Nil(n)-Modules Bifibred over Groups

Vol. 6, No. 4, 2013, 428-434

ISSN 1307-5543 – www.ejpam.com

Nil(n)-Modules Bifibred over Groups

Hasan Atik

Department of Mathematics, Science Faculty, ˙Istanbul Medeniyet University, ˙Istanbul, Turkey

Abstract. In this work, we defined a functor from category of nil(n)-modules to that of groups. Then we showed by direct calculation that the functor is both fibration and cofibration of categories.

2010 Mathematics Subject Classifications: 18D30,18A40,18A30

Key Words and Phrases: Crossed modules, Nil(n)-Modules, Pullback Crossed Modules

1. Introduction

Crossed modules were defined by Whitehead[12]as a model for homotopy connected 2-types. Some universal constructions for crossed modules, for example, the notions of pullback and induced crossed modules have been worked in[4–6]. Furthermore, for Lie algebra cases of these constructions see[8], and for commutative algebras see[10]. Induced crossed mod-ules allow detailed computations of non-abelian information on second relative homotopy calculations. By extending these constructions for two dimensional case of crossed modules, Arslan, Arvasi and Onarli in[1], have defined the notions of pullback and induced 2-crossed module.

Baues[3]defined nil(n)-modules as a model for homotopy 2-types and studied some prop-erties of nil(2)-modules which forms a base for his homotopy connected 3-types “quadratic module”. Atik has constructed pullback and induced nil(2)- modules in his thesis[2]. In this work, by using a similar way given in these cited works, we have shown that the category of nil(n)-modules is bifibred over groups in the sense of A. Grothendieck[9].

2. Nil(n)-Modules

A pre-crossed module is a group homomorphism∂ : M _→Q together with an action ofQ on M, writtenmq forq ∈Qand m∈M, satisfying the condition∂(mq) =q−1∂(m)q for all m∈M andq∈Q. This is a crossed module if in addition

x−1y−1x = (y)∂x

Email addresses:[email protected],[email protected]

We define Peiffer commutator in a pre-crossed module

x,y

=x−1y−1x(y)∂1x

Thus∂ is a crossed module if and only if_〈x,y〉=1 for allx,y∈M. In a groupG we have the lower central series

Γn+1⊂Γn⊂. . .⊂Γ1=G

whereΓ_n= Γ_n(G)is the subgroup ofGgenerated by all iterated commutators(x₁,x₂, . . . ,x_n) of length n. Here Γ2(G) is the commutator subgroup of G. Similarly we obtain the lower Peiffer central series

P_n+1⊂P_n⊂. . ._⊂P₁=M

in a pre-crossed module ∂ : M → N. Where Pn = Pn(∂) is the subgroup of M generated by all iterated Peiffer commutators_〈x1,x2, . . . ,xn〉of lengthn. The groupPn(∂)is the Peiffer subgroup ofM, this generalizes the commutator subgroup in a group. The following definition is given by Baues[3].

Definition 1. A pre-crossed module∂ :M _→N is a Peiffer nilpotent of class n if P_n+1(∂) =1, in this case we call∂ is a nil(n)-module. That is

〈x1,x2,x3· · ·,xn〉=1,

A morphism between two nil(n)-modules ∂ : M →Q and∂0 : M0 →Q0 is a pair (g,f) of homomorphisms of groups g:M _→ M0 and f :Q_→Q0 such that f∂ =∂0g and the actions preserved, i.e. g(mq) = g(m)f(q) for any m ∈ M,q ∈Q. We shall denote the category of nil(n)-modulesbyNil(n).

3. Bifibration of Categories

We recall the definition of fibration of categories from[7].

Definition 2. LetΦ:X_→Bbe a functor. A morphismϕ:Y _→X inXover u:= Φ(ϕ)is called Cartesian if and only if for allυ:K →J inBandθ:Z→X withΦ(θ) =uυthere is a unique morphismψ:Z→Y withΦ(ψ) =υandθ=ϕψ.

Z

ψ // θ

'

'

Y _ϕ //X

Φ

K

uυ

'

'

υ //J u //I

Definition 3. The functorΦ:X→Bis fibration or category fibred overBif and only if for all u:J→I inBand X ∈X_I there is a Cartesian morphismϕ:Y →X over u:such aϕis called a Cartesian lifting of X along u.

We now give the duals of the above definition.

Definition 4. LetΦ:X_→Bbe a functor. A morphismψ:Z→Y inXoverυ:= Φ(ψ)is called cocartesian if and only if for all u:J→I inBandθ:Z→X withΦ(θ) =uυthere is a unique morphismϕ:Y _→X withΦ(ϕ) =u andθ=ϕψ.

Z

ψ // θ

'

'

Y _ϕ //X

Φ

K

uυ

'

'

υ //J u //I

Definition 5. The functorΦ:X→Bis cofibration or category cofibred overBif and only if for allυ:K_→J inBand Z_∈X_K there is a Cartesian morphismψ:Z _→Z0 overυ:such aψis called a cocartesian lifting of X alongυ.

