The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module
aa r X i v : . [ m a t h . C T ] J a n The fundamental Gray 3-groupoid of a smoothmanifold and local 3-dimensional holonomy basedon a 2-crossed module
Jo˜ao Faria Martins
Departamento de Matem´aticaFaculdade de Ciˆencias e Tecnologia, Universidade Nova de LisboaQuinta da Torre, 2829-516 Caparica, Portugal
Roger Picken
Departamento de Matem´atica,Instituto Superior T´ecnico, TU LisbonAv. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]
November 2, 2018
Abstract
We define the thin fundamental Gray 3-groupoid S ( M ) of a smoothmanifold M and define (by using differential geometric data) 3-dimensionalholonomies, to be smooth strict Gray 3-groupoid maps S ( M ) → C ( H ),where H is a 2-crossed module of Lie groups and C ( H ) is the Gray 3-groupoid naturally constructed from H . As an application, we defineWilson 3-sphere observables. Key words and phrases:
Higher Gauge Theory; 3-dimensional holonomy;2-crossed module; crossed square, Gray 3-groupoid; Wilson 3-sphere.
Introduction
This is the first of a series of papers on (Gray) 3-bundles and 3-dimensionalholonomy (also called parallel transport), aimed at categorifying the notionof 2-bundles and non-abelian gerbes with connection, and their 2-dimensionalparallel transport; see [BS, SW3, BrMe, FMP2, H, MP, ACG, Wo, Bar]. Themain purpose here is to clarify the notion of 3-dimensional holonomy based on aLie 2-crossed module, extending some of the constructions in [SW1, SW2, SW3,1MP1] to the fundamental Gray 3-groupoid of a smooth manifold (which wewill construct in this article). For some related work see [SSS, J1, J2, Br].The definition of a Gray 3-groupoid appears, for example, in [Cr, KP, GPS].The first main result of this article concerns the construction of the fundamen-tal (semi-strict) Gray 3-groupoid S ( M ) of a smooth manifold M , which is notobvious. The main innovation for defining S ( M ) rests on the notion of lam-inated rank-2 homotopy, a weakening of the notion of rank-2 homotopy (see[MP, FMP1, SW3, BS]), which makes 2-dimensional holonomy based on a pre-crossed module, as opposed to a crossed module, invariant. This also permitsus to define the fundamental pre-crossed module of a smooth manifold M .The definition of a 2-crossed module is due to Conduch´e; see [Co] and [KP,MuPo, Po]. It is a complex of (not necessarily abelian) groups L δ −→ E ∂ −→ G together with left actions ⊲ by automorphisms of G on L and E , and a G equivariant function { , } : E × E → L (called the Peiffer lifting), satisfyingcertain properties (Definition 3). The extension to 2-crossed modules of Liegroups is the obvious one. It is well known that Gray 3-groupoids are modelledby 2-crossed modules (of groupoids); see [KP]. We provide a detailed descriptionof this connection in subsection 1.2.5.In the light of this connection, given a Lie 2-crossed module H , it is naturalto define a (local) 3-dimensional holonomy as being a smooth (strict) Gray3-functor S ( M ) → C ( H ), where C ( H ) is the Gray 3-groupoid, with a singleobject, constructed out of H .A differential 2-crossed module, also called a 2-crossed module of Lie alge-bras, see [AA, E], is given by a complex of Lie algebras: l δ −→ e ∂ −→ g together with left actions ⊲ by derivations of g on l and e , and a g -equivariantbilinear map { , } : e × e → l (called the Peiffer lifting), satisfying appropriateconditions (Definition 8). Any 2-crossed module of Lie groups H defines, in thenatural way, a differential 2-crossed module H , and this assignment is functorial.The second main result of this paper concerns how to define 3-dimensionalholonomies S ( M ) → C ( H ) by picking Lie-algebra valued differential forms ω ∈A ( M, g ), m ∈ A ( M, e ) and θ ∈ A ( M, l ), satisfying ∂ ( m ) = dω + [ ω, ω ] . = Ω,the curvature of ω , and δ ( θ ) = dm + ω ∧ ⊲ m . = M (see the appendix for the ∧ ⊲ notation), the exterior covariant derivative of m , called the 2-curvature 3-formof ( ω, m ). This is analogous to the construction in [BS, FMP1, SW2]. A lot ofthe proofs will make use of Chen integrals in the loop space [Ch], which is theapproach taken in [BS].The 3-curvature 4-form Θ of the triple ( ω, m, θ ), satisfying the above equa-tions, is defined as Θ = dθ + ω ∧ ⊲ θ − m ∧ { , } m (see the appendix for the ∧ { , } notation). We will prove a relation between 3-curvature and 3-dimensionalholonomy, completely analogous to that for principal G -bundles and 2-bundles2ith a structural crossed module of groups, including an Ambrose-Singer typetheorem for the triple ( ω, m, θ ). This will also prove that the 3-dimensionalholonomy is invariant under rank-3 homotopy, as long as it restricts to a lami-nated rank-2 homotopy on the boundary (see Definition 23).As in the case of 2-bundles, the 1-, 2- and 3-gauge transformations wouldbe better understood by passing to the notion of a Gray triple groupoid, which(to keep the size of this paper within limits) we will analyse in a future article,where we will also address the general definition of a Gray 3-bundle, and describethe corresponding 3-dimensional parallel transport, categorifying the results of[SW3, FMP2, MP, P].The 3-dimensional holonomy which we define in this article can be associatedto embedded oriented 3-spheres S in a manifold M , yielding a Wilson 3-sphereobservable W ( S, ω, m, θ ) ∈ ker δ ⊂ L independent of the parametrisation of S chosen, up to acting by elements of G . This will be a corollary of the invariance ofthe 3-dimensional holonomy under rank-3 homotopy, with laminated boundary.Considering the Lie 2-crossed module given by a finite type chain complex ofvector spaces (see [KP] and 1.2.6), the construction in this article will describe(locally and as a Gray 3-functor) the first three instances of the holonomy of arepresentation up to homotopy; see [AC]. Notice however that the constructionin this article is valid for any Lie group 2-crossed module. We expect thatto describe all instances of the ω -parallel transport of a representation up tohomotopy one will need Gray ω -groupoids and 2-crossed complexes. We hopeto address this in a future paper. Contents S ( M ) of a smoothmanifold M . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A (Lie group) pre-crossed module G = ( ∂ : E → G, ⊲ ) is given by a Lie group map ∂ : E → G together with a mooth left action ⊲ of G on E by automorphisms such that: ∂ ( g ⊲ e ) = g∂ ( e ) g − , for each g ∈ G and e ∈ E. The Peiffer commutators in a pre-crossed module are defined as h e, f i = ef e − (cid:0) ∂ ( e ) ⊲ f − (cid:1) , where e, f ∈ E. A pre-crossed module is said to be a crossed module if all of its Peiffer commu-tators are trivial, which is to say that: ∂ ( e ) ⊲ f = ef e − for each e, f ∈ E. Note that the map ( e, f ) ∈ E × E
7→ h e, f i ∈ E , called the Peiffer pairing, is G -equivariant: g ⊲ h e, f i = h g ⊲ e, g ⊲ f i , for each e, f ∈ E and g ∈ G. Moreover h e, f i = 1 if either e or f is 1. Therefore, the second differential ofthe Peiffer pairing defines a bilinear map h , i : e × e → e , where e = T E is theLie algebra of E . The infinitesimal counterpart of a Lie pre-crossed module is a differential pre-crossed module, also called a pre-crossed module of Lie algebras.
Definition 2 (Differential pre-crossed module)
A differential pre-crossedmodule G = ( ∂ : e → g , ⊲ ) is given by a Lie algebra map ∂ : e → g together witha left action ⊲ of g on e by derivations such that:1. ∂ ( X ⊲ v ) = [
X, ∂ ( v )] , for each v ∈ e and X ∈ g .Note that the map ( X, v ) ∈ g × e X ⊲ v ∈ e is necessarily bilinear. In additionwe have X ⊲ [ v, w ] = [ X ⊲ v, w ] + [ v, X ⊲ w ] , for each v, w ∈ e and X ∈ g (this expresses the condition that g acts on e on the left by derivations.) Therefore if G is a Lie pre-crossed module then the induced structure on Liealgebras is a differential pre-crossed module G . Reciprocally, if G is a differen-tial pre-crossed module there exists a unique Lie pre-crossed module of simplyconnected Lie groups G whose differential form is G (any action by derivationsof a Lie algebra on another Lie algebra can always be lifted to an action byautomorphisms, considering the corresponding simply connected Lie groups).In a differential pre-crossed module, the Peiffer commutators are defined as: h u, v i = [ u, v ] − ∂ ( u ) ⊲ v ∈ e ; u, v ∈ e . u, v ) ∈ e × e
7→ h u, v i ∈ e (the Peiffer pairing) is bilinear (though notnecessarily symmetric), and coincides with the second differential of the Peifferpairing in E . In addition the Peiffer pairing is g -equivariant: X ⊲ h u, v i = h X ⊲ u, v i + h u, X ⊲ v i , for each X ∈ g , and u, v ∈ e . A differential pre-crossed module is said to be a differential crossed module ifall of its Peiffer commutators vanish, which is to say that: ∂ ( u ) ⊲ v = [ u, v ]; for each u, v ∈ e . For more on crossed modules of groups, Lie groups and Lie algebras, see [B, BL,BC, BH1, BHS, FM].
We follow the conventions of [Co] for the definition of a 2-crossed module. Seealso [MuPo, KP, BG, Po, RS].
Definition 3 (2-crossed module of Lie-groups)
A 2-crossed module (of Liegroups) is given by a complex of Lie groups: L δ −→ E ∂ −→ G together with smooth left actions ⊲ by automorphisms of G on L and E (andon G by conjugation), and a G -equivariant smooth function { , } : E × E → L (called the Peiffer lifting). As before G -equivariance means a ⊲ { e, f } = { a ⊲ e, a ⊲ f } , for each a ∈ G and e, f ∈ E. These are to satisfy:1. L δ −→ E ∂ −→ G is a complex of G -modules (in other words ∂ and δ are G -equivariant and ∂ ◦ δ = 1 .)2. δ ( { e, f } ) = h e, f i , for each e, f ∈ E . Recall h e, f i = ef e − ∂ ( e ) ⊲ f − . [ l, k ] = { δ ( l ) , δ ( k ) } , for each l, k ∈ L . Here [ l, k ] = lkl − k − .4. { ef, g } = (cid:8) e, f gf − (cid:9) ∂ ( e ) ⊲ { f, g } , for each e, f, g ∈ E .5. { e, f g } = { e, f } { e, g } n h e, g i − , ∂ ( e ) ⊲ f o , where e, f, g ∈ E .6. { δ ( l ) , e } { e, δ ( l ) } = l ( ∂ ( e ) ⊲ l − ) , for each e ∈ E and l ∈ L . e ⊲ ′ l = l (cid:8) δ ( l ) − , e (cid:9) , where l ∈ L and e ∈ E. (1)It follows from the previous axioms that ⊲ ′ is a left action of E on L by au-tomorphisms (this is not entirely immediate; a proof is in [Co, BG]). There itis also shown that, together with the map δ : L → M , this defines a crossedmodule. This will be of prime importance later. For the time being, note thatcondition 5. yields: { e, f g } = { e, f } ( ∂ ( e ) ⊲ f ) ⊲ ′ { e, g } = ( δ ( { e, f } ) ⊲ ′ ( ∂ ( e ) ⊲ f ) ⊲ ′ { e, g } ) { e, f } = (cid:0) ( ef e − ) ⊲ ′ { e, g } (cid:1) { e, f } . (2)This proves that the definition of 2-crossed modules appearing here is equivalentto the one of [KP, Po]. The following simple lemma is useful here and later on: Lemma 4
In a 2-crossed module H = ( L δ −→ E ∂ −→ G, ⊲, { , } ) we have { E , e } = { e, E } = 1 L , for each e ∈ E . Proof.
Apply axioms 4 and 5 of the definition of a 2-crossed module to { E E , e } and { e, E E } .Using this together with equations 4. and 5. of the definition of a 2-crossedmodule as well as equation (2) it follows (compare also with equation (9)): Lemma 5
For each e, f ∈ E we have: { e, f } − = ∂ ( e ) ⊲ { e − , ef e − } , { e, f } − = ( ef e − ) ⊲ ′ { e, f − } , { e, f } − = ( ∂ ( e ) ⊲ f ) ⊲ ′ { e, f − } . By using condition 6. of the definition of a 2-crossed module it follows that: ∂ ( e ) ⊲ l = ( e ⊲ ′ l ) { e, δ ( l ) − } . (3)Note: (cid:0) ∂ ( e ) ⊲ f (cid:1) ⊲ ′ (cid:0) ∂ ( e ) ⊲ l (cid:1) . = ( ∂ ( e ) ⊲ l ) { ∂ ( e ) ⊲ δ ( l − ) , ∂ ( e ) ⊲ f } = ( ∂ ( e ) ⊲ l ) ∂ ( e ) ⊲ { δ ( l − ) , f } = ∂ ( e ) ⊲ ( f ⊲ ′ l ) , where we have used the fact that the Peiffer lifting { , } is G -equivariant andthat G acts on L by automorphisms. We thus have the following identity foreach e, f ∈ E and l ∈ L : (cid:0) ∂ ( e ) ⊲ f (cid:1) ⊲ ′ (cid:0) ∂ ( e ) ⊲ l (cid:1) = ∂ ( e ) ⊲ ( f ⊲ ′ l ) . (4)7e also have:( e ⊲ ′ { f, g } ) { e, ∂ ( f ) ⊲ g } = ∂ ( e ) ⊲ { f, g }{ e, ( ∂ ( f ) ⊲ g ) f g − f − } − { e, ∂ ( f ) ⊲ g } = ∂ ( e ) ⊲ { f, g } ( ∂ ( e ) ⊲ ∂ ( f ) ⊲ g ) ⊲ ′ { e, f g − f − } − = ∂ ( e ) ⊲ { f, g } (cid:0) ∂ ( e ) ⊲ ∂ ( f ) ⊲ g ) ⊲ ′ ( ∂ ( e ) ⊲ ( f g − f − ) (cid:1) ⊲ ′ { e, f gf − } = ∂ ( e ) ⊲ { f, g } (cid:0) ∂ ( e ) ⊲ δ { f, g } − (cid:1) ⊲ ′ { e, f gf − } = { e, f gf − } ∂ ( e ) ⊲ { f, g } = { ef, g } , using equation (3) in the first step, condition 5. of the definition of a 2-crossedmodule in the second, the third equation of Lemma 5 in the third. The penul-timate step follows from the fact that ( δ : L → E, ⊲ ′ ) is a crossed module.Let us display all equations we have proved (note that equation (5) appearsin [Co, page 162]): Lemma 6
In a 2-crossed module we have, for each e, f, g ∈ E : { ef, g } = ( e ⊲ ′ { f, g } ) { e, ∂ ( f ) ⊲ g } , (5) { ef, g } = (cid:8) e, f gf − (cid:9) ∂ ( e ) ⊲ { f, g } , (6) and { e, f g } = { e, f } ( ∂ ( e ) ⊲ f ) ⊲ ′ { e, g } , (7) { e, f g } = (cid:0) ( ef e − ) ⊲ ′ { e, g } (cid:1) { e, f } . (8)In particular, by using the first equation, we have (compare with Lemma 5): { e, f } − = e ⊲ ′ { e − , ∂ ( e ) ⊲ f } . (9)Consider the totally intransitive groupoid with morphisms G × L and objects G , the source and target maps being given by ( g, l ) g , and identity as g ( g, L ). As composition, we take the group multiplication in L . Consider alsothe groupoid with objects G and morphisms G × E , and source and target givenby ( g, e ) g and ( g, e ) ∂ ( e ) − g , respectively. The composition is( g ( g,e ) −−−→ ∂ ( e ) − g ( ∂ ( e ) − g,f ) −−−−−−−→ ∂ ( f ) − ∂ ( e ) − g ) = ( g ( g,ef ) −−−−→ ∂ ( ef ) − g ) . The map δ : G × L → G × E defined as δ ( g, k ) = ( g, δ ( k )) is a groupoid mapand together with the left action by automorphisms of G × E on G × L :( g, e ) ⊲ ′ ( ∂ ( e ) − g, l ) = ( g, e ⊲ ′ l ) , where g ∈ G, e ∈ E and l ∈ L defines a crossed module of groupoids; see [B1, BH1, BHS, FMPo, No], a partof what is called a braided regular crossed module in [BG].8 xample 7 The simplest non-trivial example of a Lie 2-crossed module is prob-ably the following one. Let G be a Lie group. Consider G δ −→ G ⋊ ad G ∂ −→ G, where the multiplication in the semidirect product is: ( e, f )( g, h ) = ( ef gf − , f h ) and also ∂ ( g, h ) = gh . In addition put δ ( g ) = ( g − , g ) . Here e, f, g, h ∈ G .The action of G on G is the adjoint action and on G ⋊ ad G is g ⊲ ( a, b ) =( gag − , gbg − ) . The Peiffer lifting is: { ( a, b ) , ( c, d ) } = [ bdb − , a ] , where [ x, y ] = xyx − y − . Therefore ∂ ( a, b ) ⊲ g = ( ab ) g ( ab ) − and ( a, b ) ⊲ ′ g = bgb − . To transport 2-crossed modules of Lie groups to the Lie algebras world we useLemma 4. This tells us that the second differential of the Peiffer lifting of H defines a bilinear map { , } : e × e → l , which is g -equivariant: X ⊲ { u, v } = { X ⊲ u, v } + { u, X ⊲ v } , for each X ∈ g and each u, v ∈ e . Further relations motivated by the remaining properties of the Peiffer liftinghold. These can be gathered inside the definition of a differential 2-crossedmodule (also called a ). This definition appearedin [E]. Note that our conventions are different.
