The group of strong Galois objects associated to a cocommutative Hopf quasigroup

J. Korean Math. Soc. 54 (2017), No. 2, pp. 517–543 https://doi.org/10.4134/JKMS.j160118 pISSN: 0304-9914 / eISSN: 2234-3008 THE GROUP OF STRONG GALOIS OBJECTS ASSOCIATED TO A COCOMMUTATIVE HOPF QUASIGROUP Jose N. Alonso ´ Alvarez, Ramon Gonz´ alez Rodr´ ıguez, and Jose M. Fern´ andez Vilaboa Abstract. Let Hbe a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler. Introduction Let Rbe a commutative ring with unit. The notion of Galois H-object for a commutative, cocommutative Hopf R-algebra H, which is a finitely generated projective R-module, is due to Chase and Sweedler [7]. As was pointed by Beattie [4], although the discussion of Galois H-objects in [7] is limited to commutative algebras, the main properties can be easily extended to non commutative algebras. One of more relevant is the following: if His cocommutative, the isomorphism classes of Galois H-objects form a group denoted by Gal(R, H). The product in Gal(R, H) is defined by the kernel of a suitable morphism and the class of His the identity element. This construction can be extended to symmetric closed categories with equalizers and coequalizers working with monoids instead of algebras and some of the more important properties and exact sequences involving the group Gal(R, H) were obtained in this categorical setting ([9], [13], [14]). An interesting generalization of Hopf algebras are Hopf quasigroups introduced by Klim and Majid in [8] in order to understand the structure and relevant properties of the algebraic 7-sphere. They are not associative but the lack of this property is compensated by some axioms involving the antipode. The concept of Hopf quasigroup is a particular instance of the notion of unital coassociative H-bialgebra introduced in [11] and includes the example of an Received February 20, 2016. 2010 Mathematics Subject Classification. 18D10, 17A01, 16T05, 81R50, 20N05. Key words and phrases. monoidal category, unital magma, Hopf quasigroup, (strong) Galois H-object, Galois group, normal basis. c 2017 Korean Mathematical Society 517 518 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA enveloping algebra U(L) of a Malcev algebra (see [8]) as well as the notion of quasigroup algebra RL of an I.P. loop L. Then, quasigroups unify I.P. loops and Malcev algebras in the same way that Hopf algebras unified groups and Lie algebras. In this paper we are interested to answer the following question: is it possible to extend the construction of Gal(R, H) to the situation where His a cocommutative Hopf quasigroup? in other words, can we construct in a nonassociative setting a group of Galois H-objects? The main obstacle to define the group is the lack of associativity because we must work with unital magmas, i.e., objects where there exists a non-associative product with unit. As we can see in the first section of this paper, Hopf quasi groups are examples of these algebraic structures. The paper is organized as follows. We begin introducing the notion of right H-comodule magma, where His a Hopf quasigroup, and defining the product of right H-comodule magmas. In the second section we introduce the notions of Galois H-object and strong Galois H-objects proving that, with the product defined in the first section for comodule magmas, the set of isomorphism classes forms a monoid, in the case of Galois H-objects, and a group when we work with strong Galois H-objects. In this point it appears the main difference between our Galois H-objects and the ones associated to a Hopf algebra because in the Hopf algebra setting the inverse of the class of a Galois H-object Ais the class of the opposite Galois H-object Aop, while in the quasigroup context this property fails. We only have the following: the product of Aand Aop is isomorphic to Honly as comodules. To obtain an isomorphism of magmas we need to work with strong Galois H-objects. Then, the strong condition appears in a natural way and we want to point out that in the classical case of Galois H-objects associated to a Hopf algebra Hall of them are strong. Finally, in the last section, we study the connections between Galois H-objects and invertible comodules with geometric normal basis. Throughout this paper Cdenotes a strict symmetric monoidal category with equalizers where ⊗denotes the tensor product, Kthe unit object and cthe symmetry isomorphism. We denote the class of objects of Cby |C| and for each object M∈ |C|, the identity morphism by idM:M→M. For simplicity of notation, given objects M,Nand Pin Cand a morphism f:M→N, we write P⊗ffor idP⊗fand f⊗Pfor f⊗idP. We will say