Higher dimensional representations of crossed and quadratic modules of groupoids

Categorification is a type of structural method in which a mathematical object is replaced by another categorical structure that reveals richer and generally deeper structural properties. With this motivation, our aim in the thesis is to construct a higher-dimensional categorical representation theory of algebraic homotopy type models associated with topological data. First of all, while constructing this representation structure, the crossed module structure on groupoids will be examined in detail. From a crossed module, a strict 2-groupoid structure with more than one object will be constructed. Then, the concept of the quadratic module of groupoids will be defined in detail, and why it does not form a strict 3-groupoid structure will be examined. A Gray groupoid will be obtained from the 3-categorical structure (from sestertius groupoid) constructed from the quadratic module of groupoids. Additionally, it will be shown that the quadratic module of groupoids is a strict cubical trigroupoid. Finally, the categories of length 1 and 2 chain complexes of K-vector spaces, respectively, will be examined, and a 2-linear representation of the crossed module of groupoids and a 3-linear representation of the quadratic module of groupoids will be constructed with these structures.