Representations of braided regular crossed modules

The main aim of this study is to investigate the relationship between Braided regular crossed modules, which are among high categorical structures, and Gray categories; and to demonstrate the existence of an equivalence with Gray 3-group(oid)s with a single object. Through this equivalence, the aim is to construct a regular representation and present a cat$^2$-group version of Cayley's theorem. This thesis consists of eight chapters. In the first chapter, the basic concepts are given, and in the second chapter, the connected structures between braided regular crossed modules and Gray 3-group(oid)s are given. In the third and fourth chapters, all axioms related to the transition from braided regular crossed modules to the Gray 3-group(oid) with a single object structure, and on the other hand, from Gray 3-group(oid)s with a single object to braided regular crossed modules, have been examined. In the fifth chapter, the equivalence between structures is addressed with the help of functors. In chapter six, it was shown that the structure $\mathfrak{C^2}$ is a cat$^2$-group. By using the group algebra functor, the cat$^2$-group algebra was obtained. From this resulting structure, a chain complex of length two was constructed. In chapter seven, using the 2-length chain complex constructed earlier, a lax 3-functor was defined between braided regular crossed modules and cat$^2$-group algebras automorphisms. It was explained that this 3-functor constitutes a regular representation from braided regular crossed modules to cat$^2$-groups. In the final chapter, the Cayley theorem is defined for braided regular crossed modules, and the representation of this theorem in the form of a cat$^2$-group version is obtained.