that A∈ |C| is flat if the functor A⊗ − :C → C preserves equalizers. If moreover A⊗ − reflects isomorphisms we say that Ais faithfully flat. By a unital magma in Cwe understand a triple A= (A, ηA, µA) where A is an object in Cand ηA:K→A(unit), µA:A⊗A→A(product) are morphisms in Csuch that µA◦(A⊗ηA) = idA=µA◦(ηA⊗A). If µAis associative, that is, µA◦(A⊗µA) = µA◦(µA⊗A), the unital magma will be called a monoid in C. For any unital magma Awith Awe will denote the opposite unital magma (A, ηA=ηA, µA=µA◦cA,A). Given two unital magmas (monoids) A= (A, ηA, µA) and B= (B, ηB, µB), f:A→Bis a morphism of THE GROUP OF STRONG GALOIS OBJECTS 519 unital magmas (monoids) if µB◦(f⊗f) = f◦µAand f◦ηA=ηB. By duality, a counital comagma in Cis a triple D= (D, εD, δD) where Dis an object in C and εD:D→K(counit), δD:D→D⊗D(coproduct) are morphisms in C such that (εD⊗D)◦δD=idD= (D⊗εD)◦δD. If δDis coassociative, that is, (δD⊗D)◦δD= (D⊗δD)◦δD, the counital comagma will be called a comonoid. If D= (D, εD, δD) and E= (E, εE, δE) are counital comagmas (comonoids), f:D→Eis morphism of counital magmas (comonoids) if (f⊗f)◦δD=δE◦f and εE◦f=εD. Finally note that if A,Bare unital magmas (monoids) in C, the object A⊗B is a unital magma (monoid) in Cwhere ηA⊗B=ηA⊗ηBand µA⊗B= (µA⊗µB)◦ (A⊗cB,A ⊗B).With Aewe will denote the unital magma A⊗A. In a dual way, if D,Eare counital comagmas (comonoids) in C,D⊗Eis a counital comagma (comonoid) in Cwhere εD⊗E=εD⊗εEand δD⊗E= (D⊗cD,E ⊗E)◦(δD⊗δE). 1. Comodule magmas for Hopf quasigroups This first section is devoted to the study of the notion of H-comodule magma associated to a Hopf quasigroup H. We will show that, as in the Hopf algebra setting, it is possible to define a product using suitable equalizers which induces a monoidal structure in the category of flat H-comodule magmas. The notion of Hopf quasigroup was introduced in [8] and the following is its monoidal version. Definition 1.1. A Hopf quasigroup Hin Cis a unital magma (H, ηH, µH) and a comonoid (H, εH, δH) such that the following axioms hold: (a1) εHand δHare morphisms of unital magmas. (a2) There exists λH:H→Hin C(called the antipode of H) such that: (a2-1) µH◦(λH⊗µH)◦(δH⊗H) =εH⊗H =µH◦(H⊗µH)◦(H⊗λH⊗H)◦(δH⊗H). (a2-2) µH◦(µH⊗H)◦(H⊗λH⊗H)◦(H⊗δH) =H⊗εH =µH◦(µH⊗λH)◦(H⊗δH). If His a Hopf quasigroup, the antipode is unique, antimultiplicative, anticomultiplicative and leaves the unit and the counit invariable: (1) λH◦µH=µH◦(λH⊗λH)◦cH,H , δH◦λH=cH,H ◦(λH⊗λH)◦δH, (2) λH◦ηH=ηH, εH◦λH=εH ([8], Proposition 4.2 and [10], Proposition 1). Note that by (a2), (3) µH◦(λH⊗idH)◦δH=µH◦(idH⊗λH)◦δH=εH⊗ηH. 520 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA A Hopf quasigroup His cocommutative if cH,H ◦δH=δH. In this case, as in the Hopf algebra setting, we have that λH◦λH=idH(see Proposition 4.3 of [8]). Let Hand Bbe Hopf quasigroups. We say that f:H→Bis a morphism of Hopf quasigroups if it is a morphism of unital magmas and comonoids. In this case λB◦f=f◦λH(see Proposition 1.5 of [1]). Examples 1.2. The notion of Hopf quasigroup was introduced in [8] and it can be interpreted as the linearization of the concept of quasigroup. A quasigroup is a set Qtogether with a product such that for any two elements u, v ∈Qthe equations ux =v,xu =vand uv =xhave unique solutions in Q. A quasigroup Lwhich contains an element eLsuch that ueL=u=eLufor every u∈Lis called a loop. A loop Lis said to be a loop with the inverse property (for brevity an I.P. loop) if and only if, to every element u∈L, there corresponds an element u−1∈Lsuch that the equations u−1(uv) = v= (vu)u−1hold for every v∈L. If Lis an I.P. loop, it is easy to show (see [5]) that for all u∈Lthe element u−1is unique and u−1u=eL=uu−1. Moreover, for all u,v∈L, the equality (uv)−1=v−1u−1holds. Let Rbe a commutative ring and Land I.P. loop. Then, by Proposition 4.7 of [8], we know that RL =M u∈L Ru is a cocommutative Hopf quasigroup with product given by the linear extension of the one defined in Land δRL(u) = u⊗u, εRL(u) = 1R, λRL(u) = u−1 on the basis elements. Now we briefly describe another example of Hopf quasigroup constructed working with Malcev algebras (see [12] for details). Consider a commutative and associative ring Kwith 1 2and 1 3in K. A Malcev algebra (M, [,]) over Kis a free module in K-Mod with a bilinear anticommutative operation [ , ] on Msatisfying that [J(a, b, c), a] = J(a, b, [a, c]),where J(a, b, c) = [[a, b], c]− [[a, c], b]−[a, [b, c]] is the Jacobian in a, b, c. Denote by U(M) the not necessarily associative algebra defined as the quotient of K{M}, the free non-associative algebra on a basis of M, by the ideal I(M) generated by the set {ab −ba − [a, b],(a, x, y) + (x, a, y),(x, a, y) + (x, y, a) : a, b ∈M, x, y ∈K{M}},where (x, y, z) = (xy)z−x(yz) is the usual additive associator. By Proposition 4.1 of [12] and Proposition 4.8 of [8], the diagonal map δU(M):U(M)→U(M)⊗U(M) defined by δU(M)(x) = 1 ⊗x+x⊗1 for all x∈M, and the map εU(M):U(M)→Kdefined by εU(M)(x) = 0 for all x∈M, both extended to U(M) as morphisms of unital magmas; together with the map λU(M):U(M)→U(M), defined by λU(M)(x) = −xfor all THE GROUP OF STRONG GALOIS OBJECTS 521 x∈Mand extended to U(M) as an antimultiplicative morphism, provide a cocommutative Hopf quasigroup structure on U(M). Definition 1.3. Let Hbe a Hopf quasigroup and let Abe a unital magma (monoid) with a right coaction ρA:A→A⊗H. We will say that A= (A, ρA) is a right H-comodule magma (monoid) if (A, ρA) is a right H-comodule (i.e., (ρA⊗H)◦ρA= (A⊗δH)◦ρA, (A⊗εH)◦ρA=idA), and the following identities (b1) ρA◦ηA=ηA⊗ηH, (b2) ρA◦µA=µA⊗H◦(ρA⊗ρA), hold. Obviously, if His a Hopf quasigroup, the pair H= (H, δH) is an example of right H-comodule magma. Let A,Bbe right H-comodule magmas (monoids). A morphism of right H-comodule magmas (monoids) f:A→Bis a morphism f:A→Bin Cof unital magmas (monoids) and right H-comodules, that is (f⊗H)◦ρA=ρB◦f. Remark 1.4.Note that, if His cocommutative, every endomorphism α:H→H of right H-comodule magmas is an isomorphism. Indeed: First note that by the comodule condition and the cocommutativity of Hwe have α= ((εH◦α)⊗ H)◦δH= (H⊗(εH◦α)) ◦δHand then α′= (H⊗(εH◦α◦λH)) ◦δHis the inverse of αbecause by the properties of H: α′◦α=α◦α′= (H⊗(((εH◦α)⊗(εH◦α◦λH)) ◦δH)) ◦δH = (H⊗(εH◦α◦µH◦(H⊗λH)◦δH)) ◦δH =idH. Proposition 1.5. Let Hbe a Hopf quasigroup and A,Bright H-comodule magmas. The pairs A⊗1B= (A⊗B, ρ1 A⊗B= (A⊗cH,B )◦(ρA⊗B)),A⊗2B= (A⊗B, ρ2 A⊗B=A⊗ρB)are right H-comodule magmas. Moreover A⊗1Band B⊗2Aare isomorphic right H-comodule magmas. Proof. We give the proof only for A⊗1B. The calculus for A⊗2Bare analogous and we left to the reader. First note that the object A⊗Bis a unital magma in C. On the other hand, the pair (A⊗B, ρ1 A⊗B) is a right H-comodule because trivially (A⊗B⊗εH)◦ρ1 A⊗B=idA⊗Band using the naturality of cwe obtain that (ρ1 A⊗B⊗H)◦ρ1 A⊗B= (A⊗δH)◦ρ1 A⊗B. Moreover, ρ1 A⊗B◦ηA⊗B=ηA⊗B⊗ηH and also by the naturality of cwe have ρ1 A⊗B◦µA⊗B= (µA⊗B⊗µH)◦(A⊗ B⊗cH,A⊗B⊗H)◦(ρ1 A⊗B⊗ρ1 A⊗B).Finally, cA,B is an isomorphism of right H-comodule magmas between A⊗1Band B⊗2Abecause by the naturally of cwe obtain that cA,B ◦ηA⊗B=ηB⊗A,µB⊗A◦(cA,B ⊗cA,B) = cA,B ◦µA⊗B and ρ2 B⊗A◦cA,B = (cA,B ⊗H)◦ρ1 A⊗B. Proposition 1.6. Let Hbe a cocommutative Hopf quasigroup and Aa right H-comodule magma. Then A= (A, ρA= (A⊗λH)◦ρA)is a right H-comodule magma. 522 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA Proof. Trivially (A⊗εH)◦ρA=idA. Using that His cocommutative and (1) we obtain (ρA⊗H)◦ρA= (A⊗δH)◦ρA. Moreover by (b1) of Definition 1.3 and (2), the identity ρA◦ηA=ηA⊗ηHholds. Finally, by the naturality of c, (b2) of Definition 1.3 and (1) the equality ρA◦µA=µA⊗H◦(ρA⊗ρA) follows easily.  Proposition 1.7. Let Hbe a Hopf quasigroup and A,Bright H-comodule magmas. The object A•Bdefined by the equalizer diagram ✲ ✲ ✲ A•BA⊗BA⊗B⊗H, iA•B ρ1 A⊗B ρ2 A⊗B where ρ1 A⊗Band ρ2 A⊗Bare the morphisms defined in Proposition 1.5, is a unital magma where ηA•Band µA•Bare the factorizations through iA•Bof the morphisms ηA⊗Band µA⊗B◦(iA•B⊗iA•B)respectively. Moreover, if His flat and the coaction ρA•B:A•B→A•B⊗His the factorization of ρ2 A⊗B◦iA•B through iA•B⊗H, the pair A•B= (A•B, ρA•B)is a right H-comodule magma. Proof. Trivially ρ1 A⊗B◦ηA⊗B=ηA⊗ηB⊗ηH=ρ2 A⊗B◦ηA⊗B. Therefore, there exists a unique morphism ηA•B:K→A•Bsuch that iA•B◦ηA•B=ηA⊗B. On the other hand, using the properties of ρAand ρBand the naturality of c we have ρ1 A⊗B◦µA⊗B◦(iA•B⊗iA•B) = (µA⊗B⊗µH)◦(A⊗B⊗cH,A⊗B⊗H)◦((ρ1 A⊗B◦iA•B)⊗(ρ1 A⊗B◦iA•B)) = (µA⊗B⊗µH)◦(A⊗B⊗cH,A⊗B⊗H)◦((ρ2 A⊗B◦iA•B)⊗(ρ2 A⊗B◦iA•B)) =ρ2 A⊗B◦µA⊗B◦(iA•B⊗iA•B). Then, there exists a unique morphism µA•B:A•B⊗A•B→A•Bsuch that iA•B◦µA•B=µA⊗B◦(iA•B⊗iA•B).Moreover, µA•B◦(ηA•B⊗A•B) = idA•B=µA•B◦(A•B⊗ηA•B) because iA•B◦µA•B◦(ηA•B⊗A•B) = iA•B= iA•B◦µA•B◦(A•B⊗ηA•B).Therefore, A•Bis a unital magma. Moreover, ✲ ✲ ✲ A•B⊗H A ⊗B⊗H A ⊗B⊗H⊗H iA•B⊗Hρ1 A⊗B⊗H ρ2 A⊗B⊗H is an equalizer diagram, because −⊗Hpreserves equalizers, and by the properties of ρAand ρBand the naturally of cwe obtain (ρ1 A⊗B⊗H)◦ρ2 A⊗B◦iA•B= (ρ2 A⊗B⊗H)◦ρ2 A⊗B◦iA•B.As a consequence, there exists a unique morphism ρA•B:A•B→A•B⊗Hsuch that (iA•B⊗H)◦ρA•B=ρ2 A⊗B◦iA•B. Then, the pair (A•B, ρA•B) is a right H-comodule because (iA•B⊗εH)◦ρA•B=iA•B and also (((iA•B⊗H)◦ρA•B)⊗H)◦ρA•B= (iA•B⊗δH)◦ρA•B.Finally, (b1) and (b2) of Definition 1.3 follow, by a similar reasoning, from (iA•B⊗H)◦ THE GROUP OF STRONG GALOIS OBJECTS 523 ρA•B◦ηA•B= (iA•B⊗H)◦(ηA•B⊗ηH) and (iA•B⊗H)◦ρA•B◦µA•B= (iA•B⊗H)◦µA•B⊗H◦(ρA•B⊗ρA•B).  