Definition 8 (Differential 2-crossed module)
A differential 2-crossed mod-ule is given by a complex of Lie algebras: l δ −→ e ∂ −→ g together with left actions ⊲ by derivations of g on l , e and g (on the latter by theadjoint representation), and a g -equivariant bilinear map { , } : e × e → l : X ⊲ { u, v } = { X ⊲ u, v } + { u, X ⊲ v } , for each X ∈ g and u, v ∈ e , (called the Peiffer lifting) such that:1. L δ −→ E ∂ −→ G is a complex of g -modules.2. δ ( { u, v } ) = h u, v i , for each u, v ∈ e . (Recall h u, v i = [ u, v ] − ∂ ( u ) ⊲ v .)3. [ x, y ] = { δ ( x ) , δ ( y ) } , for each x, y ∈ l . . { [ u, v ] , w } = ∂ ( u ) ⊲ { v, w } + { u, [ v, w ] } − ∂ ( v ) ⊲ { u, w } − { v, [ u, w ] } , for each u, v, w ∈ e . This is the same as: { [ u, v ] , w } = { ∂ ( u ) ⊲v, ω }−{ ∂ ( v ) ⊲u, ω }−{ u, δ { v, w }} + { v, δ { u, w }} (10) { u, [ v, w ] } = { δ { u, v } , w } − { δ { u, w } , v } , for each u, v, w ∈ e . This im-plies that ⊲ ′ defined by v ⊲ ′ x = − { δ ( x ) , v } is a left action of e on l ; seebelow.6. { δ ( x ) , v } + { v, δ ( x ) } = − ∂ ( v ) ⊲ x , for each x ∈ l and v ∈ e . Analogously to the 2-crossed module of Lie groups case we have:
Lemma 9
The action ⊲ ′ of e on l is by derivations, and together with the map δ : l → e defines a differential crossed module. We divide the proof into four claims.
Claim 1 ( u, x ) ∈ e × l u ⊲ ′ x = −{ δ ( x ) , u } is a left action of e on l . Proof.
For each u, v ∈ e and x ∈ l we have:[ u, v ] ⊲ ′ x = −{ δ ( x ) , [ u, v ] } = { δ { δ ( x ) , v } , u }−{ δ { δ ( x ) , u } , v } = u⊲ ′ ( v⊲ ′ x ) − v⊲ ′ ( u⊲ ′ x ) Claim 2
The left action ⊲ ′ of e on l is by derivations. Proof.
We need to prove that u ⊲ ′ [ x, y ] = [ u ⊲ ′ x, y ] + [ x, u ⊲ ′ y ], for each u ∈ e and x, y ∈ g . We have: u⊲ ′ [ x, y ] = −{ δ ([ x, y ]) , u } = −{ [ δ ( x ) , δ ( y )] , u } = { δ ( y ) , [ δ ( x ) , u ] }−{ δ ( x ) , [ δ ( y ) , u ] } . We have used 4 . together with the fact ∂δ = 0. The last term can be simplifiedas: { δ ( y ) , [ δ ( x ) , u ] } − { δ ( x ) , [ δ ( y ) , u ] } = { δ ( { δ ( y ) , δ ( x ) } ) , u } ) − { δ ( { δ ( y ) , u } ) , δ ( x ) } ) − { δ ( { δ ( x ) , δ ( y ) } ) , u } ) + { δ ( { δ ( x ) , u } ) , δ ( y ) } )= − [ x, u ⊲ ′ y ] − [ u ⊲ ′ x, y ] + 2 u ⊲ ′ { δ ( x ) , δ ( y ) } = − [ x, u ⊲ ′ y ] − [ u ⊲ ′ x, y ] + 2 u ⊲ ′ [ x, y ] . Therefore u ⊲ ′ [ x, y ] = − [ x, u ⊲ ′ y ] − [ u ⊲ ′ x, y ] + 2 u ⊲ ′ [ x, y ] , from which the result follows. Claim 3 δ ( x ) ⊲ ′ y = [ x, y ] , for each x, y ∈ l . Proof. δ ( x ) ⊲ ′ y = −{ δ ( y ) , δ ( x ) } = [ x, y ] , for each x, y ∈ l Claim 4 δ ( u ⊲ ′ x ) = [ u, δ ( x )] , for each x ∈ l and u ∈ e . Proof. δ ( u ⊲ ′ x ) = − δ ( { δ ( x ) , u } ) = − < δ ( x ) , u > = [ u, δ ( x )] , since ∂δ = 0. 10 .2.3 Lie 2-crossed modules and differential 2-crossed modules The definition of a differential 2-crossed module is an exact differential replica ofthe definition of a 2-crossed module of Lie groups. Straightforward calculationsprove that:
Theorem 10
Let H = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module of Liegroups. The induced chain complex of Lie algebras l δ −→ e ∂ −→ g , together with theinduced actions ⊲ of g on e and l and the second differential of the Peiffer lifting { , } defines a differential 2-crossed module H . Moreover the assignment H 7→ H is functorial. Theorem 10 can be proved by taking differentials, in the obvious way, of theequations appearing in the definition of a 2-crossed module of Lie groups. Thereis, however, a more conceptual way to prove this theorem, which guarantees thatwe can go in the opposite direction. Namely, it is well known that the categoriesof simplicial groups with Moore complex of length two and of 2-crossed modulesare equivalent, see for example [Co, P, BG]. This equivalence of categories alsoholds in the Lie algebra case, as is proved in [E]. From standard Lie theoryit follows that the categories of simplicial Lie algebras and simplicial (simplyconnected) Lie groups are equivalent. Taking Moore complexes, it thereforefollows that:
Theorem 11
Given a differential 2-crossed module H = ( l δ −→ e ∂ −→ g , ⊲, { , } ) there exists a 2-crossed module of Lie groups H whose differential form is H . We now define Gray 3-groupoids. Our conventions are slightly different fromthe ones of [KP, Cr].A (small) Gray 3-groupoid C is given by a set C of objects, a set C ofmorphisms, a set C of 2-morphisms and a set C of 3-morphisms, and maps ∂ ± i : C k → C i − , where i = 1 , . . . , k (and k = 1 , ,
3) such that:1. ∂ ± ◦ ∂ ± = ∂ ± , as maps C → C .2. ∂ ± = ∂ ± ◦ ∂ ± = ∂ ± ◦ ∂ ± , as maps C → C .3. ∂ ± = ∂ ± ◦ ∂ ± , as maps C → C .4. There exists an upwards multiplication J♮ J ′ of 3-morphisms if ∂ +3 ( J ) = ∂ − ( J ′ ), making C into a groupoid whose set of objects is C (identitiesare implicit).5. There exists a vertical compositionΓ ♮ Γ ′ = (cid:18) Γ ′ Γ (cid:19) of 2-morphisms if ∂ +2 (Γ) = ∂ − (Γ ′ ), making C into a groupoid whose setof objects is C (identities are implicit).11. There exists a vertical composition J♮ J ′ = (cid:18) J ′ J (cid:19) of 3-morphisms whenever ∂ +2 ( J ) = ∂ − ( J ′ ) making the set of 3-morphismsinto a groupoid with set of objects C and such that the boundaries ∂ ± : C → C are functors.7. The vertical and upwards compositions of 3-morphisms satisfy the inter-change law ( J♮ J ′ ) ♮ ( J ♮ J ′ ) = ( J♮ J ) ♮ ( J ′ ♮ J ′ ), whenever the compo-sitions are well defined. Combining with the previous axioms, this meansthat the vertical and upwards compositions of 3-morphisms and the ver-tical composition of 2-morphisms give C the structure of a 2-groupoid,with set of objects being C , set of morphisms C and set of 2-morphisms C . (The definition of a 2-groupoid appears for example in [HKK]. Itis well known that the categories of (small) 2-groupoids and of crossedmodules of groupoids are equivalent; see for instance [BHS, BS].)8. (Existence of whiskering by 1-morphisms) For each x, y in C we cantherefore define a 2-groupoid C ( x, y ) of all 1-, 2- and 3-morphisms b suchthat ∂ − ( b ) = x and ∂ +1 ( b ) = y . Given a 1-morphism γ with ∂ − ( γ ) = y and ∂ +1 ( γ ) = z there exists a 2-groupoid map ♮ γ : C ( x, y ) → C ( y, z ), calledright whiskering. Similarly if ∂ +1 ( γ ′ ) = x and ∂ − ( γ ′ ) = w there exists a2-groupoid map γ ′ ♮ : C ( x, y ) → C ( w, y ) , called left whiskering.9. There exists therefore a horizontal composition of γ♮ γ ′ of 1-morphisms if ∂ +1 ( γ ) = ∂ − ( γ ′ ), which is to be associative and to define a groupoid withset of objects C and set of morphisms C .10. Given γ, γ ′ ∈ C we must have: ♮ γ ◦ ♮ γ ′ = ♮ ( γ ′ γ ) γ♮ ◦ γ ′ ♮ = ( γγ ′ ) ♮ γ♮ ◦ ♮ γ ′ = ♮ γ ′ ◦ γ♮ , whenever these compositions make sense.11. We now define two horizontal compositions of 2-morphisms (cid:18) Γ ′ Γ (cid:19) = (cid:18) ∂ +2 (Γ) ♮ Γ ′ Γ ♮ ∂ − (Γ ′ ) (cid:19) = (cid:0) Γ ♮ ∂ − (Γ ′ ) (cid:1) ♮ (cid:0) ∂ +2 (Γ) ♮ Γ ′ (cid:1) and (cid:18) Γ Γ ′ (cid:19) = (cid:18) Γ ♮ ∂ +2 (Γ ′ ) ∂ − (Γ) ♮ Γ ′ (cid:19) = (cid:0) ∂ − (Γ) ♮ Γ ′ (cid:1) ♮ (cid:0) Γ ♮ ∂ +2 (Γ ′ ) (cid:1) ;12nd of 3-morphisms: (cid:18) J ′ J (cid:19) = (cid:18) ∂ +2 ( J ) ♮ J ′ J ♮ ∂ − ( J ′ ) (cid:19) = (cid:0) J♮ ∂ − ( J ′ ) (cid:1) ♮ (cid:0) ∂ +2 ( J ) ♮ J ′ (cid:1) and (cid:18) J J ′ (cid:19) = (cid:18) J ♮ ∂ +2 ( J ′ ) ∂ − ( J ) ♮ J ′ (cid:19) = (cid:0) ∂ − ( J ) ♮ J ′ (cid:1) ♮ (cid:0) J♮ ∂ +2 ( J ′ ) (cid:1) It follows from the previous axioms that they are associative. In fact theyalso define functors C ( x, y ) × C ( y, z ) → C ( x, z ), where C ( x, y ) is thecategory with objects 2-morphisms Γ with ∂ − (Γ) = x and ∂ +2 (Γ) = y and morphisms the 3-morphisms J with ∂ − ( J ) = x and ∂ +2 ( J ) = y , andupwards multiplication as composition; this follows from 7 and 8.12. (Interchange 3-cells) For any two 2-morphisms Γ and Γ ′ with ∂ +1 (Γ) = ∂ − (Γ ′ ) a 3-morphism (called an interchange 3-cell) (cid:18) Γ ′ Γ (cid:19) = ∂ − (Γ ′ ) Γ ′ −−−−−−−→ ∂ +3 (Γ ′ ) = (cid:18) Γ Γ ′ (cid:19) (2-functoriality) For any 3-morphisms Γ = ∂ − ( J ) J −→ ∂ +3 ( J ) = Γ andΓ ′ = ∂ − ( J ′ ) J ′ −→ ∂ +3 ( J ′ ) = Γ ′ , with ∂ +1 ( J ) = ∂ − ( J ′ ) the following upwardscompositions of 3-morphisms coincide: (cid:18) Γ ′ Γ (cid:19) Γ ′ −−−−−−−−→ (cid:18) Γ Γ ′ (cid:19) J J ′ −−−−−−−→ (cid:18) Γ Γ ′ (cid:19) and (cid:18) Γ ′ Γ (cid:19) J ′ J −−−−−−−→ (cid:18) Γ ′ Γ (cid:19) Γ ′ −−−−−−−−→ (cid:18) Γ Γ ′ (cid:19) . This of course means that the collection Γ ′ , for arbitrary 2-morphismsΓ and Γ ′ with ∂ +1 (Γ) = ∂ − (Γ ′ ) defines a natural transformation betweenthe two functors of 11. Note that by using the interchange condition for thevertical and upwards compositions, we only need to verify this conditionfor the case when either J or J ′ is an identity. (This is the way this axiomappears written in [KP, Cr, Be].)14. (1-functoriality) For any three 2-morphisms γ Γ −→ φ Γ ′ −→ ψ and γ ′′ Γ ′′ −−→ φ ′′ with ∂ +2 (Γ) = ∂ − (Γ ′ ) and ∂ +1 (Γ) = ∂ +1 (Γ ′ ) = ∂ − (Γ ′′ ) the followingupwards compositions of 3-morphisms coincide: ψ♮ Γ ′′ Γ ′ ♮ γ ′′ Γ ♮ γ ′′ Γ ′ ′′ Γ ♮ γ ′′ −−−−−−−−→ Γ ′ ♮ φ ′′ φ♮ Γ ′′ Γ ♮ γ ′′ Γ ′ ♮ φ ′′ Γ ′′ −−−−−−−→ Γ ′ ♮ φ ′′ Γ ♮ φ ′′ γ♮ Γ ′′ ψ♮ Γ ′′ Γ ′ ♮ γ ′′ Γ ♮ γ ′′ Γ ′ Γ ′′ −−−−−−−→ Γ ′ ♮ φ ′′ Γ ♮ φ ′′ γ♮ Γ ′′ . Furthermore an analogous equation holds with left and right whiskeringexchanged.
