# Some Results on Strict Graded Categorical Groups

Authors: Nguyen Tien Quang; Che Thi Kim Phung
Journal: Algebra
 (2013)

---

## Abstract

We present some applications of strict graded categorical groups to the construction of the obstruction of an equivariant kernel and to the classification of equivariant group extensions which are central ones. The composition of a graded categorical group and an equivariant group homomorphism is also determined.


---

## Body

## 1. Introduction

The group extension problem has an important significance in the development of modern algebra. Some notions of this problem such as crossed product, factor set, and obstruction (see [1]) are not only applied to rings or to algebraic types but also are raised to a categorical level.The theory of graded categorical groups studied by Cegarra et al. [2] can be viewed as a generalization of both the categorical group theory of Sinh [3] and the graded category theory of Fröhlich and Wall [4]. The equivariant group extension problem is one of applications of this theory.Strict graded categorical groups, with their simple structures compared to the general case, are more likely to give a lot of interesting applications. In [5] we presented an application of this notion to the classification of equivariant crossed modules. In this paper we continue to introduce some other applications. Firstly, we show that if [h] is the third invariant of the strict graded categorical group HolΓG and p:Π→OutG is an equivariant kernel; then
(1)p*h∈Obs(p).
Secondly, we classify equivariant group extensions of A by G which are central extensions by graded monoidal autofunctors of the strict graded categorical group ∫Γ‍(G,A,0). Finally, we construct the composition of a Γ-graded categorical group with a Γ-homomorphism, which is analogous to the composition of a group extension with a group homomorphism (see [1, Chapter 3]).


## 2. Preliminaries

### 2.1.  Graded Categorical Groups

We recall briefly some basic notions about graded categorical groups in [2].We regard the groupΓ as a category with one object, say *, where the morphisms are elements of Γ and the composition is the group operation. A category 𝔾 is Γ-graded if there is a functor gr:𝔾→Γ. The grading gr is said to be stable if for any object X∈Ob𝔾 and any σ∈Γ there exists an isomorphism f in 𝔾 with domain X and gr(f)=σ. A Γ-graded monoidal category𝔾=(𝔾,gr,⊗,I,a,l,r) consists of(1)
a stableΓ-graded category (𝔾,gr), Γ-graded functors ⊗:𝔾×Γ𝔾→𝔾 and I:Γ→𝔾,(2)
natural isomorphisms of grade 1aX,Y,Z:(X⊗Y)⊗Z→~X⊗(Y⊗Z), lX:I⊗X→~X and rX:X⊗I→~X such that, for all X,Y,Z,T∈Ob𝔾, the following two coherence conditions hold:
(2)aX,Y,Z⊗TaX⊗Y,Z,T=(idX⊗aY,Z,T)aX,Y⊗Z,T(aX,Y,Z⊗idT),(idX⊗lY)aX,I,Y=rX⊗idY.Agraded categorical group is a graded monoidal category 𝔾 in which every object is invertible and every morphism is an isomorphism. In this case, the subcategory Ker𝔾 consisting of all objects of 𝔾 and all morphisms of grade 1 in 𝔾 is a categorical group.If𝔾, 𝔾′ are Γ-monoidal categories, then a graded monoidal functor(F,F~,F*):𝔾→𝔾′ consists of a Γ-graded functor F:𝔾→𝔾′, natural isomorphisms of grade 1 F~X,Y:FX⊗FY→F(X⊗Y), and an isomorphism of grade 1 F*:I′→FI, such that, for all X,Y,Z∈Ob𝔾, the following coherence conditions hold:
(3)F~X,Y⊗Z(idFX⊗F~Y,Z)aFX,FY,FZ=F(aX,Y,Z)F~X⊗Y,Z(F~X,Y⊗idFZ),F(rX)F~X,I(idFX⊗F*)=rFX,F(lX)F~I,X(F*⊗idFX)=lFX.
Let (F,F~,F*), (F′,F~′,F*′) be two Γ-graded monoidal functors. A graded monoidal natural equivalenceθ:F→~F′ is a natural equivalence of functors such that all isomorphisms θX:FX→F′X are of grade 1, and for all X,Y∈Ob𝔾, the following coherence conditions hold:
(4)F~X,Y′(θX⊗θY)=θX⊗YF~X,Y,θIF*=F*′.


### 2.2. Reduced Graded Categorical Groups

The authors of [2] showed that any Γ-graded categorical group 𝔾=(𝔾,gr,⊗,I,a,l,r) determines a triple (Π,A,h), where(1)
the setΠ=π0𝔾 of 1-isomorphism classes of the objects in 𝔾 is a Γ-group,(2)
the setA=π1𝔾 of 1-automorphisms of the unit object I is a Γ-equivariant Π-module,(3)
the third invariant is an equivariant cohomology class[h]∈HΓ3(Π,A).Based on these data, they constructed aΓ-graded categorical group, denoted by ∫Γ‍(Π,A,h), which is graded monoidally equivalent to 𝔾. Below, we briefly recall this construction.Objects of∫Γ‍(Π,A,h) are elements x∈Π and its morphisms are pairs (a,σ):x→y consisting of an element a∈A and σ∈Γ such that σx=y.The composition of two morphisms(x→(a,σ)y→(b,τ)z) is given by
(5)(b,τ)(a,σ)=(b+aτ+h(x,τ,σ),τσ).The graded tensor product∫Γ‍(Π,A,h)×Γ∫Γ‍(Π,A,h)→∫Γ‍(Π,A,h) is given by
(6)(x⟶(a,σ)y)⊗(x′⟶(b,σ)y′)=(xx′→(a+yb+h(x,x′,σ),σ)yy′).The unit constrains are strict in the sense thatlx=(0,1)=rx:x→x. The associativity isomorphisms are
(7)ax,y,z=(h(x,y,z),1):(xy)z⟶x(yz).The stableΓ-grading is gr(a,σ)=σ.The unit graded functorI:Γ→∫Γ‍(Π,A,h) is given by
(8)I(*⟶σ*)=(1→(0,σ)1).
We call the Γ-graded categorical group ∫Γ‍(Π,A,h) a reduction of the Γ-graded categorical group 𝔾, simply denoted by S𝔾.


