aa r X i v : . [ m a t h . C T ] M a y Actor of categories internal to groups
Tunçar ¸SAHAN ∗ Department of Mathematics, Aksaray University, Aksaray, TURKEY
Abstract
In this study, using the Brown-Spencer theorem and in the ligth of the works of Norrie, inthe category of internal categories within groups, also called group-groupoids, we interpret thenotion of actor of a crossed module over groups. Further, we construct the action of a group-groupoid on a group-groupoid. Moreover, we give the explicit construction of semi-directproduct of two group-groupoids and of holomorph of group-groupoids.
Key Words:
Group-groupoid, actor, action, holomorph
Classification:
1. Introduction
Crossed modules were introduced by Whitehead in [26, 27, 28]. Crosed modules model all(connected) homotopy 2-types, i.e. connected spaces with π n ( X ) = 0 for n > whilst groupsmodel all (connected) homotopy 1-types. Crossed module concept generalizes the concept ofnormal subgroup and that of module since these are standart examples of crossed modules. Acrossed module consists of two group homomorphisms α : A → B and B → Aut( A ) (i.e. an actionof B on A which denoted by b · a ) satisfying (i) α ( b · a ) = b + α ( a ) − b and (ii) α ( a ) · a = a + a − a forall a, a ∈ A and b ∈ B . See [3, 4, 9, 12, 13] for applications of crossed modules in other branchesof mathematics.Brown and Spencer [5] proved that the category of group objects in the category of smallcategories, which is also called the category of group-groupoids, and the category of crossedmodules are equivalent. This equivalence makes it possible to interpret a notion or a concept ora problem given in one of the above-mentioned categories in another. See for example [19, 20].It also has been shown that group-groupoids are in fact internal categories within the categoryof groups. The study of internal category theory was continued in the work of Datuashvili [7].Cohomology theory of internal categories within certain algebraic categories which are calledcategory of groups with operations is developed in [6] (see also [23] for more information oninternal categories in categories of groups with operations). The equivalence of the categories in[5] enables us to generalize some results on group-groupoids to more general internal groupoidsfor a certain algebraic category C (see for example [1], [15], [17] and [18]). ∗ Tunçar ¸SAHAN (e-mail : [email protected]) G be a group and Aut( G ) the group of automorphisms of G . It is a well-known fact thatthere exist a canonical group homomorphism ϕ : G → Aut( G ) which assigns an element of G toits corresponding inner automorphism whose kernel is the centre of G .In the category of groups, if there exist a short exact sequence of groups / / N (cid:31) (cid:127) / / G / / / / H / / such that G acts on N , i.e. there exist a group homomorphism G → Aut( N ) , then this action giverise to the following commutative diagram / / N (cid:31) (cid:127) / / (cid:15) (cid:15) G / / / / (cid:15) (cid:15) H / / (cid:15) (cid:15) / / Inn( N ) (cid:31) (cid:127) / / Aut( N ) / / / / Out( N ) / / The main object of this presented paper is to construct, in the category of group-groupoids, anobject which is called the actor, that plays analogous role to automorphism group in the categoryof groups. Similar construction is studied for crossed modules over groups by Lue in [14] anddeveloped by Norrie in [21]. Norrie called this structure by actor crossed module.The problem is finding an analogous object to
Aut( N ) in the category of group-groupoids thatcompletes similar diagram to one given above. That object will be called actor group-groupoid.Using actor group-groupoids it is possible to define the notions of centre, semi-direct productand holomorph in the category of group-groupoids. Furthermore, we define abelian and com-plete group-groupoids using the notion of centre for group-groupoids.
2. Preliminaries
A category C consist of a class C of morphisms (or arrows), a class C of objects, initial andfinal point maps d , d : C → C , object inclusion (identity morphism) map ε : C → C and apartial composition map m : C d × d C → C denoted by m ( b, a ) = b ◦ a for all ( b, a ) ∈ C d × d C where C d × d C is the pullback of d and d . These are subject to the following. (i) d ε = d ε = 1 C ; (ii) d m = d π , d m = d π ; (iii) m (1 C × m ) = m ( m × C ) and (iv) m ( εd , C ) = m (1 C , εd ) = 1 C . C d × d C m / / C d / / d / / C ε t t In a category C , a morphism a ∈ C will be denoted by a : x → y where x = d ( a ) and y = d ( a ) .For any pair of objects x, y ∈ C , C ( x, y ) will stand for the set d − ( x ) ∩ d − ( y ) . We denote ε ( x ) with x for x ∈ C .A groupoid is a category whose morphisms are invertible up to partial composition, i.e. areisomorphisms. That is, a category C is called a groupoid if for all a ∈ C ( x, y ) there exist a uniquemorphism a − ∈ C ( y, x ) such that a − ◦ a = 1 x and a ◦ a − = 1 y .2 morphisms f = ( f , f ) of a category C to D , which is called a functor, is a pair of maps f : C → D and f : C → D such that the following diagram commutes. C d × d C m / / f × f (cid:15) (cid:15) C f (cid:15) (cid:15) d / / d / / C f (cid:15) (cid:15) ε t t D d × d D m / / D d / / d / / C ε j j It is straightforward that for a functor f = ( f , f ) : C → D , f can be defined via f as follows:Let x ∈ C . Then f ( x ) = d (1 x ) ( f ( x ) = d (1 x )) . Hence it is sufficient to use only f for a functor f = ( f , f ) . So we briefly use f without indices to denote a functor when no confusion arise. Definition 2.1.