Definition 12
A (strict) Gray functor F : C → C ′ between Gray 3-groupoids C and C ′ is given by maps C i → C ′ i ( i = 0 , . . . , preserving all compositions,identities, interchanges and boundaries, strictly. Let H = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module (of groups). We canconstruct a Gray 3-groupoid C with a single object out of H . We put C = {∗} , C = G , C = G × E and C = G × E × L . This construction appears in[KP], with different conventions, and also in [CCG, BG], in a slightly differentlanguage.As boundaries ∂ ± : C k → C = {∗} , where k = 1 , ,
3, we take the uniquepossible map. Furthermore: ∂ − ( X, e ) = X and ∂ +2 ( X, e ) = ∂ ( e ) − X. In addition put (as vertical composition):(
X, e ) ♮ ( ∂ ( e ) − X, f ) = (
X, ef ) , and also X♮ ( Y, e ) = (
XY, X ⊲ e ) and (
Y, e ) ♮ X = ( Y X, e ) . Analogously X♮ ( Y, e, l ) = (
XY, X ⊲ e, X ⊲ l ) and (
Y, e, l ) ♮ X = ( Y X, e, l ) . Looking at 3-cells, put ∂ − ( X, e, l ) = (
X, e ) and ∂ +3 ( X, e, l ) = (
X, δ ( l ) − e )and ∂ − ( X, e, l ) = X and ∂ +2 ( X, e, l ) = ∂ ( e ) − X. (Note ∂ +2 ∂ +3 ( X, e, l ) = ∂ +2 ( X, δ ( l ) − e ) = ∂ ( e ) − X = ∂ +2 ( X, e, l ), since ∂δ = 1.)As vertical composition of 3-morphisms we put: (cid:0) X, e, l (cid:1) ♮ (cid:0) ∂ ( e ) − X, f, k (cid:1) = (cid:0) ∂ ( e ) − X, f, k (cid:1)(cid:0)
X, e, l (cid:1) ! = (cid:0) X, ef, ( e ⊲ ′ k ) l (cid:1) , (cid:0) X, e, l (cid:1) ♮ (cid:0) X, δ ( l ) − e, k (cid:1) = (cid:0) X, e, lk (cid:1)
The vertical and upwards compositions of 2-cells define a 2-groupoid since (cid:0) δ : G × L → G × E, ⊲ ′ (cid:1) is a crossed module of groupoids [BHS]; see the commentsafter definition 3. Recall that the category of crossed modules of groupoids andthe category of 2-groupoids are equivalent.Let us now define the interchange 3-cells. We can see that: (cid:18) ( Y, f )( X, e ) (cid:19) = (cid:0) XY, e (cid:0) ∂ ( e ) − X (cid:1) ⊲ f (cid:1) and (cid:18) ( X, e ) (
Y, f ) (cid:19) = (cid:0) XY, (cid:0)
X ⊲ f (cid:1) e (cid:1) . We therefore take:(
X, e )
Y, f ) = (cid:16)
XY, e (cid:0) ∂ ( e ) − X (cid:1) ⊲ f, e ⊲ ′ (cid:8) e − , X ⊲ f (cid:9) − (cid:17) . (11)Note δ (cid:16) e ⊲ ′ (cid:8) e − , X ⊲ f (cid:9) − (cid:17) − e (cid:0) ∂ ( e ) − X (cid:1) ⊲ f = ee − ( X ⊲ f ) e (cid:0) ∂ ( e ) − X (cid:1) ⊲ f − e − e (cid:0) ∂ ( e ) − X (cid:1) ⊲ f = ( X ⊲ f ) e. It is easy to see that: (cid:18) ( Y, f, l )( X, e, k ) (cid:19) = (cid:18) ( ∂ ( e ) − XY, ∂ ( e ) − X ⊲ f, ∂ ( e ) − X ⊲ l ( XY, e, k ) (cid:19) = (cid:16) XY, e (cid:0) ∂ ( e ) − X (cid:1) ⊲ f, (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l (cid:1) k (cid:17) and (cid:18) ( X, e, k ) (
Y, f, l ) (cid:19) = (cid:18) ( X∂ ( f ) − Y, e, k )( XY, X ⊲ f, X ⊲ l ) (cid:19) = (cid:16) XY, ( X ⊲ f ) e, (( X ⊲ f ) ⊲ ′ k ) X ⊲ l (cid:17) . To prove condition 13. of the definition of a Gray 3-groupoid (2-functoriality)we must prove that (for each X ∈ G , e, f ∈ E and k, l ∈ L ): e ⊲ ′ { e − , X ⊲ f } − (( X ⊲ f ) ⊲ ′ k ) X ⊲ l = (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l (cid:1) k ( δ ( k ) − e ) ⊲ ′ { e − δ ( k ) , X ⊲ ( δ ( l ) − f ) } − . or, by using the fact that ( δ : L → E, ⊲ ′ ) is a crossed module: e ⊲ ′ { e − , X ⊲ f } − (( X ⊲ f ) ⊲ ′ k ) X ⊲ l = (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l (cid:1) e ⊲ ′ { e − δ ( k ) , X ⊲ ( δ ( l ) − f ) } − k. (12)15or l = 1 this is equivalent to: e ⊲ ′ { e − , X ⊲ f } − (( X ⊲ f ) ⊲ ′ k ) = e ⊲ ′ { e − δ ( k ) , X ⊲ f } − k. or: (( X ⊲ f ) ⊲ ′ k − ) e ⊲ ′ { e − , X ⊲ f } = k − e ⊲ ′ { e − δ ( k ) , X ⊲ f } which follows from equation (5) and the definition of e ⊲ ′ l = l { δ ( l − ) , e } . Notethat ∂ ◦ δ = 1 L . For k = 1 equation (12) is the same as: e ⊲ ′ { e − , X ⊲ f } − X ⊲ l = (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l (cid:1) e ⊲ ′ { e − , X ⊲ ( δ ( l ) − f ) } − , or ( X ⊲ l − ) e ⊲ ′ { e − , X ⊲ f } = e ⊲ ′ { e − , X ⊲ ( δ ( l ) − f ) } (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l − (cid:1) . This can be proved as follows, by using equation (8) e ⊲ ′ { e − , X ⊲ ( δ ( l ) − f ) } (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l − (cid:1) = (cid:0) δ ( X ⊲ l − ) e ⊲ ′ { e − , X ⊲ f } (cid:1) (cid:0) e ⊲ ′ { e − , X ⊲ δ ( l ) − } (cid:1) (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l − (cid:1) = ( X ⊲ l − ) e ⊲ ′ { e − , X ⊲ f } ( X ⊲ l ) (cid:0) e ⊲ ′ { e − , X ⊲ δ ( l ) − } (cid:1) (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l − (cid:1) where we have used the fact that ( δ : L → E, ⊲ ′ ) is a crossed module. Now note,by using 6 . of the definition of a 2-crossed module: (cid:0) e ⊲ ′ { e − , X ⊲ δ ( l ) − } (cid:1) (cid:0) e ⊲ ′ ∂ ( e ) − X ⊲ l − (cid:1) = e ⊲ ′ (cid:0) { X ⊲ δ ( l − ) , e − } − ( X ⊲ l − ) (cid:1) = ( eδ ( X ⊲ l − )) ⊲ ′ (cid:0) { X ⊲ δ ( l ) , e − } (cid:1) e ⊲ ′ (cid:0) X ⊲ l − (cid:1) , by equation (9)= ( e ⊲ ′ X ⊲ l − ) e ⊲ ′ (cid:0) { X ⊲ δ ( l ) , e − } (cid:1) , since ( δ : L → E, ⊲ ′ ) is a crossed module= e ⊲ ′ (cid:0) ( X ⊲ l − ) { X ⊲ δ ( l ) , e − } (cid:1) = X ⊲ l − , by definition of e ⊲ ′ l = l { δ ( l − ) , e } . The general case of equation (12) follows from k = 1 and l = 1 cases by theinterchange law for the upwards and vertical compositions.Let us now prove 1-functoriality (condition 14. of the definition of a Gray3-groupoid). The first condition is equivalent to:( ef ) ⊲ ′ (cid:8) f − e − , X ⊲ g (cid:9) − = ( ef ) ⊲ ′ (cid:8) f − , ∂ ( e ) − X ⊲ g (cid:9) − e ⊲ ′ (cid:8) e − , X ⊲ g (cid:9) − . This follows directly from equation (5). The second condition is equivalent to: e ⊲ ′ { e − , X ⊲ f X ⊲ g } − = e ⊲ ′ { e − , X ⊲ f } − (cid:0) ( X ⊲ f ) e (cid:1) ⊲ ′ { e − , X ⊲ g } − which follows from equation (8).We have therefore proved that any 2-crossed module H defines a Gray 3-groupoid C ( H ), with a single object. This process is reversible: a Gray 3-groupoid C together with an object x ∈ C of it defines a 2-crossed module; see[KP, Be, BG]. 16 .2.6 Example: (finite type) chain complexes Suppose A = { A n , ∂ n = ∂ } n ∈ Z is a chain complex of finite dimensional vectorspaces, such that the set of all n for which A n is not the trivial vector space isfinite. (Chain complexes like this will be called of finite type.) This constructionis analogous to the one in [KP].Let us then construct a Lie 2-crossed moduleGL( A ) = (cid:16) GL ( A ) α −→ GL ( A ) β −→ GL ( A ) , ⊲, { , } (cid:17) out of A . If A is not of finite type, then the same construction will still yield acrossed module, albeit of infinite dimensional Lie groups.The group GL ( A ) is given by all invertible chain maps f : A → A , withcomposition as product. This is a Lie group, with Lie algebra gl ( A ) given byall chain maps A → A , with bracket given by the usual commutator of chainmaps. We also denote the algebra of all chain maps A → A with compositionas product as hom ( A ).Recall that a homotopy is given by a degree-1 map s : A → A . Let hom ( A )denote the vector space of 1-homotopies. Likewise, we define an n -homotopy asbeing a degree- n map b : A → A , and denote the vector space of n -homotopies as h n +1 ( A ). Notice also that we have a complex { h n ( A ) , ∂ ′ n } , where ∂ ′ n ( b ) = ∂b − ( − n b∂ , for each b ∈ hom n ( A ). Note h ( A ) = hom ( A ) and hom ( A ) ⊂ h ( A ).We define the ∗ product of two 1-homotopies as: s ∗ t = s + t + s∂t + st∂ This defines an associative product in hom ( A ). Even though hom ( A ) is notan algebra, considering commutators this yields a Lie algebra gl ( A ), with com-mutator: [ s, t ] = st∂ + s∂t − ts∂ − t∂s. Note that gl ( A ) is the Lie algebra of the Lie group GL ( A ) of invertible elementsof hom ( A ), the identity of this latter group being the null homotopy.It is easy to see that the map β : hom ( A ) → hom ( A ) such that β ( s ) = 1 + ∂s + s∂ respects the products. This thus defines a Lie group morphism β : GL ( A ) → GL ( A ), the differential form of which is given by the Lie algebra map β ′ : gl ( A ) → gl ( A ), where β ′ ( s ) = ∂s + s∂. There is also a left action of GL ( A ) on GL ( A ) by automorphisms given by f ⊲ s = f sf − . Its differential form is given by the left action of gl ( A ) on gl ( A ) by derivationssuch that: f ⊲ s = f s − sf.