Proposition 1.8. Let Hbe a flat Hopf quasigroup and f:A→B,g:T→D morphisms of right H-comodule magmas. Then the morphism f•g:A•T→ B•D, obtained as the factorization of (f⊗g)◦iA•T:A•T→B⊗Dthrough the equalizer iB•D, is a morphism of right H-comodule magmas between A•T and B•D. Moreover, if fand gare isomorphisms, so is f•g. Proof. Using that fand gare comodule morphisms we obtain ρ1 B⊗D◦(f⊗g)◦ iA•T=ρ2 B⊗D◦(f⊗g)◦iA•Tand as a consequence there exist a unique morphism (f•g) : A•T→B•Dsuch that iB•D◦(f•g) = (f⊗g)◦iA•T. The morphism f•g is a morphism of unital magmas because iB•D◦ηB•D=iB•D◦(f•g)◦ηA•Tand for the product the equality iB•D◦µB•D◦((f•g)⊗(f•g)) = iB•D◦(f•g)◦µA•T holds. Also, it is a comodule morphism because (iB•D⊗H)◦ρB•D◦(f•g) = (iB•D⊗H)◦((f•g)⊗H)◦ρA•T. Finally, it is easy to show that, if fand gare isomorphisms, f•gis an isomorphism with inverse f−1•g−1. Proposition 1.9. Let Hbe a flat Hopf quasigroup and A,Bright H-comodule magmas. Then A•Band B•Aare isomorphic as right H-comodule magmas. Proof. First note that by the naturally of cand the properties of the equaliser morphism iA•Bwe have that ρ1 B⊗A◦cA,B ◦iA•B=ρ2 B⊗A◦cA,B ◦iA•Band then there exists a morphism τA,B :A•B→B•Asuch that iB•A◦τA,B = cA,B◦iA•B. Also there exists an unique morphism τB,A :B•A→A•Bsuch that iA•B◦τB,A =cB,A◦iB•A. Then iA•B◦τB,A◦τA,B =cB,A◦cA,B◦iA•B=iA•Band similarly iB•A◦τA,B ◦τB,A =iB•A. Thus τA,B is an isomorphism with inverse τB,A. Moreover, iB•A◦τA,B ◦ηA•B=cA,B ◦iA•B◦ηA•B=ηB⊗A=iB•A◦ηB•A and iB•A◦τA,B ◦µA•B=µB⊗A◦((cA,B ◦iA•B)⊗(cA,B ◦iA•B)) =µB⊗A◦((iB•A◦τA,B)⊗(iB•A◦τA,B)) =iB•A◦µB•A◦(τA,B ⊗τA,B). Therefore, τA,B is a morphism of unital magmas and finally it is a morphism of right H-comodules because ((iB•A◦τA,B)⊗H)◦ρA•B= (iB•A⊗H)◦ρB•A◦τA,B.  Proposition 1.10. Let Hbe a flat Hopf quasigroup and A,B,Dright Hcomodule magmas such that Aand Dare flat. Then A•(B•D)and (A•B)•D are isomorphic as right H-comodule magmas. Proof. First, note that ✲ ✲ ✲ A⊗B•D A ⊗B⊗D A ⊗B⊗D⊗H A⊗iB•D A⊗ρ1 B⊗D A⊗ρ2 B⊗D 524 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA and ✲ ✲ ✲ A•B⊗D A ⊗B⊗D A ⊗B⊗D⊗H iA•B⊗D(A⊗B⊗cH,D )◦(ρ1 A⊗B⊗D) (A⊗B⊗cH,D )◦(ρ2 A⊗B⊗D) are equalizer diagrams because Aand Dare flat and A⊗B⊗cH,D an isomorphism. On the other hand, it is easy to show that (A⊗iB•D⊗H)◦ρ1 A⊗B•D= (A⊗B⊗cH,D)◦(ρ1 A⊗B⊗D)◦(A⊗iB•D) and (A⊗iB•D⊗H)◦ρ2 A⊗B•D= (A⊗B⊗cH,D)◦(ρ2 A⊗B⊗D)◦(A⊗iB•D). Therefore (A⊗B⊗cH,D)◦(ρ1 A⊗B⊗D)◦(A⊗iB•D)◦iA•(B•D) = (A⊗B⊗cH,D)◦(ρ2 A⊗B⊗D)◦(A⊗iB•D)◦iA•(B•D) and as a consequence there exists a unique morphism h:A•(B•D)→ (A•B)⊗Dsuch that (4) (iA•B⊗D)◦h= (A⊗iB•D)◦iA•(B•D). The diagram ✲ ✲ ✲ A•(B•D) (A•B)⊗DA•B⊗D⊗H hρ1 A•B⊗D ρ2 A•B⊗D is an equalizer diagram. Indeed, it is easy to see that ρ1 A•B⊗D◦h=ρ1 A•B⊗D◦h and, if f:C→A•B⊗Dis a morphism such that ρ1 (A•B)⊗D◦f=ρ2 (A•B)⊗D◦f, we have that (A⊗ρ1 B⊗D)◦(iA•B⊗D)◦f= (A⊗ρ2 B⊗D)◦(iA•B⊗D)◦f because (A⊗ρ1 B⊗D)◦(iA•B⊗D) = (iA•B⊗D⊗H)◦ρ1 A•B⊗D and (A⊗ρ2 B⊗D)◦(iA•B⊗D) = (iA•B⊗D⊗H)◦ρ2 A•B⊗D. Then, there exists a unique morphism t:C→A⊗B•Dsuch that (A⊗iB•D)◦ t= (iA•B⊗D)◦f. The morphism tfactorizes through the equalizer iA•(B•D) because (A⊗iB•D⊗H)◦ρ1 A⊗B•D◦t= (A⊗iB•D⊗H)◦ρ2 A⊗B•D◦t and then ρ1 A⊗B•D◦t=ρ2 A⊗B•D◦t holds. Thus, there exists a unique morphism g:C→A•(B•D) satisfying the equality iA•(B•D)◦g=t. As a consequence (iA•B⊗D)◦h◦g= (A⊗iB•D)◦iA•(B•D)◦g THE GROUP OF STRONG GALOIS OBJECTS 525 = (A⊗iB•D)◦t= (iA•B⊗D)◦f and then h◦g=f. Moreover, gis the unique morphism such that h◦g=t, because if d:C→A•(B•D) satisfies h◦d=f, we obtain that iA•(B•D)◦d=t and therefore d=g. As a consequence, there exists an isomorphism nA,B,C :A•(B•D)→ (A•B)•Dsuch that (5) i(A•B)•D◦nA,B,D =h. The isomorphism nA,B,C is a morphism of unital magmas because by (4), (5) and the naturality of cwe have (iA•B⊗D)◦i(A•B)•D◦nA,B,D ◦ηA•(B•D) = (iA•B⊗D)◦h◦ηA•(B•D) = (A⊗iB•D)◦iA•(B•D)◦ηA•(B•D) =ηA⊗ηB⊗ηD = (iA•B⊗D)◦i(A•B)•D◦η(A•B)•D and (iA•B⊗D)◦i(A•B)•D◦nA,B,D ◦µA•(B•D) = (iA•B⊗D)◦h◦µA•(B•D) = (A⊗iB•D)◦iA•(B•D)◦µA•(B•D) = (A⊗iB•D)◦µA⊗(B•D)◦(iA•(B•D)⊗iA•(B•D)) =µA⊗B⊗D◦(((A⊗iB•D)◦iA•(B•D))⊗((A⊗iB•D)◦iA•(B•D))) =µA⊗B⊗D◦(((iA•B⊗D)◦h)⊗((iA•B⊗D)◦h)) = (iA•B⊗D)◦µA•B⊗D◦(h⊗h) = (iA•B⊗D)◦µA•B⊗D◦((i(A•B)•D◦nA,B,C )⊗(i(A•B)•D◦nA,B,C)) = (iA•B⊗D)◦i(A•B)•D◦µ(A•B)•D◦(nA,B,D ⊗nA,B,D). Finally, using a similar reasoning, we obtain that nA,B,C is a morphism of right H-comodules because (iA•B⊗D⊗H)◦(i(A•B)•D⊗H)◦ρ(A•B)•D◦nA,B,D = (A⊗B⊗ρD)◦(iA•B⊗D)◦i(A•B)•D◦nA,B,C = (A⊗B⊗ρD)◦(iA•B⊗D)◦h = (A⊗B⊗ρD)◦(A⊗iB•D)◦iA•(B•D) = (A⊗iB•D⊗H)◦(A⊗ρB•D)◦iA•(B•D) = (A⊗iB•D⊗H)◦(iA•(B•D)⊗H)◦ρA•(B•D) = (iA•B⊗D⊗H)◦(h⊗H)◦ρA•(B•D) = (iA•B⊗D⊗H)◦(i(A•B)•D⊗H)◦(nA,B,D ⊗H)◦ρA•(B•D). 