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

Transcription

1 The inner automorphism 3-roup o a strict 2-roup Dav Roberts and Urs Schreiber July 25, 2007 Abstract or any roup G, there is a 2-roup o inner automorphisms, INN(G). This plays the role o the universal G-bundle. Similarly, or every 2- roup G (2) there is a 3-roup INN(G (2) ) and a slihtly smaller 3-roup INN 0(G (2) ) o inner automorphisms. We construct these or G (2) any strict 2-roup, discuss how INN 0(G (2) ) can be understood as arisin rom the mappin cone o the entity on G (2) and show that it its into a short exact sequence Disc(G (2) ) INN 0 (G (2) ) ΣG (2) o strict 2-roupos. We close by indicatin how this makes INN 0(G (2) ) the universal G (2) -2-bundle. Contents 1 Introduction 2 2 n-groups in terms o roups Conventions or strict 2-roups and crossed modules Groups and 2-crossed modules Mappin cones o crossed modules Main results The exact sequence Disc(G (2) ) INN 0 (G (2) ) ΣG (2) INN 0 (G (2) ) rom a mappin cone Inner automorphism n-roups Exact sequences o strict 2-roupos Tanent 2-cateories Inner automorphisms Inner automorphism 3-roups

2 5 The 3-roup INN 0 (G (2) ) Objects Morphisms Composition Product Morphisms Composition Product Properties o INN 0 (G (2) ) Structure morphisms Strictness as a 2-roupo Trivializability Universality The correspondin 2-crossed module Relation to the mappin cone o H G Universal n-bundles Universal 1-bundles in terms o INN(G) Universal 2-bundles in terms o INN 0 (G (2) ) Introduction By deinition, an n-roup is a 1-object n-roupo. When we oret this one object, the n-roup becomes a monoal (n 1)-roupo. I this is denoted by G (n) then the correspondin 1-object n-roupo will be denoted ΣG (n). One source o n-roups are automorphisms: or any n-cateory C, the n- cateory AUT(C) : Aut ncat (C) is an (n + 1)-roup. In the special case that C ΣG (n) is itsel already a 1-object n-roupo we write AUT(G (n) ) : Aut ncat (ΣG (n) ). Then we can speak o inner automorphisms: these are essentially those n- unctors C C which come rom conjuatin any n-morphism with a iven invertible 1-morphism. In eneral there may be 1-morphisms in ΣG (n) which induce the same automorphism G (n) G (n) under conjuation. It is o interest to nevertheless distinuish these. We write INN(G (n) ) 2

3 or the (n + 1)-roup whose objects are those o G (n) and whose hiher morphisms are transormations o conjuations. We write INN 0 (G (n) ) INN(G (n) ) or the n + 1-roup o conjuations just by objects o G (n). In the present context we shall work exclusively in the 3-cateory 2Cat whose objects are strict 2-cateories, whose morphisms are strict 2-unctors, whose 2-morphisms are pseudonatural transormations and whose 3-morphisms are modiications o these: we shall study INN(G (2) ) or G (2) any strict 2-roup. (But or any n and any notion o (weak) n-cateory, the above notions o automorphism (n + 1)-roups and inner automorphism (n + 1)-roups make sense.) The basic example is the inner automorphism 2-roup o an ordinary roup G. Every strict 2-roup comes rom a crossed module H t G α Aut(H) o ordinary roups. INN(G) is the one that comes rom the crossed module G G Ad Aut(G). This 2-roup INN(G) its into the exact sequence Disc(G) INN(G) ΣG o roupos. Here Disc(G) denotes the discrete cateory over G. This is the cateory whose space o objects is the space underlyin G and whose only morphisms are entity morphisms. By takin nerves and realizin eometrically, this becomes the universal G- bundle G EG BG. We show that and how an analoous statement holds or INN 0 (G (2) ), with G (2) any strict 2-roup. Our main results are stated in 3. A brie outlook on the interpretation o INN 0 (G (2) ) as the universal G (2) -2-bundle closes the discussion in 7. We are rateul to Jim Stashe or helpul discussions and or emphasizin the importance o the mappin cone construction in the present context. We also thank Zoran Škoda or helpul comments. 3

4 2 n-groups in terms o roups Suiciently strict n-roups are equivalent to certain structures crossed modules and eneralizations theoreo involvin just collections o ordinary roups with certain structure on them. 2.1 Conventions or strict 2-roups and crossed modules It is well known that strict 2-roups are equivalent to crossed modules o ordinary roups. Deinition 1. A crossed module o roups is a diaram H t G α Aut(H) in Grp such that and H G H t Ad Aut(H) α G Id t G G α Ad. H t G Deinition 2. A strict 2-roup G (2) is any o the ollowin equivalent entities a roup object in Cat a cateory object in Grp a strict 2-roupo with a sinle object A detailed discussion can be ound in [1]. One entiies G is the roup o objects o G (2). H is the roup o morphism o G (2) startin at the entity object. t : H G is the taret homomorphism such that h : Id t(h) or all h H. 4