Let f, g : C → D be two functors. A natural transformation η : f ⇒ g from f to g isa class of morphisms η ( x ) : f ( x ) → g ( x ) in D for all x ∈ C such that η ( x ) ◦ f ( a ) = g ( a ) ◦ η ( y ) forany a ∈ C ( x, y ) , i.e. the following diagram commutes. f ( x ) η ( x ) / / f ( a ) (cid:15) (cid:15) g ( x ) g ( a ) (cid:15) (cid:15) f ( y ) η ( y ) / / g ( y ) The set of all natural transformations between functors from a category C to a category D isdenoted by Nat( C , D ) . In the case of C = D we briefly denote Nat( C , C ) by Nat( C ) .If η ( x ) : f ( x ) → g ( x ) is an isomorphism for all x ∈ C then η : f ⇒ g is called a naturalisomorphism. In this case f and g are said to be naturally isomorphic . The set of all naturalisomorphisms between functors from a category C to a category D is denoted by N ( C , D ) . In thecase of C = D we briefly denote N ( C , C ) by N ( C ) .Since each morphism in a groupoid can be invertible then we can give the following fact. Lemma 2.2.
Let C be a category and G a groupoid. Then Nat( C , G ) = N ( C , G ) , i.e. every naturaltransformation between functors from C to G is a natural isomorphism. Natural transformations can be composed in two ways: Vertical composition is a partial op-eration on the set
Nat( C , D ) of all natural transformations. Let η, ζ ∈ Nat( C , D ) with η : f ⇒ g , ζ : g ⇒ h . Then the vertical composition ζ ◦ v η : f ⇒ h can be defined and for all x ∈ C it is givenby ( ζ ◦ v η )( x ) = ζ ( x ) ◦ η ( x ) . On the other hand horizontal composition is defined as follows: Let η ′ ∈ Nat( D , E ) with η ′ : f ′ ⇒ g ′ . Then for all x ∈ C , η ′ ◦ h η : f ′ f ⇒ g ′ g is given by ( η ′ ◦ h η )( x ) = η ′ ( g ( x )) ◦ f ′ ( η ( x )) . Let η, ζ ∈ Nat( C , D ) and η ′ , ζ ′ ∈ Nat( D , E ) as in the following diagram C f ! ! ✤✤ ✤✤ (cid:11) (cid:19) η = = h ✤✤ ✤✤ (cid:11) (cid:19) ζg / / D f ′ ! ! ✤✤ ✤✤ (cid:11) (cid:19) η ′ > > h ′ ✤✤ ✤✤ (cid:11) (cid:19) ζ ′ g ′ / / E ( ζ ′ ◦ h ζ ) ◦ v ( η ′ ◦ h η ) = ( ζ ′ ◦ v η ′ ) ◦ h ( ζ ◦ v η ) As similar to the homotopy of continuous functions, the homotopy of functors is defined in [2,p.228] and it has been proven in [16] that two parallel functors are homotopic if and only if theyare naturally isomorphic.A group-groupoid G is a group object in the category of small categories. Group-groupoidsare also known as internal categories within the category of groups. So group-groupoids, alsocalled 2-groups, can be seen as 2-dimensional groups.Let G be a group-groupoid. Then G is a groupoid such that the class of morphisms G andthe class of objects G has group structures and the category structural maps d , d , ε and m aregroup homomorphisms. A group-groupoid G will be denoted by G = ( G , G ) for short when noconfusion arise. Example 2.3. (i)
Let X be a topological group. Then the set πX of all homotopy classes of pathsin X has a group-groupoid structure where the group of objects is X [5]. (ii) Let G be a group. Then ( G × G, G ) is a group-groupoid where G ( x, y ) = ( G × G )( x, y ) = { ( x, y ) } for all x, y ∈ G . (iii) Let G be a group. Then ( G, G ) is a group-groupoid where G ( x, y ) = G ( x, y ) = ( { x } , x = y ∅ , x = y for all x, y ∈ G . This group-groupoid is called discrete.A morphism of group-groupoids f = ( f , f ) is a functor such that each component of f is agroup homomorphism. Thus one can define the category of group-groupoids which is denotedby GpGd .Recall from [5] that m being a group homomorphism implies that ( b ◦ a ) + ( d ◦ c ) = ( b + d ) ◦ ( a + c ) whenever left side (hence both sides) of the equation make sense for all a, b, c, d ∈ G . This equa-tion is called the interchange law.As a consequence of the interchange law in a group-groupoid G the partial composition canbe given in terms of group operation as follow: Let a ∈ G ( x, y ) and b ∈ G ( y, z ) be two morphismin G . Then b ◦ a = ( b + 0) ◦ (1 y + ( − y + a ))= ( b ◦ y ) + (0 ◦ ( − y + a ))= b − y + a and similarly b ◦ a = a − y + b [5]. Here, if y = 0 then a + b = b + a , that is, the elements of Ker d and of Ker d are commute. Also for a ∈ G ( x, y ) , a − = 1 x − a + 1 y is the inverse of a up to thepartial composition [5].A crossed module over groups is a group homomorphism α : A → B with an (left) action of B on A denoted by b · a such that 4 CM1) α ( b · a ) = b + α ( a ) − b and (CM2) α ( a ) · a = a + a − a for all a, a ∈ A and b ∈ B . Example 2.4.
