1 Introduction
The concept of Γgraded extensions of a category was introduced by A.M. Cegarra, A.R.
Garzn, A.R. Grandjean [2] as a categorization of the notion of group extension. Most of
the problems of group extension are raised to Γgraded extension of categories.
If G is a Γgraded monoidal category, then the subcategory KerG consisting of ob jects
of G and all morphisms of grade 1 in G inherits a monoidal structure of G. When KerG is
a categorical group, G is said to be a Γgraded categorical group or a Γgraded extension
of categorical group. Since KerG is a groupoid if and only if G is, a Γgraded extension
of categorical groups can be defined as a Γgraded groupoid such that for any ob ject X
there is an ob ject X0 with an 1arrow X ⊗ X0 → 1. In [4], the authors classified Γgraded
extensions of categorical groups by cohomology with operator Γ.
Since each categorical group is equivalent to a categorical group of the type (Π, A), in
this paper we only consider Γgraded extensions of categorical groups of the type (Π, A),
regarded as a connection of the paper [1]. The main results in this paper show the element
h ∈ H3
Γ(Π, A) by directly calculating on factor sets receiving values in (Π, A), differs from
the another way authors did in [4].
The main results of the paper give a precise interpretation about the cohomology
group with thirddimensional operator Γ.
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GRADED EXTENTION OF CATEGORICAL GROUPS
OF THE TYPE (Π, A)
Nguyen Tien Quang and Nguyen Hai Son
Hanoi National University of Education
Abstract. In this paper, we present the concept of strict factor set on the group Γ with
coefficients in categorical groups of the type (Π, A). We define the element h ∈ H3
Γ
and give a new
interpretation of cohomology groups with thirddimensional operator Γ. The element h determines
a 11 correspondence between classes of factor sets on Γ with coefficients in (Π, A) and group
H3
Γ
(Π, A).
1 Introduction
The concept of Γgraded extensions of a category was introduced by A.M. Cegarra, A.R.
Garzn, A.R. Grandjean [2] as a categorization of the notion of group extension. Most of
the problems of group extension are raised to Γgraded extension of categories.
If G is a Γgraded monoidal category, then the subcategory KerG consisting of objects
of G and all morphisms of grade 1 in G inherits a monoidal structure of G. When KerG is
a categorical group, G is said to be a Γgraded categorical group or a Γgraded extension
of categorical group. Since KerG is a groupoid if and only if G is, a Γgraded extension
of categorical groups can be defined as a Γgraded groupoid such that for any object X
there is an object X ′ with an 1arrow X ⊗X ′ → 1. In [4], the authors classified Γgraded
extensions of categorical groups by cohomology with operator Γ.
Since each categorical group is equivalent to a categorical group of the type (Π, A), in
this paper we only consider Γgraded extensions of categorical groups of the type (Π, A),
regarded as a connection of the paper [1]. The main results in this paper show the element
h ∈ H3Γ(Π, A) by directly calculating on factor sets receiving values in (Π, A), differs from
the another way authors did in [4].
The main results of the paper give a precise interpretation about the cohomology
group with thirddimensional operator Γ.
2 The factor set of monoidal category of the type (Π, A)
In [3], if monoidal categories are replaced by categorical groups then we obtain the follow
ing:
Definition 1. A factor set on Γ with coefficients in ((Π, A),⊗) is a pair (θ, F ) consists
of:
 A family of monoidal equivalences
F σ = (F σ ,Φσ,Φσ0 ) : ((Π, A),⊗)→ ((Π, A),⊗)
1
 Isomorphisms of monoidal functors
θσ,τ : F σF τ
∼
−→ F στ (σ, τ ∈ Γ)
satisfying the conditions
i) F 1 = id((Π,A),×),
ii) θ1,σ = θσ,1 = idFσ (σ ∈ Γ),
iii) θσ,τγ .F σθτ,γ = θστ,γ .θσ,τF γ (σ, τ, γ ∈ Γ).
In this paper, we say that the factor set (θ, F ) is strict if Φσ0 = idI for all σ ∈ Γ.
