Cat$^1$-Objects in the Category of Crossed Modules

The main aim of this study is to investigate cat$^1$-objects within the category of crossed modules, which we call cat$^1$-crossed modules, and to characterize these structures systematically. First, fundamental notions from category theory and crossed modules are presented, followed by the definition and properties of cat$^1$-groups. Subsequently, cat$^1$-crossed modules are introduced in the category of crossed modules, and their homotopical and algebraic properties are explored. Furthermore, natural equivalences are studied in detail between cat$^1$-crossed modules and other algebraic structures such as crossed squares, cat$^2$-groups, and double group-groupoids. The results obtained represent a significant step toward a deeper understanding of higher-dimensional algebraic models.