Following homomorphisms are standart examples of crossed modules. (i)
Let X be a topological space, A ⊂ X and x ∈ A . Then the boundary map ρ : π ( X, A, x ) → π ( X, x ) from the second relative homotopy group π ( X, A, x ) to the fundamental group π ( X, x ) at x ∈ X is a crossed module. (ii) Let G be a group and N a normal subgroup of G . Then the inclusion function N inc −→ G is acrossed module where the action of G on N is conjugation. (iii) Let G be a group. Then the inner automorphism map G → Aut( G ) is a crossed module.Here the action is given by ψ · g = ψ ( g ) for ψ ∈ Aut( G ) and g ∈ G . (iv) Given any G -module, M , the trivial homomorphism M → G is a crossed G -module withthe given action of G on M .A morphism f = h f A , f B i of crossed modules from ( A, B, α ) to ( A ′ , B ′ , α ′ ) is a pair of grouphomomorphisms f A : A → A ′ and f B : B → B ′ such that f B α = α ′ f A and f A ( b · a ) = f B ( b ) · f A ( a ) for all a ∈ A and b ∈ B . A α / / f A (cid:15) (cid:15) B f B (cid:15) (cid:15) A ′ α ′ / / B ′ Crossed modules form a category with morphisms defined above. The category of crossedmodules is denoted by
XMod .Following theorem was given in [5]. We sketch the proof since we need some details later.
Theorem 2.5 ( Brown & Spencer Theorem ) . [5] The category GpGd of group-groupoids and the cate-gory
XMod of crossed modules are equivalent.Proof.
Define a functor ϕ : GpGd → XMod as follows: Let G be a group-groupoid. Then ϕ ( G ) = ( A, B, α ) is a crossed modules where A = ker s , B = G , α is the restriction of t and the action of B on A is given by x · a = 1 x + a − x .Conversely, define a functor ψ : XMod → GpGd as follows: Let ( A, B, α ) be a crossed module. Then the semi-direct product group A ⋊ B is agroup-groupoid on B where s ( a, b ) = b , t ( a, b ) = α ( a ) + b , ε ( b ) = (0 , b ) and the composition is ( a ′ , b ′ ) ◦ ( a, b ) = ( a ′ + a, b ) where b ′ = α ( a ) + b .Other details are straightforward so is omitted. ✷ . Actor of a group-groupoid Norrie [22] defined actor crossed module for a given crossed module using regular deriva-tions of that crossed module and showed how actor crossed modules provides an analogue ofautomorphisms groups of groups. In this section, in the light of the Brown & Spencer Theorem[5, Theorem 1] we define actor group-groupoid, group-groupoid actions and semi-direct productof group-groupoids. First we recall the construction of actor crossed module from [22].The notion of derivations first appears in the work of Whitehead [27] under the name ofcrossed homomorphisms (see also [22]). Let ( A, B, α ) be a crossed module. A derivation of ( A, B, α ) is a map d : B → A such that d ( b + b ) = d ( b ) + b · d ( b ) for all b, b ∈ B . Any derivation of ( A, B, α ) defines endomorphisms θ d and σ d on A and B respectively, as follows: θ d ( a ) = dα ( a ) + a and σ d ( b ) = αd ( b ) + b It is easy to see that ( θ d , σ d ) : ( A, B, α ) → ( A, B, α ) is a crossed module morphism and θ d ( d ) = d ( σ d ) . All derivations of ( A, B, α ) are denoted by Der(
B, A ) . Whitehead defined a multiplicationon Der(
B, A ) as follows: Let d , d ∈ Der(
B, A ) then d = d ◦ d where d ( b ) = d σ d ( b ) + d ( b ) (= θ d d ( b ) + d ( b )) . By this multiplication (Der(
B, A ) , ◦ ) becomes a monoid where the identity derivation is zeromorphism, i.e. d : B → A, b d ( b ) = 0 . Furthermore θ d = θ d θ d and σ d = σ d σ d . Group ofunits of Der(
B, A ) is called Whitehead group and denoted by D( B, A ) . These units are called regularderivations .Following proposition which is given in [22] is a combined result from [27] and [14]. Proposition 3.1. [22] Let ( A, B, α ) be a crossed module. Then the followings are equivalent. (i) d ∈ D( B, A ) ; (ii) θ ∈ Aut A ; (iii) σ ∈ Aut B . Actor of a crossed module ( A, B, α ) is defined in [22] to be the crossed module (D( B, A ) , Aut(
A, B, α ) , ∆) where ∆( d ) = h θ d , σ d i and the action of Aut(
A, B, α ) on D( B, A ) is given by h f, g i · d = f dg − for all h f, g i ∈ Aut(
