Luận án Hệ nhân tử trong nhóm phạm trù phân bậc

a Mac Lane ®· ®­îc N. T. Quang vµ D. D. Hanh sö dông ®Ó ph©n líp c¸c Ann-hµm tö gi÷a c¸c Ann-ph¹m trï (2009), vµ míi ®©y nhãm H3MacL(R,M) ®· ®­îc N. T. Quang sö dông ®Ó ph©n líp c¸c Ann-ph¹m trï tæng qu¸t (2013). 5.2 Song m«®un chÐo vµ E-hÖ chÝnh quy §Þnh nghÜa. i) Mét song m«®un chÐo lµ mét bé ba (B,D, d), trong ®ã D lµ K-®¹i sè kÕt hîp, B lµ D-song m«®un vµ d : B → D lµ ®ång cÊu cña c¸c D-song m«®un sao cho d(b)b′ = bd(b′), b, b′ ∈ B. ii) Mét ®ång cÊu (k1, k0) : (B,D, d)→ (B′, D′, d′) gi÷a hai song m«®un chÐo bao gåm cÆp ®ång cÊu k1 : B → B′, k0 : D → D′, trong ®ã k1 lµ ®ång cÊu nhãm, k0 lµ ®ång cÊu K-®¹i sè sao cho víi mäi x ∈ D, b ∈ B th× k0d = d ′k1, 19 k1(xb) = k0(x)k1(b), k1(bx) = k1(b)k0(x). Khi vµnh c¬ së K ®­îc lÊy lµ vµnh c¸c sè nguyªn Z th× song m«®un chÐo (B,D, d) ®­îc gäi lµ song m«®un chÐo trªn vµnh. Kh¸i niÖm E-hÖ mµ chóng t«i nªu ra d­íi ®©y ®­îc xem nh­ mét phiªn b¶n cña m«®un chÐo cho c¸c vµnh. §Þnh nghÜa. Mét E-hÖ lµ mét bé bènM = (B,D, d, θ) trong ®ã d : B → D, θ : D → MB lµ c¸c ®ång cÊu vµnh tháa m·n c¸c ®iÒu kiÖn sau víi mäi x ∈ D, b ∈ B: θ ◦ d = µ, d(θxb) = x.d(b), d(bθx) = d(b).x. E-hÖ (B,D, d, θ) ®­îc gäi lµ chÝnh quy nÕu θ lµ 1-®ång cÊu (®ång cÊu biÕn ®¬n vÞ thµnh ®¬n vÞ) vµ c¸c phÇn tö thuéc θ(D) lµ giao ho¸n. Mét ®ång cÊu (f1, f0) : (B,D, d, θ) → (B′, D′, d′, θ′) gi÷a hai E-hÖ bao gåm c¸c ®ång cÊu vµnh f1 : B → B′, f0 : D → D′ sao cho: f0d = d ′f1, f1(θxb) = θ ′ f0(x) f1(b), f1(bθx) = f1(b)θ ′ f0(x) . §Þnh lý d­íi ®©y chøng tá kh¸i niÖm E-hÖ ®­îc xem nh­ lµ sù lµm yÕu cña kh¸i niÖm song m«®un chÐo trªn vµnh. §Þnh lý 5.2. Ph¹m trï c¸c E-hÖ chÝnh quy vµ ph¹m trï c¸c song m«®un chÐo trªn vµnh lµ ®¼ng cÊu. 5.3 Ph©n líp c¸c E-hÖ chÝnh quy Tõ mçi E-hÖ chÝnh quy B → D cho tr­íc, chóng t«i x©y dùng ®­îc mét Ann-ph¹m trï chÆt chÏ AB→D gäi lµ Ann-ph¹m trï liªn kÕt víi E-hÖ chÝnh quy, vµ ng­îc l¹i. C¸c bæ ®Ò d­íi ®©y nãi lªn mèi liªn hÖ gi÷a c¸c ®ång cÊu E-hÖ chÝnh quy vµ c¸c Ann-hµm tö gi÷a c¸c Ann-ph¹m trï liªn kÕt. Bæ ®Ò 5.3. Cho ®ång cÊu (f1, f0) : (B,D, d, θ)→ (B′, D′, d′, θ′) gi÷a c¸c E-hÖ chÝnh quy. Khi ®ã: i) Tån t¹i mét hµm tö F : AB→D → AB′→D′ x¸c ®Þnh bëi F (x) = f0(x), F (b) = f1(b), x ∈ ObA, b ∈ MorA. 20 ii) C¸c ®¼ng cÊu tù nhiªn F˘x,y : F (x+ y)→ Fx+Fy, F˜x,y : F (xy)→ FxFy cïng víi F lµ mét Ann-hµm tö nÕu F˘ vµ F˜ lµ c¸c h»ng thuéc Ker d′ sao cho víi mäi x, y ∈ D, θ′Fx(F˜ ) = (F˜ )θ ′ Fy = F˜ , θ′Fx(F˘ ) = (F˘ )θ ′ Fy = F˘ + F˜ . Khi ®ã, ta nãi r»ng F lµ mét Ann-hµm tö d¹ng (f1, f0). Ann-hµm tö (F, F˘ , F˜ ) ®­îc gäi lµ ®¬n nÕu F (0) = 0, F (1) = 1 vµ F˘ , F˜ lµ nh÷ng h»ng. Víi kh¸i niÖm nµy chóng t«i ph¸t biÓu mÖnh ®Ò ®¶o cña mÖnh ®Ò trªn. Bæ ®Ò 5.4. Cho Ann-hµm tö ®¬n (F, F˘ , F˜ ) : AB→D → AB′→D′. Khi ®ã cã mét ®ång cÊu (f1, f0) : (B → D) → (B′ → D′) gi÷a c¸c E-hÖ chÝnh quy, ®­îc x¸c ®Þnh bëi f1(b) = F (b), f0(x) = F (x), víi mçi b ∈ B, x ∈ D. Bæ ®Ò 5.5. Hai Ann-hµm tö cïng d¹ng (F, F˘ , F˜ ), (F ′, F˘ ′, F˜ ′) : AB→D → AB′→D′ lµ ®ång lu©n. Hai Ann-hµm tö (F, F˘ , F˜ ), (F ′, F˘ ′, F˜ ′) ®­îc gäi lµ ®ång lu©n m¹nh nÕu chóng ®ång lu©n vµ F = F ′. HÖ qu¶ 5.6. Hai Ann-hµm tö F, F ′ : AB→D → AB′→D′ lµ ®ång lu©n m¹nh khi vµ chØ khi chóng cïng d¹ng. Ký hiÖu ph¹m trï cña c¸c Ann-ph¹m trï chÆt chÏ vµ c¸c Ann-hµm tö ®¬n bëiAnnstr. Ta ®Þnh nghÜa ph¹m trï ®ång lu©n m¹nhHoAnnstr cñaAnnstr lµ mét ph¹m trï th­¬ng víi cïng c¸c vËt nh­ng c¸c mòi tªn lµ c¸c líp ®ång lu©n m¹nh cña c¸c Ann-hµm tö ®¬n: HomHoAnnstr(A,A′) = HomAnnstr(A,A′) ®ång lu©n m¹nh . Ký hiÖu ESyst lµ ph¹m trï cã c¸c vËt lµ c¸c E-hÖ chÝnh quy vµ mòi tªn lµ c¸c ®ång cÊu E-hÖ chÝnh quy, ta cã kÕt qu¶ sau. §Þnh lý 5.7. [§Þnh lý ph©n líp] Tån t¹i mét t­¬ng ®­¬ng ph¹m trï Φ : ESyst → HoAnnstr, (B → D) 7→ AB→D (f1, f0) 7→ [F ] trong ®ã F (x) = f0(x), F (b) = f1(b), víi x ∈ ObA, b ∈ MorA. 21 5.4 Më réng vµnh kiÓu E-hÖ chÝnh quy §Þnh nghÜa. Cho E-hÖ chÝnh quy (B,D, d, θ). Mét më réng cña vµnh B bëi vµnh Q kiÓu E-hÖ B → D lµ mét biÓu ®å c¸c ®ång cÊu vµnh 0 // B j // E p // ε  Q // 0, B d // D trong ®ã dßng trªn lµ khíp, hÖ (B,E, j, θ0) lµ mét E-hÖ chÝnh quy víi θ0 lµ phÐp lÊy song tÝch, cÆp (id, ε) lµ mét ®ång cÊu cña c¸c E-hÖ chÝnh quy. Mçi më réng nh­ vËy c¶m sinh mét ®ång cÊu vµnh ψ : Q→ Coker d. Môc tiªu cña chóng t«i lµ sö dông lý thuyÕt c¶n trë cña c¸c Ann-hµm tö ®Ó nghiªn cøu tËp ExtB→D(Q,B, ψ) c¸c líp t­¬ng ®­¬ng c¸c më réng cña B bëi Q kiÓu E-hÖ chÝnh quy B → D c¶m sinh ψ. Bæ ®Ò d­íi ®©y cho thÊy c¸c Ann-hµm tö DisQ→ AB→D lµ hÖ d÷ liÖu phï hîp ®Ó x©y dùng c¸c më réng nh­ vËy. Bæ ®Ò 5.8. Cho E-hÖ chÝnh quy (B,D, d, θ) vµ ®ång cÊu vµnh ψ : Q→ Coker d. Víi mçi Ann-hµm tö (F, F˘ , F˜ ) : DisQ → A c¶m sinh cÆp (ψ, 0) ®Òu tån t¹i mét më réng EF cña B bëi Q kiÓu E-hÖ B → D c¶m sinh ®ång cÊu ψ : Q→ Coker d. §Þnh lý d­íi ®©y lµ phiªn b¶n cña lý thuyÕt Schreier cho c¸c më réng vµnh kiÓu E-hÖ chÝnh quy. §Þnh lý 5.9. [Lý thuyÕt Schreier cho c¸c më réng vµnh kiÓu E-hÖ chÝnh quy] Cã mét song ¸nh Ω : HomAnn(ψ,0)[DisQ,A]→ ExtB→D(Q,B, ψ). Do Ann-ph¹m trï thu gän cña Ann-ph¹m trï liªn kÕt AB→D cã d¹ng SA = (Cokerd,Kerd, k), trong ®ã k ∈ H3Shu(Cokerd,Kerd), nªn ®ång cÊu ψ : Q→ Cokerd c¶m sinh mét c¶n trë ψ∗k ∈ H3Shu(Q,Kerd). §Þnh lý 5.10. Cho E-hÖ chÝnh quy (B,D, d, θ) vµ ®ång cÊu vµnh ψ : Q → Cokerd. Khi ®ã sù triÖt tiªu cña ψ∗k trong H3Shu(Q,Kerd) lµ ®iÒu kiÖn cÇn vµ ®ñ ®Ó tån t¹i më réng vµnh cña B bëi Q, kiÓu E-hÖ B → D c¶m sinh ψ. H¬n n÷a, khi ψ∗k triÖt tiªu th× tån t¹i mét song ¸nh: ExtB→D(Q,B, ψ)↔ H2Shu(Q,Kerd). 22 KÕt luËn M«®un chÐo vµ nhãm ph¹m trï, mét c¸ch ®éc lËp, ®· ®­îc sö dông réng r·i trong nhiÒu khung c¶nh kh¸c nhau. C¸c kÕt qu¶ vÒ nhãm ph¹m trï cña H. X. SÝnh (1975) ®· ®­îc n©ng lªn cho nhãm ph¹m trï ph©n bËc bëi A. M. Cegarra vµ c¸c céng sù, vµ cho tr­êng hîp vµnh ph¹m trï (hay Ann-ph¹m trï) bëi N. T. Quang. Bªn c¹nh ®ã, R. Brown vµ C. Spencer (1976) ®· chØ ra r»ng m«®un chÐo cã thÓ ®­îc nghiªn cøu bëi c¸c nhãm ph¹m trï chÆt chÏ. §iÒu nµy gîi ý cho chóng t«i r»ng cã thÓ nghiªn cøu c¸c líp ph¹m trï phøc t¹p h¬n nh­: nhãm ph¹m trï ph©n bËc chÆt chÏ, Ann-ph¹m trï chÆt chÏ, ®Ó tõ ®ã nghiªn cøu c¸c cÊu tróc gÇn víi m«®un chÐo nh­: m«®un chÐo ®¼ng biÕn, E-hÖ. LuËn ¸n ®· gi¶i quyÕt vÊn ®Ò nµy víi nh÷ng kÕt qu¶ chÝnh nh­ sau: 1. X¸c ®Þnh kiÓu cña mét hµm tö monoidal gi÷a hai nhãm ph¹m trï vµ lý thuyÕt c¶n trë cña mét hµm tö. Tõ ®ã ®­a ra ®Þnh lý ph©n líp chÝnh x¸c cho ph¹m trï c¸c nhãm ph¹m trï vµ ph¹m trï c¸c nhãm ph¹m trï bÖn. 2. Ph©n líp c¸c m«®un chÐo vµ x©y dùng lý thuyÕt Schreier cho c¸c më réng nhãm kiÓu m«®un chÐo dùa trªn c¸c kÕt qu¶ cña lý thuyÕt nhãm ph¹m trï chÆt chÏ. C¸c kÕt qu¶ thu ®­îc lµ më réng c¸c kÕt qu¶ cña R. Brown vµ c¸c ®ång t¸c gi¶. 