5 α : G Aut(H) is conjuation with entity morphisms: Ad Id ( Id h t(h) ) Id α(h)(h) t(α()(h)) or all G, h H. We oten abbreviate h : α()(h). Beyond that there are 2 2 choices to be made when entiyin a strict 2-roup G (2) with a crossed module o roups. The irst choice to be made is in which order to multiply elements in G. or 1 2 and two morphisms in ΣG (2), we can either set 1 2 : 1 2 () or 1 2 : 2 1 (B). The other choice to be made is how to describe arbitrary morphisms by an element in the semirect product roup G H: every morphism o G (2) may be written as the product o one startin at the entity object with an entity morphism on some object. The choice o orderin here yields either Id h : h (R) or Id h : h (L) Here we choose the convention LB. This implies h t(h) 5

6 or all G, h H, as well as the ollowin two equations or horizontal and vertical composition in ΣG (2), expressed in terms o operations in the crossed module h 1 h 2 h 2 2h1 and 1 1 h 1 2 h 2h 1. h Groups and 2-crossed modules As we are conserin strict models in this paper, we will assume that all 3- roups are as strict as possible. This means they will be one-object Graycateories, or Gray-monos [8]. A Gray-mono is a (strict) 2-cateory M such that the product unctor M M M uses the Gray tensor product, not the usual Cartesian product o 2-cateories. Thus non-entity coherence morphisms only appear when we use the monoal structure on M. Just as a 2-roup ives rise to a crossed module, a 3-roup ives rise to a 2-crossed module. Rouhly, this is a complex o roups and a unction L M N, M M L (1) such that L M is a crossed module, and (1) measures the ailure o M N to be a crossed module. An example is when L 1, and then we have a crossed module. Now or the ormal deinition. See [3]. Deinition 3. A 2-crossed module is a normal complex o lenth 2 L 2 M 1 N o N-roups (N actin on itsel by conjuation) and an N-equivariant unction {, } : M M L, called a Peier litin, satisyin these conditions: 6

7 1. 2 {m, m } (mm m 1 )( 1m m ) 1, 2. { 2 l, 2 l } [l, l ] : ll l 1 l 1 3. (a) {m, m m } {m, m } mm m 1 {m, m } (b) {mm, m } {m, m m m 1 } 1m {m, m }, 4. {m, 2 l} ( m l)( 1m l) 1 5. n {m, m } { n m, n m }, where l, l L, m, m, m M and n N. Here m l denotes the action M L L (m, l) m l : l{ 2 l 1, m}. (2) A normal complex is one in which im is normal in ker partial or all dierentials. It ollows rom these conditions that 2 : L M is a crossed module with the action (2). To et rom a 3-roup G (3) to a 2-crossed module, we emulate the construction o a crossed module rom a 2-roup: one entities N is the roup o objects o G (3). M is the roup o 1-morphisms o G (3) startin at the entity object. L is the roup o 2-morphisms startin at the entity 1-arrow o the entity object 1 : M N is the taret homomorphism such that m : Id 1 (m) or all m M. 2 : L M is the taret homomorphism such that l : Id Id 2 (l) or all l L. The various actions arise by whiskerin, analoously to the case o a 2- roup. We will not o into the proo that this ives rise to a 2-crossed module or all 3-roups, but only in the case we are conserin. One reason to conser 2-crossed modules is that the homotopy roups o G (3) can be calculated as the homoloy o the sequence underlyin the 2-crossed module. 7

8 2.3 Mappin cones o crossed modules Another notion equivalent to 3-roups is crossed modules internal to crossed modules (more technically known as crossed squares, [7],[5]). More enerally, conser a map φ o crossed modules: Deinition 4 (nonabelian mappin cone [5]). or H 2 φ H H 1 t 2 t 1 G 2 φ G G 1 a 2-term complex o crossed modules (t i : H i G i ), we say its mappin cone is the complex o roups where and H 2 2 G 2 H 1 1 G 1, (3) 1 : ( 2, h 1 ) t 1 (h 1 )φ G ( 2 ) 2 : h 2 (t 2 (h 2 ), φ H (h 2 ) 1 ). Here G 2 acts on H 1 by way o the morphism φ G : G 2 G 1. When no structure is imposed on φ, (3) is merely a complex. However, i φ is a crossed square, the mappin cone is a 2-crossed module (oriinally shown in [2], but see [3]). We will not need to deine crossed squares here (the interested reader may consult [3]), but just note they come equipped with a map h : G 2 H 1 H 2 satisyin conditions similar to the Peier litin. The only crossed square we will see in this paper is the entity map on a crossed module H H t t with the structure map G G h : G H H (4) (, h) h h 1. (5) 8