A, B, α ) and d ∈ D( B, A ) . Actor crossed module of ( A, B, α ) is denoted by A ( A, B, α ) .Now we obtain analogous construction for group-groupoids. First we recall some preliminarydefinitions and properties for group-groupoids. Definition 3.2. [20] Let G be a group-groupoid and H a subgroupoid of G such that H is a sub-group of G . Then H is called a subgroup-groupoid of G and this denoted by H G .6 efinition 3.3. [20] Let G be a group-groupoid and N a subgroupoid of G such that N is a normalsubgroup of G . Then N is called a normal subgroup-groupoid of G and this denoted by H ✁ G . Definition 3.4. [20] Let G be a group-groupoid and H a normal subgroup-groupoid of G . Thenthe quotient group G /N is a group-groupoid on the quotient group G /N . This group-groupoidis called the quotient group-groupoid and denoted by G / N .A morphism f = ( f , f ) : G → G ′ of group-groupoids is called an isomorphism (monomor-phism, epimorphism and automorphism) if f and f are both isomorphisms (monomorphisms,epimorphisms and automorphisms). The group of all automorphisms of G is denoted by Aut( G ) .The kernel of a group-groupoid morphism f = ( f , f ) : G → G ′ is the normal subgroup-groupoid Ker f = (Ker f , Ker f ) of G . The image of f = ( f , f ) is the subgroup-groupoid Im f = (Im f , Im f ) of G ′ .Let G be a group-groupoid and H , K G . Then [ H , K ] = ([ H , K ] , [ H , K ]) is a subgroup-groupoid of G and called the commutator of H and K . In particular [ G , G ] , which is denoted by G ′ , is called the commutator subgroup-groupoid or derived subgroup-groupoid of G . It is easyto see from Definition 3.3 that G ′ is a normal subgroup-groupoid of G since [ G , G ] is a normalsubgroup of G .A natural transformation η ∈ Nat( G , H ) between group-groupoid morphisms from G to H is ausual natural transformation such that η ( x + x ) = η ( x ) + η ( x ) for all x , x ∈ G .For a group-groupoid G , Nat( G ) is a monoid where the monoid operation is horizontal com-position of natural transformations. Here the identity element of Nat( G ) is G . A natural transfor-mation η in Nat( G ) is called regular if it has an inverse up to horizontal composition. All regularnatural transformations forms a group. This group is denoted by W ( G ) . Proposition 3.5.
Let η ∈ Nat( G ) with η : f ⇒ g . Then η ∈ W ( G ) if and only if f, g ∈ Aut( G ) .Proof. If η ∈ W ( G ) then there exist η ′ ∈ Nat( G ) with η ′ : f ′ ⇒ g ′ such that η ◦ h η ′ = 1 G = η ′ ◦ h η .So f f ′ = 1 G = f ′ f and gg ′ = 1 G = g ′ g . Thus f ′ = f − and g ′ = g − , i.e. f, g ∈ Aut( G ) .Conversely, let f, g ∈ Aut( G ) . Then horizontal inverse of η is given by η − h ( x ) = (cid:2) f − (cid:0) η (cid:0) g − ( x ) (cid:1)(cid:1)(cid:3) − . Thus η ∈ W ( G ) . This completes the proof. ✷ In the following lemma we define an automorphism using regular natural transformationswhich will be useful later.
Lemma 3.6.
Let η ∈ W ( G ) with η : f ⇒ g . Then the map defined by F η : G → G a F η ( a ) = g ( a ) ◦ η ( d ( a )) = η ( d ( a )) ◦ f ( a ) is an automorphism of G . roof. First we show that F η is a group homomorphism. Let a ∈ G ( x, y ) and a ∈ G ( x , y ) .Then F η ( a + a ) = g ( a + a ) ◦ η ( x + x )= ( g ( a ) + g ( a )) ◦ ( η ( x ) + η ( x ))= ( g ( a ) ◦ η ( x )) + ( g ( a ) ◦ η ( x ))= F η ( a ) + F η ( a ) . Now let a, a ∈ G such that a = a . It is obvious that F η ( a ) = F η ( a ) if at least one of theinitial or final points of a and a is different since the initial or final points of F η ( a ) and F η ( a ) isdifferent in that case. Thus let a, a ∈ G ( x, y ) . Then F η ( a ) = g ( a ) ◦ η ( x ) = g ( a ) ◦ η ( x ) = F η ( a ) since g is an isomorphism. Hence F η is a monomorphism.Finally, let b ∈ G ( m, n ) . If we set a = g − ( b ◦ η ( f − ( m )) − ) then a ∈ G ( f − ( m ) , g − ( n )) and F η ( a ) = g ( a ) ◦ η ( f − ( m ))= ( b ◦ η ( f − ( m )) − ) ◦ η ( f − ( m ))= b. Hence F η is an epimorphism. So this completes the proof. ✷ f ( x ) η ( x ) / / f ( a ) (cid:15) (cid:15) F η ( a ) " " g ( x ) g ( a ) (cid:15) (cid:15) f ( y ) η ( y ) / / g ( y ) Remark 3.7.
