Some Results on Strict Graded Categorical Groups

We present some applications of strict graded categorical groups to the construction of the obstruction of an equivariant kernel and to the classification of equivariant group extensions which are central ones. The composition of a graded categorical group and an equivariant group homomorphism is also determined. 1. Introduction The group extension problem has an important significance in the development of modern algebra. Some notions of this problem such as crossed product, factor set, and obstruction (see [1]) are not only applied to rings or to algebraic types but also are raised to a categorical level. The theory of graded categorical groups studied by Cegarra et al. [2] can be viewed as a generalization of both the categorical group theory of Sinh [3] and the graded category theory of Fr？hlich and Wall [4]. The equivariant group extension problem is one of applications of this theory. Strict graded categorical groups, with their simple structures compared to the general case, are more likely to give a lot of interesting applications. In [5] we presented an application of this notion to the classification of equivariant crossed modules. In this paper we continue to introduce some other applications. Firstly, we show that if is the third invariant of the strict graded categorical group and is an equivariant kernel; then Secondly, we classify equivariant group extensions of by which are central extensions by graded monoidal autofunctors of the strict graded categorical group . Finally, we construct the composition of a -graded categorical group with a -homomorphism, which is analogous to the composition of a group extension with a group homomorphism (see [1, Chapter 3]). 2. Preliminaries 2.1. Graded Categorical Groups We recall briefly some basic notions about graded categorical groups in [2]. We regard the group as a category with one object, say , where the morphisms are elements of and the composition is the group operation. A category is -graded if there is a functor . The grading is said to be stable if for any object and any there exists an isomorphism in with domain and . A -graded monoidal category consists of(1)a stable -graded category , -graded functors and ,(2)natural isomorphisms of grade 1 , and such that, for all , the following two coherence conditions hold: A graded categorical group is a graded monoidal category in which every object is invertible and every morphism is an isomorphism. In this case, the subcategory consisting of all objects of and all morphisms of grade 1 in is a categorical group. If , are -monoidal categories, then