The Factor Sets of Gr-Categories of the Type (Π,A)

Transcription

1 International Journal of Algebra, Vol. 4, 2010, no. 14, The Factor Sets of Gr-Categories of the Type (Π,A) Nguyen Tien Quang Department of Mathematics, Hanoi National University of Education 136 Xuan Thuy Street, Hanoi, Vietnam Abstract Each Γ monoidal extension of a Gr category is determined by a factor set on a group Γ. In this paper, we shall show that the such factor set can be reduced and induces a 3-cocycle of Γ groups. Moreover, this correspondence determines a bijection between the set of equivalence classes of Γ monoidal extensions of Gr categories of the type (Π,A) and the cohomology group H 3 Γ (Π,A). Mathematics Subject Classification: 18D10, 20E22 Keywords: graded extension, monoidal category, factor set, crossed product, group cohomology 1 Introduction The notion of graded monoidal category was introduced by Fröhlich and Wall [4] by a generalization of some manifolds of categories with an action of a group Γ. Γ will be also regarded as a category with exactly one object, say, where the morphisms are the elements of Γ and the composition is the group operation. By a Γ grading on a category D, we shall mean a functor gr : D Γ. The grading is called stable if for any C obd, and any σ Γ, there exists a morphism u in D with domain C such that gr(u) =σ. We refer to σ as the grade of u. If(D,gr)isaΓ graded category, we define KerD to be the subcategory of all morphisms of grade 1. For any Γ graded category (D,gr), authors of [4] have considered the category Rep(D,gr) of Γ functors F : Γ D, together with natural transformations. An object of Rep(D,gr) consists an object X of D with a homomorphism Γ Aut D (C). Now if I is the unit object of D, then the homomorphism Γ Aut D (I), which is also denoted by I, is the right inverse of the graded

2 656 N. T. Quang homomorphism Aut D (I) Γ. Let U denote the normal subgroup of automorphisms of grade 1. We obtain that Aut D (I) is a split extension of U by I(Γ), where I(Γ) is isomorphic to Γ. This extension determines an action of Γ on U by σu = I(σ) u I(σ 1 ). In [1], authors have considered the Γ monoidal extensions of a monoidal category as a categorification of the group extension problem. A Γ monoidal extension of the monoidal category C is a Γ monoidal category D together with a monoidal isomorphism j : C KerD. Construction and classification problems of Γ monoidal extensions have been solved by raising the main results of Schreier-Eilenberg-MacLane on group extensions to categorical level. With notions of a factor set and a crossed product extension, authors proved that there exists a bijection Δ:H 2 (Γ, C) Ext(Γ, C) between the set of cohomology classes of factor sets on Γ with coefficients in the monoidal category C and the set of equivalence classes of Γ monoidal extensions of C. The case where C is a Gr-category (also called a categorical group) has been considered in [2]. Then, the Γ equivariant structure appears on the Π module A, where Π = Π 0 (C),A = Aut C (1) = Π 1 (C). The category Γ CG of Γ graded Gr categories is classified by the functors cl : Γ CG HΓ 3 ; : ZΓ 3 Γ CG, where ZΓ 3 is the category in which any object is a triple (Π,A,h), where (Π,A) is a Γ pair and h ZΓ 3(Π,A); and H3 Γ is the category obtained from Z3 Γ when h ZΓ 3(Π,A) is replaced with h H3 Γ (Π,A). In our opinion, the most interesting as well as the most complicated part of the proof of Classification Theorem (Theorem 3.3) of [2] is the construction of 3-cocycle induced by a Γ extension G via a skeleton category. In this paper, we expect to introduce another proof of the Classification Theorem based on the notions of a factor set and a crossed product extension when we use thorough knowledge of Gr functors. It is known that each Gr category is equivalent to a Gr category of the type (Π,A) with strict unitivity constraints in [7]. So we assert that we just have to prove the Theorem for manifolds of graded Gr categories G whose kernel KerG is a Gr category of the type (Π,A) where (Π,A) is fixed. Then, the 3-cocycle h can be determined from a factor set analogously as the determination of the obstruction in the group extension problem. The main detail of the proof of Classification Theorem can be summarized as follows: Thanks to the description of Gr functors of Gr categories of the Γ