It is easy to see that F η (1 x ) = η ( x ) for all x ∈ G . Also F ( G ) = 1 G . Lemma 3.8.
Let η, τ ∈ W ( G ) . If η = τ then F η = F τ .Proof. If η = τ then there exist at least one object x ∈ G such that η ( x ) = τ ( x ) . Then by Remark3.7 F η (1 x ) = η ( x ) = τ ( x ) = F τ (1 x ) . This completes the proof. ✷ As a result of this lemma, first of all if F η = F τ for some η, τ ∈ W ( G ) then η = τ . Also wecan say that the map which assigns a regular transformation η to an automorphism F η of G isinjective.We will now examine how this automorphism behaves under horizontal and vertical compo-sitions of natural transformations. Let η, η ′ , τ ∈ W ( G ) with η : f ⇒ g , η ′ : f ′ ⇒ g ′ and τ : g ⇒ h .Then for any a ∈ G ( x, y ) F ( τ ◦ v η ) ( a ) = F τ ( a ) ◦ η ( x ) = ( h ( a ) ◦ τ ( x )) ◦ η ( x ) or equivalently F ( τ ◦ v η ) ( a ) = τ ( y ) ◦ F η ( a ) = τ ( y ) ◦ ( η ( y ) ◦ f ( a )) . On the other side F ( η ′ ◦ h η ) ( a ) = η ′ ( g ( y )) ◦ f ′ ( F η ( a )) = F η ′ ( g ( a )) ◦ f ′ ( η ( x )) . orollary 3.9. Let η, η ′ ∈ W ( G ) . Then F ( η ′ ◦ h η ) = F η ′ ◦ F η . Proposition 3.10.
Let G be a group-groupoid. Then ( W ( G ) , Aut( G ) , e d , e d , e ε, e m ) is equipped with agroup-groupoid structure where e d ( η ) = f , e d ( η ) = g , e ε ( f ) = 1 f and η ′ e ◦ η = η ′ ◦ v η for all η : f ⇒ g, η ′ : g ⇒ h ∈ W ( G ) . W ( G ) f d × f d W ( G ) e m / / W ( G ) f d / / f d / / Aut( G ) e ε (cid:9) (cid:9) Here the interchange law comes from the well-known identiy ( η ′ ◦ v η ) ◦ h ( η ′ ◦ v η ) = ( η ′ ◦ h η ′ ) ◦ v ( η ◦ h η ) for horizontal and vertical compositiom of natural transformations whenever one side (henceboth sides) of the equation make sense. Definition 3.11.
Let G be a group-groupoid. Then the group-groupoid ( W ( G ) , Aut( G )) definedabove is called the actor group-groupoid of G and denoted by A ( G ) . Example 3.12. (i)
Let X be a topological group. Then A ( πX ) ∼ = ( H ( X ) , Aut( X )) where Aut( X ) is the set of all topological group automorphisms of X and H ( X ) is the set of all homotopiesbetween the topological group automorphisms of X . (ii) We know that G = ( G × G, G ) is a group-groupoid for a group G . In this case W ( G ) ∼ =Aut( G ) × Aut( G ) since for each pair ( f, g ) of automorphisms of G there exist exactly oneregular natural transformation form f to g . Thus A ( G ) ∼ = (Aut( G ) × Aut( G ) , Aut( G )) . (iii) Let G be a group. Then for the group-groupoid G = ( G, G ) , W ( G ) ∼ = Aut( G ) since W ( G ) contains only f for all f ∈ Aut( G ) . Thus the actor group-groupoid A ( G ) ∼ = (Aut( G ) , Aut( G )) .Let G be a group-groupoid, A ( G ) the actor group-groupoid of G and ( A, B, α ) the correspond-ing crossed module to G according to Theorem 2.5. So A = Ker d , B = G and α = d | A . Here wecan define two group homomorphisms ξ : Ker e d → D( B, A ) η ξ ( η ) = d η where d η ( x ) is given by d η ( x ) = η ( x ) − x for x ∈ B and λ : Aut( G ) → Aut(
A, B, α ) f λ ( f ) = h f | A , f i . It is easy to see that these morphisms are isomorphisms. Here the inverse of ξ is given by ξ − ( d ) = η d where η d ( x ) = η ( x ) + 1 x for all x ∈ G and the inverse of λ is given by λ − ( h f A , f B i ) = ( f A × f B , f B ) for all h f A , f B i ∈ Aut(
A, B, α ) . Theorem 3.13. h ξ, λ i : (cid:16) Ker e d , Aut( G ) , e d | Ker f d (cid:17) → (D( B, A ) , Aut(
