on graded extensions of braided categorical groups

mINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY






CHE THI KIM PHUNG

ON GRADED EXTENSIONS OF
BRAIDED CATEGORICAL GROUPS


Speciality: Algebra and Number Theory

Code: 62. 46. 01. 04




A SUMMARY OF
MATHEMATICS DOCTORAL tHESIS










NGHE AN - 2014

Work is completed at Vinh University

Supervisor:
1. Assoc. Prof. Dr. Nguyen Tien Quang
2. Assoc. Prof. Dr. Ngo Sy Tung


Reviewer 1:


Reviewer 2:


Reviewer 3:


Thesis will be defended at school-level thesis evaluating council at Vinh
University at date month year






Thesis can be found at:
1. Nguyen Thuc Hao Library and Information Center - Vinh University
2. Vietnam National Library
1
PREFACE
1. Rationale

The theory of categories with tensor products was studied by J. Bnabou
(1963) and S. MacLane (1963). They considered categories equipped with a
tensor product, an associativity constraint a and unit constraints l, r satis-
fying the commutative diagrams. These categories are called monoidal cate-
gories by S. MacLane (1963), and he gived sufficient conditions for coherence
of natural isomorphisms a, l, r. S. MacLane also showed sufficient conditions
for coherence of natural isomorphisms in symmetric monoidal categories, i. e.
monoidal categories have a commutativity constraint c which is compatible
with unit and associativity constraints. Later, the theory of monoidal cate-
gory has been concerned and developed by mathematicians in many aspects.
Monoidal categories can be “refined” to become categories with group struc-
ture if the objects are all invertible (see M. L. Laplaza (1983) and N. S. Rivano
(1972)). When the underlying categories are groupoids (i. e. morphisms are
all isomorphisms), we obtain group-like monoidal categories (see A. Fr¨ohlich
and C. T. C. Wall (1974)), or Gr-categories (see H. X. Sinh (1975)). In this
thesis, we say these categories to be categorical groups according to recently
popular documents (see P. Carrasco and A. R. Garzn (2004), A. M. Cegarra
et al. (2002)). In a case where categorical groups have a commutativity
constraint then they become Picard categories (see H. X. Sinh (1975)), or
symmetric categorical groups (see M. Bullejos et al. (1993)).
Braided monoidal categories appeared in the work of A. Joyal and R. Street
(1993) and were extensions of symmetric monoidal categories. The authors
“refined” braided monoidal categories to become braided categorical groups if
the objects are all invertible and the morphisms are all isomorphisms. They
also classified braided monoidal categories by quadratic functions (thanks to
the result of S. Eilenberg and S. Mac Lane on representations of quadratic
functions by the abelian cohomology group H
3
ab
(G, A)). Before that, symmet-

2
ric categorical groups (or Picard categories) was solved by H. X. Sinh (1975).
Note that the notion of symmetric categorical groups is a special case of one
of braided monoidal categories.
A generalization of categorical groups was graded categorical groups intro-
duced by A. Fr¨ohlich and C. T. C. Wall (1974). Then A. M. Cegarra and
E. Khmaladze (2007) studied braided graded categorical groups and graded
Picard categories. These structures are genneral cases of braided categorical
groups and Picard categories, respectively. They obtained the classification
results due to the cohomology theory of Γ-modules constructed by themselves.
According to a different research direction, some authors was interested in
the class of special categorical groups in which the constraints are all identities
and objects are all strict invertible, that is X ⊗ Y = I = Y ⊗ X. These
categories are called G-groupoids by R. Brown and C. B. Spencer (1976),
strict Gr-categories by H. X. Sinh (1978), strict categorical groups by A.
Joyal and R. Street (1993), strict 2-groups by J. C. Baez and A. D. Lauda
(2004) or 2-groups by B. Noohi (2007).
R. Brown and C. B. Spencer (1976) showed that each crossed module
is defined by a G-groupoid, and vice versa. Then the authors proved that
the category of crossed modules is equivalent to the category of G-groupoids
(Brown-Spencer equivalence).
As mentioned, G-groupoids are also called strict categorical groups, but
the category of G-groupoids is just a subcategory of the category of strict
categorical groups. N. T. Quang et al. (2014) showed a relation between
the second category and the category of crossed modules, in which Brown-
Spencer equivalence is just a special case. This result leads to applying the
results on the obstruction theory for functors and the cohomology theory to
study crossed modules. Furthermore, this approach leads to linking some
types of crossed modules with appropriate categorical algebras as well as we
shall present in Chapter 3 and Chapter 4.