3 The factor sets of Gr-categories of the type (Π,A) 657 type (Π,A) in [6], we can describe explicitly a factor set and show that the Γ equivariant structure of A is a necessary condition of a factor set. This gives us an interpretation to the existence of the functor I :Γ D. Moreover, the notion of a factor set can be reduced if the condition F 1 = id is omitted. Then we shall show that each factor set is homotopic (natural equivalent) to an almost strict factor set. Each such factor set induces a 3-cocycle of Γ groups h. This construction of h is rather simple compared to the way done in explicitly in [2]. Therefore, we obtain a bijection Ω: Γ Gr(Π,A) H 3 Γ (Π,A), where Γ Gr(Π,A) is the set of equivalence classes of Γ extensions of Grcategories of the type (Π,A). 2 Some basic notions A monoidal category is called a Gr category (or a categorical group) if every object is invertible and every morphism is an isomorphism. Each Gr category G is equivalent to a Gr category S of the type (Π,A), where Π = Π 0 (G) is the group of iso-classes of objects of G, A= Π 1 (G) = Aut(1) is a left Π module. Objects of the category S are all elements x Π, morphisms are automorphisms Aut(x) ={x} A, where x Π. The composition of two morphisms is defined by The operation is defined by (x, u) (x, v) =(x, u + v). x y = xy, (x, u) (y, v) =(xy, u + xv). The associativity constraint a is associated to a 3 cocycle (in the sense of group cohomology) ξ Z 3 (Π,A), and the unitivity constraints are strict (in the sense l x = r x = id x ). If (F, F, F ) is a monoidal functor between Gr categories, then the isomorphism F can be deduced from the pair (F, F ). Moreover, in [6], the author has described monoidal functor between Gr categories of the type (Π, A). Thanks to this description, we can prove a necessary condition of a factor set (Theorem 3.2). Definition 2.1 (See [6]). Let S =(Π,A,ξ), S =(Π,A,ξ ) be Gr-categories. A functor F : S S is called a functor of the type (ϕ, f) if F (x) =ϕ(x),f(x, u) =(ϕ(x),f(u)),

4 658 N. T. Quang and (ϕ :Π Π, f : A A ) is a pair of group homomorphisms satisfying f(xa) =ϕ(x)f(a) for x Π,a A. We have Theorem 2.2 (See [6]). Let S =(Π,A,ξ), S =(Π,A,ξ ) be Gr-categories and F =(F, F, F ) be a monoidal functor from S to S. Then, F is a functor of the type (ϕ, f). Let us recall from [1], [2], [4] some basic notions about graded extension of a Gr category. Let (G,gr) and (H,gr) be stable Γ graded categories. A graded functor F :(G,gr) (H,gr) is a functor F : G Hpreserving grades of morphisms. Suppose that F :(G,gr) (H,gr) is also a graded functor. A graded natural equivalence θ : F F is a natural equivalence of functors such that all isomorphisms θ X : FX F X are of grade 1. For Γ graded category (G,gr), let G Γ G denote the subcategory of product category G G consisting of isomorphisms which are pairs of morphisms of the same grade of G. AΓ monoidal category consists of a stable Γ graded category (D,gr) together with Γ functors : D Γ D D; I :Γ D, and natural isomorphisms of grade 1: a X,Y,Z :(X Y ) Z X (Y Z), l X : I X X, r X : X I X, where I = I( ) such that for all object X, Y, Z, T D, two coherence conditions hold: (a X,Y,Z id T ) a X,Y Z,T (id X a Y,Z,T )=a X Y,Z,T a X,Y,Z T, id X l y =(r X id Y ) a X,I,Y. If (D,gr, ) and (D,gr, ) are Γ monoidal categories, then a Γ monoidal functor from (D,gr, ) to(d,gr, ), (F, F, F ), consists of a Γ functor F : (D,gr) (D,gr ), natural isomorphism of grade 1 F X,Y : F (X Y ) FX FY, and an isomorphism of grade 1: F : FI I, such that coherence conditions hold. A morphism between two Γ monoidal functors F = (F, F, F ) and F =(F, F, F ) is a monoidal morphism θ : F F. AΓ monoidal extension of a monoidal categroy C consists of a Γ monoidal category D =(D,gr), and a monoidal isomorphism J =(J, J,Ĵ) :C KerD.

