[工程科技]SOME ALGEBRAIC APPLICATIONS OF GRADED CATEGORICAL GROUP THEORY.pdfVIP

Theory and Applications of Categories, Vol. 11, No. 10, 2003, pp. 215–251. SOME ALGEBRAIC APPLICATIONS OF GRADED CATEGORICAL GROUP THEORY ´ A.M. CEGARRA AND A.R. GARZON ABSTRACT. The homotopy classification of graded categorical groups and their homo- morphisms is applied, in this paper, to obtain appropriate treatments for diverse crossed product constructions with operators which appear in several algebraic contexts. Precise classification theorems are therefore stated for equivariant extensions by groups either of monoids, or groups, or rings, or rings-groups or algebras as well as for graded Clif- ford systems with operators, equivariant Azumaya algebras over Galois extensions of commutative rings and for strongly graded bialgebras and Hopf algebras with operators. These specialized classifications follow from the theory of graded categorical groups after identifying, in each case, adequate systems of factor sets with graded monoidal functors to suitable graded categorical groups associated to the structure dealt with. 1. Introduction Graded categorical groups provide a suitable categorical setting for the treatment of an extensive list of subjects with recognized mathematical interest. Let us briefly recall that, if Γ is a group, then a Γ-graded categorical group is a groupoid G equipped with a grading functor gr : G → Γ and with a graded monoidal structure, by graded functors ⊗ : G × G → G and I : Γ → G, such that for each object X , there is an object Γ X′ with a 1-graded arrow X ⊗ X′ → I (see Section 2 for the details). These graded categorical grou