Definition 2. (Cohomology relation of factor sets).
Let Γ is a group and C is a categorical group of the type (Π, A). We say that two
factor sets (θ, F ) and (µ,G) on Γ with cofficients in (Π, A) are cohomologous if (θ, F )
and (µ,G) are associated to the same Γmonoidal extension of C.
Then, two factor sets (θ, F ) and (µ,G) on Γ with cofficients in (Π, A) are cohomol
ogous if and only if the crossed product extensions (∆(θ, F ), j) and (∆(µ,G), j′) are Γ
equivalence.
Theorem 1. (The necessary and sufficient condition for two factor sets are cohomolo
gous).
Let Γ is a group and C is a categorical group of the type ((Π, A),⊗). Two factor sets
(θ, F ) and (µ,G) on Γ with coefficients in C are cohomologous if and only if there exists
a family of isomorphisms of monoidal functors
ϕσ : (F σ ,Φσ,Φσ0 )
∼
−→ (Gσ,Ψσ,Ψσ0 ) (σ ∈ Γ)
satisfying
ϕ1 = id(Π,A)
ϕστ .θστ = µσ,τ .ϕσGτ .F σϕτ (σ, τ ∈ Γ)
3 Cohomology of Γgroups
Let a Γ−pair (Π, A), that is, a Π−module Γ−equavariant A, cohomology groups HnΓ(Π, A)
were studied in [5]. We recall that cohomology group HnΓ(Π, A), with n ≤ 3, can be
computed as the cohomology group of the truncated cochain complex:
∼
CΓ (Π, A) : 0 −−−−→ C
1
Γ(Π, A)
∂
−−−−→ C2Γ(Π, A)
∂
−−−−→ Z3Γ(Π, A) −−−−→ 0,
in which C1Γ(Π, A) consists of normalized maps f : Π → A, C
2
Γ(Π, A) consists of nor
malized maps g : Π2 ∪ (Π × Γ) → A and Z3Γ(Π, A) consists of normalized maps h :
Π3 ∪ (Π2 × Γ) ∪ (Π× Γ2)→ A satisfying the following 3cocycle conditions:
h(x, y, zt) + h(xy, z, t) = x(h(y, z, t)) + h(x, yz, t) + h(x, y, z) (1)
σ(h(x, y, z))+h(xy, z, σ)+h(x, y, σ) = h(σ(x), σ(y), σ(z))+(σ(x))(h(y, z, σ))+h(x, yz, σ)
(2)
2
σ(h(x, y, τ)) + h(τ(x), τ(y), σ) + h(x, σ, τ) + (στ(x))(h(y, σ, τ)) = h(x, y, στ) + h(xy, σ, τ)
(3)
σ(h(x, τ, γ)) + h(x, σ, τγ) = h(x, στ, γ) + h(γ(x), σ, τ) (4)
for x, y, z, t ∈ Π; σ, τ, γ ∈ Γ.
For each f ∈ C1Γ(Π, A), the coboundary ∂f is given by:
(∂f)(x, y) = x(f(y))− f(xy) + f(y), (5)
(∂f)(x, σ) = σ(f(x))− f(σ(x)), (6)
and for g ∈ C2Γ(Π, A), ∂g is given by:
(∂g)(x, y, z) = x(g(y, z)) − g(xy, z) + g(x, yz) − g(x, y), (7)
(∂g)(x, y, σ) = σ(g(x, y)) − g(σ(x), σ(y)) − σ(x)(g(y, σ)) + g(xy, σ) − g(x, σ), (8)
(∂g)(x, σ, τ) = σ(g(x, τ)) − g(x, στ) + g(τ(x), σ). (9)
4 Graded extensions of categorical groups of the type (Π, A)
When the monoidal category C is replaced by the categorical group of the type (Π, A) we
obtain better descriptions than in the general case. For example, in [1], authors proved
that the condition i) of the definition of factor set of categorical groups of the type (Π, A)
is redundant. Now, we continue reducing" this concept in terms of other face.
