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Graded Extensions of Monoidal Categories Journal of Algebra 241, 620᎐657Ž. 2001 doi:10.1006rjabr.2001.8769, available online at http:rrwww.idealibrary.com on Graded Extensions of Monoidal Categories A. M. Cegarra, A. R. Garzon,´ and J. A. Ortega1 Departamento de Algebra,´ Facultad de Ciencias, Uni¨ersidad de Granada, View metadata, citation and similar papers at 18071core.ac.uk Granada, Spain brought to you by CORE E-mail: [email protected] provided by Elsevier - Publisher Connector Communicated by Susan Montgomery Received March 3, 2000 The long-known results of Schreier᎐Eilenberg᎐Mac Lane on group extensions are raised to a categorical level, for the classification and construction of the manifold of all graded monoidal categories, the type being given group ⌫ with 1-component a given monoidal category. Explicit application is made to the classification of strongly graded bialgebras over commutative rings. ᮊ 2001 Aca- demic Press Key Words: graded category; monoidal category; factor set; crossed product; group cohomology; nonabelian cohomology; theory of obstructions; coalgebra; strongly graded ring. 1. INTRODUCTION AND SUMMARY Graded monoidal categories were introduced by Frohlich¨ and C.T.C. Wall inwx 8 , where they presented a suitable abstract setting to study the Brauer group in equivariant situations. This paper is concerned with the analysis and classification of these graded monoidal categories, following a parallel treatment to that made inwx 2 for the non-monoidal case. In any graded monoidal category, its 1-component, or subcategory of all morphisms of grade 1, inherits a monoidal structure, and the graded category can be regarded as an extension of that monoidal subcategory by the group of grades. In developing this point of view, we are led to the problem of extending monoidal categories by groups, that is, the classifica- tion and construction of the manifold of all graded monoidal categories, the type being given group ⌫ with 1-component isomorphic to a given monoidal category Ž.C, m . This problem, including the theory of obstruc- 1 Supported by project DGICYT of Spain, PB97-0829. 620 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. GRADED EXTENSIONS OF MONOIDAL CATEGORIES 621 tions, is studied by means of factor systems and various homological invariants, generalized from Schreier᎐Eilenberg᎐Mac Lane group exten- sion theorywx 6, 17 . For the reader’s interest and due to its potential interest for ring theorists, we have included a section where we explain the meaning of our abstract results in the particular case in which graded monoidal categories are extensions of categories of modules with tensor product. Our conclu- sions are here related to strongly graded algebras in the sense of Dadew 4, x wx 14Ž also called Clifford systems in 3. , S s [␴ g ⌫ S␴ endowed with a bialgebra structure such that every ␴-component S␴ is a subcoalgebra. In Section 2, after we state a minimum of necessary notation and terminology, we analyze the structure of ⌫-monoidal extensions of a monoidal category Ž.C, m . We explain a factor set theory for these extensions; namely, a factor set of such an extension consists of a family of monoidal equivalences F ␴, ␴ g ⌫,ofŽ.C, m with itself and isomorphisms of monoidal functors ␪ ␴ , ␶ : F ␴ F␶ ª F ␴␶ , ␴ , ␶ g ⌫, satisfying a cocycle Ž.coherence condition, that is, a pseudo-representation of the group ⌫ by monoidal autoequivalences of Ž.C, m . Any factor set gives rise to a crossed product extension and any extension is isomorphic to a crossed product one. We sum up these results in a bijection ExtŽŽ⌫, C, m( .. ވ2 ŽŽ⌫, C, m .. that states how a survey of all isomorphism classes of extensions of a prescribed pair ŽŽ⌫, C, m ..is obtained in terms of non-abelian cohomol- ogy classes of 2-cocycles, or factor sets, of ⌫ with coefficients in Ž.C, m . Each ⌫-graded extension of a monoidal category Ž.C, m realizes a collective character of ⌫ in Ž.C, m , that is, a group homomorphism ဧ: ⌫ ª PicŽ.C, m of ⌫ into the group of isomorphism classes of monoidal equivalences of Ž.C, m with itself, and this leads to a problem of obstruc- tions. When a collective character is specified, it is possible that no graded monoidal category realizing the specified homomorphism can exist. One of the main goals of this paper is to obtain a necessary and sufficient condition for the existence of graded monoidal categories representing elements in Ext ဧ ŽŽ⌫, C, m: ..Ext ŽŽ⌫, C, m .., the subset of those classes of ⌫-graded extensions of Ž.C, m that realize a given collective character ဧ. This condition is formulated in terms of a certain 3-dimensional group cohomology class TŽ.