A, B, α ) , ∆) is an isomorphism ofcrossed modules. roof. Let η ∈ Ker e d . Then there exist an automorphism g in Aut( G ) such that η : 1 G ⇒ g . So e d ( η ) = g and hence λ ( e d ( η )) = λ ( g ) = h g | A , g i .On the other hand ξ ( η ) = d η and ∆( ξ ( η )) = ∆( d η ) = (cid:10) θ d η , σ d η (cid:11) . Now let compute the mor-phisms θ d η and σ d η . θ d η ( a ) = d η ( d ( a )) + a = η ( d ( a )) − d ( a ) + a = η ( d ( a )) ◦ a = g ( a ) ◦ η ( d ( a ))= g ( a ) ◦ η (0)= g ( a ) and σ d η ( x ) = d ( d η ( x )) + x = d ( η ( x ) − x ) + x = d ( η ( x )) − d (1 x ) + x = g ( x ) − x + x = g ( x ) . That is ∆( d η ) = (cid:10) θ d η , σ d η (cid:11) = h g | A , g i . So we obtain the equality λ e d | Ker f d = ∆ ξ .Now we need to show that ξ ( f · η ) = λ ( f ) · ξ ( η ) . By Theorem 2.5 f · η = 1 f ◦ h η ◦ h − f =1 f ◦ h η ◦ h f − . Then ξ ( f · η ) = d f · η = d f ◦ h η ◦ h f − and for all x ∈ Bd ( f ◦ h η ◦ h f − ) = (1 f ◦ h η ◦ h f − ) ( x ) − x = f ( η ( f − ( x ))) − x . On the other hand λ ( f ) · ξ ( η ) = h f , f i · d η = f d η f − and for all x ∈ B (cid:0) f d η f − (cid:1) ( x ) = f (cid:0) d η (cid:0) f − ( x ) (cid:1)(cid:1) = f (cid:16) η (cid:0) f − ( x ) (cid:1) − f − ( x ) (cid:17) = f (cid:0) η (cid:0) f − ( x ) (cid:1)(cid:1) − f (cid:16) f − ( x ) (cid:17) = f ( η ( f − ( x ))) − x . So ξ ( f · η ) = λ ( f ) · ξ ( η ) and hence h ξ, λ i is a crossed module morphism. We know that ξ and λ aregroup isomorphisms. So this completes the proof. ✷ Corollary 3.14.
Let G be a group-groupoid and ( A, B, α ) the corresponding crossed module to G . Then A ( A, B, α ) = (D(
B, A ) , Aut(
A, B, α ) , ∆) is isomorphic to the corresponding crossed module to A ( G ) . Theorem 3.13 states that the definition of actor group-groupoid is compatible with the onegiven in [22] by Norrie for crossed modules.
Now let us consider the group-groupoid automorphisms f x := ( f x , f x ) of G given by f x : G → G b f x ( b ) = 1 x + b − x
10n the group of objects and by f x : G → G z f x ( z ) = x + z − x on the group of morphisms for all x ∈ G . Using these automorphisms we can define a canonicalgroup-groupoid morphism ϕ = ( ϕ , ϕ ) : G → A ( G ) as follows: Let a : x → y be a morphism in G and z an object in G . Then ϕ ( a ) = η a : f x ⇒ f y where η a ( z ) = a + 1 z − a and ϕ ( z ) = f z .It is easy to see that η a ∈ W ( G ) for all a ∈ G and ϕ is a group-groupoid morphism. Here theautomorphism F η a introduced in Lemma 3.6 corresponding to η a is given by F η a ( b ) = a + b − a ,i.e. the conjugation.Note that for all x, y ∈ G , f x f y = f x + y and for all a, b ∈ G , η a ◦ h η b = η a + b . Hence the inverse ( f x ) − of f x is equal to f − x and the horizontal inverse ( η a ) − h of η a is equal to η − a .We define the centre of a group-groupoid G to be the kernel Z ( G ) of ϕ . Thus ( Z ( G )) := Ker ϕ = { a ∈ G | a + 1 x = 1 x + a for all x ∈ G } and ( Z ( G )) := Ker ϕ = { x ∈ G | a + 1 x = 1 x + a for all a ∈ G } . Here note that ( Z ( G )) is a subgroup of the centre Z ( G ) of G .Since Z ( G ) is the kernel of the group-groupoid morphism ϕ then we can give the followingLemma. Lemma 3.15.
Let G be a group-groupoid. Then Z ( G ) is a normal subgroup-groupoid of G . Definition of a centre for a group-groupoid is consistent with the categorical one given by Huq[10] since the corresponding crossed module to Z ( G ) is isomorphic to ξ ( A, B, α ) which is definedby Norrie in [22] while ( A, B, α ) is the corresponding crossed module to G . Lemma 3.16.
Let G be a group-groupoid. Then the elements of G and of ( Z ( G )) are commute.Proof. Let a ∈ ( Z ( G )) and b ∈ G . Then a − d ( a ) ∈ Ker d and b − d ( b ) ∈ Ker d so ( a − d ( a ) ) + ( b − d ( b ) ) = ( b − d ( b ) ) + ( a − d ( a ) ) . Since d ( a ) ∈ ( Z ( G )) then ( a − d ( a ) ) + ( b − d ( b ) ) = ( b − d ( b ) ) + ( a − d ( a ) )( a + b ) + ( − d ( a ) − d ( b ) ) = ( b + a ) + ( − d ( b ) − d ( a ) ) a + b = b + a This completes the proof. ✷ Analogous to the group case we can introduce the notion of abelian group-groupoid usingcentres.
Definition 3.17.
A group-groupoid G is called abelian if it coincide with its centre, i.e. G = Z ( G ) .Following is a consequence of Lemma 3.16 and Definition 3.17. Corollary 3.18.