5 The factor sets of Gr-categories of the type (Π,A) 659 If (D,J) and (D,J ) are two Γ monoidal extensions of a monoidal category C, by a morphism of extensions we mean a Γ monoidal functor F : D D such that F.J = J. We usually say that Γ monoidal extensions (D,J) and (D,J ) are equivalent, thanks to the following proposition. Proposition 2.3. Every morphism of Γ monoidal extensions of the monoidal category C is an isomorphism. Definition 2.4 (See [1]). Let Γ be a group and C be any monoidal category. By a factor set on Γ with coefficients in a monoidal category C, we shall mean a pair (θ, F) consisting of: A family of monoidal autoequivalences F σ =(F σ, F σ, F σ ):C C,σ Γ. A family of isomorphisms of monoidal functors θ σ,τ : F σ F τ F στ,σ,τ Γ satisfying the conditions: i) F 1 = id C ii) θ 1,σ = id F σ = θ σ,1, σ Γ iii) for all σ, τ, γ Γ, the following diagrams are commutative: F σ F τ F γ F σ θ τ,γ θ σ,τ F γ F στ F γ θ στ,γ F σ F τγ θ σ,τ γ F στγ, According to Theorem 2.2, any monoidal autoequivalence F σ is of the form F σ =(ϕ σ,f σ ). This remark is used frequently throughout the paper. Definition 2.5 (See [2]). Let Γ be a group, Π be a Γ group. A Γ module A is called an equivariant module on the Γ group Π if A is a Π module satisfying σ(xa) =(σx)(σa), for all σ Γ,x Π and a A. Then, we shall say that (Π,A)isaΓ pair. 3 Γ graded extension of a Gr category of the type (Π,A) According to [2], each factor set (θ, F) on Γ with coefficients in a monoidal category C determines a crossed product extension Δ(θ,F). It has the same

6 660 N. T. Quang objects as category C and its morphisms are pairs (u, σ) :A B consisting of an element σ Γ and a morphism in C, u: F σ (A) B. The composition of two morphisms A (u,σ) B (v,τ) C is defined by (v, τ) (u, σ) =(v F σ (u) (θ τ,σ A ) 1,τσ). The stable Γ grading on Δ(θ, F) is given by gr(u, σ) =σ. If (u, σ) :X X and (v, σ) :Y Y then (u, σ) (v, σ) =((u v). F σ X,Y,σ) and the unit Γ functor I :Γ Δ(θ, F) is defined by I(σ) =( F σ,σ), where I is the unit object of C. We shall begin by proving that any Γ extension of a Gr category is equivalent to a Γ extension of a Gr category of the type (Π,A). This is a consequence of the following proposition. Proposition 3.1. If G : C C is a monoidal equivalence, then each factor set (θ, F) of C induces a factor set (θ,f ) of C. Moreover, respective crossed product extensions are Γ equivalent. Proof. Let H : C Cbe a monoidal equivalence such that β : H G = id C. Let F σ be the composition H F σ G and θ σ,τ X = G(θ σ,τ HX F σ (β F τ HX)). One can verify that (θ,f ) is a factor set of C. A monoidal functor G : C C can be extended to a Γ functor Δ G :Δ(θ, F) Δ(θ,F ) as follows: for an object X of C, set Δ G X = GX; for a morphism (u, σ) :X Y, where u : F σ X Y, set Δ G (u, σ) =(G(u F σ (β X )),σ), ΔG = G, ΔG = Ĝ. One can verify that (Δ G, Δ G, Δ G )isaγ equivalence. We now prove a necessary condition of a factor set of a Gr category. Theorem 3.2. Let Γ be a group and S = S(Π,A,ξ) be a Gr-category. If (θ, F) is a factor set on Γ, with coefficients in S, then i) There exists a pair of group homomorphisms ϕ :Γ AutΠ ; f :Γ AutA, and A is equipped with a Γ equivariant Π module structure, induced by ϕ, f. ii) The condition i) in definition of the factor set can be deduced from the rest conditions.