## 2.1.  Graded Categorical Groups

We recall briefly some basic notions about graded categorical groups in [2].We regard the groupΓ as a category with one object, say *, where the morphisms are elements of Γ and the composition is the group operation. A category 𝔾 is Γ-graded if there is a functor gr:𝔾→Γ. The grading gr is said to be stable if for any object X∈Ob𝔾 and any σ∈Γ there exists an isomorphism f in 𝔾 with domain X and gr(f)=σ. A Γ-graded monoidal category𝔾=(𝔾,gr,⊗,I,a,l,r) consists of(1)
a stableΓ-graded category (𝔾,gr), Γ-graded functors ⊗:𝔾×Γ𝔾→𝔾 and I:Γ→𝔾,(2)
natural isomorphisms of grade 1aX,Y,Z:(X⊗Y)⊗Z→~X⊗(Y⊗Z), lX:I⊗X→~X and rX:X⊗I→~X such that, for all X,Y,Z,T∈Ob𝔾, the following two coherence conditions hold:
(2)aX,Y,Z⊗TaX⊗Y,Z,T=(idX⊗aY,Z,T)aX,Y⊗Z,T(aX,Y,Z⊗idT),(idX⊗lY)aX,I,Y=rX⊗idY.Agraded categorical group is a graded monoidal category 𝔾 in which every object is invertible and every morphism is an isomorphism. In this case, the subcategory Ker𝔾 consisting of all objects of 𝔾 and all morphisms of grade 1 in 𝔾 is a categorical group.If𝔾, 𝔾′ are Γ-monoidal categories, then a graded monoidal functor(F,F~,F*):𝔾→𝔾′ consists of a Γ-graded functor F:𝔾→𝔾′, natural isomorphisms of grade 1 F~X,Y:FX⊗FY→F(X⊗Y), and an isomorphism of grade 1 F*:I′→FI, such that, for all X,Y,Z∈Ob𝔾, the following coherence conditions hold:
(3)F~X,Y⊗Z(idFX⊗F~Y,Z)aFX,FY,FZ=F(aX,Y,Z)F~X⊗Y,Z(F~X,Y⊗idFZ),F(rX)F~X,I(idFX⊗F*)=rFX,F(lX)F~I,X(F*⊗idFX)=lFX.
Let (F,F~,F*), (F′,F~′,F*′) be two Γ-graded monoidal functors. A graded monoidal natural equivalenceθ:F→~F′ is a natural equivalence of functors such that all isomorphisms θX:FX→F′X are of grade 1, and for all X,Y∈Ob𝔾, the following coherence conditions hold:
(4)F~X,Y′(θX⊗θY)=θX⊗YF~X,Y,θIF*=F*′.


## 2.2. Reduced Graded Categorical Groups

The authors of [2] showed that any Γ-graded categorical group 𝔾=(𝔾,gr,⊗,I,a,l,r) determines a triple (Π,A,h), where(1)
the setΠ=π0𝔾 of 1-isomorphism classes of the objects in 𝔾 is a Γ-group,(2)
the setA=π1𝔾 of 1-automorphisms of the unit object I is a Γ-equivariant Π-module,(3)
the third invariant is an equivariant cohomology class[h]∈HΓ3(Π,A).Based on these data, they constructed aΓ-graded categorical group, denoted by ∫Γ‍(Π,A,h), which is graded monoidally equivalent to 𝔾. Below, we briefly recall this construction.Objects of∫Γ‍(Π,A,h) are elements x∈Π and its morphisms are pairs (a,σ):x→y consisting of an element a∈A and σ∈Γ such that σx=y.The composition of two morphisms(x→(a,σ)y→(b,τ)z) is given by
(5)(b,τ)(a,σ)=(b+aτ+h(x,τ,σ),τσ).The graded tensor product∫Γ‍(Π,A,h)×Γ∫Γ‍(Π,A,h)→∫Γ‍(Π,A,h) is given by
(6)(x⟶(a,σ)y)⊗(x′⟶(b,σ)y′)=(xx′→(a+yb+h(x,x′,σ),σ)yy′).The unit constrains are strict in the sense thatlx=(0,1)=rx:x→x. The associativity isomorphisms are
(7)ax,y,z=(h(x,y,z),1):(xy)z⟶x(yz).The stableΓ-grading is gr(a,σ)=σ.The unit graded functorI:Γ→∫Γ‍(Π,A,h) is given by
(8)I(*⟶σ*)=(1→(0,σ)1).
We call the Γ-graded categorical group ∫Γ‍(Π,A,h) a reduction of the Γ-graded categorical group 𝔾, simply denoted by S𝔾.


