Algebraic models and gray groupoids

This thesis consists of five chapters. In the first chapter, some basic information used in this thesis is given. In the second chapter, the notion of Gray groupoid is given and a functor is constructed from the category of simplicial groups with Moore complex of length 3 to category of Gray groupoids by using image of hypercrossed complex pairings. In the third chapter, the definition of 3-crossed modules is given and then a functor is defined from this category to that of Gray groupoids furthermore, a functor is constructed from category of crossed 3-cubes to Gray groupoids by using relation between 3-crossed modules and crossed 3-cubes given by Arvasi, Kuspinari, Uslu (2009). In the forth chapter, definitions of the category of simplicial groupoids and simplicially enriched groupoids are given and groupoid version of hypercrossed complex pairings used in this chapter is explained. After that a functor is defined step-by-step from the category of simplicially enriched groupoids with Moore complex of length 2 to Gray groupoids and it is shown that these categories are equivalent by defining adjoint functor. In the fifth chapter, notions of crossed square and 2-crossed modules of groupoids are given. Moreover, relations among categories of simplicially enriched groupoids, crossed square and 2-crossed modules of groupoids are investigated. Also, functors are defined from the category of crossed squares and 2-crossed modules of groupoids to that of Gray groupoids.