3. Nghiªn cøu nhãm ph¹m trï ph©n bËc chÆt chÏ, vµ tõ ®ã ph©n líp c¸c Γ-m«®un chÐo vµ x©y dùng lý thuyÕt Schreier cho c¸c më réng nhãm ®¼ng biÕn kiÓu Γ-m«®un chÐo. C¸c kÕt qu¶ thu ®­îc lµ phñ lªn lý thuyÕt më réng nhãm ®¼ng biÕn cña A. M. Cegarra - J. M. GarcÝa-Calcines - J. A. Ortega vµ lý thuyÕt më réng nhãm kiÓu m«®un chÐo cña R. Brown - O. Mucuk. 4. Nghiªn cøu Ann-ph¹m trï chÆt chÏ, tõ ®ã ph©n líp c¸c E-hÖ chÝnh quy vµ c¸c më réng vµnh kiÓu E-hÖ chÝnh quy. 5. Ph©n líp ph¹m trï c¸c nhãm ph¹m trï bÖn Γ-ph©n bËc nhê c¸c hÖ nh©n tö trªn Γ víi hÖ tö trong nhãm ph¹m trï bÖn kiÓu (M,N). 23 Danhmôcc¸cc«ngtr×nhcñat¸cgi¶cãliªnquan®Õn luËn ¸n 1. N. T. Quang, N. T. Thuy, P. T. Cuc, Monoidal functors between (braided) Gr-categories and their applications, East-West J. of Mathematics, 13, No 2 (2011), 163-186. 2. N. T. Quang, P. T. Cuc, Crossed bimodules over rings and Shukla cohomol- ogy, Math. Commun., 17 No. 2 (2012), 575-598. 3. N. T. Quang, P. T. Cuc, Classification of graded braided categorical groups by pseudo-functors, Journal of Science, Hue University, Vol. 77, No. 8 (2012), 59-68. 4. N. T. Quang, P. T. Cuc, N. T. Thuy, Strict Gr-categories and crossed modules, Communications of Korean Mathematical Society, Vol. 29, No.1 (2014), 9-22. 5. N. T. Quang, P. T. Cuc, Equivariant crossed bimodules and cohomology of groups with operators, arXiv: 1302.4573v1 [math.CT] 19 Feb 2013. C¸c kÕt qu¶ trong luËn ¸n ®· ®­îc b¸o c¸o vµ th¶o luËn t¹i: - Héi th¶o khoa häc vÒ Mét sè h­íng nghiªn cøu míi trong to¸n häc hiÖn ®¹i vµ øng dông, Thanh Hãa (do Tr­êng §¹i häc Hång §øc, ViÖn To¸n häc ViÖt Nam vµ Tr­êng §¹i häc s­ ph¹m Hµ Néi phèi hîp tæ chøc), 5/2011. - Héi nghÞ toµn quèc vÒ §¹i sè - H×nh häc - T«p«, Th¸i Nguyªn, 11/2011. - §¹i héi To¸n häc ViÖt Nam lÇn thø VIII, Nha Trang, 8/2013. - Xeminar Bé m«n §¹i sè - H×nh häc, Khoa To¸n, Tr­êng §¹i häc s­ ph¹m, §¹i häc HuÕ. - Xeminar Bé m«n §¹i sè, Khoa Khoa häc tù nhiªn, Tr­êng §¹i häc Hång §øc, Thanh Hãa. 24 hue university college of education pham thi cuc factor sets in graded categorical groups Major: Algebra and Number Theory Code: 62. 46. 05. 01 Doctoral dissertation on mathematics Hue - 2014 This research project done at: College of Education, Hue University Scientific advisors: 1. Assoc. Prof. Dr. Nguyen Tien Quang 2. Prof. Dr. Le Van Thuyet Reviewer 1: Reviewer 2: Reviewer 3: The dissertation has been defended under the assessment of Hue University Doctoral Assessment Committee: On: ... , ... ... ... 2014 This dissertation is accessible at: - Centre for Academy, Hue University - College of Education Library, Hue University Introduction The notion of monoidal category (or tensor category) was introduced by J. Be´nabou, S. Mac Lane, G. M. Kelly, ... in the early 60’s of last century, then it was studied by many researchers and has developed quite past. Monoidal categories can be “refined” to become categories with group structure as they are added invertible objects. Then, if the underlying cat- egory is a groupoid, (that is, every morphism is an isomorphism) we have the notion of (symmetric) categorical groups. In the case that categorical groups have a commutativity constraint one obtains the notion of symmet- ric categorical group (or Picard category). The first authors studying categorical groups are N. Saavedra Rivano, H. X. Sinh, M. L. Laplaza, ... In her thesis in 1975, H. X. Sinh described the structure of categorical groups and Picard categories, and classified them by the means of the third cohomology group of groups. This result determined the relation among the categorical group theory, cohomology of groups and the classical problem of group extensions of Schreier - Eilenberg - Mac Lane. Then, theory of categorical groups, with its generality has more and more applications. Γ-graded categorical groups were first introduced by A. Frohlich and C. T. C. Wall (1974). In 2002, A. M. Cegarra and his co-authors proved the precise theorems on the homotopy classification of graded categorical groups and their homomorphisms by the third equivariant cohomology group. Then, they are applied to show a treatment of the equivariant group extensions problem with a non-abelian kernel. Braided categorical groups were first considered by A. Joyal and R. Street (1993) as extensions of Picard categories, in which braided categori- cal groups are classified by the abelian cohomology group H3ab(M,N). The problem of homotopy classification of graded braided categorical groups and of the particular case, graded Picard categories, was completely solved by A. M. Cegarra and E. Khmaladze in 2007. In 2010, N. T. Quang introduced a new approach for the problem of classification of Γ-graded braided categorical groups based on the factor set method (or pseudo-functor method in the sense of A. Grothendieck). This 1 method can be applied effectively for Γ-graded braided categorical groups. While categorical groups are regarded as a version of the group struc- ture, then in 1988 N. T. Quang introduced the notion of Ann-category, which is a categorification of rings, here objects and morphisms of the underlying category are required to be invertible. In particular, regular Ann-categories (its symmetric constraint satisfies the condition cX,X = id for all object X) are classified by N. T. Quang thanks to the cohomology group of associative algebrasH3Shu(R,M) in the sense of Shukla. After that, the classifying problem of Ann-functors was solved by N. T. Quang and D. D. Hanh (2009) thanks to the low-dimensional cohomology groups of rings in the sense of Mac Lane, and they also showed the relation between the ring extension problem and the obstruction