Cat1-objects in the category of crossed modules

The concept of crossed modules was initially introduced in 1946 as a fundamental algebraic model for connected homotopy 2-types. This structure has found applications not only in homotopy theory but also in representation theory of groups, homology and cohomology, algebraic K-theory, and differential geometry. In the following decades, researchers such as Brown, Higgins, and Loday further developed various aspects of crossed modules, enriching these structures from both algebraic and categorical perspectives. In particular, in 1982, the consept of cat¹-groups is introduced and a natural categorical equivalence is established between crossed modules and cat¹-groups, highlighting their deep connection with homotopy types. The main aim of this thesis is to investigate cat¹-objects within the category of crossed modules and to characterize these structures systematically. First, fundamental notions from category theory, group 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, the natural equivalence between cat¹-crossed modules and cat¹-groups is studied in detail, with several illustrative examples provided. In conclusion, this thesis contributes to the theory of crossed modules and demonstrates the applicability of these structures through categorical equivalences. The results obtained represent a significant step toward a deeper understanding of higher-dimensional algebraic models and their potential applications across different areas of mathematics.