## 3. Strict Graded Categorical Groups

### 3.1. Definitions and Examples

It is well known that each crossed module of groups can be seen as a strict categorical group (see [6], Remark 3.1 in [7]). Crossed modules of groups can be enriched in some ways to become, for example, crossed bimodules over rings or equivariant crossed modules. In the former case, each crossed bimodule can be seen as a strict Ann-category [8]. In the later case, each crossed module of Γ-groups can be identified with a strictΓ-graded categorical group [5] whose definition is recalled as below.Firstly, if(F,F~,F*) is a monoidal functor between categorical groups, then the isomorphism F*:I′→FI can be deduced from F and F~, so we can omit F* when not necessary. A monoidal functor (F,F~):𝔾→𝔾′ between two categorical groups is termed regular whenever
(9)F(x)⊗F(y)=F(x⊗y),F(b)⊗F(c)=F(b⊗c),
for all x,y∈Ob𝔾, b,c∈Mor𝔾. A factor set (F,θ) on Γ with coefficients in a categorical group 𝔾 (see [9]) is regular if θσ,τ=id, and Fσ is a regular monoidal functor, for all σ,τ∈Γ.Definition 1 (see [5]).
A graded categorical group(𝔾,gr) is said to be strict if(i)
Ker𝔾 is a strict categorical group,(ii)
𝔾 induces a regular factor set (F,θ) on Γ with coefficients in a categorical group Ker𝔾.Equivalently, a graded categorical group(𝔾,gr) is strict if it is a Γ-graded extension of a strict categorical group by a regular factor set.Note 1.
In this paper we denote by + for the operation of the groupG.Example 2 (the strict graded categorical groupDisΓΠ).
ThediscreteΓ-graded categorical group DisΓΠ defined by a Γ-group Π has the elements of Π as objects, and its morphisms σ:x→y are the elements σ∈Γ with σx=y. Composition is multiplication in Γ, and the grading gr:DisΓΠ→Γ is the obvious map gr(σ)=σ. The graded tensor product is given by
(10)(x⟶σy)⊗(x′⟶σy′)=(xx′⟶σyy′)
and the graded unit I:Γ→DisΓΠ by
(11)I(*⟶σ*)=(1⟶σ1);
the associativity and unit isomorphisms are identities.Example 3 (the strict graded categorical groupHolΓG of a Γ-group G).
Firstly, observe that ifG is a Γ-group, then the group Aut G of automorphisms of G is also a Γ-group with the action
(12)(fσ)(b)=σ(f(bσ-1)),b∈G,f∈AutG.
Then, the homomorphism μ:G→AutG, b↦μb (μb is the inner automorphism of G given by conjugation with b) is a homomorphism of Γ-groups. Indeed, for all σ∈Γ, a,b∈G one has
(13)μbσ(a)=bσ+a-bσ=(b+aσ-1-b)σ=(μb(aσ-1))σ=(μbσ)(a).
For eachΓ-group G, we can construct a strict graded categorical group, denoted by HolΓG (see [2]), whose objects are elements of the Γ-group AutG. A σ-morphism α→β is a pair (u,σ), where u∈G with σα=μuβ. Composition of two morphisms α→(u,σ)β→(v,τ)γ is given by
(14)(v,τ)∘(u,σ)=(uτ+v,τσ).
The graded tensor product is(15)(α→(u,σ)β)⊗(α′→(u′,σ)β′)=(αα′→(u+β(u′),σ)ββ′),
and the graded unit I:Γ→HolΓG is defined by
(16)I(*⟶σ*)=idG⟶(0,σ)idG.
The associativity and unit isomorphisms are identities.Example 4 (strict graded categorical groups associated with an equivariant crossed module).
The notion ofΓ-crossed module is a generalization of that of crossed module of groups introduced by Whitehead [10].Definition 5 (see [5]).
LetB, D be Γ-groups. A Γ-crossed module is a quadruple (B,D,d,ϑ), where d:B→D, ϑ:D→AutB are Γ-homomorphisms satisfying the following conditions:
(17)C1·ϑd=μ,C2·d(ϑx(b))=μx(d(b)),C3·(ϑx(b))σ=ϑ(xσ)(bσ),
where σ∈Γ, x∈D, b∈B, and μx is the inner automorphism given by conjugation with x.AΓ-crossed module is also called an equivariant crossed moduleby Noohi [11].From the definition ofΓ-crossed module, it is easy to deduce the following properties: (i) Kerd is a Γ-subgroup in Z(B); (ii) Imd is both a normal subgroup in D and a Γ-group; (iii) Kerd is a left Γ-equivariant Cokerd-module under the actions
(18)sa=φx(a),σs=[xσ],a∈Kerd,x∈s∈Cokerd.The strict graded categorical group𝔾 associated with the Γ-crossed module (B,D,d,ϑ) is constructed as follows.Objects of𝔾 are the elements of the group D, and a σ-morphism x→y is a pair (b,σ), where b∈B, σ∈Γ with σx=d(b)y. Composition of two morphisms is defined by
(19)(x⟶(b,σ)y⟶(c,τ)z)=(x→(τb+c,τσ)z).The tensor product on objects is given by the multiplication in the groupD, and for two morphisms (x→(b,σ)y), (x′→(c,σ)y′), then
(20)(x⟶(b,σ)y)⊗(x′→(c,σ)y′)=(xx′→(b+ϑyc,σ)yy′).The associativity and unit constraints of the tensor product are strict.The graded functorgr:𝔾→Γ is given by
(21)(b,σ)⟼σ
and the graded unit functor I:Γ→𝔾 by
(22)I(*⟶σ*)=(1→(0,σ)1).


### 3.2. Equivariant Kernelp:Π→OutG and Strict Γ-Categorical Group HolΓ(G)

The notion ofequivariant kernel was introduced in [2]. It is a triple (Π,G,p), where Π, G are Γ-groups and p:Π→AutG/InG is a homomorphism of Γ-groups. Then ZG is Γ-equivariant OutG-module under the action [f]a=f(a). The obstruction of (Π,G,p) is an element Obs(p)∈HΓ3(Π,ZG), defined as follows (see [2, page 996]). For each x∈Π, choose an automorphism φx of G, φx∈p(x), in particular φ1=idG. Since
(23)p(xy)=p(x)p(y),p(x)σ=p(xσ),===============x,y∈Π,σ∈Γ,
there exist elements f(x,y),f(x,σ)∈G such that
(24)φxφy=μf(x,y)φxy,φxσ=μf(x,σ)φ(σx),
where μu is the inner automorphism of Γ-group G induced by u∈G. The pair (f,φ) therefore induces an element k∈ZΓ3(Π,ZG) defined by the following relations. The associative law for a product φxφyφz leads to
(25)φxf(y,z)+f(x,yz)=k(x,y,z)+f(x,y)+f(xy,z).
The relations
(26)(φxφy)σ=φxσφyσ,(φxσ)τ=φxτσ
imply
(27)f(x,σ)+φ(xσ)(f(y,σ))+f(xσ,yσ)=k(x,y,σ)+fσ(x,y)+f(xy,σ)(28)τf(x,σ)+f(xσ,τ)=k(x,τ,σ)+f(x,τσ)
According to [2], k∈ZΓ3(Π,ZG). The cohomology class [k] is called the obstruction of p, denoted by Obs(p).Theorem 4.1 in [2] states that equivariant extensions G by Π inducing p exist if and only if Obs(p)=0. By Theorem 4.2 in [2], there is a bijection
(29)ExtΓ(Π,G)≃Hom[DisΓΠ,HolΓG].The following theorem describes the invariants of the graded categorical groupHolΓG and shows a relation between third invariant of HolΓG and Obs(p).Theorem 6.
Let(Π,G,p) be an equivariant kernel. Then the invariants of the strict graded categorical group HolΓG are(i)
π0(HolΓG)=OutG, π1(HolΓG)=ZG,(ii)
h∈ZΓ3(OutG,ZG) with p*h∈Obs(p).Proof.
(i) It is obvious.
(ii) According to Section 3 in [2], there is a Γ-monoidal equivalence
(30)(H,H~):∫Γ‍(OutG,ZG,h)⟶HolΓG.
Let us observe that each morphism (a,σ):r→s in ∫Γ‍(OutG,ZG,h) is written in the form
(31)r→(0,σ)rσ→(a,1)s.
Let
(32)H(r→(0,σ)rσ)=(H(r)→(g(r,σ),σ)Hσ(r)).
SinceH(r→(a,1)r)=(a,1), for r∈OutG, a∈ZG (see [12, Proposition 13]), the γ-monoidal functor (H,H~) defines a function f by
(33)f(x,y)=H~px,py=(p*H~)(x,y),f(x,σ)=g(px,σ)=(p*g)(x,σ).
Set φ=H∘p:Π→OutG→AutG, and the pair (f,φ) determines a function k∈Obs(p) by relations (25)–(28) which can be written in the form k=δf, even though f does not take values in ZG.
The compatibility of(H,H~) with the associativity constraints leads to h=δH~. Thus, for x,y,z∈Π, we obtain
(34)(p*h)(x,y,z)=h(px,py,pz)=(δH~)(px,py,pz)=(δf)(x,y,z)=(25)k(x,y,z).
Next, we show that(p*h)(x,y,σ)=k(x,y,σ).
Since(H,H~) is a ⊗-functor, the following diagram commutes:
(35)
where u=(r→(0,σ)r′), v=(s→(0,σ)s′).
Due to the relations (5)–(15) of the ⊗ product and the composition in ∫Γ‍(OutG,ZG,h) and HolΓG, one has
(36)H(u⊗v)∘H~r,s=(Hr,sσ+g(rs,σ)+h(r,s,σ),σ),H~r′,s′∘(H(u)⊗H(v))=(g(r,σ)+H(r′)g(s′,σ)+H~r′,s′,σ).
Let us note that(37)Hr,s=Hpx,py=f(x,y),h(r,s,σ)=(p*h)(x,y,σ),g(r,σ)=g(p(x),σ)=(p*g)(x,σ)=f(x,σ),H(r′)=H(rσ)=Hσ(r)=Hσ(p(x))=φσx.
Therefore,
(38)fσ(x,y)+f(xy,σ)+(p*h)(x,y,σ)=f(x,σ)+φxσf(y,σ)+f(xσ,yσ).
This shows that (p*h)(x,y,σ)=(27)k(x,y,σ).
Finally, we prove that(p*h)(x,τ,σ)=k(x,τ,σ) based on the functorial property of H. Indeed, consider the composition
(39)r→(0,σ)s→(0,τ)t.
On one hand,
(40)H[(0,τ)∘(0,σ)]=(5)H[(h(r,τ,σ),τσ)]=H[h(r,τ,σ),1)∘(0,τσ)]=H[(h(r,τ,σ),1)]∘H[(0,τσ)]=(h(r,τ,σ),1)∘(g(r,τσ),τσ)=(14)(g(r,τσ)+h(r,τ,σ),τσ).
On the other hand,
(41)H(0,τ)∘H(0,σ)=H(r)→(g(r,σ),σ)H(s)→(g(s,τ),τ)H(t)=(14)H(r)→(τg(r,σ)+g(σr,τ),τσ)H(t).
Again, h(r,τ,σ)∈ZG, the functorial property of H, implies
(42)h(r,τ,σ)+g(r,τσ)=gτ(r,σ)+g(rσ,τ),
or h(r,τ,σ)=δg. Clearly,
(43)(p*h)(x,τ,σ)=h(px,τ,σ)=δg(px,τ,σ)=δf(x,τ,σ)=(28)k(x,τ,σ).


