LIMITS IN THE CATEGORY OF 2-GENERALIZED CROSSED MODULES

In a previous study, limits were calculated within the category of 2-crossed modules of groups over a fixed group R, where the group actions involved were specifically taken to be conjugation actions. While this framework provides a rich algebraic structure for constructing certain homotopical and categorical structures, the limitation of conjugation-based actions restricts the flexibility of this approach. In this paper, we extend this framework by introducing the notion of 2-generalized crossed modules (2GCM), which generalize the structure of 2-crossed modules by allowing more flexible and arbitrary group actions, rather than restricting them to conjugation actions. Furthermore, we prove that the category of 2-generalized crossed modules is finitely complete, meaning that it possesses all finite limits, such as products and equalizers. This property is important for higher-level categorical analysis and supports the application of 2-generalized crossed modules in both theoretical and applied contexts, particularly in higher-dimensional algebra and homotopy theory.