The idea of R. Brown and C. B. Spencer was developed for braided crossed
modules and braided strict categorical groups by A. Joyal and R. Street
3
(1993). But A. Joyal and R. Street just stopped in mutual determining be-
tween these two structures. A problem is whether or not an Brown-Spencer
equivalence for these subjects. We think that the problem needs to be treated.
Besides braided crossed modules, there are different types of crossed mod-
ules concerned by mathematicians such as: abelian crossed modules (see P.
Carrasco and his co-authors (2002)), Γ-crossed modules and braided Γ-crossed
modules (see B. Noohi (2011)). According to N. T. Quang et al. (2014), we
hope that we can connect these types of crossed modules with corresponding
categorical algebras, and obtain categorical equivalences for these subjects.
According to a different approach, crossed modules have a close relation
with the problem of group extensions. The problem of group extensions of the
type of a crossed module was introduced by P. Dedecker (1964), and treated
by R. Brown and O. Mucuk (1994). Therefore, we think that we can study the
problem of group extensions of the type of certain a crossed module among
types of crossed modules which are mentioned.
For the above reasons, we have chosen the topic for the thesis that is: “On
graded extensions of braided categorical groups”.
2. Objective of the research
The objective of the thesis is to study the structure of categorical al-
gebras such as: graded Picard categories, strict graded categorical groups,
braided strict graded categorical groups and braided strict categorical groups.
Then we classify braided Γ-crossed modules, braided crossed modules, abelian
crossed modules and present Schreier theory for Γ-module extensions, abelian
extensions of the type of an abelian crossed module, Γ-module extensions of
the type of an abelian Γ-crossed module and central extensions of equivariant
groups.
3. Subject of the research

Braided categorical groups, braided graded categorical groups, types of
crossed modules and the group extension problem of the type of a crossed
module.
4
4. Scope of the research
The thesis studies the strict and symmetric properties in braided categor-
ical groups and braided graded categorical groups to classify types of crossed
modules and treat group extension problems of the type of a certain crossed
module.
5. Methodology of the research
We use theoretical research method during the thesis. Technically, we use
the following three methods:
- Use the theory of factor sets to study the structure of categorical algebras;
- Use the obstruction theory of functors to treat the problem of extension;
- Use categorical algebras to classify corresponding type of crossed modules.
6. Contribution of the thesis
The results of thesis have been published or acceptted on the international
magazines. Therefore, they have scientific significance and contribution of
material for persons interested in the related issues.
7. Organization of the research
7.1. Overview of the research
A. M. Cegarra and E. Khmaladze (2007) constructed the symmetric co-
homology groups of Γ-modules H
n
Γ,s
(M, N). Then they applied the 2nd and
3rd dimension cohomology groups to classify Γ-module extensions and graded
Picard categories, respectively.
The first content of the thesis is to study graded Picard categories by the
method of factor sets as well as N. T. Quang (2010) treated to Γ-graded cate-

gorical groups. We prove that any graded Picard category P is equivalent to
a crossed product extension of a factor set with coefficients in the reduced Pi-
card category of type (π
0
P, π
1
P), and show that each above factor set induces
a Γ-module structure on abelian groups π
0
P, π
1
P and induces a normalized
3-cocycle h ∈ Z
3
Γ,s
(π
0
P, π
1
P). As an application of the theory of graded
Picard categories, we classify Γ-module extensions due to symmetric graded
5
monoidal functors. Thanks to these results, we obtained the classification
of graded Picard categories and the cohomology classification of Γ-module
extensions of A. M. Cegarra and E. Khmaladze (2007).
The notion of crossed modules was introduced by J. H. C. Whitehead
(1949). A. Joyal and R. Street (1993) studied braided crossed modules which
are more refined than crossed modules. In 2004, from the notion of crossed
modules, P. Carrasco et al. (2002) considered a case where groups have the
commutative property and gived the notion of abelian crossed modules. They

proved that the category of abelian crossed modules is equivalent to the cat-
egory of right modules over the ring of matrices. In 2011, B. Noohi equipped
with an Γ-action on groups and group homomorphisms in the notion of crossed
modules, braided crossed modules and gived the notion of Γ-crossed modules,
braided Γ-crossed modules when he compared the different methods to com-
pute cohomology groups with coefficients in a crossed module. However, in
this paper, the author did not mention the classification of these types of cross
modules. In 2013, N. T. Quang and P. T. Cuc constructed strict graded cat-
egorical groups used to classify Γ-crossed modules and the equivariant group
extension problem of the type of a Γ-crossed module. This extension is a
generalization of an equivariant group extension (see A. M. Cegarra et al.
(2002)) and a group extension of the type of a crossed module.
The second content of the thesis is to construct morphisms in the category
of braided crossed modules. Such each morphism consists a homomorphism
(f
1
, f
0
) : M → M

of braided crossed modules and an element of the group
of abelian 2-cocycles Z
2
ab
(π
0
M, π
1
M


). Then we prove that the category of
braided crossed modules is equivalent to the category of braided strict cat-
egorical groups. Morphisms in the latter are symmetric monoidal functors
(F,

F ) : P → P

which preserve a tensor operation and

F
x,y
=

F
y,x
for all
x, y ∈ Ob(P). If braided crossed modules are abelian crossed modules, then
strict categorical groups are strict Picard categories. Then we establish a cat-
egorical equivalence between the category of abelian crossed modules and the
category of strict Picard categories, and treat the abelian extension problem
6
of the type of an abelian crossed module.
The third content of the thesis is to introduce braided strict graded cate-
gorical groups associated to braided Γ-crossed modules. Then we study a re-
lation between homomorphisms of braided Γ-crossed modules and symmetric
graded monoidal functors of braided strict graded categorical groups associ-
ated to braided Γ-crossed modules. This leads to a categorical equivalence
between the category of braided Γ-crossed modules and one of braided strict
graded categorical groups. We also treat the Γ-module extension problem of
the type of an abelian Γ-crossed module.