9 3 Main results 3.1 The exact sequence Disc(G (2) ) INN 0 (G (2) ) ΣG (2) We construct the 3-roup INN 0 (G (2) ) or G (2) any strict 2-roup, and show that it plays the role o the universal principal G (2) -bundle in that INN 0 (G (2) ) is equivalent to the trivial 3-roup (hence contractible ). INN 0 (G (2) ) its into the short exact sequence o strict 2-roupos. Disc(G (2) ) 3.2 INN 0 (G (2) ) rom a mappin cone INN 0 (G (2) ) ΣG (2) We show that the 3-roup INN 0 (G (2) ) comes rom a 2-crossed module which is the mappin cone o H G H G H t Id H t, G Id G the entity map o the crossed module (t : H G) which determines G (2). Notice that this harmonizes with the analoous result or Lie 2-alebras discussed in [6]. 4 Inner automorphism n-roups An automorphism o an n-roup G (n) is simply an automorphism o the n- cateory ΣG (n). We want to say that such an automorphism q is inner i it is equivalent to the entity automorphism ΣG (n) Id q ΣG (n). 9

10 A useul way to think o the n-roupo o inner automorphisms is in terms o what we call tanent cateories, a sliht variation o the concept o comma cateories. Tanent cateories in eneral happen to live in interestin exact sequences. In order to be able to talk about these, we irst quickly set up a our deinitions or exact sequences o strict 2-roupos. Remember that we work entirely within the Gray cateory whose objects are strict 2-roupos, whose morphisms are strict 2-unctors, whose 2-morphisms are pseudonatural transormations and whose 3-morphisms are modiications o these. 4.1 Exact sequences o strict 2-roupos Inner automorphism n-roups turn out to live in interestin exact sequences o (n + 1)-roups. Thereore we want to talk about eneralizations o exact sequences o roups to the world o n-roupos. Since or our purposes here only strict 2-roupos matter, we shall be content with just usin a deinition applicable to that case. Deinition 5 (exact sequence o strict 2-roupos). A collection o composable morphisms 1 C 0 1 C 1 n C n o strict 2-cateories C i is called an exact sequence i, as ordinary maps between spaces o 2-morphisms, 0 is injective n is surjective the imae o i is the preimae under i+1 o the collection o all entity 2-morphisms on entity 1-morphisms in Mor 2 (C i+1 ), or all 1 i < n. In order to make this harmonize with our distinction between n-roups G (n) and the correspondin 1-object n-roupos ΣG (n) we add to that Deinition 6 (exact sequences o strict 2-roups). A collection o composable morphisms 1 G 0 1 G 1 n G n o strict 2-roups is called an exact sequence i the correspondin chain ΣG 0 Σ 1 ΣG 1 Σ 1 Σ n ΣG n is an exact sequence o strict 2-roupos. Remark. Ordinary exact sequences o roups are thus precisely correspond to exact sequences o strict 2-roups all whose morphisms are entities. 10

11 4.2 Tanent 2-cateories We present a simple but useul way describe 2-cateories o morphisms with coincin source. Deinition 7 (the point). The point is the strict 2-cateory pt : {} with a sinle object and no nontrivial morphisms. We shall careully distinuish this rom the strict 2-roupo pt : { }, consistin o two objects connected by a 1-isomorphism. The 2-roupo pt miht be called the at point. It is o course equivalent to the point but not isomorphic. We ix one injection i : pt pt once and or all. It is useul to think o morphisms i : : pt C rom the at point to some codomain C as labeled by the correspondin imae o the ordinary point pt C. pt C Deinition 8 (tanent 2-bundle). Given any strict 2-cateory C, we deine its tanent 2-bundle T C Hom 2Cat (pt, C) to be that sub 2-cateory o morphisms rom the at point into C which collapses to a 0-cateory when pulled back alon the ixed inclusion i : pt pt : the 11

12 morphisms h in T C are all those or which pt pt. pt h C pt C The tanent 2-bundle is a disjoint union T C T x C x Obj(C) o tanent 2-cateories at each object x o C. In this way it is a 2-bundle p : T C Disc(C) over the space o objects o C. As beits a tanent bundle, the tanent 2-bundle has a canonical section e Id : Disc(C) T C which sends every object o C to the Identity morphism on it. Example (slice cateories). or C any 1-roupo, i.e. a strict 2-roupo with only entity 2-morphisms, its tanent 1-cateory is the comma cateory T C ((Disc(C) C) Id C ). This is the disjoint union o all co-over cateories on all objects o C T C (a C) a Obj(C) Objects o T C are morphisms : a b in C, and morphisms T C are commutin trianles b h in a h b in C. 12

13 Example (strict tanent 2-roupos). The example which we are mainly interested in is that where C is a strict 2-roupo. or a any object in C, an object o T a C is a morphism q a b. A 1-morphism in T a C is a illed trianle q b a q b in C. inally, a 2-morphism in T a C looks like q b a L. q b The composition o these 2-morphisms is the obvious one. We ive a detailed description or the case the C ΣG (2) in 5. Proposition 1. or any strict 2-cateory C, its tanent 2-bundle T C its into an exact sequence Mor(C) T C C o strict 2-cateories. Here Mor(C) : Disc(Mor(C)) is the 1-cateory o morphisms o C, rearded as a strict 2-cateory with only entity 2-morphisms. Proo. The strict inclusion 2-unctor on the let is ( h ) a h b h b or, : a b any two parallel morphisms in C and h any 2-morphism between them. 13