7 The factor sets of Gr-categories of the type (Π,A) 661 Proof. i) According to Theorem 1.1, any autoequivalence F σ,σ Γ, of a factor set is of the form: (ϕ σ,f σ ):S S. Since F σ x,y : F σ (xy) F σ x.f σ y, σ Γ is a morphism in (Π,A), we have F σ (xy) =F σ x.f σ y, x, y Π. This shows that ϕ σ = F σ is an endomorphism of the group Π. Furthermore, since F σ is an equivalence, ϕ σ is an automorphism of the group Π, i.e., ϕ σ AutΠ. On the other hand, since θx σ,τ : F σ F τ x F στ x is a morphism in (Π,A), we have Thus, ϕ σ ϕ τ = ϕ στ. This proves that (F σ F τ )(x) =F στ (x), x Π. ϕ :Γ AutΠ, σ ϕ σ is a group homomorphism. Then ϕ 1 = ϕ(1) = id Π. Now, for convenience, for all σ Γ,x Π,a A, let us denote σx = ϕ σ x, σa = f σ a (1) Let F σ x,y =(σ(xy), f σ (x, y)), where f σ :Π 2 A and F σ =(1,c σ ):F σ 1 1 are maps. Since F σ is a monoidal functor, we have (σx) f σ (y, z) f σ (xy, z)+ f σ (x, yz) f σ (x, y) =σ(ξ(x, y, z)) ξ(σx,σy, σz), (2) σ(x)c σ + f σ (x, 1) = 0, (3) c σ + f σ (1,x)=0. (4) We now consider isomorphisms of monoidal functors θ σ,τ =(θx σ,τ ), where (θx σ,τ )=(ϕ σ x, t σ,τ (x)) : F σ F τ (x) F στ (x), where t σ,τ :Π A are maps. According to the notion of natural transformation of a monoidal functor, we have the following equations: f σ f τ = f στ, (5) (στx)t σ,τ (y) t σ,τ (xy)+t σ,τ (x) = f στ (x, y) f σ (F τ x, F τ y) σ( f τ (x, y)), (6)

8 662 N. T. Quang σ(c τ )+c σ c στ = t σ,τ (1). (7) From the equality (5), we are led to a group homomorphism f :Γ AutA, given by f(σ) = f σ and therefore f 1 = f(1) = id A. Then, sine F σ is a functor of the type (ϕ σ,f σ ),ϕ σ (xb) =ϕ σ (x)f σ (b), i.e., A is a Γ equivariant Π module with Γ actions (1). ii) From the condition ii) in the definition of a factor set, we are led to t σ,1 = t 1,σ =0. From the equality (6), for τ = 1, we obtain f x,y 1 = 0, i.e., F x,y 1 = id. From the equation (4), for σ = 1, we have c1 = 0, i.e., F 1 = id. Thus, F 1 is an identity monoidal functor. This completes the proof. Definition 3.3. Let Γ be a group and C be a Gr category of the type (Π,A). Two factor sets (θ, F) and (μ, G) on Γ with coefficients in C are cohomologous if there exists a family of isomorphisms of monoidal functors u σ :(F σ, F σ, F σ ) (G σ, G σ, Ĝσ ), σ Γ satisfying u 1 = id (Π,A), u στ.θ στ = μ σ,τ.u σ G τ.f σ u τ, σ,τ Γ. Remark 3.4. If two factor sets (θ, F), (μ, G) are cohomologous, then F σ = G σ,σ Γ. Indeed, from the definition of cohomologous factor sets, there exists a family of isomorphisms u σ :(F σ, F σ, F σ ) (G σ, G σ Ĝ σ ), σ Γ. Since u σ x : F σ x G σ x is a morphism in (Π,A), we have G σ x = F σ x. Furthermore, for any a A, by the commutative diagram F σ x u σ x G σ x F σ (x,a) G σ (x,a) F σ u σ x x G σ x, we have F σ (x, a) =G σ (x, a). We call a factor set (θ, F) almost strict if F σ = id I for all σ Γ.

