on graded extensions of braided categorical groups

Objective of the research The objective of the thesis is to study the structure of categorical gebras such as: graded Picard categories, strict graded categorical groups,braided strict g

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

Supervisor:

1 Assoc Prof Dr Nguyen Tien Quang

2 Assoc Prof Dr Ngo Sy Tung

Thesis can be found at:

1 Nguyen Thuc Hao Library and Information Center - Vinh University

2 Vietnam National Library

1 Rationale

The theory of categories with tensor products was studied by J Bnabou(1963) and S MacLane (1963) They considered categories equipped with atensor 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 conditionsfor coherence of natural isomorphisms in symmetric monoidal categories, i e.monoidal categories have a commutativity constraint c which is compatiblewith 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 areall isomorphisms), we obtain group-like monoidal categories (see A Fr¨ohlichand C T C Wall (1974)), or Gr-categories (see H X Sinh (1975)) In thisthesis, we say these categories to be categorical groups according to recentlypopular documents (see P Carrasco and A R Garzn (2004), A M Cegarra

et al (2002)) In a case where categorical groups have a commutativityconstraint then they become Picard categories (see H X Sinh (1975)), orsymmetric 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 ifthe objects are all invertible and the morphisms are all isomorphisms Theyalso classified braided monoidal categories by quadratic functions (thanks tothe result of S Eilenberg and S Mac Lane on representations of quadraticfunctions by the abelian cohomology group Hab3 (G, A)) Before that, symmet-

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 duced by A Fr¨ohlich and C T C Wall (1974) Then A M Cegarra and

intro-E Khmaladze (2007) studied braided graded categorical groups and gradedPicard categories These structures are genneral cases of braided categoricalgroups and Picard categories, respectively They obtained the classificationresults due to the cohomology theory of Γ-modules constructed by themselves.According to a different research direction, some authors was interested inthe class of special categorical groups in which the constraints are all identitiesand objects are all strict invertible, that is X ⊗ Y = I = Y ⊗ X Thesecategories 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 thatthe 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, butthe category of G-groupoids is just a subcategory of the category of strictcategorical groups N T Quang et al (2014) showed a relation betweenthe second category and the category of crossed modules, in which Brown-Spencer equivalence is just a special case This result leads to applying theresults on the obstruction theory for functors and the cohomology theory tostudy crossed modules Furthermore, this approach leads to linking sometypes of crossed modules with appropriate categorical algebras as well as weshall present in Chapter 3 and Chapter 4

The idea of R Brown and C B Spencer was developed for braided crossedmodules and braided strict categorical groups by A Joyal and R Street

(1993) But A Joyal and R Street just stopped in mutual determining tween these two structures A problem is whether or not an Brown-Spencerequivalence 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-crossedmodules (see B Noohi (2011)) According to N T Quang et al (2014), wehope that we can connect these types of crossed modules with correspondingcategorical algebras, and obtain categorical equivalences for these subjects.According to a different approach, crossed modules have a close relationwith the problem of group extensions The problem of group extensions of thetype of a crossed module was introduced by P Dedecker (1964), and treated

be-by R Brown and O Mucuk (1994) Therefore, we think that we can study theproblem of group extensions of the type of certain a crossed module amongtypes of crossed modules which are mentioned

For the above reasons, we have chosen the topic for the thesis that is: “Ongraded extensions of braided categorical groups”

2 Objective of the research

The objective of the thesis is to study the structure of categorical 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, abeliancrossed modules and present Schreier theory for Γ-module extensions, abelianextensions of the type of an abelian crossed module, Γ-module extensions ofthe type of an abelian Γ-crossed module and central extensions of equivariantgroups

al-3 Subject of the research

Braided categorical groups, braided graded categorical groups, types ofcrossed modules and the group extension problem of the type of a crossedmodule

