arXiv subjects
Publications and source records attributed to Che Thi Kim Phung.
In this paper, we state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of obstructions for symmetric monoidal functors and symmetric cohomology groups are applied to show a treatment of the group extension problem of the type of an abelian crossed module.
If $Γ$ is a group, then braided $Γ$-crossed modules are classified by braided strict $Γ$-graded categorial groups. The Schreier theory obtained for $Γ$-module extensions of the type of an abelian $Γ$-crossed module is a generalization of the theory of $Γ$-module extensions.
There are different categorizations of the definition of a {\it ring} such as {\it Ann-category} (see N. T. Quang [6]), {\it ring category} (see M. Kapranov and V.Voevodsky [2]),... The main result of this paper is to prove that every axiom of the definition of a {\it ring category}, without the axiom $x_{0}=y_{0},$ can be deduced from the axiomatics of an {\it Ann-category}.