Lemma 2. Let C is a categorical group of the type (Π, A). Any factor set (θ, F ) on Γ with
coefficients in C is cohomologous to a strict factor set (µ,G).
Proof. For each σ ∈ Γ, consider a family of isomorphisms in (Π, A):
ϕσX =
{
idFσx if x 6= 1
(Φσ0 )
−1
if x = 1
(10)
Since F 1 = id((Π,A),⊗) we have
ϕ1 = id((Π,A),⊗) (11)
Then, we define Gσ in a unique way such that ϕσ : F σ → Gσ is a natural transformation
by setting:
Gσx =
{
F σx if x 6= 1
1 = F σ1 if x = 1
(12)
Gσ(x
f
−→ x) = (ϕσxF
σ(f)ϕσx
−1 : Gσx
Gσ(f)
−→ Gσx) (13)
ΨσX,Y = ϕ
σ
x⊗yΦ
σ
X,Y (ϕ
σ
x ⊗ ϕ
σ
y )
−1; Ψσ0 = ϕ
σ
IΦ
σ
0 = idI (14)
For such a setting, clearly we have Gσx = F σx for all x ∈ G and
Gσ = (Gσ,Ψσ,Ψσ0 ) : ((Π, A),⊗) → ((Π, A),⊗)
is a monoidal equivalence. In particular, we have:
Ψσ0 = idI (15)
3
This states the strictness of the family of functors Gσ , as well as of the factor set (µ,G).
Now, we set
µσ,τ = ϕστ .θσ,τ .(F σ(ϕτ ))−1(ϕσGτ )−1 (16)
for all σ, τ ∈ G. Clearly µσ,τ is a isomorphism of monoidal functors: GσGτ → Gστ due to
ϕσ; θσ,τ are monoidal isofunctors and F σ, Gτ are monoidal equivalences.
We will prove that the family of µσ,τ satisfy the conditions ii) and iii) of the definition of
factor set.
First, we have
ϕστ .θσ,τ = µσ,τ .ϕσGτ .F σϕτ for all σ, τ ∈ Γ (17)
For τ = 1 we have
ϕσ.θσ,1 = µσ,1.ϕσG1.F σϕ1
⇐⇒ ϕσ.idFσ = µ
σ,1.ϕσ .F σ(id(Π,A)) = µ
σ,1ϕσ
=⇒ µσ,1 = idGσ .
Analogously, µ1,σ = idGσ . Thus
µσ,1 = µ1,σ = idGσ (18)
To verify the condition iii) we will show that µσ,τ (x) = θσ,τ (x) and µσ,τ (1) = id.
Indeed, for all x ∈ G;x 6= 1, we have
(ϕστ .θσ,τ )x = ϕσ,τx = idFστx.θ
στx
(µσ,τ .ϕσGτ .F σϕτ )x = µσ,τx.ϕσ(Gτx).F σϕτx
= µσ,τx.idFσF τx.F
σ(idF τx)
= µσ,τx.idFσF τx.idFσF τx
= µσ,τx.idFσF τx
(since F σ : Hom(x, x)→ Hom(F σx, F σx) and F σ : G→ G are isomorphisms.)
It follows that idFστx.θ
σ,τx = µσ,τx.idFσF τx or
θσ,τx = µσ,τx (19)
for all x 6= 1.
In the case x = 1, we have
(ϕστ .θσ,τ )1 = ϕστ (1).θσ,τ (1) = (Φστ0 )
−1.θσ,τ (1)
(µσ,τ .ϕσGτ .F σϕτ )1 = µσ,τ (1).ϕσ(Gτ (1)).F σ(ϕτ (1))
= µσ,τ (1).ϕσ(1).F σ((Φτ0)
−1)
= µσ,τ (1).(Φσ0 )
−1.F σ((Φτ0)
−1)
It follows that
µσ,τ (1).(Φσ0 )
−1.F σ((Φτ0)
−1) = (Φστ0 )
−1.θσ,τ (1)
or
µσ,τ (1) = (Φστ0 )
−1.θσ,τ (1).F σ((Φτ0)).(Φ
σ
0 ) = idI
4
by the commutativity of the diagram:
F σF τ1
1
F στ1
?
