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, [...]... result on the classification of Γ-module extensions; - Construct a categorical equivalence between the category of braided crossed modules and the category of braided strict categorical groups; Construct a categorical equivalence between the category of abelian crossed modules and the category of strict Picard categories; - Give the notion of braided strict Γ- graded categorical groups and construct a categorical. .. 1) = (f1 (b), 1), F (x → σx) = (ϕ(px, σ), σ) and Fx,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 extensions 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. .. equivariant group extensions which are central extensions 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... introducing braided strict 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 categorical groups The following definition introduces the notion of braided. .. a categorical equivalence between the category of braided strict graded categorical groups and the category of braided Γ-crossed modules; - State and treat the problem of abelian extensions of the type of an abelian crossed module; - Apply strict graded categorical groups to the classification of equivariant group extensions which are central ones LIST OF THE AUTHOR’S ARTICLES RELATED TO THE THESIS... 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... 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... 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 Extc (Π, A) the set of equivΓ 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... 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;... 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 . 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. 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, . 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