theory of Ann-functors. Recently (2013), the classifying problem of Ann-categories in the general case is completely dealt with by N. T. Quang. Crossed modules of groups were introduced in 1949 by J. H. C. White- head in his work on representations of homotopy 2-types without the aid of the category theory. In 1976, R. Brown and C. Spencer showed that every crossed module is defined by a G-groupoid, (that is, a strict categor- ical group) and vice versa, and hence crossed modules can be studied by means of category theory. This determined a relation between the category theory and crossed modules, a basic concept of algebraic topology. The problem of group extensions of the type of a crossed module, a generalized version of classical group extension, introduced by P. Dedeker in 1964, was solved by R. Brown and O. Mucuk (1994), in which they give an exposition of a proof of this theorem using the methods of crossed complexes analogous to those of chain complexes as in standard homo- logical algebra. An another generalized version of group extension is the equivariant group extension stated by Cegarra et al thanks to the graded categorical group theory. Crossed modules of groups of J. H. C. Whitehead (1949) were gener- alized in many different ways. In 2002, H. -J. Baues introduced crossed modules over k-algebras (k is a field). Then, H. -J. Baues and T. Pirashvili (2004) replaced the field k with a commutative ring K and called crossed modules over K-algebras crossed bimodules. In particular, when K = Z we 2 obtain the notion of crossed bimodule over rings. Crossed modules over groups can be defined over rings in a different way under the name of E-systems. Their particular case, regular E-systems, are just crossed bimodules over rings, hence the notion of an E-system is weaker than that of a crossed bimodule over rings. Analogous to crossed modules over groups, we represent regular E-systems in the form of strict Ann- categories, and then classify the category of regular E-systems. Moreover, we introduce and solve the ring extension problem of the type of a regular E-system, which can be seen as an application of the notion of E-system and the Ann-category theory. A different version of crossed module over groups is the notion of Γ- equivariant crossed module (or Γ-crossed module). This notion was intro- duced by B. Noohi in 2011 to compare the different ways of defining group cohomology with coefficients in a crossed module. Having this notion, we introduce the notion of strict categorical group to represent Γ-crossed module, state and solve the equivariant group problem of the type of a Γ-crossed module. Behind the introductory and conclusion sections, the thesis is organized into 5 chapters, as follows. Chapter 1, Preliminaries, contains the notions and well-known results of the theory of categories with structures, which are used in the next chapters. Chapter 2, Classification of monoidal functors of type (ϕ, f) and ap- plications, contains the following contents. Firstly, we describe monoidal functors between categorical groups of type (Π, A), state the obstruction theory and the classification theorem of these functors. Thus, the theo- rems of classifying of categorical groups and braided categorical groups are stated, and we introduce an algebraic application of the obstruction theory of monoidal functors concerning to the group extension problem. Also, we prove the classification theorem of braided categorical groups by means of factor sets. Chapter 3, Strict categorical groups and group extensions of the type of a crossed module, states the relation among crossed modules, strict cate- gorical groups and the problem of group extensions of the type of a crossed 3 module. We show the relation between homomorphisms of crossed mod- ules and monoidal functors of strict categorical groups associated to those crossed modules, hence we obtain the classification theorem of crossed mod- ules which extends the well-known result of R. Brown and C. Spencer. We also obtain the result of the group extensions problem of the type of a crossed module thanks to the theory of strict categorical groups. Chapter 4, Strict graded categorical groups and equivariant group ex- tensions of the type of a Γ-crossed module, states the theory of equivariant group extensions of the type of a Γ-crossed module, which generalizes both the theory of group extensions of the type of a crossed module of R. Brown and the one of equivariant group extensions A. M. Cegarra. We also rep- resent Γ-crossed modules in the form of strict graded categorical groups in order to classify Γ-crossed modules. Chapter 5, Strict Ann-categories and ring extensions of the type of a regular E-system, is devoting to state E-systems, as well as their relation with the well-known notions and to seek applications concerning to ex- tension problem. We introduce the notion of E-systems and of regular E-systems as a ring version of crossed modules of groups, in which regular E-systems are represented in the form of strict Ann-categories, moreover regular E-systems are just crossed bimodules over rings. We introduce and solve the ring extension problem of the type of a regular E-system, which is regarded as an application of the notion of E-system and the Ann-category theory. The numbering of chapters, sections, theorems, propositions, ... in this summary is the same as in the thesis. The results of the thesis are stated in 5 articles consisting 4 articles published (where 3 are on international journals listed in MathSciNet) and one preprint article. 4 Chapter 1 Preliminaries In this chapter, we recall some notions and basic results concerning to categorical groups, graded categorical groups, graded braided categorical groups, and Ann-categories. 1.1 Graded (braided) categorical groups 1.1.1 Categorical groups A categorical group (G,⊗, I, a, l, r) is a monoidal category in which every object is invertible (in the sense that for each object X there exists an object Y such that X ⊗ Y ' I ' Y ⊗X) and the underlying category is a groupoid, that is, every morphism is an isomorphism. If for each object X there exists an object Y such that X ⊗ Y = I = Y ⊗X and the associativity a, unit l, r constraints are identities, then G is termed a strict categorical group. 