θσ,τ1
H
HY
Fσ(Φτ
0
).Φσ
0
Φστ
0
Now, we check the condition iii) of the definition of factor set:
For x 6= 1, we have
(µστ,γ .µσ,τGγ)x = µστ,γx.µσ,τGγx
= θστ,γx.θσ,τF γx
= θσ,τγx.F σθτ,γx
= µσ,τγx.Gσµτ,γx
= (µσ,τγ .Gσµτ,γ)x
(since if x 6= 1 and x
f
−→ x then Gσ(f) = F σ(f))
For x = 1, we have:
(µστ,γ .µσ,τGγ)(1) = µστ,γ(1).µσ,τGγ(1) = µστ,γ(1).µσ,τ (1) = id1
(µσ,τγ .Gσµτ,γ)(1) = µσ,τγ(1).Gσµτ,γ(1) = id1.G
σ(id1) = id1.id1 = id1
Thus
µστ,γ .µσ,τGγ = µσ,τγ .Gσµτ,γ (20)
for all σ, τ, γ ∈ Γ
Thus, the factor set (µ,G) defined by (4.3), (4.4), (4.5), (4.6) is a strict factor set on Γ with
coefficients in ((Π, A),⊗) and cohomologous with (θ, F ) by Theorem 2.3. This completes
the proof.
As we know, each categorical group is equivalent to a categorical group of the type
(Π, A). Moreover, in this categorical group, the associative constraint ax,y,z is a normalized
3cocycle (in the sense of group cohomoogy) and the unit constraint is strict (in the sense
lx = rx = idx). Therefore, hereafter writing categorical groups of the type (Π, A) means
categorical groups of the type (Π, A), having above properties.
In [4], authors were shown that each Γgraded extension G of categorical groups C in
duces a structure pi0(G)module Γequivariant pi1(G) and an element h
G ∈ Z3Γ(pi0(G), pi1(G))
by constructing the skeletal category and choosing isomorphisms. Now, we will prove that
result in the case C is a categorical group of the type (Π, A) by other way.
First, we recall a result in [1,Lemma 2.7]:
Lemma 3. Let Γ is a group and (Π, A) is a monoidal category. If F σ = (F σ,Φσ,Φσ0 ), σ ∈
Γ is a family of monoidal autoequivalences of (Π, A) together with a family of isomorphisms
of monoidal functors θσ,τ : F σF τ → F στ , (σ, τ ∈ Γ) Then the functor F σ = (F σ, fσ) sat
isfy the following properties:
i) The correspondence σ F σ0 define a homomorphism of groups
F : Γ→ AutΠ (21)
and hence F 1(x) = x for all x ∈ Π (F 1 = idΠ).
5
ii) fσ : A→ A is automorphisms of groups satisfying the equality:
fσ(xb) = F σ(x)fσ(b),∀x ∈ Π,∀b ∈ A. (22)
In particular, f1 is a isomorphism of Πmodules.
Now, we set
σx = F σx, σa = fσa for all σ ∈ Γ, x ∈ Π, a ∈ A (23)
Then, it follows from the above lemma that A is a Πmodule Γequivariant. Indeed, by
(4.12) we have F σ(xy) = F σxF σy or σ(xy) = σx.σ for all σ ∈ Γ;x, y ∈ Π, so Π is a
Γgroup. Also by Lemma 4.2, since fσ : A → A is a group automorphism, fσ(a + b) =
fσ(a)fσ(b) or σ(a + b) = σa + σb for all σ ∈ Γ; a, b ∈ A, and so A is a Γgroup. Finally,
from the equation (4.13) we have σ(xb) = σx.σb for all σ ∈ Γ; b ∈ A or A is a Πmodule
Γequivariant. All in all, we have the following lemma:
Lemma 4. Let C is a categorical group of the type (Π, A) and (θ, F ) is a factor set on Γ
with coefficients in (Π, A). Then it induces a structure of Πmodule Γequivariant on A.