14 The strict surjection 2-unctor on the riht is q b a K L b b k k L b. The imae o the injection is precisely the preimae under the surjection o the entity 2-morphism on the entity 1-morphisms. This means the sequence is exact. 4.3 Inner automorphisms Oten, or G any roup, inner and outer automorphisms are rearded as sittin in a short exact sequence Inn(G) Aut(G) Out(G) o ordinary roups. But we will ind shortly that we ouht to be reardin the conjuation automorphisms by two roup elements which dier by an element in the center o the roup as dierent inner automorphisms. So adoptin this point o view or ordinary roups, one ets instead the exact sequence Z(G) Inn (G) Aut(G) Out(G). O course this means settin Inn (G) G, which seems to make this step rather ill motivated. But it turns out that this deeneracy o concepts is a coincence o low dimensions and will be lited as we pass to inner automorphisms o hiher roups. irst recall the standard deinitions o center and automorphism o 2-roupos: Deinition 9. Given any strict 2-roupo C, the automorphism 3-roup AUT(C) : Aut 2Cat (C) is the 2-roupo o equivalences on C; the center o C Z(C) : ΣAUT(Id C ) is the (suspended) automorphism 2-roup o the entity on C. 14

15 Example. The automorphism 2-roup o any ordinary roup G (rearded as a 2-roup Disc(G) with only entity morphisms) AUT(G) : AUT(ΣG) is that comin rom the crossed module The center G Ad Aut(G) Z(G) : Z(ΣG) Id Aut(G). o any ordinary roup is indeed the ordinary center o the roup, rearded as a 1-object cateory. To these two standard deinitions, we add the ollowin one, which is supposed to be the proper 2-cateorical eneralization o the concept o inner automorphisms. Deinition 10 (inner automorphisms). Given any strict 2-roupo C, the tanent 2-roupo INN(C) : T IdC (Mor(2Cat)) is, called the 2-roupo o inner automorphisms o C, and as such thouht o as bein equipped with the monoal structure inherited rom End(C). I the transormation startin at the entity is denoted q, it makes ood sense to call the inner automorphism bein the taret o that transormation Ad q : Id C q C. Ad q A bion o this orm is an object in INN(C). The product o two such objects is the horizontal composition o these bions in 2Cat. We shall spell this out in reat detail or the case C ΣG (2) in 5. Proposition 2. or C any strict 2-cateory, we have canonical morphisms Z(C) INN(C) AUT(C) o strict 2-cateories whose composition sends everythin to the entity 2- morphism on the entity 1-morphism on the entity automorphism o C. Moreover, this sits inse the exact sequence rom proposition 1 as where C : Mor(2Cat). Z(C) INN(C) AUT(C) Mor(C) T C C, 15

16 Proo. Recall that a morphism in Z(C) is a transormation o the orm Id C q C. Ad qid This ives the obvious inclusion Z(G) INN(G). The morphism INN(G) AUT(G) maps Id C q C C Ad q C. Ad q Remark. One would now want to deine and construct the cokernel OUT(C) o the morphism INN(C) AUT(C) and then say that Z(C) INN(C) AUT(C) OUT(C) is an exact sequence o 3-roups. But here we do not urther conser this. 4.4 Inner automorphism 3-roups. Now we apply the eneral concept o inner automorphisms to 2-roups. The ollowin deinition just establishes the appropriate shorthand notation. Deinition 11. or G (2) a strict 2-roup, we write INN(G (2) ) : INN(ΣG (2) ) or its 3-roup o inner automorphisms. In eneral this notation could be ambiuous, since one miht want to conser the inner automorphisms o just the 1-roupo underlyin G (2). However, in the present context this will never occur and usin the above deinition makes a couple o expressions more maniestly appear as eneralizations o amiliar ones. Example. that or G an ordinary roup, rearded as a discrete 2-roup, one inds INN(G) : T IdΣG (ncat) T (ΣG) 16

17 is the codiscrete roupo over the elements o G. Its nature as a roupo is maniest rom its realization as INN(G) T (ΣG). But it is also a (strict) 2-roup. The monoal structure is that comin rom its realization as INN(G) : T IdΣG (ncat). The crossed module correspondin to this strict 2-roup is G Id G Ad Aut(G). The main point o interest or us is the eneralization o this act to strict 2-roups. One issue that one needs to be aware o then is that the above isomorphism T IdΣG (ncat) T (ΣG) becomes a mere inclusion. Proposition 3. or G (2) any strict 2-roup, we have an inclusion T ΣG (2) T IdΣG(2) (Mor(2Cat)) o strict 2-roupos. This realizes T ΣG (2) as a sub 2-roupo o T IdΣG(2) (Mor(2Cat)). Proo. The inclusion is essentially ixed by its action on objects: we deine that an object in T ΣG (2), which is a morphism q in ΣG, is sent to the conjuation automorphism Ad q : ΣG (2) ΣG (2) h q 1 h q The transormation Id ΣG (2) q ΣG (2). Ad q 17