The last content of the thesis is to apply strict graded categorical groups
to the proof that if h is the third invariant of the strict Γ-graded categorical
group Hol
Γ
G and p : Π → Out G is an equivariant kernel then p
∗
(h) is
an obstruction of p, and the classification of equivariant group extensions
A → E → Π with A ⊂ ZE by Γ-graded monoidal auto-functors of the
graded categorical group

Γ
(Π, A, 0). Besides, we construct a strict Γ-graded
categorical group which is the composition of a strict graded categorical group
and a Γ-homomorphism. This result is an extension of the pull-back structure
of S. MacLane (1963) in the construction of a group extension Eγ from a group
extension E and a homomorphism γ.
7.2. The organization of the research
Besides the sections of preface, general conclusions, list of the author’s ar-
ticles related to the thesis, the thesis is organized into five chapters. Chapter
1 presents the basic knowledge used in the next chapters. Chapter 2 studies
graded Picard categories by the method of factor sets. Chapter 3 studies
braided strict categorical groups used to classify braided crossed modules,
abelian crossed modules and abelian extensions of the type of an abelian
crossed module. Chapter 4 constructs braided strict graded categorical groups
used to classify braided Γ-crossed modules and Γ-module extensions of the
type of an abelian Γ-crossed module. Chapter 5 studies strict graded cate-
gorical groups related to the problem of equivariant group extensions.
7
CHAPTER 1

PRELIMINARIES
In this chapter, we present some notions and basic results concerning to
monoidal categories, braided categorical groups, Picard categories, graded
categorical groups, cohomology of Γ-modules, braided graded categorical groups
and graded Picard categories. This basic knowledge will be used in the next
chapters.
Section 1.1 recalls the notions about monoidal categories, monoidal func-
tors, homotopies and categorical groups.
Section 1.2 recalls the notions about braided categorical groups, Picard
categories, symmetric monoidal functors, reduced braided categorical groups,
and presents two results on the obstruction of functors of type (ϕ, f) to be-
come symmetric monoidal functors.
Section 1.3 recalls the notions about graded monoidal categories, graded
categorical groups and graded categorical groups of type (Π, A, h).
Section 1.4 recalls briefly about the low-dimension abelian (symmetric)
cohomology groups of Γ-modules.
Section 1.5 recalls the notions about braided (symmetric) graded monoidal
categories, braided (symmetric) graded categorical groups, symmetric graded
monoidal functors and braided graded categorical groups of type (M, N, h).
The last part of the section will present two results on the obstruction of
graded functors of type (ϕ, f) to become symmetric graded monoidal functors.
8
CHAPTER 2
FACTOR SETS IN GRADED PICARD CATEGORIES
In this chapter, we describe symmetric factor sets on Γ with coefficients in
Picard categories in order to interpret the symmetric cohomology group H
3
Γ,s
of Γ-modules and, classify Γ-module extensions thanks to symmetric Γ-graded
monoidal functors. The results of this chapter are based on the paper [1].

2.1. Factor sets with coefficients in Picard categories
We denote by Pic the category of Picard categories and symmetric monoidal
functors between them and by Z
3
s
a full subcategory of the category Pic,
which is defined as follows. A object of Z
3
s
is a Picard category P =

(M, N, h),
where M, N are abelian groups and h = (ξ, η) ∈ Z
3
s
(M, N) with ξ : M
3
→ N,
η : M
2
→ N. A morphism

(M, N, h) →

(M

, N

, h


) is a symmet-
ric monoidal functor (F,

F ), where F is a pair of group homomorphisms
ϕ : M → M

and f : N → N

,

F is associated to a function g : M
2
→ N

such that f
∗
(h) = ϕ
∗
(h

) + ∂g ∈ Z
3
s
(M, N

).
2.1.1 Definition. A symmetric factor set on Γ with coefficients in a Picard
category P (or a pseudo-functor F : Γ → Pic) consists of a family of sym-
metric monoidal auto-equivalences F
σ

: P → P, σ ∈ Γ, and isomorphisms
between symmetric monoidal functors θ
σ,τ
: F
σ
F
τ
→ F
στ
, σ, τ ∈ Γ, satisfy-
ing the conditions:
i) F
1
= id
P
,
ii) θ
1,σ
= id
F
σ
= θ
σ,1
, σ ∈ Γ,
9
iii) for all σ, τ, γ ∈ Γ, the following diagram commutes
F
σ
F
τ

F
γ
θ
σ,τ
F
γ
−−−−→ F
στ
F
γ
F
σ
θ
τ,γ






θ
στ,γ
F
σ
F
τγ
θ
σ,τ γ
−−−→ F
στ γ

.
. (2.1.1)
We write F = (P, F
σ
, θ
σ,τ
) and denote simply by (F, θ).
A symmetric factor set F is said to be almost strict if F
σ
∗
= id for all
σ ∈ Γ.
2.1.5 Theorem. Let P be a Γ-graded Picard category and Ker P(h) be the
reduced Picard category of Ker P. Then there is a factor set F with coefficients
in Ker P(h) such that P is equivalent to ∆F .
2.2. Factor sets with coefficients in Picard categories

