The homotopy classification of graded categorical groups and their homomorphisms 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 Clifford 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.
Keywords: graded categorical group, graded monoidal functor, equivariant group cohomology, crossed product, factor set, monoid extension, group extension, ring-group extension, Clifford system, Azumaya algebra, graded bialgebra, graded Hopf algebra
2000 MSC: 18D10, 20J06, 20M10, 20M50, 16S35, 16W50, 16H05, 16W30
Theory and Applications of Categories
, Vol. 11, 2003,
No. 10, pp 215-251.