### 3.3. Classification of Central Extensions

Denote byExtΓc(Π,A) the set of all equivalence classes of equivariant extensions A↣E↠Π, where A⊂ZE. One can classify these extensions by the autofunctors of Γ-graded categorical group ∫Γ‍(Π,A,0).Theorem 7 (the schreier theory for central extensions of equivariant groups).
LetΠ be a Γ-group, and let A be a Γ-equivariant Π-module. There is a bijection
(44)ExtΓc(Π,A)≃EndΓid(∫Γ‍(Π,A,0)),
where EndΓid is the set of homotopy classes of monoidal Γ-functors F from ∫Γ‍(Π,A,0) to itself satisfying
(45)F(x)=x,x∈Π,F(b,1)=(b,1),b∈A.Proof.
Let(F,F~)∈EndΓid(∫Γ‍(Π,A,0)). Then (F,F~) defines a function
(46)φ:(Π×Π)∪(Π×Γ)⟶A
by
(47)(φ(x,y),1)=F~x,y,(φ(x,σ),σ)=F(x→(0,σ)xσ),
with φ(x,1Γ)=0. The compatibilities of (F,F~) with the constraints imply φ(x,1)=0=φ(1,y) and
(48)φ(x,y)+φ(xy,z)=x(φ(y,z))+φ(x,yz).
The naturalness of F~ and the functority of F give, respectively,
(49)σφ(x,y)+φ(xy,σ)=φ(x,σ)+(xσ)φ(y,σ)+φ(xσ,yσ),(50)φ(x,τσ)=φσ(x,τ)+φ(xτ,σ),
for all x,y,z∈G, σ,τ∈Γ. With the function φ, we can construct an equivariant crossed product extension Eφ=A×φΠ in which the operation of Eφ is
(51)(a,x)+(b,y)=(a+xb+φ(x,y),xy).
The relation (48) and the normality of φ show that Eφ is a group. By the relations (49), (50), Eφ is a Γ-group under the Γ-action
(52)σ(a,x)=(aσ+φ(x,σ),xσ).
Then we obtain an exact sequence
(53)0⟶A⟶iEφ⟶qΠ⟶1,
where i(a)=(a,1), q(a,x)=x. Further, it is easy to see that A⊂Z(Eφ).
Conversely, suppose that(54)0⟶A⟶iE⟶qΠ⟶1
is an equivariant extension with A⊂Z(E). For each x∈Π, choose a representative ux∈E, u1=(0). Then the system {ux∣x∈Π} induces a function φ taking values in A such that
(55)ux+uy=φ(x,y)+uxy,σux=φ(x,σ)+uσx.
The rules
(56)ux+(uy+uz)=(ux+uy)+uz,σ(ux+uy)=(ux)σ+(uy)σ,τ(uxσ)=uxτσ
imply the relations (48)–(50), respectively. Thus, we can define a Γ-monoidal functor (F,F~)∈EndΓ(∫Γ‍(Π,A,0)) as follows:
(57)Fx=x,F(b,1)=(b,1),F(x→(0,σ)xσ)=(φ(x,σ),σ),F~x,y=(φ(x,y),1).
Now, two extensions inExtΓc(Π,A) are equivalent if and only if two corresponding graded monoidal functors are homotopic. Indeed, let F,F′:∫Γ‍(Π,A,0)→∫Γ‍(Π,A,0) be two homotopic Γ-graded monoidal functors in which the homotopy ε:F→F′ is given by
(58)εx=(x→(e(x),1)x),x∈Π,
where e:Π→A. From the definition of a homotopy and from the relations (5), (6), it follows that
(59)e(1)=0,φ(x,σ)+e(xσ)=eσ(x)+φ′(x,σ),φ(x,y)+e(xy)=e(x)+xe(y)+φ′(x,y),
where x,y∈Π, σ∈Γ. Then, the corresponding
(60)β:Eφ⟶Eφ′(a,x)⟼(a+e(x),x)
is an equivalence between two central extensions if and only if equalities (59) hold.
Conversely, ifβ:Eφ→Eφ′ is an isomorphism, then
(61)β(a,x)=(a+e(x),x),
where e:Π→A is a function with e(1)=0. Thus, it follows that εx=(e(x),1) is a homotopy of F and F′, as we see by retracting our steps.
Finally, each central extensionE is equivalent to one Eφ via the isomorphism α:a+ux↦(a,x).