(M, N, h)
The following theorem shows necessary conditions of a factor set with
coefficients in a Picard category.
2.2.1 Theorem. Let Γ be a group and S =

(M, N, ξ, η) be a Picard cate-
gory. Then
(i) Each almost strict factor set F = (S, F
σ
, θ
σ,τ
) : Γ → Z
3

s
induces a
Γ-module structure on M, N and a normalized 3-cocycle h
F
∈ Z
3
Γ,s
(M, N);
(ii) In the definition of factor sets, the condition F
1
= id
S
can be deduced
from other conditions.
The following result is deduced from Theorem 2.1.5 and Theorem 2.2.1.
2.2.2 Theorem. Each graded Picard category P induces a Γ-module struc-
ture on M = π
0
P, N = π
1
P and a normalized 3-cocycle h
F
∈ Z
3
Γ,s
(M, N).
In the following definition, we introduce the notion of Γ-graded Picard cat-
egories whose pre-sticks are of type (M, N). Then the classification problem
of Γ-graded Picard categories will be treated on Γ-graded Picard categories
whose pre-sticks are of type (M, N).

10
2.2.8 Definition. Let M and N be Γ-modules. We say that a Γ-graded
Picard category P has a pre-stick of type (M, N) if there exists a pair of
isomorphisms of Γ-modules
(p, q) : (M, N) → (π
0
P, π
1
P).
Obviously, any Γ-graded functor of Γ-graded Picard categories whose pre-
sticks are of type (M, N) is a Γ-graded equivalence.
With the preparation of the above, we obtain the classification of Γ-graded
Picard categories of A. M. Cegarra and E. Khmaladze (2007).
2.2.9 Theorem. There exists a bijection
Γ
Pic[M, N] ↔ H
3
Γ,s
(M, N),
where
Γ
Pic[M, N] is the set of equivalence classes of Γ-graded Picard cate-
gories whose pre-sticks are of type (M, N).
2.3. Γ-module extensions
This section is dedicated to presenting the classification of Γ-module exten-
sions due to symmetric Γ-graded monoidal functors of two Γ-graded Picard
categories Dis
Γ
M and Red
Γ

N.
2.3.1 Definition. A Γ-module extension of N by M is a short exact sequence
of Γ-modules and Γ-homomorphisms
E : 0 → N
i
−→ B
p
−→ M → 0. (2.3.1)
Denote by Ext
ZΓ
(M, N) the set of all equivalence classes of Γ-module ex-
tensions of N by M.
The discrete Γ-graded Picard category Dis
Γ
M is defined by
Dis
Γ
M =

Γ
(M, 0, 0).
Thus, Dis
Γ
M has the elements of M as objects and their morphisms σ : x → y
are the elements σ ∈ Γ with σx = y. Composition is the multiplication in
11
Γ. The grading gr : Dis
Γ
M → Γ is given by gr(σ) = σ. The graded tensor
product on objects is the operation in M, and on morphisms is given by

(x
σ
−→ y) ⊗ (x

σ
−→ y

) = (x + x

σ
−→ y + y

).
The graded unit functor I : Γ → Dis
Γ
M is given by
I(∗
σ
−→ ∗) = (0
σ
−→ 0).
The associativity, commutativity and unit isomorphisms are identities.
The reduced Γ-graded Picard category Red
Γ
N is given by
Red
Γ
N =

Γ

(0, N, 0).
Thus, Red
Γ
N has only one object, denoted by ∗, and morphisms are pairs
(n, σ) with n ∈ N and σ ∈ Γ. Composition of two morphisms is given by
(n, σ)(m, τ ) = (n + σm, στ).
The grading gr : Red
Γ
N → Γ is given by gr(n, σ) = σ. The graded tensor
product is defined by
(n, σ) ⊗ (m, σ) = (n + m, σ).
The graded unit functor I : Γ → Red
Γ
N is given by I(σ) = (0, σ). The
associativity, commutativity and unit isomorphisms are identities.
Denote by Hom
Γ,s
[Dis
Γ
M, Red
Γ
N] the set of all homotopy classes of graded
symmetric monoidal functors from Dis
Γ
M to Red
Γ
N.
2.3.2 Theorem. (Schreier theory for Γ-module extensions) There is a bijec-
tion
Ω : Ext

ZΓ
(M, N) → Hom
Γ,s
[Dis
Γ
M, Red
Γ
N].
12
CHAPTER 3
BRAIDED CROSSED MODULES AND
BRAIDED STRICT CATEGORICAL GROUPS
This chapter is devoted to defining morphisms in the category of braided
crossed modules and proving that this category is equivalent to the category of
braided strict categorical groups. We also establish a categorical equivalence
between the category of abelian crossed modules and the category of strict
Picard categories, and treat the problem of abelian extensions of the type of
an abelian crossed module. The results of this chapter are based on papers
[2] and [4].
3.1. Braided crossed modules and braided strict categorical groups
According to A. Joyal and R. Street (1993), each braided crossed module is
defined by a braided strict categorical group, and vice versa. In this section,
we construct a categorical equivalence between the category of braided crossed
modules and the category of braided strict categorical groups.
First, we show some properties and examples of braided crossed modules.
3.1.5 Proposition. Let M be a braided crossed module. Then
(i) η(x, 1) = η(1, y) = 0;
(ii) Ker d is a subgroup of ZB;
(iii) Coker d is an abelian group;
(iv) The homomorphism ϑ induces the identity on Ker d, and hence the

action of Coker d on Ker d, given by sa = ϑ
x
(a) with a ∈ Ker d and x ∈ s ∈
Coker d, is trivial.
3.1.6 Example. Let N be a normal subgroup of a group G such that the
quotient group G/N is abelian, in other words, let N be a normal subgroup
13
in G which contains the derived group (or the commutator subgroup) of G.
Then, (N, G, i, µ, η) is a braided crossed module, where i : N → G is an
inclusion, µ : G → Aut N is defined by conjugation and η : G × G → N is
given by η(x, y) = [x, y].
3.1.7 Definition. Let M = (B, D, d, ϑ, η) and M