17t is easy to see that we have defined a pre-crossed module of Lie groups and ofLie algebras. Moreover, this yields a crossed module if the chain complex is oflength 2, as we will see below.By definition a 3-track will be an element of hom ( A ) = h ( A ) /∂ ′ ( h ( A )),therefore it will be a 2-homotopy up to a 3-homotopy. Considering the sum of3-tracks, defines an abelian Lie group GL ( A ) whose Lie algebra gl ( A ) is givenby the vector space of 3-tracks, with trivial commutator.The (well defined) map α : GL ( A ) → GL ( A ) such that α ( b ) = − ∂b + b∂ is a group morphism. We also have a left action of GL ( A ) on GL ( A ) byautomorphisms defined as f ⊲ a = f af − . This defines a complex of Lie groups acted on by GL ( A ):GL ( A ) α −→ GL ( A ) β −→ GL ( A ) . Its differential form is: gl ( A ) α ′ −→ gl ( A ) β ′ −→ gl ( A ) , where gl ( A ) is the vector space hom ( A ) with trivial commutator, and α ′ = α .Note that gl ( A ) acts on the left on gl ( A ) by derivations as f ⊲ b = f b − bf .To define a 2-crossed module we now need to specify the Peiffer lifting. Wecan see that given s, t ∈ GL ( A ) we have s ∗ t ∗ s − = β ( s ) tβ ( s ) − − ∂stβ ( s ) − + st∂ + st∂s − ∂ = β ( s ) tβ ( s ) − + α ( st ) β ( s ) − = ( β ( s ) tβ ( s ) − ) ∗ ( α ( st ) β ( s ) − )= ( β ( s ) tβ ( s ) − ) ∗ α ( stβ ( s ) − )In particular the Peiffer pairing is: h s, t i = ( β ( s ) ⊲ t ) ∗ α (cid:0) stβ ( s ) − (cid:1) ∗ (cid:0) β ( s ) ⊲ t − (cid:1) . This can still be simplified. Let a ∈ GL ( A ). We can see that t ∗ α ( a ) ∗ t − = α (cid:0) aβ ( t ) − (cid:1) = α ( a ) β ( t ) − . This follows by applying the penultimate equation, noting that: t ∗ α ( a ) ∗ t − = ( β ( t ) α ( a ) β ( t ) − ) ∗ α (cid:0) tα ( a ) β ( t ) − (cid:1) = α ( β ( t ) aβ ( t ) − ) ∗ α (cid:0) tα ( a ) β ( t ) − (cid:1) = α ( β ( t ) a + tα ( a ) (cid:1) β ( t ) − = α ( ∂ta + ta∂ + a ) β ( t ) − = α ( a ) β ( t ) − h s, t i = α (cid:0) stβ ( t ) − β ( s ) − (cid:1) , and we thus have the following candidate for the role of the Peiffer lifting: { s, t } = stβ ( t ) − β ( s ) − , where s, t ∈ GL ( A ). Its differential form is: { s, t } = st, where s, t ∈ gl ( A ).Routine calculations prove that we have indeed defined 2-crossed modulesof Lie groups and of Lie algebras. The fact we are considering 3-tracks (2-homotopies up to 3-homotopies), instead of simply 2-homotopies, is used severaltimes to prove this. Let G = ( ∂ : E → G, ⊲ ) be a Lie group crossed module. Let us build the differ-ential 2-crossed module associated with the automorphism 2-crossed module of G . In the case of crossed modules of groups, the construction of this 2-crossedmodule appears in [BG, RS, N], in the latter in the language of crossed squares.The extension to crossed modules of Lie groups is straightforward.Let G = ( ∂ : e → g , ⊲ ) be the differential crossed module associated to G .Let us then construct a 2-crossed module of Lie algebras (cid:0) gl ( G ) → gl ( G ) → gl ( G ) , ⊲, { } (cid:1) . The Lie algebra gl ( G ) is given by all chain maps f = ( f , f ) : G → G ,which, termwise, are Lie algebra derivations: f ([ u, v ]) = [ f ( u ) , v ] + [ u, f ( v )] , for each u, v ∈ e and f ([ x, y ]) = [ f ( x ) , y ] + [ x, f ( y )] , for each x, y ∈ g , satisfying additionally: f ( x ⊲ v ) = f ( x ) ⊲ v + x ⊲ f ( v ) , for each x ∈ g and v ∈ e . The Lie algebra structure is given by the termwise commutator of derivations.The Lie algebra gl ( G ) is given by all pairs ( x, s ), where s : g → e is a linearmap such that: s ([ x, y ]) = x ⊲ s ( y ) − y ⊲ s ( x ) , (in other words s : g → e is a derivation, and we put s ∈ Der( g , e )) and x ∈ g .The Lie algebra structure on gl ( G ) = g ⋉ Der( g , e ) is given by a semidirectproduct, as we now explain. 19he commutator of two derivations s, t ∈ Der( g , e ) is:[ s, t ] = s∂t − t∂s. It is easy to see that this is also a derivation. The crossed module relationsare used several times to prove this. (This would not be true if a pre-crossedmodule was used.)There exists a left action of gl ( G ) on the Lie algebra of derivations s : g → e given by: ( f , f ) ⊲ s = f s − sf . We also have a Lie algebra map q : Der( g , e ) → gl ( G ) given by q ( s ) = ∂s + s∂. This defines a differential crossed module.There is another crossed module of Lie algebras that can be constructedfrom G . This is provided by the map q ′ = ( q ′ , q ′ ) : g → gl ( G ) which associatesto each x ∈ g the inner derivation f x : g → g such that f x ( y ) = [ x, y ] , for each y ∈ g , and the derivation e → e such that v x ⊲ v . The action of gl ( G ) on g is ( f , f ) ⊲ x = f ( x ), where x ∈ g .In particular, we also have an action of g on Der( g , e ), provided by the map q ′ : g → gl ( G ) and the already given action of gl ( G ) on Der( g , e ). Thereforewe can put a Lie algebra structure on gl ( G ) given by the semidirect product g ⋉ Der( g , e ). In particular[( x, s ) , ( y, t )] = ([ x, y ] , x ⊲ t − y ⊲ s + [ s, t ]) . The boundary map β ′ : gl ( G ) → gl ( G ) is: β ′ ( a, s ) = q ′ ( a ) + q ( s ) . By the above, this is a Lie algebra map. We also define f ⊲ ( a, s ) = ( f ⊲ a, f ⊲ s ),which defines a differential pre-crossed module. The Peiffer pairing is given by: h ( x, s ) , ( y, t ) i = − (cid:0) ∂s ( y ) , F s ( y ) (cid:1) Here, given e ∈ e , the map F e : g → e is F e ( x ) = x ⊲ e .The Lie algebra gl ( G ) is given by e . The boundary map α ′ : gl ( G ) → gl ( G )is α ′ ( e ) = ( ∂e, F e ).The Peiffer lifting is defined as: { ( x, s ) , ( y, t ) } = − s ( y ) . Therefore: { α ′ ( e ) , α ′ ( f ) } = { ( ∂e, F e ) , ( ∂f, F f ) } = − F e ( ∂f ) = − ∂f ⊲ e = − [ f, e ] = [ e, f ] . Furthermore the action of gl ( G ) on e is ( f , f ) ⊲ e = f ( e ).The (rest of the) straightforward proof that this defines a differential 2-crossed module is left to the reader. Note that given a Lie group G we candefine a differential crossed module (id : g → g , ad). The automorphism 2-crossed module of it is exactly the differential 2-crossed module associated tothe 2-crossed module of Example 7. 20 The thin fundamental Gray 3-groupoid of asmooth manifold
Let M be a smooth manifold. We denote D n = [0 , n . Definition 13 ( n -path) Let n be a positive integer. An n -path is given by asmooth map α : D n = D × D n − → M for which there exists an ǫ > suchthat α ( x , x , . . . x n ) = α (0 , x , . . . x n ) if x ≤ ǫ , and analogously for any otherface of D n , of any dimension. We will abbreviate this condition as saying that α has a product structure close to the boundary of the n -cube. We also supposethat α (0 × D n − ) and α (1 × D n − ) each consist of just a single point. Given an n -path and an i ∈ { , . . . , n } we can define ( n − ∂ − i ( α ) and ∂ + i ( α ) by restricting f to D i − × { } × D n − i and D i − × { } × D n − i . Note that ∂ ± ( α ) are necessarily constant ( n − n -paths α and β with ∂ + i ( α ) = ∂ − i ( β ) we consider the obvious concatenation α♮ i b , which given theproduct structure condition of α and β is also an n -path; see examples below. Note that a 1-path is given by a smooth path γ : [0 , → M such that there existsan ǫ > γ is constant in [0 , ǫ ] ∪ [1 − ǫ, γ has a sitting instant (we are using theterminology of [CP]). Given a 1-path γ , define the source and target of γ as ∂ − ( γ ) = γ (0) and ∂ +1 ( γ ) = γ (1), respectively.Given two 1-paths γ and φ with ∂ +1 ( γ ) = ∂ − ( φ ), their concatenation γφ = γ♮ φ is the usual one ( γφ )( t ) = ( γ (2 t ) , if t ∈ [0 , / φ (2 t − , if t ∈ [1 / , , → M such thatthere exists an ǫ > t, s ) = Γ(0 ,
0) if 0 ≤ t ≤ ǫ and s ∈ [0 , t, s ) = Γ(1 ,
0) if 1 − ǫ ≤ t ≤ s ∈ [0 , t, s ) = Γ( t,
0) if 0 ≤ s ≤ ǫ and t ∈ [0 , t, s ) = Γ( t,
1) if 1 − ǫ ≤ s ≤ t ∈ [0 , Definition 14 (Rank-1 homotopy)
Two 1-paths φ and γ are said to be rank-1 homotopic (and we write φ ∼ = γ ) if there exists a 2-path Γ such that: . ∂ − (Γ) = γ and ∂ +2 (Γ) = φ .2. Rank( D Γ( v )) ≤ , ∀ v ∈ [0 , . Here D denotes derivative. Note that if γ and φ are rank-1 homotopic, then they have the same initialand end-points. Given the product structure condition on 2-paths, it followsthat rank-1 homotopy is an equivalence relation. Given a 1-path γ , the equiva-lence class to which it belongs is denoted by [ γ ], or simply γ , when there is noambiguity.We denote the set of 1-paths of M by S ( M ). The quotient of S ( M ) by therelation of thin homotopy is denoted by S ( M ). We call the elements of S ( M )1-tracks.It is easy to prove that the concatenations of 1-tracks (defined in the obviousway from the concatenation of 1-paths) together with the source and target maps σ, τ : S ( M ) → M , defines a groupoid S ( M ) whose set of morphisms is S ( M )and whose set of objects is M . (For details see [CP].) Definition 15
Let ∗ ∈ M be a base point. The group π ( M, ∗ ) is defined asbeing the set of 1-tracks [ γ ] ∈ S ( M ) starting and ending at ∗ , with the groupoperation being the concatenation of 1-tracks. Two 2-paths Γ and Γ ′ are saidto be strong rank-2 homotopic (and we write Γ ∼ = s Γ ′ ) if there exists a 3-path J : D → M such that:1. We have ∂ − ( J ) = Γ and ∂ +3 ( J ) = Γ ′ .2. The restrictions ∂ ± ( J ) restrict to rank-1 homotopies ∂ ± (Γ) → ∂ ± (Γ ′ ) .3. Rank( D J ( v )) ≤ for any v ∈ [0 , . Due to the fact that any 3-path has a product structure close to its boundary,it follows that strong rank 2-homotopy is an equivalence relation.We denote by S ( M ) the set of all 2-paths of M . The quotient of S ( M )by the relation of strong rank-2 homotopy is denoted by S s ( M ). We call theelements of S s ( M ) strong 2-tracks.This notion of strong rank-2 homotopy was used in [MP, SW2, SW3, BS,FMP1, FMP2, M], and it behaves very nicely with respect to 2-dimensionalholonomy based on a crossed module. For this reason, it is too strong for ourpurposes in this article. Therefore we define now a weaker version of rank-2homotopy. 22 .2.2 Laminated 2-Tracks (the laminated rank-2 homotopy equiva-lence relation)Definition 17 (Laminated rank-2 homotopy) Two 2-paths Γ and Γ ′ aresaid to be laminated rank-2 homotopic (and we write Γ ∼ = l Γ ′ ) if there exists a3-path J : D → M (say J ( t, s, x )) such that:1. We have ∂ − ( J ) = Γ and ∂ +3 ( J ) = Γ ′ , in other words J ( t, s,
0) = Γ( t, s ) and J ( t, s,
1) = Γ ′ ( t, s ) for each s, t ∈ [0 , .2. The restrictions ∂ ± ( J ) (in other words J ( t, , x ) and J ( t, , x ) ) restrict torank-1 homotopies ∂ ± (Γ) → ∂ ± (Γ ′ ) . Rank( D J ( v )) ≤ for any v ∈ [0 , .4. For each < s, x < , at least one of the following conditions holds (up toa set of Lebesgue measure zero):(a) Laminatedness
For either ζ = s or ζ = x the following conditionholds for all t ∈ [0 , : Rank( D ( t,ζ ) J ( t, s, x )) ≤ . (b) Path space thinness:
There exist non-zero constants a and b suchthat a ∂∂s J ( t, s, x ) + b ∂∂x J ( t, s, x ) = 0 for each t ∈ [0 , . Once again, since any 3-path has a product structure close to its boundary, itfollows that laminated rank-2 homotopy is an equivalence relation.The quotient of S ( M ), the set of 2-paths of M , by the relation of laminatedrank-2 homotopy is denoted by S l ( M ). We call the elements of S l ( M ) lami-nated 2-tracks. Note that the boundaries ∂ ± i : S ( M ) → S i − ( M ) descend toboundaries ∂ ± i : S l ( M ) → S i − ( M ) and ∂ ± i : S s ( M ) → S i − ( M ); here i = 1 , There exists a map S ( M ) → S ( M ) sending a path γ to the 2-path id( γ )(frequently written simply as γ ) such that id( γ )( t, s ) = γ ( t ) , for each s, t ∈ [0 , . It descends to maps id : S ( M ) → S s ( M ) and id : S ( M ) → S l ( M ). The following lemma will be needed for proving the consistency of the verti-cal composition of laminated and strong 2-tracks. It generalises Lemma 52 of[FMP1].
Lemma 18
Let f : ∂ ( D ) → M be a smooth map such that: . The restriction of f to any face of ∂D has a product structure close tothe boundary of it.2. The restrictions f (0 , s, x ) and f (1 , s, x ) are constant.3. We have Rank( D f ( v )) ≤ , ∀ v ∈ ∂D .Then f extends to a map g : D → M defining a laminated rank-2 homotopyconnecting ∂ − ( f ) = ∂ − ( g ) and ∂ +3 ( f ) = ∂ +3 ( g ) . Proof.
Let D = { ( t, s, x ) : − ≤ t, s, x ≤ } and S be its boundary; a smoothmanifold with corners. According to the proof of Lemma 52 of [FMP1], the map f : S → M factors as f = p ◦ φ , where φ : S → N is a smooth map, N beinga contractible manifold, and p : N → M being a smooth local diffeomorphism.In particular conditions 1,2 and 3 hold for φ .Choose a contraction c : N × [0 , → N of N to a point ∗ of it. We cansuppose that, at each end of [0 , c has a sitting instant, in other words, thatthere exists a positive ǫ such that c ( x, t ) = x if t ∈ [0 , ǫ ] and c ( x, t ) = ∗ if t ∈ [1 − ǫ, x ∈ N .Consider the map g : D → M defined as: g ( t, s, x ) = p (cid:18) c (cid:18) φ (cid:16) ( t, s, x ) | ( t, s, x ) | (cid:17) , − | ( t, s, x ) | (cid:19)(cid:19) where | ( t, s, x ) | = max ( | t | , | s | , | x | ). The map g is smooth and has a productstructure close to the boundary of D . All this follows from the product struc-ture condition for φ on the faces of S and the sitting instant condition on c ,which take care of the arguments where ( t, s, x )
7→ | ( t, s, x ) | is not smooth.The map g : D → M extends f : ∂D → M , and therefore we now only needto prove that it is a laminated rank-2 homotopy. Note that it follows triviallythat g is a rank-2 homotopy, as in the proof of of Lemma 52 of [FMP1].Define | ( s, x ) | = max( | s | , | x | ). Let ( s, x ) ∈ [ − , . If ( s, x ) is a pointwhere | ( s, x ) | is smooth then either ∂∂x | ( s, x ) | = 0 or ∂∂s | ( s, x ) | = 0. Suppose ∂∂x | ( s, x ) | = 0, from which it follows that ∂∂x | ( t, s, x ) | = 0, for each t ∈ [0 , D ( t,x ) g ( t, s, x )) ≤ , for each t ∈ [0 , . (13)Given a t ∈ [0 , ∂∂t | ( t, s, x ) | = 0 or not. In the first case (13)follows from the fact Rank( D f ( v )) ≤ , ∀ v ∈ S together with ∂∂x | ( t, s, x ) | = 0,for each t ∈ [0 , f is constantwhen t = 1 or t = −
1, which makes g depend only on t in a neighbourhood of( t, s, x ).The same argument is valid when ∂∂s | ( s, x ) | = 0. The remaining points ( s, x )have measure zero. 24 .2.5 Vertical composition of 2-tracks Recall that we can vertically compose any two 2-paths Γ and Γ ′ with ∂ +2 (Γ) = ∂ − (Γ ′ ). Denote it by Γ ♮ Γ ′ , and represent it graphically as:Γ ♮ Γ ′ = ∂ +2 (Γ ′ ) x Γ ′ ∂ − (Γ ′ ) = ∂ +2 (Γ) x Γ ∂ − (Γ) = Γ ′ Γ . Suppose [Γ] and [Γ ′ ] are (laminated or strong 2-tracks) such that ∂ +2 ([Γ]) = ∂ − ([Γ]). Choose a rank-1 homotopy H connecting ∂ +2 (Γ) and ∂ − (Γ ′ ). Define[Γ] ♮ [Γ ′ ] = [Γ ♮ H♮ Γ ′ ]. By using Lemma 18 we obtain the following, not entirelytrivial, result: Lemma 19
The vertical composition of (laminated or strong) 2-tracks is welldefined (does not depend on any of the choices made).