9 The factor sets of Gr-categories of the type (Π,A) 663 Lemma 3.5. Let S be a Gr category of the type (Π,A). Any factor set (θ, F) on Γ with cofficients in S is cohomologous to an almost strict factor set (μ, G). Proof. For each σ Γ, consider a family of isomorphisms in S: { u σ x = id F σ x if x 1, ( F σ ) 1 if x =1, where 1 σ Γ, and u 1 = id. Then, we define G σ uniquely such that u σ : G σ F σ is a natural transformation by setting G σ = F σ and For such setting, clearly we have G σ x,y =(uσ x uσ y ) 1 F σ x,y (u σ xy ); Ĝσ = id I. G σ =(G σ, G σ, Ĝσ ):S S is a monoidal equivalence. Since Ĝσ = id I, the factor set (μ, G) is almost strict. Now, we can choose μ σ,τ : G σ G τ G στ the natural transformation which makes the following diagram μ σ,τ G σ G τ G στ F στ G σ u τ (8) G σ F τ u σ F τ θ σ,τ F σ F τ commute, for all σ, τ Γ. Clearly, μ σ,τ is a isomorphism of monoidal functors. One can verify that the family of μ σ,τ satisfies the condition ii) of the definition of a factor set. We now prove that they satisfy the condition iii). Consider the following diagram u στ G σ G τ G γ μ σ,τ G γ G στ G γ Gσ G τ u γ G σ G τ F γ (I) μ σ,τ F γ G στ u γ G στ F γ G σ μ τ,γ (VI) G σ u τ F γ G σ F τ F γ u σ F τ F γ (II) F σ F τ F γ θ σ,τ F γ u στ F γ (VII) F στ F γ μ στ,γ G σ F τγ Gσ θ τ,γ G σ u τγ (III) u σ F τγ F σ F τγ (V) (IV) θ σ,τ γ θ στ,γ F στγ u στγ G σ F τγ μ σ,τ γ G στγ

10 664 N. T. Quang In this diagram, the region (I) commutes thanks to the naturality of μ σ,τ ; the regions (II), (V), (VI), (VII) commute thanks to (8); the region (III) commutes thanks to the naturality of u σ ; the region (IV) commutes thanks to the definition of factor set (θ, F). So the perimeter commutes. This completes the proof. 4 Classification theorem Γ pair (Π,A) is given. In [3], cohomology groups HΓ n (Π,A), with n 3, can be regarded as group cohomologies of the struncated complex: CΓ (Π,A):0 C 1 Γ (Π,A) C 2 Γ (Π,A) ZΓ 3 (Π,A) 0, where CΓ 1(Π,A) consists of all normalized maps f :Π A, C2 Γ (Π,A) consists of all normalized maps g :Π 2 (Π Γ) A and ZΓ 3 (Π,A) consists of all normalized maps h :Π 3 (Π 2 Γ) (Π Γ 2 ) A satisfying conditions of a 3 cocycle: h(x,y, zt)+h(xy, z, t) = x(h(y, z, t)) + h(x,yz, t)+h(x, y, z), (9) σh(x, y, z)+h(xy, z, σ)+h(x, y, σ) =h(σx,σy, σz)+(σx)h(y, z, σ)+h(x,yz, σ), (10) σh(x, y, τ)+h(τx,τy,σ)+h(x, σ, τ)+(στx)h(y, σ, τ) =h(x,y, στ)+h(xy, σ, τ), (11) σh(x, τ, γ)+h(x, σ, τγ) =h(x, στ, γ)+h(γx,σ,τ), (12) for all x, y, z, t Π; σ, τ, γ Γ. For each g CΓ 2 (Π,A), g is given by ( g)(x, y, z) =xg(y, z) g(xy, z)+g(x, yz) g(x, y), (13) ( g)(x, y, σ) =σg(x, y) g(σx,σy) (σx)g(y, σ)+g(xy, σ) g(x, σ), (14) ( g)(x, σ, τ) =σg(x, τ) g(x, στ)+g(τx,σ). (15) We now show that each factor set on Γ induces a 3 cocycle of Γ groups.