= (B

, D

, d

, ϑ

, η

) be
two braided crossed modules. A homomorphism of braided crossed modules
M and M

consists of group homomorphisms f
1
: B → B


and f
0
: D → D

such that:
(H
1
) f
0
d = d

f
1
;
(H
2
) f
1
(ϑ
x
b) = ϑ

f
0
(x)
f
1
(b);
(H

3
) f
1
(η(x, y)) = η

(f
0
(x), f
0
(y))
for all x, y ∈ D, b ∈ B.
3.1.8 Lemma. Let (f
1
, f
0
) : M → M

be a homomorphism of braided crossed
modules M and M

. Let P and P

be braided strict categorical groups associ-
ated to M and M

, respectively.
(i) There exists the functor F : P → P

defined by F (x) = f
0

(x) and
F (b) = f
1
(b) for all x ∈ Ob(P), b ∈ Mor(P);
(ii) Natural isomorphisms

F
x,y
: F (x)F (y) → F (xy) together with F is
a symmetric monoidal functor if and only if

F
x,y
= p
∗
ϕ(x, y) = ϕ(px, py),
where p : D → Coker d is a canonical projection and ϕ is an abelian 2-cocycle
of the group Z
2
ab
(Coker d, Ker d

).
We determine the category BrCross whose objects are braided crossed
modules and morphisms are triples (f
1
, f
0
, ϕ), where (f
1

, f
0
) : M → M

is a
homomorphism of braided crossed modules and ϕ ∈ Z
2
ab
(Coker d, Ker d

).
3.1.8 Definition. Let P and P

be two braided strict categorical groups.
Then a symmetric monoidal functor (F,

F ) : P → P

is termed regular if
(B
1
) F (x) ⊗ F (y) = F (x ⊗ y);
(B
2
) F (b) ⊗ F (c) = F (b ⊗ c);
(B
3
)

F

x,y
=

F
y,x
for all x, y ∈ Ob(P) and b, c ∈ Mor(P).
14
3.1.10 Lemma. Let P, P

be corresponding braided strict categorical groups
associated to braided crossed modules M, M

, and let (F,

F ) : P → P

be a
regular symmetric monoidal functor. Then, the triple (f
1
, f
0
, ϕ), where
f
1
(b) = F (b), f
0
(x) = F (x), ϕ(px, py) =

F
x,y

for b ∈ B and x, y ∈ D, is a morphism in BrCross.
Denote by BrGr
∗
the category of braided strict categorical groups and reg-
ular symmetric monoidal functors and denote by p : D → Coker d a canonical
projection, we obtain the following classification result.
3.1.11 Theorem. (Classification Theorem) There exists an equivalence
Φ : BrCross −→ BrGr
∗
,
M −→ P
M
(f
1
, f
0
, ϕ) −→ (F,

F )
where F (x) = f
0
(x), F(b) = f
1
(b),

F
x,y
= ϕ(px, py) for x, y ∈ D and b ∈ B.
3.2. Abelian crossed modules and strict Picard categories
The following notion of abelian crossed modules was studied in the work

of P. Carrasco et al. (2002).
3.2.1 Definition. A crossed module M = (B, D, d, ϑ) is said to be abelian
when B, D are abelian and ϑ = 0.
Note that an abelian crossed module is also a particular case of a braided
crossed module. For convenience, M is also denoted by (B, D, d) or simply
by B → D.
We say that a strict Picard category is a symmetric strict categorical group
whose commutativity constraint is an identity, that is c = id.
Section 3.1 construct a categorical equivalence for braided crossed modules
M and braided strict categorical groups P
M
. Now if M = (B, D, d) is an
abelian crossed module then P
M
associated to M is a strict Picard category.
15
We determine the category AbCross whose objects are abelian crossed
modules and morphisms are triples (f
1
, f
0
, ϕ), where (f
1
, f
0
) : (B, D, d) →
(B

, D


, d

) is a homomorphism of abelian crossed modules and ϕ is a sym-
metric 2-cocycle of the group Z
2
s
(Coker d, Ker d

).
3.2.5 Definition. Let P and P

be strict Picard categories. A symmetric
monoidal functor (F,

F ) : P → P

is termed regular if F(x)⊗F (y) = F (x⊗y)
for x, y ∈ Ob(P).
Let Picstr denote the category of strict Picard categories and regular
symmetric monoidal functors, we obtain the following theorem.
3.2.7 Theorem. (Classification Theorem) There exists an equivalence
Φ : AbCross −→ Picstr,
(B → D) −→ P
B→D
(f
1
, f
0
, ϕ) −→ (F,


