = | N G (2) | and, b y definition, |N G (2) | = B G (2) . Moreov er, since each row of Dec 1 N G (2) is contractible, | Dec 1 N G (2) | is also contractible. 4 This r elates to our co nstruction by the fact that Dec 1 N G (2) is the nerve of a double gro upoid (in fact a cat 2 -group [23]) constructed from the crossed squar e H id / / t   H t   G id / / G . A Crossed squ ares As noted above, cro ssed squares were introduced in [18]. W e include the de fini- tion for c ompleteness: Definition 18 (cr o ssed squar e) . A cr osse d squar e is a c ommutative squar e of P -gr oups L f / / u   M v   N g / / P (with P acting on itself by c onjugatio n, actions denote d by a b ) and a fun ction M × N → L such that 1. f and u ar e P - e quivariant, and N − → P M − → P L − → P ar e cr osse d mo dules, 4 The realization of a bi simplicial set can also be calculated as fir st taking the ordinary realization of the rows then realizing the resulting s i mplicial space - this r esults in a space homeomorphic to the previous description. 52 2. f ( h ( x, y )) = x g ( y ) x − 1 , u ( h ( x, y )) = v ( x ) y y − 1 , 3. h ( f ( z ) , y ) = z g ( y ) z − 1 , h ( x, u ( z )) = v ( x ) z z − 1 , 4. h ( xx ′ , y ) = v ( x ) h ( x ′ , y ) h ( x, y ) , h ( x, y y ′ ) = h ( x, y ) g ( y ) h ( x, y ′ ) , 5. h is P - e quivariant for t he diagonal action of P on M × N . It follows fr om the definition that L → M and L → N a r e cr ossed modules where M , N a ct on L via the maps to P . 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Math. 4 7 , pp. 806–81 0 ( 1946) School of Ma thema tical Sciences, The University of Adelaide, SA, 5005, Australia E-mail addr ess : drob erts@ma ths.adelaide.edu.au Dep ar tment Ma thema tik, U niversit ¨ at Hamburg, Bund esstraße 55, 20146 Ha mbur g, Germany E-mail addr ess : urs. schreib er@math.uni-hamburg.de 55