11 The factor sets of Gr-categories of the type (Π,A) 665 Proposition 4.1. Each almost strict factor set (θ, F) on Γ with coefficients in a Gr category S =(Π,A,ξ) induces an element h Z 3 Γ (Π,A). Proof. Suppose F σ =(F σ, F σ,id). Then, F σ x,y is associated to f :Π 2 Γ A. The family of isomorphisms of monoidal functors θ σ,τ is associated to a function t :Π Γ 2 A. From functions ξ, f,t, we determine a function h as follows: where h = ξ f t, in the sense h :Π 3 (Π 2 Γ) (Π Γ 2 ) A, h Π 3= ξ; h Π 2 Γ= f; and h Π Γ 2= t. (16) The above-determined h is a 3 cocycle of Γ groups. Indeed, 3 cocycle of ξ leads to (9). Equations (3), (6) turn into (10), (11). The cocycle condition θ στ,γ.θ σ,τ F γ = θ σ,τ γ.f σ θ τ,γ leads to equation (12). However, we need prove the normalized property of h. First, since the unitivity constraints of (Π,A) are strict and the factor set (θ, F) is almost strict, equations (3), (4), (7) turn into: h(x, 1,σ)= f σ (x, 1) = 0 = f σ (1,x)=h(1,x,σ), h(1,σ,τ)=t σ,τ (1) = 0. Since F 1 = id, we have h(x, y, 1 Γ )= f 1 (x, y) =0. Thanks to the normalized property of associativity constraint ξ, we have h(1,y,z)=h(x, 1,z)=h(x, y, 1) = 0. Thanks to ii) in the definition of a factor set, we have h(x, 1 Γ,τ)=h(x, σ, 1 Γ )= 0. Let h (θ,f ) = h, we have h (θ,f ) ZΓ 3 (Π,A). This completes the proof. Proposition 4.2. Each element h (μ,g) HΓ 3 (Π,A) determines a Γ monoidal extension of a Gr category of the type (Π,A). Proof. According to the definition of cocycle of Γ groups, we have functions (16). Hence, we may determine the factor set (θ, F) : F σ x = σx; F σ (x, c) =(σx,σc), F σ σ = id 1, F x,y =(σ(xy), f(x, y, σ)), θ στ x =(στx, t(x, σ, τ)), for all σ, τ Γ,x Π,c A. Clearly, the these factor set induces h. In order to prove the Classification Theorem, we need the following lemma.