See [BH1, BHS, HKK, FMP1, BHKP] for similar constructions.It is easy to see that the vertical composition of strong and laminated 2-tracks is associative. In the laminated case, we will need to use the path-spacethinness condition in definition 17. We have:
Proposition 20
The vertical composition of strong or laminated 2-tracks de-fines categories with morphisms S s ( M ) and S l ( M ) , respectively, and objects S ( M ) . The source and target maps are ∂ − and ∂ +2 . The identities are as in2.2.3. Let Γ be a 2-path. Let also γ be a 1-path, such that ∂ +1 (Γ) = ∂ − ( γ ) . The rightwhiskering of Γ with γ is by definition the 2-path Γ ♮ γ . = Γ ♮ id( γ ); see 2.2.3.We analogously define left whiskering γ ′ ♮ Γ if ∂ +1 ( γ ) = ∂ − (Γ), and whiskeringof n -paths by 1-paths for arbitrary n .It is easy to show that these whiskerings descend to an action of the groupoid S ( M ) on the categories S l ( M ) and S s ( M ). The main part of the proof isto show that, in the laminated case, [Γ] ♮ [ γ ] . = [Γ ♮ γ ] does not depend onthe representatives Γ and γ chosen. Suppose we have a rank 1-homotopy H connecting γ and γ and a laminated rank-2 homotopy J connecting Γ andΓ . Then ( J♮ γ ) ♮ (id(Γ ) ♮ id( H )) is a laminated rank-2 homotopy connectingΓ ♮ γ and Γ ♮ γ . Here id(Γ )( t, s, x ) = Γ ( t, s ) and id( H )( t, s, x ) = H ( t, x ),where t, s, x ∈ [0 , Γ ′ γ γ ′ Γ γ ′ x γ γ Γ ′ x ˆΓ ′ x Γ γ ′ x γ γ Γ ′ x ˆΓ ′ x Γ γ ′ x γ γ Γ ′ x ˆΓ ′ x Γ Γ ′ γ γ ′ x = x = x = x =0 x =1 Figure 1: Slices of Γ ′ for x = 0, x = 1 / x = 1 / x = 3 / x = 1. Let Γ and Γ ′ be two 2-paths with ∂ +1 (Γ) = ∂ − (Γ ′ ). We can consider the obvioushorizontal concatenation Γ ♮ Γ ′ , denoted by ΓΓ ′ . There are however two othernatural ways to define the horizontal composition of Γ and Γ ′ . These are:Γ ♮ − Γ ′ = (cid:0) Γ ♮ ∂ − (Γ ′ ) (cid:1) ♮ (cid:0) ∂ +2 (Γ) ♮ Γ ′ (cid:1) = Γ ′ Γ ! and Γ ♮ +1 Γ ′ = (cid:0) ∂ − (Γ) ♮ Γ ′ (cid:1) ♮ (cid:0) Γ ♮ ∂ +2 (Γ ′ ) (cid:1) = Γ Γ ′ ! It is easy to see that these three horizontal compositions descend to the quo-tient S s ( M ) of S ( M ) under strong rank-2 homotopy and they all coincide. Inaddition the interchange law between the horizontal and vertical compositionsholds. In fact we have: Theorem 21
The horizontal and vertical composition of strong 2-tracks definesa 2-groupoid S s ( M ) with objects given by the set of points of M , 1-morphismsgiven by S ( M ) and 2-morphisms by S s ( M ) . See [FMP1, FMP2, MP, SW2] for details. The definition of a 2-groupoid ap-pears, for example, in [HKK]. For related constructions see [BH1, BHS, BH3,BHS, HKK].
Let us now look at the behaviour of the horizontal composition under the rela-tion of laminated rank-2 homotopy. We can see that ♮ does not descend to thequotient. However ♮ − and ♮ +1 do descend, even though they do not coincide.Given 2-paths Γ and Γ ′ with ∂ +1 (Γ) = ∂ − (Γ ′ ), let J = Γ ′ be the 3-pathwhose typical slices as x varies appear in figure 1. We have put γ = ∂ +2 (Γ), γ = ∂ − (Γ), γ ′ x ( t ) = Γ ′ ( t, x ), Γ ′ x ( t, s ) = Γ ′ ( t, xs ) and ˆΓ ′ x ( t, s ) = Γ ′ ( t, x + s (1 − x )),where t, s, x ∈ [0 , ′ x ♮ ˆΓ ′ x = Γ ′ , for each x ∈ [0 , ′ is well defined up to reparametrisationsin the x -direction. Below (see subsection 2.3) we define an equivalence relationon 3-paths which solves this ambiguity.26ote that ∂ − (Γ ′ ) = Γ ♮ − Γ ′ and ∂ +3 (Γ ′ ) = Γ ♮ +1 Γ ′ . This will give us theinterchange 3-cells; see 1.2.4.
Definition 22 (Good 3-path)
A 3-path ( t, s, x ) J ( t, s, x ) is called good if ∂ ± ( J ) each are independent of x . Definition 23 (rank-3 homotopy (with laminated boundary))
We willsay that two good 3-paths J and J ′ are rank-3 homotopic (with laminated bound-ary), and we write J ∼ = J ′ if there exists a 4-path ( t, s, x, u ) ∈ D W ( t, s, x, u ) ∈ M such that:1. We have ∂ − ( W ) = J and ∂ +4 ( W ) = J ′ , in other words W ( t, s, x,
0) = J ( t, s, x ) and W ( t, s, x,
1) = J ′ ( t, s, x ) , where t, s, x, u ∈ [0 , .2. The restriction W ( t, , x, u ) is independent of x and defines a rank-1 ho-motopy connecting W ( t, , x, and W ( t, , x, (each independent of x ,therefore identified with paths in M ), and the same for the restriction W ( t, , x, u ) .3. The restriction W ( t, s, , u ) defines a laminated rank-2 homotopy connect-ing J ( t, s, and J ′ ( t, s, , and analogously for W ( t, s, , u ) .4. For each v ∈ [0 , we have Rank ( D W ( v )) ≤ . We denote S ( M ) as being the set of all good 3-paths up to rank-3 homotopy(with laminated boundary). The elements of S ( M ) will be called 3-tracks.Notice that the interchange 3-track Γ ′ of two 2-tracks with ∂ +1 (Γ) = ∂ − (Γ ′ ) gives rise to a well defined element of S ( M ). In addition, all boundaries ∂ ± i , i = 1 , , ∂ ± : S ( M ) → M , ∂ ± : S ( M ) →S ( M ) and ∂ ± : S ( M ) → S ( M ), which are the maps needed to get the struc-ture of a Gray 3-groupoid (1.2.4.)The following lemma will be useful later. The proof is achieved by using afilling argument very similar to the proof of Theorem A of [BH2]. Lemma 24
Two good 3-paths J and J ′ are rank-3 homotopic (with laminatedboundary) if, and only if, there exists a 4-path W : D → M satisfying theconditions 1,3,4 of definition 23 but with condition replaced by:
2’ The restriction W ( t, , x, u ) defines a laminated rank-2 homotopy whichconnects the 2-paths W ( t, , x,
0) and W ( t, , x, W ( t, , x, u ). Proof.
One of the implications (2 = ⇒ ′ ) is immediate. Let us prove thereciprocal. We will just discuss how to deal with the condition on the s = 1face, since the other face is dealt with analogously.27f W is a homotopy connecting J and J ′ as in the statement of the lemma,we substitute W by W ♮ V , where V is defined in the following way: con-sider a smooth retraction r : D → ∂D \ { x = 1 } . Define a smooth function U : (cid:0) ∂D \ { x = 1 } (cid:1) × [0 , → M , as:1. U ( t, , x, u ) = W ( t, , x, u ), the right hand side is a laminated rank-2 ho-motopy.2. In (cid:0) ∂ ( D ) \ ( { x = 1 } ∪ { s = 0 } ) (cid:1) × [0 ,
1] we put U ( t, s, x, u ) = W ( t, , , u ).Note W ( t, , , u ) is a rank-1 homotopy.Finally, let V ( t, s, x, u ) = U ( r ( t, s, x ) , u )). Then W ♮ V is a rank-3 homotopywith laminated boundary, and will give us, after a very minor adjustment (mak-ing use of the path-space thinness condition), a rank-3 homotopy with laminatedboundary connecting J and J ′ . Let J and J ′ be good 3-paths with ∂ +2 ( J ) = ∂ − ( J ′ ). Recall that we can performtheir vertical composition J♮ J ′ . If [ J ] and [ J ′ ] are such that ∂ +2 ([ J ]) = ∂ − ([ J ′ ]),then choosing a rank-1 homotopy H connecting ∂ +2 ( J ) and ∂ − ( J ′ ) permits us toput [ J ] ♮ [ J ′ ] . = [ J♮ id( H ) ♮ J ′ ], where id( H )( t, s, x ) = H ( t, s ), for each t, s, x ∈ [0 , Let J and J ′ be good 3-paths with ∂ +3 ( J ) = ∂ − ( J ′ ). Consider the composition J♮ J ′ , called upwards composition. Suppose that we have ∂ +3 ( J ) ∼ = l ∂ − ( J ′ ).Choose a laminated rank-2 homotopy H connecting ∂ +3 ( J ) and ∂ − ( J ′ ), andput: [ J ] ♮ [ J ′ ] . = [ J♮ H♮ J ′ ]. The proof of the following essential lemma (veryinspired by [BH2, BHS]) will be very similar to the proof of Lemma 24 and willmake use of it. Lemma 25
This upwards composition is well defined in S ( M ) . Proof.
Suppose J ∼ = J and J ′ ∼ = J ′ , and choose rank-3 homotopies with lam-inated boundary A and A ′ yielding these equivalences. Choose also laminatedrank-2 homotopies with H connecting ∂ +3 ( J ) and ∂ − ( J ′ ) and H connecting ∂ +3 ( J ) and ∂ − ( J ′ ).Consider the map W : D → M defined in the following way: we fill the x = 1 and x = 0 faces of D with the laminated rank-2 homotopies ∂ − ( A ′ ) and ∂ +3 ( A ). Then we fill the u = 0 and u = 1 faces of D with H and H ′ . Theboundary of the s = 0 face of D will define a rank-1 homotopy U ( t, v ) connect-ing H ( t, ,
0) with itself. Explicitly it is given by the following concatenation of28ank-1 homotopies: H ( t, , H ( t, ,x ) −−−−−→ H ( t, , A ′ ( t, , ,u ) −−−−−−−→ A ′ ( t, , ,
1) = H ′ ( t, , H ′ ( t, , − x ) −−−−−−−−→ H ′ ( t, , A ( t, , , − u ) −−−−−−−−→ H ( t, , s = 0 face of D can therefore be filled with alaminated rank-2 homotopy. To extend W to the rest of D put W ( t, s, x, u ) = W ( t, r ( s, x, u )), where r is a smooth retraction of D onto ∂D \ { s = 1 } .Then A♮ W ♮ A ′ satisfies the conditions of Lemma 24, which implies thereexists a rank-3 homotopy with laminated boundary connecting J♮ H♮ J ′ and J ♮ H ♮ J ′ . The treatment is entirely similar to what was presented in 2.2.6. S ( M ) of a smoothmanifold M Combining all of the above we have:
Theorem 26
Let M be a smooth manifold. The sets of 1-tracks, laminated2-tracks and 3-tracks can be arranged into a Gray 3-groupoid S ( M ) = (cid:0) M, S ( M ) , S l ( M ) , S ( M ) (cid:1) whose set of objects is M . Proof.
