Cat¹-Objects in the Category of Crossed Modules

The main aim of this study is to investigate cat¹-objects within the category of crossed modules, which we call cat¹-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 cat¹-groups. Subsequently, cat¹-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 cat¹-crossed modules and other algebraic structures such as crossed squares, cat²-groups, and double group-groupoids. The results obtained represent a significant step toward a deeper understanding of higher-dimensional algebraic models.