2-crossed modules of R-algebroid and related structures

This study consists of four chapters. In the first chapter, concepts frequently used in the thesis are presented. The concept of a 2-crossed module defined on categorical structures considered as single objects, such as group, algebra, and Lie algebra, has been generalized to groupoids, which are a multi-object categorical structure. In the second chapter, a 2-crossed module defined on R-algebroids, which can be considered a multi-object algebra, has been investigated alongside its relationship with simplicial R-algebroids. That is, an equivalence has been established between the category of 2-crossed modules of R-algebroids and the category of simplicial R-algebroids. In the third chapter, the crossed square of R-algebroids has been defined. Additionally, a functor has been defined from the category of crossed squares to the category of simplicial R-algebroids, as well as a functor from the category of simplicial R-algebroids to the category of crossed squares. The aim is to establish an equivalence between categories using these functors. In the last chapter, functors have been defined between the category of 2-crossed modules of R-algebroids defined in the previous chapters and the category of crossed squares.