We need to verify the conditions in 1.2.4. Conditions 1. to 3. are trivial.Note that ∂ ± , applied to good 3-paths, can be naturally regarded as mappingto 1-paths.The difficult bit (existence of compositions) of conditions 4. to 10. arealready proved, and all the rest follows straightforwardly by the definition oflaminated and rank-3 homotopy (with laminated boundary), as in the construc-tion in [FMP1, FMP2, MP, SW2]. We have already proved the existence of aninterchange 3-cell; 2.2.8. The fact that it verifies conditions 13. and 14. followssince both sides of each equation can be connected by rank-3 homotopies witha laminated boundary. Fix a smooth manifold M . We will make use of Chen’s definition of differentialforms in the smooth space of smooth paths in M , as well as iterated integralsof differential forms; see [Ch]. For conventions see the Appendix.The main result of this section is: 29 heorem 27 Consider a 2-crossed module H = ( L δ −→ E ∂ −→ G, ⊲, { , } ) withassociated differential 2-crossed module ( l δ −→ e ∂ −→ g , ⊲, { , } ) . Let M be a smoothmanifold. Consider differential forms ω ∈ A ( M, g ) , m ∈ A ( M, e ) and θ ∈A ( M, l ) such that δ ( θ ) = M and ∂ ( m ) = Ω , where Ω = dω + [ ω, ω ] = dω + ω ∧ ad ω and M = dm + ω ∧ ⊲ m denote the curvature of ω and 2-curvature3-form of the pair ( m, ω ) ; see [FMP1, FMP2, BS, SW2, SW3].Then we can define a (smooth) strict Gray 3-groupoid functor (definition 12) ( ω,m,θ ) H : S ( M ) → C ( H ) , where C ( H ) is the Gray 3-groupoid constructed from H (and whose sets of objectsand 1,2 and 3-morphisms are smooth manifolds.) The explicit description of H appears in 3.3.3. At the level of 1-paths H co-incides with the usual holonomy of a non-abelian 1-form, whereas for crossedmodules it yields the 2-dimensional holonomy of [SW2, BS, FMP2]. Definition 28
Given ω, m and θ as above, the 3-curvature 4-form Θ of ( ω, m, θ ) is given by Θ = dθ + ω ∧ ⊲ θ − m ∧ { , } m ; see the appendix for this notation. Here m ∧ { , } m is the antisymmetrisation of { m, m } . Let G be a Lie group with Lie algebra g and A : R → g be a smooth map. Let F A ( t , t ) be the solution of the differential equation: ∂∂t F A ( t , t ) = F A ( t , t ) A ( t ) , with F A ( t , t ) = 1 G It thus follows that ∂∂t F A ( t , t ) = − A ( t ) F A ( t , t ) . Moreover F A ( t , t + t ′ ) = F A ( t , t ) F A ( t, t ′ ) . More generally, suppose that ( t, s ) ∈ R A s ( t ) ∈ g is a smooth map. It iswell known (and not difficult to prove) that: ∂∂s F A s ( t , t ) = Z tt F A s ( t , t ′ ) ∂∂s A s ( t ′ ) F A s ( t ′ , t ) dt ′ . Let ω be a 1-form in the manifold M . Let γ : [0 , → M be a smoothcurve in M , and A be given by γ ∗ ( ω ) = A ( ˙ γ ). Put g ωγ ( t , t ) = F A ( t , t ). Note g ωγ ( t , t + t ′ ) = g ωγ ( t , t ) g ωγ ( t, t ′ ) and g ωγ ( t, s ) = g ωγ ( s, t ) − .30uppose s ∈ I γ s is a smooth one-parameter family of smooth curves in M . In other words the map Γ : [0 , → M such that Γ( t, s ) = γ s ( t ) for each s, t ∈ [0 ,
1] is smooth. We have: ∂∂s g ωγ s ( a, b )= Z ba g ωγ s ( a, t ) ∂∂s ω (cid:18) ∂∂t γ s ( t ) (cid:19) g ωγ s ( t, b ) dt = Z ba g ωγ s ( a, t ) (cid:18) dω (cid:18) ∂∂s γ s ( t ) , ∂∂t γ s ( t ) (cid:19) + ∂∂t ω (cid:18) ∂∂s γ s ( t ) (cid:19)(cid:19) g ωγ s ( t, b ) dt = g ωγ s ( a, b ) Z ba g ωγ s ( b, t ) (cid:18) dω (cid:18) ∂∂s γ s ( t ) , ∂∂t γ s ( t ) (cid:19) + ∂∂t ω (cid:18) ∂∂s γ s ( t ) (cid:19)(cid:19) g ωγ s ( t, b ) dt Now note that (integrating by parts): Z ba ( g ωγ s ( t, b )) − ∂∂t ω (cid:18) ∂∂s γ s ( t ) (cid:19) g ωγ s ( t, b ) dt = ( g ωγ s ( t, b )) − ω (cid:18) ∂∂s γ ( t, s ) (cid:19) ( g ωγ s ( t, b )) (cid:12)(cid:12)(cid:12)(cid:12) t = bt = a + Z ba ( g ωγ s ( t, b )) − (cid:20) ω (cid:18) ∂∂s γ s ( t ) (cid:19) , ω (cid:18) ∂∂t γ s ( t ) (cid:19)(cid:21) g ωγ s ( t, b ) dt And therefore, putting Ω = dω + [ ω, ω ] as being the curvature of ω (and where[ ω, ω ]( X, Y ) = [ ω ( X ) , ω ( Y )]), we get the following very well known lemma: Lemma 29
Let ω ∈ A ( M, g ) be a g -valued 1-form in M . We have: ∂∂s g ωγ s ( a, b ) = − g ωγ s ( a, b ) Z ba ( g ωγ s ( t, b )) − Ω (cid:18) ∂∂t γ s ( t ) , ∂∂s γ s ( t ) (cid:19) g ωγ s ( t, b ) dt + g ωγ s ( a, b ) ( g ωγ s ( t, b )) − ω (cid:18) ∂∂s γ ( t, s ) (cid:19) ( g ωγ s ( t, b )) (cid:12)(cid:12)(cid:12)(cid:12) t = bt = a ! . (14) This can also be written as: ∂∂s g ωγ s ( a, b ) − = g ωγ s ( a, b ) − Z ba g ωγ s ( a, t )Ω (cid:18) ∂∂t γ s ( t ) , ∂∂s γ s ( t ) (cid:19) g ωγ s ( t, a ) − dt − g ωγ s ( a, b ) − g ωγ s ( a, t ) ω (cid:18) ∂∂s γ ( t, s ) (cid:19) ( g ωγ s ( a, t )) − (cid:12)(cid:12)(cid:12)(cid:12) t = bt = a ! . (15)We thus arrive at the following well known result; see [CP, MP, FMP1, BS]. Corollary 30
One dimensional holonomy based on a Lie group is invariantunder rank-1 homotopy. More precisely, if γ and γ ′ are rank-1 homotopic 1-paths then g ωγ (0 ,
1) = g ωγ ′ (0 , . .1.1 A useful lemma of Baez and Schreiber Suppose that the group G has a left action ⊲ , by automorphisms, on the Liegroup E , with Lie algebra e . Let ∗ , ∗ ′ ∈ M . We define P ( M, ∗ , ∗ ′ ) to be thespace of all smooth paths γ : [0 , → M that start at ∗ and finish at ∗ ′ .Let U be an open set of M . Let f : ( t, x ) ∈ [0 , × U 7→ γ x ( t ) = f t ( x ) ∈ M define a smooth map F : U → P ( M, ∗ , ∗ ′ ), which is the same as saying that f is smooth. Note that we have γ x (0) = ∗ and γ x (1) = ∗ ′ , ∀ x ∈ U . Let A be a( n + 1)-form in M with values in e . Let ω be a 1-form in M with values in g .Put D ω A = dA + ω ∧ ⊲ A , the exterior covariant derivative of A .Consider the twisted iterated integral: I b g ωγ x ⊲ f ∗ ( A ) = Z b g ωγ x (0 , t ) ⊲ (cid:16) ι ∂∂t f ∗ ( A ) (cid:17) dt ∈ A n ( U , e ) . (16)By using the first equation of the previous lemma we have d ( g ωγ x (0 , t )) = − (cid:18)Z t ( g ωγ x (0 , t ′ )) ⊲ ad ι ∂∂t f ∗ (Ω) dt ′ (cid:19) g ωγ x (0 , t ) + g ωγ x (0 , t ) f ∗ ( ω ) . Here the map ( t, x ) g ωγ x (0 , t ) ∈ G should be seen (through the action ⊲ ) astaking values in the vector space of linear maps e → e . Applying equation (23)from the Appendix it thus follows: d I b g ωγ x ⊲ f ∗ ( A )= − I b g ωγ x ⊲ D ω f ∗ ( A ) − I b (cid:0) g ωγ x ⊲ ad f ∗ (Ω) (cid:1) ∗ ⊲ (cid:0) g ωγ x ⊲ f ∗ ( A ) (cid:1) + f ∗ b (cid:0) g ωγ x ⊲ A (cid:1) We have used the identity ι X ( α ∧ β ) = ι X ( α ) ∧ β + J ( α ) ∧ ι X ( β ), valid for anytwo forms α and β . Recall J ( α ) = ( − n α , where n is the degree of α .We define the following form in the loop space P ( M, ∗ , ∗ ′ ): I ω A = I (cid:0) g ωγ (0 , t ) (cid:1) ⊲ A, and analogously for iterated integrals. We are following the notation (but notthe conventions) of [BS]. In other words: F ∗ (cid:18)I ω A (cid:19) = I g ωF ( x ) (0 , t ) ⊲ f ∗ ( A ) dt, for each plot F : U → P ( M, ∗ , ∗ ′ ), the map f : I × U → M being f ( t, x ) = F ( x )( t ), where x ∈ U and t ∈ I .We thus have the following very useful lemma which appeared in [BS].32 emma 31 (Baez-Schreiber) Let the Lie group G act on the Lie group E byautomorphisms. Consider a smooth manifold M . Let A be a ( n + 1) -form in M with values in e . Let ω be a 1-form in M with values in g . We have: d I ω A = − I ω D ω A − I ω Ω ∗ ⊲ A, (17) where D ω A = dA + ω ∧ ⊲ A is the exterior covariant derivative of A with respect to ω . Suppose that G = ( ∂ : E → G, ⊲ ) is a (as usual Lie) pre-crossed module. Let G = ( ∂ : e → g , ⊲ ) be the associated differential pre-crossed module. Let ω ∈A ( M, g ) be a g -valued smooth 1-form in M . Let m ∈ A ( M, e ) be an arbitrary e -valued 2-form in M . (Later we will put the restriction ∂ ( m ) = Ω, whereΩ = dω + [ ω, ω ] = dω + ω ∧ ⊲ w is the curvature of ω ). See the Appendix fornotation.Given a smooth map Γ : ( t, s ) ∈ [0 , γ s ( t ) ∈ M , defining therefore asmooth map (plot) s ∈ [0 , ˆΓ γ s ∈ P ( M, ∗ , ∗ ′ ), define e ( ω,m )Γ ( s , s ) as thesolution of the differential equation: dds e ( ω,m )Γ ( s , s ) = e ( ω,m )Γ ( s , s ) Z g ωγ s (0 , t ) ⊲ m (cid:18) ∂∂t γ s ( t ) , ∂∂s γ s ( t ) (cid:19) dt, (18)with initial condition e ( ω,m )Γ ( s , s ) = 1 E . In other words dds e ( ω,m )Γ ( s , s ) = e ( ω,m )Γ ( s , s )ˆΓ ∗ (cid:18)I ω m (cid:19) ;see 3.1.1 and 5.2 for this notation. Note that (by using Lemma 31) the curvatureof the 1-form H ω m (in the path space P ( M, ∗ , ∗ ′ )) is d I ω m + (cid:20)I ω m, I ω m (cid:21) = − I ω M − I ω Ω ∗ ⊲ m + (cid:20)I ω m, I ω m (cid:21) , where M = dm + ω ∧ ⊲ m is defined as being the 2-curvature 3-form of ( ω, m ).By using Lemma 29, it thus follows that: Theorem 32
Let M be a smooth manifold. Suppose G = ( ∂ : E → G, ⊲ ) isa Lie pre-crossed module, with associated differential pre-crossed module G =( ∂ : e → g , ⊲ ) . Let ω ∈ A ( M, g ) and m ∈ A ( M, e ) . Let M = dm + ω ∧ ⊲ m bethe 2-curvature of ( ω, m ) . Suppose that ( t, s, x ) ∈ [0 , Γ x ( t, s ) = Γ s ( t, x ) = γ xs ( t ) = J ( t, s, x ) ∈ M is smooth. Suppose that γ xs (0) = ∗ and γ xs (1) = ∗ ′ for ach s, x ∈ [0 , . Let ˆ J : [0 , → P ( M, ∗ , ∗ ′ ) be the associated plot, ( s, x ) γ xs ,We have: ddx e ( ω,m )Γ x (0 , a )= Z a e ( ω,m )Γ x (0 , s ) ˆ J ∗ (cid:18)I ω M + I ω Ω ∗ ⊲ m − (cid:20)I ω m, I ω m (cid:21)(cid:19) (cid:18) ∂∂s , ∂∂x (cid:19) e ( ω,m )Γ x ( s, a ) ds + e ( ω,m )Γ x (0 , a ) ˆΓ a ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19) − ˆΓ ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19) e ( ω,m )Γ x (0 , a ) . Explicitly: ddx e ( ω,m )Γ x (0 , a )= Z a Z e ( ω,m )Γ x (0 , s ) (cid:18) g ωγ xs (0 , t ) ⊲ M (cid:18) ∂∂t γ xs ( t ) , ∂∂s γ xs ( t ) , ∂∂x γ xs ( t ) (cid:19)(cid:19) e ( ω,m )Γ x ( s, a ) dtds + Z a Z e ( ω,m )Γ x (0 , s ) (cid:18)Z t g ωγ xs (0 , t ′ ) ⊲ ad Ω (cid:18) ∂∂t ′ γ xs ( t ′ ) , ∂∂s γ xs ( t ′ ) (cid:19) dt ′ (cid:19) g ωγ xs (0 , t ) ⊲m (cid:18) ∂∂t γ xs ( t ) , ∂∂x γ xs ( t ) (cid:19) e ( ω,m )Γ x ( s, a ) dtds − Z a Z e ( ω,m )Γ x (0 , s ) (cid:18)Z t g ωγ xs (0 , t ′ ) ⊲ ad Ω (cid:18) ∂∂t ′ γ xs ( t ′ ) , ∂∂x γ xs ( t ′ ) (cid:19) dt ′ (cid:19) g ωγ xs (0 , t ) ⊲m (cid:18) ∂∂t γ xs ( t ) , ∂∂s γ xs ( t ) (cid:19) e ( ω,m )Γ x ( s, a ) dtds − Z a e ( ω,m )Γ x (0 , s ) h Z g ωγ xs (0 , t ′ ) ⊲ m (cid:18) ∂∂t ′ γ xs ( t ′ ) , ∂∂s γ xs ( t ′ ) (cid:19) dt ′ , Z g ωγ xs (0 , t ) ⊲ m (cid:18) ∂∂t γ xs ( t ) , ∂∂x γ xs ( t ) (cid:19) dt i e ( ω,m )Γ x ( s, a ) ds + e ( ω,m )Γ x (0 , a ) Z g ωγ xa (0 , t ) ⊲ m (cid:18) ∂∂t γ xa ( t ) , ∂∂x γ xa ( t ) (cid:19) dt − (cid:18)Z g ωγ x (0 , t ) ⊲ m (cid:18) ∂∂t γ x ( t ) , ∂∂x γ x ( t ) (cid:19) dt (cid:19) e ( ω,m )Γ x (0 , a )Now note that for any smooth functions f and g we have [ R f ( t ) dt, R g ( t ′ ) dt ′ ] = R [ f ( t ) , R t g ( t ′ ) dt ′ ] dt + R [ R t f ( t ′ ) dt ′ , g ( t )] dt , and use it in the fourth term. Con-sider also the definition of the Peiffer commutators h u, v i = [ u, v ] − ∂ ( u ) ⊲v, where u, v ∈ e . We obtain the following result: Corollary 33
Under the conditions of the previous theorem, and, furthermore, ssuming that ∂ ( m ) = Ω , we have: ∂∂x e ( ω,m )Γ x (0 , a )= Z a e ( ω,m )Γ x (0 , s ) ˆ J ∗ (cid:18)I ω M − I ω m ∗ h , i m (cid:19) (cid:18) ∂∂s , ∂∂x (cid:19) e ( ω,m )Γ x ( s, a ) ds + e ( ω,m )Γ x (0 , a ) ˆΓ a ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19) − ˆΓ ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19) e ( ω,m )Γ x (0 , a ) , which can also be written as: ∂∂x e ( ω,m )Γ x (0 , a ) − = − e ( ω,m )Γ x (0 , a ) − Z a e ( ω,m )Γ x (0 , s ) ˆ J ∗ (cid:18)I ω M − I ω m ∗ h , i m (cid:19) (cid:18) ∂∂s , ∂∂x (cid:19) e ( ω,m )Γ x ( s, ds − ˆΓ a ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19) e ( ω,m )Γ x (0 , a ) − + e ( ω,m )Γ x (0 , a ) − ˆΓ ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19) . (19) Explicitly (looking at the first expression): ddx e ( ω,m )Γ x (0 , a )= Z a Z e ( ω,m )Γ x (0 , s ) (cid:18) g ωγ xs (0 , t ) ⊲ M (cid:18) ∂∂t γ xs ( t ) , ∂∂s γ xs ( t ) , ∂∂x γ xs ( t ) (cid:19)(cid:19) e ( ω,m )Γ x ( s, a ) dtds − Z a Z e ( ω,m )Γ x (0 , s ) D Z t g ωγ xs (0 , t ′ ) ⊲ m (cid:18) ∂∂t ′ , ∂∂s γ xs ( t ′ ) (cid:19) dt ′ ,g ωγ xs (0 , t ) ⊲ m (cid:18) ∂∂t γ xs ( t ) , ∂∂x γ xs ( t ) (cid:19) e ( ω,m )Γ x ( s, a ) dt E ds + Z a Z e ( ω,m )Γ x (0 , s ) D Z t g ωγ xs (0 , t ′ ) ⊲m (cid:18) ∂∂t ′ γ xs ( t ′ ) , ∂∂x γ xs ( t ′ ) (cid:19) dt ′ ,g ωγ xs (0 , t ) ⊲ m (cid:18) ∂∂t γ xs ( t ) , ∂∂s γ xs ( t ) (cid:19) e ( ω,m )Γ x ( s, a ) dt E ds + e ( ω,m )Γ x (0 , a ) Z g ωγ xa (0 , t ) ⊲ m (cid:18) ∂∂t γ xa ( t ) , ∂∂x γ xa ( t ) (cid:19) dt − (cid:18)Z g ωγ x (0 , t ) ⊲ m (cid:18) ∂∂t γ x ( t ) , ∂∂x γ x ( t ) (cid:19) dt (cid:19) e ( ω,m )Γ x (0 , a ) . Note that if ( ∂ : E → G, ⊲ ) is a crossed module then all terms involving Peiffercommutators vanish. This will be of prime importance now and later. Using this last result it follows that:
Corollary 34
The two dimensional holonomy based on a pre-crossed module ( ∂ : E → G, ⊲ ) is invariant under laminated rank-2 homotopy. More precisely, f Γ , Γ ′ are 2-paths which are laminated rank-2 homotopic, and m ∈ A ( M, e ) and ω ∈ A ( M, g ) are such that ∂ ( m ) = Ω = dω + [ ω, ω ] then e ( ω,m )Γ (0 ,
1) = e ( ω,m )Γ ′ (0 , . Two dimensional holonomy based on a crossed module (where, bydefinition, the Peiffer commutators vanish) is invariant under strong rank-2homotopy. Proof. If J ( t, s, x ) = γ xs ( t ) is a laminated rank-2 homotopy then the right handside of the last equation in the previous lemma vanishes. If J is a strong rank-2homotopy, then in principle only the first term vanishes, but this is compensatedby the fact that the Peiffer pairing is zero. Fix a pre-crossed module ( ∂ : E → G, ⊲ ), with associated differential 2-crossedmodule ( ∂ : e → g , ⊲ ). The next result follows directly from the unicity theoremfor ordinary differential equations, together with equations (15) and (18). Lemma 35 (Non-abelian Green Theorem)
Let Γ be a 2-path in M . Let m ∈ A ( M, e ) and ω ∈ A ( M, g ) be such that ∂ ( m ) = Ω = dω + [ ω, ω ] . We thenhave: ∂ (cid:16) e ( ω,m )Γ ( s, s ′ ) (cid:17) − g ωγ s = g ωγ s ′ , for each s, s ′ ∈ [0 , . Recall that we put Γ( t, s ) = γ s ( t ) , for each s, t ∈ [0 , and g ωγ s . = g ωγ s (0 , . Combining corollaries 30 and 34 with the Non-abelian Green Theorem itfollows that (recall the notation of 2.3.4):
Theorem 36
Given ( ω, m ) as in the previous lemma, the assignments: γ ∈ S ( M ) g ωγ ∈ G, and Γ ∈ S l ( M ) (cid:16) g ω∂ − (Γ) , e ( ω,m )Γ (cid:17) ∈ G × E, where g ωγ . = g ωγ (0 , and e ( ω,m )Γ . = e ( ω,m )Γ (0 , satisfy all the axioms for a mor-phisms of Gray 3-groupoids S ( M ) → C ( H ) which do not involve 3-morphisms(is short it defines a morphism of sesquigroupoids, [S]). Proof.