A group-groupoid G is abelian if and only if G (hence also G ) is an abelian group.
11y Lemma 3.16 and Corollary 3.18 we can give the following corollary.
Corollary 3.19.
A group-groupoid G is abelian if and only if a + 1 x = 1 x + a for all a ∈ G and x ∈ G . Corollary 3.20.
Let G be a group-groupoid. Then G / G ′ is an abelian group-groupoid where G ′ is thecommutator subgroup-groupoid of G . The quotient group-groupoid given above is called the abelianization of G and denoted by G ab = G / G ′ . Example 3.21. (i)
Let X be an abelian topological group. Then ( πX, X ) is an abelian group-groupoid. (ii) G = ( G × G, G ) is an abelian group-groupoid if and only if G is an abelian group. (iii) If G is an abelian group then the group-groupoid G = ( G, G ) is abelian.Abelian group-groupoids forms a full subcategory of GpGd and this subcategory is denotedby
AbGpGd .The inner actor I ( G ) of G is the image Im ϕ of the group-groupoid morphism ϕ : G → A ( G ) .One can see that I ( G ) is a normal subgroup-groupoid of A ( G ) . Here, by the first isomorphism the-orem for group-groupoids [20], G /Z ( G ) ∼ = I ( G ) . Also the quotient group-groupoid A ( G ) / I ( G ) iscalled the outer actor of G and denoted by O ( G ) . Hence we obtain the short exact sequence / / Z ( G ) ker ϕ / / G ϕ / / A ( G ) coker ϕ / / O ( G ) / / of group-groupoids since O ( G ) is the cokernel of ϕ .By the definition of centre of a group-groupoid G , if Z ( G ) = , i.e. the group-groupoid withone morphism, then ϕ : G → A ( G ) is a monomorphism of group-groupoids. In this case G ∼ = I ( G ) and can be consider as a normal subgroup-groupoid of A ( G ) . Proposition 3.22.
Let G be a group-groupoid. If Z ( G ) is trivial group-groupoid then so is Z ( A ( G )) .Proof. Let η ∈ ( Z ( A ( G ))) . Then η ◦ h η ′ = η ′ ◦ h η for all η ′ ∈ W ( G ) by Lemma 3.16. In particular η ◦ h η a = η a ◦ h η for any a ∈ G . Then for all b ∈ G F ( η ◦ h η a ) ( b ) = F ( η a ◦ h η ) ( b ) F η ( F η a ( b )) = F η a ( F η ( b )) F η ( a + b − a ) = a + ( F η ( b )) − aF η ( a ) + F η ( b ) − F η ( a ) = a + ( F η ( b )) − a and hence F η ( b ) + ( − F η ( a ) + a ) = ( − F η ( a ) + a ) + ( F η ( b )) . Because of the fact that F η is an isomorphism ( − F η ( a ) + a ) ∈ ( Z ( G )) . Since Z ( G ) is trivial then F η ( a ) = a for all a ∈ G .On the other hand we already know that F ( G )( a ) = a for all a ∈ G . So F η = F ( G ) . Then byLemma 3.8, η = 1 G . Hence Z ( A ( G )) is also trivial. ✷
12y Proposition 3.22 for a given group-groupoid G with Z ( G ) = , we can construct a sequenceof group-groupoids G / / A ( G ) / / A ( A ( G )) = A ( G ) / / A ( A ( A ( G ))) = A ( G ) / / · · · in which each group-groupoid embeds as a normal subgroup-groupoid in its successor. Thissequence is called the actor tower of G . Definition 3.23.
A group-groupoid is called complete if Z ( G ) = and the canonical morphism ϕ : G → A ( G ) is an isomorphism. Corollary 3.24. If G is a complete group-groupoid then G ∼ = I ( G ) = A ( G ) . We know that if Z ( G ) = then ϕ : G → A ( G ) is a monomorphism. So a group-groupoid iscomplete if and only if Z ( G ) = and the canonical morphism ϕ : G → A ( G ) is an epimorphism. Lemma 3.25.
