On the simplicity crossed polymodules

Abstract

The conception of crossed modules was first expressed by Whitehead when he was working on combinatorial theory. This concept has many uses in various fields, such as category theory, algebra, and k-theory. Moreover, one of the equations that plays a very important role in mathematical problems is the Yang-Baxter equation. In fact, although it doesn't seem like it, these equations play an effective role in studies such as particle physics, statistical mechanics, quantum field theory, and quantum groups. We use crossed modules to solve them. In crossed modules, the Actor was defined by Alp. Nilpotent, Solvable, $n$-Complete, and Representations of crossed modules were studied by Dehghanizadeh and Davvaz. In studies of group theory, a simplicial group is an object. Davvaz and Alp studied simplicial polygroups, and the generalized Moore complexes. They proved the existence of a functor from the category cat$^1$-polygroups to the category of groups and, furthermore, conversely. In this paper, we provide simplicity-crossed polymodules and some of their properties. We have also presented a simple crossed polymodules theorem.