18 connectin this to the entity is iven by the component map ( ) Id q q 1 q q. In eneral one could conser transormations whose component maps involve here a non-entity 2-morphism. The inclusion we are describin picks out excactly those transormations whose component map only involves entity 2-morphisms. The crucial point to realize now is the orm o the component maps o morphisms Ad q q Id ΣG(2) Ad q Ad q in T IdΣG(2) (Mor(2Cat)). The correspondin component map equation is q q Id q Ad q Ad () Ad q q q. Id q Ad q 18

19 Solvin this or Ad shows that this is iven by conjuation q 1 q Ad : ( ) q 1 q with a morphism in T (ΣG (2) ). And each such morphism in T (ΣG (2) ) yields a morphism in T IdΣG(2) (Mor(2Cat)) this way. inally, 2-morphisms in T IdΣG(2) (Mor(2Cat)) between these morphisms ΣG (2) come rom component maps Ad q Ad L Ad q Ad K ΣG (2) L Mor 2 (ΣG (2) ). k A suicient condition or these component maps to solve the required condition or modiications o pseudonatural transormations is that they make q K L k 2-commute. But this deines a 2-morphism in T ΣG (2). And each such 2- morphism in T (ΣG (2) ) yields a 2-morphism in T IdΣG(2) (Mor(2Cat)) this way. The crucial point is that by the embeddin T ΣG (2) T IdΣG(2) (Mor(2Cat)) 19

20 the ormer 2-cateory inherits the monoal structure o the latter and hence becomes a 3-roup in its own riht. This 3-roup is the object o interest here. 5 The 3-roup INN 0 (G (2) ) Deinition 12 (INN 0 (G (2) )). or G (2) any strict 2-roup, the 3-roup INN 0 (G (2) ) is, as a 2-roupo, iven by INN 0 (G (2) ) : T ΣG (2) and equipped with the monoal structure inherited rom the embeddin o proposition 3. We now describe INN 0 (G (2) ) or G (2) comin rom the crossed module H t G α Aut(H) in more detail, in particular spellin out the monoal structure. We extract the operations in the crossed module correspondin to the various compositions in INN 0 (G (2) ) and then inally entiy the 2-crossed module encoded by this. 5.1 Objects The objects o INN 0 (G (2) ) are exactly the objects o G (2), hence the elements o G: Obj(INN(G (2) )) G. The product o two objects in INN(G (2) ) is just the product in G. 5.2 Morphisms The morphisms h in INN(G (2) ) are Mor(INN 0 (G (2) )) h,, h G, H h t( ) {(, ; ), G, H}. 20

21 5.2.1 Composition The composition o two such morphisms q 1 1 q 2 q 1 2. is in terms o roup labels iven by q q Product Horizontal composition o automorphisms ΣG (2) ΣG (2) ives the product in the 3-roup INN(G (2) ) Let whiskerin o pseudonatural transormations Ad q ΣG (2) Ad ΣG (2) Ad ΣG (2) Ad q amounts to the operation q q q on the correspondin trianles. 21

22 Riht whiskerin o pseudonatural transormations Ad q ΣG (2) Ad Ad q ΣG (2) ΣG (2) Ad q amounts to the operation q q 1 q 1 on the correspondin trianles. Since 2Cat is a Gray cateory, the horizontal composition o pseudonatural transormations Ad q1 Ad q2 ΣG (2) 1 ΣG (2) 2 Ad ΣG (2) Ad q 1 Ad q 2 is ambiuous. We shall aree to read this as ΣG (2) Ad q1 Ad 1 ΣG (2) Ad q2 ΣG (2) Ad q2. Ad q 1 ΣG (2) ΣG (2) Ad2 ΣG (2) Ad q 1 Ad q 2 22

23 The correspondin operation on trianles labelled in the crossed module is q 2 q 1 q 1 q 2 1q q q q 2q 1 q 21 2 q 2 1q q q 21 2q 2 1q 1 2 The non-entitcal isomorphism which relates this to the other possible way to evaluate the horizontal composition o pseudonatural transormations ives rise to the Peier litin o the correspondin 2-crossed module. This is discussed in

24 5.3 2-Morphisms The 2-morphisms in INN 0 (G (2) ) are iven by diarams q K L. k In terms o the roup labels this means that L H satisies L K 1. (6) Composition The horizontal composition o 2-morphisms in INN 0 (G (2) ) is iven by q 1 1 L 1 k 1 q 1 1 K 1 q 2 G J 2 1 L 2L1 2 k 2k 1 G 2 k 2 1, J K 2 2 K 1 2 K 2 2 L 2 k 2 and vertical composition by q q L 1 L 2 L 2L 1 (Notice that these compositions do o horizontally and vertically, respectively, once we rotate such that the bions have the standard orientation.) 24

25 Notice that whiskerin alon 1-morphisms ΣG (2) Ad q1 Ad q L Ad q Ad q2 ΣG (2) acts on the component maps as p G q p K L k K L k G, K K k G and q p K L k K L k G, K G k K. G 25