532 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA where the first equality follows by the definition, the second, the fourth and the fifth ones by the naturality and symmetry of c, the third and the ninth ones by (ii) of Lemma 2.5, the sixth one by the cocommutativity of Hand, finally, the eighth and the tenth ones by the naturality of c. Then, there exists a unique morphism βA,B :A•B⊗H→A•B⊗A•B such that (11) (iA•B⊗A•B)◦βA,B =αA,B and then (12) (iA•B⊗iA•B)◦βA,B = Γ−1 A⊗B◦(iA•B⊗δH). The morphism βA,B satisfies (iA•B⊗H)◦γA•B◦βA,B = (µA⊗B⊗H)◦(iA•B⊗((iA•B⊗H)◦ρA•B)) ◦βA,B = (µA⊗B⊗H)◦(iA•B⊗((A⊗ρB)◦iA•B)) ◦βA,B = (µA⊗γB)◦(A⊗cB,A ⊗B)◦Γ−1 A⊗B◦(iA•B⊗δH) = (((A⊗εH)◦γA◦γ−1 A)⊗B⊗H)◦(A⊗cB,H ⊗H)◦(iA•B⊗δH) =iA•B⊗H and by the cocommutativity of Hwe have (iA•B⊗iA•B)◦βA,B ◦γA•B = Γ−1 A⊗B◦(iA•B⊗δH)◦(µA•B⊗H)◦(A•B⊗ρA•B) = Γ−1 A⊗B◦(µA⊗B⊗δH)◦(iA•B⊗((A⊗ρB)◦iA•B)) = Γ−1 A⊗B◦(A⊗B⊗cH,H )◦(µA⊗γB⊗H)◦(A⊗cB,A ⊗ρB) ◦(iA•B⊗iA•B) = (A⊗cA,B ⊗B)◦(γ−1 A⊗B⊗B)◦(A⊗cB,H ⊗B) ◦(µA⊗B⊗(cB,H ◦ρB)) ◦(A⊗cB,A ⊗B)◦(iA•B⊗iA•B) = (A⊗cA,B ⊗B)◦(γ−1 A⊗B⊗B)◦(µA⊗cB,H ⊗B) ◦(A⊗cB,A ⊗H⊗B)◦(iA•B⊗((ρA⊗B)◦iA•B)) = (((A⊗cA,B)◦((γ−1 A◦γA)⊗B)) ⊗B)◦(A⊗cB,A ⊗B)◦(iA•B⊗iA•B) =iA•B⊗iA•B. Taking into account that His flat and that A•Bis faithfully flat we obtain that βA,B is the inverse of the canonical morphism γA•B. Now we assume that Aand Bare strong Galois H-objects. To prove that A•Bis a strong Galois H-object we only need to show that fA•B:H→(A•B)e is a morphism of unital magmas. If fAand fBare morphisms of unital magmas, by the properties of iA•Band the naturality of cwe have (iA•B⊗iA•B)◦fA•B◦ηH= (A⊗cA,B ⊗B)◦((fA◦ηH)⊗(fB◦ηH)) THE GROUP OF STRONG GALOIS OBJECTS 533 =ηA⊗ηB⊗ηA⊗ηB= (iA•B⊗iA•B)◦η(A•B)e and (iA•B⊗iA•B)◦µ(A•B)e◦(fA•B⊗fA•B) = (A⊗cA,B ⊗B)◦((µAe◦(fA⊗fA)) ⊗(µBe◦(fB⊗fB)) ◦δH⊗H = (A⊗cA,B ⊗B)◦(fA⊗fB)◦δH◦µH= (iA•B⊗iA•B)◦fA•B◦µH. Therefore, fA•B◦ηH=η(A•B)eand µ(A•B)e◦(fA•B⊗fA•B) = fA•B◦µH. Proposition 2.7. Let Hbe a cocommutative Hopf quasigroup and Aa Galois H-object. Then the right H-comodule magma Adefined in Proposition 1.6 is a Galois H-object. Moreover, if Ais strong so is A. Proof. To prove that Ais a Galois H-object we only need to show that γAis an isomorphism. We begin by proving the following identity: (13) (A⊗(µH◦cH,H ◦(λH⊗H))) ◦(ρA⊗H)◦γA=γA◦cA,A. Indeed: (A⊗(µH◦cH,H ◦(λH⊗H))) ◦(ρA⊗H)◦γA = (µA⊗(µH◦(H⊗µH)◦(cH,H ⊗H)◦(H⊗cH,H )◦(cH,H ⊗H) ◦(λH⊗λH⊗H)◦(H⊗δH))) ◦(A⊗cH,A ⊗H)◦(ρA⊗ρA) = (µA⊗(µH◦(λH⊗µH)◦(δH⊗H)◦cH,H ◦(λH⊗H))) ◦(A⊗cH,A ⊗H)◦(ρA⊗ρA) = (µA⊗H)◦(A⊗cH,A)◦(ρA⊗A) =γA◦cA,A, where the first equality follows by (b2) of Definition 1.3, (1) and the naturality of c, the second one by the cocommutativity of Hand the naturality of c, the third one by (a2-1) of Definition 1.1 and the last one by the symmetry and naturality of c. Define the morphism γ′ A:A⊗H→A⊗Aby (14) γ′ A=cA,A ◦γ−1 A◦(A⊗(µH◦cH,H )) ◦(ρA⊗H). Then, by (13), the naturality of c, the cocommutativity of Hand (a2-2) of Definition 1.1, we have the following: γA◦γ′ A =γA◦cA,A ◦γ−1 A◦(A⊗(µH◦cH,H )) ◦(ρA⊗H) = (A⊗(µH◦cH,H ◦(λH⊗H))) ◦(ρA⊗(µH◦cH,H )) ◦(ρA⊗H) = (A⊗(µH◦(µH⊗H)◦(H⊗cH,H )◦(cH,H ⊗H)◦(λH⊗cH,H ) ◦(δH⊗H))) ◦(ρA⊗H) = (A⊗((µH◦(µH⊗H)◦(H⊗λH⊗H)◦(H⊗δH)) ◦cH,H )) ◦(ρA⊗H) =idA⊗H. 534 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA Moreover, by a similar reasoning but using (a2-1) of Definition 1.1 instead of (a2-2) we obtain γ′ A◦γA =cA,A ◦γ−1 A◦((µA◦cA,A)⊗(µH◦(H⊗µH)◦(H⊗λH⊗H) ◦(δH⊗H)◦cH,H )) ◦(A⊗cH,A ⊗H)◦(ρA⊗ρA) =cA,A ◦γ−1 A◦γA◦cA,A =idA⊗A. Therefore, γAis an isomorphism and Aa Galois H-object. Finally, it is easy to show that fA=cA,A ◦fA. Then, if fAis a morphism of unital magmas, so is fA. Thus if Ais strong, Ais strong.  Proposition 2.8. Let Hbe a cocommutative flat Hopf quasigroup and Aa Galois H-object. Then A•Ais isomorphic to Has right H-comodules. Moreover, if Ais strong, the previous isomorphism is a morphism of right H-comodule magmas. Proof. First note that, by Proposition 2.4, we know that (7) is an equalizer diagram and then so is ✲ ✲ ✲ HA⊗H⊗H ηA⊗HρA⊗H A⊗ηH⊗H A⊗H because His flat. For the morphism γA◦iA•A:A•A:→A⊗Hwe have the following: (ρA⊗H)◦γA◦iA•A = (µA⊗H⊗λH)◦(ρA⊗((A⊗δH)◦ρA)) ◦iA•A = (µA⊗H⊗λH)◦(A⊗H⊗((A⊗δH)◦ρA)) ◦(A⊗(cA,H ◦(A⊗λH)◦ρA)) ◦iA•A = (µA⊗µH⊗λH)◦(A⊗((A⊗H⊗δH)◦(A⊗cH,H ) ◦(A⊗((H⊗λH)◦δH)))) ◦(A⊗ρA)◦iA•A = (µA⊗((µH◦((λH⊗H)◦δH)⊗λH)◦δH)) ◦(A⊗ρA)◦iA•A = (A⊗ηH⊗H)◦γA◦iA•A, where the first equality follows by the naturality of cand (b2) of Definition 1.3, the second one because ρ1 A⊗A◦iA•A=ρ2 A⊗A◦iA•A, the third one relies on the symmetry and the naturality of c, the fourth one follows by (1) and the last one by (3). Therefore, there exists an unique morphism hA:A•A→Hsuch that (15) (ηA⊗H)◦hA=γA◦iA•A. THE GROUP OF