4 Scope of the research

The thesis studies the strict and symmetric properties in braided ical groups and braided graded categorical groups to classify types of crossedmodules and treat group extension problems of the type of a certain crossedmodule

categor-5 Methodology of the research

We use theoretical research method during the thesis Technically, we usethe 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 internationalmagazines Therefore, they have scientific significance and contribution ofmaterial 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 homology groups of Γ-modules HΓ,sn (M, N ) Then they applied the 2nd and3rd dimension cohomology groups to classify Γ-module extensions and gradedPicard categories, respectively

co-The first content of the thesis is to study graded Picard categories by themethod 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 card category of type (π0P, π1P), and show that each above factor set induces

Pi-a Γ-module structure on Pi-abeliPi-an groups π0P, π1P and induces a normalized3-cocycle h ∈ ZΓ,s3 (π0P, π1P) As an application of the theory of gradedPicard categories, we classify Γ-module extensions due to symmetric graded

monoidal functors Thanks to these results, we obtained the classification

of graded Picard categories and the cohomology classification of Γ-moduleextensions 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 whichare more refined than crossed modules In 2004, from the notion of crossedmodules, P Carrasco et al (2002) considered a case where groups have thecommutative property and gived the notion of abelian crossed modules Theyproved 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 equippedwith an Γ-action on groups and group homomorphisms in the notion of crossedmodules, 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, inthis paper, the author did not mention the classification of these types of crossmodules In 2013, N T Quang and P T Cuc constructed strict graded cat-egorical groups used to classify Γ-crossed modules and the equivariant groupextension problem of the type of a Γ-crossed module This extension is ageneralization 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(f1, f0) : M → M0 of braided crossed modules and an element of the group

of abelian 2-cocycles Zab2 (π0M, π1M0) Then we prove that the category ofbraided crossed modules is equivalent to the category of braided strict cat-egorical groups Morphisms in the latter are symmetric monoidal functors(F,F ) :e P → P0 which preserve a tensor operation and Fex,y = Fey,x for all

x, y ∈ Ob(P) If braided crossed modules are abelian crossed modules, thenstrict categorical groups are strict Picard categories Then we establish a cat-egorical equivalence between the category of abelian crossed modules and thecategory of strict Picard categories, and treat the abelian extension problem

of the type of an abelian crossed module.

The third content of the thesis is to introduce braided strict graded gorical groups associated to braided Γ-crossed modules Then we study a re-lation between homomorphisms of braided Γ-crossed modules and symmetricgraded monoidal functors of braided strict graded categorical groups associ-ated to braided Γ-crossed modules This leads to a categorical equivalencebetween the category of braided Γ-crossed modules and one of braided strictgraded categorical groups We also treat the Γ-module extension problem ofthe type of an abelian Γ-crossed module

cate-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 categoricalgroup HolΓG 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 thegraded categorical group RΓ(Π, A, 0) Besides, we construct a strict Γ-gradedcategorical group which is the composition of a strict graded categorical groupand 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 groupextension E and a homomorphism γ

7.2 The organization of the research

Besides the sections of preface, general conclusions, list of the author’s ticles related to the thesis, the thesis is organized into five chapters Chapter

ar-1 presents the basic knowledge used in the next chapters Chapter 2 studiesgraded Picard categories by the method of factor sets Chapter 3 studiesbraided strict categorical groups used to classify braided crossed modules,abelian crossed modules and abelian extensions of the type of an abeliancrossed module Chapter 4 constructs braided strict graded categorical groupsused to classify braided Γ-crossed modules and Γ-module extensions of thetype of an abelian Γ-crossed module Chapter 5 studies strict graded cate-gorical groups related to the problem of equivariant group extensions

CHAPTER 1PRELIMINARIES

In this chapter, we present some notions and basic results concerning tomonoidal categories, braided categorical groups, Picard categories, gradedcategorical groups, cohomology of Γ-modules, braided graded categorical groupsand graded Picard categories This basic knowledge will be used in the nextchapters