26 There is one more type o whiskerin possible with 2-morphisms, Ad q ΣG (2) ΣG (2) k L ΣG (2) Ad q, ΣG (2) which acts in the ollowin way on the components: q pq G p K Ad p L kad p, K L k where p G, K K kp G, L kp G 1 L p G. and p qp K L k q K Ad q q L Ad qk, G 26

27 where G q, K G q K. An important case o this is: K L k. t( ) L 1 L k Product The whiskerin alon objects ΣG (2) Ad q ΣG (2) k L ΣG (2) Ad q ΣG (2) ives the product o objects with 2-morphisms in the 3-roup INN(G (2) ). Its action on 2-morphisms, which we have already disucssed, extends in a simple way to 3-morphisms: let whiskerin alon an object acts as q q K L k K L, k 27

28 while riht whiskerin alon an object acts as q K L k q K k 1 L. k 1 To calculate the product o a pair o 2-morphisms, we use the act that a 2-morphism is uniquely determined by its source and taret. q 1 q 2q 1 2Ad q2 1 1 K 1 1 L 1 k 1 q 2 K L k 2Ad q2 k 1, 2 K 2 2 L 2 k 2 with 2 2q 2 1 K K 2 k 2q 2 K 1 L L 2 2q 2 L 1 6 Properties o INN 0 (G (2) ) 6.1 Structure morphisms We have deined Σ(INN 0 (G (2) )) essentially as a sub 3-cateory o 2Cat. The latter is a Gray cateory, in that it is a 3-cateory which is strict except or the exchane law or composition o 2-morphisms. Accordinly, also Σ(INN 0 (G (2) )) is strict except or the exchane law or 2-morphisms. This means that as a mere 2-roupo (orettin the monoal structure) INN 0 (G (2) ) is strict Strictness as a 2-roupo Proposition 4. The underlyin 2-roupo o INN 0 (G (2) ) is strict. 28

29 Proo. This ollows rom the rules or horizontal and vertical composition o 2-morphisms in INN 0 (G (2) ) displayed in and the act that G (2) itsel is a strict 2-roup, by assumption. But the product 2-unctor on INN 0 (G (2) ) respects horizontal composition in INN 0 (G (2) ) only weakly. In the lanuae o 2-roups, this corresponds to a ailure o the Peier entity 6.2 Trivializability Proposition 5. The 2-roupo INN 0 (G (2) ) is connected, π 0 (INN(G 1 )) 1. Proo. or any two objects q and q there is the morphism q q q q 1 Proposition 6. The Hom-roupos o the 2-cateory INN 0 (G (2) ) are codiscrete, meanin that they have precisely one morphism or every ordered pair o objects. Proo. By equation (6) there is at most one 2-morphism between any parallel pair o morphisms in INN 0 (G (2) ). or there to be any such 2-morphism at all, the two roup elements and k in the diaram above (6) have to satisy k 1 im(t). But by usin the source-taret matchin condition or and K one readily sees that this is always the case. Theorem 1. The 3-roup INN 0 (G (2) ) is equivalent to the trivial 3-roup. I G (2) is a Lie 2-roup, then INN 0 (G (2) ) is equivalent to the trivial Lie 3-roup even as a Lie 3-roup. Proo. Equivalence o 3-roups G (3), G (3) is, by deinition, that o the correspondin 1-object 3-roupos ΣG (3), ΣG (3). or showin equivalence with the trivial 3-roup, it suices to exhibit a pseudonatural transormation o 3- unctors Σ(INN0(G (2) )) I Σ(INN0(G (2) )), where I Σ(INN0(G (2) )) sends everythin to the entity on the sinle object o ΣINN 0 (G (2) ). Such a transormation is obtained by sendin the sinle object 29

30 to the entity 1-morphism on that object and sendin any 1-morphism q to the 2-morphism q rom prop 5. By prop 6 this implies the existence o a unique assinment o a 3-morphism to any 2-morphism such that we do indeed obtain the component map o a pseudonatural transormation o 3-unctors. By construction, this is clearly smooth when G (2) is Lie. 6.3 Universality Theorem 2. We have a short exact sequence o strict 2-roupos Disc(G (2) ) INN 0 (G (2) ) ΣG (2). Proo. This is just proposition 1, ater noticin that Mor(ΣG (2) ) G (2). So the strict inclusion 2-unctor on the let is ( h ) h h, while the strict surjection 2-unctor on the riht is q K L k k L. 6.4 The correspondin 2-crossed module We now extract the structure o a 2-crossed module rom INN 0 (G (2) ). irst, 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 ) be subroups o the 1- and 2-morphisms o INN 0 (G (2) ) respectively. 30

31 Proposition 7. The roup o 1-morphisms in INN 0 (G (2) ) startin at the entity object orm the semirect product roup Mor I 1 G H under the entitication Id (, ) in that Id Id 1 1 Id Proo. Use composition in ΣG (2). We have the obvious roup homomorphism which is just the restriction o the taret map 1 : Mor I 1 Obj : Obj(INN 0 (G (2) )) iven by Id 1 : t( ). This and the ollowin constructions are to be compared with deinition 4. There is an obvious action on Mor I 1 : 31