STRONG GALOIS OBJECTS 535 The morphism hAis a right comodule morphism because by the cocommutativity of H, (1) and the comodule properties of A, we have ηA⊗((hA⊗H)◦ρA•A) = ((γA◦iA•A)⊗H)◦ρA•A = ((µA◦cA,A)⊗((λH⊗λH)◦δH)) ◦(A⊗ρA)◦iA•A = ((µA◦cA,A)⊗((λH⊗λH)◦cH,H ◦δH)) ◦(A⊗ρA)◦iA•A = ((µA◦cA,A)⊗(δH◦λH)) ◦(A⊗ρA)◦iA•A = (A⊗δH)◦γA◦iA•A =ηA⊗(δH◦hA) and using that ηA⊗H⊗His an equalizer morphism we obtain (h⊗H)◦ρA•A= δH◦hA. On the other hand, for fA:H→A⊗Awe have the following (γA⊗H)◦ρ1 A⊗A◦fA = (µA⊗λH⊗H)◦(cA,A ⊗H⊗H)◦(A⊗ρA⊗H)◦(A⊗cH,A) ◦(ρA⊗A)◦cA,A ◦fA = (µA⊗λH⊗H)◦(A⊗cH,A ⊗H)◦(ρA⊗ρA)◦fA = (µA⊗((λH⊗H)◦γ−1 H◦γH)) ◦(A⊗cH,A ⊗H)◦(ρA⊗ρA)◦fA = (A⊗(λH◦µH)⊗H)◦(ρA⊗((λH⊗H)◦δH)) ◦γA◦fA =ηA⊗(((λH◦λH)⊗H)◦δH) =ηA⊗δH, where the first equality follows because fA=cA,A ◦fA, the second one by the symmetry and the naturality of c. In the third one we used that His a Galois H-object and the fourth and the sixth ones are a consequence of (b1) of Definition 1.3. Finally, in the fifth one we applied that Ais a Galois H-object, and the last one relies on the cocommutativity of H. Also (γA⊗H)◦ρ2 A⊗A◦fA = ((µA◦cA,A)⊗((λH⊗λH)◦δH)) ◦(A⊗ρA)◦cA,A ◦fA = (µA⊗((λH⊗λH)◦δH)) ◦(A⊗cH,A)◦(ρA⊗A)◦fA = (µA⊗((εH⊗((λH⊗λH)◦δH)) ◦cH,H ◦((µH◦(µH⊗λH) ◦(H⊗δH)) ⊗H))) ◦(A⊗cH,A ⊗δH)◦(ρA⊗ρA)◦fA = (A⊗((λH⊗λH)◦δH)) ◦(A⊗µH)◦((ρA◦ηA)⊗λH) =ηA⊗δH, where the first equality follows by (b1) of Definition 1.3 and the comodule properties of A, the second one by the naturality of c, the third one by (a2-2) of Definition 1.1 and the counit properties, the fourth one by (b2) of Definition 536 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA 1.3 and in the last one we used that Ais a Galois H-object, (b1) of Definition 1.3 and the cocommutativity of H. Then, ρ1 A⊗A◦fA=ρ2 A⊗A◦fAand, as a consequence, there exists a unique morphism h′ A:H→A•Asuch that (16) iA•A◦h′ A=fA. Therefore, by (15) and (16) we have iA•A◦h′ A◦hA=fA◦hA=γ−1 A◦(ηA⊗H)◦hA=γ−1 A◦γA◦iA•A=iA•A and (ηA⊗H)◦hA◦h′ A=γA◦iA•A◦h′ A=γA◦fA=γA◦γ−1 A◦(ηA⊗H) = (ηA⊗H). Then, h′ A◦hA=idA•Aand hA◦h′ A=idHand his an isomorphism. Finally, assume that Ais strong. By (16) and the equality fA=cA,A ◦fA we obtain that h′ Ais a morphism of unital magmas. Then hAis a morphism of unital magmas and the proof is finished.  Remark 2.9.Note that, in the Hopf algebra setting, for any Galois H-object A, the morphism hAobtained in the previous proposition is a morphism of monoids because this property can be deduced from the associativity of the product defined in A. In the Hopf quasigroup world this proof does not work because Ais a magma. Theorem 2.10. Let Hbe a cocommutative faithfully flat Hopf quasigroup. The set of isomorphism classes of Galois H-objects is a commutative monoid. Moreover, the set of isomorphism classes of strong Galois H-objects is a commutative group. Proof. Let GalC(H) be the set of isomorphism classes of Galois H-objects. For a Galois H-object Awe denote its class in GalC(H) by [A]. By by Propositions 2.6 and 1.8, the product (17) [A]·[B] = [A•B] is well-defined. By Propositions 1.10, 1.9 and 1.11 we obtain that GalC(H) is a commutative monoid with unit [H]. If we denote by Gals C(H) the set of isomorphism classes of strong Galois H-objects, with the product defined in (17) for Galois H-objects, Gals C(H) is a commutative group because by (ii) of Proposition 2.6 the product of strong Galois H-objects is a strong Galois H-object, by Example 2.2 we know that H is a strong Galois H-object and by Propositions 2.7 and 2.8, the inverse of [A] in Gals C(H) is [A].  Definition 2.11. Let Hbe a cocommutative faithfully flat Hopf quasigroup. If Ais a (strong) Galois H-object, we will say that Ahas a normal basis if (A, ρA) is isomorphic to (H, δH) as right H-comodules. We denote by nAthe H-comodule isomorphism between Aand H. THE GROUP OF STRONG GALOIS OBJECTS 537 Obviously, NC(H), the set of isomorphism classes of Galois H-objects with normal basis, is a submonoid of GalC(H) because H= (H, δH) is a Galois Hobject with normal basis and if A,Bare Galois H-objects with normal basis and associated isomorphisms nA,nBrespectively, then A•Bis a Galois H-object with normal basis and associated H-comodule isomorphism nA•B=rH◦nA•nB where nA•nBis defined as in Proposition 1.8 and rHis the isomorphism defined in Proposition 1.11. Moreover, for a strong Galois H-object with normal basis A, with associated isomorphism nA, we have that A= (A, ρA) is also a strong Galois H-object with normal basis, where nA=λH◦nA, and then, if we denote by Ns C(H) the set of isomorphism classes of strong Galois H-objects with normal basis, Ns C(H) is a subgroup of Gals C(H). Note that, if His a Hopf algebra we have that Gals C(H) = GalC(H) and