Next, we have the following theorem:
Theorem 5. Any strict factor set (θ, F ) on Γ with coefficients in C = (Π, A) induces an
element h ∈ Z3Γ(Π, A) in the sense cohomology of Γgroups [5]).
Proof. Suppose F σ = (F σ,Φσ,Φσ0 ). Then, we can write
Φσx,y = (F
σ(xy), gσ(x, y)) = (F σ(xy), g(x, y, σ)) for g : Π2 × Γ→ A is a funtion.
And
Φσ0 = (1, a
σ) = (1, k(a, σ)) for g : Π× Γ→ Ais a funtion.
For the family of isomorphisms of monoidal functors θσ,τ = (θσ,τx ), we are able to write
θσ,τx = (F
στx, tσ,τ (x)) = (F στx, t(x, σ, τ)
for t : Π× Γ2 → A.
From functions a, g, t , in which a is associative constraint of (Π, A), we determine the
function h as follow:
h : Π3 × (Π2 × Γ)× (Π× Γ2)→ A
such that
h Π3= a; h Π2×Γ= g; and h Π×Γ2= t
From the relations obtained in the proof of Theorem 2.8 in [1] and the definition of Π
module Γequivariant A with actions (4.14), we have:
−σx.h(y, z, σ) + h(xy, z, σ) + h(x, y, σ) − h(x, yz, σ) = h(σx, σy, σz) − σ(h(x, y, z)) (24)
(στ)x.h(y, σ, τ) + h(xy, σ, τ) − h(x, σ, τ) = h(x, y, στ) − h(τx, τx, σ) − σh(x, y, τ) (25)
Moreover, since the relations of 3cocycle ax,y,z, we obtain
xh(y, z, t) − h(xy, z, t) + h(x, yz, t) − h(x, y, zt) + h(x, y, z) = 0 (26)
6
The cocycle condition
θστ,γ .θσ,τF γ = θσ,τγ .F σθτ,γ
yields
h(x, στ, γ) + h(γx, σ, τ) = h(x, σ, τγ) + σh(x, τ, γ) (27)
for all x, y, z ∈ Π; σ, τ ∈ Γ; a, b ∈ A. It follows that h satisfies the relations of a 3
cocycle in Z3Γ(Π, A). However, we have to prove the normalized property of h.
First, since the normalized property of associative constraint a, and the proof of Theorem
2.8 [1], we obtain the relations:
h(x, y, 1Γ) = −Φ
1(x, y) = 0 (28)
h(1, y, z) = h(x, 1, z) = h(x, y, 1) = 0 (29)
h(x, 1Γ, tau) = h(x, σ, 1Γ) = 0 (30)
Now, we have to prove:
h(1, σ, τ) = 0 (31)
h(x, 1, σ) = h(1, x, σ) = 0 (32)
Also by the proof of Theorem 2.8 [1], we have the relation fσ(aτ )−aσ−aστ = −h(1, σ, τ),
on the other hand the factor set is strict, that is Φσ0 = id or a
σ = 0 for all σ ∈ Γ so we obtain
(4.22). Furthermore, if we write the unit constraints of (Π, A) as the form rx = (x, r(x))
and lx = (x, l(x)), then from the proof of Theorem 2.8 [1] we have:
−r(x) + fσ(r(x)) = F σ(x).aσ + h(x, 1, σ), −l(x) + fσ(r(x)) = aσ + h(x, 1, σ).
And since the strict property in terms of unit of category (Π, A), which yields r(x) =
l(x) = 0, we obtain (4.23).
Set h(θ,F ) = h, we have h(θ,F ) ∈ Z3Γ(Π, A). This completes the proof of theorem.
.
Assume that (µ,H) is other strict factor sets on Γ with coefficients in ((Π, A),⊗) which is
cohomologous to (θ, F ). Then, the structure of Gmodule Γequivariant A and the element
h(µ,H) ∈ H3Γ(Π, A) determined by (µ,H) have the following property:
Proposition 6. Let two strict factor sets on Γ with coefficients in categorical of the type
(Π, A), (µ,H) and (θ, F ) are cohomologous . Then, they determine the same structure of
Πmodule Γequivariant on A and h(θ,F ), h(µ,H) are cohomologous.
