EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
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, İstanbul Medeniyet University, İstanbul, 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 of Q
on M , written mq for q ∈ Q and m ∈ M , satisfying the condition ∂ (mq) = q−1∂ (m)q for all
m ∈ M and q ∈Q. This is a crossed module if in addition

x−1 y−1 x = (y)∂ x

Email addresses: hasan.atik@medeniyet.edu.tr, hasanatik@yahoo.com

http://www.ejpam.com 428 c© 2013 EJPAM All rights reserved.



H. Atik / Eur. J. Pure Appl. Math, 6 (2013), 428-434 429

We define Peiffer commutator in a pre-crossed module



x , y
�

= x−1 y−1 x(y)∂1 x

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

Γn+1 ⊂ Γn ⊂ . . .⊂ Γ1 = G

where Γn = Γn(G) is the subgroup of G generated by all iterated commutators (x1, x2, . . . , xn)
of length n. Here Γ2(G) is the commutator subgroup of G. Similarly we obtain the lower
Peiffer central series

Pn+1 ⊂ Pn ⊂ . . .⊂ 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 length n. The group Pn(∂ ) is the Peiffer
subgroup of M , 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 Pn+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 ∂ ′ : M ′ → Q′ is a pair (g, f ) of
homomorphisms of groups g : M → M ′ and f : Q → Q′ such that f ∂ = ∂ ′g 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)-modules by Nil(n).

3. Bifibration of Categories

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

Definition 2. Let Φ : X→ B be a functor. A morphism ϕ : Y → X in X over u := Φ(ϕ) is called
Cartesian if and only if for all υ : K → J in B and θ : Z → X with Φ(θ) = uυ there is a unique
morphism ψ : Z → Y with Φ(ψ) = υ and θ = ϕψ.

Z
ψ

//

θ

''
Y ϕ

// X

Φ

��K

uυ

''
υ

// J u
// I

A morphism α : Z → Y is called vertical (with respect to Φ) if and only if Φ(α) is an iden-
tity isomorphism in B. In particular, for I ∈ B we write XI, called the fibre over I , for the
subcategory of X consisting of those morphisms α with Φ(α) = idI ,



H. Atik / Eur. J. Pure Appl. Math, 6 (2013), 428-434 430

Definition 3. The functor Φ : X→ B is fibration or category fibred over B if and only if for all
u : J → I in B and X ∈ XI there is a Cartesian morphism ϕ : Y → X over u : such a ϕ is called a
Cartesian lifting of X along u.

We now give the duals of the above definition.

Definition 4. Let Φ : X→ B be a functor. A morphism ψ : Z → Y in X over υ := Φ(ψ) is called
cocartesian if and only if for all u : J → I in B and θ : Z → X with Φ(θ) = uυ there is a unique
morphism ϕ : Y → X with Φ(ϕ) = u and θ = ϕψ.

Z
ψ

//

θ

''
Y ϕ

// X

Φ

��K

uυ

''
υ

// J u
// I

Definition 5. The functor Φ : X→ B is cofibration or category cofibred over B if and only if for
all υ : K → J in B and Z ∈ XK there is a Cartesian morphism ψ : Z → Z ′ over υ : such a ψ is
called a cocartesian lifting of X along υ.

Proposition 1. Let Φ : X→ B be a fibration of categories. Thenψ : Z → Y in X over υ : K → J in
B is cocartesian if and only if for all θ ′ : Z → X ′ over υ there is a unique morphism ψ′ : Y → X ′

in XJ with θ ′ =ψ′ψ.

Corollary 1. Let Φ : X→ B be a fibration of categories which has a left adjoint and suppose that
X admits pushouts. Then Φ is 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 functor ΦN : Nil(n)→ Grp in which (M → N) −→ N. This
forgetful functor is fibred.

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

σ∗(M)

β1

��

σ1 // M

∂

��
P σ

// Q

(1)

where σ1 : σ∗(M)→ P is given by σ1(p, m) = m and β1 : σ∗(M)→ P is given by β1(p, m) = p
for all (p, m) ∈ σ∗(M). The action of p′ ∈ P on (p, m) ∈ σ∗(M) can be given by

(p, m)p
′
= (p′−1pp′, mσ(p

′)).



H. Atik / Eur. J. Pure Appl. Math, 6 (2013), 428-434 431

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

β1((p, m)p
′
) = β1(p

′−1pp′, mσ(p
′)) = p′−1pp′ = p′−1β1(p, m)p′.

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), (p3, 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

−1m1m2
σ(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
−1m2

−1m1m2
∂ (m1)), (p4, m4)〉 . . .〉, (pn, mn)〉

=〈. . . 〈(1, m2
∂ (m1)

−1
m1
−1m2m1)(p3

−1, m3
−1)(1, m1

−1m2
−1m1m2

∂ (m1))

(p3, m3), (p4, m4)〉 . . .〉, (pn, mn)〉

=〈. . . 〈(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, m2〉
−1m3

−1〈m1, m2〉m3
∂1(〈m1,m2〉)), (p4, 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 ∂1 is a nil(n)-
module then (〈m1, m2, m3, . . . mn〉) = 1, it gives the following result: β1 is nil(n)-module.