What is left to prove is entirely analogous to the proof of Theorem 39of [FMP1].
Remark 37 If ( ∂ : E → G, ⊲ ) is a crossed module then the assignments: γ ∈ S ( M ) g ωγ ∈ G, and Γ ∈ S s ( M ) (cid:16) g ω∂ − (Γ) , e ( ω,m )Γ (cid:17) ∈ G × E, define a morphism of 2-groupoids; see 2.2.7. This result appears in [BS, SW2,FMP1, FMP2]. .3 Three-Dimensional holonomy based on a 2-crossed mod-ule Let ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module, of Lie groups, and let ( l δ −→ e ∂ −→ g , ⊲, { , } ) be the associated differential 2-crossed module. Recall that we have aleft action ⊲ ′ of E on L defined as e ⊲ ′ l = l { δ ( l ) − , e } , where e ∈ E and l ∈ L .The differential form of this action is v ⊲ ′ x = −{ δ ( x ) , v } , where v ∈ e and x ∈ l .Together with the boundary map δ : L → E , this defines a crossed module.This permits us to reduce the analysis of 3-dimensional holonomy based on a2-crossed module to the analysis of a 2-dimensional holonomy in the path space,based on a crossed module. Compare with 3.2.1 and [FMP1, FMP2, SW2, BS].Suppose we are given forms ω ∈ A ( M, g ), m ∈ A ( M, e ) and θ ∈ A ( M, l ) . We suppose that ∂ ( m ) = Ω = dω + [ ω, ω ] and δ ( θ ) = M = dm + ω ∧ ⊲ m . Notethat ∂δ ( θ ) = d Ω + ω ∧ ad Ω = 0, as it should, by the Bianchi identity. We definethe 3-curvature 4-form Θ of ( ω, m, θ ) asΘ = dθ + ω ∧ ⊲ θ − m ∧ { , } m, where the 4-form m ∧ { , } m is the antisymmetrisation of the contravariant tensor6 { m, m } . See the Appendix for notation.Let J : [0 , → M be a good 3-path, Definition 22. Put J ( t, s, x ) =Γ x ( t, s ) = γ xs ( t ) , for each t, s, x ∈ [0 , J : [0 , → P ( M, ∗ , ∗ ′ ) isdefined as ( s, x ) γ xs ; see subsection 5.2. Here ∗ = ∂ − ( J ) and ∗ ′ = ∂ +1 ( J ).We define l ( ω,m,θ ) J ( x , x ) = l J ( x , x ) as being the solution of the differentialequation: ∂∂x l J ( x , x )= − l J ( x , x ) Z e ( ω,m )Γ x (0 , s ) ⊲ ′ ˆ J ∗ (cid:18)I ω θ − I ω m ∗ { , } m (cid:19) (cid:18) ∂∂s , ∂∂x (cid:19) ds (20)with initial condition l J ( x , x ) = 1 L . Compare with (19). Let ∗ , ∗ ′ ∈ M . Consider the following ( e -valued and l -valued) forms in the pathspace P ( M, ∗ , ∗ ′ ), of smooth paths in M that start in ∗ and finish in ∗ ′ : A = I ω mB = − I ω θ + I ω m ∗ { , } m, and note, see above, that δ ( B ) coincides with the curvature dA + [ A, A ] = − H ω M + H ω m ∗ <,> m of A . 37et us calculate the 2-curvature 3-form C = dB + A ∧ ⊲ ′ B (in the path space)of the pair ( A, B ). Recall that ( δ : l → e , ⊲ ′ ) is a differential crossed module.First of all note that, by using the results of 3.1.1 and the Appendix, it followsthat (taking into account that X ⊲ { u, v } = { X ⊲ u, v } + { u, X ⊲ v } )): d I ω m ∗ { , } m = − I ω D ω m ∗ { , } m + I ω m ∗ { , } D ω m − I ω m ∧ { , } m − I ω Ω ∗ ⊲ (cid:16) m ∗ { , } m (cid:17) + I ω m ∗ { , } (Ω ∗ ⊲ m )Thus (by using 3.1.1 again:) C = I ω D ω θ + I ω Ω ∗ ⊲ θ − I ω M ∗ { , } m + I ω m ∗ { , } M − I ω m ∧ { , } m − I ω Ω ∗ ⊲ (cid:16) m ∗ { , } m (cid:17) + I ω m ∗ { , } (Ω ∗ ⊲ m ) − I ω m ∧ ⊲ ′ I ω θ + I ω m ∧ ⊲ ′ I ω m ∗ { , } m. Recall that M = D ω m . Then note that, since v ⊲ ′ x = −{ δ ( x ) , v } for each v ∈ e and x ∈ l : I ω m ∧ ⊲ ′ I ω θ = − I ω m ∧ { , } op I ω δ ( θ ) = − I ω m ∧ { , } op I ω M = − I M ∧ { , } I ω m. Therefore: I ω M ∗ { , } m + I ω m ∧ ⊲ ′ I ω θ = − I ω m ∗ { , } op M . By using condition 6 of the definition of a differential 2-crossed module togetherwith δ ( θ ) = M and ∂ ( m ) = Ω: I ω Ω ∗ ⊲ θ + I ω m ∗ { , } M + I ω m ∗ { , } op M = 0 . Therefore: C = I ω D ω θ − I ω m ∧ { , } m − I ω Ω ∗ ⊲ (cid:16) m ∗ { , } m (cid:17) + I ω m ∗ { , } (Ω ∗ ⊲ m )+ I ω m ∧ ⊲ ′ I ω m ∗ { , } m. By using condition 5. of the definition of a differential 2-crossed module follows: I ω m ∧ ⊲ ′ I ω m ∗ { , } m . = − (cid:18)I ω m ∗ h , i m (cid:19) ∧ { , } I ω m = − I ω m ∗ { , } op (cid:16) m ∗ h , i m (cid:17) − I ω m ∗ { , } (cid:16) m ∗ [ , ] m (cid:17) . h u, v i . = [ u, v ] − ∂ ( u ) ⊲ v for each u, v ∈ e : C = I ω D ω θ − I ω m ∧ { , } m − I ω Ω ∗ ⊲ (cid:16) m ∗ { , } m (cid:17) − I ω m ∗ { , } op (cid:0) m ∗ <,> m (cid:1) − I ω m ∗ { , } (cid:0) m ∗ <,> m (cid:1) . By using now condition 6. of the definition of a 2-crossed module it followsthat the 2-curvature 3-form C = dA + A ∧ ⊲ ′ B of the pair ( A, B ) is C = I ω D ω θ − I ω m ∧ { , } m . = I ω Θ , recall that Θ = D ω θ − m ∧ { , } m = dθ + ω ∧ ⊲ θ − m ∧ { , } m denotes the 3-curvature4-form of ( ω, m, θ ). By using Corollary 33 for the crossed module ( δ : L → E, ⊲ ′ )(or [FMP1, FMP2]) follows: Theorem 38
Let M be a smooth manifold and ∗ , ∗ ′ ∈ M . Let ( L δ −→ E ∂ −→ G, ⊲, { , }} be a 2-crossed module, and let ( l δ −→ e ∂ −→ g , ⊲, { , } ) be the associateddifferential 2-crossed module. Choose forms ω ∈ A ( M, g ) , m ∈ A ( M, e ) and θ ∈ A ( M, l ) such that δ ( θ ) = M = D ω m and ∂ ( m ) = Ω , the curvature of ω .Let W : D → M be a smooth map such that ∂ +1 ( W ) = ∗ ′ and ∂ − ( W ) = ∗ ,defining therefore a plot ˆ W : [0 , → P ( M, ∗ , ∗ ′ ) . Let W ( t, s, x, u ) = J u ( t, s, x ) = J x ( t, s, u ) = Γ ux ( t, s ) , where t, s, x, u ∈ [0 , . Suppose also that J u is good (defi-nition 22) for all u . We have: ∂∂u l ( ω,m,θ ) J u (0 , Z [0 , l ( ω,m,θ ) J u (0 , x ) (cid:18) e ( ω,m )Γ ux (0 , s ) ⊲ ′ ˆ W ∗ (cid:18)I ω Θ (cid:19) (cid:18) ∂∂s , ∂∂x , ∂∂u (cid:19)(cid:19) l ( ω,m,θ ) J u ( x, dsdx − l ( ω,m,θ ) J u (0 , Z e ( ω,m )Γ u (0 , s ) ⊲ ′ ˆ J ∗ (cid:18)I ω θ − Z ω m ∗ { , } m (cid:19) (cid:18) ∂∂s , ∂∂u (cid:19) ds + (cid:18)Z e ( ω,m )Γ u (0 , s ) ⊲ ′ ˆ J ∗ (cid:18)I ω θ − I ω m ∗ { , } m (cid:19) (cid:18) ∂∂s , ∂∂u (cid:19) ds (cid:19) l ( ω,m,θ ) J u (0 , . Corollary 39
Three dimensional holonomy based on a 2-crossed module is in-variant under rank-3 holonomy, restricting to laminated rank-2 holonomy inthe boundary. More precisely if J and J ′ are good 3-paths which are rank-3homotopic (with laminated boundary) then l ( ω,m,θ ) J (0 ,
1) = l ( ω,m,θ ) J ′ (0 , , as long as δ ( θ ) = M = D ω m and ∂ ( m ) = Ω , the curvature of ω . From now on we will usually abbreviate l ( ω,m,θ ) J (0 ,
1) = l ( ω,m,θ ) J , as we didfor g ωγ . = g ωγ (0 ,
1) and e ( ω,m )Γ . = e ( ω,m )Γ (0 , .3.2 The holonomy of the interchange 3-cells Fix a 2-crossed module ( L δ −→ E ∂ −→ G, ⊲, { , }} with associated differential 2-crossed module ( l δ −→ e ∂ −→ g , ⊲, { , } ). As before, consider differential forms ω ∈A ( M, g ), m ∈ A ( M, e ) and θ ∈ A ( M, l ) such that δ ( θ ) = M and ∂ ( m ) = Ω,where as usual Ω = dω + [ ω, ω ] = dω + ω ∧ ad ω and M = dm + ω ∧ ⊲ m .Let Γ and Γ ′ be two 2-paths with ∂ +1 (Γ) = ∂ − (Γ). Let ∗ = ∂ − (Γ), ∗ ′ = ∂ +1 (Γ)and ∗ ′′ = ∂ +1 (Γ ′ ). Theorem 40 (The holonomy of the interchange 3-cell)
We have l ( ω,m,θ )Γ ′ = e ( ω,m )Γ ⊲ ′ (cid:26)(cid:16) e ( ω,m )Γ (cid:17) − , g ω∂ − (Γ) ⊲ e ( ω,m )Γ ′ (cid:27) − . Proof.