### 3.4. Composition of aΓ-Categorical Group with a Γ-Homomorphism

It is well known that for a given extension(62)E:0⟶A⟶iB⟶qΠ→1,
where A is Abelian, and a given homomorphism γ:Π′→Π, there is an extension E′ of A by Π′ with E′=Eγ. The extension E′ is just the pull pack of the pair (q,γ). We state an analogous result on strict Γ-categorical groups in the following proposition.Proposition 8.
Letℍ be a strict graded categorical group with three invariants Π, C, h, and let p:Π′→Π be an equivariant homomorphism. Then there exists a strict graded categorical group 𝔾 which is equivalent to the graded categorical group ∫Γ‍(Π′,C,h′), where C is viewed as a G′-module under the actions xc=p(x)c, x∈Π′, c∈C, and [h′]=[p*h].Proof.
The strict graded categorical group𝔾 is constructed as follows:
(63)Ob𝔾={(x,X)∣x∈Π′,X∈p(x)}.
A σ-morphism (x,X)→(x,Y) is a triple (x,u,σ), where u:X→Y is a σ-morphism in ℍ. The composition of two morphisms (x,X)→(x,u,σ)(x,Y)→(x,v,τ)(x,Z) is given by
(64)(x,v,τ)∘(x,u,σ)=(x,v∘u,τσ).
The tensor products on objects and on morphisms in𝔾 are given by
(65)(x,X)⊗(y,Y)=(xy,X⊗Y),(x,u,σ)⊗(y,v,σ)=(xy,u⊗v,σ).
For each morphism(x,u,σ) in 𝔾, one has
(66)(x,u,σ)-1=(x,u-1,σ-1).
The graded functor gr:𝔾→Γ is given by
(67)(x,u,σ)⟼σ
and the graded unit functor I:Γ→𝔾 by
(68)σ⟼((1,I)→(1,idI,σ)(1,I)).
The unit object of𝔾 is (1,I), where I is the unit object of ℍ. The associativity and unit constraints are identities.
Suppose that the graded categorical groupℍ induces a regular factor set {Fℍσ,σ∈Γ}. Then, 𝔾 induces a regular factor set {F𝔾σ,σ∈Γ} by
(69)F𝔾σ(x,X)=(x,FℍσX),F𝔾σ(x,u,τ)=(x,Fℍσ(u),τ).
Thus,𝔾 is a strict graded categorical group.
We define a pair(φ,f) of maps by
(70)λ:π0𝔾⟶Π′,f:π1𝔾⟶π1ℍ=C,[(x,X)]⟼x,(1,c,σ)⟼(c,σ).
Then, λ is an isomorphism of Γ-groups and f is an isomorphism of Γ-equivariant Π′-module, where the Π′-module structure of C is xc=p(x)c,x∈Π′,c∈C.
The graded monoidal functor(F,F~):𝔾→ℍ given by
(71)F(x,X)=X,F(x,u,σ)=(u,σ),F~=id
induces one SF=(ϕ,ϕ~):S𝔾→Sℍ, where Sℍ=∫Γ‍(Π,C,h). Then,
(72)ϕ[(x,X)]=F0[(x,X)]=[F(x,X)]=[X]=p(x)=pλ[(x,X)],ϕ(1,c,σ)=F1(1,c,σ)=γF(1,I)-1F(1,c,σ)=γI-1(c,σ)=(c,σ)=f(1,c,σ),
where (c,σ) is a morphism in Sℍ. Thus, SF is a functor of type (pλ,f).
Suppose thath𝔾∈ZΓ3(π0𝔾,π1𝔾). According to Theorem 3.2 in [2], the obstruction of pair (pλ,f) vanishes in H3(π0𝔾,π1ℍ)=H3(π0𝔾,C); that is,
(73)(pλ)*h=f*h𝔾+δϕ~.
Now, we denote h′=f*h𝔾, and then the pair (J,J~), where J=(λ,f), J~=id, is a graded monoidal functor from S𝔾 to 𝕁=∫Γ‍(Π′,C,h′). The composition
(74)𝔾⟶(G,G~)S𝔾⟶(J,J~)𝕁
is an equivalence from 𝔾 to 𝕁.
Finally, we prove thath′ is in the same cohomology class with p*h. Let K=(λ-1,f-1):𝕁→S𝔾; then K together with K~=id is a graded monoidal functor, and the composition
(75)(ϕ,ϕ~)∘(K,K~):𝕁⟶Sℍ
is a graded monoidal functor making the following diagram commute:
(76)
Since ϕ∘K is a graded monoidal functor of type (p,id), its obstruction vanishes. This implies that p*h-h′=∂g, and that means that [h′]=[p*h].IfΓ=1, we obtain Proposition 14 in [12].The graded categorical group𝔾 is called a composition of ℍ with p, denoted by 𝔾=ℍ∘p. It follows that 𝔾 can be seen as a contravariant functor in terms of the invariant π0𝔾 with π1𝔾 fixed. Indeed,
(77)𝔾∘idπ0𝔾=𝔾,𝔾∘(p∘p′)=(𝔾∘p)∘p′.