Section 1.1 recalls the notions about monoidal categories, monoidal tors, homotopies and categorical groups

func-Section 1.2 recalls the notions about braided categorical groups, Picardcategories, 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, gradedcategorical 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 monoidalcategories, braided (symmetric) graded categorical groups, symmetric gradedmonoidal functors and braided graded categorical groups of type (M, N, h).The last part of the section will present two results on the obstruction ofgraded functors of type (ϕ, f ) to become symmetric graded monoidal functors

CHAPTER 2FACTOR SETS IN GRADED PICARD CATEGORIES

In this chapter, we describe symmetric factor sets on Γ with coefficients inPicard categories in order to interpret the symmetric cohomology group HΓ,s3

of Γ-modules and, classify Γ-module extensions thanks to symmetric Γ-gradedmonoidal 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 monoidalfunctors between them and by Z3s a full subcategory of the category Pic,which is defined as follows A object of Z3s is a Picard categoryP = R(M, N, h),where M, N are abelian groups and h = (ξ, η) ∈ Zs3(M, N ) with ξ : M3 → N ,

η : M2 → N A morphism R(M, N, h) → R(M0, N0, h0) is a ric monoidal functor (F,F ), where F is a pair of group homomorphismse

symmet-ϕ : M → M0 and f : N → N0, F is associated to a function g : Me 2 → N0such that f∗(h) = ϕ∗(h0) + ∂g ∈ Zs3(M, N0)

2.1.1 Definition A symmetric factor set on Γ with coefficients in a Picardcategory P (or a pseudo-functor F : Γ → Pic) consists of a family of sym-metric monoidal auto-equivalences Fσ : P → P, σ ∈ Γ, and isomorphismsbetween symmetric monoidal functors θσ,τ : FσFτ → Fστ, σ, τ ∈ Γ, satisfy-ing the conditions:

i) F1 = idP,

ii) θ1,σ = idFσ = θσ,1, σ ∈ Γ,

iii) for all σ, τ, γ ∈ Γ, the following diagram commutes

We write F = (P, Fσ, θσ,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 thereduced 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 R (M, N, h)The following theorem shows necessary conditions of a factor set withcoefficients in a Picard category

2.2.1 Theorem Let Γ be a group and S = R(M, N, ξ, η) be a Picard gory Then

cate-(i) Each almost strict factor set F = (S, Fσ, θσ,τ) : Γ → Z3s induces aΓ-module structure on M, N and a normalized 3-cocycle hF ∈ ZΓ,s3 (M, N );(ii) In the definition of factor sets, the condition F1 = idS can be deducedfrom 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 = π0P, N = π1P and a normalized 3-cocycle hF ∈ ZΓ,s3 (M, N )

In the following definition, we introduce the notion of Γ-graded Picard egories whose pre-sticks are of type (M, N ) Then the classification problem

cat-of Γ-graded Picard categories will be treated on Γ-graded Picard categorieswhose pre-sticks are of type (M, N )

2.2.8 Definition Let M and N be Γ-modules We say that a Γ-gradedPicard category P has a pre-stick of type (M, N ) if there exists a pair ofisomorphisms of Γ-modules

2.3 Γ-module extensions

This section is dedicated to presenting the classification of Γ-module sions due to symmetric Γ-graded monoidal functors of two Γ-graded Picardcategories DisΓM and RedΓN

exten-2.3.1 Definition A Γ-module extension of N by M is a short exact sequence

of Γ-modules and Γ-homomorphisms

Γ The grading gr : DisΓM → Γ is given by gr(σ) = σ The graded tensorproduct on objects is the operation in M , and on morphisms is given by

(x −→ y) ⊗ (xσ 0 σ−→ y0) = (x + x0 σ−→ y + y0)

The graded unit functor I : Γ → DisΓM is given by

I(∗−→ ∗) = (0σ −→ 0).σThe associativity, commutativity and unit isomorphisms are identities

The reduced Γ-graded Picard category RedΓN is given by