Proposition 1. LetΦ:X_→Bbe a fibration of categories. Thenψ:Z_→Y inXoverυ:K_→J in Bis cocartesian if and only if for allθ0:Z→X0overυthere is a unique morphismψ0:Y →X0 in X_J withθ0=ψ0ψ.

Corollary 1. LetΦ:X_→Bbe a fibration of categories which has a left adjoint and suppose that X admits pushouts. ThenΦis also a cofibration.

For detailed information about bifibration categories we advise carefull reading of T.Streicher

[11].

4. Nil(n)-Modules Bifibred over Groups

Proposition 2. We have a forgetful functorΦN :Nil(n)→Grpin which(M →N)−→N . This

forgetful functor is fibred.

Suppose that∂ :M →Qis a nil(n)-module andσ:P→Qis a homomorphism of groups. Takeσ∗(M) ={(p,m) :∂(m) =σ(p)} as the fiber product of∂ andσ. Thus we have the following pullback diagram

σ∗_(M) β1

σ1 // M

∂

P _σ //Q

(1)

whereσ1:σ∗(M)→Pis given byσ1(p,m) =mandβ1:σ∗(M)→Pis given byβ1(p,m) =p for all(p,m)∈σ∗(M). The action ofp0∈Pon(p,m)∈σ∗(M)can be given by

This action obviously is a group action ofPonσ∗(M)and according to this action,β1becomes a nil(n)-module. Indeed,β1 is a pre-crossed module since for all(p,m)∈σ∗(M),

β1((p,m)p0) =β1(p0−1pp0,mσ(p0)) =p0−1pp0=p0−1β1(p,m)p0. Moreover,for(p1,m1),(p2,m2), . . . ,(pn,mn)∈σ∗(M), we have

〈. . ._〈〈(p1,m1),(p2,m2)〉,(p3,m3)〉, . . .〉,(pn,mn)〉

=〈. . ._〈〈(p₁,m1)−1(p₂,m₂)−1(p₁,m1)(p₂,m₂)β1(p1,m1)_,(_p

3,m3)〉,(p4,m4)〉, . . .〉,(pn,mn)〉

=〈. . .〈〈(p1−1,m1−1)(p2−1,m2−1)(p1,m1)(p2,m2)p1,(p3,m3)〉,(p4,m4)〉. . .〉,(pn,mn)〉

=〈. . ._〈〈(1,m1−1m2−1m1m2σ(p1)),(p3,m3)〉,(p4,m4)〉. . .〉,(pn,mn)〉

=〈. . .〈〈(1,m1−1m2−1m1m2∂(m1)),(p3,m3)〉,(p4,m4)〉. . .〉,(pn,mn)〉

=〈. . ._〈(1,m1−1m2−1m1m2∂(m1))

−1

(p3−1,m3−1), (1,m1−1m2−1m1m2∂(m1))(p3,m3)β1(1,m1

−1_m

2−1m1m2∂(m1))_,(_p

4,m4)〉. . .〉,(pn,mn)〉

=〈. . ._〈(1,m2∂(m1)

−1

m1−1m2m1)(p3−1,m3−1)(1,m1−1m2−1m1m2∂(m1)) (p₃,m3),(p₄,m4)_〉. . ._〉,(p_n,m_n)_〉

=〈. . ._〈(1,_〈m1,m2〉−1)(p3−1,m3−1)(1,〈m1,m2〉)(p3,m3),(p4,m4)〉. . .〉,(pn,mn)〉

=〈. . ._〈(1,_〈m1,m2〉−1m3−1〈m1,m2〉m3),(p4,m4)〉. . .〉,(pn,mn)〉

=〈. . ._〈(1,_〈m₁,m_2〉−1m₃−1_〈m₁,m_2〉m₃∂1(〈m1,m2〉))_,(_p

4,m4)〉. . .〉,(pn,mn)〉

=〈. . .〈(1,〈〈m1,m2〉,m3〉),(p4,m4)〉. . .〉,(pn,mn)〉.

if we continue calculations in this way, we obtain;(1,_〈m1,m2,m3, . . .mn〉). Since∂1is a nil(n)-module then(〈m1,m2,m3, . . .mn〉) =1, it gives the following result:β1is nil(n)-module.

Thusβ1:σ∗(M)→Pis a nil(n)-module. In the diagram (1), the pair of homomorphisms (σ1,σ) is a nil(n)-module morphism. This diagram is commutative since

∂ σ1(p,m) =∂(m) =σ(p) =σβ1(p,m)forp∈Pandm∈M. We have

σ1((p,m)p0) =σ1((p0)−1pp0,mσ(p0)) =mσ(p0)=σ1(p,m)σ(p0)

for all (p,m)∈σ∗(M) and p∈P. Therefore we have a pullback nil(n)-module. Further we will show thatσ1 is a Cartesian morphism overσ. Let(υ1,υ0):(K1→K0)→(σ∗(M)→P)

be homomorphism of nil(n)-modules andθ :K1→ P be a unique nil(n)-module morphism.