## 3.1. Definitions and Examples

It is well known that each crossed module of groups can be seen as a strict categorical group (see [6], Remark 3.1 in [7]). Crossed modules of groups can be enriched in some ways to become, for example, crossed bimodules over rings or equivariant crossed modules. In the former case, each crossed bimodule can be seen as a strict Ann-category [8]. In the later case, each crossed module of Γ-groups can be identified with a strictΓ-graded categorical group [5] whose definition is recalled as below.Firstly, if(F,F~,F*) is a monoidal functor between categorical groups, then the isomorphism F*:I′→FI can be deduced from F and F~, so we can omit F* when not necessary. A monoidal functor (F,F~):𝔾→𝔾′ between two categorical groups is termed regular whenever
(9)F(x)⊗F(y)=F(x⊗y),F(b)⊗F(c)=F(b⊗c),
for all x,y∈Ob𝔾, b,c∈Mor𝔾. A factor set (F,θ) on Γ with coefficients in a categorical group 𝔾 (see [9]) is regular if θσ,τ=id, and Fσ is a regular monoidal functor, for all σ,τ∈Γ.Definition 1 (see [5]).
A graded categorical group(𝔾,gr) is said to be strict if(i)
Ker𝔾 is a strict categorical group,(ii)
𝔾 induces a regular factor set (F,θ) on Γ with coefficients in a categorical group Ker𝔾.Equivalently, a graded categorical group(𝔾,gr) is strict if it is a Γ-graded extension of a strict categorical group by a regular factor set.Note 1.
In this paper we denote by + for the operation of the groupG.Example 2 (the strict graded categorical groupDisΓΠ).
ThediscreteΓ-graded categorical group DisΓΠ defined by a Γ-group Π has the elements of Π as objects, and its morphisms σ:x→y are the elements σ∈Γ with σx=y. Composition is multiplication in Γ, and the grading gr:DisΓΠ→Γ is the obvious map gr(σ)=σ. The graded tensor product is given by
(10)(x⟶σy)⊗(x′⟶σy′)=(xx′⟶σyy′)
and the graded unit I:Γ→DisΓΠ by
(11)I(*⟶σ*)=(1⟶σ1);
the associativity and unit isomorphisms are identities.Example 3 (the strict graded categorical groupHolΓG of a Γ-group G).
Firstly, observe that ifG is a Γ-group, then the group Aut G of automorphisms of G is also a Γ-group with the action
(12)(fσ)(b)=σ(f(bσ-1)),b∈G,f∈AutG.
Then, the homomorphism μ:G→AutG, b↦μb (μb is the inner automorphism of G given by conjugation with b) is a homomorphism of Γ-groups. Indeed, for all σ∈Γ, a,b∈G one has
(13)μbσ(a)=bσ+a-bσ=(b+aσ-1-b)σ=(μb(aσ-1))σ=(μbσ)(a).
For eachΓ-group G, we can construct a strict graded categorical group, denoted by HolΓG (see [2]), whose objects are elements of the Γ-group AutG. A σ-morphism α→β is a pair (u,σ), where u∈G with σα=μuβ. Composition of two morphisms α→(u,σ)β→(v,τ)γ is given by
(14)(v,τ)∘(u,σ)=(uτ+v,τσ).
The graded tensor product is(15)(α→(u,σ)β)⊗(α′→(u′,σ)β′)=(αα′→(u+β(u′),σ)ββ′),
and the graded unit I:Γ→HolΓG is defined by
(16)I(*⟶σ*)=idG⟶(0,σ)idG.
The associativity and unit isomorphisms are identities.Example 4 (strict graded categorical groups associated with an equivariant crossed module).
The notion ofΓ-crossed module is a generalization of that of crossed module of groups introduced by Whitehead [10].Definition 5 (see [5]).
LetB, D be Γ-groups. A Γ-crossed module is a quadruple (B,D,d,ϑ), where d:B→D, ϑ:D→AutB are Γ-homomorphisms satisfying the following conditions:
(17)C1·ϑd=μ,C2·d(ϑx(b))=μx(d(b)),C3·(ϑx(b))σ=ϑ(xσ)(bσ),
where σ∈Γ, x∈D, b∈B, and μx is the inner automorphism given by conjugation with x.AΓ-crossed module is also called an equivariant crossed moduleby Noohi [11].From the definition ofΓ-crossed module, it is easy to deduce the following properties: (i) Kerd is a Γ-subgroup in Z(B); (ii) Imd is both a normal subgroup in D and a Γ-group; (iii) Kerd is a left Γ-equivariant Cokerd-module under the actions
(18)sa=φx(a),σs=[xσ],a∈Kerd,x∈s∈Cokerd.The strict graded categorical group𝔾 associated with the Γ-crossed module (B,D,d,ϑ) is constructed as follows.Objects of𝔾 are the elements of the group D, and a σ-morphism x→y is a pair (b,σ), where b∈B, σ∈Γ with σx=d(b)y. Composition of two morphisms is defined by
(19)(x⟶(b,σ)y⟶(c,τ)z)=(x→(τb+c,τσ)z).The tensor product on objects is given by the multiplication in the groupD, and for two morphisms (x→(b,σ)y), (x′→(c,σ)y′), then
(20)(x⟶(b,σ)y)⊗(x′→(c,σ)y′)=(xx′→(b+ϑyc,σ)yy′).The associativity and unit constraints of the tensor product are strict.The graded functorgr:𝔾→Γ is given by
(21)(b,σ)⟼σ
and the graded unit functor I:Γ→𝔾 by
(22)I(*⟶σ*)=(1→(0,σ)1).