Proof. By Theorem 2.3, there exists ϕσ : (F σ,Φσ,Φσ0 ) → (H
σ,Ψσ,Ψσ0 ) (σ ∈ Γ) is a
family of isomorphisms of functors such that
ϕ1 = id((Π,A),⊗), ϕ
στ .θσ,τ = µσ,τ .ϕσHτ .F σϕτ (σ, τ ∈ Γ).
We set ϕσx = (F σx, g(x, σ)) for all x ∈ Π, σ ∈ Γ.
Since ϕσx : F σx→ Hσx is a morphism in (Π, A) we have:
Hσx = F σx = σx (33)
7
is an action of Γ on Π.
Furthermore, for any a ∈ A, Hσ = (Hσ , gσ), F σ = (F σ , fσ), by the commutativity of the
diagram
F σx Hσx
F σ Hσx

ϕσx
?
Fσ(x,a)
?
Hσ(x,a)

ϕσx
we have
g(x, σ) + fσ(a) = gσ(a) + g(x, σ)⇒ fσ(a) = gσ(a) = σa (34)
is an action of Γ on A. It follows that (µ,H) and (θ, F ) determine the same structure of
Πmodule Γequivariant A.
Now, we prove that elements h(θ,F ) and h(µ,H) are cohomologous.
We denote h(µ,H) = h′. Hence, by determining of h(µ,H) in Theorem 4.4 we have
(σ(xy), h′(x, y, σ)) = Ψσx,y; (στx, h
′(x, σ, τ) = µσ,τx; for all x, y ∈ Π, σ, τ ∈ Γ
According to our setting, g : Π × Γ → A. It determines a extending 2cochain of g, also
denoted by g with g : Π2 → A is the null map.
Since (ϕστ .θσ,τ )x = (µσ,τ .ϕσHτ .F σϕτ )x we have
g(x, στ) − h(x, σ, τ) = −h′(x, σ, τ) + g(τx, σ) + σg(x, τ) (35)
By ϕσx⊗yΨ
σ
x,y = Ψ
σ
x,y.(ϕ
σx⊗ ϕσy) we have
−h′(x, y, σ) − h(x, y, σ) − h(x, y, σ) = g(x, σ) + σxg(y, σ) − g(xy, σ) (36)
By Φσ0 = Ψ
σ
0 = id we have ϕ
σ
1 = Ψ
σ
0 (Φ
σ
0 )
−1 = id1. Thus
g(1, σ) = 0 for all σ ∈ Γ (37)
Since ϕ1 = id((Π,A),⊗)
g(x, 1Γ) = 0 for all x ∈ G (38)
By the determining g and relations (4.26)(4.29), we have g ∈ C2Γ(Π, A) and h
(θ,F ) −
h(µ,H) = ∂g. This completes the proof of proposition.
Therefore, each factor set (θ, F ) on Γ with coefficients in categorical groups of the
type (Π, A) determines a structure of Gmodule Γequivariant A and an elemnet h(θ,F ) ∈
H3Γ(Π, A) uniquely.
Now, we consider the problem: Giving a categorical group of the type (Π, A), in which
A is a Πmodule Γequivariant, and h ∈ Z3Γ(Π, A), does there exist a factor set (θ, F ) on
Γ with coefficients in (Π, A) such that cl(h(θ,F )) = cl(h) ∈ H3Γ(Π, A)?