Thus β1 : σ∗(M)→ P is 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) for p ∈ P and m ∈ M . We have

σ1((p, m)p
′
) = σ1((p

′)−1pp′, mσ(p
′)) = mσ(p

′) = σ1(p, m)σ(p
′)

for all (p, m) ∈ σ∗(M) and p ∈ P. Therefore we have a pullback nil(n)-module. Further we
will show that σ1 is a Cartesian morphism over σ. Let (υ1,υ0) : (K1 → K0)→ (σ∗(M)→ P)
be homomorphism of nil(n)-modules and θ : 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=σ′

%%
υ0

// P σ
// Q

where υ1 : K1 → σ∗(M) is given by υ1(k1) = (υ0∂1k1,θk1). Since σ(υ0∂1k1) = ∂ θk1 = ∂m
so ψ is a well defined homomorphism.



H. Atik / Eur. J. Pure Appl. Math, 6 (2013), 428-434 432

Proposition 3. The functor ΦN : Nil(n)→ Grp is 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 set M ×Q. Let S be a subgroup of f∗(M)
generated by the following relations: (m, m′ ∈ M , q ∈Q)

1. (m, q)(m′, q)(mm′, 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−1 f µ(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) = f µ(m) for all m ∈ M . The action of Q on f ∗(M)/S can be given by

((m, q)S)q
′
= (m, qq

′
)S for m ∈ M and q, q′ ∈ Q. By using this action, we have the following

result.

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

Proof. Since

µ(((m, q)S)q
′

) =µ((m, qq
′
)S)

=(qq
′
)
−1

f µ(m)qq
′
= (q

′
)−1(q−1 f µ(m)q)q

′

=(q
′
)−1µ((m, q)S)q

′
,

for all m ∈ M and q, q′ ∈Q, µ is a pre-crossed module.
Further, for all (m, q)S, (m′, q)S, . . . , (m(n), q)S ∈ f∗(M)/S,

〈. . .〈〈(m, q)S, (m′, q)S〉, (m′′, q)S〉, . . .〉(m(n), q)S〉

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

=〈. . . 〈〈(m, q)S(m′, q)S(m−1, q)S((m′−1, q)S)q
−1 f µ(m)q, (m′′, q)S〉, . . .〉(m(n), q)S〉

=〈. . . 〈〈(mm′m−1, q)S((m′−1, qq−1 f µ(m)q)S, (m′′, q)S〉, . . .〉(m(n), q)S〉

=〈. . . 〈〈(mm′m−1, q)S((m′−1)µ(m), q)S, (m′′, q)S〉, . . .〉(m(n), q)S〉

=〈. . . 〈〈(mm′m−1(m′−1)µ(m), q)S, (m′′, q)S〉, . . .〉(m(n), q)S〉



REFERENCES 433

=〈. . . 〈〈(〈m, m′〉, q)S, (m′′, q)S〉 . . .〉(m(n), q)S〉

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

=〈. . . 〈(〈m, m′〉, q)S(m′′, q)S(〈m, m′〉, q)−1S((m′′, q)S−1)q
′−1 f µ(〈m,m′〉)q, . . .〉(m(n), q)S〉

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

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

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

=〈. . . 〈(〈〈m, m′〉, m′′〉, q)S, . . .〉(m(n), q)S〉

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

=(〈m1, m2, m3, . . . mn〉, q)S ∼= S

Thus we have that µ is 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)

and µθ(m) = µ((m, 1)S) = f µ(m) for all m ∈ M and p ∈ P. Then one can easily show that θ
is a cocartesian morphism over f .

References

[1] U. E. Arslan, Z. Arvasi, and G. Onarli. Induced two-crossed modules, arXiv:1107.4291v1
[math.AT] 21 Jul 2011.

[2] H. Atik. Categorical Structures of Quadratic Modules of Commutative Algebras, Ph.D. The-
sis, Osmangazi Universitesi. 2012.

[3] H.J. Baues. Combinatorial homotopy and 4-dimensional complexes, Walter de Gruyter, 15,
380 pages, (1991).

[4] R. Brown and P. J. Higgins. Colimit-theorems for relative homotopy groups, Journal of
Pure and Applied Algebra, Vol. 22, 11-41, (1981).

[5] R. Brown and P. J. Higgins. On the connection between the second relative homotopy
groups of some related spaces, Procedings of the London Mathematical Society, (3) 36
(2) (1978)193-212.

[6] R. Brown, P. J. Higgins, and R. Sivera. Nonabelian algebraic topology: filtered spaces,
crossed complexes, cubical higher homotopy groupoids, http://www.bangor.ac.uk/
~mas010/pdffiles/rbrsbookb-e231109.pdf.



REFERENCES 434

[7] R. Brown and R. Sivera. Algebraic colimit calculations in homotopy theory using fibred
and cofibred categories, Theory and Applications of Categories, 22 (2009) 222-251.

[8] J.M. Casas and M. Ladra. Colimits in the crossed modules category in Lie algebras, Geor-
gian Mathematical Journal, V7 N3, 461-474, 2000.

[9] A. Grothendieck. Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes
in Mathematics, Volume 79. Springer-Verlag, Berlin (1968).

[10] T. Porter. Some categorical results in the theory of crossed modules in commutative algebras,
Journal of Algebra, 109, pp 415-429, (1987).

[11] T. Streicher. Fibred categories à la Bénabou, http://www.mathematik.
tu-darmstadt.de/~streicher/FIBR/FibLec.pdf, pp 1-85, (1999).

[12] J.H.C. Whitehead. Combinatorial homotopy II, Bulletin of the American Mathematical
Society, 55, pp 453-496, (1949).