32 1 1 (7) This action almost ives us a crossed module Mor I 1 Obj. But not quite, since the Peier entity holds only up to 3- isomorphism. To see this, let 1 H h t(h)h. or the Peier entity to hold we need the action (7) to be equal to the adjoint action o the 2-cell (h, H; ). To see that this ails, irst notice that the inverse o the approriate 2-cell consered as an element in the roup Mor I 1 is H h t(h)h Thereore the conjuation is 1 h 1 H 1 h 1 t(h) 1 h

33 h 1 H 1 1 h 1 t( ) H h h 1 H 1 1 h 1 t( ) H h h 1 h 1 H 1 1 h 1 H h h t( ) H 1h hh 1 H t( ) 1 hh 1 (8) 1 (9) t( ) 1 33

34 Thouh the Peier entity does not hold, both actions ive rise to 2-cells with the same source and taret, and hence deine a 3-cell P. Denote the 2-cell (8) by (c, C; ) and the 2-cell (9) by (a, A; ) (or conjuation and action respectively). C A c P a Then P A 1 C 1 ( H h hh 1 H 1) 1 ( t(h)h H hh 1 H 1) 1 ( H hh 1 H 1) H hh 1 H 1 However, what we really want is the Peier litin, which will be a 3-cell with source the entity 2-cell. Hence, Proposition 8. The roup o 2-morphisms in INN 0 (G (2) ) startin at the entity arrow on the entity object orm the roup Mor I 2 H under the entitication L t(l) L in that 34

35 L 1 t(l 1) L 2L 1 t(l 2L 1) L 2 t(l 2) Proo. Use the multiplication o 2-morphisms. So, we whisker the 3-cell (P ; a, A; ) above with the inverse o (a, A; ): a 1 A 1 a 1 P, t(p )ca 1 A C a P c and the back ace is necessarily P 1. Deinition 13 (Peier litin). Deine the map {, } : Mor I 1 MorI 1 MorI 2 by H h, P, P H hh 1 H 1 t(p ) Now deine the homomorphism 2 : Mor I 2 MorI 1 35

36 by Id 2 : L t(l) L 1 t(l), Id which is aain the restriction o the taret map. Note there is an action o Obj on Mor I 2: L t(l) L t(l) t(l) 1 (10) Clearly 2 1 is the constant map at the entity, and im 2 is a normal subroup o ker 1, so L t( L) Mor I 2 2 Mor I 1 1 Obj (11) is a sequence. We let the action o Obj on the other two roups be as described above in (7) and (10), and the maps 2 and 1 are clearly equivariant or this action. Proposition 9. The map {, } does indeed satisy the properties o a Peier litin, and (11) is a 2- crossed module. Proo. The irst condition holds by deinition, the second and the last one are easy to check. The others are tedious. It is easy, usin the crossed module properties o H G, to calculate that the actions o Mor I 1 on Mor I 2 as deined rom INN 0 (G (2) ) and as deined via {, } are the same. Since im 2 ker 1, 2 is injective and 1 is onto, this shows that (11) has trivial homoloy and proves us with another proo that INN 0 (G (2) ) is contractible. 36

37 6.4.1 Relation to the mappin cone o H G Given a crossed square L M u v N with structure map h : N M L, Conduché [3]ives the Peier litin o the mappin cone L N M P as P {(, h), (k, l)} h(k 1, h). Recall rom 2.3 that the entity map on t : H G is a crossed square with h(, h) h h 1, so the mappin cone is a 2-crossed module H 2 G H 1 G, where and with Peier litin d 2 (h) (t(h), h 1 ), d 1 (, h) t(h), {( 1, h 1 ), ( 2, h 2 )} h h 1 1. which is what we ound or INN 0 (G (2) ). More precisely, Deinition 14. A morphism ψ o 2-crossed modules is a map o the underlyin complexes 2 L 1 1 M 1 N 1 ψ L ψ M ψ N L 2 2 M 2 1 N 2 such that ψ L, ψ M and ψ N are equivariant or the N- and M-actions, and {ψ M ( ), ψ M ( )} 2 ψ L ({, } 1 ). 37

38 Usin propositions 7 and 8, we have a map Mor I 2 2 Mor I 1 1 Obj H d 2 G H d1 G and the actions and Peier litin aree, so Proposition 10. The 2-crossed module associated to INN 0 (G ( 2)) is isomorphic to the mappin cone o the entity map on the crossed module associated to G (2). 7 Universal n-bundles In order to put the relevance o the 3-roup INN 0 (G (2) ) in perspective, we urther illuminate our statement, 3.1, that INN 0 (G (2) ) plays the role o the universal G (2) -bundle. An exhaustive discussion will be iven elsewhere. 7.1 Universal 1-bundles in terms o INN(G) Let π : Y X be a ood cover o a space X and write Y [2] : Y X Y or the correspondin roupo. Deinition 15 (G-cocycles). A G-(1-)cocycle on X is a unctor : Y [2] ΣG. This unctor can be understood as arisin rom a choice π P t Y G o trivialization o a principal riht G-bundle P X (which is essentially just a map to G) as : π 2t π 1t 1, by noticin that G-equivariant isomorphisms are in bijection with elements o G actin rom the let. G G (x, y) : h (x, y)h 38