1.1.2 Reduced categorical groups and canonical equivalences Each categorical group G determines three invariants: a group Π, a left Π-module A and a 3-cocycle k ∈ Z3(Π, A). Then, one constructs a categorical group SG which is monoidally equivalent to G thanks to canonical equivalences, and SG is called a reduction of the categorical group G. It is said that SG is of type (Π, A, k), or simply of type (Π, A). 1.1.3 Graded categorical groups A Γ-graded monoidal category G = (G, gr,⊗, I, a, l, r) consists of: i) a stable Γ-graded category (G, gr), Γ-graded functors⊗ : G×ΓG→ G 5 and I : Γ→ G, ii) natural isomorphisms of grade 1 aX,Y,Z : (X ⊗ Y )⊗ Z ∼→ X ⊗ (Y ⊗ Z), lX : I ⊗X ∼→ X, rX : X ⊗ I ∼→ X such that the coherence conditions of a monoidal category hold. A graded categorical group is a graded monoidal category G in which every object is invertible and every morphism is an isomorphism. 1.1.4 Graded braided categorical groups A braided categorical groupG is one equipped with a braiding constraint which is compatible with the unit and associativity constraints. A Γ-graded braided categorical group (G, gr) is a Γ-graded monoidal category satisfying the coherence conditions of a braided categorical group. 1.2 Ann-categories 1.2.1 Ann-categories Ann-categories are those simulate the structure of rings introduced by N. T. Quang in 1988, in which an Ann-category is a category A together with two bifunctors ⊕,⊗ : A × A → A satisfying some similar conditions to in a ring. In particular, if all of its constraints respects to ⊕,⊗ are identities, then one obtains the notion of strict Ann-category. 1.2.2 Ann-functors An Ann-functor is a functor F between two Ann-categories such that F is a symmetric monoidal functor respects to ⊕, also a monoidal functor respects to ⊗ and compatible with the distributivity constraints. 1.2.3 Reduced Ann-categories N. T. Quang proved that each Ann-category determines three invari- ants: a ring R, an R-bimodule M and an element h ∈ Z3MacL(R,M). Thus, he constructed a reduced Ann-category SA = (R,M, h) which is equivalent to A, called an Ann-category of type (R,M). 6 Chapter 2 Classification of monoidal functors of type (ϕ, f ) and applications In this chapter, we describe forms of monoidal functors between reduced categorical groups and state some of their applications. 2.1 Cohomology classification of monoidal functors of type (ϕ, f ) The following proposition is given by H. X. Sinh (1975). Proposition 2.1. Let (F, F˜ ) : G → G′ be a monoidal functor. Then, (F, F˜ ) induces a pair of group homomorphisms F0 : pi0G→ pi0G′, [X] 7→ [FX], F1 : pi1G→ pi1G′, u 7→ γ−1FI (Fu), satisfying F1(su) = F0(s)F1(u), where the isomorphism γX(u) is given by: γX(u) = lX ◦ (u⊗ id) ◦ l−1X . Our first result is to strengthen Proposition 2.1 by Proposition 2.4 thanks to the fact that each monoidal functor (F, F˜ ) : G → G′ induces a monoidal functor SG → SG′. We need two following lemmas. Lemma 2.2. Let G,G′ be ⊗-categories with, respectively, unit constraints (I, l, r) and (I ′, l′, r′). Let (F, F˜ , F∗) : G → G′ be an ⊗-functor which is compatible with the unit constraints. Then, γ−1FI (Fu) = F −1 ∗ F (u)F∗. Lemma 2.3. Under the hypothesis of Lemma 2.2, we have Fγ X (u) = γ FX (γ−1 FI Fu). 7 Let S,S′ be categorical groups of type (Π, A, h) and (Π, A, h′), respec- tively. A functor F : S→ S′ is called a functor of type (ϕ, f) if F (x) = ϕ(x), F (x, a) = (ϕ(x), f(a)), where ϕ : Π → Π′, f : A → A′ are group homomorphisms satisfying f(xa) = ϕ(x)f(a) for x ∈ Π, a ∈ A. Proposition 2.4. Each monoidal functor (F, F˜ ) : G → G′ induces a monoidal functor SF : SG → SG′ of type (ϕ, f), in which ϕ = F0, f = F1. Further, SF = G ′FH, where H,G′ are canonical equivalences. Proposition 2.5. Every monoidal functor (F, F˜ ) : S→ S′ is a functor of type (ϕ, f). We refer to Hom(ϕ,f)[S,S′] as the set of homotopy classes of monoidal functors of type (ϕ, f) from S = (Π, A, h) to S′ = (Π, A′, h′). Then the function k = ϕ∗h′ − f∗h is called an obstruction of the functor F . Theorem 2.6. The functor F : S→ S′ of type (ϕ, f) is a monoidal functor if and only if its obstruction k vanishes in H3(Π, A′). Then, there exist bijections: i) Hom(ϕ,f)[S,S′]↔ H2(Π, A′), ii) Aut(F )↔ Z1(Π, A′). 2.2 Classification of categorical groups Let CG be a category whose objects are categorical groups, and whose morphisms are monoidal functors between them. Denote by H3Gr for the category whose objects are triples (Π, A, h) where h ∈ H3(Π, A), a mor- phism (ϕ, f) : (Π, A, h) → (Π′, A′, h′) is a pair (ϕ, f) such that there is a function g : Π2 → A′ so that (ϕ, f, g) is a monoidal functor (Π, A, h) → (Π′, A′, h′). Theorem 2.7 (Classification Theorem). There is a classifying functor d : CG → H3Gr G 7→ (pi0G, pi1G, hG) (F, F˜ ) 7→ (F0, F1) which has the following properties: i) dF is an isomorphism if and only if F is an equivalence. 8 ii) d is surjective on objects. iii) d is full, but not faithful. For (ϕ, f) : dG → dG′ there is a bijection d : Hom(ϕ,f)[G,G′]→ H2(pi0G, pi1G′). Let Π be a group and A be a Π-module. We say that a categorical group G has a pre-stick of type (Π, A) if there exists a pair of group isomorphisms p : Π→ pi0G, q : A→ pi1G which are compatible with the action of modules q(su) = p(s)q(u), where s ∈ Π, u ∈ A. Denote by CG[Π, A] the set of equivalence classes of categorical groups whose pre-sticks are of type (Π, A). We have the following result. Theorem 2.8. There exists a bijection Γ : CG[Π, A]→ H3(Π, A), [G] 7→ q−1∗ p∗hG. 2.3 Classification of braided categorical groups Let B = (B,⊗, I, a, l, r, c) be a braided categorical group. Analogously to categorical groups, one constructs a reduced one SB = (M,N, h, η) (where η is a function induced by the braiding constraint). Corrolary 2.9. Each braided monoidal functor (F, F˜ ) : S→ S′ is a triple (ϕ, f, g), where ϕ∗(h′, η′)− f∗(h, η) = ∂ab(g). Let H3BGr denote a category whose objects are triples (M,N, (h, η)), where (h, η) ∈ H3ab(M,N), and BCG denote a category whose objects are braided categorical groups. Theorem 2.10. [Classification Theorem] There is a classifying functor d : BCG → H3BGr B 7→ (pi0B, pi1B, (h, η)B) (F, F˜ ) 7→ (F0, F1) which has the following properties: i) dF is an isomorphism if and only if F is an equivalence. ii) d is surjective on objects. iii) d is full, but not faithful. For (ϕ, f) : dB→ dB′ we have 9 HomBr(ϕ,f)[B,B′] ∼= H2ab(pi0B, pi1B′), where HomBr(ϕ,f)[B,B′] is the set of homotopy classes of braided monoidal functors from B to B′ inducing the pair (ϕ, f). We write BCG[M,N ] for the set of equivalence classes of braided cate- gorical groups whose pre-sticks are of type (M,N). We obtains a similar result to Theorem 2.8. Theorem 2.11. There exists a bijection Γ : BCG[M,N ]→ H3ab(M,N), [B] 7→ q−1∗ p∗(h, η)B. 2.4 Classification of graded braided categorical groups by factor sets The notion of factor set was introduced by A. Grothendieck in 1971 and then was developed by many authors (A. M. Cegarra, N. T. Quang, ...). For a group Γ and a category C, let Psd(Γ,C) be the category of (normalized) factor sets from Γ to C, and let ΓBCG be the category of Γ-graded braided categorical groups. Theorem 2.13. For a group Γ, there is an isomorphism: ΓBCG ' Psd(Γ,BCG). 2.5 Applications on classical group extensions 2.5.1 The categorical group of an abstract kernel An abstract kernel is a triple (Π, G, ψ), where ψ : Π → AutG/InG is a group homomorphism. The obstruction of (Π, G, ψ) is an element k ∈ H3(Π, ZG). We can construct a strict categorical group AutG, whose objects are elements of the group of automorphisms AutG and whose mor- phisms are given by Hom(α, β) = {c ∈ G|α = µc ◦ β}. Its three invariants are: AutG/InG,ZG and ψ∗h, where ψ∗h = k. Using this result we prove that each categorical group is equivalent to a strict one. This proof is different from that done by H. X. Sinh (1978). 2.5.2 Monoidal functors and the group extension problem In this subsection, we used the results of the categorical group theory to obtain those of the classical group extension problem. 10 Chapter 3 Strict categorical groups and group extensions of the type of a crossed module In this chapter we represent the equivalence of the category of crossed modules and that of G-groupoids by means of strict categorical groups, hence we obtain the classification theorem of crossed modules which ex- tends the well-known result of R. Brown and C. Spencer (1976). 3.1 Categorical group associated to a crossed module Definition. A crossed module is a quadruple M = (B,D, d, θ) (or B d→ D, B → D), where d : B → D, θ : D → AutB are group homomorphisms satisfying the following relations: C1. θd = µ, C2. d(θx(b)) = µx(d(b)), x ∈ D, b ∈ B, where µx is an inner automorphism given by conjugation with x. Proposition 3.1. Let M = (B,D, d, θ) be a crossed module. i) Kerd ⊂ Z(B), ii) Imd is a normal subgroup in D, iii) The homomorphism θ induces one ϕ : D → Aut(Kerd) by ϕx = θx|Kerd, iv) Kerd is a left Cokerd-module under the action: sa = ϕx(a). For any crossed module (B,D, d, θ) we construct a strict categorical group PB→D called the associated one to the crossed module, and vice versa. 3.2 Classification of crossed modules We obtain the following results on the relation between homomorphisms of crossed modules and monoidal functors of associated categorical groups. 11 Lemma 3.2. Let (f1, f0) : (B,D, d, θ) → (B′, D′, d′, θ′) be a homomor- phism of crossed modules. Let P,P′ be the two categorical groups associated to the crossed modules (B,D, d, θ) and (B′, D′, d′, θ′), respectively. i) There exists a functor F : P→ P′ defined by F (x) = f0(x), F (b) = f1(b), for x ∈ ObP, b ∈ MorP. ii) Natural isomorphisms F˜x,y : F (x)F (y) → F (xy) together with F is a monoidal functor if and only if F˜x,y = ϕ(x, y), where ϕ ∈ Z2(Coker d,Ker d′). Denote by Cross the category whose objects are crossed modules and whose morphisms are triples (f1, f0, ϕ), where (f1, f0) is a homomorphism of crossed modules and ϕ ∈ Z2(Cokerd,Kerd′). A monoidal functor (F, F˜ ) : P→ P′ is called regular if: S1. F (x)⊗ F (y) = F (x⊗ y), for all x, y ∈ ObP. S2. F (b)⊗ F (c) = F (b⊗ c), for all b, c ∈ MorP. Lemma 3.3. Let P, P′ be corresponding categorical groups associated to the crossed modules (B,D, d, θ), (B′, D′, d′, θ′), and (F, F˜ ) : P → P′ be a regular monoidal functor. Then, the triple (f1, f0, ϕ), where f1(b) = F (b), f0(x) = F (x), ϕ(x, y) = F˜x,y, for b ∈ B, x ∈ D, x ∈ Coker d, is a morphism in the category Cross. We write Grstr for the category of strict categorical groups and regular monoidal functors. One obtains the following result. Theorem 3.4 (Classification Theorem). There exists an equivalence Φ : Cross → Grstr, (B → D) 7→ PB→D (f1, f0, ϕ) 7→ (F, F˜ ) where F (x) = f0(x), F (b) = f1(b), F˜x,y = ϕ(x, y), for x, y ∈ D, b ∈ B. 3.3. Group extension of the type of a crossed module: obstruction theory and classification theorem Definition. Let M = (B d→ D) be a crossed module and Q be a group. An extension of B by Q of type M is a diagram of group homomorphisms E : 0 // B j // E p // ε  Q // 1, B d // D 12 where the top row is exact, the quadruple (B,E, j, θ0) is a crossed module, θ0 is given by conjugation, (idB, ε) is a homomorphism of crossed modules. Each such extension induces a homomorphism ψ : Q → Coker d. Our objective is to study the set ExtB→D(Q,B, ψ) of equivalence classes of extensions of B by Q of type B → D inducing ψ : Q→ Cokerd. Let DisQ denote the categorical group of type (Q, 0, 0) (it is just the categorical group associated to the crossed module (0, Q, 0, 0)). The fol- lowing lemma shows that monoidal functors DisQ→ P are the appropriate systems of data to construct the manifold of all such group extensions. Lemma 3.5. Let B → D be a crossed module and ψ : Q → Coker d be a group homomorphism. Then, for each monoidal functor (F, F˜ ) : DisQ→ P with F (1) = 1 and inducing the pair (ψ, 0) : (Q, 0)→ (Cokerd,Kerd), there exists an extension EF of B by Q of type B → D inducing ψ. Theorem 3.6 (Schreier theory for group extensions of the type of a crossed module). There is a bijection Ω : Hom(ψ,0)[DisQ,PB→D]→ ExtB→D(Q,B, ψ). Let P = PB→D be the categorical group associated to the crossed mod- ule B → D. Since pi0P = Coker d and pi1P = Ker d, its reduced category SP is of form SP = (Cokerd,Kerd, k), k ∈ H3(Cokerd,Kerd). Then, the ho- momorphism ψ : Q → Cokerd induces an obstruction ψ∗k ∈ Z3(Q,Kerd). Thus, one obtains the theorem on the existence and classification of group extensions of the type of a crossed module. Theorem 3.7. Let (B,D, d, θ) be a crossed module and ψ : Q → Cokerd be a group homomorphism. Then, the vanishing of ψ∗k in H3(Q,Kerd) is necessary and sufficient for there to exist an extension of B by Q of type B → D inducing ψ. Further, if ψ∗k vanishes, then the equivalence classes of such extensions are bijective with H2(Q,Kerd). 