F )
where F (x) = f
0
(x), F (b) = f
1
(b) and

F
x
1
,x
2
= ϕ(s
1
, s
2
) for x ∈ D, b ∈ B,
x
i
∈ s
i
∈ Coker d, i = 1, 2.
3.3. Abelian extensions of the type of an abelian crossed module
R. Brown and O. Mucuk (1994) interpreted the existence and the cohomol-
ogy classification of group extensions of the type of a crossed module by using
the methods of crossed complexes. Recently, N. T. Quang et al. (2013) used
the obstruction theory of monoidal functors to treat the problem of group
extensions of the type of a crossed module.
This section present a version of the above results for a case of abelian
crossed modules.

3.3.1 Definition. Let M = (B, D, d) be an abelian crossed module, and let
Q be an abelian group. An abelian extension of B by Q of type M is the
diagram of group homomorphisms
16
E : 0
//
B
j
//
E
p
//
ε

Q
//
0,
B
d
//
D
(3.3.1)
where the top row is exact and (id
B
, ε) is a homomorphism of abelian crossed
modules.
Each extension E induces a homomorphism ψ : Q → Coker d. Our ob-
jective is to study the set Ext
ab
B→D

(Q, B, ψ) of equivalence classes of abelian
extensions of B by Q of type B
d
→ D inducing ψ.
The following theorem present the classification of abelian extensions of
the type of an abelian crossed module due to symmetric monoidal functors.
Denote by Hom
P ic
(ψ,0)
[Dis
s
Q, P
B→D
] the set of homotopy classes of symmetric
monoidal functors of type (ψ, 0) from Dis
s
Q to P
B→D
.
3.3.3 Theorem. (Schreier Theory for abelian extensions of the type of an
abelian crossed module) Let B → D be an abelian crossed module, Q be an
abelian group and ψ : Q → Coker d be a group homomorphism. Then there
exists a bijection
Ω : Hom
P ic
(ψ,0)
[Dis
s
Q, P
B→D

] → Ext
ab
B→D
(Q, B, ψ)
if one of the above sets is nonempty.
Since the reduced Picard category of P
B→D
is P(h) =

(Coker d, Ker d, h),
h ∈ Z
3
s
(Coker d, Ker d), the homomorphism (ψ, 0) : (Q, 0) → (Coker d, Ker d)
induces an obstruction ψ
∗
h ∈ Z
3
s
(Q, Ker d). With this notion of obstruction,
we have:
3.3.4 Theorem. Let (B, D, d) be an abelian crossed module, and let ψ : Q →
Coker d be a homomorphism of abelian groups. Then, the vanishing of
ψ
∗
h
in H
3
s
(Q, Ker d) is necessary and sufficient for there to exist an extension of

B by Q of type B → D inducing ψ. Further, if ψ
∗
h vanishes, then the set of
equivalence classes of such extensions is bijective with H
2
s
(Q, Ker d).
17
CHAPTER 4
BRAIDED Γ-CROSSED MODULES AND
BRAIDED STRICT GRADED CATEGORICAL GROUPS
The first part of the chapter is devoted to introducing braided strict Γ-
graded categorical groups associated to braided Γ-crossed modules. Then we
classify braided Γ-crossed modules and state Schreier theory for Γ-module
extensions of the type of an abelian Γ-crossed module. The results of this
chapter are based on the paper [4].
4.1. Braided Γ-crossed modules and braided strict graded categori-
cal groups
The following definition introduces the notion of braided strict graded cat-
egorical groups associated to braided Γ-crossed modules. First, we say that
a symmetric factor set (see Definition 2.1.1, Chapter 2) is regular if for all
σ, τ ∈ Γ, then F
σ
is a regular symmetric monoidal functor and θ
σ,τ
= id.
4.1.5 Definition. A braided Γ-graded categorical group P is said to be strict
if
(i) Ker P is a braided strict categorical group;
(ii) P induces a regular symmetric factor set (F, θ) on Γ with coefficients

in Ker P.
We prove that we can construct a braided strict Γ-graded categorical group
P
M
from a given braided Γ-crossed module M, and vice versa.
Now we present a relation between homomorphisms of braided Γ-crossed
modules and symmetric Γ-graded monoidal functors of braided strict Γ-graded
categorical groups associated to braided Γ-crossed modules.
18
First, we note that each morphism x
(b,σ)
−−−→ y in P
M
is written in the form
x
(0,σ)
−−−→ σx
(b,1)
−−→ y
and then each graded symmetric monoidal functor (F,

F ) : P
M
→ P
M

defines
a function f : D
2
∪ (D × Γ) → B


by
(f(x, y), 1) =

F
x,y
, (f(x, σ), σ) = F (x
(0,σ)
→ σx).
4.1.7 Lemma. Let (f
1
, f
0
) : M → M

be a homomorphism of braided Γ-
crossed modules M and M

. Let P and P

be braided strict graded categorical
groups associated to M and M

. Then there is a symmetric graded monoidal
functor (F,

F ) : P → P

such that F (x) = f
0

(x), F (b, 1) = (f
1
(b), 1) if and
only if f = p
∗
ϕ, where ϕ ∈ Z
2
Γ,s
(Coker d, Ker d