## 3.2. Equivariant Kernelp:Π→OutG and Strict Γ-Categorical Group HolΓ(G)

The notion ofequivariant kernel was introduced in [2]. It is a triple (Π,G,p), where Π, G are Γ-groups and p:Π→AutG/InG is a homomorphism of Γ-groups. Then ZG is Γ-equivariant OutG-module under the action [f]a=f(a). The obstruction of (Π,G,p) is an element Obs(p)∈HΓ3(Π,ZG), defined as follows (see [2, page 996]). For each x∈Π, choose an automorphism φx of G, φx∈p(x), in particular φ1=idG. Since
(23)p(xy)=p(x)p(y),p(x)σ=p(xσ),===============x,y∈Π,σ∈Γ,
there exist elements f(x,y),f(x,σ)∈G such that
(24)φxφy=μf(x,y)φxy,φxσ=μf(x,σ)φ(σx),
where μu is the inner automorphism of Γ-group G induced by u∈G. The pair (f,φ) therefore induces an element k∈ZΓ3(Π,ZG) defined by the following relations. The associative law for a product φxφyφz leads to
(25)φxf(y,z)+f(x,yz)=k(x,y,z)+f(x,y)+f(xy,z).
The relations
(26)(φxφy)σ=φxσφyσ,(φxσ)τ=φxτσ
imply
(27)f(x,σ)+φ(xσ)(f(y,σ))+f(xσ,yσ)=k(x,y,σ)+fσ(x,y)+f(xy,σ)(28)τf(x,σ)+f(xσ,τ)=k(x,τ,σ)+f(x,τσ)
According to [2], k∈ZΓ3(Π,ZG). The cohomology class [k] is called the obstruction of p, denoted by Obs(p).Theorem 4.1 in [2] states that equivariant extensions G by Π inducing p exist if and only if Obs(p)=0. By Theorem 4.2 in [2], there is a bijection
(29)ExtΓ(Π,G)≃Hom[DisΓΠ,HolΓG].The following theorem describes the invariants of the graded categorical groupHolΓG and shows a relation between third invariant of HolΓG and Obs(p).Theorem 6.
Let(Π,G,p) be an equivariant kernel. Then the invariants of the strict graded categorical group HolΓG are(i)
π0(HolΓG)=OutG, π1(HolΓG)=ZG,(ii)
h∈ZΓ3(OutG,ZG) with p*h∈Obs(p).Proof.
(i) It is obvious.
(ii) According to Section 3 in [2], there is a Γ-monoidal equivalence
(30)(H,H~):∫Γ‍(OutG,ZG,h)⟶HolΓG.
Let us observe that each morphism (a,σ):r→s in ∫Γ‍(OutG,ZG,h) is written in the form
(31)r→(0,σ)rσ→(a,1)s.
Let
(32)H(r→(0,σ)rσ)=(H(r)→(g(r,σ),σ)Hσ(r)).
SinceH(r→(a,1)r)=(a,1), for r∈OutG, a∈ZG (see [12, Proposition 13]), the γ-monoidal functor (H,H~) defines a function f by
(33)f(x,y)=H~px,py=(p*H~)(x,y),f(x,σ)=g(px,σ)=(p*g)(x,σ).
Set φ=H∘p:Π→OutG→AutG, and the pair (f,φ) determines a function k∈Obs(p) by relations (25)–(28) which can be written in the form k=δf, even though f does not take values in ZG.
The compatibility of(H,H~) with the associativity constraints leads to h=δH~. Thus, for x,y,z∈Π, we obtain
(34)(p*h)(x,y,z)=h(px,py,pz)=(δH~)(px,py,pz)=(δf)(x,y,z)=(25)k(x,y,z).
Next, we show that(p*h)(x,y,σ)=k(x,y,σ).
Since(H,H~) is a ⊗-functor, the following diagram commutes:
(35)
where u=(r→(0,σ)r′), v=(s→(0,σ)s′).
Due to the relations (5)–(15) of the ⊗ product and the composition in ∫Γ‍(OutG,ZG,h) and HolΓG, one has
(36)H(u⊗v)∘H~r,s=(Hr,sσ+g(rs,σ)+h(r,s,σ),σ),H~r′,s′∘(H(u)⊗H(v))=(g(r,σ)+H(r′)g(s′,σ)+H~r′,s′,σ).
Let us note that(37)Hr,s=Hpx,py=f(x,y),h(r,s,σ)=(p*h)(x,y,σ),g(r,σ)=g(p(x),σ)=(p*g)(x,σ)=f(x,σ),H(r′)=H(rσ)=Hσ(r)=Hσ(p(x))=φσx.
Therefore,
(38)fσ(x,y)+f(xy,σ)+(p*h)(x,y,σ)=f(x,σ)+φxσf(y,σ)+f(xσ,yσ).
This shows that (p*h)(x,y,σ)=(27)k(x,y,σ).
Finally, we prove that(p*h)(x,τ,σ)=k(x,τ,σ) based on the functorial property of H. Indeed, consider the composition
(39)r→(0,σ)s→(0,τ)t.
On one hand,
(40)H[(0,τ)∘(0,σ)]=(5)H[(h(r,τ,σ),τσ)]=H[h(r,τ,σ),1)∘(0,τσ)]=H[(h(r,τ,σ),1)]∘H[(0,τσ)]=(h(r,τ,σ),1)∘(g(r,τσ),τσ)=(14)(g(r,τσ)+h(r,τ,σ),τσ).
On the other hand,
(41)H(0,τ)∘H(0,σ)=H(r)→(g(r,σ),σ)H(s)→(g(s,τ),τ)H(t)=(14)H(r)→(τg(r,σ)+g(σr,τ),τσ)H(t).
Again, h(r,τ,σ)∈ZG, the functorial property of H, implies
(42)h(r,τ,σ)+g(r,τσ)=gτ(r,σ)+g(rσ,τ),
or h(r,τ,σ)=δg. Clearly,
(43)(p*h)(x,τ,σ)=h(px,τ,σ)=δg(px,τ,σ)=δf(x,τ,σ)=(28)k(x,τ,σ).


## 3.3. Classification of Central Extensions

Denote byExtΓc(Π,A) the set of all equivalence classes of equivariant extensions A↣E↠Π, where A⊂ZE. One can classify these extensions by the autofunctors of Γ-graded categorical group ∫Γ‍(Π,A,0).Theorem 7 (the schreier theory for central extensions of equivariant groups).
LetΠ be a Γ-group, and let A be a Γ-equivariant Π-module. There is a bijection
(44)ExtΓc(Π,A)≃EndΓid(∫Γ‍(Π,A,0)),
where EndΓid is the set of homotopy classes of monoidal Γ-functors F from ∫Γ‍(Π,A,0) to itself satisfying
(45)F(x)=x,x∈Π,F(b,1)=(b,1),b∈A.Proof.
Let(F,F~)∈EndΓid(∫Γ‍(Π,A,0)). Then (F,F~) defines a function
(46)φ:(Π×Π)∪(Π×Γ)⟶A
by
(47)(φ(x,y),1)=F~x,y,(φ(x,σ),σ)=F(x→(0,σ)xσ),
with φ(x,1Γ)=0. The compatibilities of (F,F~) with the constraints imply φ(x,1)=0=φ(1,y) and
(48)φ(x,y)+φ(xy,z)=x(φ(y,z))+φ(x,yz).
The naturalness of F~ and the functority of F give, respectively,
(49)σφ(x,y)+φ(xy,σ)=φ(x,σ)+(xσ)φ(y,σ)+φ(xσ,yσ),(50)φ(x,τσ)=φσ(x,τ)+φ(xτ,σ),
for all x,y,z∈G, σ,τ∈Γ. With the function φ, we can construct an equivariant crossed product extension Eφ=A×φΠ in which the operation of Eφ is
(51)(a,x)+(b,y)=(a+xb+φ(x,y),xy).
The relation (48) and the normality of φ show that Eφ is a group. By the relations (49), (50), Eφ is a Γ-group under the Γ-action
(52)σ(a,x)=(aσ+φ(x,σ),xσ).
Then we obtain an exact sequence
(53)0⟶A⟶iEφ⟶qΠ⟶1,
where i(a)=(a,1), q(a,x)=x. Further, it is easy to see that A⊂Z(Eφ).
Conversely, suppose that(54)0⟶A⟶iE⟶qΠ⟶1
is an equivariant extension with A⊂Z(E). For each x∈Π, choose a representative ux∈E, u1=(0). Then the system {ux∣x∈Π} induces a function φ taking values in A such that
(55)ux+uy=φ(x,y)+uxy,σux=φ(x,σ)+uσx.
The rules
(56)ux+(uy+uz)=(ux+uy)+uz,σ(ux+uy)=(ux)σ+(uy)σ,τ(uxσ)=uxτσ
imply the relations (48)–(50), respectively. Thus, we can define a Γ-monoidal functor (F,F~)∈EndΓ(∫Γ‍(Π,A,0)) as follows:
(57)Fx=x,F(b,1)=(b,1),F(x→(0,σ)xσ)=(φ(x,σ),σ),F~x,y=(φ(x,y),1).
Now, two extensions inExtΓc(Π,A) are equivalent if and only if two corresponding graded monoidal functors are homotopic. Indeed, let F,F′:∫Γ‍(Π,A,0)→∫Γ‍(Π,A,0) be two homotopic Γ-graded monoidal functors in which the homotopy ε:F→F′ is given by
(58)εx=(x→(e(x),1)x),x∈Π,
where e:Π→A. From the definition of a homotopy and from the relations (5), (6), it follows that
(59)e(1)=0,φ(x,σ)+e(xσ)=eσ(x)+φ′(x,σ),φ(x,y)+e(xy)=e(x)+xe(y)+φ′(x,y),
where x,y∈Π, σ∈Γ. Then, the corresponding
(60)β:Eφ⟶Eφ′(a,x)⟼(a+e(x),x)
is an equivalence between two central extensions if and only if equalities (59) hold.
Conversely, ifβ:Eφ→Eφ′ is an isomorphism, then
(61)β(a,x)=(a+e(x),x),
where e:Π→A is a function with e(1)=0. Thus, it follows that εx=(e(x),1) is a homotopy of F and F′, as we see by retracting our steps.
Finally, each central extensionE is equivalent to one Eφ via the isomorphism α:a+ux↦(a,x).