Let J = Γ ′ be the interchange 3-path. Let γ = ∂ − (Γ). Let F ( s )and F ′ ( s ) be the paths γ s and γ ′ s , respectively, for each s ∈ [0 , F : [0 , → P ( M, ∗ , ∗ ′ ) and F ′ : [0 , → P ( M, ∗ ′ , ∗ ′′ ). As usualwe put γ s ( t ) = Γ( t, s ) for each s, t ∈ [0 , γ ′ s ( t ). Put e ( x ) = e ( ω,m )Γ (0 , x ), e = e (1) and f ( x ) = e ( ω,m )Γ ′ (0 , x ). Also put l ( x ) = l ( ω,m,θ ) J (0 , x ). Asusual g γ s = g ωγ s . Recall g γ s = ∂ ( e ( s )) − g γ . By a straightforward explicit calculation, using equation (20), and the ex-plicit form of J = Γ ′ , indicated by figure 1, we have: ddx l ( x )= l ( x ) Z ( g γ ⊲ f ( x )) e ( s ) ⊲ ′ (cid:26) F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19) , g γ s ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) ds = l ( x )( g γ ⊲ f ( x )) ⊲ ′ Z e ( s ) ⊲ ′ (cid:26) F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19) , g γ s ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) ds. We have used the fact that the interchange 3-cell has derivative of rank ≤ Z e ( s ) ⊲ ′ (cid:26) F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19) , g γ s ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) ds = Z (cid:26) e ( s ) F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19) , g γ s ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) ds − Z (cid:26) e ( s ) , ∂ (cid:18) F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19)(cid:19) g γ s ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) ds. Since, by definition: dds e ( s ) = e ( s ) F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19) and dds e − ( s ) = − F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19) e − ( s ) , and also ∂ ( e − ( s )) g γ = g γ s (Lemma 35), thus dds g γ s = − ∂ (cid:18) F ∗ (cid:18)I ω m (cid:19)(cid:19) g γ s ,
40t follows, by the Leibnitz rule, and Lemma 4: Z e ( s ) ⊲ ′ n F ∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂s (cid:19) , g γ s ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19) o ds = Z ∂∂s (cid:26) e ( s ) , g γ s ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) = (cid:26) e, g γ ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) . Therefore: ddx l ( x ) = l ( x )( g γ ⊲ f ( x )) ⊲ ′ (cid:26) e, g γ ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) . (21)On the other hand, by equation (8) and the definition of f ( x ) we have: ddx (cid:8) e − , g γ ⊲ f ( x ) (cid:9) = (cid:26) e − , g γ ⊲ f ( x ) g γ ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) = (cid:0) e − (cid:0) g γ ⊲ f ( x ) (cid:1) e (cid:1) ⊲ ′ (cid:26) e − , g γ ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) (cid:8) e − , g γ ⊲ f ( x ) (cid:9) , thus in particular: ddx e ⊲ ′ (cid:8) e − , g γ ⊲ f ( x ) (cid:9) − = − (cid:16) e ⊲ ′ (cid:8) e − , g γ ⊲ f ( x ) (cid:9) − (cid:17) ( g γ ⊲ f ( x )) ⊲ ′ e ⊲ ′ (cid:26) e − , g γ ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) = e ⊲ ′ { e, g γ ⊲ f ( x ) } − ( g γ ⊲ f ( x )) ⊲ ′ (cid:26) e, ∂ ( e − ) g γ ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) = e ⊲ ′ { e, g γ ⊲ f ( x ) } − ( g γ ⊲ f ( x )) ⊲ ′ (cid:26) e, g γ ⊲ F ′∗ (cid:18)I ω m (cid:19) (cid:18) ∂∂x (cid:19)(cid:27) , taking into account equation (5) and Lemma 4 for the second step, and Lemma35 for the third. By comparing with equation (21) and applying the unicitytheorem for ordinary differential equations finishes the proof of the theorem, byusing Lemma 4 again. Fix a 2-crossed module ( L δ −→ E ∂ −→ G, ⊲, { , }} with associated differential 2-crossed module ( l δ −→ e ∂ −→ g , ⊲, { , } ).Similarly to the Non-abelian Green Theorem, the following follows directlyfrom the unicity theorem for ordinary differential equations:41 emma 41 (Non-abelian Stokes Theorem) Let J be a good 3-path in M .Consider differential forms ω ∈ A ( M, g ) , m ∈ A ( M, e ) and θ ∈ A ( M, l ) suchthat δ ( θ ) = M and ∂ ( m ) = Ω . We then have: ∂ (cid:16) l ( ω,m,θ ) J ( x, x ′ ) (cid:17) − e ( ω,m )Γ x = e ( ω,m )Γ x ′ recall that we put J ( t, s, x ) = Γ x ( t, s ) , for each t, s, x ∈ [0 , . In addition e ( ω,m )Γ x . = e ( ω,m )Γ x (0 , . By combining all that was done in this section, it thus follows that theassignment γ ∈ S ( M ) g ωγ ∈ G, Γ ∈ S l ( M ) (cid:16) g ω∂ − (Γ) , e ( ω,m )Γ (cid:17) and J ∈ S ( M ) (cid:16) g ω∂ − ( J ) , e ( ω,m ) ∂ − ( J ) , l ( ω,m,θ ) J (cid:17) defines a strict Gray 3-groupoid map (definition 12) ( ω,m,θ ) H : S ( M ) → C ( H ) , where C ( H ) is the Gray 3-groupoid constructed from G (see 1.2.5) and S ( M )is the fundamental Gray 3-groupoid of M (see 2.3.4). What is left to prove isentirely similar to the proof of Theorem 39 of [FMP1]; see 3.2.1. Let M be a manifold. Let J be a 3-path with J ( ∂D ) = {∗ ′ } . Given a 1-path γ with ∂ − ( γ ) = ∗ and ∂ +1 ( γ ) = ∗ ′ , let γ.J be the 3-path obtained byfilling { z ∈ D , | z | ≤ / } with J and the rest of the cube D with γ , in theobvious way. This corresponds to the standard way of defining the action of thefundamental groupoid of M on the homotopy groups π ( M, ∗ ′ ). Note that γ.J is well defined up to rank-3 homotopy, with laminated boundary.Let ( L δ −→ E ∂ −→ G, ⊲, { , }} be a 2-crossed module with associated differential2-crossed module ( l δ −→ e ∂ −→ g , ⊲, { , } ). Consider differential forms ω ∈ A ( M, g ), m ∈ A ( M, e ) and θ ∈ A ( M, l ), as usual such that δ ( θ ) = M and ∂ ( m ) = Ω. Lemma 42
We have: l ( ω,m,θ ) γ.J = g ωγ ⊲ l ( ω,m,θ ) J . Proof.
Follows by the definition of l ( ω,m,θ ) J , noting that what was added to J has derivative of rank ≤
1, together with the identity g ⊲ ( e ⊲ ′ l ) = ( g ⊲ e ) ⊲ ′ ( g ⊲ l ) , where g ∈ G, e ∈ E and l ∈ L, valid in any 2-crossed module; see equation (4).42 .2 The definition of Wilson 3-sphere observables Let S ⊂ M be an oriented 3-sphere S embedded in M . Consider an orientationpreserving parametrisation J : D /∂D = S → S ⊂ M of S . Define W ( S, ω, m, θ ) = l ( ω,m,θ ) J ∈ ker δ ⊂ L, (recall the Non-abelian Stokes Theorem), called the Wilson 3-sphere observable.We have: Theorem 43
The Wilson 3-sphere observable W does not depend on the parametri-sation J of S chosen up to acting by elements of G . If S ∗ denotes S with thereversed orientation we have W ( S ∗ , ω, m, θ ) = ( W ( S, ω, m, θ )) − . Proof.
The second statement is immediate from Theorem 27. Let us provethe first. Let
J, J ′ : D /∂D be orientation preserving parametrisations of S .Let J ( ∂D ) = ∗ and J ′ ( ∂D ) = ∗ ′ . Consider an isotopy W : D × I → S connecting J and J ′ , recall that the oriented mapping class group of S istrivial. Let γ ( x ) = W ( ∂D , x ), a smooth path in M . Obviously (by using W ) J is rank-3 homotopic, with laminated boundary, to γ.J ′ ; see subsection 4.1 forthis notation. The result follows from Lemma 42 together with Corollary 39. Fix a manifold M . Given a vector space U , we denote the vector space of U -valued differential n -forms in M as A n ( M, U ). Let V and W be vector spaces.Suppose we have a bilinear map B : ( u, v ) ∈ U × V u ∗ v ∈ W . If we aregiven U and V valued forms α ∈ A a ( M, U ) and β ∈ A b ( M, V ) we define the W -valued ( a + b )-form α ∧ B β in M (also denoted α ∧ ∗ β ) as: α ∧ B β = ( a + b )! a ! b ! Alt( α ⊗ B β ) ∈ A a + b ( M, W ) . Here α ⊗ B β is the covariant tensor B ◦ ( α × β ) and Alt denotes the naturalprojection from the vector space of W -valued covariant tensor fields in M ontothe vector space of W -valued differential forms in M . Note that if B op denotesthe bilinear map V × U → W such that B op ( v, u ) = B ( u, v ) then β ∧ B op α = ( − ab α ∧ B α. On the other hand if we define J ( ω ) = ( − m ω , where ω is an m -form, we willcontinue to have d ( α ∧ β ) = ( dα ) ∧ B β + J ( α ) ∧ B dβ . For details see [Ch, BT]. Let U be an open set in some R n . Let ω , . . . , ω n be forms in A n i +1 ( R × U , W i ), for i = 1 , . . . , k , where W i are vector spaces.Suppose we are given bilinear maps W i × W i +1 → W i +1 , say ( v, w ) v ∗ w .43iven −∞ < a < b < + ∞ , we define α = H ba ω dt = H ba ω as being the W -valued n -form in U such that α ( v , . . . , v n ) = Z ba ω (cid:18) ∂∂t , v , . . . , v n (cid:19) dt. Having defined the W m -valued ( n + . . . + n m )-form H ba ω ∗ . . . ∗ ω m dt , then the W m +1 -valued ( n + . . . + n m + n m +1 )-form H ba ω ∗ . . . ∗ ω m +1 dt is I ba ω ∗ . . . ∗ ω m +1 = Z ba (cid:18)I ta ω ∗ . . . ∗ ω m dt ′ (cid:19) ∧ ∗ ι ∂∂t ( ω m +1 ) dt, where ι X ( ω ) denotes the contraction of a form ω with a vector field X . Some-times parentheses may be inserted to denote the order in which we apply bilinearmaps (if not the order above).For simplicity (but with enough generality for this article), suppose that allforms ω i vanish when t = 0 and t = 1. Given b ∈ [0 , ω b = r ∗ b ( ω ), where r b ( x ) = ( b, x ) for each x ∈ U . As in [Ch] we can prove that: d I b ω = − Z b ( ι ∂∂t dω ) dt + Z b ddt ω t dt = ω b − ω − Z b ( ι ∂∂t dω ) dt (22)= ω b − I b ( dω ) dt. (23)In general in the case when V i = R we have (by induction): d I b ω ∗ . . . ∗ ω m dt = m X i =1 ( − i I b Jω ∗ . . . ∗ Jω i − ∗ ( dω i ) ∗ ω i +1 ∗ . . . ∗ ω m + m − X i =1 ( − i I b Jω ∗ . . . ∗ Jω i − ∗ ( Jω i ∧ ∗ ω i +1 ) ∗ ω i +2 ∗ . . . ∗ ω m − ( − m I b J ( ω ) ∗ . . . ∗ J ( ω m − ) ! ∧ ∗ r ∗ b ( ω m ) (24)For a proof see [Ch].From now on we use the following convention I ω ∗ . . . ∗ ω n = I ω ∗ . . . ∗ ω n dt. Let M be a smooth manifold. Let P ( M, ∗ , ∗ ′ ) denote the space of all smoothcurves γ : [0 , → M that start in ∗ and finish at ∗ ′ . Recall that a plot is amap F : U → P ( M, ∗ , ∗ ′ ), such that the associated map F ′ : I × U → M givenby F ′ ( t, x ) = F ( x )( t ) is smooth; see [Ch]. (Here U is an open set in some R n .)44y definition a p -form α in P ( M, ∗ , ∗ ′ ) is given by a rule which associatesa p -form F ∗ ( α ) in U to each plot F : U → P ( M, ∗ , ∗ ′ ), satisfying the followingcompatibility condition: for any smooth map g : U ′ → U , where U ′ is an openset in some R n , we have ( F ◦ g ) ∗ ( ω ) = g ∗ ( F ∗ ( ω )), in U ′ . The sum, exteriorproduct and exterior derivative of forms in the space of curves are defined as: F ∗ ( α + β ) = F ∗ ( α ) + F ∗ ( β ) , F ∗ ( α ∧ β ) = F ∗ ( α ) ∧ F ∗ ( β ) , F ∗ ( dα ) = dF ∗ ( α ) , respectively, for each plot F : U → P ( M, ∗ , ∗ ′ ).Let ω , . . . , ω n be forms in M of degrees a i + 1, for i = 1 , . . . , n . Then wedefine a a + . . . + a n -form H ω ∗ . . . ∗ ω n in the path space P ( M, ∗ , ∗ ′ ) by putting: F ∗ (cid:18)I ω ∗ . . . ∗ ω n (cid:19) = I ( F ′ ) ∗ ( ω ) ∗ . . . ∗ ( F ′ ) ∗ ( ω n ) . Note that if we are given a smooth map f : [0 , × [0 , n → M , such that f ( { } × [0 , n ) = ∗ and f ( { } × [0 , n ) = ∗ ′ , then we have an associated plotˆ f : [0 , n → P ( M, ∗ , ∗ ′ ), sometimes denoted by F : [0 , n → P ( M, ∗ , ∗ ′ ). Acknowledgements
The first author was supported by the Centro de Matem´atica da Universidade doPorto , financed by
Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia (FCT) through the programmes POCTI and POSI, with Portuguese andEuropean Community structural funds. This work was partially supported bythe
Programa Operacional Ciˆencia e Inova¸c˜ao 2010 , financed by FCT and cofi-nanced by the European Community fund FEDER, in part through the researchproject Quantum Topology POCI/MAT/60352/2004.We would like to thank Tim Porter, Marco Zambom, Urs Schreiber, BranislavJurˇco and Bj¨orn Gohla for useful discussions and / or comments.
References K -theory (Luminy, 1983). J. PureAppl. Algebra 34 (1984), no. 2-3, 155–178.[Cr] Crans S.E.: A tensor product for Gray -categories. Theory Appl. Categ.5 (1999), No. 2, 12–69 (electronic).[E] Ellis G.J.: Homotopical aspects of Lie algebras. J. Austral. Math. Soc.Ser. A 54 (1993), no. 3, 393–419.[FM] Faria Martins J.: The fundamental crossed module of the comple-ment of a knotted surface, to appear in Trans AMS. arXiv:0801.3921v1[math.GT].[FMP1] Faria Martins J.; Picken R.: On two-dimensional holonomy, to appearin Trans AMS. arXiv:0710.4310v2 [math.DG].[FMP2] Faria Martins J.; Picken R.: A Cubical Set Approach to 2-Bundles withConnection and Wilson Surfaces, arXiv:0808.3964v2 [math.CT][FMPo] Faria Martins J.; Porter T.: On Yetter’s invariant and an extensionof the Dijkgraaf-Witten invariant to categorical groups. Theory Appl.Categ. 18 (2007), No. 4, 118–150 (electronic).[GPS] Gordon R.; Power A. J.; Street R.: Coherence for tricategories. Mem.Amer. Math. Soc. 117 (1995), no. 558.[HKK] Hardie K. A.; Kamps K. H.; Kieboom R. W.: A Homotopy 2-Groupoidof a Hausdorff Space. Papers in honour of Bernhard Banaschewski(Cape Town, 1996). Appl. Categ. Structures 8 (2000), no. 1-2, 209–214.[Hi] Higgins, Philip J.: Thin elements and commutative shells in cubical ω -categories. Theory Appl. Categ. 14 (2005), No. 4, 60–74 (electronic).[H] Hitchin N.: Lectures on special Lagrangian submanifolds. WinterSchool on Mirror Symmetry, Vector Bundles and Lagrangian Submani-folds (Cambridge, MA, 1999), 151–182, AMS/IP Stud. Adv. Math., 23,Amer. Math. Soc., Providence, RI, 2001.[J1] Jurˇco B.: Nonabelian bundle 2-gerbes. arXiv:0911.1552v2 [math.DG].[J2] Jurˇco B.: Differential geometry of 2-crossed module bundle 2-gerbes.See: http://branislav.jurco.googlepages.com/ K -Theory 25 (2002), no. 4, 373–409.[K] Knapp A.W.: Lie groups beyond an introduction. Progress in Mathe-matics, 140. Birkh¨auser Boston, Inc., Boston, MA, 1996.[MP] Mackaay M.; Picken R.F.: Holonomy and parallel transport for abeliangerbes. Adv. Math. 170 (2002), no. 2, 287-219. .[M] Mackaay M.: A note on the holonomy of connections in twisted bundles.Cah. Topol. G´eom. Diff´er. Cat´eg. 44 (2003), no. 1, 39–62.[MuPo] Mutlu A.; Porter T.: Freeness conditions for 2-crossed modules andcomplexes. Theory Appl. Categ. 4 (1998), No. 8, 174–194 (electronic)[No] Noohi B.: Notes on 2-groupoids, 2-groups, and crossed-modules. Ho-mology, Homotopy and Applications, Vol. 9 (2007), no. 1, 75–106.[N] Norrie K.: Actions and automorphisms of crossed modules. Bulletin dela Soci´et´e Math´ematique de France, 118 no. 2 (1990), p. 129-146.[P] Picken R.F.; TQFT and gerbes, Algebraic and Geometric Topology, 4(2004) 243–272[Po] Porter T.: Crossed Menagerie: an introduction to crossed gad-getry and cohomology in algebra and topology. Online notes: http://ncatlab.org/timporter/files/menagerie9.pdfhttp://ncatlab.org/timporter/files/menagerie9.pdf