Clearly, by the construction of categories Γextension crossed product (∆(θ, F ), j) and
the category
∫
Γ(Π, A, h
(θ,F )) in [3], [4] we obtain (∆(θ, F ), j) ≡
∫
Γ(Π, A, h
(θ,F )) (where j
is embedding functor j : ((Π, A),⊗)) ↪→ (∆(θ, F ), j)). Hence, the structure Πmodule Γ
equivariant A determined as above coincides with the structure Πmodule Γequivariant
A determined from Γextension of the categorical group ((Π, A),⊗) in [4]. Similarly, with
suitable choice of isomorphisms, hθ,F and h∆(θ,F ) are also coincident. The above comment
8
also shows that if we give a categorical group of the type (Π, A), a structure Πmodule
Γequivariant A and h ∈ Z3Γ(Π, A) such that h(x, y, z) = ax,y,z for all x, y, z ∈ Π then there
exists a factor set (θ, F ) on Γ with coefficients in ((Π, A),⊗) and h(θ,F ) = h. In fact, we
have the following proposition:
Proposition 7. Let C is a categorical group of the type (Π, A) with ax,y,z is the asso
ciative constraint, and A is a Πmodule Γequivariant and element h ∈ Z3Γ(Π, A) such
that h(x, y, z) = ax,y,z + ∂g vi g ∈ C
2(Π, A) (that is h Π3 is cohomologous with
a in the sense of group cohomology). Then, there exists a factor set (θ, F ) on Γ with
coefficients in C such that (θ, F ) induces a structure Πmodule Γequivariant on A and
cl(h(θ,F )) = cl(h) ∈ H3Γ(Π, A).
Proof. First, we remark that h(x, y, z) = ax,y,z + ∂g for g ∈ C
2(Π, A), so we have the
monoidal equivalence:
j = (id,Φ,Φ0) : ((Π, A),⊗, ax,y,z , (I, id, id)) → ((Π, A),⊗, h(x, y, z), (I, id, id)) (39)
for Φx,y = (xy, g(x, y)); Φ0 = idI ; j(x) = x; j(x, a) = (x, a). Thus, because (
∫
Γ(Π, A, h), id)
is Γextension of the category ((Π, A),⊗, h(x, y, z), (I, id, id)), (
∫
Γ(Π, A, h), j) is Γextension
of ((Π, A),⊗, ax,y,z , (I, id, id)). Therefore, by the determining of the factor set (θ, F ) on Γ
with coefficients in ((Π, A),⊗, ax,y,z , (I, id, id)) in [3] with Υx = (0, σ), we have
i) A family of monoidal autoequivalences F σ = (F σ,Φσ,Φσ0 ) : ((Π, A),⊗) → ((Π, A),⊗) for
F σx = σx; F σ(x, a) = (σx, σa) that is fσ(a) = σa, Φσ0 = id1, Φ
σ
x,y = (σ(xy), h(x, y, σ)+
σg(x, y) − g(σx, σy))
ii) Isomorphisms of functors θστ : F σF τ → F στ for θστx = (στx, h(x, σ, τ)). Then, if we
set g′(x, y) = g(x, y), g′(x, σ) = 0 for all x, y ∈ G,σ ∈ Γ then we obtain g′ ∈ C2Γ(Π, A)
and h− h(θ,F ) = ∂g′ or cl(h) = cl(h(θ,F )). This completes the proof of proposition.
Theorem 8. There exists a bijection between the set of cohomology classes of factor sets
on Γ with coefficients in a categorical group of the type (Π, A) and a sub group of H3Γ(Π, A)
consisting of h satisfying h ∗ (x, y, z) = cl(ax,y,z) ∈ H
3(Π, A). in which a is associative
constrain of (Π, A).
Corollary 9. There exists a bijection between the set of cohomology classes of factor sets
on Γ with coefficients in all categorical groups of the type (Π, A) and the group H3Γ(Π, A)
H3Γ(Π, A).
REFERENCES
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type (Π, A). Journal of Science No.1, Hanoi University of Education.
[2] A.M. Ceggara, A.R.Garzn and A. R. Grandjean, 2001. Graded extensions of cate
gories. Journal of Algebra. 241, pp. 620657.
[3] A.M. Ceggara, A.R.Garzn and J.A.Ortega, 2001. Graded extensions of monoidal
categories. Journal of Algebra. 241, pp. 620657.
[4] A. M. Cegarra, J. M. Garca  Calcines and J. A. Ortega, 2002. On grade categorical
groups and equivariant group extensions. Canad. J. Math. Vol. 54(5), pp. 970  997.
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