12 666 N. T. Quang Lemma 4.3. Let (μ, G), (θ, F) be two almost strict factor sets on Γ with coefficients in a Gr category S of the type (Π,A) and be cohomologous. Then, they determine the same Π module Γ equivariant structure on A and 3 cocyles inducing h (θ,f ), h (μ,g) are cohomologous. Proof. Since (θ,f), (μ, G) are cohomologous, there exists a family of isomorphisms of functors u σ : F σ G σ,σ Γ. According to Remark 3.4, F σ = G σ,σ Γ. Then, they induce the same Π-module Γ-equivariant structure on A thanks to relation (1). Now, we prove that h (θ,f ) and h (μ,g) are cohomologous. Denote h (μ,g) = h. Hence, by the determination of h (μ,g) referred in Proposition 4.1, for all x, y Π,σ,τ Γ, we have G σ x,y =(σ(xy),h (x, y, σ)) ; μ σ,τ x =(στx, h (x, σ, τ)), Let u :Π Γ A be the function defined by u(x, σ) =u σ x. It determines an extending 2 cochain g of u, where g Π 2 :Π 2 A is the null map. Since (u στ.θ σ,τ )x =(μ σ,τ.u σ G τ.f σ u τ )x, we have g(x, στ)+h(x, σ, τ) =h (x, σ, τ)+g(τx,σ)+σg(x, τ). (17) Since G σ x,y uσ x y =(uσ x uσ y ) F x,y σ, we have h (x, y, σ) h(x, y, σ) =g(x, σ)+(σx)g(y, σ) g(xy, σ). (18) Since F σ = Ĥσ = id, we have u σ 1 = id 1. Hence, g(1,σ)=0, for all σ Γ. Since u 1 = id ((Π,A), ), we have g(x, 1 Γ )=0, for all x G. From the determination of g and relations (17) - (18), we have g CΓ 2(Π,A) and h(θ,f ) h (μ,g) = g. This completes the proof. We can simply the equivalence classification problem of Γ extensions of Gr categories by classifying Γ extensions of Gr categories with the first two invariants as the following definition. Definition 4.4. Let Π be a Γ group, A be a Γ equivariant Π module. We say that Γ monoidal extension G of a Gr category has a pre-stick of the type (Π,A) if there exists a pair (p, q) :(Π,A) (Π 0 (G), Π 1 (G)), where p :Π Π 0 (G) is an equivariant isomorphism (to make Π 1 (G) bea Π module) and q : A Π 1 (G) is an isomorphism of Γ equivariant Π modules. Obviously, each Γ functor between two Γ extensions whose pre-sticks are of the type (Π,A)is a Γ equivalence.

13 The factor sets of Gr-categories of the type (Π,A) 667 Theorem 4.5. There exists a bijection Ω: Γ Gr(Π,A) H 3 Γ(Π,A), where Γ Gr(Π,A) is the set of equivalence classes of Γ monoidal extensions whose pre-sticks are of the type (Π,A). Proof. Each element of Γ Gr(Π,A) may be represented by a crossed product extension Δ(θ, F) of a Gr category S =(Π,A,ξ). According to Lemma 3.5, it is possible to assume that (θ, F) is almost strict. Then, (θ, F) induces a 3 cocycle h = h (θ,f ) (Proposition 4.1). According to Lemma 4.3, the correspondence cl(θ, F) cl(h (θ,f ) ) is a map. According to Proposition 4.2, this correspondence is a surjection. We now prove that it is an injection. Let Δ and Δ be two crossed product extensions of Gr categories S = S(Π,A,ξ), S = S (Π,A,ξ ) by factor sets (θ, F), (θ,f ). Moreover, 3 cocycles inducing h, h are cohomologous. We shall prove that two Γ extensions Δ and Δ are equivalent. According to the determination of h, h, we have h Π 3= ξ,h Π 3= ξ and ξ = ξ + δg, where g :Π 2 A is a function. Then, there exists a Gr equivalence of two Gr categories, (K, K, K) :S S, where K x,y =(,g(x, y)). According to Proposition 3.1, we may extend the Gr functor (K, K, K) toaγ equivalence So Ω is an injection. (Δ K, Δ K, Δ K ):Δ Δ References [1] A. M. Cegarra, A. R. Garzón and J. A. Ortega, Graded extensions of monoidal categories, J. Algebra. 241 (2) (2001), [2] A. M. Cegarra, J. M. García - Calcines and J. A. Ortega, On grade categorical groups and equivariant group extensions, Canad. J. Math. 54 (5) (2002), [3] A. M. Cegarra, J. M. García - Calcines, J. A. Ortega, Cohomology of groups with operators, Homology Homotopy Appl. 4 (1) (2002), [4] A. Fröhlich and C. T. C. Wall, Graded monoidal categories, Compositio Math. 28 (1974), [5] S. Mac Lane, Homology, Springer-Verlag, Berlin-New York, 1967.

14 668 N. T. Quang [6] N. T. Quang, On Gr-functors between Gr-categories: Obstruction theory for Gr-functors of the type (ϕ, f), arxiv: v2 [math.ct] 18 Apr [7] H. X. Sinh, Gr-catégories, Thèse de doctorat (Université Paris VII, 1975). Received: January, 2010