), and p : D → Coker d is a
canonical projection.
We define the category
Γ
BrCross whose objects are braided Γ-crossed
modules and morphisms are triples (f
1
, f
0
, ϕ), where (f
1
, f
0
) : M → M

is a
homomorphism of braided Γ-crossed modules, and ϕ ∈ Z
2
Γ,s
(Coker d, Ker d


).
4.1.8 Definition. A graded symmetric monoidal functor (F,

F ) : P → P

is
termed regular if (F,

F ) is a regular symmetric monoidal functor and satisfies
(B
4
) F (σx) = σF (x);
(B
5
) F (σb) = σF (b)
where x ∈ Ob(P), σ ∈ Γ, and b is a morphism of grade 1 in P.
4.1.9 Lemma. Let P, P

be corresponding braided strict Γ-graded categor-
ical groups associated to braided Γ-crossed modules M,M

, and let (F,

F ) :
P → P

be a regular symmetric graded monoidal functor. Then, the triple
(f
1

, f
0
, ϕ), where
(i) f
0
(x) = F (x), (f
1
(b), 1) = F (b, 1) for σ ∈ Γ, b ∈ B, x ∈ D;
(ii) p
∗
ϕ = f,
is a morphism in the category
Γ
BrCross.
Denote by
Γ
BrGr
∗
the category of braided strict Γ-graded categorical
groups and regular graded symmetric monoidal functors, we obtain the fol-
lowing result.
19
4.1.10 Theorem. (Classification Theorem) There exists an equivalence
Φ :
Γ
BrCross −→
Γ
BrGr
∗
,

M −→ P
M
(f
1
, f
0
, ϕ) −→ (F,

F )
where F (x) = f
0
(x), F (b, 1) = (f
1
(b), 1), F (x
(0,σ)
→ σx) = (ϕ(px, σ), σ) and

F
x,y
= (ϕ(px, py), 1) for all x ∈ D, b ∈ B, σ ∈ Γ.
4.2. Γ-module extensions of the type of an abelian Γ-crossed module
This section is devoted to presenting the classification of Γ-module exten-
sions of the type of an abelian Γ-crossed module. This extension is a general
case of a Γ-module extension (see Section 2.3) and an abelian extension of the
type of an abelian crossed module (see Section 3.2). First, we state the notion
of strict Γ-graded Picard categories and one of abelian Γ-crossed modules.
From the definition of braided Γ-crossed modules, by an abelian Γ-crossed
module, we shall mean a braided Γ-crossed module (B, D, d, ϑ, η) that ϑ =
0, η = 0. The construction of braided strict Γ-graded categorical groups from
braided Γ-crossed modules ensures that each abelian Γ-crossed module M

define a graded category P
M
whose Ker P
M
is a strict Picard category. We
say that P
M
is a strict Γ-graded Picard category.
4.2.1 Definition. Let M = (B, D, d) be an abelian Γ-crossed module, and
let Q be a Γ-module. A Γ-module extension of B by Q of type M is the
diagram of Γ-homomorphisms
E : 0
//
B
j
//
E
p
//
ε

Q
//
0,
B
d
//
D
(4.2.1)
where the top row is exact and (id

B
, ε) : (B, E, j) → (B, D, d) is a homomor-
phism of abelian Γ-crossed modules.
Each extension E induces a homomorphism of Γ-modules ψ : Q → Coker d.
This section studies the set Ext
M
ZΓ
(Q, B, ψ) of equivalence classes of Γ-module
20
extensions of B by Q of the type of an abelian Γ-crossed module M inducing
ψ : Q → Coker d due to the obstruction theory for graded symmetric Γ-graded
monoidal functors of strict Γ-graded Picard categories Dis
Γ,s
Q and P
M
, we
obtain the following results.
4.2.3 Theorem. (Schreier Theory for Γ-module extensions of the type of an
abelian Γ-crossed module) Let M = (B, D, d) be an abelian Γ-crossed module,
Q be a Γ-module and ψ : Q → Coker d be a homomorphism of Γ-modules.
Then there exists a bijection
Ω : Hom
(ψ,0)
[Dis
Γ,s
Q, P
M
] → Ext
M
ZΓ

(Q, B, ψ),
where Hom
(ψ,0)
[Dis
Γ,s
Q, P
M
] denotes the set of homotopy classes of sym-
metric graded monoidal functors of type (ψ, 0) between strict graded Picard
categories Dis
Γ,s
Q and P
M
.
Since the reduced Γ-graded Picard category of P
M
is
P(h) =

Γ
(Coker d, Ker d, h), h ∈ Z
3
Γ,s
(Coker d, Ker d),
the Γ-homomorphism (ψ, 0) : (Q, 0) → (Coker d, Ker d) induces an obstruc-
tion ψ
∗
h ∈ Z
3
Γ,s

