Graded extention of categorical groups of the type (π, A)

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 third-dimensional operator Γ. The element h determines a 1-1 correspondence between classes of factor sets on Γ with coefficients in (Π, A) and group H3 Γ (Π, A). 1 Introduction The concept of Γ-graded 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 Γ-graded extension of categories. If G is a Γ-graded 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 Γ-graded categorical group or a Γ-graded extension of categorical group. Since KerG is a groupoid if and only if G is, a Γ-graded extension of categorical groups can be defined as a Γ-graded groupoid such that for any object X there is an object X ′ with an 1-arrow X ⊗X ′ → 1. In [4], the authors classified Γ-graded 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 Γ-graded 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 third-dimensional 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 Γ-monoidal 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 Γ-groups 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 3-cocycle 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 3-cocycle (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 Γ-graded extension G of categorical groups C in- duces a structure pi0(G)-module Γ-equivariant 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 Π-modules. 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 Π-module Γ-equivariant. 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 Γ-group. 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 Γ-group. Finally, from the equation (4.13) we have σ(xb) = σx.σb for all σ ∈ Γ; b ∈ A or A is a Π-module Γ-equivariant. 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 Π-module Γ-equivariant 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 Γ-groups [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 Γ-equivariant 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 3-cocycle 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 G-module Γ-equivariant 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 Π-module Γ-equivariant 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 Π-module Γ-equivariant 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 2-cochain 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 G-module Γ-equivariant 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 Π-module Γ-equivariant, 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 Γ-extension 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 Π-module Γ- equivariant A determined as above coincides with the structure Π-module Γ-equivariant A determined from Γ-extension 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 Π-module Γ-equivariant 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 Π-module Γ-equivariant 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 Π-module Γ-equivariant 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 Γ-extension of the category ((Π, A),⊗, h(x, y, z), (I, id, id)), ( ∫ Γ(Π, A, h), j) is Γ-extension 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. 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