## 3.4. Composition of aΓ-Categorical Group with a Γ-Homomorphism

It is well known that for a given extension(62)E:0⟶A⟶iB⟶qΠ→1,
where A is Abelian, and a given homomorphism γ:Π′→Π, there is an extension E′ of A by Π′ with E′=Eγ. The extension E′ is just the pull pack of the pair (q,γ). We state an analogous result on strict Γ-categorical groups in the following proposition.Proposition 8.
Letℍ be a strict graded categorical group with three invariants Π, C, h, and let p:Π′→Π be an equivariant homomorphism. Then there exists a strict graded categorical group 𝔾 which is equivalent to the graded categorical group ∫Γ‍(Π′,C,h′), where C is viewed as a G′-module under the actions xc=p(x)c, x∈Π′, c∈C, and [h′]=[p*h].Proof.
The strict graded categorical group𝔾 is constructed as follows:
(63)Ob𝔾={(x,X)∣x∈Π′,X∈p(x)}.
A σ-morphism (x,X)→(x,Y) is a triple (x,u,σ), where u:X→Y is a σ-morphism in ℍ. The composition of two morphisms (x,X)→(x,u,σ)(x,Y)→(x,v,τ)(x,Z) is given by
(64)(x,v,τ)∘(x,u,σ)=(x,v∘u,τσ).
The tensor products on objects and on morphisms in𝔾 are given by
(65)(x,X)⊗(y,Y)=(xy,X⊗Y),(x,u,σ)⊗(y,v,σ)=(xy,u⊗v,σ).
For each morphism(x,u,σ) in 𝔾, one has
(66)(x,u,σ)-1=(x,u-1,σ-1).
The graded functor gr:𝔾→Γ is given by
(67)(x,u,σ)⟼σ
and the graded unit functor I:Γ→𝔾 by
(68)σ⟼((1,I)→(1,idI,σ)(1,I)).
The unit object of𝔾 is (1,I), where I is the unit object of ℍ. The associativity and unit constraints are identities.
Suppose that the graded categorical groupℍ induces a regular factor set {Fℍσ,σ∈Γ}. Then, 𝔾 induces a regular factor set {F𝔾σ,σ∈Γ} by
(69)F𝔾σ(x,X)=(x,FℍσX),F𝔾σ(x,u,τ)=(x,Fℍσ(u),τ).
Thus,𝔾 is a strict graded categorical group.
We define a pair(φ,f) of maps by
(70)λ:π0𝔾⟶Π′,f:π1𝔾⟶π1ℍ=C,[(x,X)]⟼x,(1,c,σ)⟼(c,σ).
Then, λ is an isomorphism of Γ-groups and f is an isomorphism of Γ-equivariant Π′-module, where the Π′-module structure of C is xc=p(x)c,x∈Π′,c∈C.
The graded monoidal functor(F,F~):𝔾→ℍ given by
(71)F(x,X)=X,F(x,u,σ)=(u,σ),F~=id
induces one SF=(ϕ,ϕ~):S𝔾→Sℍ, where Sℍ=∫Γ‍(Π,C,h). Then,
(72)ϕ[(x,X)]=F0[(x,X)]=[F(x,X)]=[X]=p(x)=pλ[(x,X)],ϕ(1,c,σ)=F1(1,c,σ)=γF(1,I)-1F(1,c,σ)=γI-1(c,σ)=(c,σ)=f(1,c,σ),
where (c,σ) is a morphism in Sℍ. Thus, SF is a functor of type (pλ,f).
Suppose thath𝔾∈ZΓ3(π0𝔾,π1𝔾). According to Theorem 3.2 in [2], the obstruction of pair (pλ,f) vanishes in H3(π0𝔾,π1ℍ)=H3(π0𝔾,C); that is,
(73)(pλ)*h=f*h𝔾+δϕ~.
Now, we denote h′=f*h𝔾, and then the pair (J,J~), where J=(λ,f), J~=id, is a graded monoidal functor from S𝔾 to 𝕁=∫Γ‍(Π′,C,h′). The composition
(74)𝔾⟶(G,G~)S𝔾⟶(J,J~)𝕁
is an equivalence from 𝔾 to 𝕁.
Finally, we prove thath′ is in the same cohomology class with p*h. Let K=(λ-1,f-1):𝕁→S𝔾; then K together with K~=id is a graded monoidal functor, and the composition
(75)(ϕ,ϕ~)∘(K,K~):𝕁⟶Sℍ
is a graded monoidal functor making the following diagram commute:
(76)
Since ϕ∘K is a graded monoidal functor of type (p,id), its obstruction vanishes. This implies that p*h-h′=∂g, and that means that [h′]=[p*h].IfΓ=1, we obtain Proposition 14 in [12].The graded categorical group𝔾 is called a composition of ℍ with p, denoted by 𝔾=ℍ∘p. It follows that 𝔾 can be seen as a contravariant functor in terms of the invariant π0𝔾 with π1𝔾 fixed. Indeed,
(77)𝔾∘idπ0𝔾=𝔾,𝔾∘(p∘p′)=(𝔾∘p)∘p′.



---

*Source: 306978-2013-08-26.xml*