Ns C(H) = NC(H). Therefore, in the associative setting we recover the classical group of Galois H-objects. Remark 2.12.In this remark we use some classical results of algebraic K-theory (see [3] for the details). Let G(C, H) and Gs(C, H) be the categories of Galois H-objects and strong Galois H-objects, respectively. Then, by Proposition 1.13 these categories are symmetric monoidal and then they are categories with product. The Grothendieck group of G(C, H) is the abelian group generated by the isomorphisms classes of objects Aof G(C, H) module the relations [A•B] = [A]·[B]. This group will be denoted by K0G(C, H) and, by the general theory of Grothendieck groups, we know that for A,Bin G(C, H), [A] = [B] in K0G(C, H) if and only if there exists a Din G(C, H) such that A•Dis isomorphic in G(C, H) to B•D. The unit of K0G(C, H) is [H]. In a similar way we can define K0Gs(C, H), but in this case K0Gs(C, H) = Gals C(H) because the set of isomorphism classes of objects of Gs(C, H) is a group. The inclusion functor i:Gs(C, H)→G(C, H) is a product preserving functor and then we have a group morphism K0i:Gals C(H)→K0G(C, H). If [A]∈ Ker(K0i) we have that [A] = [H] in K0G(C, H). Then there exists Din G(C, H) such that A•D∼ =H•D∼ =Din G(C, H). As a consequence A•D•D∼ =D•Din G(C, H). Then, By Proposition 2.8, A∼ =Has right H-comodules. Therefore Ais a strong Galois H-object with normal basis and Ker(K0i) is a subgroup of Ns C(H). The full subcategory H={H}of Gs(C, H) is cofinal because, for all Ain Gs(C, H), A•A∼ =Has right H-comodule magmas. Therefore, the Whitehead group of Gs(C, H) is isomorphic to the Whitehead group of H. Therefore, K1Gs(C, H)∼ =AutGs(C,H)(H). The group AutGs(C,H)(H) admits a good explanation in terms of grouplike elements of a suitable Hopf quasigroup if His finite, that is, if there exists an object H∗in Cand an adjunction H⊗ − ⊣ H∗⊗ −. For this adjunction we will denote with aH:idC→H∗⊗H⊗ − and bH:H⊗H∗⊗ − → idCthe unit and the counit respectively. The object H∗will be called the dual of H. 538 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA A Hopf coquasigroup Din Cis a monoid (D, ηD, µD) and a counital comagma (D, εD, δD) such that the following axioms hold: (d1) εDand δDare morphisms of monoids. (d2) There exists λD:D→Din C(called the antipode of D) such that: (d2-1) (µD⊗D)◦(λD⊗δD)◦δD =ηD⊗D = (µD⊗D)◦(D⊗((λD⊗D)◦δD)) ◦δD. (d2-2) (D⊗µD)◦(δD⊗λD)◦δD =D⊗ηD = (D⊗µD)◦(((D⊗λD)◦δD)⊗D)◦δD. As in the case of quasigroups, the antipode is unique, antimultiplicative, anticomultiplicative, leaves the unit and the counit invariable and satisfies (3). If Dis a Hopf coquasigroup we define G(D) as the set of morphisms h: K→Dsuch that δD◦h=h⊗hand εD◦h=idK. If Dis commutative, G(D) with the convolution h∗g=µD◦(h⊗g) is a commutative group, called the group of grouplike morphisms of D. Note that the unit element of G(D) is ηD and the inverse of h∈G(D) is h−1=λD◦h. It is easy to show that, if His a finite cocommutative Hopf quasigroup, its dual H∗is a commutative finite Hopf coquasigroup where: ηH∗= (H∗⊗εH)◦aH, µH∗= (H∗⊗bH)◦(H∗⊗H⊗bH⊗H∗)◦(H∗⊗δH⊗H∗⊗H∗) ◦(aH⊗H∗⊗H∗)), εH∗=bH◦(ηH⊗H∗), δH∗= (H∗⊗H∗⊗(bH◦(µH⊗H∗))) ◦(H∗⊗aH⊗H⊗H∗)◦(aH⊗H∗)) and the antipode is (H∗⊗bH)◦(H∗⊗λH⊗H∗)◦(aH⊗H∗). The groups G(H∗) and AutGs(C,H)(H) are isomorphic. The proof is equal to the one given in Proposition 3.7 of [14]. If α∈AutGs(C,H)(H), the morphism zα= (H∗⊗(εH◦α)) ◦aHis in G(H∗). Then, we define the map AutGs(C,H)(H)→G(H∗) by z(α) = zα. On the other hand, if h∈G(H∗), then xh= (H⊗bH)◦(δH⊗h) : H→His a morphism of Galois H-objects and then, by Remark 1.4, it is an isomorphism, that is xh∈AutGs(C,H)(H). The map x:G(H∗)→AutGs(C,H)(H) defined by x(h) = xhis the inverse of z. Therefore, K1Gs(C, H)∼ =G(H∗). Finally, Ns(C, H) is the subcategory of Gs(C, H) whose objects are the strong Galois H-objects with normal basis, note that H={H}it is also cofinal in Ns(C, H) and then K1Ns(C, H)∼ =G(H∗). THE GROUP OF STRONG GALOIS OBJECTS 539 3. Invertible comodules with geometric normal basis This section is devoted to study the connections between Galois H-objects and invertible comodules with geometric normal basis. First of all, we introduce the notion of invertible comodule with geometric normal basis which is a generalization to the non associative setting of the one defined by Caenepeel in [6]. Definition 3.1. Let Hbe a cocommutative faithfully flat Hopf quasigroup. A right H-comodule M= (M, ρM) is called invertible with geometric normal basis if there exist a faithfully flat unital magma Sand an isomorphism hM: S⊗M→S⊗Hof right H-comodules such that hMis almost lineal, that is (18) hM= (µS⊗H)◦(S⊗(hM◦(ηS⊗M))). A morphism