ဧ , referred to here as the Teichmuller¨ obstruction of ဧ. In Section 3, it is shown that any collective character ဧ determines a character ဧ*: ⌫ ª AutŽŽZ C, m ..* of the group ⌫ in the abelian group of the units of the monoid center of monoidal category Ž.C, m . Hence, n ZŽ.C, m *isa⌫-module and the cohomology groups Hဧ ŽŽ⌫, Z C, m ..*, n G 0, are defined. Section 4 is dedicated to the construction of the 3 Teichmuller¨ obstruction map ဧ ¬ TŽ.ဧ g H ဧ ŽŽ⌫, Z C, m ..* and Section 5 622 CEGARRA, GARZON´ , AND ORTEGA includes two main results, namely: The set Ext ဧ ŽŽ⌫, C, m ..is non-empty if and only if its obstruction vanishes, and If ဧ is unobstructed, then Ext ဧ ŽŽ⌫, C, m ..is a principal homogeneous 2 space under the abelian group Hဧ ŽŽ⌫, Z C, m ..*. We conclude Section 5 by exhibiting a non-realizable collective charac- ter of the cyclic group of order two in the monoidal category of representa- tions of the dihedral group D10 over a field. The last section is devoted to applying the general results to graded m extensions of Ž.RkM, , the monoidal category of modules over a k-bialge- bra R. Our main objective here is the classification of strongly graded bialgebras, which follows from the observation that strongly ⌫-graded s [ bialgebras S ␴ g ⌫ S␴ whose coefficient bialgebra S1 is isomorphic to a given bialgebra R, are in natural and essentially bijective correspondence ⌫ m ⌫ with -monoidal extensions of Ž.RkM, , for any group . Along the way, we show some interesting interpretations such as that of the center of m Ž.RkM, in terms of group-like central elements of the bialgebra or that m of the Picard group of Ž.RkM, in terms of invertible Ž.R, R -coalgebras. 2. GRADED MONOIDAL CATEGORIES AND CROSSED PRODUCTS Hereafter we will regard a group ⌫ as a category with exactly one object, say ), where the morphisms, all being invertible, are the members of ⌫ and the composition law is the group composition operation. A ⌫-grading on a category D wx8 is a functor gr: D ª ⌫. For any morphism f in D, we refer to grŽ.f as the grade of f.A⌫-functor is a functor preserving grades of morphisms, and a natural transformation of ⌫-functors is a natural transformation, ␪: F ª FЈ such that all morphisms ␪ ª Ј g X : FX F X, X D, are of grade 1. A ⌫-graded category Ž.D, gr is called stable if, for any object X g D ; and any ␴ g ⌫, there exists an isomorphism f: X ª Y in D with source X and such that grŽ.f s ␴ . In other words, the grading gr: D ª ⌫ is a cofibration in the sense of Grothendieckwx 9 . For ⌫-graded categories Ž.ŽD, gr and DЈ, gr . , we write D =⌫ DЈ for the pullback of the gradings. Monoidal categories have been used extensively in the literature and we refer towx 11, 12, 16 , for general background about them. DEFINITION 2.1wx 8 . A ⌫-monoidal category Ž.ŽD, gr, ms D, gr, m , a, I, l, r.Ž.is a stable ⌫-graded category D, gr together with ⌫-functors m: D =⌫ D ª D, I: ⌫ ª D, GRADED EXTENSIONS OF MONOIDAL CATEGORIES 623 and natural isomorphisms of grade 1Ž called, respectively, the associativity and the left-unit, and right-unit constraints. s m m ª; m m a aX , Y , Z : Ž.X Y Z X Ž.Y Z , s m ª;;s m ª l l XX: I X X, r r : X I X, where I s IŽ.) such that, for all objects X, Y, Z g D, two coherence conditions hold m и и m s и Ž.Ž.id XYa , Z, TXa , YmZ, TXa , Y , ZTid a X, Y , ZmTXa mY , Z, T , m и s m Ž.id XYl a X, I, YXr id Y . If Ž.Ž.D, gr,m and DЈ, gr, m are ⌫-monoidal categories, then a ⌫-mon- s ⌽ ⌽ ⌫ oidal functor between them, F Ž.F, , 0 , consists of a -functor F: D ª DЈ, natural isomorphisms of grade 1 ⌽ s ⌽ m ª; m X , Y : FXŽ.Y FX FY, and an isomorphism of grade 1Ž natural with respect to the elements ␴ g ⌫. ⌽ ª; 0 : FI I such that, for all objects X, Y, Z g D, coherence conditions hold и⌽ m и⌽ aFX, FY, FZŽ. X, YFZXid mY , Z s m ⌽и⌽и Ž.Ž.id FX Y, ZX, YmZXFa, Y , Z , и m ⌽и⌽ s и⌽m и⌽ s rFXŽ.id FX 0 X , IXFXFr Ž.Ž.; l 0 id FX I, XXFl Ž.. The ⌫-monoidal functor F is called strict when each of the isomor- ⌽ ⌽ phisms X, Y , 0 is an identity. ⌫ ⌽ ⌽ Ј ⌽Ј ⌽X A morphism between -monoidal functors Ž.ŽF, , 00and F , , . is a natural transformation of ⌫-functors ␪: F ª FЈ such that, for all objects X, Y g D, coherence condition holds ␪ m ␪ и⌽ s ⌽X и ␪ ⌽X и ␪ s ⌽ Ž.XY X, YX, YXmY , 0 I 0 .
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