Then we have the following commutative diagram

K1 ∂1 / / θ & &

σ∗_(M)

σ1 / / _M K0

συ0=σ0

%

%

υ0

/

/_P

σ //Q

whereυ1:K1 →σ∗(M)is given byυ1(k1) = (υ0∂1k1,θk1). Sinceσ(υ0∂1k1) =∂ θk1=∂m

Proposition 3. The functorΦ_N:Nil(n)_→Grpis cofibred.

Let µ : M → P be a nil(2)-module and f : P →Q be a homomorphism of groups. Let f_∗(M) =F(M_×Q)be a free group generated by the setM_×Q. LetSbe a subgroup of f_∗(M) generated by the following relations: (m,m0∈M,q∈Q)

1. (m,q)(m0,q)(mm0,q)−1_∈S 2. (mp,q)(m,f(p)q)−1∈S

Now, consider the following diagram

M

µ

θ _{// f} ∗(M)/S

µ

P

f //Q

in which µ : f_∗(M)/S → Q is given by µ((m,q)S) = q−1fµ(m)q and θ : M → f_∗(M)/S is given by θ(m) = (m, 1)S for m _∈ M and q _∈ Q. This diagram is commutative, since

µθ(m) =µ((m, 1)S) = fµ(m) for allm∈M. The action ofQon f∗(M)/S can be given by

((m,q)S)q0 = (m,qq0)S form∈M andq,q0∈Q. By using this action, we have the following result.

Proposition 4. The homomorphismµ: f_∗(M)/S _→Q given byµ((m,q)S) =q−1fµ(m)q, as defined above, is an induced nil(n)-module by the homomorphism of groups f : P →Q of the nil(n)-moduleµ:M→P.

Proof. Since

µ(((m,q)S)q

0

) =µ((m,qq0)S)

=(qq0)−1fµ(m)qq0= (q0)−1(q−1fµ(m)q)q0

=(q0)−1µ((m,q)S)q0,

for allm∈M andq,q0∈Q,µis a pre-crossed module. Further, for all(m,q)S,(m0,q)S, . . . ,(m(n),q)S_∈f_∗(M)/S,

〈. . .〈〈(m,q)S,(m0,q)S〉,(m00,q)S〉, . . .〉(m(n),q)S〉

=〈. . ._〈(m,q)S(m0,q)S(m,q)S−1((m0,q)S−1)µ(m,q)S,(m00,q)S〉, . . ._〉(m(n),q)S〉

=〈. . .〈〈(m,q)S(m0,q)S(m−1,q)S((m0−1,q)S)q−1fµ(m)q,(m00,q)S〉, . . .〉(m(n),q)S〉

=〈. . ._〈〈(mm0m−1,q)S((m0−1,qq−1fµ(m)q)S,(m00,q)S〉, . . ._〉(m(n),q)S〉

=〈. . .〈〈(mm0m−1,q)S((m0−1)µ(m),q)S,(m00,q)S〉, . . .〉(m(n),q)S〉

=〈. . ._〈〈(_〈m,m0〉,q)S,(m00,q)S〉. . ._〉(m(n),q)S〉

=〈. . .〈(〈m,m0〉,q)S(m00,q)S(〈m,m0〉,q)−1S((m00,q)S−1)µ(〈m,m0〉,q)S, . . .〉(m(n),q)S〉

=〈. . ._〈(_〈m,m0_〉,q)S(m00,q)S(_〈m,m0_〉,q)−1S((m00,q)S−1)q0−1fµ(〈m,m0〉)q, . . ._〉(m(n),q)S_〉

=〈. . .〈(〈m,m0〉,q)S(m00,q)S(〈m,m0〉−1,q)S((m00−1,q)S), . . .〉(m(n),q)S〉

=〈. . ._〈(_〈m,m0〉m00〈m,m0〉−1(m00−1,q)S), . . ._〉(m(n),q)S〉

=〈. . .〈(〈m,m0〉m00〈m,m0〉−1(m00−1)µ(〈m,m0〉),q)S, . . .〉(m(n),q)S〉

=〈. . ._〈(_〈〈m,m0〉,m00〉,q)S, . . ._〉(m(n),q)S〉

=〈. . .〈(1,q)S, . . .〉(m(n),q)S〉 ..

.

=(〈m1,m2,m3, . . .mn〉,q)S∼=S

Thus we have thatµis a nil(n)-module. Now, we will show that(θ,f)is a nil(n)-module morphism. We have

θ(mp) = (mp, 1)S=m,f(p)1)S= ((m, 1)S)f(p)=θ(m)f(p)

andµθ(m) =µ((m, 1)S) = fµ(m)for allm_∈M andp_∈P. Then one can easily show thatθ is a cocartesian morphism over f.

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