Let G be a 1-transitive group-groupoid, i.e. G ( x, y ) is singleton for all x, y ∈ G . Then G is complete if and only if G is a complete group. Let G be a group-groupoid with Z ( G ) = . If A n ( G ) is complete for some n ∈ N then the actortower stops, that is, for all k ∈ N with k > n , A n ( G ) ∼ = A k ( G ) . In this case the actor tower of G is afinite sequence G / / A ( G ) / / A ( G ) / / · · · / / A n ( G ) . An action of a group-groupoid H on a group-groupoid G is defined to be a group-groupoidmorphism θ : H → A ( G ) . Here note that A ( G ) acts on G where θ = 1 A ( G ) . I ( G ) also acts on G andon any normal subgroup-groupoid N of G . Thus G acts on any of its normal subgroup-groupoids.For a given action θ : H → A ( G ) one can obtain group actions of H on G (by Lemma 3.6) andof H on G . These actions are compatible with the group-groupoid structural maps, that is for all a ∈ G ( x , x ) , a ′ ∈ G ( x ′ , x ′ ) with x = x ′ and b ∈ H ( y , y ) , b ′ ∈ H ( y ′ , y ′ ) with y = y ′ (i) d ( b · a ) = d ( b ) · d ( a ) = y · x , (ii) d ( b · a ) = d ( b ) · d ( a ) = y · x , (iii) y · x = ε ( y · x ) = ε ( y ) · ε ( x ) = 1 y · x and (iv) ( b ′ ◦ b ) · ( a ′ ◦ a ) = ( b ′ · a ′ ) ◦ ( b · a ) .Here note that, by (iii) , the action of H on G can be obtained from that of H on G . So it issufficient to consider only the action of H on G .An extension of a group-groupoid N by a group-groupoid H as given in [25] is a sequence ofgroup-groupoid morphisms / / N (cid:31) (cid:127) i / / G p / / / / H / / s t t where i is a monomorphism, p is an epimorphism, ker p = i ( N ) and ps = 1 H . An action of G on N induces the following commutative diagram: 13 / / N (cid:31) (cid:127) / / (cid:15) (cid:15) G / / / / (cid:15) (cid:15) H / / (cid:15) (cid:15) / / I ( N ) (cid:31) (cid:127) / / A ( N ) / / / / O ( N ) / / . Let the group-groupoid H acts on the group-groupoid G . So we have a group-groupoid mor-phism θ : H → A ( G ) . Then there exist group actions θ of H on G and θ of H on G . Usingthese actions we can construct the semi-direct product groups G ⋊ θ H and G ⋊ θ H . Moreover, ( G ⋊ θ H , G ⋊ θ H ) has a natural group-groupoid structure. We call this group-groupoid thesemi-direct product of G and H relative to θ and denote this group-groupoid by G ⋊ θ H or brieflyby G ⋊ H when no confusion arise.Analogous to group case we can define internal semi-direct product of group-groupoids. Let G be a group-groupoid with a normal subgroup-groupoid N and a subgroup-groupoid M . If (i) G = N + M and (ii) N ∩ M = 0 then there exist a morphism of group-groupoids τ : M → A ( N ) , i.e. an action of M on N definedas follows: m · n = m + n − m for all m ∈ M and n ∈ N . Then the semi-direct product group-groupoid N ⋊ τ M obtained fromthe action τ is isomorphic to the group-groupoid G .The notion of crossed modules over group-groupoids is introduced in [25] using split exten-sions of group-groupoids by the method used in [23]. We recall the definition from [25]. Definition 3.26. [25] A crossed module over group-groupoids consist of group-groupoid mor-phisms α : G → H and θ : H → A ( G ) , i.e. a group-groupoid action of H on G , such that α : G → H is a crossed module over groups. G α / / d (cid:15) (cid:15) d (cid:15) (cid:15) H d (cid:15) (cid:15) d (cid:15) (cid:15) G ε I I α / / H ε U U We know that for a group A , the inner automorphism map A → Aut( A ) forms a crossedmodule structure over groups. Analogously we can obtain this fact for group-groupoids. Proposition 3.27.
Let G be a group-groupoid. Then the canonical group-groupoid morphism ϕ : G →A ( G ) is equipped with the structure of crossed module over group-groupoids.Proof. The proof is straightforward and is omitted. ✷ .3 Holomorph of a group-groupoid A concept in group theory which arose in connection with the following problem: "Is it pos-sible to include any given group A as a normal subgroup in some other group so that all theautomorphisms of A are restrictions of inner automorphisms of this large group?". This problemhas been answered positively and this large group is called the holomorph of A [8, 11].The holomorph of a group is a group which simultaneously contains copies of the group andits automorphism group. The holomorph provides interesting examples of groups, and allowsone to treat group elements and group automorphisms in a uniform context. In group theory, fora group A , the holomorph of A denoted Hol( A ) can be described as a semi-direct product group A ⋊ Aut( A ) or as a permutation group.The problem mentioned above has also positive answer for group-groupoids. Statement of theproblem for group-groupoids is as follows: "Is it possible to include any given group-groupoid G as a normal subgroup-groupoid in some other group-groupoid so that all the actors of G arerestrictions of inner actors of this large group-groupoid?". This large group-groupoid is the holo-morph of G . Definition 3.28.
The holomorph
Hol( G ) of a group-groupoid G is the semi-direct product group-groupoid G ⋊ A ( G ) where the action θ : A ( G ) → A ( G ) is the conjugation. Hol( G ) = G ⋊ A ( G ) = ( G ⋊ W ( G ) , G ⋊ Aut( G )) Definition 3.29.
A subgroup-groupoid H of a group-groupoid G is called characteristic in G ifrestriction defines a group-groupoid morphism A ( G ) → A ( H ) .Characteristic subgroup-groupoids are subgroup-groupoids which are invariant under all in-ner actors. In [24] it has been shown that there is a bijection between the set of characteristicsubgroups of a group A and the set of all normal subgroups of Hol( A ) contained in A . Now wegive a similar result for group-groupoids. Lemma 3.30.
A subgroup-groupoid of G is characteristic if and only if its image in Hol( G ) is a normalsubgroup-groupoid. Example 3.31.
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