(Q, Ker d). With this notion of obstruction, we have:
4.2.4 Theorem. Let M = (B, D, d) be an abelian Γ-crossed module, and let
ψ : Q → Coker d be a homomorphism of Γ-modules. Then the vanishing of
ψ
∗
h in H
3
Γ,s
(Q, Ker d) is necessary and sufficient for there to exist an extension
E of type M inducing ψ. Further, if ψ
∗
h vanishes, then the set of equivalence
classes of such extensions is bijective with H
2
Γ,s
(Q, Ker d).
21
CHAPTER 5
EQUIVARIANT GROUP EXTENSIONS
AND STRICT GRADED CATEGORICAL GROUPS
In this chapter, we show a relation between the third invariant of the strict
graded categorical group Hol
Γ
G and the obstruction of an equivariant kernel
(Π, G, p). We also classify equivariant group extensions which are central ex-
tensions and determine a strict graded categorical group from a strict graded
categorical group and a Γ-homomorphism. The results of this chapter are
based on the paper [3].
5.1. Strict graded categorical groups
The following notion of strict graded categorical groups was introduced by

N. T. Quang and P. T. Cuc (2013).
5.1.1 Definition. A graded categorical group G is said to be strict if
(i) Ker G is a strict categorical group;
(ii) G induces a regular factor set (F, θ) on Γ with coefficients in a categor-
ical group Ker G.
Now we present three examples on strict graded categorical groups.
5.1.2 Example. For each Γ-group Π, we can construct a strict Γ-graded
categorical group Dis
Γ
Π like the construction of a Γ-graded Picard category
Dis
Γ
M (see Section 2.3) without the commutativity of the operation.
5.1.3 Example. For each Γ-group G, we can construct a strict graded cate-
gorical group, denoted by Hol
Γ
G, whose objects are elements of the Γ-group
Aut G. A morphism of graded σ is a pair (u, σ) : α → β such that σα = µ
u
β,
22
where u ∈ G, σ ∈ Γ. Composition of two morphisms α
(u,σ)
−−−→ β
(v,τ )
−−−→ γ is
given by
(v, τ) ◦ (u, σ) = (τu + v, τσ). (5.1.1)
The tensor product on objects is composition of automorphisms in the Γ-
group Aut G, and on morphisms is given by

(α
(u,σ)
−−−→ β) ⊗ (α

(u

,σ)
−−−→ β

) = (αα

(u+β(u

),σ)
−−−−−−−→ ββ

). (5.1.2)
The unit graded functor I : Γ → Hol
Γ
G is defined by
I(∗
σ
−→ ∗) = id
G
(0,σ)
−→ id
G
.
The associativity and unit isomorphisms are identities.
5.1.4 Example. Strict graded categorical groups associated to Γ-crossed mod-

ules (This example is taken from the work of N. T. Quang and P. T. Cuc
(2013)).
5.2. Equivariant kernels
The notion of equivariant kernels was introduced by A. M. Cegarra et al.
(2002). It is a triple (Π, G, p), where Π, G are Γ-groups and p : Π → Out G is a
homomorphism of Γ-groups. Then ZG is a Γ-equivariant OutG-module under
the action [f]a = f(a). The obstruction of an equivariant kernel (Π, G, p),
denoted by Obs(p), is an element of the group H
3
Γ
(Π, ZG).
The following theorem describes the invariants of the graded categorical
group Hol
Γ
G and shows a relation between the third invariant of Hol
Γ
G and
Obs(p).
5.2.1 Theorem. Let (Π, G, p) be an equivariant kernel. Then the invariants
of the strict graded categorical group Hol
Γ
G are:
(i) π
0
(Hol
Γ
G) = Out G, π
1
(Hol
Γ

G) = ZG;
(ii) h ∈ Z
3
Γ
(Out G, ZG) sao cho p
∗
h ∈ Obs(p).
23
5.3. Classification of central extensions of equivariant groups
In this section, we classify Γ-equivariant group extensions A  E  Π
which have A ⊂ ZE due to graded monoidal auto-functors of the Γ-graded
categorical group

Γ
(Π, A, 0). First, we denote by Ext
c
Γ
(Π, A) the set of equiv-
alence classes of Γ-equivariant group extensions of A by Π which are central
extensions.
5.3.1 Theorem. (Schreier Theory for central extensions of equivariant groups)
Let Π be a Γ-group and A be a Γ-equivariant Π-module. There is a bijection
Ext
c
Γ
(Π, A) ↔ End
id
Γ



Γ
(Π, A, 0)

,
where End
id
Γ


Γ
(Π, A, 0)

is the set of homotopy classes of graded monoidal
functors (F,

F ) from

Γ
(Π, A, 0) to itself satisfying
F (x) = x, x ∈ Π, F (b, 1) = (b, 1), b ∈ A.
5.4. Composition of a Γ-categorical group with a Γ-homomorphism
According to the reference “Homology” of S. MacLane (1963), for each
homomorphism γ : Π

→ Π and extension
E : 0 → A
i
−→ B
q
−→ Π → 1,

where A is an abelian group, there exists an extension E

of A by Π

such
that E

= Eγ. In the following theorem, we consider the similar problem to
a case of strict graded categorical groups.
5.4.1 Theorem. Let H be a strict graded categorical group with three invari-
ants Π, C, h and let p : Π

→ Π be an equivariant homomorphism. Then
there exists a strict graded categorical group G which is equivalent to the
graded categorical group

Γ
(Π

, C, h

), where C is a Π

-module under the ac-
tion xc = p(x)c, x ∈ Π

, c ∈ C and h

= p
∗

h.