39 Observation 1 (G-bundles as morphisms o sequences o roupos). Given a G-cocycle on X as above, its pullback alon the exact sequence which we write as Disc(G) INN(G) ΣG, Y Disc(G) Y [2] INN(G) [2] Y X, Disc(G) INN(G) ΣG {} produces the bundle o roupos Y [2] INN(G) Y [2] which plays the role o the total space o the G-bundle classiied by. This should be compared with the simplicial constructions described, or instance, in [4]. Remark. Usin the act that INN(G) is a 2-roup, and usin the injection Disc(G) INN(G) we naturally obtain the G-action on Y [2] INN(G). Remark. Notice that this is closely related to the interated Atiyah sequence AdP P G P X X o roupos over X X comin rom the G-principal bundle P X: AdP P G P X X Y Disc(G) Y [2] INN(G) [2] Y X. Disc(G) INN(G) ΣG {} We now make precise in which sense, in turn, Y [2] INN(G) plays the role o the total space o the G-bundle characterized by the cocycle. 39

40 To reobtain the G-bundle P X rom the roupo Y [2] INN(G) we orm the pushout o Y [2] INN(G) taret Y G. (12) source Y G Proposition 11. I is the cocycle classiyin a G-bundle P on X, then the pushout o 12 is (up to isomorphism) that very G-bundle P. Proo. Conser the square Y [2] INN(G) taret Y G source t 1 π P, Y G t 1 π P P where t : π P Y G is the local trivialization o P which ives rise to the transition unction. Then the diaram commutes by the very deinition o. Since t is an isomorphism and since π P P is locally an isomorphism, it ollows that this is the universal pushout. 7.2 Universal 2-bundles in terms o INN 0 (G (2) ) Now let G (2) be any strict 2-roup. Let Y [3] be the 2-roupo whose 2- morphisms are triples o lits to Y o points in X. A principal G (2) -2-bundle [9, 10] has local trivializations characterized by 2-unctors : Y [3] ΣG (2). Deinition 16 (G (2) -cocycles). A G (2) -(2-)cocycle on X is a 2-unctor : Y [3] ΣG (2). (Instead o 2-unctors on Y [3] one could use pseudo unctors on Y [2].) As beore, we can pull these back alon our exact sequence o 2-roupos 3.1 Disc(G (2) ) INN(G (2) ) ΣG (2) 40

41 to obtain Y Disc(G (2) ) Y [3] INN(G (2) ) [3] Y X. Disc(G (2) ) INN(G (2) ) ΣG (2) {} o We reconstruct the total 2-space o the 2-bundle by ormin the weak pushout Y [3] INN(G (2) ) taret Y G (2) source. (13) Y G (2) Here source and taret are deined relative to the inclusion Y Disc(G (2) ) Y [2] INN(G (2) ). This means that or a iven 1-morphism q 1 :(x,y) x q 2 y in Y [3] INN(G (2) ) (or any x, y Y with π(x) π(y) and or (x, y) the correspondin component o the iven 2-cocycle) which we may equivalently rewrite as q 1 x 1 1 q 2 y 41

42 the source in this sense is q 1 x 1 1 q 2, rearded as a morphism in Y G (2), while the taret is q 1 q 2 y rearded as a morphism in Y G (2). This way the transition unction (x, y) acts on the copies o G (2) which appear as the trivialized ibers o the G (2) -bundle. Bartels [9][proo o prop. 22] ives a reconstruction o total space o principal 2-bundle rom their 2-cocycles which is closely related to Y [3] INN(G (2) ). We end by sayin that INN(G (2) ) is a special case o a much more eneral construction in 2-bundles [11], and that the results o this paper will, in a uture paper, be connected to similar results usin simplicial roups and simplicial universal bundles. Reerences [1] J. Baez, A. Lauda, Hiher-Dimensional Alebra V: 2-Groups, Th. and App. o Cat. 12 (2004), , available as math/ [2] D. Conduché, Unpublished letter to R. Brown and J.-L. Loday, 198? [3] D. Conduché, Simplicial crossed modules and mappin cones, Georian Mathematical Journal [4] B. Jurčo, Crossed Module Bundle Gerbes; Classiication, Strin Group and Dierential Geometry, available as math/ [5] J.-L. Loday, Spaces with initely many homotopy roups, JPAA [6] U. Schreiber, J. Stashe, in preparation [7] D. Guin-Waléry and J.-L. Loday, Obstructions à l Excision en K-théorie Alébrique, Spriner Lecture Notes in Math., 854, (1981),

43 [8] B.Day, R. Street, Monoal bicateories and Hop alebros, Adv. Math 129, (1997) [9] T. Bartels, 2-Bundles, PhD Thesis, available as math/ [10] I. Baković, Biroupo bitorsors, PhD Thesis [11] D. M. Roberts, work in proress 43