13 Chapter 4 Strict graded categorical groups and equivariant group extensions of the type of a Γ-crossed module In this chapter we introduce the notion of strict graded categorical group to represent the notion of Γ-crossed module, then classify Γ-crossed modules and state the theory of equivariant group extensions of the type of a Γ-crossed module. 4.1 Equivariant group cohomology theory of Cegarra Theory of equivariant group cohomology of A. M. Cegarra and his co- authors (2002) will be used to prove the result on classification Γ-graded monoidal functors of type (ϕ, f) and to classify equivariant group exten- sions of the type of a Γ-crossed module. The equivariant cohomology groups are denoted by H iΓ(Π, A), i = 1, 2, 3. 4.2 Reduced graded categorical groups and graded monoidal functors of type (ϕ, f ) In this subsection we construct the reduced graded categorical group of a given one, and classify Γ-graded monoidal functors of type (ϕ, f). 4.2.1 Construction of reduced categorical groups by means of skeletal categories For any Γ-graded categorical group G, A. M. Cegarra and his co-authors construct a Γ-graded categorical group, denoted by ∫ Γ(Π, A, h), and they state (without proof) that it is monoidally equivalent to G. We prove this through the following proposition. 14 Proposition 4.1. Γ-functor (HΓ, H˜Γ, id) : ∫ Γ(Π, A, h)→ G defined by HΓ(s) = Xs HΓ(r (a,σ)→ s) = (Xr γˆXs(a)◦Υ(r,σ)−−−−−−−→ Xs) (H˜Γ)r,s = i −1 Xr⊗Xs, where σr = s, is a Γ-graded monoidal equivalence. 4.2.2 Construction of the reduced categorical group by means of factor sets In this subsection, using the method of factor sets developed by N. T. Quang (2010) we show that for each Γ-graded categorical group G, one can construct one, dented by ∆F , which is monoidally equivalent to G. Further, ∆F is just the Γ-graded categorical group ∫Γ(Π, A, h). 4.2.3 Classification of graded monoidal functors of type (ϕ, f) The result on the classification of graded monoidal functors of type (ϕ, f) is stated in the following proposition. Proposition 4.5. Let G,G′, S = (Π, A, h), S′ = (Π′, A′, h′) be Γ-graded categorial groups. i) Each Γ-graded functor (F, F˜ ) : G → G′ induces a Γ-graded monoidal functor SF : SG → SG′ of type (ϕ, f), in which ϕ = F0, f = F1 are given by F0 : pi0G→ pi0G′, [X] 7→ [FX], F1 : pi1G→ pi1G′, u 7→ γˆ−1FI (Fu). Further, SF = G ′ ΓFHΓ, where HΓ, G ′ Γ are canonical Γ-graded equivalences. ii) Each Γ-graded monoidal functor (F, F˜ ) : S→ S′ is one of type (ϕ, f). iii) Γ-graded functor F : S → S′ of type (ϕ, f) is realizable, that is, it in- duces a Γ-graded monoidal functor, if and only if its obstruction ξ vanishes in H3Γ(Π, A ′). Then, there is a bijection Hom(ϕ,f)[S,S′]↔ H2Γ(Π, A′). 4.3 Γ-crossed modules and associated graded cate- gorical groups Definition. Let B,D be Γ-groups. A Γ-crossed module is a quadruple M = (B,D, d, θ) where d : B → D, θ : D → AutB are Γ-homomorphisms 15 satisfying the following conditions: C1. θd = µ, C2. d(θx(b)) = µx(d(b)), C3. σ(θx(b)) = θσx(σb), where σ ∈ Γ, x ∈ D, b ∈ B, µx is the inner automorphism given by conju- gation with x. The notion of strict Γ-graded categorical group introduced below is to represent Γ-crossed modules. Firstly, a factor set F = (G, F σ, ησ,τ) on Γ with coefficients in a cate- gorical group G is termed regular if ησ,τ = id and F σ is a regular monoidal functor, for all σ, τ ∈ Γ. Definition. A graded categorical group (P, gr) is said to be strict if: i) KerP is a strict categorical group,, ii) P induces a regular factor set F on Γ with coefficients in the categorical group KerP. We proved that from a given Γ-crossed moduleM one can construct a strict Γ-graded categorical group PM associated to M, and vice versa. 4.4 Classification of Γ-crossed modules The following lemmas state the relation between homomorphisms of Γ- crossed modules and graded monoidal functors of corresponding associated graded categorical groups. Lamma 4.7. Let (f1, f0) :M→M′ be a homomorphism of Γ-crossed mod- ules. Then, there exists a Γ-graded monoidal functor (F, F˜ ) : PM → PM′ defined by F (x) = f0(x), F (b, 1) = (f1(b), 1) if and only if f = p ∗ϕ, where ϕ ∈ Z2Γ(Coker d, Ker d′), and p : D → Coker d is a canonical projection. Let ΓCross denote a category whose objects are Γ-crossed modules and whose morphisms are triples (f1, f0, ϕ), where (f1, f0) :M→M′ is a homomorphism of Γ-crossed modules and ϕ ∈ Z2Γ(Coker d,Ker d′). A Γ-graded monoidal functor (F, F˜ ) : P → P′ between two strict Γ- graded categorical groups is called regular if: S1. F (x⊗ y) = F (x)⊗ F (y), S2. F (b⊗ c) = F (b)⊗ F (c), S3. F (σb) = σF (b), 16 S4. F (σx) = σF (x), for x, y ∈ ObP, and b, c are morphisms of grade 1 in P. Let p : D → Coker d be the canonical projection, one has: Lemma 4.8. Let P and P′ be corresponding strict Γ-graded categorical groups associated to Γ-crossed modules M and M′, and let (F, F˜ ) : P → P′ be a regular Γ-graded monoidal functor. Then, the triple (f1, f0, ϕ), where i) f0(x) = F (x), (f1(b), 1) = F (b, 1), σ ∈ Γ, b ∈ B, x, y ∈ D, ii) p∗ϕ = f , is a morphism in the category ΓCross. Denote by ΓGrstr the category of strict Γ-graded categorical groups and regular Γ-graded monoidal functors, we have the following result. Theorem 4.9. [Classification Theorem] There exists an equivalence Φ : ΓCross → ΓGrstr, (B → D) 7→ PB→D (f1, f0, ϕ) 