between two invertible right H-comodules with normal basis is a morphism of right H-comodules. Note that, if Sis a monoid, hMis a morphism of left S-modules, for ϕS⊗M= µS⊗Mand ϕS⊗H=µS⊗H, if and only if (18) holds. Then in the Hopf algebra setting this definition is the one introduced by Caenepeel in [6]. Example 3.2. Let Hbe a cocommutative faithfully flat Hopf quasigroup and let A= (A, ρA) be a Galois H-object. Then A= (A, ρA) is an invertible right H-comodule with geometric normal basis because hA=γAis an isomorphism of right H-comodules and trivially γAis almost lineal. In particular, H= (H, δH) is an example of invertible right H-comodule with geometric normal basis. Proposition 3.3. Let Hbe a cocommutative faithfully flat Hopf quasigroup and M,Nbe invertible right H-comodules with geometric normal basis. Then the right H-comodule M•N= (M•N, ρM•N), where M•Nand ρM•Nare defined as in Proposition 1.7, is a right H-comodule with geometric normal basis. Proof. Let S,Rand hM,hNbe the faithfully flat unital magmas and the isomorphisms of right H-comodules associated to Mand Nrespectively. Then T=S⊗Ris faithfully flat. On the other hand, ✲ ✲ ✲ T⊗M•N T ⊗M⊗N T ⊗M⊗N⊗H T⊗iM•N T⊗ρ1 M⊗N T⊗ρ2 M⊗N and ✲ ✲ ✲ T⊗H T ⊗H⊗H T ⊗H⊗H⊗H T⊗δH T⊗ρ1 H⊗H T⊗ρ2 H⊗H are equalizer diagrams and for the morphism gM⊗N=(S⊗cR,H ⊗H)◦(hM⊗hN)◦(S⊗cR,M ⊗N):S⊗R⊗M⊗N→S⊗R⊗H⊗H 540 J. N. A. ´ ALVAREZ, R. G. RODR´ IGUEZ, AND J. M. F. VILABOA we have that (S⊗R⊗ρ1 H⊗H)◦gM⊗N◦(S⊗R⊗iM•N) = (S⊗cH,R ⊗cH,H )◦(S⊗H⊗cH,R ⊗H)◦(((S⊗δH)◦hM)⊗hN) ◦(S⊗cR,M ⊗N)◦(S⊗R⊗iM•N) = (S⊗cH,R ⊗cH,H )◦(S⊗H⊗cH,R ⊗H)◦(((hM⊗H)◦(S⊗ρM)) ⊗hN) ◦(S⊗cR,M ⊗N)◦(S⊗R⊗iM•N) = (((S⊗cH,R ⊗H)◦(hM⊗hN)) ⊗H)◦(S⊗cR,M ⊗N⊗H) ◦(S⊗R⊗((M⊗cH,N )◦(ρM⊗N)◦iM•N)) = (((S⊗cH,R ⊗H)◦(hM⊗hN)) ⊗H)◦(S⊗cM,R ⊗((M⊗ρN)◦iM•N)) = (S⊗R⊗ρ2 H⊗H)◦gM⊗N◦(S⊗R⊗iM•N), where the first and the third equalities follow by the naturality of c, the second and the fifth ones by the comodule morphism condition for hMand hN respectively and finally the fourth one by the properties of iM•N. Therefore, there exists a unique morphism hM•N:T⊗M•N→T⊗Hsuch that (19) (T⊗δH)◦hM•N=gM⊗N◦(T⊗iM•N). Moreover, if we define the morphism g′ M⊗N=(S⊗cM,R⊗N)◦(h−1 M⊗h−1 N)◦(S⊗cR,H ⊗H):S⊗R⊗H⊗H→S⊗R⊗M⊗N by the naturality of c, the comodule morphism condition for h−1 Mand h−1 Nand the cocommutativity of H, the following equalities hold (S⊗R⊗ρ1 M⊗N)◦g′ M⊗N◦(S⊗R⊗δH) = (S⊗cM,R ⊗cH,N )◦(S⊗M⊗cH,R ⊗N)◦(((S⊗ρM)◦h−1 M)⊗h−1 N) ◦(S⊗cH,R ⊗H)◦(S⊗R⊗δH) = (S⊗cM,R ⊗cH,N )◦(S⊗M⊗cH,R ⊗N)◦(((h−1 M⊗H)◦(S⊗δH)) ⊗h−1 N) ◦(S⊗cH,R ⊗H)◦(S⊗R⊗δH) = (g′ M⊗N⊗H)◦(S⊗R⊗((H⊗δH)◦δH)) = (S⊗R⊗ρ2 M⊗N)◦g′ M⊗N◦(S⊗R⊗δH). As a consequence, there exists a unique morphism h′ M•N:T⊗H→T⊗M•N such that (20) (T⊗iM•N)◦h′ M•N=g′ M⊗N◦(T⊗δH). Thus, by (19) and (20) hM•N◦h′ M•N= (T⊗((εH⊗H)◦δH)) ◦hM•N◦h′ M•N = (T⊗εH⊗H)◦gM⊗N◦(T⊗iM•N)◦h′ M•N = (T⊗εH⊗H)◦gM⊗N◦g′ M⊗N◦(T⊗δH) = idT⊗H THE GROUP OF STRONG GALOIS OBJECTS 541 and (T⊗iM•N)◦h′ M•N◦hM•N=g′ M⊗N◦(T⊗δH)◦hM•N =g′ M⊗N◦gM⊗N◦(T⊗iM•N) = T⊗iM•N and then hM•Nis an isomorphism with inverse h−1 M•N=h′ M•N. The morphism hM•Nis a morphism of right H-comodules because (hM•N⊗H)◦(T⊗ρM•N) = (T⊗((H⊗εH)◦δH)⊗H)◦(hM•N⊗H)◦(T⊗ρM•N) = (T⊗H⊗εH⊗H)◦(gM⊗N⊗H)◦(T⊗((iM•N⊗H)◦ρM•N)) = (T⊗H⊗εH⊗H)◦(gM⊗N⊗H)◦(T⊗((M⊗ρN)◦iM•N)) = (T⊗H⊗((H⊗εH)◦δH)) ◦gM⊗N◦(T⊗iM•N) = (T⊗δH)◦hM•N, where the first equality follows by the counit property, the second and the last ones by (19), the third one the properties of ρM•Nand the fourth one by the comodule condition for hN. Finally, we will prove that hM•Nis almost lineal. Indeed: (µT⊗H)◦(T⊗(hM•N◦(ηT⊗M•N))) = (µT⊗((εH⊗H)◦δH)) ◦(T⊗(hM•N◦(ηT⊗M•N))) = (µS⊗R⊗εH⊗H)◦(S⊗R⊗(gM⊗N◦(ηS⊗ηR⊗iM•N))) = (S⊗εH⊗R⊗H)◦(((µS⊗H)◦(S⊗(hM◦(ηS⊗M)))) ⊗((µR⊗H) ◦(R⊗(hN◦(ηR⊗N))))) ◦(S⊗cR,M ⊗N)◦(S⊗R⊗iM•N) = (T⊗εH⊗H)◦gM⊗N◦(T⊗iM•N) = (T⊗((εH⊗H)◦δH)) ◦hM•N =hM•N. In the last equalities, the first and the sixth ones follow by the properties of the counit, the second and the fifth ones by (19), the third one is a consequence of the naturality of cand the fourth one relies on the almost lineal condition for hMand hN. As a direct consequence of this proposition we have the following theorem. Theorem 3.4. Let Hbe a cocommutative faithfully flat Hopf quasigroup. If we denote by Pgnb(K, H)the category whose objects are the invertible right H-comodules with geometric normal basis and whose morphisms are the morphisms of right H-comodules between them, Pgnb(K, H)with the product defined in the previous proposition is a symmetric monoidal category where the unit object is Hand the symmetry isomorphisms, the left, right an associative constraints are defined as in Proposition 1.13. Moreover, the set of isomorphism classes in Pgnb(K, H)is a monoid that we will denote by P icgnb(K, H).