7→ (F, F˜ ) in which F (x) = f0(x), F (b, 1) = (f1(b), 1), and F (x (0,σ)→ σx) = (ϕ(px, σ), σ), F˜x,y = (ϕ(px, py), 1), for x, y ∈ D, b ∈ B, σ ∈ Γ. 4.5 Equivariant group extensions of the type of a Γ- crossed modules: obstruction theory and classi- fication theorem In this section we develop a theory of equivariant group extensions of the type of a Γ-crossed module which extends both group extension theory of the type of a crossed module of P. Dedeker - R. Brown and equivariant group extension theory A. M. Cegarra. Definition. Let B d→ D be a Γ-crossed module and Q be a Γ-group. An equivariant group extension of B by Q of type B d−→ D is a diagram of Γ-homomorphisms E 0 // B j // E p // ε  Q // 1, B d // D 17 where the top row is exact, the family (B,E, j, θ0) is a Γ-crossed module in which θ0 is given by conjugation, and (id, ε) is a homomorphism of Γ-crossed modules. Each such extension induces a Γ-homomorphism ψ : Q → Cokerd. Our objective is to study the set ExtΓB→D(Q,B, ψ) of equivalence classes of equivariant extensions of B by Q of type B → D inducing ψ : Q→ Cokerd. Let DisΓQ denote the strict Γ-graded categorical group associated to the Γ-crossed module (0, Q, 0, 0). The following lemma shows that Γ- graded monoidal functors DisΓQ → PB→D are the appropriate system of data to construct all of such extensions. Lemma 4.10. Let B d→ D be a Γ-crossed module, and let ψ : Q→ Coker d be a Γ-homomorphism. For each Γ-graded monoidal functor (F, F˜ ) : DisΓQ→ PB→D, which satisfies F (1) = 1 and induces a pair of Γ-homomorphisms (ψ, 0) : (Q, 0) → (Coker d,Ker d), there exists an equivariant group exten- sion EF of B by Q of type B → D inducing ψ. Theorem 4.11. [Schreier theory for equivariant group extensions of the type of a Γ-crossed module] There is a bijection Ω : Hom(ψ,0)[DisΓQ,PB→D]→ ExtΓB→D(Q,B, ψ). One obtains the following consequence of equivariant group extensions. Lemma 4.12. For Γ-groups B and Q, there exists a bijection HomΓ[DisΓQ,HolΓB]→ ExtΓ(Q,B). Since the reduced graded categorical group of PB→D is SP = (Cokerd,Kerd, h), h ∈ Z3Γ(Cokerd,Kerd), Γ-homomorphism ψ : Q → Cokerd induces an obstruction ψ∗h ∈ Z3Γ(Q,Kerd). Under this notion of obstruction, we state the following theorem. Theorem 4.13. Let (B,D, d, θ) be a Γ-crossed module, and let ψ : Q → Cokerd be a Γ-homomorphism. Then, the vanishing of ψ∗h in H3Γ(Q,Kerd) is necessary and sufficient for there to exist an equivariant extension of B by Q of type B → D inducing ψ. Further, if ψ∗h vanishes, then the equivalence classes of such extensions are bijective with H2Γ(Q,Kerd). 18 Chapter 5 Strict Ann-categories and ring extensions of the type of a regular E-system In this chapter we introduce a ring version of crossed modules over groups of J. H. C. Whitehead, called E-systems, classify regular E-systems and solve the ring extension problem of the type of a regular E-system 5.1 The theory of ring cohomology of Mac Lane and of Shukla The cohomology group H3Shu(R,M) in the sense of Shukla (in which the ring R is regarded as a Z-algebra) is used by N. T. Quang to classify regular Ann-categories (1988). Then, the cohomology group H2MacL(R,M) of rings in the sense of Mac Lane is used by N. T. Quang and D. D. Hanh to classify Ann-functors between Ann-categories (2009), and recently the group H3MacL(R,M) is used by N. T. Quang to classify Ann-categories in the general case (2013). 5.2 Crossed bimodules and regular E-systems Definition. i) A crossed bimodule is a triple (B,D, d), where D is an asso- ciative K-algebra, B is a D-bimodule and d : B → D is a homomorphism of D-bimodules such that d(b)b′ = bd(b′), b, b′ ∈ B. ii) A morphism (k1, k0) : (B,D, d) → (B′, D′, d′) of crossed bimodules is a pair k1 : B → B′, k0 : D → D′, where k1 is a group homomorphism, k0 is a K-algebra homomorphism such that for all x ∈ D, b ∈ B, k0d = d ′k1, k1(xb) = k0(x)k1(b), k1(bx) = k1(b)k0(x). 19 When the base ring K is the ring of integers Z, then a crossed bimodule (B,D, d) is called a crossed bimodule over rings. The notion of E-system introduced below can be seen as a version of the concept of a crossed module over rings. Definition. An E-system is a quadruple M = (B,D, d, θ) where d : B → D, θ : D → MB are the ring homomorphisms satisfying the following conditions for all x ∈ D, b ∈ B: θ ◦ d = µ, d(θxb) = x.d(b), d(bθx) = d(b).x. An E-system (B,D, d, θ) is regular if θ is a 1-homomorphism (a homo- morphism carries the identity to the identity), and the elements of θ(D) are permutable. A morphism (f1, f0) : (B,D, d, θ)→ (B′, D′, d′, θ′) of E-systems consists of ring homomorphisms f1 : B → B′, f0 : D → D′ such that f0d = d ′f1, f1(θxb) = θ ′ f0(x) f1(b), f1(bθx) = f1(b)θ ′ f0(x) . By the following theorem, the notion of an E-system can be seen as a weaken version of the notion of a crossed bimodule over rings. Theorem 5.2. The categories of regular E-systems and of crossed bimodules over rings are isomorphic. 5.3 Classification of regular E-systems For a given E-system B → D we can construct a strict Ann-category AB→D, called the Ann-category associated to the E-system, and vice versa. The following lemmas show the relation between homomorphisms of regular E-systems and Ann-functors of associated Ann-categories. Lemma 5.3. Let (f1, f0) : (B,D, d, θ) → (B′, D′, d′, θ′) be a morphism of regular E-systems. i) There is a functor F : AB→D → AB′→D′ defined by F (x) = f0(x), F (b) = f1(b), x ∈ ObA, b ∈ MorA. ii) The isomorphisms F˘x,y : F (x + y) → Fx + Fy, F˜x,y : F (xy) → FxFy together with the functor F is an Ann-functor if and only if F˘ and F˜ are 20 constants in Ker d′ and for all x, y ∈ D, θ′Fx(F˜ ) = (F˜ )θ ′ Fy = F˜ , θ′Fx(F˘ ) = (F˘ )θ ′ Fy = F˘ + F˜ . Then, we say that F is an Ann-functor of form (f1, f0). An Ann-functor (F, F˘ , F˜ ) is single if F (0) = 0, F (1) = 1 and F˘ , F˜ are constants. Then we state the converse of Lemma 5.3. Lemma 5.4. Let (F, F˘ , F˜ ) : AB→D → AB′→D′be a single Ann-functor. Then, there is a morphism of regular E-systems (f1, f0) : (B → D)→ (B′ → D′) given by f1(b) = F (b), f0(x) = F (x), for b ∈ B, x ∈ D. Lemma 5.5. Two Ann-functors (F, F˘ , F˜ ), (F ′, F˘ ′, F˜ ′) : AB→D → AB′→D′ of the same form are homotopic. Two Ann-functors (F, F˘ , F˜ ), (F ′, F˘ ′, F˜ ′) are strong homotopic if they are homotopic and F = F ′. Corollary 5.6. Two Ann-functors F, F ′ : AB→D → AB′→D′ are strong ho- motopic if and only if they are of the same form. We write Annstr or the category of strict Ann-categories and their sin- gle Ann-functors. We can define the strong homotopy category HoAnnstr of Annstr to be the quotient category with the same objects, but mor- phisms are strong homotopy classes of single Ann-functors: HomHoAnnstr(A,A′) = HomAnnstr(A,A′) strong homotopies . Denote ESyst the category whose objects are regular E-systems and whose morphisms are homomorphisms of regular E-systems, we obtain the follow- ing result. Theorem 5.7. [Classification Theorem] There exists an equivalence of cat- egories Φ : ESyst → HoAnnstr, (B → D) 7→ AB→D (f1, f0) 7→ [F ] where F (x) = f0(x), F (b) = f1(b), for x ∈ ObA, b ∈ MorA. 21 5.4 Ring extensions of the type of an E-system Definition. Let (B,D, d, θ) be a regular E-system. A ring extension of B by Q of type B → D is a diagram of ring homomorphisms 0 // B j // E p // ε  Q // 0, B d // D where the top row is exact, the quadruple (B,E, j, θ0) is an regular E- system where θ0 is given by the bimultiplication type, and the pair (id, ε) is a morphism of regular E-systems. Each such extension induces a ring homomorphism ψ : Q → Coker d. We use the obstruction theory of Ann-functors to study the set ExtB→D(Q, B, ψ) of equivalence class of extensions of B by Q of type regular E-system B → D inducing ψ. The following lemma shows that the Ann-functors DisQ → AB→D are the appropriate data to construct such extensions. Lemma 5.8. Let (B,D, d, θ) be a regular E-system, ψ : Q → Coker d be a ring homomorphism. Then, for each Ann-functor (F, F˘ , F˜ ) : DisQ → A inducing the pair (ψ, 0) there exists an extension EF of B by Q of type B → D inducing ψ : Q→ Coker d. The following theorem is a version of Schreier theory for ring extensions of the type of a regular E-system. Theorem 5.9. [Schreier theory for ring extensions of the type of a regular E-system] There is a bijection Ω : HomAnn(ψ,0)[DisQ,A]→ ExtB→D(Q,B, ψ). Since the reduced Ann-category of the associated Ann-category AB→D is of form SA = (Cokerd,Kerd, k), where k ∈ H3Shu(Cokerd,Kerd), the homo- morphism ψ : Q→ Cokerd induces an obstruction ψ∗k ∈ H3Shu(Q,Kerd). Theorem 5.10. Let (B,D, d, θ) be a regular E-system, ψ : Q → Cokerd be a ring homomorphism. Then, the vanishing of ψ∗k in H3Shu(Q,Kerd) is necessary and sufficient for there to exist a ring extension of B by Q of type B → D inducing ψ. Further, if ψ∗k vanishes then there is a bijection ExtB→D(Q,B, ψ)↔ H2Shu(Q,Kerd). 22 Conclusion Crossed modules and categorical groups have been used widely and in- dependently, and in various contexts. The results on categorical groups of H. X. Sinh (1975) are raised to the graded categorical groups by A. M. Ce- garra and his co-authors, and to the categorical rings (or Ann-categories) by N. T. Quang. Besides, R. Brown and C. Spencer (1976) showed that crossed modules can be studied by means of strict categorical groups. This suggests us to study the more complex categorical algebras, such as: strict graded categorical groups, strict Ann-categories, and then to study the structures which are analogous to crossed modules, such as: equivariant crossed modules, E-systems. The main results of this thesis are the follow- ings: 1. We describe type of a monoidal functor between two categorical groups and the obstruction theory of a functor. Therefore, the precise theorems on the classification categorical groups and on that of braided categorical groups are stated. 2. Based on the results of the strict categorical group theory, we classify crossed modules and construct the Schreier theory for group extensions of the type of a crossed module. Our results extend the results of R. Brown and his co-authors. 3. We study strict graded categorical groups to classify equivariant crossed modules and construct the Schreier theory for equivariant group extensions of the type of a Γ-crossed module. Our results contain the theory of equivariant group exteniosn of A. M. Cegarra - J. M. Garca-Calcines - J. A. Ortega and the theory of group extensions of the type of a crossed modules of R. Brown - O. Mucuk. 4. We study strict Ann-categories to classify crossed bimodules over rings and to classify ring extensions of the type of a regular E-system. 5. We classify the category of Γ-graded braided categorical groups by means of factor sets on the group Γ with coefficients in the category of braided categorical groups of type (M, N). 23 List of the author's articles related to the thesis 1. N. T. Quang, N. T. Thuy, P. T. Cuc, Monoidal functors between (braided) Gr-categories and their applications, East-West J. of Mathemat- ics, 13, No 2 (2011), 163-186. 2. N. T. Quang, P. T. Cuc, Crossed bimodules over rings and Shukla cohomology, Math. Commun., 17 No. 2 (2012), 575-598. 3. N. T. Quang, P. T. Cuc, Classification of graded braided categorical groups by pseudo-functors, Journal of Science, Hue University, Vol. 77, No. 8 (2012), 59-68. 4. N. T. Quang, P. T. Cuc, N. T. Thuy, Strict Gr-categories and crossed modules, Communications of Korean Mathematical Society, Vol. 29, No.1 (2014), 9-22. 5. N. T. Quang, P. T. Cuc, Equivariant crossed bimodules and cohomology of groups with operators, arXiv: 1302.4573v1 [math.CT] 19 Feb 2013. The results of the thesis have reported and discussed in: - Science workshop on “Some of new research directions in modern math- ematics and their applications”, Thanhhoa (by Hongduc University, Insti- tute of Mathematics and Hanoi National University of Education jointly organized), May 2011. - National conference on Algebra - Geometry - Topology, Thainguyen, November 2011. - 8th Mathematical Congress, Nhatrang, August 2013. - Seminar at Dept. of Algebra - Geometry, Mathematical Faculty, Hue University’s College of Education. - Seminar at Dept. of Algebra, Natural Science Faculty, Hongduc Univer- sity, Thanhhoa. 24